Subgroups — C₇×#60717 #13237933

Order 1792.

1870 conjugacy classes of subgroups.

The subgroup lattice has 518 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	2	yes	maximal, complemented
2	C₇×#3183	C₂	1	4	yes	maximal, complemented
3	C₇×#3177	C₂	1	2	yes	maximal, complemented
4	C₇×#3176	C₂	1	2	yes	maximal, complemented
5	C₇×#3174	C₂	1	4	yes	maximal, complemented
6	C₇×#3168	C₂	1	4	yes	maximal, complemented
7	C₂×C₇×#499	C₂	1	1	yes	maximal, characteristic, complemented
8	C₇×#3147	C₂	1	1	yes	maximal, characteristic, complemented
9	C₇×#3142	C₂	1	2	yes	maximal, complemented
10	C₇×#3061	C₂	1	2	yes	maximal, complemented
11	C₇×#3060	C₂	1	2	yes	maximal, complemented
12	C₇×#2952	C₂	1	2	yes	maximal, complemented
13	C₇×#2951	C₂	1	2	yes	maximal, complemented
14	C₄×C₇×#131	C₂	1	1	yes	maximal, characteristic, complemented
15	C₂×C₇×#135	C₂²	1	2	yes	—
16	C₂×C₇×#135	C₂²	1	2	yes	—
17	C₂×C₇×#131	C₂²	1	1	yes	characteristic
18	C₂×C₇×#131	C₂²	1	1	yes	characteristic
19	C₇×#553	—	2	8	no	—
20	C₇×#553	—	2	4	no	—
21	C₇×#552	—	2	4	no	—
22	C₇×#541	C₂²	1	4	yes	—
23	C₇×#540	C₂²	1	4	yes	—
24	C₇×#537	C₂²	1	4	yes	—
25	C₇×#532	C₂²	1	2	yes	—
26	C₇×#529	C₂²	1	8	yes	—
27	C₂×C₇×#125	C₂²	1	2	yes	—
28	C₇×#520	C₂²	1	4	yes	—
29	C₄×C₇×#41	C₂²	1	1	yes	characteristic
30	C₇×#499	C₂²	1	2	yes	—
31	C₇×#499	C₂²	1	2	yes	—
32	C₇×#499	C₂²	1	2	yes	—
33	C₇×#499	C₂²	1	2	yes	—
34	C₇×#495	C₂²	1	2	yes	—
35	C₇×#495	C₂²	1	2	yes	—
36	C₇×#494	C₂²	1	1	yes	characteristic
37	C₇×#493	C₂²	1	1	yes	characteristic
38	C₇×#492	C₂²	1	2	yes	—
39	C₇×#491	C₂²	1	4	yes	—
40	C₇×#490	C₂²	1	4	yes	—
41	C₇×#488	C₂²	1	8	yes	—
42	C₇×#470	C₂²	1	4	yes	—
43	C₇×#469	C₂²	1	4	yes	—
44	C₇×#468	C₂²	1	8	yes	—
45	C₇×#467	C₂²	1	4	yes	—
46	C₇×#466	C₂²	1	4	yes	—
47	C₇×#465	C₂²	1	8	yes	—
48	C₇×#442	C₂²	1	4	yes	—
49	C₇×#441	C₂²	1	4	yes	—
50	C₇×#440	C₂²	1	8	yes	—
51	C₄×C₇×Q₁₆	C₂²	1	4	yes	—
52	C₄×C₇×SD₁₆	C₂²	1	8	yes	—
53	C₄×C₇×D₈	C₂²	1	4	yes	—
54	C₂×C₇×#107	C₂²	1	2	yes	—
55	C₂×C₇×#106	C₂²	1	2	yes	—
56	C₇×#417	C₂²	1	4	yes	—
57	C₇×#417	C₂²	1	4	yes	—
58	C₇×#405	C₂²	1	2	yes	—
59	C₇×#405	C₂²	1	2	yes	—
60	C₇×#405	C₂²	1	1	yes	characteristic
61	C₇×#405	C₂²	1	1	yes	characteristic
62	C₄×C₇×M₁₆	C₂²	1	2	yes	—
63	C₄×C₇×M₁₆	C₂²	1	1	yes	characteristic
64	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
65	C₂×C₄×C₇×C₈	C₂²	1	2	yes	—
66	#60717	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
67	C₂×C₇×#41	C₂³	1	4	yes	—
68	C₂×C₇×#41	C₂³	1	1	yes	characteristic
69	C₇×#135	—	2	8	no	—
70	C₇×#135	—	2	8	no	—
71	C₂×C₇×Q₁₆	C₂³	1	2	yes	—
72	C₂×C₇×Q₁₆	C₂³	1	2	yes	—
73	C₂×C₇×SD₁₆	C₂³	1	4	yes	—
74	C₂×C₇×SD₁₆	C₂³	1	4	yes	—
75	C₂×C₇×D₈	C₂³	1	2	yes	—
76	C₂×C₇×D₈	C₂³	1	2	yes	—
77	C₇×#131	—	2	4	no	—
78	C₇×#131	—	2	4	no	—
79	C₂×C₇×M₁₆	C₂³	1	2	yes	—
80	C₂×C₇×M₁₆	C₂³	1	2	yes	—
81	C₂×C₇×M₁₆	C₂³	1	1	yes	characteristic
82	C₂×C₇×M₁₆	C₂³	1	1	yes	characteristic
83	C₂²×C₇×C₈	C₂³	1	2	yes	—
84	C₂²×C₇×C₈	C₂³	1	2	yes	—
85	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
86	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
87	C₇×C₄⋊Q₈	C₂³	1	2	yes	—
88	C₇×#127	C₂³	1	2	yes	—
89	C₇×#126	—	2	8	no	—
90	C₇×#126	—	2	8	no	—
91	C₇×#125	—	2	8	no	—
92	C₇×#125	—	2	4	no	—
93	C₇×#125	C₂³	1	4	yes	—
94	C₇×#125	C₂³	1	4	yes	—
95	C₇×#124	C₂³	1	4	yes	—
96	C₇×#123	—	2	8	no	—
97	C₇×#123	—	2	8	no	—
98	C₇×C₂²⋊Q₈	—	2	4	no	—
99	C₇×C₂²⋊Q₈	C₂³	1	8	yes	—
100	C₇×#121	—	2	4	no	—
101	C₇×#121	—	2	8	no	—
102	C₇×#121	—	2	4	no	—
103	C₇×#121	—	2	8	no	—
104	C₇×#121	C₂³	1	8	yes	—
105	C₄×C₇×Q₈	D₄	1	2	yes	—
106	C₄×C₇×Q₈	C₂³	1	8	yes	—
107	C₄×C₇×D₄	—	2	8	no	—
108	C₄×C₇×D₄	—	2	8	no	—
109	C₄×C₇×D₄	—	2	4	no	—
110	C₄×C₇×D₄	D₄	1	4	yes	—
111	C₄×C₇×D₄	D₄	1	2	yes	—
112	C₄×C₇×D₄	C₂³	1	8	yes	—
113	C₇×#117	—	2	4	no	—
114	C₇×#117	—	2	8	no	—
115	C₇×#117	C₂³	1	1	yes	characteristic
116	C₇×#117	C₂³	1	2	yes	—
117	C₂×C₇×C₄⋊C₄	—	2	4	no	—
118	C₂×C₇×C₄⋊C₄	—	2	4	no	—
119	C₂×C₇×C₄⋊C₄	C₂³	1	4	yes	—
120	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
121	C₂×C₄²×C₇	C₂³	1	2	yes	—
122	C₇×#107	C₂³	1	4	yes	—
123	C₇×#107	C₂³	1	4	yes	—
124	C₇×#106	C₂³	1	4	yes	—
125	C₇×#106	C₂³	1	4	yes	—
126	C₇×#103	C₂³	1	8	yes	—
127	C₇×#103	C₂³	1	8	yes	—
128	C₇×#102	C₂³	1	8	yes	—
129	C₇×#102	C₂³	1	8	yes	—
130	C₇×#97	C₂³	1	2	yes	—
131	C₇×#97	C₂³	1	2	yes	—
132	C₇×#97	C₂³	1	2	yes	—
133	C₄×C₇×C₈	C₂³	1	2	yes	—
134	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
135	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
136	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
137	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
138	C₄×C₇×C₈	C₂³	1	2	yes	—
139	C₄×C₇×C₈	C₂³	1	2	yes	—
140	#3512	C₂×C₇	1	2	yes	complemented
141	#3183	C₂×C₇	1	4	yes	complemented
142	#3177	C₂×C₇	1	2	yes	complemented
143	#3176	C₂×C₇	1	2	yes	complemented
144	#3174	C₂×C₇	1	4	yes	complemented
145	#3168	C₂×C₇	1	4	yes	complemented
146	C₂×#499	C₂×C₇	1	1	yes	characteristic, complemented
147	#3147	C₂×C₇	1	1	yes	characteristic, complemented
148	#3142	C₂×C₇	1	2	yes	complemented
149	#3061	C₂×C₇	1	2	yes	complemented
150	#3060	C₂×C₇	1	2	yes	complemented
151	#2952	C₂×C₇	1	2	yes	complemented
152	#2951	C₂×C₇	1	2	yes	complemented
153	C₄×#131	C₂×C₇	1	1	yes	characteristic, complemented
154	C₇×#41	—	2	2	no	—
155	C₇×#41	—	2	2	no	—
156	C₇×#41	—	2	8	no	—
157	C₇×#41	—	2	8	no	—
158	C₂×C₇×Q₈	C₂×D₄	1	1	yes	characteristic
159	C₂×C₇×Q₈	C₂⁴	1	4	yes	—
160	C₂×C₇×D₄	—	2	4	no	—
161	C₂×C₇×D₄	—	2	8	no	—
162	C₂×C₇×D₄	—	2	4	no	—
163	C₂×C₇×D₄	C₂×D₄	1	2	yes	—
164	C₂×C₇×D₄	C₂×D₄	1	1	yes	characteristic
165	C₂×C₇×D₄	C₂⁴	1	4	yes	—
166	C₂²×C₄×C₇	—	2	4	no	—
167	C₂²×C₄×C₇	—	2	4	no	—
168	C₂²×C₄×C₇	—	2	8	no	—
169	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
170	C₂²×C₄×C₇	C₂×D₄	1	4	yes	—
171	C₂²×C₄×C₇	C₂⁴	1	2	yes	—
172	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
173	C₇×Q₁₆	—	2	4	no	—
174	C₇×Q₁₆	—	2	4	no	—
175	C₇×SD₁₆	—	2	8	no	—
176	C₇×SD₁₆	—	2	8	no	—
177	C₇×D₈	—	2	4	no	—
178	C₇×D₈	—	2	4	no	—
179	C₇×M₁₆	—	2	2	no	—
180	C₇×M₁₆	—	2	2	no	—
181	C₇×M₁₆	—	2	4	no	—
182	C₇×M₁₆	—	2	4	no	—
183	C₂×C₇×C₈	—	2	4	no	—
184	C₂×C₇×C₈	—	2	4	no	—
185	C₂×C₇×C₈	#41	1	4	yes	—
186	C₂×C₇×C₈	#41	1	4	yes	—
187	C₂×C₇×C₈	C₂⁴	1	2	yes	—
188	C₂×C₇×C₈	C₂⁴	1	2	yes	—
189	C₂×C₇×C₈	C₂⁴	1	1	yes	characteristic
190	C₂×C₇×C₈	C₂⁴	1	1	yes	characteristic
191	C₂×C₇×C₈	C₂⁴	1	1	yes	characteristic
192	C₂×C₇×C₈	C₂⁴	1	1	yes	characteristic
193	C₇×C₄⋊C₄	—	2	4	no	—
194	C₇×C₄⋊C₄	—	2	8	no	—
195	C₇×C₄⋊C₄	—	2	8	no	—
196	C₇×C₄⋊C₄	—	2	8	no	—
197	C₇×C₄⋊C₄	—	2	8	no	—
198	C₇×C₄⋊C₄	—	2	4	no	—
199	C₇×C₄⋊C₄	C₂×D₄	1	2	yes	—
200	C₇×C₄⋊C₄	C₂×D₄	1	4	yes	—
201	C₇×C₄⋊C₄	C₂⁴	1	8	yes	—
202	C₇×C₂²⋊C₄	—	2	8	no	—
203	C₇×C₂²⋊C₄	—	2	4	no	—
204	C₇×C₂²⋊C₄	—	2	8	no	—
205	C₇×C₂²⋊C₄	—	2	8	no	—
206	C₇×C₂²⋊C₄	—	2	4	no	—
207	C₇×C₂²⋊C₄	C₂×D₄	1	2	yes	—
208	C₇×C₂²⋊C₄	C₂×D₄	1	4	yes	—
209	C₄²×C₇	—	2	8	no	—
210	C₄²×C₇	C₂×D₄	1	4	yes	—
211	C₄²×C₇	C₂×D₄	1	2	yes	—
212	C₄²×C₇	C₂⁴	1	1	yes	characteristic
213	C₄²×C₇	C₂⁴	1	2	yes	—
214	C₄²×C₇	C₂⁴	1	1	yes	characteristic
215	C₂×#135	C₂²×C₇	1	2	yes	—
216	C₂×#135	C₂²×C₇	1	2	yes	—
217	C₂×#131	C₂²×C₇	1	1	yes	characteristic
218	C₂×#131	C₂²×C₇	1	1	yes	characteristic
219	#553	—	2	8	no	—
220	#553	—	2	4	no	—
221	#552	—	2	4	no	—
222	#541	C₂²×C₇	1	4	yes	—
223	#540	C₂²×C₇	1	4	yes	—
224	#537	C₂²×C₇	1	4	yes	—
225	#532	C₂²×C₇	1	2	yes	—
226	#529	C₂²×C₇	1	8	yes	—
227	C₂×#125	C₂²×C₇	1	2	yes	—
228	#520	C₂²×C₇	1	4	yes	—
229	C₄×#41	C₂²×C₇	1	1	yes	characteristic
230	#499	C₂²×C₇	1	2	yes	—
231	#499	C₂²×C₇	1	2	yes	—
232	#499	C₂²×C₇	1	2	yes	—
233	#499	C₂²×C₇	1	2	yes	—
234	#495	C₂²×C₇	1	2	yes	—
235	#495	C₂²×C₇	1	2	yes	—
236	#494	C₂²×C₇	1	1	yes	characteristic
237	#493	C₂²×C₇	1	1	yes	characteristic
238	#492	C₂²×C₇	1	2	yes	—
239	#491	C₂²×C₇	1	4	yes	—
240	#490	C₂²×C₇	1	4	yes	—
241	#488	C₂²×C₇	1	8	yes	—
242	#470	C₂²×C₇	1	4	yes	—
243	#469	C₂²×C₇	1	4	yes	—
244	#468	C₂²×C₇	1	8	yes	—
245	#467	C₂²×C₇	1	4	yes	—
246	#466	C₂²×C₇	1	4	yes	—
247	#465	C₂²×C₇	1	8	yes	—
248	#442	C₂²×C₇	1	4	yes	—
249	#441	C₂²×C₇	1	4	yes	—
250	#440	C₂²×C₇	1	8	yes	—
251	C₄×Q₁₆	C₂²×C₇	1	4	yes	—
252	C₄×SD₁₆	C₂²×C₇	1	8	yes	—
253	C₄×D₈	C₂²×C₇	1	4	yes	—
254	C₂×#107	C₂²×C₇	1	2	yes	—
255	C₂×#106	C₂²×C₇	1	2	yes	—
256	#417	C₂²×C₇	1	4	yes	—
257	#417	C₂²×C₇	1	4	yes	—
258	#405	C₂²×C₇	1	2	yes	—
259	#405	C₂²×C₇	1	2	yes	—
260	#405	C₂²×C₇	1	1	yes	characteristic
261	#405	C₂²×C₇	1	1	yes	characteristic
262	C₄×M₁₆	C₂²×C₇	1	1	yes	characteristic
263	C₄×M₁₆	C₂²×C₇	1	2	yes	—
264	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
265	C₂×C₄×C₈	C₂²×C₇	1	2	yes	—
266	C₂³×C₇	—	2	4	no	—
267	C₂³×C₇	C₂²×D₄	1	2	yes	—
268	C₂³×C₇	C₂²×D₄	1	1	yes	characteristic
269	C₇×Q₈	—	2	2	no	—
270	C₇×Q₈	—	2	4	no	—
271	C₇×Q₈	—	2	4	no	—
272	C₇×D₄	—	2	4	no	—
273	C₇×D₄	—	2	2	no	—
274	C₇×D₄	—	2	4	no	—
275	C₇×D₄	—	2	4	no	—
276	C₇×D₄	—	4	8	no	—
277	C₇×D₄	—	8	4	no	—
278	C₇×D₄	—	8	8	no	—
279	C₂×C₄×C₇	—	2	2	no	—
280	C₂×C₄×C₇	—	2	2	no	—
281	C₂×C₄×C₇	—	2	4	no	—
282	C₂×C₄×C₇	—	2	8	no	—
283	C₂×C₄×C₇	—	4	4	no	—
284	C₂×C₄×C₇	—	4	2	no	—
285	C₂×C₄×C₇	—	4	8	no	—
286	C₂×C₄×C₇	—	8	8	no	—
287	C₂×C₄×C₇	—	8	4	no	—
288	C₂×C₄×C₇	C₂²×D₄	1	4	yes	—
289	C₂×C₄×C₇	C₂²×D₄	1	2	yes	—
290	C₂×C₄×C₇	C₂²×D₄	1	2	yes	—
291	C₂×C₄×C₇	C₂²×D₄	1	2	yes	—
292	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
293	C₂×C₄×C₇	C₂×#41	1	2	yes	—
294	C₂×C₄×C₇	C₂×#41	1	2	yes	—
295	C₂×C₄×C₇	C₂⁵	1	1	yes	characteristic
296	C₇×C₈	—	2	2	no	—
297	C₇×C₈	—	2	2	no	—
298	C₇×C₈	C₂×#41	1	2	yes	—
299	C₇×C₈	C₂×#41	1	2	yes	—
300	C₇×C₈	C₂×#41	1	2	yes	—
301	C₇×C₈	C₂×#41	1	2	yes	—
302	C₂×#41	C₂³×C₇	1	4	yes	—
303	C₂×#41	C₂³×C₇	1	1	yes	characteristic
304	#135	—	2	8	no	—
305	#135	—	2	8	no	—
306	C₂×Q₁₆	C₂³×C₇	1	2	yes	—
307	C₂×Q₁₆	C₂³×C₇	1	2	yes	—
308	C₂×SD₁₆	C₂³×C₇	1	4	yes	—
309	C₂×SD₁₆	C₂³×C₇	1	4	yes	—
310	C₂×D₈	C₂³×C₇	1	2	yes	—
311	C₂×D₈	C₂³×C₇	1	2	yes	—
312	#131	—	2	4	no	—
313	#131	—	2	4	no	—
314	C₂×M₁₆	C₂³×C₇	1	1	yes	characteristic
315	C₂×M₁₆	C₂³×C₇	1	1	yes	characteristic
316	C₂×M₁₆	C₂³×C₇	1	2	yes	—
317	C₂×M₁₆	C₂³×C₇	1	2	yes	—
318	C₂²×C₈	C₂³×C₇	1	2	yes	—
319	C₂²×C₈	C₂³×C₇	1	2	yes	—
320	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
321	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
322	C₄⋊Q₈	C₂³×C₇	1	2	yes	—
323	#127	C₂³×C₇	1	2	yes	—
324	#126	—	2	8	no	—
325	#126	—	2	8	no	—
326	#125	—	2	8	no	—
327	#125	—	2	4	no	—
328	#125	C₂³×C₇	1	4	yes	—
329	#125	C₂³×C₇	1	4	yes	—
330	#124	C₂³×C₇	1	4	yes	—
331	#123	—	2	8	no	—
332	#123	—	2	8	no	—
333	C₂²⋊Q₈	—	2	4	no	—
334	C₂²⋊Q₈	C₂³×C₇	1	8	yes	—
335	#121	—	2	4	no	—
336	#121	—	2	8	no	—
337	#121	—	2	4	no	—
338	#121	—	2	8	no	—
339	#121	C₂³×C₇	1	8	yes	—
340	C₄×Q₈	C₇×D₄	1	2	yes	—
341	C₄×Q₈	C₂³×C₇	1	8	yes	—
342	C₄×D₄	—	2	8	no	—
343	C₄×D₄	—	2	8	no	—
344	C₄×D₄	—	2	4	no	—
345	C₄×D₄	C₇×D₄	1	4	yes	—
346	C₄×D₄	C₇×D₄	1	2	yes	—
347	C₄×D₄	C₂³×C₇	1	8	yes	—
348	#117	—	2	4	no	—
349	#117	—	2	8	no	—
350	#117	C₂³×C₇	1	1	yes	characteristic
351	#117	C₂³×C₇	1	2	yes	—
352	C₂×C₄⋊C₄	—	2	4	no	—
353	C₂×C₄⋊C₄	—	2	4	no	—
354	C₂×C₄⋊C₄	C₂³×C₇	1	4	yes	—
355	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
356	C₂×C₄²	C₂³×C₇	1	2	yes	—
357	#107	C₂³×C₇	1	4	yes	—
358	#107	C₂³×C₇	1	4	yes	—
359	#106	C₂³×C₇	1	4	yes	—
360	#106	C₂³×C₇	1	4	yes	—
361	#103	C₂³×C₇	1	8	yes	—
362	#103	C₂³×C₇	1	8	yes	—
363	#102	C₂³×C₇	1	8	yes	—
364	#102	C₂³×C₇	1	8	yes	—
365	#97	C₂³×C₇	1	2	yes	—
366	#97	C₂³×C₇	1	2	yes	—
367	#97	C₂³×C₇	1	2	yes	—
368	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
369	C₄×C₈	C₂³×C₇	1	2	yes	—
370	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
371	C₄×C₈	C₂³×C₇	1	2	yes	—
372	C₄×C₈	C₂³×C₇	1	2	yes	—
373	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
374	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
375	C₂²×C₇	—	2	2	no	—
376	C₂²×C₇	—	2	4	no	—
377	C₂²×C₇	—	2	2	no	—
378	C₂²×C₇	—	4	4	no	—
379	C₂²×C₇	—	4	4	no	—
380	C₂²×C₇	—	4	4	no	—
381	C₂²×C₇	#532	1	2	yes	—
382	C₂²×C₇	C₂³×D₄	1	1	yes	characteristic
383	C₄×C₇	—	2	2	no	—
384	C₄×C₇	—	2	4	no	—
385	C₄×C₇	—	2	2	no	—
386	C₄×C₇	—	2	2	no	—
387	C₄×C₇	—	4	8	no	—
388	C₄×C₇	—	4	4	no	—
389	C₄×C₇	#532	1	2	yes	—
390	C₄×C₇	C₂²×#41	1	1	yes	characteristic
391	C₄×C₇	C₂²×#41	1	1	yes	characteristic
392	#41	—	2	2	no	—
393	#41	—	2	2	no	—
394	#41	—	2	8	no	—
395	#41	—	2	8	no	—
396	C₂×Q₈	C₂×C₇×D₄	1	1	yes	characteristic
397	C₂×Q₈	C₂⁴×C₇	1	4	yes	—
398	C₂×D₄	—	2	4	no	—
399	C₂×D₄	—	2	8	no	—
400	C₂×D₄	—	2	4	no	—
401	C₂×D₄	C₂×C₇×D₄	1	2	yes	—
402	C₂×D₄	C₂×C₇×D₄	1	1	yes	characteristic
403	C₂×D₄	C₂⁴×C₇	1	4	yes	—
404	C₂²×C₄	—	2	4	no	—
405	C₂²×C₄	—	2	4	no	—
406	C₂²×C₄	—	2	8	no	—
407	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
408	C₂²×C₄	C₂×C₇×D₄	1	4	yes	—
409	C₂²×C₄	C₂⁴×C₇	1	2	yes	—
410	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
411	Q₁₆	—	2	4	no	—
412	Q₁₆	—	2	4	no	—
413	SD₁₆	—	2	8	no	—
414	SD₁₆	—	2	8	no	—
415	D₈	—	2	4	no	—
416	D₈	—	2	4	no	—
417	M₁₆	—	2	2	no	—
418	M₁₆	—	2	2	no	—
419	M₁₆	—	2	4	no	—
420	M₁₆	—	2	4	no	—
421	C₂×C₈	—	2	4	no	—
422	C₂×C₈	—	2	4	no	—
423	C₂×C₈	C₇×#41	1	4	yes	—
424	C₂×C₈	C₇×#41	1	4	yes	—
425	C₂×C₈	C₂⁴×C₇	1	2	yes	—
426	C₂×C₈	C₂⁴×C₇	1	2	yes	—
427	C₂×C₈	C₂⁴×C₇	1	1	yes	characteristic
428	C₂×C₈	C₂⁴×C₇	1	1	yes	characteristic
429	C₂×C₈	C₂⁴×C₇	1	1	yes	characteristic
430	C₂×C₈	C₂⁴×C₇	1	1	yes	characteristic
431	C₄⋊C₄	—	2	4	no	—
432	C₄⋊C₄	—	2	8	no	—
433	C₄⋊C₄	—	2	8	no	—
434	C₄⋊C₄	—	2	8	no	—
435	C₄⋊C₄	—	2	8	no	—
436	C₄⋊C₄	—	2	4	no	—
437	C₄⋊C₄	C₂×C₇×D₄	1	2	yes	—
438	C₄⋊C₄	C₂×C₇×D₄	1	4	yes	—
439	C₄⋊C₄	C₂⁴×C₇	1	8	yes	—
440	C₂²⋊C₄	—	2	8	no	—
441	C₂²⋊C₄	—	2	4	no	—
442	C₂²⋊C₄	—	2	8	no	—
443	C₂²⋊C₄	—	2	8	no	—
444	C₂²⋊C₄	—	2	4	no	—
445	C₂²⋊C₄	C₂×C₇×D₄	1	2	yes	—
446	C₂²⋊C₄	C₂×C₇×D₄	1	4	yes	—
447	C₄²	—	2	8	no	—
448	C₄²	C₂×C₇×D₄	1	4	yes	—
449	C₄²	C₂×C₇×D₄	1	2	yes	—
450	C₄²	C₂⁴×C₇	1	1	yes	characteristic
451	C₄²	C₂⁴×C₇	1	2	yes	—
452	C₄²	C₂⁴×C₇	1	1	yes	characteristic
453	C₂×C₇	—	2	2	no	—
454	C₂×C₇	—	4	2	no	—
455	C₂×C₇	—	8	4	no	—
456	C₂×C₇	C₂×#532	1	1	yes	characteristic
457	C₂×C₇	#3584	1	2	yes	—
458	C₂³	—	2	4	no	—
459	C₂³	C₂²×C₇×D₄	1	2	yes	—
460	C₂³	C₂²×C₇×D₄	1	1	yes	characteristic
461	Q₈	—	2	2	no	—
462	Q₈	—	2	4	no	—
463	Q₈	—	2	4	no	—
464	D₄	—	2	4	no	—
465	D₄	—	2	4	no	—
466	D₄	—	2	4	no	—
467	D₄	—	2	2	no	—
468	D₄	—	4	8	no	—
469	D₄	—	8	4	no	—
470	D₄	—	8	8	no	—
471	C₂×C₄	—	2	2	no	—
472	C₂×C₄	—	2	2	no	—
473	C₂×C₄	—	2	4	no	—
474	C₂×C₄	—	2	8	no	—
475	C₂×C₄	—	4	4	no	—
476	C₂×C₄	—	4	2	no	—
477	C₂×C₄	—	4	8	no	—
478	C₂×C₄	—	8	8	no	—
479	C₂×C₄	—	8	4	no	—
480	C₂×C₄	C₂²×C₇×D₄	1	4	yes	—
481	C₂×C₄	C₂²×C₇×D₄	1	2	yes	—
482	C₂×C₄	C₂²×C₇×D₄	1	2	yes	—
483	C₂×C₄	C₂²×C₇×D₄	1	2	yes	—
484	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
485	C₂×C₄	C₂×C₇×#41	1	2	yes	—
486	C₂×C₄	C₂×C₇×#41	1	2	yes	—
487	C₂×C₄	C₂⁵×C₇	1	1	yes	characteristic
488	C₈	—	2	2	no	—
489	C₈	—	2	2	no	—
490	C₈	C₂×C₇×#41	1	2	yes	—
491	C₈	C₂×C₇×#41	1	2	yes	—
492	C₈	C₂×C₇×#41	1	2	yes	—
493	C₈	C₂×C₇×#41	1	2	yes	—
494	C₇	#60717	1	1	yes	characteristic, direct product factor, Sylow
495	C₂²	—	2	2	no	—
496	C₂²	—	2	4	no	—
497	C₂²	—	2	2	no	—
498	C₂²	—	4	4	no	—
499	C₂²	—	4	4	no	—
500	C₂²	—	4	4	no	—
501	C₂²	C₇×#532	1	2	yes	—
502	C₂²	C₂³×C₇×D₄	1	1	yes	characteristic
503	C₄	—	2	2	no	—
504	C₄	—	2	4	no	—
505	C₄	—	2	2	no	—
506	C₄	—	2	2	no	—
507	C₄	—	4	8	no	—
508	C₄	—	4	4	no	—
509	C₄	C₇×#532	1	2	yes	—
510	C₄	C₂²×C₇×#41	1	1	yes	characteristic
511	C₄	C₂²×C₇×#41	1	1	yes	characteristic
512	C₂	—	2	2	no	—
513	C₂	—	4	2	no	—
514	C₂	—	8	4	no	—
515	C₂	C₂×C₇×#532	1	1	yes	characteristic
516	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×#3183	C₂	semidirect
3. C₇×#3177	C₂	semidirect
4. C₇×#3176	C₂	semidirect
5. C₇×#3174	C₂	semidirect
6. C₇×#3168	C₂	semidirect
7. C₂×C₇×#499	C₂	semidirect
8. C₇×#3147	C₂	semidirect
9. C₇×#3142	C₂	semidirect
10. C₇×#3061	C₂	semidirect
11. C₇×#3060	C₂	semidirect
12. C₇×#2952	C₂	semidirect
13. C₇×#2951	C₂	semidirect
14. C₄×C₇×#131	C₂	semidirect
66. #60717	C₇	direct (N × H)
140. #3512	C₂×C₇	semidirect
141. #3183	C₂×C₇	semidirect
142. #3177	C₂×C₇	semidirect
143. #3176	C₂×C₇	semidirect
144. #3174	C₂×C₇	semidirect
145. #3168	C₂×C₇	semidirect
146. C₂×#499	C₂×C₇	semidirect
147. #3147	C₂×C₇	semidirect
148. #3142	C₂×C₇	semidirect
149. #3061	C₂×C₇	semidirect
150. #3060	C₂×C₇	semidirect
151. #2952	C₂×C₇	semidirect
152. #2951	C₂×C₇	semidirect
153. C₄×#131	C₂×C₇	semidirect
494. C₇	#60717	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×#3183	2
C₇×#3177	2
C₇×#3176	2
C₇×#3174	2
C₇×#3168	2
C₂×C₇×#499	2
C₇×#3147	2
C₇×#3142	2
C₇×#3061	2
C₇×#3060	2
C₇×#2952	2
C₇×#2951	2
C₄×C₇×#131	2
#60717	7

Subgroups — C7×#60717 #13237933

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#60717 #13237933