Subgroups — C₇×#23016 #13200232

Order 1792.

880 conjugacy classes of subgroups.

The subgroup lattice has 546 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#3035	C₂	1	1	yes	maximal, characteristic
2	C₂×C₇×#444	C₂	1	2	yes	maximal, complemented
3	C₇×#2935	C₂	1	1	yes	maximal, characteristic, complemented
4	C₇×#2906	C₂	1	1	yes	maximal, characteristic, complemented
5	C₇×#1962	C₂	1	1	yes	maximal, characteristic, complemented
6	C₇×#1950	C₂	1	1	yes	maximal, characteristic, complemented
7	C₇×#1938	C₂	1	2	yes	maximal, complemented
8	C₇×#1876	C₂	1	2	yes	maximal, complemented
9	C₇×#1866	C₂	1	2	yes	maximal, complemented
10	C₇×#1767	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×C₇×#104	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₇×#135	C₂²	1	2	yes	—
13	C₂×C₇×#131	C₂²	1	2	yes	—
14	C₂×C₇×#131	C₂²	1	2	yes	—
15	C₇×#532	C₂²	1	1	yes	characteristic
16	C₄×C₇×#41	C₂²	1	1	yes	characteristic
17	C₇×#464	C₄	1	2	yes	complemented
18	C₇×#463	C₄	1	2	yes	complemented
19	C₇×#462	C₄	1	1	yes	characteristic, complemented
20	C₇×#461	C₄	1	1	yes	characteristic, complemented
21	C₇×#460	C₄	1	1	yes	characteristic, complemented
22	C₇×#459	C₄	1	1	yes	characteristic, complemented
23	C₇×#444	—	2	8	no	—
24	C₇×#436	—	2	1	no	—
25	C₇×#435	—	2	2	no	—
26	C₇×C₈×D₄	—	2	1	no	—
27	C₂×C₇×#108	C₂²	1	2	yes	—
28	C₇×#424	—	2	2	no	—
29	C₇×#424	C₂²	1	2	yes	—
30	C₇×#423	—	2	1	no	—
31	C₇×#423	C₂²	1	1	yes	characteristic
32	C₂×C₇×C₄⋊C₈	—	2	1	no	—
33	C₂×C₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
34	C₂×C₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
35	C₂×C₇×#104	C₂²	1	1	yes	characteristic
36	C₂×C₇×#104	C₂²	1	2	yes	—
37	C₂×C₇×#104	C₂²	1	2	yes	—
38	C₂×C₇×#104	C₂²	1	1	yes	characteristic
39	C₇×#416	C₂²	1	2	yes	—
40	C₇×#408	C₂²	1	2	yes	—
41	C₄×C₇×M₁₆	C₂²	1	1	yes	characteristic
42	C₂×C₇×#97	C₂²	1	2	yes	—
43	C₄³×C₇	C₂²	1	1	yes	characteristic
44	C₇×#341	C₂²	1	2	yes	—
45	C₇×#339	C₂²	1	1	yes	characteristic
46	C₇×#339	C₂²	1	1	yes	characteristic
47	C₇×#336	C₂²	1	2	yes	—
48	#23016	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
49	C₂×C₇×#41	C₂³	1	2	yes	—
50	C₂×C₇×#41	C₂³	1	1	yes	characteristic
51	C₇×#135	—	2	4	no	—
52	C₇×#135	—	2	4	no	—
53	C₂×C₇×Q₁₆	C₂×C₄	1	2	yes	—
54	C₂×C₇×SD₁₆	C₂×C₄	1	2	yes	—
55	C₂×C₇×SD₁₆	C₂×C₄	1	2	yes	—
56	C₂×C₇×D₈	C₂×C₄	1	2	yes	—
57	C₇×#131	—	2	4	no	—
58	C₇×#131	—	2	4	no	—
59	C₇×#131	—	4	2	no	—
60	C₇×#131	—	4	2	no	—
61	C₂×C₇×M₁₆	—	2	2	no	—
62	C₂×C₇×M₁₆	—	2	2	no	—
63	C₂×C₇×M₁₆	C₂³	1	1	yes	characteristic
64	C₂×C₇×M₁₆	C₂³	1	2	yes	—
65	C₂×C₇×M₁₆	C₂³	1	1	yes	characteristic
66	C₂²×C₇×C₈	—	2	2	no	—
67	C₂²×C₇×C₈	—	2	2	no	—
68	C₂²×C₇×C₈	C₂³	1	2	yes	—
69	C₂²×C₇×C₈	C₂³	1	2	yes	—
70	C₇×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
71	C₇×#127	C₂×C₄	1	1	yes	characteristic
72	C₇×#124	C₂×C₄	1	2	yes	—
73	C₇×#121	—	2	1	no	—
74	C₇×#121	—	2	1	no	—
75	C₄×C₇×Q₈	C₂×C₄	1	1	yes	characteristic
76	C₄×C₇×Q₈	C₂×C₄	1	1	yes	characteristic
77	C₄×C₇×D₄	—	2	1	no	—
78	C₄×C₇×D₄	—	2	1	no	—
79	C₄×C₇×D₄	—	2	1	no	—
80	C₄×C₇×D₄	—	2	1	no	—
81	C₄×C₇×D₄	C₂×C₄	1	1	yes	characteristic
82	C₄×C₇×D₄	C₂×C₄	1	1	yes	characteristic
83	C₇×#117	—	2	1	no	—
84	C₇×#117	C₂³	1	1	yes	characteristic
85	C₂×C₄²×C₇	—	2	1	no	—
86	C₂×C₄²×C₇	—	2	1	no	—
87	C₂×C₄²×C₇	—	2	1	no	—
88	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
89	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
90	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
91	C₇×#108	—	2	4	no	—
92	C₇×C₄⋊C₈	—	2	2	no	—
93	C₇×C₄⋊C₈	—	2	2	no	—
94	C₇×C₄⋊C₈	—	2	1	no	—
95	C₇×C₄⋊C₈	—	2	1	no	—
96	C₇×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
97	C₇×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
98	C₇×C₄⋊C₈	C₂×C₄	1	2	yes	—
99	C₇×C₄⋊C₈	D₄	1	1	yes	characteristic
100	C₇×C₄⋊C₈	D₄	1	1	yes	characteristic
101	C₇×C₄⋊C₈	D₄	1	2	yes	—
102	C₇×#104	—	2	2	no	—
103	C₇×#104	—	2	4	no	—
104	C₇×#104	—	4	2	no	—
105	C₇×#104	D₄	1	4	yes	complemented
106	C₇×#103	C₂×C₄	1	2	yes	—
107	C₇×#103	C₂×C₄	1	2	yes	—
108	C₇×#102	C₂×C₄	1	2	yes	—
109	C₇×#102	C₂×C₄	1	2	yes	—
110	C₇×C₂²⋊C₈	—	2	2	no	—
111	C₇×C₂²⋊C₈	—	2	2	no	—
112	C₇×#97	—	2	4	no	—
113	C₇×#97	—	2	1	no	—
114	C₄×C₇×C₈	—	2	1	no	—
115	#3035	C₂×C₇	1	1	yes	characteristic
116	C₂×#444	C₂×C₇	1	2	yes	complemented
117	#2935	C₂×C₇	1	1	yes	characteristic, complemented
118	#2906	C₂×C₇	1	1	yes	characteristic, complemented
119	#1962	C₂×C₇	1	1	yes	characteristic, complemented
120	#1950	C₂×C₇	1	1	yes	characteristic, complemented
121	#1938	C₂×C₇	1	2	yes	complemented
122	#1876	C₂×C₇	1	2	yes	complemented
123	#1866	C₂×C₇	1	2	yes	complemented
124	#1767	C₂×C₇	1	1	yes	characteristic, complemented
125	C₄×#104	C₂×C₇	1	1	yes	characteristic, complemented
126	C₇×#41	—	2	1	no	—
127	C₇×#41	—	2	1	no	—
128	C₇×#41	—	2	2	no	—
129	C₇×#41	—	2	2	no	—
130	C₇×#41	—	2	2	no	—
131	C₇×#41	—	2	2	no	—
132	C₇×#41	C₂×D₄	1	2	yes	—
133	C₇×#41	#41	1	2	yes	—
134	C₂×C₇×Q₈	C₂²×C₄	1	2	yes	—
135	C₂×C₇×Q₈	C₂²×C₄	1	1	yes	characteristic
136	C₂×C₇×D₄	—	2	1	no	—
137	C₂×C₇×D₄	—	2	2	no	—
138	C₂×C₇×D₄	—	2	1	no	—
139	C₂×C₇×D₄	—	2	1	no	—
140	C₂×C₇×D₄	C₂²×C₄	1	2	yes	—
141	C₂×C₇×D₄	C₂²×C₄	1	1	yes	characteristic
142	C₂²×C₄×C₇	—	2	1	no	—
143	C₂²×C₄×C₇	—	2	1	no	—
144	C₂²×C₄×C₇	—	2	1	no	—
145	C₂²×C₄×C₇	—	2	1	no	—
146	C₂²×C₄×C₇	—	2	2	no	—
147	C₂²×C₄×C₇	—	2	1	no	—
148	C₂²×C₄×C₇	#41	1	1	yes	characteristic
149	C₂²×C₄×C₇	#41	1	1	yes	characteristic
150	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
151	C₇×Q₁₆	—	2	4	no	—
152	C₇×SD₁₆	—	2	4	no	—
153	C₇×SD₁₆	—	2	4	no	—
154	C₇×D₈	—	2	4	no	—
155	C₇×M₁₆	—	2	1	no	—
156	C₇×M₁₆	—	2	2	no	—
157	C₇×M₁₆	—	2	1	no	—
158	C₇×M₁₆	—	2	2	no	—
159	C₇×M₁₆	—	2	1	no	—
160	C₇×M₁₆	—	4	4	no	—
161	C₇×M₁₆	—	4	4	no	—
162	C₇×M₁₆	C₂×D₄	1	2	yes	—
163	C₂×C₇×C₈	—	2	2	no	—
164	C₂×C₇×C₈	—	2	2	no	—
165	C₂×C₇×C₈	—	2	1	no	—
166	C₂×C₇×C₈	—	2	2	no	—
167	C₂×C₇×C₈	—	2	2	no	—
168	C₂×C₇×C₈	—	2	1	no	—
169	C₂×C₇×C₈	—	2	2	no	—
170	C₂×C₇×C₈	—	4	4	no	—
171	C₂×C₇×C₈	—	4	4	no	—
172	C₂×C₇×C₈	C₂²×C₄	1	2	yes	—
173	C₂×C₇×C₈	C₂²×C₄	1	2	yes	—
174	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
175	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
176	C₇×C₄⋊C₄	—	2	1	no	—
177	C₇×C₄⋊C₄	—	2	1	no	—
178	C₇×C₄⋊C₄	—	2	1	no	—
179	C₇×C₄⋊C₄	—	2	1	no	—
180	C₇×C₄⋊C₄	C₂²×C₄	1	1	yes	characteristic
181	C₇×C₄⋊C₄	C₂²×C₄	1	1	yes	characteristic
182	C₇×C₂²⋊C₄	—	2	1	no	—
183	C₇×C₂²⋊C₄	—	2	1	no	—
184	C₇×C₂²⋊C₄	—	2	1	no	—
185	C₇×C₂²⋊C₄	—	2	2	no	—
186	C₇×C₂²⋊C₄	—	2	2	no	—
187	C₄²×C₇	—	2	1	no	—
188	C₄²×C₇	—	2	1	no	—
189	C₄²×C₇	—	2	1	no	—
190	C₄²×C₇	—	2	1	no	—
191	C₄²×C₇	—	2	1	no	—
192	C₄²×C₇	—	2	1	no	—
193	C₄²×C₇	—	4	2	no	—
194	C₄²×C₇	—	4	2	no	—
195	C₄²×C₇	C₂²×C₄	1	1	yes	characteristic
196	C₄²×C₇	C₂²×C₄	1	1	yes	characteristic
197	C₄²×C₇	C₂×D₄	1	2	yes	—
198	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
199	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
200	C₂×#135	C₂²×C₇	1	2	yes	—
201	C₂×#131	C₂²×C₇	1	2	yes	—
202	C₂×#131	C₂²×C₇	1	2	yes	—
203	#532	C₂²×C₇	1	1	yes	characteristic
204	C₄×#41	C₂²×C₇	1	1	yes	characteristic
205	#464	C₄×C₇	1	2	yes	complemented
206	#463	C₄×C₇	1	2	yes	complemented
207	#462	C₄×C₇	1	1	yes	characteristic, complemented
208	#461	C₄×C₇	1	1	yes	characteristic, complemented
209	#460	C₄×C₇	1	1	yes	characteristic, complemented
210	#459	C₄×C₇	1	1	yes	characteristic, complemented
211	#444	—	2	8	no	—
212	#436	—	2	1	no	—
213	#435	—	2	2	no	—
214	C₈×D₄	—	2	1	no	—
215	C₂×#108	C₂²×C₇	1	2	yes	—
216	#424	—	2	2	no	—
217	#424	C₂²×C₇	1	2	yes	—
218	#423	—	2	1	no	—
219	#423	C₂²×C₇	1	1	yes	characteristic
220	C₂×C₄⋊C₈	—	2	1	no	—
221	C₂×C₄⋊C₈	C₂²×C₇	1	1	yes	characteristic
222	C₂×C₄⋊C₈	C₂²×C₇	1	1	yes	characteristic
223	C₂×#104	C₂²×C₇	1	1	yes	characteristic
224	C₂×#104	C₂²×C₇	1	2	yes	—
225	C₂×#104	C₂²×C₇	1	2	yes	—
226	C₂×#104	C₂²×C₇	1	1	yes	characteristic
227	#416	C₂²×C₇	1	2	yes	—
228	#408	C₂²×C₇	1	2	yes	—
229	C₄×M₁₆	C₂²×C₇	1	1	yes	characteristic
230	C₂×#97	C₂²×C₇	1	2	yes	—
231	C₄³	C₂²×C₇	1	1	yes	characteristic
232	#341	C₂²×C₇	1	2	yes	—
233	#339	C₂²×C₇	1	1	yes	characteristic
234	#339	C₂²×C₇	1	1	yes	characteristic
235	#336	C₂²×C₇	1	2	yes	—
236	C₂³×C₇	—	2	1	no	—
237	C₂³×C₇	—	2	2	no	—
238	C₂³×C₇	C₂×#41	1	1	yes	characteristic
239	C₇×Q₈	—	2	1	no	—
240	C₇×Q₈	—	2	2	no	—
241	C₇×Q₈	—	2	2	no	—
242	C₇×Q₈	C₄×D₄	1	2	yes	—
243	C₇×D₄	—	2	2	no	—
244	C₇×D₄	—	2	2	no	—
245	C₇×D₄	—	2	2	no	—
246	C₇×D₄	—	2	1	no	—
247	C₇×D₄	—	2	2	no	—
248	C₇×D₄	—	4	2	no	—
249	C₇×D₄	—	4	2	no	—
250	C₇×D₄	—	4	2	no	—
251	C₇×D₄	—	8	1	no	—
252	C₇×D₄	C₄×D₄	1	2	yes	—
253	C₂×C₄×C₇	—	2	1	no	—
254	C₂×C₄×C₇	—	2	2	no	—
255	C₂×C₄×C₇	—	2	2	no	—
256	C₂×C₄×C₇	—	2	2	no	—
257	C₂×C₄×C₇	—	2	2	no	—
258	C₂×C₄×C₇	—	2	1	no	—
259	C₂×C₄×C₇	—	2	1	no	—
260	C₂×C₄×C₇	—	2	2	no	—
261	C₂×C₄×C₇	—	2	2	no	—
262	C₂×C₄×C₇	—	4	1	no	—
263	C₂×C₄×C₇	—	4	1	no	—
264	C₂×C₄×C₇	—	4	1	no	—
265	C₂×C₄×C₇	—	4	1	no	—
266	C₂×C₄×C₇	—	4	1	no	—
267	C₂×C₄×C₇	—	4	2	no	—
268	C₂×C₄×C₇	—	4	2	no	—
269	C₂×C₄×C₇	—	4	2	no	—
270	C₂×C₄×C₇	—	8	1	no	—
271	C₂×C₄×C₇	C₄×D₄	1	1	yes	characteristic
272	C₂×C₄×C₇	C₄×D₄	1	1	yes	characteristic
273	C₂×C₄×C₇	C₄×D₄	1	1	yes	characteristic
274	C₂×C₄×C₇	C₄×D₄	1	1	yes	characteristic
275	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
276	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
277	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
278	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
279	C₂×C₄×C₇	H(5,𝔽₂)	1	1	yes	characteristic
280	C₂×C₄×C₇	2₋¹⁺⁴	1	1	yes	characteristic
281	C₇×C₈	—	2	2	no	—
282	C₇×C₈	—	2	2	no	—
283	C₇×C₈	—	2	2	no	—
284	C₇×C₈	—	2	2	no	—
285	C₇×C₈	—	2	2	no	—
286	C₇×C₈	—	4	2	no	—
287	C₇×C₈	—	4	1	no	—
288	C₂×#41	C₂³×C₇	1	2	yes	—
289	C₂×#41	C₂³×C₇	1	1	yes	characteristic
290	#135	—	2	4	no	—
291	#135	—	2	4	no	—
292	C₂×Q₁₆	C₂×C₄×C₇	1	2	yes	—
293	C₂×SD₁₆	C₂×C₄×C₇	1	2	yes	—
294	C₂×SD₁₆	C₂×C₄×C₇	1	2	yes	—
295	C₂×D₈	C₂×C₄×C₇	1	2	yes	—
296	#131	—	2	4	no	—
297	#131	—	2	4	no	—
298	#131	—	4	2	no	—
299	#131	—	4	2	no	—
300	C₂×M₁₆	—	2	2	no	—
301	C₂×M₁₆	—	2	2	no	—
302	C₂×M₁₆	C₂³×C₇	1	2	yes	—
303	C₂×M₁₆	C₂³×C₇	1	1	yes	characteristic
304	C₂×M₁₆	C₂³×C₇	1	1	yes	characteristic
305	C₂²×C₈	—	2	2	no	—
306	C₂²×C₈	—	2	2	no	—
307	C₂²×C₈	C₂³×C₇	1	2	yes	—
308	C₂²×C₈	C₂³×C₇	1	2	yes	—
309	C₄⋊Q₈	C₂×C₄×C₇	1	1	yes	characteristic
310	#127	C₂×C₄×C₇	1	1	yes	characteristic
311	#124	C₂×C₄×C₇	1	2	yes	—
312	#121	—	2	1	no	—
313	#121	—	2	1	no	—
314	C₄×Q₈	C₂×C₄×C₇	1	1	yes	characteristic
315	C₄×Q₈	C₂×C₄×C₇	1	1	yes	characteristic
316	C₄×D₄	—	2	1	no	—
317	C₄×D₄	—	2	1	no	—
318	C₄×D₄	—	2	1	no	—
319	C₄×D₄	—	2	1	no	—
320	C₄×D₄	C₂×C₄×C₇	1	1	yes	characteristic
321	C₄×D₄	C₂×C₄×C₇	1	1	yes	characteristic
322	#117	—	2	1	no	—
323	#117	C₂³×C₇	1	1	yes	characteristic
324	C₂×C₄²	—	2	1	no	—
325	C₂×C₄²	—	2	1	no	—
326	C₂×C₄²	—	2	1	no	—
327	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
328	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
329	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
330	#108	—	2	4	no	—
331	C₄⋊C₈	—	2	2	no	—
332	C₄⋊C₈	—	2	2	no	—
333	C₄⋊C₈	—	2	1	no	—
334	C₄⋊C₈	—	2	1	no	—
335	C₄⋊C₈	C₂×C₄×C₇	1	1	yes	characteristic
336	C₄⋊C₈	C₂×C₄×C₇	1	1	yes	characteristic
337	C₄⋊C₈	C₂×C₄×C₇	1	2	yes	—
338	C₄⋊C₈	C₇×D₄	1	1	yes	characteristic
339	C₄⋊C₈	C₇×D₄	1	1	yes	characteristic
340	C₄⋊C₈	C₇×D₄	1	2	yes	—
341	#104	—	2	2	no	—
342	#104	—	2	4	no	—
343	#104	—	4	2	no	—
344	#104	C₇×D₄	1	4	yes	complemented
345	#103	C₂×C₄×C₇	1	2	yes	—
346	#103	C₂×C₄×C₇	1	2	yes	—
347	#102	C₂×C₄×C₇	1	2	yes	—
348	#102	C₂×C₄×C₇	1	2	yes	—
349	C₂²⋊C₈	—	2	2	no	—
350	C₂²⋊C₈	—	2	2	no	—
351	#97	—	2	4	no	—
352	#97	—	2	1	no	—
353	C₄×C₈	—	2	1	no	—
354	C₂²×C₇	—	2	1	no	—
355	C₂²×C₇	—	2	1	no	—
356	C₂²×C₇	—	2	2	no	—
357	C₂²×C₇	—	4	1	no	—
358	C₂²×C₇	—	4	1	no	—
359	C₂²×C₇	—	4	2	no	—
360	C₂²×C₇	—	4	2	no	—
361	C₂²×C₇	—	4	2	no	—
362	C₂²×C₇	C₂×C₄×D₄	1	1	yes	characteristic
363	C₂²×C₇	#547	1	1	yes	characteristic
364	C₂²×C₇	#553	1	1	yes	characteristic
365	C₄×C₇	—	2	2	no	—
366	C₄×C₇	—	2	1	no	—
367	C₄×C₇	—	2	1	no	—
368	C₄×C₇	—	2	1	no	—
369	C₄×C₇	—	4	2	no	—
370	C₄×C₇	—	4	2	no	—
371	C₄×C₇	—	4	2	no	—
372	C₄×C₇	—	4	1	no	—
373	C₄×C₇	—	4	1	no	—
374	C₄×C₇	#444	1	2	yes	complemented
375	C₄×C₇	C₂×C₄×D₄	1	1	yes	characteristic
376	C₄×C₇	#520	1	1	yes	characteristic
377	C₄×C₇	D₄²	1	1	yes	characteristic
378	C₄×C₇	#548	1	1	yes	characteristic
379	#41	—	2	1	no	—
380	#41	—	2	1	no	—
381	#41	—	2	2	no	—
382	#41	—	2	2	no	—
383	#41	—	2	2	no	—
384	#41	—	2	2	no	—
385	#41	C₂×C₇×D₄	1	2	yes	—
386	#41	C₇×#41	1	2	yes	—
387	C₂×Q₈	C₂²×C₄×C₇	1	2	yes	—
388	C₂×Q₈	C₂²×C₄×C₇	1	1	yes	characteristic
389	C₂×D₄	—	2	1	no	—
390	C₂×D₄	—	2	2	no	—
391	C₂×D₄	—	2	1	no	—
392	C₂×D₄	—	2	1	no	—
393	C₂×D₄	C₂²×C₄×C₇	1	2	yes	—
394	C₂×D₄	C₂²×C₄×C₇	1	1	yes	characteristic
395	C₂²×C₄	—	2	1	no	—
396	C₂²×C₄	—	2	1	no	—
397	C₂²×C₄	—	2	1	no	—
398	C₂²×C₄	—	2	1	no	—
399	C₂²×C₄	—	2	2	no	—
400	C₂²×C₄	—	2	1	no	—
401	C₂²×C₄	C₇×#41	1	1	yes	characteristic
402	C₂²×C₄	C₇×#41	1	1	yes	characteristic
403	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
404	Q₁₆	—	2	4	no	—
405	SD₁₆	—	2	4	no	—
406	SD₁₆	—	2	4	no	—
407	D₈	—	2	4	no	—
408	M₁₆	—	2	1	no	—
409	M₁₆	—	2	2	no	—
410	M₁₆	—	2	1	no	—
411	M₁₆	—	2	2	no	—
412	M₁₆	—	2	1	no	—
413	M₁₆	—	4	4	no	—
414	M₁₆	—	4	4	no	—
415	M₁₆	C₂×C₇×D₄	1	2	yes	—
416	C₂×C₈	—	2	1	no	—
417	C₂×C₈	—	2	2	no	—
418	C₂×C₈	—	2	2	no	—
419	C₂×C₈	—	2	2	no	—
420	C₂×C₈	—	2	2	no	—
421	C₂×C₈	—	2	1	no	—
422	C₂×C₈	—	2	2	no	—
423	C₂×C₈	—	4	4	no	—
424	C₂×C₈	—	4	4	no	—
425	C₂×C₈	C₂²×C₄×C₇	1	2	yes	—
426	C₂×C₈	C₂²×C₄×C₇	1	2	yes	—
427	C₂×C₈	C₂×C₇×D₄	1	2	yes	—
428	C₂×C₈	C₂×C₇×D₄	1	2	yes	—
429	C₄⋊C₄	—	2	1	no	—
430	C₄⋊C₄	—	2	1	no	—
431	C₄⋊C₄	—	2	1	no	—
432	C₄⋊C₄	—	2	1	no	—
433	C₄⋊C₄	C₂²×C₄×C₇	1	1	yes	characteristic
434	C₄⋊C₄	C₂²×C₄×C₇	1	1	yes	characteristic
435	C₂²⋊C₄	—	2	1	no	—
436	C₂²⋊C₄	—	2	1	no	—
437	C₂²⋊C₄	—	2	1	no	—
438	C₂²⋊C₄	—	2	2	no	—
439	C₂²⋊C₄	—	2	2	no	—
440	C₄²	—	2	1	no	—
441	C₄²	—	2	1	no	—
442	C₄²	—	2	1	no	—
443	C₄²	—	2	1	no	—
444	C₄²	—	2	1	no	—
445	C₄²	—	2	1	no	—
446	C₄²	—	4	2	no	—
447	C₄²	—	4	2	no	—
448	C₄²	C₂²×C₄×C₇	1	1	yes	characteristic
449	C₄²	C₂²×C₄×C₇	1	1	yes	characteristic
450	C₄²	C₂×C₇×D₄	1	2	yes	—
451	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
452	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
453	C₂×C₇	—	2	1	no	—
454	C₂×C₇	—	2	1	no	—
455	C₂×C₇	—	4	2	no	—
456	C₂×C₇	—	8	2	no	—
457	C₂×C₇	#2361	1	1	yes	characteristic
458	C₂×C₇	C₂×#444	1	1	yes	characteristic
459	C₂×C₇	#2956	1	1	yes	characteristic
460	C₂³	—	2	1	no	—
461	C₂³	—	2	2	no	—
462	C₂³	C₂×C₇×#41	1	1	yes	characteristic
463	Q₈	—	2	1	no	—
464	Q₈	—	2	2	no	—
465	Q₈	—	2	2	no	—
466	Q₈	C₄×C₇×D₄	1	2	yes	—
467	D₄	—	2	2	no	—
468	D₄	—	2	2	no	—
469	D₄	—	2	2	no	—
470	D₄	—	2	1	no	—
471	D₄	—	2	2	no	—
472	D₄	—	4	2	no	—
473	D₄	—	4	2	no	—
474	D₄	—	4	2	no	—
475	D₄	—	8	1	no	—
476	D₄	C₄×C₇×D₄	1	2	yes	—
477	C₂×C₄	—	2	1	no	—
478	C₂×C₄	—	2	2	no	—
479	C₂×C₄	—	2	2	no	—
480	C₂×C₄	—	2	2	no	—
481	C₂×C₄	—	2	2	no	—
482	C₂×C₄	—	2	1	no	—
483	C₂×C₄	—	2	1	no	—
484	C₂×C₄	—	2	2	no	—
485	C₂×C₄	—	2	2	no	—
486	C₂×C₄	—	4	1	no	—
487	C₂×C₄	—	4	1	no	—
488	C₂×C₄	—	4	1	no	—
489	C₂×C₄	—	4	1	no	—
490	C₂×C₄	—	4	1	no	—
491	C₂×C₄	—	4	2	no	—
492	C₂×C₄	—	4	2	no	—
493	C₂×C₄	—	4	2	no	—
494	C₂×C₄	—	8	1	no	—
495	C₂×C₄	C₄×C₇×D₄	1	1	yes	characteristic
496	C₂×C₄	C₄×C₇×D₄	1	1	yes	characteristic
497	C₂×C₄	C₄×C₇×D₄	1	1	yes	characteristic
498	C₂×C₄	C₄×C₇×D₄	1	1	yes	characteristic
499	C₂×C₄	C₂³×C₄×C₇	1	1	yes	characteristic
500	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
501	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
502	C₂×C₄	C₂×C₇×#41	1	1	yes	characteristic
503	C₂×C₄	C₇×H(5,𝔽₂)	1	1	yes	characteristic
504	C₂×C₄	C₇×2₋¹⁺⁴	1	1	yes	characteristic
505	C₈	—	2	2	no	—
506	C₈	—	2	2	no	—
507	C₈	—	2	2	no	—
508	C₈	—	2	2	no	—
509	C₈	—	2	2	no	—
510	C₈	—	4	2	no	—
511	C₈	—	4	1	no	—
512	C₇	#23016	1	1	yes	characteristic, direct product factor, Sylow
513	C₂²	—	2	1	no	—
514	C₂²	—	2	1	no	—
515	C₂²	—	2	2	no	—
516	C₂²	—	4	1	no	—
517	C₂²	—	4	1	no	—
518	C₂²	—	4	2	no	—
519	C₂²	—	4	2	no	—
520	C₂²	—	4	2	no	—
521	C₂²	C₂×C₄×C₇×D₄	1	1	yes	characteristic
522	C₂²	C₇×#547	1	1	yes	characteristic
523	C₂²	C₇×#553	1	1	yes	characteristic
524	C₄	—	2	1	no	—
525	C₄	—	2	1	no	—
526	C₄	—	2	1	no	—
527	C₄	—	2	2	no	—
528	C₄	—	4	2	no	—
529	C₄	—	4	2	no	—
530	C₄	—	4	1	no	—
531	C₄	—	4	1	no	—
532	C₄	—	4	2	no	—
533	C₄	C₇×#444	1	2	yes	complemented
534	C₄	C₂×C₄×C₇×D₄	1	1	yes	characteristic
535	C₄	C₇×#520	1	1	yes	characteristic
536	C₄	C₇×D₄²	1	1	yes	characteristic
537	C₄	C₇×#548	1	1	yes	characteristic
538	C₂	—	2	1	no	—
539	C₂	—	2	1	no	—
540	C₂	—	4	2	no	—
541	C₂	—	8	2	no	—
542	C₂	C₇×#2361	1	1	yes	characteristic
543	C₂	C₂×C₇×#444	1	1	yes	characteristic
544	C₂	C₇×#2956	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₇	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₇
socle	C₂²×C₇	C₂×C₄×D₄
prime radical	C₇×#3035	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₇ ◁ C₂×C₇×C₄⋊C₈ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₇×#444	C₂	semidirect
3. C₇×#2935	C₂	semidirect
4. C₇×#2906	C₂	semidirect
5. C₇×#1962	C₂	semidirect
6. C₇×#1950	C₂	semidirect
7. C₇×#1938	C₂	semidirect
8. C₇×#1876	C₂	semidirect
9. C₇×#1866	C₂	semidirect
10. C₇×#1767	C₂	semidirect
11. C₄×C₇×#104	C₂	semidirect
17. C₇×#464	C₄	semidirect
18. C₇×#463	C₄	semidirect
19. C₇×#462	C₄	semidirect
20. C₇×#461	C₄	semidirect
21. C₇×#460	C₄	semidirect
22. C₇×#459	C₄	semidirect
48. #23016	C₇	direct (N × H)
105. C₇×#104	D₄	semidirect
116. C₂×#444	C₂×C₇	semidirect
117. #2935	C₂×C₇	semidirect
118. #2906	C₂×C₇	semidirect
119. #1962	C₂×C₇	semidirect
120. #1950	C₂×C₇	semidirect
121. #1938	C₂×C₇	semidirect
122. #1876	C₂×C₇	semidirect
123. #1866	C₂×C₇	semidirect
124. #1767	C₂×C₇	semidirect
125. C₄×#104	C₂×C₇	semidirect
205. #464	C₄×C₇	semidirect
206. #463	C₄×C₇	semidirect
207. #462	C₄×C₇	semidirect
208. #461	C₄×C₇	semidirect
209. #460	C₄×C₇	semidirect
210. #459	C₄×C₇	semidirect
344. #104	C₇×D₄	semidirect
374. C₄×C₇	#444	semidirect
512. C₇	#23016	direct (N × H)
533. C₄	C₇×#444	semidirect

Maximal subgroups

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

subgroup	index
C₇×#3035	2
C₂×C₇×#444	2
C₇×#2935	2
C₇×#2906	2
C₇×#1962	2
C₇×#1950	2
C₇×#1938	2
C₇×#1876	2
C₇×#1866	2
C₇×#1767	2
C₄×C₇×#104	2
#23016	7

Subgroups — C7×#23016 #13200232

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#23016 #13200232