Subgroups — C₇×#24774 #13201990

Order 1792.

882 conjugacy classes of subgroups.

The subgroup lattice has 748 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×#472	C₂	1	1	yes	maximal, characteristic
2	C₂×C₇×#471	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇×#3040	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₇×#463	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄×C₇×AGL(1,ℤ/8)	C₂	1	1	yes	maximal, characteristic, complemented
6	C₇×#2906	C₂	1	1	yes	maximal, characteristic, complemented
7	C₇×#2064	C₂	1	1	yes	maximal, characteristic, complemented
8	C₇×#2063	C₂	1	1	yes	maximal, characteristic, complemented
9	C₇×#2046	C₂	1	1	yes	maximal, characteristic, complemented
10	C₇×#2045	C₂	1	1	yes	maximal, characteristic, complemented
11	C₇×#1967	C₂	1	1	yes	maximal, characteristic, complemented
12	C₇×#1950	C₂	1	1	yes	maximal, characteristic, complemented
13	C₇×#1908	C₂	1	1	yes	maximal, characteristic, complemented
14	C₇×#1907	C₂	1	1	yes	maximal, characteristic, complemented
15	C₇×#1800	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×C₇×#137	C₂²	1	1	yes	characteristic
17	C₂×C₇×AGL(1,ℤ/8)	C₂²	1	1	yes	characteristic
18	C₂×C₇×AGL(1,ℤ/8)	C₂²	1	1	yes	characteristic
19	C₂²×C₇×SD₁₆	C₂²	1	1	yes	characteristic
20	C₂²×C₇×D₈	C₂²	1	1	yes	characteristic
21	C₂×C₇×#131	C₂²	1	1	yes	characteristic
22	C₂×C₇×#131	C₂²	1	1	yes	characteristic
23	C₂×C₇×#124	C₂²	1	1	yes	characteristic
24	C₄×C₇×#41	C₂²	1	1	yes	characteristic
25	C₂×C₄×C₇×D₄	C₂²	1	1	yes	characteristic
26	C₇×#472	—	2	4	no	—
27	C₇×#471	—	2	4	no	—
28	C₇×#464	—	2	1	no	—
29	C₇×#464	—	2	1	no	—
30	C₇×#463	—	2	1	no	—
31	C₇×#463	—	2	1	no	—
32	C₇×#463	—	2	1	no	—
33	C₇×#463	—	2	1	no	—
34	C₇×#463	—	2	1	no	—
35	C₇×#463	—	2	1	no	—
36	C₇×#442	—	2	1	no	—
37	C₇×#440	—	2	1	no	—
38	C₄×C₇×SD₁₆	—	2	1	no	—
39	C₄×C₇×D₈	—	2	1	no	—
40	C₂×C₇×#108	C₂²	1	1	yes	characteristic
41	C₂×C₇×#108	C₂²	1	1	yes	characteristic
42	C₇×#428	C₂²	1	1	yes	characteristic
43	C₇×#424	—	2	1	no	—
44	C₇×#424	—	2	1	no	—
45	C₇×#424	C₂²	1	1	yes	characteristic
46	C₇×#424	C₂²	1	1	yes	characteristic
47	C₇×#423	—	2	1	no	—
48	C₇×#423	C₂²	1	1	yes	characteristic
49	C₇×#423	C₂²	1	1	yes	characteristic
50	C₂×C₇×C₄⋊C₈	—	2	1	no	—
51	C₂×C₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
52	C₇×#419	C₂²	1	1	yes	characteristic
53	C₇×#418	C₂²	1	1	yes	characteristic
54	C₇×#418	C₂²	1	1	yes	characteristic
55	C₇×#417	C₂²	1	1	yes	characteristic
56	C₂×C₇×#103	C₂²	1	1	yes	characteristic
57	C₂×C₇×#102	C₂²	1	1	yes	characteristic
58	C₂×C₇×#101	C₂²	1	1	yes	characteristic
59	C₂×C₇×#101	C₂²	1	1	yes	characteristic
60	C₂×C₇×#100	C₂²	1	1	yes	characteristic
61	C₂×C₇×#100	C₂²	1	1	yes	characteristic
62	C₂×C₇×#100	C₂²	1	1	yes	characteristic
63	C₂×C₇×#100	C₂²	1	1	yes	characteristic
64	C₄×C₇×M₁₆	C₂²	1	1	yes	characteristic
65	C₇×#343	C₂²	1	1	yes	characteristic
66	C₇×#343	C₂²	1	1	yes	characteristic
67	C₇×#341	C₂²	1	1	yes	characteristic
68	C₇×#341	C₂²	1	1	yes	characteristic
69	#24774	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
70	C₂×C₇×#41	C₂³	1	1	yes	characteristic
71	C₂²×C₇×Q₈	C₂³	1	1	yes	characteristic
72	C₂²×C₇×D₄	C₂³	1	1	yes	characteristic
73	C₂²×C₇×D₄	C₂³	1	1	yes	characteristic
74	C₂³×C₄×C₇	—	2	1	no	—
75	C₇×#137	—	2	2	no	—
76	C₇×#137	—	2	2	no	—
77	C₇×AGL(1,ℤ/8)	—	2	2	no	—
78	C₇×AGL(1,ℤ/8)	—	2	2	no	—
79	C₇×AGL(1,ℤ/8)	—	2	2	no	—
80	C₇×AGL(1,ℤ/8)	D₄	1	4	yes	—
81	C₂×C₇×Q₁₆	—	2	1	no	—
82	C₂×C₇×SD₁₆	—	2	1	no	—
83	C₂×C₇×SD₁₆	—	2	1	no	—
84	C₂×C₇×SD₁₆	—	2	2	no	—
85	C₂×C₇×SD₁₆	—	2	1	no	—
86	C₂×C₇×SD₁₆	—	2	1	no	—
87	C₂×C₇×SD₁₆	—	2	1	no	—
88	C₂×C₇×SD₁₆	—	4	1	no	—
89	C₂×C₇×D₈	—	2	1	no	—
90	C₂×C₇×D₈	—	2	2	no	—
91	C₂×C₇×D₈	—	2	1	no	—
92	C₂×C₇×D₈	—	2	1	no	—
93	C₂×C₇×D₈	—	2	1	no	—
94	C₂×C₇×D₈	—	4	1	no	—
95	C₇×#131	—	2	2	no	—
96	C₇×#131	—	2	2	no	—
97	C₇×#131	—	4	1	no	—
98	C₇×#131	—	4	1	no	—
99	C₂×C₇×M₁₆	—	2	1	no	—
100	C₂×C₇×M₁₆	—	2	1	no	—
101	C₂×C₇×M₁₆	C₂³	1	1	yes	characteristic
102	C₂×C₇×M₁₆	C₂³	1	1	yes	characteristic
103	C₂×C₇×M₁₆	C₂³	1	1	yes	characteristic
104	C₂×C₇×M₁₆	C₂³	1	1	yes	characteristic
105	C₂×C₇×M₁₆	C₂³	1	1	yes	characteristic
106	C₂×C₇×M₁₆	C₂³	1	1	yes	characteristic
107	C₂²×C₇×C₈	—	2	1	no	—
108	C₂²×C₇×C₈	—	2	1	no	—
109	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
110	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
111	C₇×#124	—	2	1	no	—
112	C₇×#124	—	2	1	no	—
113	C₇×#124	—	2	1	no	—
114	C₇×#124	—	2	1	no	—
115	C₄×C₇×Q₈	D₄	1	1	yes	characteristic
116	C₄×C₇×Q₈	D₄	1	1	yes	characteristic
117	C₄×C₇×D₄	—	2	1	no	—
118	C₄×C₇×D₄	—	2	1	no	—
119	C₄×C₇×D₄	—	2	1	no	—
120	C₄×C₇×D₄	—	2	1	no	—
121	C₄×C₇×D₄	—	2	1	no	—
122	C₄×C₇×D₄	—	2	1	no	—
123	C₄×C₇×D₄	D₄	1	1	yes	characteristic
124	C₄×C₇×D₄	D₄	1	1	yes	characteristic
125	C₇×#117	—	2	1	no	—
126	C₇×#117	C₂³	1	1	yes	characteristic
127	C₂×C₇×C₄⋊C₄	C₂³	1	1	yes	characteristic
128	C₂×C₇×C₂²⋊C₄	—	2	1	no	—
129	C₂×C₇×C₂²⋊C₄	—	2	1	no	—
130	C₂×C₇×C₂²⋊C₄	—	2	1	no	—
131	C₂×C₄²×C₇	—	2	1	no	—
132	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
133	C₇×#108	—	2	2	no	—
134	C₇×#108	—	2	2	no	—
135	C₇×#107	—	2	1	no	—
136	C₇×#106	—	2	1	no	—
137	C₇×C₄⋊C₈	—	2	1	no	—
138	C₇×C₄⋊C₈	—	2	1	no	—
139	C₇×C₄⋊C₈	—	2	1	no	—
140	C₇×C₄⋊C₈	—	2	1	no	—
141	C₇×C₄⋊C₈	—	2	1	no	—
142	C₇×C₄⋊C₈	—	2	1	no	—
143	C₇×C₄⋊C₈	—	2	1	no	—
144	C₇×C₄⋊C₈	—	2	1	no	—
145	C₇×C₄⋊C₈	D₄	1	1	yes	characteristic
146	C₇×C₄⋊C₈	D₄	1	1	yes	characteristic
147	C₇×C₄⋊C₈	D₄	1	1	yes	characteristic
148	C₇×C₄⋊C₈	D₄	1	1	yes	characteristic
149	C₇×#103	—	2	1	no	—
150	C₇×#103	—	2	1	no	—
151	C₇×#103	—	2	1	no	—
152	C₇×#103	—	2	1	no	—
153	C₇×#103	—	2	1	no	—
154	C₇×#102	—	2	1	no	—
155	C₇×#102	—	2	1	no	—
156	C₇×#102	—	2	1	no	—
157	C₇×#102	—	2	1	no	—
158	C₇×#102	—	2	1	no	—
159	C₇×#102	—	2	1	no	—
160	C₇×#102	—	2	1	no	—
161	C₇×#101	—	2	2	no	—
162	C₇×#101	—	4	1	no	—
163	C₇×#100	—	2	2	no	—
164	C₇×#100	—	2	2	no	—
165	C₇×#100	—	2	2	no	—
166	C₇×#100	—	4	1	no	—
167	C₇×#97	—	2	1	no	—
168	C₄×C₇×C₈	—	2	1	no	—
169	C₂×#472	C₂×C₇	1	1	yes	characteristic
170	C₂×#471	C₂×C₇	1	1	yes	characteristic, complemented
171	#3040	C₂×C₇	1	1	yes	characteristic, complemented
172	C₂×#463	C₂×C₇	1	1	yes	characteristic, complemented
173	C₄×AGL(1,ℤ/8)	C₂×C₇	1	1	yes	characteristic, complemented
174	#2906	C₂×C₇	1	1	yes	characteristic, complemented
175	#2064	C₂×C₇	1	1	yes	characteristic, complemented
176	#2063	C₂×C₇	1	1	yes	characteristic, complemented
177	#2046	C₂×C₇	1	1	yes	characteristic, complemented
178	#2045	C₂×C₇	1	1	yes	characteristic, complemented
179	#1967	C₂×C₇	1	1	yes	characteristic, complemented
180	#1950	C₂×C₇	1	1	yes	characteristic, complemented
181	#1908	C₂×C₇	1	1	yes	characteristic, complemented
182	#1907	C₂×C₇	1	1	yes	characteristic, complemented
183	#1800	C₂×C₇	1	1	yes	characteristic, complemented
184	C₂⁴×C₇	—	2	1	no	—
185	C₂⁴×C₇	—	2	1	no	—
186	C₇×#41	—	2	1	no	—
187	C₇×#41	—	2	1	no	—
188	C₇×#41	C₂×D₄	1	2	yes	—
189	C₇×#41	#41	1	2	yes	—
190	C₂×C₇×Q₈	—	2	1	no	—
191	C₂×C₇×Q₈	—	2	1	no	—
192	C₂×C₇×Q₈	—	2	1	no	—
193	C₂×C₇×Q₈	—	2	1	no	—
194	C₂×C₇×Q₈	—	2	1	no	—
195	C₂×C₇×Q₈	—	2	1	no	—
196	C₂×C₇×Q₈	C₂×D₄	1	1	yes	characteristic
197	C₂×C₇×D₄	—	2	1	no	—
198	C₂×C₇×D₄	—	2	1	no	—
199	C₂×C₇×D₄	—	2	1	no	—
200	C₂×C₇×D₄	—	2	1	no	—
201	C₂×C₇×D₄	—	2	1	no	—
202	C₂×C₇×D₄	—	2	1	no	—
203	C₂×C₇×D₄	—	2	1	no	—
204	C₂×C₇×D₄	—	2	1	no	—
205	C₂×C₇×D₄	—	2	1	no	—
206	C₂×C₇×D₄	—	2	1	no	—
207	C₂×C₇×D₄	—	2	1	no	—
208	C₂×C₇×D₄	—	2	1	no	—
209	C₂×C₇×D₄	C₂×D₄	1	2	yes	—
210	C₂×C₇×D₄	C₂×D₄	1	1	yes	characteristic
211	C₂²×C₄×C₇	—	2	1	no	—
212	C₂²×C₄×C₇	—	2	1	no	—
213	C₂²×C₄×C₇	—	2	1	no	—
214	C₂²×C₄×C₇	—	2	1	no	—
215	C₂²×C₄×C₇	—	2	1	no	—
216	C₂²×C₄×C₇	—	4	1	no	—
217	C₂²×C₄×C₇	—	4	1	no	—
218	C₂²×C₄×C₇	—	8	2	no	—
219	C₂²×C₄×C₇	#41	1	1	yes	characteristic
220	C₂²×C₄×C₇	#41	1	1	yes	characteristic
221	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
222	C₇×Q₁₆	—	4	2	no	—
223	C₇×SD₁₆	—	2	4	no	—
224	C₇×SD₁₆	—	4	2	no	—
225	C₇×SD₁₆	—	4	2	no	—
226	C₇×SD₁₆	—	4	2	no	—
227	C₇×SD₁₆	—	4	2	no	—
228	C₇×D₈	—	2	4	no	—
229	C₇×D₈	—	4	2	no	—
230	C₇×D₈	—	4	2	no	—
231	C₇×D₈	—	4	2	no	—
232	C₇×M₁₆	—	2	1	no	—
233	C₇×M₁₆	—	2	1	no	—
234	C₇×M₁₆	—	2	1	no	—
235	C₇×M₁₆	—	2	1	no	—
236	C₇×M₁₆	—	2	1	no	—
237	C₇×M₁₆	—	2	1	no	—
238	C₇×M₁₆	—	2	1	no	—
239	C₇×M₁₆	—	2	1	no	—
240	C₇×M₁₆	—	2	1	no	—
241	C₇×M₁₆	—	2	1	no	—
242	C₇×M₁₆	—	2	1	no	—
243	C₇×M₁₆	—	4	2	no	—
244	C₇×M₁₆	—	4	2	no	—
245	C₇×M₁₆	C₂×D₄	1	2	yes	—
246	C₂×C₇×C₈	—	2	1	no	—
247	C₂×C₇×C₈	—	2	1	no	—
248	C₂×C₇×C₈	—	2	1	no	—
249	C₂×C₇×C₈	—	2	1	no	—
250	C₂×C₇×C₈	—	2	1	no	—
251	C₂×C₇×C₈	—	2	1	no	—
252	C₂×C₇×C₈	—	2	1	no	—
253	C₂×C₇×C₈	—	2	1	no	—
254	C₂×C₇×C₈	—	2	1	no	—
255	C₂×C₇×C₈	—	2	1	no	—
256	C₂×C₇×C₈	—	4	2	no	—
257	C₂×C₇×C₈	—	4	2	no	—
258	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
259	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
260	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
261	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
262	C₇×C₄⋊C₄	—	2	1	no	—
263	C₇×C₄⋊C₄	—	2	1	no	—
264	C₇×C₄⋊C₄	—	2	1	no	—
265	C₇×C₄⋊C₄	—	2	1	no	—
266	C₇×C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
267	C₇×C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
268	C₇×C₂²⋊C₄	—	2	1	no	—
269	C₇×C₂²⋊C₄	—	2	1	no	—
270	C₇×C₂²⋊C₄	—	2	1	no	—
271	C₇×C₂²⋊C₄	—	4	1	no	—
272	C₇×C₂²⋊C₄	—	4	1	no	—
273	C₇×C₂²⋊C₄	—	4	1	no	—
274	C₇×C₂²⋊C₄	—	4	1	no	—
275	C₇×C₂²⋊C₄	—	4	1	no	—
276	C₇×C₂²⋊C₄	—	4	1	no	—
277	C₄²×C₇	—	2	1	no	—
278	C₄²×C₇	—	2	1	no	—
279	C₄²×C₇	—	2	1	no	—
280	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
281	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
282	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
283	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
284	C₂×#137	C₂²×C₇	1	1	yes	characteristic
285	C₂×AGL(1,ℤ/8)	C₂²×C₇	1	1	yes	characteristic
286	C₂×AGL(1,ℤ/8)	C₂²×C₇	1	1	yes	characteristic
287	C₂²×SD₁₆	C₂²×C₇	1	1	yes	characteristic
288	C₂²×D₈	C₂²×C₇	1	1	yes	characteristic
289	C₂×#131	C₂²×C₇	1	1	yes	characteristic
290	C₂×#131	C₂²×C₇	1	1	yes	characteristic
291	C₂×#124	C₂²×C₇	1	1	yes	characteristic
292	C₄×#41	C₂²×C₇	1	1	yes	characteristic
293	C₂×C₄×D₄	C₂²×C₇	1	1	yes	characteristic
294	#472	—	2	4	no	—
295	#471	—	2	4	no	—
296	#464	—	2	1	no	—
297	#464	—	2	1	no	—
298	#463	—	2	1	no	—
299	#463	—	2	1	no	—
300	#463	—	2	1	no	—
301	#463	—	2	1	no	—
302	#463	—	2	1	no	—
303	#463	—	2	1	no	—
304	#442	—	2	1	no	—
305	#440	—	2	1	no	—
306	C₄×SD₁₆	—	2	1	no	—
307	C₄×D₈	—	2	1	no	—
308	C₂×#108	C₂²×C₇	1	1	yes	characteristic
309	C₂×#108	C₂²×C₇	1	1	yes	characteristic
310	#428	C₂²×C₇	1	1	yes	characteristic
311	#424	—	2	1	no	—
312	#424	—	2	1	no	—
313	#424	C₂²×C₇	1	1	yes	characteristic
314	#424	C₂²×C₇	1	1	yes	characteristic
315	#423	—	2	1	no	—
316	#423	C₂²×C₇	1	1	yes	characteristic
317	#423	C₂²×C₇	1	1	yes	characteristic
318	C₂×C₄⋊C₈	—	2	1	no	—
319	C₂×C₄⋊C₈	C₂²×C₇	1	1	yes	characteristic
320	#419	C₂²×C₇	1	1	yes	characteristic
321	#418	C₂²×C₇	1	1	yes	characteristic
322	#418	C₂²×C₇	1	1	yes	characteristic
323	#417	C₂²×C₇	1	1	yes	characteristic
324	C₂×#103	C₂²×C₇	1	1	yes	characteristic
325	C₂×#102	C₂²×C₇	1	1	yes	characteristic
326	C₂×#101	C₂²×C₇	1	1	yes	characteristic
327	C₂×#101	C₂²×C₇	1	1	yes	characteristic
328	C₂×#100	C₂²×C₇	1	1	yes	characteristic
329	C₂×#100	C₂²×C₇	1	1	yes	characteristic
330	C₂×#100	C₂²×C₇	1	1	yes	characteristic
331	C₂×#100	C₂²×C₇	1	1	yes	characteristic
332	C₄×M₁₆	C₂²×C₇	1	1	yes	characteristic
333	#343	C₂²×C₇	1	1	yes	characteristic
334	#343	C₂²×C₇	1	1	yes	characteristic
335	#341	C₂²×C₇	1	1	yes	characteristic
336	#341	C₂²×C₇	1	1	yes	characteristic
337	C₂³×C₇	—	2	1	no	—
338	C₂³×C₇	—	2	2	no	—
339	C₂³×C₇	—	4	1	no	—
340	C₂³×C₇	—	4	1	no	—
341	C₂³×C₇	—	4	1	no	—
342	C₂³×C₇	—	4	1	no	—
343	C₂³×C₇	—	4	1	no	—
344	C₂³×C₇	—	8	1	no	—
345	C₂³×C₇	—	8	1	no	—
346	C₂³×C₇	—	8	1	no	—
347	C₂³×C₇	—	8	1	no	—
348	C₂³×C₇	C₂×#41	1	1	yes	characteristic
349	C₇×Q₈	—	2	1	no	—
350	C₇×Q₈	—	4	1	no	—
351	C₇×Q₈	—	4	1	no	—
352	C₇×Q₈	—	4	1	no	—
353	C₇×Q₈	—	4	1	no	—
354	C₇×Q₈	#121	1	2	yes	—
355	C₇×D₄	—	2	2	no	—
356	C₇×D₄	—	2	2	no	—
357	C₇×D₄	—	2	2	no	—
358	C₇×D₄	—	2	1	no	—
359	C₇×D₄	—	2	2	no	—
360	C₇×D₄	—	4	1	no	—
361	C₇×D₄	—	4	1	no	—
362	C₇×D₄	—	4	1	no	—
363	C₇×D₄	—	4	1	no	—
364	C₇×D₄	—	4	1	no	—
365	C₇×D₄	—	4	1	no	—
366	C₇×D₄	#121	1	2	yes	—
367	C₂×C₄×C₇	—	2	1	no	—
368	C₂×C₄×C₇	—	2	2	no	—
369	C₂×C₄×C₇	—	2	2	no	—
370	C₂×C₄×C₇	—	2	1	no	—
371	C₂×C₄×C₇	—	2	1	no	—
372	C₂×C₄×C₇	—	4	1	no	—
373	C₂×C₄×C₇	—	4	1	no	—
374	C₂×C₄×C₇	—	4	2	no	—
375	C₂×C₄×C₇	—	4	1	no	—
376	C₂×C₄×C₇	—	4	1	no	—
377	C₂×C₄×C₇	—	4	1	no	—
378	C₂×C₄×C₇	—	4	1	no	—
379	C₂×C₄×C₇	—	8	1	no	—
380	C₂×C₄×C₇	—	8	2	no	—
381	C₂×C₄×C₇	—	8	2	no	—
382	C₂×C₄×C₇	—	8	1	no	—
383	C₂×C₄×C₇	#121	1	1	yes	characteristic
384	C₂×C₄×C₇	#121	1	1	yes	characteristic
385	C₂×C₄×C₇	#121	1	1	yes	characteristic
386	C₂×C₄×C₇	#121	1	1	yes	characteristic
387	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
388	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
389	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
390	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
391	C₂×C₄×C₇	H(5,𝔽₂)	1	1	yes	characteristic
392	C₂×C₄×C₇	2₋¹⁺⁴	1	1	yes	characteristic
393	C₇×C₈	—	2	2	no	—
394	C₇×C₈	—	2	1	no	—
395	C₇×C₈	—	2	1	no	—
396	C₇×C₈	—	2	1	no	—
397	C₇×C₈	—	2	1	no	—
398	C₇×C₈	—	4	1	no	—
399	C₇×C₈	—	4	1	no	—
400	C₇×C₈	—	4	1	no	—
401	C₇×C₈	—	4	1	no	—
402	C₇×C₈	—	4	1	no	—
403	C₂×#41	C₂³×C₇	1	1	yes	characteristic
404	C₂²×Q₈	C₂³×C₇	1	1	yes	characteristic
405	C₂²×D₄	C₂³×C₇	1	1	yes	characteristic
406	C₂²×D₄	C₂³×C₇	1	1	yes	characteristic
407	C₂³×C₄	—	2	1	no	—
408	#137	—	2	2	no	—
409	#137	—	2	2	no	—
410	AGL(1,ℤ/8)	—	2	2	no	—
411	AGL(1,ℤ/8)	—	2	2	no	—
412	AGL(1,ℤ/8)	—	2	2	no	—
413	AGL(1,ℤ/8)	C₇×D₄	1	4	yes	—
414	C₂×Q₁₆	—	2	1	no	—
415	C₂×SD₁₆	—	2	1	no	—
416	C₂×SD₁₆	—	2	1	no	—
417	C₂×SD₁₆	—	2	2	no	—
418	C₂×SD₁₆	—	2	1	no	—
419	C₂×SD₁₆	—	2	1	no	—
420	C₂×SD₁₆	—	2	1	no	—
421	C₂×SD₁₆	—	4	1	no	—
422	C₂×D₈	—	2	1	no	—
423	C₂×D₈	—	2	2	no	—
424	C₂×D₈	—	2	1	no	—
425	C₂×D₈	—	2	1	no	—
426	C₂×D₈	—	2	1	no	—
427	C₂×D₈	—	4	1	no	—
428	#131	—	2	2	no	—
429	#131	—	2	2	no	—
430	#131	—	4	1	no	—
431	#131	—	4	1	no	—
432	C₂×M₁₆	—	2	1	no	—
433	C₂×M₁₆	—	2	1	no	—
434	C₂×M₁₆	C₂³×C₇	1	1	yes	characteristic
435	C₂×M₁₆	C₂³×C₇	1	1	yes	characteristic
436	C₂×M₁₆	C₂³×C₇	1	1	yes	characteristic
437	C₂×M₁₆	C₂³×C₇	1	1	yes	characteristic
438	C₂×M₁₆	C₂³×C₇	1	1	yes	characteristic
439	C₂×M₁₆	C₂³×C₇	1	1	yes	characteristic
440	C₂²×C₈	—	2	1	no	—
441	C₂²×C₈	—	2	1	no	—
442	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
443	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
444	#124	—	2	1	no	—
445	#124	—	2	1	no	—
446	#124	—	2	1	no	—
447	#124	—	2	1	no	—
448	C₄×Q₈	C₇×D₄	1	1	yes	characteristic
449	C₄×Q₈	C₇×D₄	1	1	yes	characteristic
450	C₄×D₄	—	2	1	no	—
451	C₄×D₄	—	2	1	no	—
452	C₄×D₄	—	2	1	no	—
453	C₄×D₄	—	2	1	no	—
454	C₄×D₄	—	2	1	no	—
455	C₄×D₄	—	2	1	no	—
456	C₄×D₄	C₇×D₄	1	1	yes	characteristic
457	C₄×D₄	C₇×D₄	1	1	yes	characteristic
458	#117	—	2	1	no	—
459	#117	C₂³×C₇	1	1	yes	characteristic
460	C₂×C₄⋊C₄	C₂³×C₇	1	1	yes	characteristic
461	C₂×C₂²⋊C₄	—	2	1	no	—
462	C₂×C₂²⋊C₄	—	2	1	no	—
463	C₂×C₂²⋊C₄	—	2	1	no	—
464	C₂×C₄²	—	2	1	no	—
465	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
466	#108	—	2	2	no	—
467	#108	—	2	2	no	—
468	#107	—	2	1	no	—
469	#106	—	2	1	no	—
470	C₄⋊C₈	—	2	1	no	—
471	C₄⋊C₈	—	2	1	no	—
472	C₄⋊C₈	—	2	1	no	—
473	C₄⋊C₈	—	2	1	no	—
474	C₄⋊C₈	—	2	1	no	—
475	C₄⋊C₈	—	2	1	no	—
476	C₄⋊C₈	—	2	1	no	—
477	C₄⋊C₈	—	2	1	no	—
478	C₄⋊C₈	C₇×D₄	1	1	yes	characteristic
479	C₄⋊C₈	C₇×D₄	1	1	yes	characteristic
480	C₄⋊C₈	C₇×D₄	1	1	yes	characteristic
481	C₄⋊C₈	C₇×D₄	1	1	yes	characteristic
482	#103	—	2	1	no	—
483	#103	—	2	1	no	—
484	#103	—	2	1	no	—
485	#103	—	2	1	no	—
486	#103	—	2	1	no	—
487	#102	—	2	1	no	—
488	#102	—	2	1	no	—
489	#102	—	2	1	no	—
490	#102	—	2	1	no	—
491	#102	—	2	1	no	—
492	#102	—	2	1	no	—
493	#102	—	2	1	no	—
494	#101	—	2	2	no	—
495	#101	—	4	1	no	—
496	#100	—	2	2	no	—
497	#100	—	2	2	no	—
498	#100	—	2	2	no	—
499	#100	—	4	1	no	—
500	#97	—	2	1	no	—
501	C₄×C₈	—	2	1	no	—
502	C₂²×C₇	—	2	1	no	—
503	C₂²×C₇	—	2	1	no	—
504	C₂²×C₇	—	2	2	no	—
505	C₂²×C₇	—	4	1	no	—
506	C₂²×C₇	—	4	1	no	—
507	C₂²×C₇	—	4	2	no	—
508	C₂²×C₇	—	8	2	no	—
509	C₂²×C₇	—	8	2	no	—
510	C₂²×C₇	—	8	1	no	—
511	C₂²×C₇	—	8	1	no	—
512	C₂²×C₇	—	8	1	no	—
513	C₂²×C₇	—	8	1	no	—
514	C₂²×C₇	—	8	1	no	—
515	C₂²×C₇	—	16	1	no	—
516	C₂²×C₇	—	16	1	no	—
517	C₂²×C₇	C₂×#121	1	1	yes	characteristic
518	C₂²×C₇	#547	1	1	yes	characteristic
519	C₂²×C₇	#553	1	1	yes	characteristic
520	C₄×C₇	—	2	2	no	—
521	C₄×C₇	—	2	1	no	—
522	C₄×C₇	—	2	1	no	—
523	C₄×C₇	—	2	1	no	—
524	C₄×C₇	—	2	1	no	—
525	C₄×C₇	—	4	1	no	—
526	C₄×C₇	—	4	1	no	—
527	C₄×C₇	—	8	1	no	—
528	C₄×C₇	—	8	1	no	—
529	C₄×C₇	C₂×#121	1	1	yes	characteristic
530	C₄×C₇	#537	1	1	yes	characteristic
531	C₄×C₇	D₄²	1	1	yes	characteristic
532	C₄×C₇	#548	1	1	yes	characteristic
533	C₂⁴	—	2	1	no	—
534	C₂⁴	—	2	1	no	—
535	#41	—	2	1	no	—
536	#41	—	2	1	no	—
537	#41	C₂×C₇×D₄	1	2	yes	—
538	#41	C₇×#41	1	2	yes	—
539	C₂×Q₈	—	2	1	no	—
540	C₂×Q₈	—	2	1	no	—
541	C₂×Q₈	—	2	1	no	—
542	C₂×Q₈	—	2	1	no	—
543	C₂×Q₈	—	2	1	no	—
544	C₂×Q₈	—	2	1	no	—
545	C₂×Q₈	C₂×C₇×D₄	1	1	yes	characteristic
546	C₂×D₄	—	2	1	no	—
547	C₂×D₄	—	2	1	no	—
548	C₂×D₄	—	2	1	no	—
549	C₂×D₄	—	2	1	no	—
550	C₂×D₄	—	2	1	no	—
551	C₂×D₄	—	2	1	no	—
552	C₂×D₄	—	2	1	no	—
553	C₂×D₄	—	2	1	no	—
554	C₂×D₄	—	2	1	no	—
555	C₂×D₄	—	2	1	no	—
556	C₂×D₄	—	2	1	no	—
557	C₂×D₄	—	2	1	no	—
558	C₂×D₄	C₂×C₇×D₄	1	2	yes	—
559	C₂×D₄	C₂×C₇×D₄	1	1	yes	characteristic
560	C₂²×C₄	—	2	1	no	—
561	C₂²×C₄	—	2	1	no	—
562	C₂²×C₄	—	2	1	no	—
563	C₂²×C₄	—	2	1	no	—
564	C₂²×C₄	—	2	1	no	—
565	C₂²×C₄	—	4	1	no	—
566	C₂²×C₄	—	4	1	no	—
567	C₂²×C₄	—	8	2	no	—
568	C₂²×C₄	C₇×#41	1	1	yes	characteristic
569	C₂²×C₄	C₇×#41	1	1	yes	characteristic
570	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
571	Q₁₆	—	4	2	no	—
572	SD₁₆	—	2	4	no	—
573	SD₁₆	—	4	2	no	—
574	SD₁₆	—	4	2	no	—
575	SD₁₆	—	4	2	no	—
576	SD₁₆	—	4	2	no	—
577	D₈	—	2	4	no	—
578	D₈	—	4	2	no	—
579	D₈	—	4	2	no	—
580	D₈	—	4	2	no	—
581	M₁₆	—	2	1	no	—
582	M₁₆	—	2	1	no	—
583	M₁₆	—	2	1	no	—
584	M₁₆	—	2	1	no	—
585	M₁₆	—	2	1	no	—
586	M₁₆	—	2	1	no	—
587	M₁₆	—	2	1	no	—
588	M₁₆	—	2	1	no	—
589	M₁₆	—	2	1	no	—
590	M₁₆	—	2	1	no	—
591	M₁₆	—	2	1	no	—
592	M₁₆	—	4	2	no	—
593	M₁₆	—	4	2	no	—
594	M₁₆	C₂×C₇×D₄	1	2	yes	—
595	C₂×C₈	—	2	1	no	—
596	C₂×C₈	—	2	1	no	—
597	C₂×C₈	—	2	1	no	—
598	C₂×C₈	—	2	1	no	—
599	C₂×C₈	—	2	1	no	—
600	C₂×C₈	—	2	1	no	—
601	C₂×C₈	—	2	1	no	—
602	C₂×C₈	—	2	1	no	—
603	C₂×C₈	—	2	1	no	—
604	C₂×C₈	—	2	1	no	—
605	C₂×C₈	—	4	2	no	—
606	C₂×C₈	—	4	2	no	—
607	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
608	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
609	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
610	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
611	C₄⋊C₄	—	2	1	no	—
612	C₄⋊C₄	—	2	1	no	—
613	C₄⋊C₄	—	2	1	no	—
614	C₄⋊C₄	—	2	1	no	—
615	C₄⋊C₄	C₂×C₇×D₄	1	1	yes	characteristic
616	C₄⋊C₄	C₂×C₇×D₄	1	1	yes	characteristic
617	C₂²⋊C₄	—	2	1	no	—
618	C₂²⋊C₄	—	2	1	no	—
619	C₂²⋊C₄	—	2	1	no	—
620	C₂²⋊C₄	—	4	1	no	—
621	C₂²⋊C₄	—	4	1	no	—
622	C₂²⋊C₄	—	4	1	no	—
623	C₂²⋊C₄	—	4	1	no	—
624	C₂²⋊C₄	—	4	1	no	—
625	C₂²⋊C₄	—	4	1	no	—
626	C₄²	—	2	1	no	—
627	C₄²	—	2	1	no	—
628	C₄²	—	2	1	no	—
629	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
630	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
631	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
632	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
633	C₂×C₇	—	2	1	no	—
634	C₂×C₇	—	2	1	no	—
635	C₂×C₇	—	4	2	no	—
636	C₂×C₇	—	8	2	no	—
637	C₂×C₇	—	16	1	no	—
638	C₂×C₇	#2416	1	1	yes	characteristic
639	C₂×C₇	#3067	1	1	yes	characteristic
640	C₂×C₇	#3068	1	1	yes	characteristic
641	C₂³	—	2	1	no	—
642	C₂³	—	2	2	no	—
643	C₂³	—	4	1	no	—
644	C₂³	—	4	1	no	—
645	C₂³	—	4	1	no	—
646	C₂³	—	4	1	no	—
647	C₂³	—	4	1	no	—
648	C₂³	—	8	1	no	—
649	C₂³	—	8	1	no	—
650	C₂³	—	8	1	no	—
651	C₂³	—	8	1	no	—
652	C₂³	C₂×C₇×#41	1	1	yes	characteristic
653	Q₈	—	2	1	no	—
654	Q₈	—	4	1	no	—
655	Q₈	—	4	1	no	—
656	Q₈	—	4	1	no	—
657	Q₈	—	4	1	no	—
658	Q₈	C₇×#121	1	2	yes	—
659	D₄	—	2	2	no	—
660	D₄	—	2	2	no	—
661	D₄	—	2	2	no	—
662	D₄	—	2	1	no	—
663	D₄	—	2	2	no	—
664	D₄	—	4	1	no	—
665	D₄	—	4	1	no	—
666	D₄	—	4	1	no	—
667	D₄	—	4	1	no	—
668	D₄	—	4	1	no	—
669	D₄	—	4	1	no	—
670	D₄	C₇×#121	1	2	yes	—
671	C₂×C₄	—	2	1	no	—
672	C₂×C₄	—	2	2	no	—
673	C₂×C₄	—	2	2	no	—
674	C₂×C₄	—	2	1	no	—
675	C₂×C₄	—	2	1	no	—
676	C₂×C₄	—	4	1	no	—
677	C₂×C₄	—	4	1	no	—
678	C₂×C₄	—	4	2	no	—
679	C₂×C₄	—	4	1	no	—
680	C₂×C₄	—	4	1	no	—
681	C₂×C₄	—	4	1	no	—
682	C₂×C₄	—	4	1	no	—
683	C₂×C₄	—	8	1	no	—
684	C₂×C₄	—	8	2	no	—
685	C₂×C₄	—	8	2	no	—
686	C₂×C₄	—	8	1	no	—
687	C₂×C₄	C₇×#121	1	1	yes	characteristic
688	C₂×C₄	C₇×#121	1	1	yes	characteristic
689	C₂×C₄	C₇×#121	1	1	yes	characteristic
690	C₂×C₄	C₇×#121	1	1	yes	characteristic
691	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
692	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
693	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
694	C₂×C₄	C₂×C₇×#41	1	1	yes	characteristic
695	C₂×C₄	C₇×H(5,𝔽₂)	1	1	yes	characteristic
696	C₂×C₄	C₇×2₋¹⁺⁴	1	1	yes	characteristic
697	C₈	—	2	2	no	—
698	C₈	—	2	1	no	—
699	C₈	—	2	1	no	—
700	C₈	—	2	1	no	—
701	C₈	—	2	1	no	—
702	C₈	—	4	1	no	—
703	C₈	—	4	1	no	—
704	C₈	—	4	1	no	—
705	C₈	—	4	1	no	—
706	C₈	—	4	1	no	—
707	C₇	#24774	1	1	yes	characteristic, direct product factor, Sylow
708	C₂²	—	2	1	no	—
709	C₂²	—	2	1	no	—
710	C₂²	—	2	2	no	—
711	C₂²	—	4	1	no	—
712	C₂²	—	4	1	no	—
713	C₂²	—	4	2	no	—
714	C₂²	—	8	2	no	—
715	C₂²	—	8	2	no	—
716	C₂²	—	8	1	no	—
717	C₂²	—	8	1	no	—
718	C₂²	—	8	1	no	—
719	C₂²	—	8	1	no	—
720	C₂²	—	8	1	no	—
721	C₂²	—	16	1	no	—
722	C₂²	—	16	1	no	—
723	C₂²	C₂×C₇×#121	1	1	yes	characteristic
724	C₂²	C₇×#547	1	1	yes	characteristic
725	C₂²	C₇×#553	1	1	yes	characteristic
726	C₄	—	2	2	no	—
727	C₄	—	2	1	no	—
728	C₄	—	2	1	no	—
729	C₄	—	2	1	no	—
730	C₄	—	2	1	no	—
731	C₄	—	4	1	no	—
732	C₄	—	4	1	no	—
733	C₄	—	8	1	no	—
734	C₄	—	8	1	no	—
735	C₄	C₂×C₇×#121	1	1	yes	characteristic
736	C₄	C₇×#537	1	1	yes	characteristic
737	C₄	C₇×D₄²	1	1	yes	characteristic
738	C₄	C₇×#548	1	1	yes	characteristic
739	C₂	—	2	1	no	—
740	C₂	—	2	1	no	—
741	C₂	—	4	2	no	—
742	C₂	—	8	2	no	—
743	C₂	—	16	1	no	—
744	C₂	C₇×#2416	1	1	yes	characteristic
745	C₂	C₇×#3067	1	1	yes	characteristic
746	C₂	C₇×#3068	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₇	C₂×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₇
prime radical	C₂×C₇×#472	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂×C₄²×C₇ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
2. C₂×C₇×#471	C₂	semidirect
3. C₇×#3040	C₂	semidirect
4. C₂×C₇×#463	C₂	semidirect
5. C₄×C₇×AGL(1,ℤ/8)	C₂	semidirect
6. C₇×#2906	C₂	semidirect
7. C₇×#2064	C₂	semidirect
8. C₇×#2063	C₂	semidirect
9. C₇×#2046	C₂	semidirect
10. C₇×#2045	C₂	semidirect
11. C₇×#1967	C₂	semidirect
12. C₇×#1950	C₂	semidirect
13. C₇×#1908	C₂	semidirect
14. C₇×#1907	C₂	semidirect
15. C₇×#1800	C₂	semidirect
69. #24774	C₇	direct (N × H)
170. C₂×#471	C₂×C₇	semidirect
171. #3040	C₂×C₇	semidirect
172. C₂×#463	C₂×C₇	semidirect
173. C₄×AGL(1,ℤ/8)	C₂×C₇	semidirect
174. #2906	C₂×C₇	semidirect
175. #2064	C₂×C₇	semidirect
176. #2063	C₂×C₇	semidirect
177. #2046	C₂×C₇	semidirect
178. #2045	C₂×C₇	semidirect
179. #1967	C₂×C₇	semidirect
180. #1950	C₂×C₇	semidirect
181. #1908	C₂×C₇	semidirect
182. #1907	C₂×C₇	semidirect
183. #1800	C₂×C₇	semidirect
707. C₇	#24774	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₇×#472	2
C₂×C₇×#471	2
C₇×#3040	2
C₂×C₇×#463	2
C₄×C₇×AGL(1,ℤ/8)	2
C₇×#2906	2
C₇×#2064	2
C₇×#2063	2
C₇×#2046	2
C₇×#2045	2
C₇×#1967	2
C₇×#1950	2
C₇×#1908	2
C₇×#1907	2
C₇×#1800	2
#24774	7

Subgroups — C7×#24774 #13201990

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#24774 #13201990