Subgroups — C₇×#61321 #13238537

Order 1792.

2126 conjugacy classes of subgroups.

The subgroup lattice has 526 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×Q₈×#41	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₇×#550	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇×#3368	C₂	1	6	yes	maximal, complemented
4	C₇×#3366	C₂	1	6	yes	maximal, complemented
5	C₇×#3362	C₂	1	2	yes	maximal, complemented
6	C₇×#3358	C₂	1	2	yes	maximal, complemented
7	C₇×#3138	C₂	1	3	yes	maximal, complemented
8	C₇×#3077	C₂	1	3	yes	maximal, complemented
9	C₇×#3038	C₂	1	3	yes	maximal, complemented
10	C₇×Q₈×M₁₆	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×C₇×AGL(1,ℤ/8)	C₂	1	3	yes	maximal, complemented
12	C₂²×C₇×#41	—	2	1	no	—
13	C₂×C₇×AGL(1,ℤ/8)	C₂²	1	3	yes	—
14	C₇×Q₈²	C₂²	1	1	yes	characteristic, complemented
15	C₇×#557	C₂²	1	3	yes	complemented
16	C₇×#554	—	2	3	no	—
17	C₇×#554	C₂²	1	3	yes	complemented
18	C₇×#550	—	2	1	no	—
19	C₇×#550	—	2	3	no	—
20	C₇×#550	C₂²	1	2	yes	—
21	C₇×#550	C₂²	1	6	yes	—
22	C₇×D₄×Q₈	—	2	1	no	—
23	C₇×D₄×Q₈	C₂²	1	1	yes	characteristic, complemented
24	C₇×#533	—	2	3	no	—
25	C₇×#533	C₂²	1	3	yes	—
26	C₂×C₇×#127	C₂²	1	3	yes	—
27	C₂×C₇×#121	—	2	3	no	—
28	C₄×C₇×#41	C₂²	1	3	yes	—
29	C₂×C₄×C₇×Q₈	—	2	1	no	—
30	C₂×C₄×C₇×Q₈	C₂²	1	1	yes	characteristic
31	C₂×C₄×C₇×D₄	C₂²	1	3	yes	—
32	C₇×#490	C₂²	1	12	yes	complemented
33	C₇×#486	C₂²	1	6	yes	complemented
34	C₇×#486	C₂²	1	6	yes	complemented
35	C₇×#478	C₂²	1	12	yes	complemented
36	C₇×#476	C₂²	1	6	yes	complemented
37	C₇×#474	C₂²	1	6	yes	complemented
38	C₇×#460	C₂²	1	6	yes	complemented
39	C₇×#460	C₂²	1	6	yes	complemented
40	C₇×#459	C₂²	1	6	yes	complemented
41	C₇×#459	C₂²	1	6	yes	complemented
42	C₇×#446	C₂²	1	6	yes	complemented
43	C₇×C₈×Q₈	C₂²	1	2	yes	complemented
44	C₇×#442	C₂²	1	6	yes	complemented
45	C₇×#440	C₂²	1	6	yes	complemented
46	C₄×C₇×SD₁₆	C₂²	1	6	yes	complemented
47	C₄×C₇×D₈	C₂²	1	6	yes	complemented
48	C₇×#428	C₂²	1	3	yes	—
49	C₇×#423	C₂²	1	3	yes	—
50	C₇×#418	C₂²	1	3	yes	—
51	C₇×#418	C₂²	1	6	yes	—
52	C₇×#417	C₂²	1	1	yes	characteristic
53	C₄×C₇×M₁₆	C₂²	1	3	yes	—
54	#61321	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
55	C₂×C₇×#41	—	2	6	no	—
56	C₂×C₇×#41	—	2	2	no	—
57	C₂×C₇×#41	—	4	8	no	—
58	C₂×C₇×#41	C₂³	1	1	yes	characteristic
59	C₂²×C₇×Q₈	—	2	1	no	—
60	C₂²×C₇×Q₈	D₄	1	1	yes	characteristic
61	C₂²×C₇×D₄	—	2	3	no	—
62	C₂²×C₇×D₄	—	2	3	no	—
63	C₂²×C₇×D₄	C₂³	1	3	yes	—
64	C₂³×C₄×C₇	—	2	3	no	—
65	C₇×AGL(1,ℤ/8)	—	2	12	no	—
66	C₂×C₇×SD₁₆	C₂³	1	6	yes	—
67	C₂×C₇×D₈	C₂³	1	6	yes	—
68	C₂×C₇×M₁₆	C₂³	1	1	yes	characteristic
69	C₂×C₇×M₁₆	C₂³	1	3	yes	—
70	C₇×C₄⋊Q₈	—	2	3	no	—
71	C₇×C₄⋊Q₈	—	2	3	no	—
72	C₇×C₄⋊Q₈	C₂³	1	3	yes	—
73	C₇×C₄⋊Q₈	C₂³	1	3	yes	—
74	C₇×#127	—	2	3	no	—
75	C₇×#127	—	2	3	no	—
76	C₇×#127	C₂³	1	6	yes	—
77	C₇×#127	C₂³	1	6	yes	—
78	C₇×#125	—	2	6	no	—
79	C₇×#125	C₂³	1	6	yes	—
80	C₇×C₂²⋊Q₈	—	2	3	no	—
81	C₇×C₂²⋊Q₈	—	2	3	no	—
82	C₇×C₂²⋊Q₈	—	2	3	no	—
83	C₇×C₂²⋊Q₈	—	2	3	no	—
84	C₇×C₂²⋊Q₈	—	2	6	no	—
85	C₇×#121	—	2	6	no	—
86	C₇×#121	—	2	12	no	—
87	C₇×#121	—	2	6	no	—
88	C₄×C₇×Q₈	—	2	1	no	—
89	C₄×C₇×Q₈	—	2	3	no	—
90	C₄×C₇×Q₈	—	2	3	no	—
91	C₄×C₇×Q₈	—	2	3	no	—
92	C₄×C₇×Q₈	D₄	1	1	yes	characteristic
93	C₄×C₇×Q₈	D₄	1	3	yes	—
94	C₄×C₇×Q₈	C₂³	1	3	yes	—
95	C₄×C₇×Q₈	C₂³	1	3	yes	—
96	C₄×C₇×Q₈	C₂³	1	1	yes	characteristic
97	C₄×C₇×Q₈	C₂³	1	3	yes	—
98	C₄×C₇×D₄	—	2	3	no	—
99	C₄×C₇×D₄	—	2	3	no	—
100	C₄×C₇×D₄	—	2	3	no	—
101	C₄×C₇×D₄	—	2	3	no	—
102	C₄×C₇×D₄	C₂³	1	6	yes	—
103	C₄×C₇×D₄	C₂³	1	6	yes	—
104	C₄×C₇×D₄	C₂³	1	3	yes	—
105	C₄×C₇×D₄	C₂³	1	3	yes	—
106	C₇×#117	—	2	3	no	—
107	C₇×#117	C₂³	1	3	yes	—
108	C₂×C₇×C₄⋊C₄	—	2	3	no	—
109	C₂×C₇×C₄⋊C₄	D₄	1	3	yes	—
110	C₂×C₇×C₄⋊C₄	C₂³	1	1	yes	characteristic
111	C₂×C₇×C₂²⋊C₄	—	2	3	no	—
112	C₂×C₄²×C₇	—	2	3	no	—
113	C₂×C₄²×C₇	C₂³	1	3	yes	—
114	C₇×#107	C₂³	1	6	yes	—
115	C₇×#106	C₂³	1	6	yes	—
116	C₇×C₄⋊C₈	C₂³	1	6	yes	—
117	C₇×C₄⋊C₈	C₂³	1	6	yes	—
118	C₇×#103	C₂³	1	2	yes	—
119	C₇×#102	C₂³	1	12	yes	—
120	C₇×#102	C₂³	1	12	yes	—
121	C₇×#102	C₂³	1	6	yes	—
122	C₇×#102	C₂³	1	6	yes	—
123	C₇×#102	C₂³	1	2	yes	—
124	C₇×#97	C₂³	1	6	yes	—
125	C₄×C₇×C₈	C₂³	1	6	yes	—
126	Q₈×#41	C₂×C₇	1	1	yes	characteristic, complemented
127	C₂×#550	C₂×C₇	1	1	yes	characteristic, complemented
128	#3368	C₂×C₇	1	6	yes	complemented
129	#3366	C₂×C₇	1	6	yes	complemented
130	#3362	C₂×C₇	1	2	yes	complemented
131	#3358	C₂×C₇	1	2	yes	complemented
132	#3138	C₂×C₇	1	3	yes	complemented
133	#3077	C₂×C₇	1	3	yes	complemented
134	#3038	C₂×C₇	1	3	yes	complemented
135	Q₈×M₁₆	C₂×C₇	1	1	yes	characteristic, complemented
136	C₄×AGL(1,ℤ/8)	C₂×C₇	1	3	yes	complemented
137	C₂⁴×C₇	—	2	3	no	—
138	C₇×#41	—	2	2	no	—
139	C₇×#41	—	4	24	no	—
140	C₇×#41	—	4	8	no	—
141	C₇×#41	#41	1	4	yes	complemented
142	C₂×C₇×Q₈	—	2	4	no	—
143	C₂×C₇×Q₈	—	2	3	no	—
144	C₂×C₇×Q₈	—	2	3	no	—
145	C₂×C₇×Q₈	—	4	4	no	—
146	C₂×C₇×Q₈	C₂×D₄	1	3	yes	—
147	C₂×C₇×Q₈	C₂×D₄	1	1	yes	characteristic
148	C₂×C₇×Q₈	C₂⁴	1	1	yes	characteristic
149	C₂×C₇×D₄	—	2	1	no	—
150	C₂×C₇×D₄	—	2	3	no	—
151	C₂×C₇×D₄	—	2	6	no	—
152	C₂×C₇×D₄	—	2	6	no	—
153	C₂×C₇×D₄	—	2	6	no	—
154	C₂×C₇×D₄	—	2	6	no	—
155	C₂×C₇×D₄	—	2	6	no	—
156	C₂×C₇×D₄	—	2	6	no	—
157	C₂×C₇×D₄	—	4	12	no	—
158	C₂×C₇×D₄	—	8	6	no	—
159	C₂×C₇×D₄	C₂⁴	1	6	yes	—
160	C₂×C₇×D₄	C₂⁴	1	1	yes	characteristic
161	C₂²×C₄×C₇	—	2	1	no	—
162	C₂²×C₄×C₇	—	2	1	no	—
163	C₂²×C₄×C₇	—	2	3	no	—
164	C₂²×C₄×C₇	—	2	6	no	—
165	C₂²×C₄×C₇	—	2	6	no	—
166	C₂²×C₄×C₇	—	2	3	no	—
167	C₂²×C₄×C₇	—	4	12	no	—
168	C₂²×C₄×C₇	C₂×D₄	1	3	yes	—
169	C₂²×C₄×C₇	C₂×D₄	1	3	yes	—
170	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
171	C₇×SD₁₆	—	2	12	no	—
172	C₇×D₈	—	2	12	no	—
173	C₇×M₁₆	—	2	6	no	—
174	C₇×M₁₆	—	2	2	no	—
175	C₂×C₇×C₈	C₂⁴	1	2	yes	—
176	C₂×C₇×C₈	C₂⁴	1	6	yes	—
177	C₇×C₄⋊C₄	—	2	6	no	—
178	C₇×C₄⋊C₄	—	2	3	no	—
179	C₇×C₄⋊C₄	—	2	1	no	—
180	C₇×C₄⋊C₄	—	2	3	no	—
181	C₇×C₄⋊C₄	—	2	3	no	—
182	C₇×C₄⋊C₄	—	2	3	no	—
183	C₇×C₄⋊C₄	—	2	6	no	—
184	C₇×C₄⋊C₄	C₂×D₄	1	3	yes	—
185	C₇×C₄⋊C₄	C₂×D₄	1	6	yes	—
186	C₇×C₄⋊C₄	C₂×D₄	1	3	yes	—
187	C₇×C₄⋊C₄	C₂⁴	1	3	yes	—
188	C₇×C₄⋊C₄	C₂⁴	1	3	yes	—
189	C₇×C₄⋊C₄	C₂⁴	1	2	yes	—
190	C₇×C₂²⋊C₄	—	2	3	no	—
191	C₇×C₂²⋊C₄	—	2	6	no	—
192	C₇×C₂²⋊C₄	—	2	6	no	—
193	C₇×C₂²⋊C₄	—	2	3	no	—
194	C₇×C₂²⋊C₄	—	2	3	no	—
195	C₄²×C₇	—	2	3	no	—
196	C₄²×C₇	—	2	3	no	—
197	C₄²×C₇	—	2	3	no	—
198	C₄²×C₇	C₂×D₄	1	3	yes	—
199	C₄²×C₇	C₂×D₄	1	3	yes	—
200	C₄²×C₇	C₂⁴	1	3	yes	—
201	C₄²×C₇	C₂⁴	1	3	yes	—
202	C₂²×#41	—	2	1	no	—
203	C₂×AGL(1,ℤ/8)	C₂²×C₇	1	3	yes	—
204	Q₈²	C₂²×C₇	1	1	yes	characteristic, complemented
205	#557	C₂²×C₇	1	3	yes	complemented
206	#554	—	2	3	no	—
207	#554	C₂²×C₇	1	3	yes	complemented
208	#550	—	2	1	no	—
209	#550	—	2	3	no	—
210	#550	C₂²×C₇	1	2	yes	—
211	#550	C₂²×C₇	1	6	yes	—
212	D₄×Q₈	—	2	1	no	—
213	D₄×Q₈	C₂²×C₇	1	1	yes	characteristic, complemented
214	#533	—	2	3	no	—
215	#533	C₂²×C₇	1	3	yes	—
216	C₂×#127	C₂²×C₇	1	3	yes	—
217	C₂×#121	—	2	3	no	—
218	C₄×#41	C₂²×C₇	1	3	yes	—
219	C₂×C₄×Q₈	—	2	1	no	—
220	C₂×C₄×Q₈	C₂²×C₇	1	1	yes	characteristic
221	C₂×C₄×D₄	C₂²×C₇	1	3	yes	—
222	#490	C₂²×C₇	1	12	yes	complemented
223	#486	C₂²×C₇	1	6	yes	complemented
224	#486	C₂²×C₇	1	6	yes	complemented
225	#478	C₂²×C₇	1	12	yes	complemented
226	#476	C₂²×C₇	1	6	yes	complemented
227	#474	C₂²×C₇	1	6	yes	complemented
228	#460	C₂²×C₇	1	6	yes	complemented
229	#460	C₂²×C₇	1	6	yes	complemented
230	#459	C₂²×C₇	1	6	yes	complemented
231	#459	C₂²×C₇	1	6	yes	complemented
232	#446	C₂²×C₇	1	6	yes	complemented
233	C₈×Q₈	C₂²×C₇	1	2	yes	complemented
234	#442	C₂²×C₇	1	6	yes	complemented
235	#440	C₂²×C₇	1	6	yes	complemented
236	C₄×SD₁₆	C₂²×C₇	1	6	yes	complemented
237	C₄×D₈	C₂²×C₇	1	6	yes	complemented
238	#428	C₂²×C₇	1	3	yes	—
239	#423	C₂²×C₇	1	3	yes	—
240	#418	C₂²×C₇	1	6	yes	—
241	#418	C₂²×C₇	1	3	yes	—
242	#417	C₂²×C₇	1	1	yes	characteristic
243	C₄×M₁₆	C₂²×C₇	1	3	yes	—
244	C₂³×C₇	—	2	1	no	—
245	C₂³×C₇	—	2	6	no	—
246	C₂³×C₇	—	4	6	no	—
247	C₂³×C₇	—	4	6	no	—
248	C₂³×C₇	—	4	6	no	—
249	C₂³×C₇	C₂²×D₄	1	1	yes	characteristic
250	C₇×Q₈	—	2	6	no	—
251	C₇×Q₈	—	2	1	no	—
252	C₇×Q₈	—	4	6	no	—
253	C₇×Q₈	—	8	3	no	—
254	C₇×Q₈	AGL(1,ℤ/8)	1	4	yes	complemented
255	C₇×Q₈	C₂×#41	1	2	yes	complemented
256	C₇×D₄	—	2	4	no	—
257	C₇×D₄	—	2	6	no	—
258	C₇×D₄	—	2	1	no	—
259	C₇×D₄	—	2	6	no	—
260	C₇×D₄	—	4	12	no	—
261	C₇×D₄	—	4	6	no	—
262	C₇×D₄	—	4	12	no	—
263	C₇×D₄	—	8	6	no	—
264	C₇×D₄	—	8	6	no	—
265	C₇×D₄	C₂×#41	1	2	yes	complemented
266	C₂×C₄×C₇	—	2	1	no	—
267	C₂×C₄×C₇	—	2	4	no	—
268	C₂×C₄×C₇	—	2	6	no	—
269	C₂×C₄×C₇	—	2	3	no	—
270	C₂×C₄×C₇	—	2	3	no	—
271	C₂×C₄×C₇	—	2	2	no	—
272	C₂×C₄×C₇	—	4	3	no	—
273	C₂×C₄×C₇	—	4	12	no	—
274	C₂×C₄×C₇	—	4	2	no	—
275	C₂×C₄×C₇	—	4	12	no	—
276	C₂×C₄×C₇	—	4	6	no	—
277	C₂×C₄×C₇	—	8	3	no	—
278	C₂×C₄×C₇	C₂²×D₄	1	3	yes	—
279	C₂×C₄×C₇	C₂²×D₄	1	3	yes	—
280	C₂×C₄×C₇	C₂²×D₄	1	3	yes	—
281	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
282	C₂×C₄×C₇	C₂²×D₄	1	3	yes	—
283	C₂×C₄×C₇	C₂×#41	1	2	yes	—
284	C₂×C₄×C₇	2₋¹⁺⁴	1	2	yes	—
285	C₂×C₄×C₇	C₂⁵	1	1	yes	characteristic
286	C₇×C₈	—	2	6	no	—
287	C₇×C₈	—	2	2	no	—
288	C₂×#41	—	2	6	no	—
289	C₂×#41	—	2	2	no	—
290	C₂×#41	—	4	8	no	—
291	C₂×#41	C₂³×C₇	1	1	yes	characteristic
292	C₂²×Q₈	—	2	1	no	—
293	C₂²×Q₈	C₇×D₄	1	1	yes	characteristic
294	C₂²×D₄	—	2	3	no	—
295	C₂²×D₄	—	2	3	no	—
296	C₂²×D₄	C₂³×C₇	1	3	yes	—
297	C₂³×C₄	—	2	3	no	—
298	AGL(1,ℤ/8)	—	2	12	no	—
299	C₂×SD₁₆	C₂³×C₇	1	6	yes	—
300	C₂×D₈	C₂³×C₇	1	6	yes	—
301	C₂×M₁₆	C₂³×C₇	1	3	yes	—
302	C₂×M₁₆	C₂³×C₇	1	1	yes	characteristic
303	C₄⋊Q₈	—	2	3	no	—
304	C₄⋊Q₈	—	2	3	no	—
305	C₄⋊Q₈	C₂³×C₇	1	3	yes	—
306	C₄⋊Q₈	C₂³×C₇	1	3	yes	—
307	#127	—	2	3	no	—
308	#127	—	2	3	no	—
309	#127	C₂³×C₇	1	6	yes	—
310	#127	C₂³×C₇	1	6	yes	—
311	#125	—	2	6	no	—
312	#125	C₂³×C₇	1	6	yes	—
313	C₂²⋊Q₈	—	2	3	no	—
314	C₂²⋊Q₈	—	2	3	no	—
315	C₂²⋊Q₈	—	2	3	no	—
316	C₂²⋊Q₈	—	2	3	no	—
317	C₂²⋊Q₈	—	2	6	no	—
318	#121	—	2	6	no	—
319	#121	—	2	12	no	—
320	#121	—	2	6	no	—
321	C₄×Q₈	—	2	1	no	—
322	C₄×Q₈	—	2	3	no	—
323	C₄×Q₈	—	2	3	no	—
324	C₄×Q₈	—	2	3	no	—
325	C₄×Q₈	C₇×D₄	1	1	yes	characteristic
326	C₄×Q₈	C₇×D₄	1	3	yes	—
327	C₄×Q₈	C₂³×C₇	1	3	yes	—
328	C₄×Q₈	C₂³×C₇	1	3	yes	—
329	C₄×Q₈	C₂³×C₇	1	1	yes	characteristic
330	C₄×Q₈	C₂³×C₇	1	3	yes	—
331	C₄×D₄	—	2	3	no	—
332	C₄×D₄	—	2	3	no	—
333	C₄×D₄	—	2	3	no	—
334	C₄×D₄	—	2	3	no	—
335	C₄×D₄	C₂³×C₇	1	6	yes	—
336	C₄×D₄	C₂³×C₇	1	6	yes	—
337	C₄×D₄	C₂³×C₇	1	3	yes	—
338	C₄×D₄	C₂³×C₇	1	3	yes	—
339	#117	—	2	3	no	—
340	#117	C₂³×C₇	1	3	yes	—
341	C₂×C₄⋊C₄	—	2	3	no	—
342	C₂×C₄⋊C₄	C₇×D₄	1	3	yes	—
343	C₂×C₄⋊C₄	C₂³×C₇	1	1	yes	characteristic
344	C₂×C₂²⋊C₄	—	2	3	no	—
345	C₂×C₄²	—	2	3	no	—
346	C₂×C₄²	C₂³×C₇	1	3	yes	—
347	#107	C₂³×C₇	1	6	yes	—
348	#106	C₂³×C₇	1	6	yes	—
349	C₄⋊C₈	C₂³×C₇	1	6	yes	—
350	C₄⋊C₈	C₂³×C₇	1	6	yes	—
351	#103	C₂³×C₇	1	2	yes	—
352	#102	C₂³×C₇	1	2	yes	—
353	#102	C₂³×C₇	1	6	yes	—
354	#102	C₂³×C₇	1	6	yes	—
355	#102	C₂³×C₇	1	12	yes	—
356	#102	C₂³×C₇	1	12	yes	—
357	#97	C₂³×C₇	1	6	yes	—
358	C₄×C₈	C₂³×C₇	1	6	yes	—
359	C₂²×C₇	—	2	1	no	—
360	C₂²×C₇	—	2	1	no	—
361	C₂²×C₇	—	2	2	no	—
362	C₂²×C₇	—	4	1	no	—
363	C₂²×C₇	—	4	1	no	—
364	C₂²×C₇	—	4	6	no	—
365	C₂²×C₇	—	4	6	no	—
366	C₂²×C₇	—	4	6	no	—
367	C₂²×C₇	—	8	12	no	—
368	C₂²×C₇	#548	1	2	yes	—
369	C₂²×C₇	C₂³×D₄	1	1	yes	characteristic
370	C₄×C₇	—	2	3	no	—
371	C₄×C₇	—	2	3	no	—
372	C₄×C₇	—	2	2	no	—
373	C₄×C₇	—	2	3	no	—
374	C₄×C₇	—	4	3	no	—
375	C₄×C₇	—	4	2	no	—
376	C₄×C₇	—	4	3	no	—
377	C₄×C₇	#548	1	2	yes	—
378	C₄×C₇	C₂×AGL(1,ℤ/8)	1	6	yes	—
379	C₄×C₇	C₂²×#41	1	1	yes	characteristic
380	C₄×C₇	C₂×2₋¹⁺⁴	1	1	yes	characteristic
381	C₂⁴	—	2	3	no	—
382	#41	—	2	2	no	—
383	#41	—	4	24	no	—
384	#41	—	4	8	no	—
385	#41	C₇×#41	1	4	yes	complemented
386	C₂×Q₈	—	2	4	no	—
387	C₂×Q₈	—	2	3	no	—
388	C₂×Q₈	—	2	3	no	—
389	C₂×Q₈	—	4	4	no	—
390	C₂×Q₈	C₂×C₇×D₄	1	3	yes	—
391	C₂×Q₈	C₂×C₇×D₄	1	1	yes	characteristic
392	C₂×Q₈	C₂⁴×C₇	1	1	yes	characteristic
393	C₂×D₄	—	2	1	no	—
394	C₂×D₄	—	2	3	no	—
395	C₂×D₄	—	2	6	no	—
396	C₂×D₄	—	2	6	no	—
397	C₂×D₄	—	2	6	no	—
398	C₂×D₄	—	2	6	no	—
399	C₂×D₄	—	2	6	no	—
400	C₂×D₄	—	2	6	no	—
401	C₂×D₄	—	4	12	no	—
402	C₂×D₄	—	8	6	no	—
403	C₂×D₄	C₂⁴×C₇	1	6	yes	—
404	C₂×D₄	C₂⁴×C₇	1	1	yes	characteristic
405	C₂²×C₄	—	2	1	no	—
406	C₂²×C₄	—	2	1	no	—
407	C₂²×C₄	—	2	3	no	—
408	C₂²×C₄	—	2	6	no	—
409	C₂²×C₄	—	2	6	no	—
410	C₂²×C₄	—	2	3	no	—
411	C₂²×C₄	—	4	12	no	—
412	C₂²×C₄	C₂×C₇×D₄	1	3	yes	—
413	C₂²×C₄	C₂×C₇×D₄	1	3	yes	—
414	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
415	SD₁₆	—	2	12	no	—
416	D₈	—	2	12	no	—
417	M₁₆	—	2	6	no	—
418	M₁₆	—	2	2	no	—
419	C₂×C₈	C₂⁴×C₇	1	2	yes	—
420	C₂×C₈	C₂⁴×C₇	1	6	yes	—
421	C₄⋊C₄	—	2	6	no	—
422	C₄⋊C₄	—	2	3	no	—
423	C₄⋊C₄	—	2	1	no	—
424	C₄⋊C₄	—	2	3	no	—
425	C₄⋊C₄	—	2	3	no	—
426	C₄⋊C₄	—	2	3	no	—
427	C₄⋊C₄	—	2	6	no	—
428	C₄⋊C₄	C₂×C₇×D₄	1	3	yes	—
429	C₄⋊C₄	C₂×C₇×D₄	1	6	yes	—
430	C₄⋊C₄	C₂×C₇×D₄	1	3	yes	—
431	C₄⋊C₄	C₂⁴×C₇	1	3	yes	—
432	C₄⋊C₄	C₂⁴×C₇	1	3	yes	—
433	C₄⋊C₄	C₂⁴×C₇	1	2	yes	—
434	C₂²⋊C₄	—	2	3	no	—
435	C₂²⋊C₄	—	2	6	no	—
436	C₂²⋊C₄	—	2	6	no	—
437	C₂²⋊C₄	—	2	3	no	—
438	C₂²⋊C₄	—	2	3	no	—
439	C₄²	—	2	3	no	—
440	C₄²	—	2	3	no	—
441	C₄²	—	2	3	no	—
442	C₄²	C₂×C₇×D₄	1	3	yes	—
443	C₄²	C₂×C₇×D₄	1	3	yes	—
444	C₄²	C₂⁴×C₇	1	3	yes	—
445	C₄²	C₂⁴×C₇	1	3	yes	—
446	C₂×C₇	—	2	2	no	—
447	C₂×C₇	—	4	2	no	—
448	C₂×C₇	—	8	6	no	—
449	C₂×C₇	C₂×#548	1	1	yes	characteristic
450	C₂×C₇	C₂²×AGL(1,ℤ/8)	1	1	yes	characteristic
451	C₂×C₇	#3585	1	1	yes	characteristic
452	C₂³	—	2	1	no	—
453	C₂³	—	2	6	no	—
454	C₂³	—	4	6	no	—
455	C₂³	—	4	6	no	—
456	C₂³	—	4	6	no	—
457	C₂³	C₂²×C₇×D₄	1	1	yes	characteristic
458	Q₈	—	2	1	no	—
459	Q₈	—	2	6	no	—
460	Q₈	—	4	6	no	—
461	Q₈	—	8	3	no	—
462	Q₈	C₇×AGL(1,ℤ/8)	1	4	yes	complemented
463	Q₈	C₂×C₇×#41	1	2	yes	complemented
464	D₄	—	2	4	no	—
465	D₄	—	2	6	no	—
466	D₄	—	2	1	no	—
467	D₄	—	2	6	no	—
468	D₄	—	4	12	no	—
469	D₄	—	4	6	no	—
470	D₄	—	4	12	no	—
471	D₄	—	8	6	no	—
472	D₄	—	8	6	no	—
473	D₄	C₂×C₇×#41	1	2	yes	complemented
474	C₂×C₄	—	2	1	no	—
475	C₂×C₄	—	2	4	no	—
476	C₂×C₄	—	2	6	no	—
477	C₂×C₄	—	2	3	no	—
478	C₂×C₄	—	2	3	no	—
479	C₂×C₄	—	2	2	no	—
480	C₂×C₄	—	4	3	no	—
481	C₂×C₄	—	4	12	no	—
482	C₂×C₄	—	4	2	no	—
483	C₂×C₄	—	4	12	no	—
484	C₂×C₄	—	4	6	no	—
485	C₂×C₄	—	8	3	no	—
486	C₂×C₄	C₂²×C₇×D₄	1	3	yes	—
487	C₂×C₄	C₂²×C₇×D₄	1	3	yes	—
488	C₂×C₄	C₂²×C₇×D₄	1	3	yes	—
489	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
490	C₂×C₄	C₂²×C₇×D₄	1	3	yes	—
491	C₂×C₄	C₂×C₇×#41	1	2	yes	—
492	C₂×C₄	C₇×2₋¹⁺⁴	1	2	yes	—
493	C₂×C₄	C₂⁵×C₇	1	1	yes	characteristic
494	C₈	—	2	6	no	—
495	C₈	—	2	2	no	—
496	C₇	#61321	1	1	yes	characteristic, direct product factor, Sylow
497	C₂²	—	2	1	no	—
498	C₂²	—	2	1	no	—
499	C₂²	—	2	2	no	—
500	C₂²	—	4	1	no	—
501	C₂²	—	4	1	no	—
502	C₂²	—	4	6	no	—
503	C₂²	—	4	6	no	—
504	C₂²	—	4	6	no	—
505	C₂²	—	8	12	no	—
506	C₂²	C₇×#548	1	2	yes	—
507	C₂²	C₂³×C₇×D₄	1	1	yes	characteristic
508	C₄	—	2	3	no	—
509	C₄	—	2	3	no	—
510	C₄	—	2	3	no	—
511	C₄	—	2	2	no	—
512	C₄	—	4	2	no	—
513	C₄	—	4	3	no	—
514	C₄	—	4	3	no	—
515	C₄	C₇×#548	1	2	yes	—
516	C₄	C₂×C₇×AGL(1,ℤ/8)	1	6	yes	—
517	C₄	C₂²×C₇×#41	1	1	yes	characteristic
518	C₄	C₂×C₇×2₋¹⁺⁴	1	1	yes	characteristic
519	C₂	—	2	2	no	—
520	C₂	—	4	2	no	—
521	C₂	—	8	6	no	—
522	C₂	C₂×C₇×#548	1	1	yes	characteristic
523	C₂	C₂²×C₇×AGL(1,ℤ/8)	1	1	yes	characteristic
524	C₂	C₇×#3585	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₇	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₇
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂×C₄×C₇×Q₈ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₇×Q₈×#41	C₂	semidirect
2. C₂×C₇×#550	C₂	semidirect
3. C₇×#3368	C₂	semidirect
4. C₇×#3366	C₂	semidirect
5. C₇×#3362	C₂	semidirect
6. C₇×#3358	C₂	semidirect
7. C₇×#3138	C₂	semidirect
8. C₇×#3077	C₂	semidirect
9. C₇×#3038	C₂	semidirect
10. C₇×Q₈×M₁₆	C₂	semidirect
11. C₄×C₇×AGL(1,ℤ/8)	C₂	semidirect
14. C₇×Q₈²	C₂²	semidirect
15. C₇×#557	C₂²	semidirect
17. C₇×#554	C₂²	semidirect
23. C₇×D₄×Q₈	C₂²	semidirect
32. C₇×#490	C₂²	semidirect
33. C₇×#486	C₂²	semidirect
34. C₇×#486	C₂²	semidirect
35. C₇×#478	C₂²	semidirect
36. C₇×#476	C₂²	semidirect
37. C₇×#474	C₂²	semidirect
38. C₇×#460	C₂²	semidirect
39. C₇×#460	C₂²	semidirect
40. C₇×#459	C₂²	semidirect
41. C₇×#459	C₂²	semidirect
42. C₇×#446	C₂²	semidirect
43. C₇×C₈×Q₈	C₂²	semidirect
44. C₇×#442	C₂²	semidirect
45. C₇×#440	C₂²	semidirect
46. C₄×C₇×SD₁₆	C₂²	semidirect
47. C₄×C₇×D₈	C₂²	semidirect
54. #61321	C₇	direct (N × H)
126. Q₈×#41	C₂×C₇	semidirect
127. C₂×#550	C₂×C₇	semidirect
128. #3368	C₂×C₇	semidirect
129. #3366	C₂×C₇	semidirect
130. #3362	C₂×C₇	semidirect
131. #3358	C₂×C₇	semidirect
132. #3138	C₂×C₇	semidirect
133. #3077	C₂×C₇	semidirect
134. #3038	C₂×C₇	semidirect
135. Q₈×M₁₆	C₂×C₇	semidirect
136. C₄×AGL(1,ℤ/8)	C₂×C₇	semidirect
141. C₇×#41	#41	semidirect
204. Q₈²	C₂²×C₇	semidirect
205. #557	C₂²×C₇	semidirect
207. #554	C₂²×C₇	semidirect
213. D₄×Q₈	C₂²×C₇	semidirect
222. #490	C₂²×C₇	semidirect
223. #486	C₂²×C₇	semidirect
224. #486	C₂²×C₇	semidirect
225. #478	C₂²×C₇	semidirect
226. #476	C₂²×C₇	semidirect
227. #474	C₂²×C₇	semidirect
228. #460	C₂²×C₇	semidirect
229. #460	C₂²×C₇	semidirect
230. #459	C₂²×C₇	semidirect
231. #459	C₂²×C₇	semidirect
232. #446	C₂²×C₇	semidirect
233. C₈×Q₈	C₂²×C₇	semidirect
234. #442	C₂²×C₇	semidirect
235. #440	C₂²×C₇	semidirect
236. C₄×SD₁₆	C₂²×C₇	semidirect
237. C₄×D₈	C₂²×C₇	semidirect
254. C₇×Q₈	AGL(1,ℤ/8)	semidirect
255. C₇×Q₈	C₂×#41	semidirect
265. C₇×D₄	C₂×#41	semidirect
385. #41	C₇×#41	semidirect
462. Q₈	C₇×AGL(1,ℤ/8)	semidirect
463. Q₈	C₂×C₇×#41	semidirect
473. D₄	C₂×C₇×#41	semidirect
496. C₇	#61321	direct (N × H)

Maximal subgroups

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

subgroup	index
C₇×Q₈×#41	2
C₂×C₇×#550	2
C₇×#3368	2
C₇×#3366	2
C₇×#3362	2
C₇×#3358	2
C₇×#3138	2
C₇×#3077	2
C₇×#3038	2
C₇×Q₈×M₁₆	2
C₄×C₇×AGL(1,ℤ/8)	2
#61321	7

Subgroups — C7×#61321 #13238537

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#61321 #13238537