Subgroups — C₇×#60724 #13237940

Order 1792.

1874 conjugacy classes of subgroups.

The subgroup lattice has 678 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#3512	C₂	1	1	yes	maximal, characteristic, complemented
2	C₇×#3510	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇×#3183	C₂	1	4	yes	maximal, complemented
4	C₇×#3179	C₂	1	4	yes	maximal, complemented
5	C₇×#3178	C₂	1	4	yes	maximal, complemented
6	C₇×#3168	C₂	1	4	yes	maximal, complemented
7	C₇×#3166	C₂	1	1	yes	maximal, characteristic, complemented
8	C₇×#3153	C₂	1	1	yes	maximal, characteristic, complemented
9	C₇×#3131	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₇×#488	C₂	1	1	yes	maximal, characteristic, complemented
11	C₇×#3063	C₂	1	2	yes	maximal, complemented
12	C₇×#3062	C₂	1	2	yes	maximal, complemented
13	C₇×#2947	C₂	1	2	yes	maximal, complemented
14	C₇×#2946	C₂	1	2	yes	maximal, complemented
15	C₄×C₇×#131	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×C₇×#137	C₂²	1	2	yes	—
17	C₂×C₇×AGL(1,ℤ/8)	C₂²	1	2	yes	—
18	C₂×C₇×#131	C₂²	1	2	yes	—
19	C₇×#559	—	2	2	no	—
20	C₇×#556	—	2	2	no	—
21	C₇×#553	—	2	4	no	—
22	C₇×#553	—	2	2	no	—
23	C₇×#552	—	2	2	no	—
24	C₇×#551	—	2	4	no	—
25	C₇×#542	C₂²	1	4	yes	—
26	C₇×#541	C₂²	1	2	yes	complemented
27	C₇×#540	C₂²	1	2	yes	complemented
28	C₇×#537	C₂²	1	2	yes	—
29	C₇×#536	C₂²	1	1	yes	characteristic
30	C₇×#535	C₂²	1	1	yes	characteristic
31	C₇×#533	C₂²	1	1	yes	characteristic
32	C₇×#532	C₂²	1	1	yes	characteristic
33	C₇×#529	C₂²	1	4	yes	—
34	C₇×#529	C₂²	1	4	yes	complemented
35	C₂×C₇×#125	C₂²	1	1	yes	characteristic
36	C₂×C₇×#124	C₂²	1	1	yes	characteristic
37	C₇×#520	C₂²	1	2	yes	—
38	C₇×#519	C₂²	1	1	yes	characteristic
39	C₇×#518	C₂²	1	1	yes	characteristic
40	C₄×C₇×#41	C₂²	1	1	yes	characteristic
41	C₇×#501	C₂²	1	4	yes	complemented
42	C₇×#499	C₂²	1	4	yes	complemented
43	C₇×#497	C₂²	1	2	yes	—
44	C₇×#496	C₂²	1	2	yes	—
45	C₇×#495	C₂²	1	4	yes	—
46	C₇×#488	C₂²	1	2	yes	—
47	C₇×#488	C₂²	1	4	yes	—
48	C₇×#488	C₂²	1	2	yes	—
49	C₇×#488	C₂²	1	2	yes	complemented
50	C₇×#488	C₂²	1	2	yes	complemented
51	C₇×#487	C₂²	1	2	yes	complemented
52	C₇×#486	C₂²	1	2	yes	complemented
53	C₇×#470	C₂²	1	4	yes	complemented
54	C₇×#469	C₂²	1	4	yes	—
55	C₇×#468	C₂²	1	4	yes	—
56	C₇×#468	C₂²	1	4	yes	complemented
57	C₇×#467	C₂²	1	4	yes	complemented
58	C₇×#466	C₂²	1	4	yes	—
59	C₇×#465	C₂²	1	4	yes	—
60	C₇×#465	C₂²	1	4	yes	complemented
61	C₇×#442	C₂²	1	4	yes	—
62	C₇×#441	C₂²	1	4	yes	complemented
63	C₇×#440	C₂²	1	4	yes	—
64	C₇×#440	C₂²	1	4	yes	complemented
65	C₄×C₇×Q₁₆	C₂²	1	4	yes	complemented
66	C₄×C₇×SD₁₆	C₂²	1	4	yes	—
67	C₄×C₇×SD₁₆	C₂²	1	4	yes	complemented
68	C₄×C₇×D₈	C₂²	1	4	yes	—
69	C₇×#428	C₂²	1	2	yes	—
70	C₇×#428	C₂²	1	2	yes	—
71	C₇×#416	C₂²	1	2	yes	—
72	C₇×#416	C₂²	1	2	yes	—
73	C₂×C₇×#103	C₂²	1	2	yes	—
74	C₂×C₇×#102	C₂²	1	2	yes	—
75	C₇×#405	C₂²	1	2	yes	—
76	C₇×#405	C₂²	1	4	yes	complemented
77	C₄×C₇×M₁₆	C₂²	1	1	yes	characteristic
78	C₄×C₇×M₁₆	C₂²	1	2	yes	complemented
79	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
80	C₂×C₄×C₇×C₈	C₂²	1	2	yes	complemented
81	#60724	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
82	C₂×C₇×#41	C₂³	1	2	yes	—
83	C₂×C₇×#41	C₂³	1	1	yes	characteristic
84	C₂²×C₇×Q₈	C₂³	1	1	yes	characteristic
85	C₂²×C₇×D₄	C₂³	1	1	yes	characteristic
86	C₇×#137	—	2	8	no	—
87	C₇×AGL(1,ℤ/8)	—	2	8	no	—
88	C₂×C₇×Q₁₆	C₂³	1	4	yes	—
89	C₂×C₇×SD₁₆	C₂³	1	4	yes	—
90	C₂×C₇×SD₁₆	C₂³	1	4	yes	—
91	C₂×C₇×D₈	C₂³	1	4	yes	—
92	C₇×#131	—	2	8	no	—
93	C₂×C₇×M₁₆	C₂³	1	4	yes	—
94	C₂×C₇×M₁₆	C₂³	1	2	yes	—
95	C₂²×C₇×C₈	C₂³	1	4	yes	—
96	C₂²×C₇×C₈	C₂³	1	2	yes	—
97	C₇×C₄⋊Q₈	—	2	1	no	—
98	C₇×C₄⋊Q₈	C₂³	1	1	yes	characteristic
99	C₇×C₄⋊Q₈	C₂³	1	2	yes	—
100	C₇×#127	—	2	1	no	—
101	C₇×#127	C₂³	1	1	yes	characteristic
102	C₇×#126	—	2	4	no	—
103	C₇×#126	—	2	4	no	—
104	C₇×#126	—	2	4	no	—
105	C₇×#126	—	2	4	no	—
106	C₇×#125	—	2	4	no	—
107	C₇×#125	—	2	2	no	—
108	C₇×#125	C₂³	1	2	yes	—
109	C₇×#125	C₂³	1	2	yes	—
110	C₇×#125	C₂³	1	2	yes	—
111	C₇×#124	—	2	1	no	—
112	C₇×#124	—	2	1	no	—
113	C₇×#124	—	2	4	no	—
114	C₇×#124	—	2	2	no	—
115	C₇×#124	C₂³	1	2	yes	—
116	C₇×#124	C₂³	1	2	yes	—
117	C₇×#124	C₂³	1	2	yes	—
118	C₇×#123	—	2	2	no	—
119	C₇×#123	—	2	4	no	—
120	C₇×#123	—	2	4	no	—
121	C₇×#123	—	2	4	no	—
122	C₇×#123	—	2	4	no	—
123	C₇×C₂²⋊Q₈	—	2	2	no	—
124	C₇×C₂²⋊Q₈	—	2	2	no	—
125	C₇×C₂²⋊Q₈	—	2	4	no	—
126	C₇×C₂²⋊Q₈	C₂³	1	4	yes	—
127	C₇×C₂²⋊Q₈	C₂³	1	4	yes	—
128	C₇×#121	—	2	2	no	—
129	C₇×#121	—	2	4	no	—
130	C₇×#121	—	2	2	no	—
131	C₇×#121	—	2	4	no	—
132	C₇×#121	C₂³	1	4	yes	—
133	C₇×#121	C₂³	1	4	yes	—
134	C₇×#120	—	2	2	no	—
135	C₄×C₇×Q₈	—	2	2	no	—
136	C₄×C₇×Q₈	—	2	2	no	—
137	C₄×C₇×Q₈	D₄	1	2	yes	—
138	C₄×C₇×Q₈	C₂³	1	4	yes	—
139	C₄×C₇×Q₈	C₂³	1	4	yes	—
140	C₄×C₇×D₄	—	2	4	no	—
141	C₄×C₇×D₄	—	2	4	no	—
142	C₄×C₇×D₄	—	2	2	no	—
143	C₄×C₇×D₄	—	2	4	no	—
144	C₄×C₇×D₄	—	2	2	no	—
145	C₄×C₇×D₄	D₄	1	4	yes	—
146	C₄×C₇×D₄	D₄	1	2	yes	—
147	C₄×C₇×D₄	C₂³	1	4	yes	—
148	C₄×C₇×D₄	C₂³	1	4	yes	—
149	C₇×#117	—	2	2	no	—
150	C₇×#117	—	2	2	no	—
151	C₇×#117	—	2	4	no	—
152	C₇×#117	—	2	4	no	—
153	C₇×#117	C₂³	1	1	yes	characteristic
154	C₇×#117	C₂³	1	1	yes	characteristic
155	C₇×#117	C₂³	1	1	yes	characteristic
156	C₇×#117	C₂³	1	2	yes	—
157	C₂×C₇×C₄⋊C₄	—	2	2	no	—
158	C₂×C₇×C₄⋊C₄	—	2	2	no	—
159	C₂×C₇×C₄⋊C₄	C₂³	1	2	yes	—
160	C₂×C₇×C₂²⋊C₄	—	2	2	no	—
161	C₂×C₇×C₂²⋊C₄	—	2	2	no	—
162	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
163	C₂×C₄²×C₇	C₂³	1	2	yes	—
164	C₇×#107	C₂³	1	4	yes	—
165	C₇×#107	C₂³	1	4	yes	—
166	C₇×#106	C₂³	1	4	yes	—
167	C₇×#106	C₂³	1	4	yes	—
168	C₇×#103	C₂³	1	2	yes	—
169	C₇×#103	C₂³	1	4	yes	—
170	C₇×#103	C₂³	1	2	yes	—
171	C₇×#103	C₂³	1	4	yes	—
172	C₇×#103	C₂³	1	2	yes	—
173	C₇×#103	C₂³	1	2	yes	—
174	C₇×#102	C₂³	1	2	yes	—
175	C₇×#102	C₂³	1	4	yes	—
176	C₇×#102	C₂³	1	4	yes	—
177	C₇×#102	C₂³	1	2	yes	—
178	C₇×#102	C₂³	1	2	yes	—
179	C₇×#102	C₂³	1	2	yes	—
180	C₇×#97	C₂³	1	2	yes	—
181	C₇×#97	C₂³	1	4	yes	—
182	C₄×C₇×C₈	C₂³	1	2	yes	—
183	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
184	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
185	C₄×C₇×C₈	C₂³	1	2	yes	—
186	C₄×C₇×C₈	C₂³	1	4	yes	—
187	#3512	C₂×C₇	1	1	yes	characteristic, complemented
188	#3510	C₂×C₇	1	1	yes	characteristic, complemented
189	#3183	C₂×C₇	1	4	yes	complemented
190	#3179	C₂×C₇	1	4	yes	complemented
191	#3178	C₂×C₇	1	4	yes	complemented
192	#3168	C₂×C₇	1	4	yes	complemented
193	#3166	C₂×C₇	1	1	yes	characteristic, complemented
194	#3153	C₂×C₇	1	1	yes	characteristic, complemented
195	#3131	C₂×C₇	1	1	yes	characteristic, complemented
196	C₂×#488	C₂×C₇	1	1	yes	characteristic, complemented
197	#3063	C₂×C₇	1	2	yes	complemented
198	#3062	C₂×C₇	1	2	yes	complemented
199	#2947	C₂×C₇	1	2	yes	complemented
200	#2946	C₂×C₇	1	2	yes	complemented
201	C₄×#131	C₂×C₇	1	1	yes	characteristic, complemented
202	C₂⁴×C₇	—	2	1	no	—
203	C₇×#41	—	2	4	no	—
204	C₇×#41	—	2	4	no	—
205	C₇×#41	—	2	4	no	—
206	C₂×C₇×Q₈	—	2	1	no	—
207	C₂×C₇×Q₈	—	2	2	no	—
208	C₂×C₇×Q₈	—	2	4	no	—
209	C₂×C₇×Q₈	C₂×D₄	1	1	yes	characteristic
210	C₂×C₇×Q₈	C₂⁴	1	2	yes	—
211	C₂×C₇×Q₈	C₂⁴	1	2	yes	—
212	C₂×C₇×D₄	—	2	2	no	—
213	C₂×C₇×D₄	—	2	4	no	—
214	C₂×C₇×D₄	—	2	4	no	—
215	C₂×C₇×D₄	—	2	1	no	—
216	C₂×C₇×D₄	—	2	2	no	—
217	C₂×C₇×D₄	—	2	4	no	—
218	C₂×C₇×D₄	C₂×D₄	1	2	yes	—
219	C₂×C₇×D₄	C₂×D₄	1	1	yes	characteristic
220	C₂×C₇×D₄	C₂⁴	1	2	yes	—
221	C₂×C₇×D₄	C₂⁴	1	2	yes	—
222	C₂²×C₄×C₇	—	2	1	no	—
223	C₂²×C₄×C₇	—	2	2	no	—
224	C₂²×C₄×C₇	—	2	2	no	—
225	C₂²×C₄×C₇	—	2	4	no	—
226	C₂²×C₄×C₇	—	2	4	no	—
227	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
228	C₂²×C₄×C₇	C₂×D₄	1	4	yes	—
229	C₂²×C₄×C₇	C₂⁴	1	2	yes	—
230	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
231	C₇×Q₁₆	—	2	8	no	—
232	C₇×SD₁₆	—	2	8	no	—
233	C₇×SD₁₆	—	2	8	no	—
234	C₇×D₈	—	2	8	no	—
235	C₇×M₁₆	—	2	8	no	—
236	C₇×M₁₆	#41	1	8	yes	—
237	C₂×C₇×C₈	—	2	4	no	—
238	C₂×C₇×C₈	—	2	8	no	—
239	C₂×C₇×C₈	C₂⁴	1	2	yes	—
240	C₂×C₇×C₈	C₂⁴	1	2	yes	—
241	C₂×C₇×C₈	C₂⁴	1	4	yes	—
242	C₇×C₄⋊C₄	—	2	2	no	—
243	C₇×C₄⋊C₄	—	2	4	no	—
244	C₇×C₄⋊C₄	—	2	4	no	—
245	C₇×C₄⋊C₄	—	2	4	no	—
246	C₇×C₄⋊C₄	—	2	2	no	—
247	C₇×C₄⋊C₄	—	2	4	no	—
248	C₇×C₄⋊C₄	—	2	4	no	—
249	C₇×C₄⋊C₄	—	2	4	no	—
250	C₇×C₄⋊C₄	—	2	2	no	—
251	C₇×C₄⋊C₄	—	2	2	no	—
252	C₇×C₄⋊C₄	—	2	2	no	—
253	C₇×C₄⋊C₄	C₂×D₄	1	2	yes	—
254	C₇×C₄⋊C₄	C₂×D₄	1	4	yes	—
255	C₇×C₄⋊C₄	C₂⁴	1	4	yes	—
256	C₇×C₄⋊C₄	C₂⁴	1	2	yes	—
257	C₇×C₄⋊C₄	C₂⁴	1	2	yes	—
258	C₇×C₂²⋊C₄	—	2	1	no	—
259	C₇×C₂²⋊C₄	—	2	1	no	—
260	C₇×C₂²⋊C₄	—	2	4	no	—
261	C₇×C₂²⋊C₄	—	2	4	no	—
262	C₇×C₂²⋊C₄	—	2	4	no	—
263	C₇×C₂²⋊C₄	—	2	4	no	—
264	C₇×C₂²⋊C₄	—	2	4	no	—
265	C₇×C₂²⋊C₄	—	2	2	no	—
266	C₇×C₂²⋊C₄	—	2	4	no	—
267	C₇×C₂²⋊C₄	—	2	4	no	—
268	C₇×C₂²⋊C₄	—	2	4	no	—
269	C₇×C₂²⋊C₄	—	2	2	no	—
270	C₇×C₂²⋊C₄	C₂×D₄	1	2	yes	—
271	C₇×C₂²⋊C₄	C₂×D₄	1	4	yes	—
272	C₄²×C₇	—	2	4	no	—
273	C₄²×C₇	—	2	4	no	—
274	C₄²×C₇	—	2	1	no	—
275	C₄²×C₇	—	2	1	no	—
276	C₄²×C₇	C₂×D₄	1	4	yes	—
277	C₄²×C₇	C₂×D₄	1	2	yes	—
278	C₄²×C₇	C₂⁴	1	1	yes	characteristic
279	C₄²×C₇	C₂⁴	1	2	yes	—
280	C₄²×C₇	C₂⁴	1	1	yes	characteristic
281	C₂×#137	C₂²×C₇	1	2	yes	—
282	C₂×AGL(1,ℤ/8)	C₂²×C₇	1	2	yes	—
283	C₂×#131	C₂²×C₇	1	2	yes	—
284	#559	—	2	2	no	—
285	#556	—	2	2	no	—
286	#553	—	2	4	no	—
287	#553	—	2	2	no	—
288	#552	—	2	2	no	—
289	#551	—	2	4	no	—
290	#542	C₂²×C₇	1	4	yes	—
291	#541	C₂²×C₇	1	2	yes	complemented
292	#540	C₂²×C₇	1	2	yes	complemented
293	#537	C₂²×C₇	1	2	yes	—
294	#536	C₂²×C₇	1	1	yes	characteristic
295	#535	C₂²×C₇	1	1	yes	characteristic
296	#533	C₂²×C₇	1	1	yes	characteristic
297	#532	C₂²×C₇	1	1	yes	characteristic
298	#529	C₂²×C₇	1	4	yes	—
299	#529	C₂²×C₇	1	4	yes	complemented
300	C₂×#125	C₂²×C₇	1	1	yes	characteristic
301	C₂×#124	C₂²×C₇	1	1	yes	characteristic
302	#520	C₂²×C₇	1	2	yes	—
303	#519	C₂²×C₇	1	1	yes	characteristic
304	#518	C₂²×C₇	1	1	yes	characteristic
305	C₄×#41	C₂²×C₇	1	1	yes	characteristic
306	#501	C₂²×C₇	1	4	yes	complemented
307	#499	C₂²×C₇	1	4	yes	complemented
308	#497	C₂²×C₇	1	2	yes	—
309	#496	C₂²×C₇	1	2	yes	—
310	#495	C₂²×C₇	1	4	yes	—
311	#488	C₂²×C₇	1	2	yes	—
312	#488	C₂²×C₇	1	4	yes	—
313	#488	C₂²×C₇	1	2	yes	—
314	#488	C₂²×C₇	1	2	yes	complemented
315	#488	C₂²×C₇	1	2	yes	complemented
316	#487	C₂²×C₇	1	2	yes	complemented
317	#486	C₂²×C₇	1	2	yes	complemented
318	#470	C₂²×C₇	1	4	yes	complemented
319	#469	C₂²×C₇	1	4	yes	—
320	#468	C₂²×C₇	1	4	yes	—
321	#468	C₂²×C₇	1	4	yes	complemented
322	#467	C₂²×C₇	1	4	yes	complemented
323	#466	C₂²×C₇	1	4	yes	—
324	#465	C₂²×C₇	1	4	yes	—
325	#465	C₂²×C₇	1	4	yes	complemented
326	#442	C₂²×C₇	1	4	yes	—
327	#441	C₂²×C₇	1	4	yes	complemented
328	#440	C₂²×C₇	1	4	yes	—
329	#440	C₂²×C₇	1	4	yes	complemented
330	C₄×Q₁₆	C₂²×C₇	1	4	yes	complemented
331	C₄×SD₁₆	C₂²×C₇	1	4	yes	—
332	C₄×SD₁₆	C₂²×C₇	1	4	yes	complemented
333	C₄×D₈	C₂²×C₇	1	4	yes	—
334	#428	C₂²×C₇	1	2	yes	—
335	#428	C₂²×C₇	1	2	yes	—
336	#416	C₂²×C₇	1	2	yes	—
337	#416	C₂²×C₇	1	2	yes	—
338	C₂×#103	C₂²×C₇	1	2	yes	—
339	C₂×#102	C₂²×C₇	1	2	yes	—
340	#405	C₂²×C₇	1	2	yes	—
341	#405	C₂²×C₇	1	4	yes	complemented
342	C₄×M₁₆	C₂²×C₇	1	1	yes	characteristic
343	C₄×M₁₆	C₂²×C₇	1	2	yes	complemented
344	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
345	C₂×C₄×C₈	C₂²×C₇	1	2	yes	complemented
346	C₂³×C₇	—	2	2	no	—
347	C₂³×C₇	—	2	2	no	—
348	C₂³×C₇	—	4	2	no	—
349	C₂³×C₇	—	8	2	no	—
350	C₂³×C₇	C₂²×D₄	1	2	yes	—
351	C₂³×C₇	C₂²×D₄	1	1	yes	characteristic
352	C₇×Q₈	—	2	2	no	—
353	C₇×Q₈	—	2	2	no	—
354	C₇×Q₈	—	2	4	no	—
355	C₇×Q₈	—	2	2	no	—
356	C₇×Q₈	—	4	2	no	—
357	C₇×Q₈	—	8	2	no	—
358	C₇×D₄	—	2	4	no	—
359	C₇×D₄	—	2	2	no	—
360	C₇×D₄	—	2	2	no	—
361	C₇×D₄	—	2	4	no	—
362	C₇×D₄	—	2	2	no	—
363	C₇×D₄	—	4	4	no	—
364	C₇×D₄	—	4	2	no	—
365	C₇×D₄	—	8	2	no	—
366	C₇×D₄	—	8	4	no	—
367	C₇×D₄	—	8	4	no	—
368	C₂×C₄×C₇	—	2	4	no	—
369	C₂×C₄×C₇	—	2	2	no	—
370	C₂×C₄×C₇	—	2	2	no	—
371	C₂×C₄×C₇	—	2	2	no	—
372	C₂×C₄×C₇	—	2	2	no	—
373	C₂×C₄×C₇	—	2	4	no	—
374	C₂×C₄×C₇	—	4	4	no	—
375	C₂×C₄×C₇	—	4	2	no	—
376	C₂×C₄×C₇	—	4	2	no	—
377	C₂×C₄×C₇	—	4	4	no	—
378	C₂×C₄×C₇	—	8	4	no	—
379	C₂×C₄×C₇	—	8	4	no	—
380	C₂×C₄×C₇	—	8	2	no	—
381	C₂×C₄×C₇	C₂²×D₄	1	4	yes	—
382	C₂×C₄×C₇	C₂²×D₄	1	2	yes	—
383	C₂×C₄×C₇	C₂²×D₄	1	2	yes	—
384	C₂×C₄×C₇	C₂²×D₄	1	2	yes	—
385	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
386	C₂×C₄×C₇	C₂×#41	1	4	yes	—
387	C₂×C₄×C₇	C₂⁵	1	1	yes	characteristic
388	C₇×C₈	—	2	2	no	—
389	C₇×C₈	—	2	2	no	—
390	C₇×C₈	C₂×#41	1	8	yes	—
391	C₂×#41	C₂³×C₇	1	2	yes	—
392	C₂×#41	C₂³×C₇	1	1	yes	characteristic
393	C₂²×Q₈	C₂³×C₇	1	1	yes	characteristic
394	C₂²×D₄	C₂³×C₇	1	1	yes	characteristic
395	#137	—	2	8	no	—
396	AGL(1,ℤ/8)	—	2	8	no	—
397	C₂×Q₁₆	C₂³×C₇	1	4	yes	—
398	C₂×SD₁₆	C₂³×C₇	1	4	yes	—
399	C₂×SD₁₆	C₂³×C₇	1	4	yes	—
400	C₂×D₈	C₂³×C₇	1	4	yes	—
401	#131	—	2	8	no	—
402	C₂×M₁₆	C₂³×C₇	1	4	yes	—
403	C₂×M₁₆	C₂³×C₇	1	2	yes	—
404	C₂²×C₈	C₂³×C₇	1	2	yes	—
405	C₂²×C₈	C₂³×C₇	1	4	yes	—
406	C₄⋊Q₈	—	2	1	no	—
407	C₄⋊Q₈	C₂³×C₇	1	1	yes	characteristic
408	C₄⋊Q₈	C₂³×C₇	1	2	yes	—
409	#127	—	2	1	no	—
410	#127	C₂³×C₇	1	1	yes	characteristic
411	#126	—	2	4	no	—
412	#126	—	2	4	no	—
413	#126	—	2	4	no	—
414	#126	—	2	4	no	—
415	#125	—	2	4	no	—
416	#125	—	2	2	no	—
417	#125	C₂³×C₇	1	2	yes	—
418	#125	C₂³×C₇	1	2	yes	—
419	#125	C₂³×C₇	1	2	yes	—
420	#124	—	2	1	no	—
421	#124	—	2	1	no	—
422	#124	—	2	4	no	—
423	#124	—	2	2	no	—
424	#124	C₂³×C₇	1	2	yes	—
425	#124	C₂³×C₇	1	2	yes	—
426	#124	C₂³×C₇	1	2	yes	—
427	#123	—	2	2	no	—
428	#123	—	2	4	no	—
429	#123	—	2	4	no	—
430	#123	—	2	4	no	—
431	#123	—	2	4	no	—
432	C₂²⋊Q₈	—	2	2	no	—
433	C₂²⋊Q₈	—	2	2	no	—
434	C₂²⋊Q₈	—	2	4	no	—
435	C₂²⋊Q₈	C₂³×C₇	1	4	yes	—
436	C₂²⋊Q₈	C₂³×C₇	1	4	yes	—
437	#121	—	2	2	no	—
438	#121	—	2	4	no	—
439	#121	—	2	2	no	—
440	#121	—	2	4	no	—
441	#121	C₂³×C₇	1	4	yes	—
442	#121	C₂³×C₇	1	4	yes	—
443	#120	—	2	2	no	—
444	C₄×Q₈	—	2	2	no	—
445	C₄×Q₈	—	2	2	no	—
446	C₄×Q₈	C₇×D₄	1	2	yes	—
447	C₄×Q₈	C₂³×C₇	1	4	yes	—
448	C₄×Q₈	C₂³×C₇	1	4	yes	—
449	C₄×D₄	—	2	4	no	—
450	C₄×D₄	—	2	4	no	—
451	C₄×D₄	—	2	2	no	—
452	C₄×D₄	—	2	4	no	—
453	C₄×D₄	—	2	2	no	—
454	C₄×D₄	C₇×D₄	1	4	yes	—
455	C₄×D₄	C₇×D₄	1	2	yes	—
456	C₄×D₄	C₂³×C₇	1	4	yes	—
457	C₄×D₄	C₂³×C₇	1	4	yes	—
458	#117	—	2	2	no	—
459	#117	—	2	2	no	—
460	#117	—	2	4	no	—
461	#117	—	2	4	no	—
462	#117	C₂³×C₇	1	1	yes	characteristic
463	#117	C₂³×C₇	1	1	yes	characteristic
464	#117	C₂³×C₇	1	1	yes	characteristic
465	#117	C₂³×C₇	1	2	yes	—
466	C₂×C₄⋊C₄	—	2	2	no	—
467	C₂×C₄⋊C₄	—	2	2	no	—
468	C₂×C₄⋊C₄	C₂³×C₇	1	2	yes	—
469	C₂×C₂²⋊C₄	—	2	2	no	—
470	C₂×C₂²⋊C₄	—	2	2	no	—
471	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
472	C₂×C₄²	C₂³×C₇	1	2	yes	—
473	#107	C₂³×C₇	1	4	yes	—
474	#107	C₂³×C₇	1	4	yes	—
475	#106	C₂³×C₇	1	4	yes	—
476	#106	C₂³×C₇	1	4	yes	—
477	#103	C₂³×C₇	1	4	yes	—
478	#103	C₂³×C₇	1	4	yes	—
479	#103	C₂³×C₇	1	2	yes	—
480	#103	C₂³×C₇	1	2	yes	—
481	#103	C₂³×C₇	1	2	yes	—
482	#103	C₂³×C₇	1	2	yes	—
483	#102	C₂³×C₇	1	4	yes	—
484	#102	C₂³×C₇	1	4	yes	—
485	#102	C₂³×C₇	1	2	yes	—
486	#102	C₂³×C₇	1	2	yes	—
487	#102	C₂³×C₇	1	2	yes	—
488	#102	C₂³×C₇	1	2	yes	—
489	#97	C₂³×C₇	1	2	yes	—
490	#97	C₂³×C₇	1	4	yes	—
491	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
492	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
493	C₄×C₈	C₂³×C₇	1	2	yes	—
494	C₄×C₈	C₂³×C₇	1	2	yes	—
495	C₄×C₈	C₂³×C₇	1	4	yes	—
496	C₂²×C₇	—	2	2	no	—
497	C₂²×C₇	—	2	4	no	—
498	C₂²×C₇	—	2	2	no	—
499	C₂²×C₇	—	4	2	no	—
500	C₂²×C₇	—	4	4	no	—
501	C₂²×C₇	—	4	2	no	—
502	C₂²×C₇	—	4	2	no	—
503	C₂²×C₇	—	4	2	no	—
504	C₂²×C₇	—	16	2	no	—
505	C₂²×C₇	#532	1	2	yes	—
506	C₂²×C₇	C₂³×D₄	1	1	yes	characteristic
507	C₄×C₇	—	2	2	no	—
508	C₄×C₇	—	2	4	no	—
509	C₄×C₇	—	2	2	no	—
510	C₄×C₇	—	2	2	no	—
511	C₄×C₇	—	4	4	no	—
512	C₄×C₇	—	4	2	no	—
513	C₄×C₇	—	4	2	no	—
514	C₄×C₇	—	4	2	no	—
515	C₄×C₇	—	4	2	no	—
516	C₄×C₇	#532	1	2	yes	—
517	C₄×C₇	C₂²×#41	1	2	yes	—
518	C₂⁴	—	2	1	no	—
519	#41	—	2	4	no	—
520	#41	—	2	4	no	—
521	#41	—	2	4	no	—
522	C₂×Q₈	—	2	1	no	—
523	C₂×Q₈	—	2	2	no	—
524	C₂×Q₈	—	2	4	no	—
525	C₂×Q₈	C₂×C₇×D₄	1	1	yes	characteristic
526	C₂×Q₈	C₂⁴×C₇	1	2	yes	—
527	C₂×Q₈	C₂⁴×C₇	1	2	yes	—
528	C₂×D₄	—	2	2	no	—
529	C₂×D₄	—	2	4	no	—
530	C₂×D₄	—	2	4	no	—
531	C₂×D₄	—	2	1	no	—
532	C₂×D₄	—	2	2	no	—
533	C₂×D₄	—	2	4	no	—
534	C₂×D₄	C₂×C₇×D₄	1	2	yes	—
535	C₂×D₄	C₂×C₇×D₄	1	1	yes	characteristic
536	C₂×D₄	C₂⁴×C₇	1	2	yes	—
537	C₂×D₄	C₂⁴×C₇	1	2	yes	—
538	C₂²×C₄	—	2	1	no	—
539	C₂²×C₄	—	2	2	no	—
540	C₂²×C₄	—	2	2	no	—
541	C₂²×C₄	—	2	4	no	—
542	C₂²×C₄	—	2	4	no	—
543	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
544	C₂²×C₄	C₂×C₇×D₄	1	4	yes	—
545	C₂²×C₄	C₂⁴×C₇	1	2	yes	—
546	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
547	Q₁₆	—	2	8	no	—
548	SD₁₆	—	2	8	no	—
549	SD₁₆	—	2	8	no	—
550	D₈	—	2	8	no	—
551	M₁₆	—	2	8	no	—
552	M₁₆	C₇×#41	1	8	yes	—
553	C₂×C₈	—	2	4	no	—
554	C₂×C₈	—	2	8	no	—
555	C₂×C₈	C₂⁴×C₇	1	2	yes	—
556	C₂×C₈	C₂⁴×C₇	1	2	yes	—
557	C₂×C₈	C₂⁴×C₇	1	4	yes	—
558	C₄⋊C₄	—	2	2	no	—
559	C₄⋊C₄	—	2	4	no	—
560	C₄⋊C₄	—	2	4	no	—
561	C₄⋊C₄	—	2	4	no	—
562	C₄⋊C₄	—	2	2	no	—
563	C₄⋊C₄	—	2	4	no	—
564	C₄⋊C₄	—	2	4	no	—
565	C₄⋊C₄	—	2	4	no	—
566	C₄⋊C₄	—	2	2	no	—
567	C₄⋊C₄	—	2	2	no	—
568	C₄⋊C₄	—	2	2	no	—
569	C₄⋊C₄	C₂×C₇×D₄	1	2	yes	—
570	C₄⋊C₄	C₂×C₇×D₄	1	4	yes	—
571	C₄⋊C₄	C₂⁴×C₇	1	4	yes	—
572	C₄⋊C₄	C₂⁴×C₇	1	2	yes	—
573	C₄⋊C₄	C₂⁴×C₇	1	2	yes	—
574	C₂²⋊C₄	—	2	1	no	—
575	C₂²⋊C₄	—	2	1	no	—
576	C₂²⋊C₄	—	2	4	no	—
577	C₂²⋊C₄	—	2	4	no	—
578	C₂²⋊C₄	—	2	4	no	—
579	C₂²⋊C₄	—	2	4	no	—
580	C₂²⋊C₄	—	2	4	no	—
581	C₂²⋊C₄	—	2	2	no	—
582	C₂²⋊C₄	—	2	4	no	—
583	C₂²⋊C₄	—	2	4	no	—
584	C₂²⋊C₄	—	2	4	no	—
585	C₂²⋊C₄	—	2	2	no	—
586	C₂²⋊C₄	C₂×C₇×D₄	1	2	yes	—
587	C₂²⋊C₄	C₂×C₇×D₄	1	4	yes	—
588	C₄²	—	2	4	no	—
589	C₄²	—	2	4	no	—
590	C₄²	—	2	1	no	—
591	C₄²	—	2	1	no	—
592	C₄²	C₂×C₇×D₄	1	4	yes	—
593	C₄²	C₂×C₇×D₄	1	2	yes	—
594	C₄²	C₂⁴×C₇	1	1	yes	characteristic
595	C₄²	C₂⁴×C₇	1	2	yes	—
596	C₄²	C₂⁴×C₇	1	1	yes	characteristic
597	C₂×C₇	—	2	2	no	—
598	C₂×C₇	—	4	2	no	—
599	C₂×C₇	—	8	2	no	—
600	C₂×C₇	—	8	2	no	—
601	C₂×C₇	C₂×#532	1	1	yes	characteristic
602	C₂×C₇	#3584	1	2	yes	—
603	C₂³	—	2	2	no	—
604	C₂³	—	2	2	no	—
605	C₂³	—	4	2	no	—
606	C₂³	—	8	2	no	—
607	C₂³	C₂²×C₇×D₄	1	2	yes	—
608	C₂³	C₂²×C₇×D₄	1	1	yes	characteristic
609	Q₈	—	2	2	no	—
610	Q₈	—	2	2	no	—
611	Q₈	—	2	4	no	—
612	Q₈	—	2	2	no	—
613	Q₈	—	4	2	no	—
614	Q₈	—	8	2	no	—
615	D₄	—	2	4	no	—
616	D₄	—	2	2	no	—
617	D₄	—	2	4	no	—
618	D₄	—	2	2	no	—
619	D₄	—	2	2	no	—
620	D₄	—	4	4	no	—
621	D₄	—	4	2	no	—
622	D₄	—	8	2	no	—
623	D₄	—	8	4	no	—
624	D₄	—	8	4	no	—
625	C₂×C₄	—	2	4	no	—
626	C₂×C₄	—	2	2	no	—
627	C₂×C₄	—	2	2	no	—
628	C₂×C₄	—	2	2	no	—
629	C₂×C₄	—	2	2	no	—
630	C₂×C₄	—	2	4	no	—
631	C₂×C₄	—	4	4	no	—
632	C₂×C₄	—	4	2	no	—
633	C₂×C₄	—	4	2	no	—
634	C₂×C₄	—	4	4	no	—
635	C₂×C₄	—	8	4	no	—
636	C₂×C₄	—	8	4	no	—
637	C₂×C₄	—	8	2	no	—
638	C₂×C₄	C₂²×C₇×D₄	1	4	yes	—
639	C₂×C₄	C₂²×C₇×D₄	1	2	yes	—
640	C₂×C₄	C₂²×C₇×D₄	1	2	yes	—
641	C₂×C₄	C₂²×C₇×D₄	1	2	yes	—
642	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
643	C₂×C₄	C₂×C₇×#41	1	4	yes	—
644	C₂×C₄	C₂⁵×C₇	1	1	yes	characteristic
645	C₈	—	2	2	no	—
646	C₈	—	2	2	no	—
647	C₈	C₂×C₇×#41	1	8	yes	—
648	C₇	#60724	1	1	yes	characteristic, direct product factor, Sylow
649	C₂²	—	2	2	no	—
650	C₂²	—	2	4	no	—
651	C₂²	—	2	2	no	—
652	C₂²	—	4	2	no	—
653	C₂²	—	4	4	no	—
654	C₂²	—	4	2	no	—
655	C₂²	—	4	2	no	—
656	C₂²	—	4	2	no	—
657	C₂²	—	16	2	no	—
658	C₂²	C₇×#532	1	2	yes	—
659	C₂²	C₂³×C₇×D₄	1	1	yes	characteristic
660	C₄	—	2	2	no	—
661	C₄	—	2	4	no	—
662	C₄	—	2	2	no	—
663	C₄	—	2	2	no	—
664	C₄	—	4	4	no	—
665	C₄	—	4	2	no	—
666	C₄	—	4	2	no	—
667	C₄	—	4	2	no	—
668	C₄	—	4	2	no	—
669	C₄	C₇×#532	1	2	yes	—
670	C₄	C₂²×C₇×#41	1	2	yes	—
671	C₂	—	2	2	no	—
672	C₂	—	4	2	no	—
673	C₂	—	8	2	no	—
674	C₂	—	8	2	no	—
675	C₂	C₂×C₇×#532	1	1	yes	characteristic
676	C₂	C₇×#3584	1	2	yes	—

	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 ◁ C₄×C₇×#41 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₇×#3512	C₂	semidirect
2. C₇×#3510	C₂	semidirect
3. C₇×#3183	C₂	semidirect
4. C₇×#3179	C₂	semidirect
5. C₇×#3178	C₂	semidirect
6. C₇×#3168	C₂	semidirect
7. C₇×#3166	C₂	semidirect
8. C₇×#3153	C₂	semidirect
9. C₇×#3131	C₂	semidirect
10. C₂×C₇×#488	C₂	semidirect
11. C₇×#3063	C₂	semidirect
12. C₇×#3062	C₂	semidirect
13. C₇×#2947	C₂	semidirect
14. C₇×#2946	C₂	semidirect
15. C₄×C₇×#131	C₂	semidirect
26. C₇×#541	C₂²	semidirect
27. C₇×#540	C₂²	semidirect
34. C₇×#529	C₂²	semidirect
41. C₇×#501	C₂²	semidirect
42. C₇×#499	C₂²	semidirect
49. C₇×#488	C₂²	semidirect
50. C₇×#488	C₂²	semidirect
51. C₇×#487	C₂²	semidirect
52. C₇×#486	C₂²	semidirect
53. C₇×#470	C₂²	semidirect
56. C₇×#468	C₂²	semidirect
57. C₇×#467	C₂²	semidirect
60. C₇×#465	C₂²	semidirect
62. C₇×#441	C₂²	semidirect
64. C₇×#440	C₂²	semidirect
65. C₄×C₇×Q₁₆	C₂²	semidirect
67. C₄×C₇×SD₁₆	C₂²	semidirect
76. C₇×#405	C₂²	semidirect
78. C₄×C₇×M₁₆	C₂²	semidirect
80. C₂×C₄×C₇×C₈	C₂²	semidirect
81. #60724	C₇	direct (N × H)
187. #3512	C₂×C₇	semidirect
188. #3510	C₂×C₇	semidirect
189. #3183	C₂×C₇	semidirect
190. #3179	C₂×C₇	semidirect
191. #3178	C₂×C₇	semidirect
192. #3168	C₂×C₇	semidirect
193. #3166	C₂×C₇	semidirect
194. #3153	C₂×C₇	semidirect
195. #3131	C₂×C₇	semidirect
196. C₂×#488	C₂×C₇	semidirect
197. #3063	C₂×C₇	semidirect
198. #3062	C₂×C₇	semidirect
199. #2947	C₂×C₇	semidirect
200. #2946	C₂×C₇	semidirect
201. C₄×#131	C₂×C₇	semidirect
291. #541	C₂²×C₇	semidirect
292. #540	C₂²×C₇	semidirect
299. #529	C₂²×C₇	semidirect
306. #501	C₂²×C₇	semidirect
307. #499	C₂²×C₇	semidirect
314. #488	C₂²×C₇	semidirect
315. #488	C₂²×C₇	semidirect
316. #487	C₂²×C₇	semidirect
317. #486	C₂²×C₇	semidirect
318. #470	C₂²×C₇	semidirect
321. #468	C₂²×C₇	semidirect
322. #467	C₂²×C₇	semidirect
325. #465	C₂²×C₇	semidirect
327. #441	C₂²×C₇	semidirect
329. #440	C₂²×C₇	semidirect
330. C₄×Q₁₆	C₂²×C₇	semidirect
332. C₄×SD₁₆	C₂²×C₇	semidirect
341. #405	C₂²×C₇	semidirect
343. C₄×M₁₆	C₂²×C₇	semidirect
345. C₂×C₄×C₈	C₂²×C₇	semidirect
648. C₇	#60724	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×#3512	2
C₇×#3510	2
C₇×#3183	2
C₇×#3179	2
C₇×#3178	2
C₇×#3168	2
C₇×#3166	2
C₇×#3153	2
C₇×#3131	2
C₂×C₇×#488	2
C₇×#3063	2
C₇×#3062	2
C₇×#2947	2
C₇×#2946	2
C₄×C₇×#131	2
#60724	7

Subgroups — C7×#60724 #13237940

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#60724 #13237940