Subgroups — C₇×#60945 #13238161

Order 1792.

1866 conjugacy classes of subgroups.

The subgroup lattice has 584 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×#529	C₂	1	2	yes	maximal, complemented
2	C₇×#3250	C₂	1	4	yes	maximal, complemented
3	C₇×#3232	C₂	1	4	yes	maximal, complemented
4	C₇×#3211	C₂	1	4	yes	maximal, complemented
5	C₇×#3189	C₂	1	1	yes	maximal, characteristic, complemented
6	C₇×#3187	C₂	1	2	yes	maximal, complemented
7	C₇×#3186	C₂	1	1	yes	maximal, characteristic
8	C₇×#3168	C₂	1	4	yes	maximal, complemented
9	C₇×#3108	C₂	1	4	yes	maximal, complemented
10	C₇×#3105	C₂	1	4	yes	maximal, complemented
11	C₂×C₇×#431	C₂	1	1	yes	maximal, characteristic, complemented
12	C₇×#529	—	2	2	no	—
13	C₇×#529	—	2	2	no	—
14	C₇×#529	—	2	2	no	—
15	C₇×#529	—	2	2	no	—
16	C₇×#529	C₂²	1	4	yes	—
17	C₇×#529	C₂²	1	4	yes	—
18	C₇×#529	C₂²	1	4	yes	complemented
19	C₇×#529	C₂²	1	4	yes	complemented
20	C₂×C₇×#126	—	2	2	no	—
21	C₂×C₇×#125	C₂²	1	2	yes	complemented
22	C₂×C₇×#124	C₂²	1	2	yes	complemented
23	C₂×C₇×#123	—	2	2	no	—
24	C₂×C₇×C₂²⋊Q₈	C₂²	1	2	yes	—
25	C₂×C₇×#121	C₂²	1	2	yes	—
26	C₂×C₄×C₇×Q₈	C₂²	1	2	yes	complemented
27	C₂×C₄×C₇×D₄	—	2	2	no	—
28	C₂×C₄×C₇×D₄	C₂²	1	2	yes	complemented
29	C₂×C₇×#117	—	2	2	no	—
30	C₂²×C₄²×C₇	C₂²	1	1	yes	characteristic
31	C₇×#499	C₂²	1	4	yes	complemented
32	C₇×#495	C₂²	1	4	yes	complemented
33	C₇×#488	C₂²	1	8	yes	complemented
34	C₇×#484	C₂²	1	4	yes	complemented
35	C₇×#483	C₂²	1	4	yes	complemented
36	C₇×#481	C₂²	1	4	yes	—
37	C₇×#480	C₂²	1	4	yes	—
38	C₇×#479	C₂²	1	8	yes	complemented
39	C₇×#478	C₂²	1	8	yes	complemented
40	C₇×#467	C₂²	1	4	yes	complemented
41	C₇×#466	C₂²	1	4	yes	—
42	C₇×#465	C₂²	1	8	yes	complemented
43	C₇×#464	C₂²	1	8	yes	complemented
44	C₇×#463	C₂²	1	8	yes	complemented
45	C₇×#451	C₂²	1	4	yes	complemented
46	C₇×#450	C₂²	1	4	yes	complemented
47	C₇×#448	C₂²	1	4	yes	—
48	C₇×#447	C₂²	1	4	yes	—
49	C₄×C₇×Q₁₆	C₂²	1	4	yes	complemented
50	C₄×C₇×SD₁₆	C₂²	1	8	yes	complemented
51	C₄×C₇×D₈	C₂²	1	4	yes	complemented
52	C₇×#431	C₂²	1	4	yes	complemented
53	C₇×#431	C₂²	1	4	yes	complemented
54	C₂×C₇×C₄⋊C₈	C₂²	1	2	yes	complemented
55	C₂×C₇×C₂²⋊C₈	C₂²	1	1	yes	characteristic
56	C₂×C₇×C₂²⋊C₈	C₂²	1	1	yes	characteristic
57	C₂×C₄×C₇×C₈	C₂²	1	2	yes	complemented
58	#60945	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
59	C₂²×C₇×Q₈	C₂³	1	2	yes	—
60	C₂²×C₇×D₄	—	2	2	no	—
61	C₂²×C₇×D₄	C₂³	1	2	yes	—
62	C₂³×C₄×C₇	—	2	2	no	—
63	C₂³×C₄×C₇	D₄	1	1	yes	characteristic
64	C₂³×C₄×C₇	D₄	1	1	yes	characteristic
65	C₂³×C₄×C₇	C₂³	1	1	yes	characteristic
66	C₂×C₇×Q₁₆	C₂³	1	4	yes	—
67	C₂×C₇×SD₁₆	C₂³	1	8	yes	—
68	C₂×C₇×D₈	C₂³	1	4	yes	—
69	C₂²×C₇×C₈	C₂³	1	2	yes	—
70	C₂²×C₇×C₈	C₂³	1	2	yes	—
71	C₇×#126	—	2	4	no	—
72	C₇×#126	—	2	4	no	—
73	C₇×#126	—	2	4	no	—
74	C₇×#126	—	2	4	no	—
75	C₇×#125	—	2	2	no	—
76	C₇×#125	—	2	2	no	—
77	C₇×#125	C₂³	1	4	yes	—
78	C₇×#125	C₂³	1	4	yes	—
79	C₇×#124	—	2	2	no	—
80	C₇×#124	—	2	2	no	—
81	C₇×#124	C₂³	1	4	yes	—
82	C₇×#124	C₂³	1	4	yes	—
83	C₇×#123	—	2	4	no	—
84	C₇×#123	—	2	4	no	—
85	C₇×#123	—	2	4	no	—
86	C₇×#123	—	2	4	no	—
87	C₇×C₂²⋊Q₈	—	2	2	no	—
88	C₇×C₂²⋊Q₈	—	2	2	no	—
89	C₇×C₂²⋊Q₈	C₂³	1	4	yes	—
90	C₇×C₂²⋊Q₈	C₂³	1	4	yes	—
91	C₇×#121	—	2	2	no	—
92	C₇×#121	—	2	2	no	—
93	C₇×#121	C₂³	1	4	yes	—
94	C₇×#121	C₂³	1	4	yes	—
95	C₄×C₇×Q₈	—	2	2	no	—
96	C₄×C₇×Q₈	—	2	2	no	—
97	C₄×C₇×Q₈	C₂³	1	4	yes	—
98	C₄×C₇×Q₈	C₂³	1	4	yes	—
99	C₄×C₇×D₄	—	2	4	no	—
100	C₄×C₇×D₄	—	2	4	no	—
101	C₄×C₇×D₄	—	2	4	no	—
102	C₄×C₇×D₄	—	2	4	no	—
103	C₄×C₇×D₄	—	2	2	no	—
104	C₄×C₇×D₄	—	2	2	no	—
105	C₄×C₇×D₄	C₂³	1	4	yes	—
106	C₄×C₇×D₄	C₂³	1	4	yes	—
107	C₇×#117	—	2	4	no	—
108	C₇×#117	—	2	4	no	—
109	C₇×#117	—	2	4	no	—
110	C₇×#117	—	2	4	no	—
111	C₂×C₇×C₄⋊C₄	—	2	2	no	—
112	C₂×C₇×C₄⋊C₄	—	2	2	no	—
113	C₂×C₇×C₄⋊C₄	—	2	2	no	—
114	C₂×C₇×C₄⋊C₄	—	2	2	no	—
115	C₂×C₇×C₄⋊C₄	C₂³	1	2	yes	—
116	C₂×C₇×C₄⋊C₄	C₂³	1	2	yes	—
117	C₂×C₇×C₂²⋊C₄	—	2	2	no	—
118	C₂×C₇×C₂²⋊C₄	—	2	2	no	—
119	C₂×C₇×C₂²⋊C₄	—	2	2	no	—
120	C₂×C₇×C₂²⋊C₄	—	2	2	no	—
121	C₂×C₇×C₂²⋊C₄	—	2	2	no	—
122	C₂×C₄²×C₇	—	2	2	no	—
123	C₂×C₄²×C₇	D₄	1	2	yes	—
124	C₂×C₄²×C₇	D₄	1	2	yes	—
125	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
126	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
127	C₂×C₄²×C₇	C₂³	1	2	yes	—
128	C₂×C₄²×C₇	C₂³	1	2	yes	—
129	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
130	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
131	C₇×#107	C₂³	1	4	yes	—
132	C₇×#107	C₂³	1	4	yes	—
133	C₇×#106	C₂³	1	8	yes	—
134	C₇×C₄⋊C₈	C₂³	1	8	yes	—
135	C₇×#103	C₂³	1	8	yes	—
136	C₇×#103	C₂³	1	8	yes	—
137	C₇×#102	C₂³	1	8	yes	—
138	C₇×#102	C₂³	1	8	yes	—
139	C₇×C₂²⋊C₈	C₂³	1	4	yes	—
140	C₇×C₂²⋊C₈	C₂³	1	4	yes	—
141	C₄×C₇×C₈	C₂³	1	4	yes	—
142	C₄×C₇×C₈	C₂³	1	4	yes	—
143	C₂×#529	C₂×C₇	1	2	yes	complemented
144	#3250	C₂×C₇	1	4	yes	complemented
145	#3232	C₂×C₇	1	4	yes	complemented
146	#3211	C₂×C₇	1	4	yes	complemented
147	#3189	C₂×C₇	1	1	yes	characteristic, complemented
148	#3187	C₂×C₇	1	2	yes	complemented
149	#3186	C₂×C₇	1	1	yes	characteristic
150	#3168	C₂×C₇	1	4	yes	complemented
151	#3108	C₂×C₇	1	4	yes	complemented
152	#3105	C₂×C₇	1	4	yes	complemented
153	C₂×#431	C₂×C₇	1	1	yes	characteristic, complemented
154	C₂⁴×C₇	—	2	2	no	—
155	C₂⁴×C₇	C₂×D₄	1	1	yes	characteristic
156	C₂×C₇×Q₈	—	2	2	no	—
157	C₂×C₇×Q₈	—	4	4	no	—
158	C₂×C₇×Q₈	C₂⁴	1	4	yes	—
159	C₂×C₇×D₄	—	2	4	no	—
160	C₂×C₇×D₄	—	2	4	no	—
161	C₂×C₇×D₄	—	2	2	no	—
162	C₂×C₇×D₄	—	4	4	no	—
163	C₂×C₇×D₄	—	4	8	no	—
164	C₂×C₇×D₄	C₂⁴	1	4	yes	—
165	C₂²×C₄×C₇	—	2	4	no	—
166	C₂²×C₄×C₇	—	2	2	no	—
167	C₂²×C₄×C₇	—	2	2	no	—
168	C₂²×C₄×C₇	—	2	2	no	—
169	C₂²×C₄×C₇	—	2	4	no	—
170	C₂²×C₄×C₇	—	2	4	no	—
171	C₂²×C₄×C₇	—	4	2	no	—
172	C₂²×C₄×C₇	—	4	2	no	—
173	C₂²×C₄×C₇	—	4	8	no	—
174	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
175	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
176	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
177	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
178	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
179	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
180	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
181	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
182	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
183	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
184	C₂²×C₄×C₇	C₂⁴	1	2	yes	—
185	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
186	C₇×Q₁₆	—	4	4	no	—
187	C₇×SD₁₆	—	4	8	no	—
188	C₇×D₈	—	4	4	no	—
189	C₂×C₇×C₈	—	4	2	no	—
190	C₂×C₇×C₈	—	4	2	no	—
191	C₂×C₇×C₈	C₂⁴	1	4	yes	—
192	C₂×C₇×C₈	C₂⁴	1	4	yes	—
193	C₇×C₄⋊C₄	—	2	4	no	—
194	C₇×C₄⋊C₄	—	2	4	no	—
195	C₇×C₄⋊C₄	—	2	4	no	—
196	C₇×C₄⋊C₄	—	2	4	no	—
197	C₇×C₄⋊C₄	—	2	4	no	—
198	C₇×C₄⋊C₄	—	2	4	no	—
199	C₇×C₄⋊C₄	—	2	4	no	—
200	C₇×C₄⋊C₄	—	2	4	no	—
201	C₇×C₄⋊C₄	—	2	2	no	—
202	C₇×C₄⋊C₄	—	2	2	no	—
203	C₇×C₄⋊C₄	C₂⁴	1	4	yes	—
204	C₇×C₄⋊C₄	C₂⁴	1	4	yes	—
205	C₇×C₂²⋊C₄	—	2	4	no	—
206	C₇×C₂²⋊C₄	—	2	4	no	—
207	C₇×C₂²⋊C₄	—	2	4	no	—
208	C₇×C₂²⋊C₄	—	2	4	no	—
209	C₇×C₂²⋊C₄	—	2	4	no	—
210	C₇×C₂²⋊C₄	—	2	4	no	—
211	C₇×C₂²⋊C₄	—	2	4	no	—
212	C₇×C₂²⋊C₄	—	2	4	no	—
213	C₇×C₂²⋊C₄	—	2	4	no	—
214	C₇×C₂²⋊C₄	—	2	4	no	—
215	C₄²×C₇	—	2	4	no	—
216	C₄²×C₇	—	2	4	no	—
217	C₄²×C₇	C₂×D₄	1	2	yes	—
218	C₄²×C₇	C₂×D₄	1	2	yes	—
219	C₄²×C₇	C₂×D₄	1	2	yes	—
220	C₄²×C₇	C₂×D₄	1	2	yes	—
221	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
222	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
223	C₄²×C₇	C₂×D₄	1	2	yes	—
224	C₄²×C₇	C₂⁴	1	1	yes	characteristic
225	C₄²×C₇	C₂⁴	1	2	yes	—
226	C₄²×C₇	C₂⁴	1	1	yes	characteristic
227	#529	—	2	2	no	—
228	#529	—	2	2	no	—
229	#529	—	2	2	no	—
230	#529	—	2	2	no	—
231	#529	C₂²×C₇	1	4	yes	—
232	#529	C₂²×C₇	1	4	yes	—
233	#529	C₂²×C₇	1	4	yes	complemented
234	#529	C₂²×C₇	1	4	yes	complemented
235	C₂×#126	—	2	2	no	—
236	C₂×#125	C₂²×C₇	1	2	yes	complemented
237	C₂×#124	C₂²×C₇	1	2	yes	complemented
238	C₂×#123	—	2	2	no	—
239	C₂×C₂²⋊Q₈	C₂²×C₇	1	2	yes	—
240	C₂×#121	C₂²×C₇	1	2	yes	—
241	C₂×C₄×Q₈	C₂²×C₇	1	2	yes	complemented
242	C₂×C₄×D₄	—	2	2	no	—
243	C₂×C₄×D₄	C₂²×C₇	1	2	yes	complemented
244	C₂×#117	—	2	2	no	—
245	C₂²×C₄²	C₂²×C₇	1	1	yes	characteristic
246	#499	C₂²×C₇	1	4	yes	complemented
247	#495	C₂²×C₇	1	4	yes	complemented
248	#488	C₂²×C₇	1	8	yes	complemented
249	#484	C₂²×C₇	1	4	yes	complemented
250	#483	C₂²×C₇	1	4	yes	complemented
251	#481	C₂²×C₇	1	4	yes	—
252	#480	C₂²×C₇	1	4	yes	—
253	#479	C₂²×C₇	1	8	yes	complemented
254	#478	C₂²×C₇	1	8	yes	complemented
255	#467	C₂²×C₇	1	4	yes	complemented
256	#466	C₂²×C₇	1	4	yes	—
257	#465	C₂²×C₇	1	8	yes	complemented
258	#464	C₂²×C₇	1	8	yes	complemented
259	#463	C₂²×C₇	1	8	yes	complemented
260	#451	C₂²×C₇	1	4	yes	complemented
261	#450	C₂²×C₇	1	4	yes	complemented
262	#448	C₂²×C₇	1	4	yes	—
263	#447	C₂²×C₇	1	4	yes	—
264	C₄×Q₁₆	C₂²×C₇	1	4	yes	complemented
265	C₄×SD₁₆	C₂²×C₇	1	8	yes	complemented
266	C₄×D₈	C₂²×C₇	1	4	yes	complemented
267	#431	C₂²×C₇	1	4	yes	complemented
268	#431	C₂²×C₇	1	4	yes	complemented
269	C₂×C₄⋊C₈	C₂²×C₇	1	2	yes	complemented
270	C₂×C₂²⋊C₈	C₂²×C₇	1	1	yes	characteristic
271	C₂×C₂²⋊C₈	C₂²×C₇	1	1	yes	characteristic
272	C₂×C₄×C₈	C₂²×C₇	1	2	yes	complemented
273	C₂³×C₇	—	2	4	no	—
274	C₂³×C₇	—	4	1	no	—
275	C₂³×C₇	—	4	1	no	—
276	C₂³×C₇	—	4	4	no	—
277	C₂³×C₇	—	4	4	no	—
278	C₂³×C₇	—	4	4	no	—
279	C₂³×C₇	#135	1	4	yes	—
280	C₂³×C₇	C₂²×D₄	1	1	yes	characteristic
281	C₂³×C₇	C₂²×D₄	1	2	yes	—
282	C₇×Q₈	—	2	8	no	—
283	C₇×Q₈	—	4	4	no	—
284	C₇×D₄	—	2	8	no	—
285	C₇×D₄	—	4	4	no	—
286	C₇×D₄	—	4	8	no	—
287	C₇×D₄	—	4	8	no	—
288	C₂×C₄×C₇	—	2	4	no	—
289	C₂×C₄×C₇	—	2	4	no	—
290	C₂×C₄×C₇	—	2	2	no	—
291	C₂×C₄×C₇	—	2	2	no	—
292	C₂×C₄×C₇	—	2	4	no	—
293	C₂×C₄×C₇	—	2	2	no	—
294	C₂×C₄×C₇	—	2	2	no	—
295	C₂×C₄×C₇	—	2	4	no	—
296	C₂×C₄×C₇	—	2	4	no	—
297	C₂×C₄×C₇	—	2	4	no	—
298	C₂×C₄×C₇	—	2	4	no	—
299	C₂×C₄×C₇	—	4	2	no	—
300	C₂×C₄×C₇	—	4	1	no	—
301	C₂×C₄×C₇	—	4	1	no	—
302	C₂×C₄×C₇	—	4	2	no	—
303	C₂×C₄×C₇	—	4	8	no	—
304	C₂×C₄×C₇	—	4	4	no	—
305	C₂×C₄×C₇	—	4	4	no	—
306	C₂×C₄×C₇	—	4	4	no	—
307	C₂×C₄×C₇	—	4	8	no	—
308	C₂×C₄×C₇	C₂²×D₄	1	2	yes	—
309	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
310	C₂×C₄×C₇	C₂²×D₄	1	2	yes	—
311	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
312	C₂×C₄×C₇	C₂²×D₄	1	2	yes	—
313	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
314	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
315	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
316	C₂×C₄×C₇	C₂⁵	1	1	yes	characteristic
317	C₇×C₈	—	2	4	no	—
318	C₇×C₈	—	2	4	no	—
319	C₂²×Q₈	C₂³×C₇	1	2	yes	—
320	C₂²×D₄	—	2	2	no	—
321	C₂²×D₄	C₂³×C₇	1	2	yes	—
322	C₂³×C₄	—	2	2	no	—
323	C₂³×C₄	C₇×D₄	1	1	yes	characteristic
324	C₂³×C₄	C₇×D₄	1	1	yes	characteristic
325	C₂³×C₄	C₂³×C₇	1	1	yes	characteristic
326	C₂×Q₁₆	C₂³×C₇	1	4	yes	—
327	C₂×SD₁₆	C₂³×C₇	1	8	yes	—
328	C₂×D₈	C₂³×C₇	1	4	yes	—
329	C₂²×C₈	C₂³×C₇	1	2	yes	—
330	C₂²×C₈	C₂³×C₇	1	2	yes	—
331	#126	—	2	4	no	—
332	#126	—	2	4	no	—
333	#126	—	2	4	no	—
334	#126	—	2	4	no	—
335	#125	—	2	2	no	—
336	#125	—	2	2	no	—
337	#125	C₂³×C₇	1	4	yes	—
338	#125	C₂³×C₇	1	4	yes	—
339	#124	—	2	2	no	—
340	#124	—	2	2	no	—
341	#124	C₂³×C₇	1	4	yes	—
342	#124	C₂³×C₇	1	4	yes	—
343	#123	—	2	4	no	—
344	#123	—	2	4	no	—
345	#123	—	2	4	no	—
346	#123	—	2	4	no	—
347	C₂²⋊Q₈	—	2	2	no	—
348	C₂²⋊Q₈	—	2	2	no	—
349	C₂²⋊Q₈	C₂³×C₇	1	4	yes	—
350	C₂²⋊Q₈	C₂³×C₇	1	4	yes	—
351	#121	—	2	2	no	—
352	#121	—	2	2	no	—
353	#121	C₂³×C₇	1	4	yes	—
354	#121	C₂³×C₇	1	4	yes	—
355	C₄×Q₈	—	2	2	no	—
356	C₄×Q₈	—	2	2	no	—
357	C₄×Q₈	C₂³×C₇	1	4	yes	—
358	C₄×Q₈	C₂³×C₇	1	4	yes	—
359	C₄×D₄	—	2	4	no	—
360	C₄×D₄	—	2	4	no	—
361	C₄×D₄	—	2	4	no	—
362	C₄×D₄	—	2	4	no	—
363	C₄×D₄	—	2	2	no	—
364	C₄×D₄	—	2	2	no	—
365	C₄×D₄	C₂³×C₇	1	4	yes	—
366	C₄×D₄	C₂³×C₇	1	4	yes	—
367	#117	—	2	4	no	—
368	#117	—	2	4	no	—
369	#117	—	2	4	no	—
370	#117	—	2	4	no	—
371	C₂×C₄⋊C₄	—	2	2	no	—
372	C₂×C₄⋊C₄	—	2	2	no	—
373	C₂×C₄⋊C₄	—	2	2	no	—
374	C₂×C₄⋊C₄	—	2	2	no	—
375	C₂×C₄⋊C₄	C₂³×C₇	1	2	yes	—
376	C₂×C₄⋊C₄	C₂³×C₇	1	2	yes	—
377	C₂×C₂²⋊C₄	—	2	2	no	—
378	C₂×C₂²⋊C₄	—	2	2	no	—
379	C₂×C₂²⋊C₄	—	2	2	no	—
380	C₂×C₂²⋊C₄	—	2	2	no	—
381	C₂×C₂²⋊C₄	—	2	2	no	—
382	C₂×C₄²	—	2	2	no	—
383	C₂×C₄²	C₇×D₄	1	2	yes	—
384	C₂×C₄²	C₇×D₄	1	2	yes	—
385	C₂×C₄²	C₇×D₄	1	1	yes	characteristic
386	C₂×C₄²	C₇×D₄	1	1	yes	characteristic
387	C₂×C₄²	C₂³×C₇	1	2	yes	—
388	C₂×C₄²	C₂³×C₇	1	2	yes	—
389	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
390	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
391	#107	C₂³×C₇	1	4	yes	—
392	#107	C₂³×C₇	1	4	yes	—
393	#106	C₂³×C₇	1	8	yes	—
394	C₄⋊C₈	C₂³×C₇	1	8	yes	—
395	#103	C₂³×C₇	1	8	yes	—
396	#103	C₂³×C₇	1	8	yes	—
397	#102	C₂³×C₇	1	8	yes	—
398	#102	C₂³×C₇	1	8	yes	—
399	C₂²⋊C₈	C₂³×C₇	1	4	yes	—
400	C₂²⋊C₈	C₂³×C₇	1	4	yes	—
401	C₄×C₈	C₂³×C₇	1	4	yes	—
402	C₄×C₈	C₂³×C₇	1	4	yes	—
403	C₂²×C₇	—	2	2	no	—
404	C₂²×C₇	—	2	1	no	—
405	C₂²×C₇	—	2	2	no	—
406	C₂²×C₇	—	2	1	no	—
407	C₂²×C₇	—	4	4	no	—
408	C₂²×C₇	—	4	4	no	—
409	C₂²×C₇	—	4	4	no	—
410	C₂²×C₇	—	4	4	no	—
411	C₂²×C₇	—	8	8	no	—
412	C₂²×C₇	C₂×#135	1	4	yes	—
413	C₂²×C₇	C₂×#135	1	2	yes	—
414	C₂²×C₇	C₂³×D₄	1	1	yes	characteristic
415	C₄×C₇	—	2	1	no	—
416	C₄×C₇	—	2	2	no	—
417	C₄×C₇	—	2	1	no	—
418	C₄×C₇	—	2	2	no	—
419	C₄×C₇	—	2	1	no	—
420	C₄×C₇	—	2	2	no	—
421	C₄×C₇	—	2	1	no	—
422	C₄×C₇	—	4	4	no	—
423	C₄×C₇	—	4	4	no	—
424	C₄×C₇	—	4	4	no	—
425	C₄×C₇	#575	1	2	yes	—
426	C₄×C₇	#585	1	2	yes	—
427	C₂⁴	—	2	2	no	—
428	C₂⁴	C₂×C₇×D₄	1	1	yes	characteristic
429	C₂×Q₈	—	2	2	no	—
430	C₂×Q₈	—	4	4	no	—
431	C₂×Q₈	C₂⁴×C₇	1	4	yes	—
432	C₂×D₄	—	2	4	no	—
433	C₂×D₄	—	2	4	no	—
434	C₂×D₄	—	2	2	no	—
435	C₂×D₄	—	4	4	no	—
436	C₂×D₄	—	4	8	no	—
437	C₂×D₄	C₂⁴×C₇	1	4	yes	—
438	C₂²×C₄	—	2	4	no	—
439	C₂²×C₄	—	2	2	no	—
440	C₂²×C₄	—	2	2	no	—
441	C₂²×C₄	—	2	2	no	—
442	C₂²×C₄	—	2	4	no	—
443	C₂²×C₄	—	2	4	no	—
444	C₂²×C₄	—	4	2	no	—
445	C₂²×C₄	—	4	2	no	—
446	C₂²×C₄	—	4	8	no	—
447	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
448	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
449	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
450	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
451	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
452	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
453	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
454	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
455	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
456	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
457	C₂²×C₄	C₂⁴×C₇	1	2	yes	—
458	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
459	Q₁₆	—	4	4	no	—
460	SD₁₆	—	4	8	no	—
461	D₈	—	4	4	no	—
462	C₂×C₈	—	4	2	no	—
463	C₂×C₈	—	4	2	no	—
464	C₂×C₈	C₂⁴×C₇	1	4	yes	—
465	C₂×C₈	C₂⁴×C₇	1	4	yes	—
466	C₄⋊C₄	—	2	4	no	—
467	C₄⋊C₄	—	2	4	no	—
468	C₄⋊C₄	—	2	4	no	—
469	C₄⋊C₄	—	2	4	no	—
470	C₄⋊C₄	—	2	4	no	—
471	C₄⋊C₄	—	2	4	no	—
472	C₄⋊C₄	—	2	4	no	—
473	C₄⋊C₄	—	2	4	no	—
474	C₄⋊C₄	—	2	2	no	—
475	C₄⋊C₄	—	2	2	no	—
476	C₄⋊C₄	C₂⁴×C₇	1	4	yes	—
477	C₄⋊C₄	C₂⁴×C₇	1	4	yes	—
478	C₂²⋊C₄	—	2	4	no	—
479	C₂²⋊C₄	—	2	4	no	—
480	C₂²⋊C₄	—	2	4	no	—
481	C₂²⋊C₄	—	2	4	no	—
482	C₂²⋊C₄	—	2	4	no	—
483	C₂²⋊C₄	—	2	4	no	—
484	C₂²⋊C₄	—	2	4	no	—
485	C₂²⋊C₄	—	2	4	no	—
486	C₂²⋊C₄	—	2	4	no	—
487	C₂²⋊C₄	—	2	4	no	—
488	C₄²	—	2	4	no	—
489	C₄²	—	2	4	no	—
490	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
491	C₄²	C₂×C₇×D₄	1	2	yes	—
492	C₄²	C₂×C₇×D₄	1	2	yes	—
493	C₄²	C₂×C₇×D₄	1	2	yes	—
494	C₄²	C₂×C₇×D₄	1	2	yes	—
495	C₄²	C₂×C₇×D₄	1	2	yes	—
496	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
497	C₄²	C₂⁴×C₇	1	1	yes	characteristic
498	C₄²	C₂⁴×C₇	1	1	yes	characteristic
499	C₄²	C₂⁴×C₇	1	2	yes	—
500	C₂×C₇	—	2	4	no	—
501	C₂×C₇	—	2	2	no	—
502	C₂×C₇	—	8	4	no	—
503	C₂×C₇	#3449	1	1	yes	characteristic
504	C₂×C₇	C₂²×#135	1	1	yes	characteristic
505	C₂×C₇	C₂×#575	1	1	yes	characteristic
506	C₂³	—	2	4	no	—
507	C₂³	—	4	1	no	—
508	C₂³	—	4	1	no	—
509	C₂³	—	4	4	no	—
510	C₂³	—	4	4	no	—
511	C₂³	—	4	4	no	—
512	C₂³	C₇×#135	1	4	yes	—
513	C₂³	C₂²×C₇×D₄	1	1	yes	characteristic
514	C₂³	C₂²×C₇×D₄	1	2	yes	—
515	Q₈	—	2	8	no	—
516	Q₈	—	4	4	no	—
517	D₄	—	2	8	no	—
518	D₄	—	4	4	no	—
519	D₄	—	4	8	no	—
520	D₄	—	4	8	no	—
521	C₂×C₄	—	2	4	no	—
522	C₂×C₄	—	2	4	no	—
523	C₂×C₄	—	2	2	no	—
524	C₂×C₄	—	2	2	no	—
525	C₂×C₄	—	2	2	no	—
526	C₂×C₄	—	2	4	no	—
527	C₂×C₄	—	2	4	no	—
528	C₂×C₄	—	2	2	no	—
529	C₂×C₄	—	2	4	no	—
530	C₂×C₄	—	2	4	no	—
531	C₂×C₄	—	2	4	no	—
532	C₂×C₄	—	4	1	no	—
533	C₂×C₄	—	4	2	no	—
534	C₂×C₄	—	4	1	no	—
535	C₂×C₄	—	4	2	no	—
536	C₂×C₄	—	4	8	no	—
537	C₂×C₄	—	4	4	no	—
538	C₂×C₄	—	4	4	no	—
539	C₂×C₄	—	4	4	no	—
540	C₂×C₄	—	4	8	no	—
541	C₂×C₄	C₂²×C₇×D₄	1	2	yes	—
542	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
543	C₂×C₄	C₂²×C₇×D₄	1	2	yes	—
544	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
545	C₂×C₄	C₂²×C₇×D₄	1	2	yes	—
546	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
547	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
548	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
549	C₂×C₄	C₂⁵×C₇	1	1	yes	characteristic
550	C₈	—	2	4	no	—
551	C₈	—	2	4	no	—
552	C₇	#60945	1	1	yes	characteristic, direct product factor, Sylow
553	C₂²	—	2	2	no	—
554	C₂²	—	2	1	no	—
555	C₂²	—	2	2	no	—
556	C₂²	—	2	1	no	—
557	C₂²	—	4	4	no	—
558	C₂²	—	4	4	no	—
559	C₂²	—	4	4	no	—
560	C₂²	—	4	4	no	—
561	C₂²	—	8	8	no	—
562	C₂²	C₂×C₇×#135	1	4	yes	—
563	C₂²	C₂×C₇×#135	1	2	yes	—
564	C₂²	C₂³×C₇×D₄	1	1	yes	characteristic
565	C₄	—	2	1	no	—
566	C₄	—	2	2	no	—
567	C₄	—	2	1	no	—
568	C₄	—	2	2	no	—
569	C₄	—	2	1	no	—
570	C₄	—	2	2	no	—
571	C₄	—	2	1	no	—
572	C₄	—	4	4	no	—
573	C₄	—	4	4	no	—
574	C₄	—	4	4	no	—
575	C₄	C₇×#575	1	2	yes	—
576	C₄	C₇×#585	1	2	yes	—
577	C₂	—	2	4	no	—
578	C₂	—	2	2	no	—
579	C₂	—	8	4	no	—
580	C₂	C₇×#3449	1	1	yes	characteristic
581	C₂	C₂²×C₇×#135	1	1	yes	characteristic
582	C₂	C₂×C₇×#575	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₇	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₇
socle	C₂²×C₇	C₂³×D₄
prime radical	C₇×#3186	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₇ ◁ C₂²×C₄²×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₇×#529	C₂	semidirect
2. C₇×#3250	C₂	semidirect
3. C₇×#3232	C₂	semidirect
4. C₇×#3211	C₂	semidirect
5. C₇×#3189	C₂	semidirect
6. C₇×#3187	C₂	semidirect
8. C₇×#3168	C₂	semidirect
9. C₇×#3108	C₂	semidirect
10. C₇×#3105	C₂	semidirect
11. C₂×C₇×#431	C₂	semidirect
18. C₇×#529	C₂²	semidirect
19. C₇×#529	C₂²	semidirect
21. C₂×C₇×#125	C₂²	semidirect
22. C₂×C₇×#124	C₂²	semidirect
26. C₂×C₄×C₇×Q₈	C₂²	semidirect
28. C₂×C₄×C₇×D₄	C₂²	semidirect
31. C₇×#499	C₂²	semidirect
32. C₇×#495	C₂²	semidirect
33. C₇×#488	C₂²	semidirect
34. C₇×#484	C₂²	semidirect
35. C₇×#483	C₂²	semidirect
38. C₇×#479	C₂²	semidirect
39. C₇×#478	C₂²	semidirect
40. C₇×#467	C₂²	semidirect
42. C₇×#465	C₂²	semidirect
43. C₇×#464	C₂²	semidirect
44. C₇×#463	C₂²	semidirect
45. C₇×#451	C₂²	semidirect
46. C₇×#450	C₂²	semidirect
49. C₄×C₇×Q₁₆	C₂²	semidirect
50. C₄×C₇×SD₁₆	C₂²	semidirect
51. C₄×C₇×D₈	C₂²	semidirect
52. C₇×#431	C₂²	semidirect
53. C₇×#431	C₂²	semidirect
54. C₂×C₇×C₄⋊C₈	C₂²	semidirect
57. C₂×C₄×C₇×C₈	C₂²	semidirect
58. #60945	C₇	direct (N × H)
143. C₂×#529	C₂×C₇	semidirect
144. #3250	C₂×C₇	semidirect
145. #3232	C₂×C₇	semidirect
146. #3211	C₂×C₇	semidirect
147. #3189	C₂×C₇	semidirect
148. #3187	C₂×C₇	semidirect
150. #3168	C₂×C₇	semidirect
151. #3108	C₂×C₇	semidirect
152. #3105	C₂×C₇	semidirect
153. C₂×#431	C₂×C₇	semidirect
233. #529	C₂²×C₇	semidirect
234. #529	C₂²×C₇	semidirect
236. C₂×#125	C₂²×C₇	semidirect
237. C₂×#124	C₂²×C₇	semidirect
241. C₂×C₄×Q₈	C₂²×C₇	semidirect
243. C₂×C₄×D₄	C₂²×C₇	semidirect
246. #499	C₂²×C₇	semidirect
247. #495	C₂²×C₇	semidirect
248. #488	C₂²×C₇	semidirect
249. #484	C₂²×C₇	semidirect
250. #483	C₂²×C₇	semidirect
253. #479	C₂²×C₇	semidirect
254. #478	C₂²×C₇	semidirect
255. #467	C₂²×C₇	semidirect
257. #465	C₂²×C₇	semidirect
258. #464	C₂²×C₇	semidirect
259. #463	C₂²×C₇	semidirect
260. #451	C₂²×C₇	semidirect
261. #450	C₂²×C₇	semidirect
264. C₄×Q₁₆	C₂²×C₇	semidirect
265. C₄×SD₁₆	C₂²×C₇	semidirect
266. C₄×D₈	C₂²×C₇	semidirect
267. #431	C₂²×C₇	semidirect
268. #431	C₂²×C₇	semidirect
269. C₂×C₄⋊C₈	C₂²×C₇	semidirect
272. C₂×C₄×C₈	C₂²×C₇	semidirect
552. C₇	#60945	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×#529	2
C₇×#3250	2
C₇×#3232	2
C₇×#3211	2
C₇×#3189	2
C₇×#3187	2
C₇×#3186	2
C₇×#3168	2
C₇×#3108	2
C₇×#3105	2
C₂×C₇×#431	2
#60945	7

Subgroups — C7×#60945 #13238161

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#60945 #13238161