Subgroups — C₇×#60236 #13237452

Order 1792.

2212 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×#532	C₂	1	1	yes	maximal, characteristic
2	C₇×#2981	C₂	1	4	yes	maximal, complemented
3	C₇×#2979	C₂	1	4	yes	maximal, complemented
4	C₇×#2975	C₂	1	2	yes	maximal, complemented
5	C₇×#2974	C₂	1	4	yes	maximal, complemented
6	C₇×#2972	C₂	1	2	yes	maximal, complemented
7	C₇×#2935	C₂	1	4	yes	maximal, complemented
8	C₇×#2931	C₂	1	4	yes	maximal, complemented
9	C₇×#2920	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₇×#431	C₂	1	1	yes	maximal, characteristic, complemented
11	C₇×#2880	C₂	1	4	yes	maximal, complemented
12	C₂²×C₇×#41	C₂²	1	2	yes	—
13	C₂×C₇×#131	C₂²	1	8	yes	complemented
14	C₇×#532	C₄	1	2	yes	—
15	C₇×#532	C₄	1	2	yes	—
16	C₇×#532	C₄	1	2	yes	—
17	C₇×#532	C₄	1	2	yes	—
18	C₇×#532	C₂²	1	2	yes	—
19	C₇×#532	C₂²	1	2	yes	—
20	C₇×#532	C₂²	1	2	yes	—
21	C₇×#532	C₂²	1	2	yes	—
22	C₂×C₇×C₄⋊Q₈	C₄	1	1	yes	characteristic
23	C₂×C₇×#127	C₄	1	1	yes	characteristic
24	C₂×C₇×#124	C₄	1	2	yes	—
25	C₂×C₇×#121	C₄	1	2	yes	—
26	C₂×C₇×#121	C₄	1	2	yes	—
27	C₂×C₄×C₇×D₄	C₂²	1	2	yes	—
28	C₂×C₄×C₇×D₄	C₂²	1	2	yes	—
29	C₂²×C₄²×C₇	C₂²	1	1	yes	characteristic
30	C₇×#436	C₂²	1	8	yes	complemented
31	C₇×#436	C₂²	1	8	yes	complemented
32	C₇×#436	C₂²	1	8	yes	complemented
33	C₇×#435	C₂²	1	16	yes	complemented
34	C₇×C₈×D₄	C₂²	1	8	yes	complemented
35	C₇×C₈×D₄	C₂²	1	8	yes	complemented
36	C₇×C₈×D₄	C₂²	1	8	yes	complemented
37	C₇×#432	C₂²	1	2	yes	—
38	C₇×#432	C₂²	1	2	yes	complemented
39	C₇×#431	C₂²	1	1	yes	characteristic
40	C₇×#431	C₂²	1	2	yes	—
41	C₇×#431	C₂²	1	1	yes	characteristic
42	C₇×#431	C₂²	1	2	yes	—
43	C₇×#431	C₂²	1	2	yes	complemented
44	C₇×#431	C₂²	1	2	yes	complemented
45	C₇×#431	C₂²	1	2	yes	complemented
46	C₇×#423	C₂²	1	2	yes	complemented
47	C₂×C₇×C₄⋊C₈	C₂²	1	2	yes	complemented
48	C₇×#408	C₂²	1	8	yes	complemented
49	C₇×#408	C₂²	1	16	yes	complemented
50	C₇×#407	C₂²	1	2	yes	—
51	C₇×#407	C₂²	1	8	yes	—
52	C₂×C₇×C₂²⋊C₈	C₂²	1	2	yes	—
53	C₂×C₇×C₂²⋊C₈	C₂²	1	8	yes	—
54	C₄×C₇×M₁₆	C₂²	1	2	yes	complemented
55	C₂×C₄×C₇×C₈	C₂²	1	2	yes	complemented
56	#60236	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
57	C₂×C₇×#41	—	4	8	no	—
58	C₂×C₇×#41	C₂×C₄	1	4	yes	—
59	C₂×C₇×#41	C₂×C₄	1	4	yes	—
60	C₂×C₇×#41	C₂³	1	4	yes	—
61	C₂×C₇×#41	C₂³	1	4	yes	—
62	C₂²×C₇×Q₈	C₂×C₄	1	2	yes	—
63	C₂²×C₇×D₄	C₂×C₄	1	2	yes	—
64	C₂²×C₇×D₄	C₂×C₄	1	4	yes	—
65	C₂²×C₇×D₄	C₂×C₄	1	2	yes	—
66	C₂²×C₇×D₄	C₂×C₄	1	2	yes	—
67	C₂³×C₄×C₇	C₂×C₄	1	1	yes	characteristic
68	C₂³×C₄×C₇	C₂×C₄	1	1	yes	characteristic
69	C₂³×C₄×C₇	C₂³	1	1	yes	characteristic
70	C₂³×C₄×C₇	C₂³	1	4	yes	—
71	C₇×#131	—	4	16	no	—
72	C₂×C₇×M₁₆	C₂³	1	4	yes	—
73	C₂×C₇×M₁₆	C₂³	1	16	yes	—
74	C₂²×C₇×C₈	C₂³	1	4	yes	—
75	C₂²×C₇×C₈	C₂³	1	16	yes	—
76	C₇×C₄⋊Q₈	C₂×C₄	1	2	yes	—
77	C₇×C₄⋊Q₈	C₂×C₄	1	2	yes	—
78	C₇×C₄⋊Q₈	C₂×C₄	1	2	yes	—
79	C₇×C₄⋊Q₈	C₂×C₄	1	2	yes	—
80	C₇×#127	C₂×C₄	1	2	yes	—
81	C₇×#127	C₂×C₄	1	2	yes	—
82	C₇×#127	C₂×C₄	1	2	yes	—
83	C₇×#127	C₂×C₄	1	2	yes	—
84	C₇×#124	C₂×C₄	1	4	yes	—
85	C₇×#124	C₂×C₄	1	4	yes	—
86	C₇×#124	C₂×C₄	1	4	yes	—
87	C₇×#124	C₂×C₄	1	4	yes	—
88	C₇×#121	C₂×C₄	1	4	yes	—
89	C₇×#121	C₂×C₄	1	4	yes	—
90	C₇×#121	C₂×C₄	1	4	yes	—
91	C₇×#121	C₂×C₄	1	4	yes	—
92	C₇×#121	C₂×C₄	1	4	yes	—
93	C₇×#121	C₂×C₄	1	4	yes	—
94	C₇×#121	C₂×C₄	1	4	yes	—
95	C₇×#121	C₂×C₄	1	4	yes	—
96	C₄×C₇×D₄	C₂×C₄	1	4	yes	—
97	C₄×C₇×D₄	C₂×C₄	1	4	yes	—
98	C₄×C₇×D₄	C₂×C₄	1	4	yes	—
99	C₄×C₇×D₄	C₂×C₄	1	4	yes	—
100	C₄×C₇×D₄	C₂³	1	4	yes	—
101	C₄×C₇×D₄	C₂³	1	4	yes	—
102	C₄×C₇×D₄	C₂³	1	4	yes	—
103	C₄×C₇×D₄	C₂³	1	4	yes	—
104	C₂×C₇×C₄⋊C₄	C₂×C₄	1	2	yes	—
105	C₂×C₇×C₄⋊C₄	C₂×C₄	1	2	yes	—
106	C₂×C₇×C₂²⋊C₄	C₂×C₄	1	4	yes	—
107	C₂×C₇×C₂²⋊C₄	C₂×C₄	1	4	yes	—
108	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
109	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
110	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
111	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
112	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
113	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
114	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
115	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
116	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
117	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
118	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
119	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
120	C₇×C₄⋊C₈	C₂³	1	2	yes	—
121	C₇×C₄⋊C₈	C₂³	1	4	yes	—
122	C₇×C₄⋊C₈	C₂³	1	2	yes	—
123	C₇×C₄⋊C₈	C₂³	1	4	yes	—
124	C₇×C₄⋊C₈	C₂³	1	4	yes	—
125	C₇×C₂²⋊C₈	C₂³	1	2	yes	—
126	C₇×C₂²⋊C₈	C₂³	1	4	yes	—
127	C₇×C₂²⋊C₈	C₂³	1	2	yes	—
128	C₇×C₂²⋊C₈	C₂³	1	4	yes	—
129	C₇×C₂²⋊C₈	C₂³	1	16	yes	—
130	C₇×C₂²⋊C₈	C₂³	1	4	yes	—
131	C₇×C₂²⋊C₈	C₂³	1	16	yes	—
132	C₇×#97	C₂³	1	4	yes	—
133	C₄×C₇×C₈	C₂³	1	2	yes	—
134	C₄×C₇×C₈	C₂³	1	4	yes	—
135	C₄×C₇×C₈	C₂³	1	2	yes	—
136	C₄×C₇×C₈	C₂³	1	4	yes	—
137	C₂×#532	C₂×C₇	1	1	yes	characteristic
138	#2981	C₂×C₇	1	4	yes	complemented
139	#2979	C₂×C₇	1	4	yes	complemented
140	#2975	C₂×C₇	1	2	yes	complemented
141	#2974	C₂×C₇	1	4	yes	complemented
142	#2972	C₂×C₇	1	2	yes	complemented
143	#2935	C₂×C₇	1	4	yes	complemented
144	#2931	C₂×C₇	1	4	yes	complemented
145	#2920	C₂×C₇	1	1	yes	characteristic, complemented
146	C₂×#431	C₂×C₇	1	1	yes	characteristic, complemented
147	#2880	C₂×C₇	1	4	yes	complemented
148	C₂⁴×C₇	C₂²×C₄	1	1	yes	characteristic
149	C₂⁴×C₇	C₂²×C₄	1	4	yes	—
150	C₇×#41	—	2	16	no	—
151	C₇×#41	—	4	8	no	—
152	C₇×#41	—	4	8	no	—
153	C₇×#41	—	4	8	no	—
154	C₂×C₇×Q₈	—	4	4	no	—
155	C₂×C₇×Q₈	C₂²×C₄	1	4	yes	—
156	C₂×C₇×Q₈	C₂²×C₄	1	4	yes	—
157	C₂×C₇×D₄	—	2	8	no	—
158	C₂×C₇×D₄	—	4	4	no	—
159	C₂×C₇×D₄	—	4	4	no	—
160	C₂×C₇×D₄	—	4	8	no	—
161	C₂×C₇×D₄	C₂²×C₄	1	4	yes	—
162	C₂×C₇×D₄	C₂²×C₄	1	4	yes	—
163	C₂×C₇×D₄	C₂²×C₄	1	4	yes	—
164	C₂×C₇×D₄	C₂²×C₄	1	8	yes	—
165	C₂×C₇×D₄	C₂²×C₄	1	4	yes	—
166	C₂×C₇×D₄	C₂²×C₄	1	4	yes	—
167	C₂×C₇×D₄	C₂²×C₄	1	8	yes	—
168	C₂×C₇×D₄	C₂²×C₄	1	4	yes	—
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	8	no	—
173	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
174	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
175	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
176	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
177	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
178	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
179	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
180	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
181	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
182	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
183	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
184	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
185	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
186	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
187	C₂²×C₄×C₇	C₂²×C₄	1	4	yes	—
188	C₂²×C₄×C₇	C₂²×C₄	1	8	yes	—
189	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
190	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
191	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
192	C₂²×C₄×C₇	C₂⁴	1	8	yes	—
193	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
194	C₇×M₁₆	—	2	8	no	—
195	C₇×M₁₆	—	4	16	no	—
196	C₂×C₇×C₈	—	2	8	no	—
197	C₂×C₇×C₈	—	4	16	no	—
198	C₂×C₇×C₈	C₂⁴	1	4	yes	—
199	C₂×C₇×C₈	C₂⁴	1	4	yes	—
200	C₂×C₇×C₈	C₂⁴	1	8	yes	—
201	C₇×C₄⋊C₄	C₂²×C₄	1	4	yes	—
202	C₇×C₄⋊C₄	C₂²×C₄	1	4	yes	—
203	C₇×C₄⋊C₄	C₂²×C₄	1	4	yes	—
204	C₇×C₄⋊C₄	C₂²×C₄	1	4	yes	—
205	C₇×C₂²⋊C₄	C₂²×C₄	1	8	yes	—
206	C₇×C₂²⋊C₄	C₂²×C₄	1	8	yes	—
207	C₇×C₂²⋊C₄	C₂²×C₄	1	8	yes	—
208	C₇×C₂²⋊C₄	C₂²×C₄	1	8	yes	—
209	C₄²×C₇	C₂²×C₄	1	1	yes	characteristic
210	C₄²×C₇	C₂²×C₄	1	1	yes	characteristic
211	C₄²×C₇	C₂²×C₄	1	1	yes	characteristic
212	C₄²×C₇	C₂²×C₄	1	1	yes	characteristic
213	C₄²×C₇	C₂²×C₄	1	1	yes	characteristic
214	C₄²×C₇	C₂²×C₄	1	1	yes	characteristic
215	C₄²×C₇	C₂²×C₄	1	1	yes	characteristic
216	C₄²×C₇	C₂²×C₄	1	1	yes	characteristic
217	C₄²×C₇	C₂²×C₄	1	1	yes	characteristic
218	C₄²×C₇	C₂²×C₄	1	1	yes	characteristic
219	C₄²×C₇	C₂²×C₄	1	1	yes	characteristic
220	C₄²×C₇	C₂²×C₄	1	1	yes	characteristic
221	C₄²×C₇	C₂⁴	1	1	yes	characteristic
222	C₄²×C₇	C₂⁴	1	1	yes	characteristic
223	C₄²×C₇	C₂⁴	1	1	yes	characteristic
224	C₄²×C₇	C₂⁴	1	1	yes	characteristic
225	C₂²×#41	C₂²×C₇	1	2	yes	—
226	C₂×#131	C₂²×C₇	1	8	yes	complemented
227	#532	C₄×C₇	1	2	yes	—
228	#532	C₄×C₇	1	2	yes	—
229	#532	C₄×C₇	1	2	yes	—
230	#532	C₄×C₇	1	2	yes	—
231	#532	C₂²×C₇	1	2	yes	—
232	#532	C₂²×C₇	1	2	yes	—
233	#532	C₂²×C₇	1	2	yes	—
234	#532	C₂²×C₇	1	2	yes	—
235	C₂×C₄⋊Q₈	C₄×C₇	1	1	yes	characteristic
236	C₂×#127	C₄×C₇	1	1	yes	characteristic
237	C₂×#124	C₄×C₇	1	2	yes	—
238	C₂×#121	C₄×C₇	1	2	yes	—
239	C₂×#121	C₄×C₇	1	2	yes	—
240	C₂×C₄×D₄	C₂²×C₇	1	2	yes	—
241	C₂×C₄×D₄	C₂²×C₇	1	2	yes	—
242	C₂²×C₄²	C₂²×C₇	1	1	yes	characteristic
243	#436	C₂²×C₇	1	8	yes	complemented
244	#436	C₂²×C₇	1	8	yes	complemented
245	#436	C₂²×C₇	1	8	yes	complemented
246	#435	C₂²×C₇	1	16	yes	complemented
247	C₈×D₄	C₂²×C₇	1	8	yes	complemented
248	C₈×D₄	C₂²×C₇	1	8	yes	complemented
249	C₈×D₄	C₂²×C₇	1	8	yes	complemented
250	#432	C₂²×C₇	1	2	yes	—
251	#432	C₂²×C₇	1	2	yes	complemented
252	#431	C₂²×C₇	1	1	yes	characteristic
253	#431	C₂²×C₇	1	2	yes	—
254	#431	C₂²×C₇	1	1	yes	characteristic
255	#431	C₂²×C₇	1	2	yes	—
256	#431	C₂²×C₇	1	2	yes	complemented
257	#431	C₂²×C₇	1	2	yes	complemented
258	#431	C₂²×C₇	1	2	yes	complemented
259	#423	C₂²×C₇	1	2	yes	complemented
260	C₂×C₄⋊C₈	C₂²×C₇	1	2	yes	complemented
261	#408	C₂²×C₇	1	8	yes	complemented
262	#408	C₂²×C₇	1	16	yes	complemented
263	#407	C₂²×C₇	1	2	yes	—
264	#407	C₂²×C₇	1	8	yes	—
265	C₂×C₂²⋊C₈	C₂²×C₇	1	2	yes	—
266	C₂×C₂²⋊C₈	C₂²×C₇	1	8	yes	—
267	C₄×M₁₆	C₂²×C₇	1	2	yes	complemented
268	C₂×C₄×C₈	C₂²×C₇	1	2	yes	complemented
269	C₂³×C₇	—	2	2	no	—
270	C₂³×C₇	—	2	2	no	—
271	C₂³×C₇	—	2	8	no	—
272	C₂³×C₇	—	4	1	no	—
273	C₂³×C₇	—	4	4	no	—
274	C₂³×C₇	—	4	4	no	—
275	C₂³×C₇	C₂³×C₄	1	8	yes	—
276	C₂³×C₇	C₂³×C₄	1	1	yes	characteristic
277	C₂³×C₇	C₂³×C₄	1	1	yes	characteristic
278	C₂³×C₇	C₂³×C₄	1	1	yes	characteristic
279	C₇×Q₈	—	2	8	no	—
280	C₇×Q₈	—	4	4	no	—
281	C₇×D₄	—	2	16	no	—
282	C₇×D₄	—	2	8	no	—
283	C₇×D₄	—	2	8	no	—
284	C₇×D₄	—	2	8	no	—
285	C₇×D₄	—	2	8	no	—
286	C₇×D₄	—	4	4	no	—
287	C₇×D₄	—	4	4	no	—
288	C₇×D₄	—	4	8	no	—
289	C₂×C₄×C₇	—	2	2	no	—
290	C₂×C₄×C₇	—	2	2	no	—
291	C₂×C₄×C₇	—	2	2	no	—
292	C₂×C₄×C₇	—	2	2	no	—
293	C₂×C₄×C₇	—	2	2	no	—
294	C₂×C₄×C₇	—	2	2	no	—
295	C₂×C₄×C₇	—	2	2	no	—
296	C₂×C₄×C₇	—	2	2	no	—
297	C₂×C₄×C₇	—	2	2	no	—
298	C₂×C₄×C₇	—	2	2	no	—
299	C₂×C₄×C₇	—	2	2	no	—
300	C₂×C₄×C₇	—	2	16	no	—
301	C₂×C₄×C₇	—	2	2	no	—
302	C₂×C₄×C₇	—	2	2	no	—
303	C₂×C₄×C₇	—	4	1	no	—
304	C₂×C₄×C₇	—	4	1	no	—
305	C₂×C₄×C₇	—	4	1	no	—
306	C₂×C₄×C₇	—	4	4	no	—
307	C₂×C₄×C₇	—	4	8	no	—
308	C₂×C₄×C₇	#131	1	4	yes	complemented
309	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
310	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
311	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
312	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
313	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
314	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
315	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
316	C₂×C₄×C₇	C₂³×C₄	1	8	yes	—
317	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
318	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
319	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
320	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
321	C₂×C₄×C₇	H(5,𝔽₂)	1	4	yes	—
322	C₂×C₄×C₇	C₂⁵	1	1	yes	characteristic
323	C₇×C₈	—	2	4	no	—
324	C₇×C₈	—	2	4	no	—
325	C₇×C₈	—	2	8	no	—
326	C₂×#41	—	4	8	no	—
327	C₂×#41	C₂×C₄×C₇	1	4	yes	—
328	C₂×#41	C₂×C₄×C₇	1	4	yes	—
329	C₂×#41	C₂³×C₇	1	4	yes	—
330	C₂×#41	C₂³×C₇	1	4	yes	—
331	C₂²×Q₈	C₂×C₄×C₇	1	2	yes	—
332	C₂²×D₄	C₂×C₄×C₇	1	2	yes	—
333	C₂²×D₄	C₂×C₄×C₇	1	2	yes	—
334	C₂²×D₄	C₂×C₄×C₇	1	4	yes	—
335	C₂²×D₄	C₂×C₄×C₇	1	2	yes	—
336	C₂³×C₄	C₂×C₄×C₇	1	1	yes	characteristic
337	C₂³×C₄	C₂×C₄×C₇	1	1	yes	characteristic
338	C₂³×C₄	C₂³×C₇	1	1	yes	characteristic
339	C₂³×C₄	C₂³×C₇	1	4	yes	—
340	#131	—	4	16	no	—
341	C₂×M₁₆	C₂³×C₇	1	4	yes	—
342	C₂×M₁₆	C₂³×C₇	1	16	yes	—
343	C₂²×C₈	C₂³×C₇	1	4	yes	—
344	C₂²×C₈	C₂³×C₇	1	16	yes	—
345	C₄⋊Q₈	C₂×C₄×C₇	1	2	yes	—
346	C₄⋊Q₈	C₂×C₄×C₇	1	2	yes	—
347	C₄⋊Q₈	C₂×C₄×C₇	1	2	yes	—
348	C₄⋊Q₈	C₂×C₄×C₇	1	2	yes	—
349	#127	C₂×C₄×C₇	1	2	yes	—
350	#127	C₂×C₄×C₇	1	2	yes	—
351	#127	C₂×C₄×C₇	1	2	yes	—
352	#127	C₂×C₄×C₇	1	2	yes	—
353	#124	C₂×C₄×C₇	1	4	yes	—
354	#124	C₂×C₄×C₇	1	4	yes	—
355	#124	C₂×C₄×C₇	1	4	yes	—
356	#124	C₂×C₄×C₇	1	4	yes	—
357	#121	C₂×C₄×C₇	1	4	yes	—
358	#121	C₂×C₄×C₇	1	4	yes	—
359	#121	C₂×C₄×C₇	1	4	yes	—
360	#121	C₂×C₄×C₇	1	4	yes	—
361	#121	C₂×C₄×C₇	1	4	yes	—
362	#121	C₂×C₄×C₇	1	4	yes	—
363	#121	C₂×C₄×C₇	1	4	yes	—
364	#121	C₂×C₄×C₇	1	4	yes	—
365	C₄×D₄	C₂×C₄×C₇	1	4	yes	—
366	C₄×D₄	C₂×C₄×C₇	1	4	yes	—
367	C₄×D₄	C₂×C₄×C₇	1	4	yes	—
368	C₄×D₄	C₂×C₄×C₇	1	4	yes	—
369	C₄×D₄	C₂³×C₇	1	4	yes	—
370	C₄×D₄	C₂³×C₇	1	4	yes	—
371	C₄×D₄	C₂³×C₇	1	4	yes	—
372	C₄×D₄	C₂³×C₇	1	4	yes	—
373	C₂×C₄⋊C₄	C₂×C₄×C₇	1	2	yes	—
374	C₂×C₄⋊C₄	C₂×C₄×C₇	1	2	yes	—
375	C₂×C₂²⋊C₄	C₂×C₄×C₇	1	4	yes	—
376	C₂×C₂²⋊C₄	C₂×C₄×C₇	1	4	yes	—
377	C₂×C₄²	C₂×C₄×C₇	1	1	yes	characteristic
378	C₂×C₄²	C₂×C₄×C₇	1	1	yes	characteristic
379	C₂×C₄²	C₂×C₄×C₇	1	1	yes	characteristic
380	C₂×C₄²	C₂×C₄×C₇	1	1	yes	characteristic
381	C₂×C₄²	C₂×C₄×C₇	1	1	yes	characteristic
382	C₂×C₄²	C₂×C₄×C₇	1	1	yes	characteristic
383	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
384	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
385	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
386	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
387	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
388	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
389	C₄⋊C₈	C₂³×C₇	1	2	yes	—
390	C₄⋊C₈	C₂³×C₇	1	4	yes	—
391	C₄⋊C₈	C₂³×C₇	1	2	yes	—
392	C₄⋊C₈	C₂³×C₇	1	4	yes	—
393	C₄⋊C₈	C₂³×C₇	1	4	yes	—
394	C₂²⋊C₈	C₂³×C₇	1	2	yes	—
395	C₂²⋊C₈	C₂³×C₇	1	4	yes	—
396	C₂²⋊C₈	C₂³×C₇	1	2	yes	—
397	C₂²⋊C₈	C₂³×C₇	1	4	yes	—
398	C₂²⋊C₈	C₂³×C₇	1	16	yes	—
399	C₂²⋊C₈	C₂³×C₇	1	4	yes	—
400	C₂²⋊C₈	C₂³×C₇	1	16	yes	—
401	#97	C₂³×C₇	1	4	yes	—
402	C₄×C₈	C₂³×C₇	1	2	yes	—
403	C₄×C₈	C₂³×C₇	1	4	yes	—
404	C₄×C₈	C₂³×C₇	1	2	yes	—
405	C₄×C₈	C₂³×C₇	1	4	yes	—
406	C₂²×C₇	—	2	1	no	—
407	C₂²×C₇	—	2	1	no	—
408	C₂²×C₇	—	2	1	no	—
409	C₂²×C₇	—	2	1	no	—
410	C₂²×C₇	—	2	1	no	—
411	C₂²×C₇	—	2	8	no	—
412	C₂²×C₇	—	2	1	no	—
413	C₂²×C₇	—	2	8	no	—
414	C₂²×C₇	—	2	1	no	—
415	C₂²×C₇	—	2	8	no	—
416	C₂²×C₇	—	4	4	no	—
417	C₂²×C₇	—	8	8	no	—
418	C₂²×C₇	#518	1	2	yes	—
419	C₂²×C₇	C₂×#131	1	2	yes	—
420	C₂²×C₇	C₂⁴×C₄	1	1	yes	characteristic
421	C₄×C₇	—	2	1	no	—
422	C₄×C₇	—	2	1	no	—
423	C₄×C₇	—	2	8	no	—
424	C₄×C₇	—	2	1	no	—
425	C₄×C₇	—	2	1	no	—
426	C₄×C₇	—	2	1	no	—
427	C₄×C₇	—	2	1	no	—
428	C₄×C₇	—	2	1	no	—
429	C₄×C₇	#518	1	2	yes	—
430	C₄×C₇	C₂×#131	1	2	yes	—
431	C₄×C₇	C₂×#131	1	2	yes	—
432	C₄×C₇	#568	1	2	yes	—
433	C₄×C₇	C₂×H(5,𝔽₂)	1	2	yes	—
434	C₂⁴	C₂²×C₄×C₇	1	1	yes	characteristic
435	C₂⁴	C₂²×C₄×C₇	1	4	yes	—
436	#41	—	2	16	no	—
437	#41	—	4	8	no	—
438	#41	—	4	8	no	—
439	#41	—	4	8	no	—
440	C₂×Q₈	—	4	4	no	—
441	C₂×Q₈	C₂²×C₄×C₇	1	4	yes	—
442	C₂×Q₈	C₂²×C₄×C₇	1	4	yes	—
443	C₂×D₄	—	2	8	no	—
444	C₂×D₄	—	4	4	no	—
445	C₂×D₄	—	4	4	no	—
446	C₂×D₄	—	4	8	no	—
447	C₂×D₄	C₂²×C₄×C₇	1	4	yes	—
448	C₂×D₄	C₂²×C₄×C₇	1	4	yes	—
449	C₂×D₄	C₂²×C₄×C₇	1	4	yes	—
450	C₂×D₄	C₂²×C₄×C₇	1	8	yes	—
451	C₂×D₄	C₂²×C₄×C₇	1	4	yes	—
452	C₂×D₄	C₂²×C₄×C₇	1	4	yes	—
453	C₂×D₄	C₂²×C₄×C₇	1	8	yes	—
454	C₂×D₄	C₂²×C₄×C₇	1	4	yes	—
455	C₂²×C₄	—	2	4	no	—
456	C₂²×C₄	—	2	4	no	—
457	C₂²×C₄	—	4	2	no	—
458	C₂²×C₄	—	4	8	no	—
459	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
460	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
461	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
462	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
463	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
464	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
465	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
466	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
467	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
468	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
469	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
470	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
471	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
472	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
473	C₂²×C₄	C₂²×C₄×C₇	1	4	yes	—
474	C₂²×C₄	C₂²×C₄×C₇	1	8	yes	—
475	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
476	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
477	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
478	C₂²×C₄	C₂⁴×C₇	1	8	yes	—
479	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
480	M₁₆	—	2	8	no	—
481	M₁₆	—	4	16	no	—
482	C₂×C₈	—	2	8	no	—
483	C₂×C₈	—	4	16	no	—
484	C₂×C₈	C₂⁴×C₇	1	4	yes	—
485	C₂×C₈	C₂⁴×C₇	1	4	yes	—
486	C₂×C₈	C₂⁴×C₇	1	8	yes	—
487	C₄⋊C₄	C₂²×C₄×C₇	1	4	yes	—
488	C₄⋊C₄	C₂²×C₄×C₇	1	4	yes	—
489	C₄⋊C₄	C₂²×C₄×C₇	1	4	yes	—
490	C₄⋊C₄	C₂²×C₄×C₇	1	4	yes	—
491	C₂²⋊C₄	C₂²×C₄×C₇	1	8	yes	—
492	C₂²⋊C₄	C₂²×C₄×C₇	1	8	yes	—
493	C₂²⋊C₄	C₂²×C₄×C₇	1	8	yes	—
494	C₂²⋊C₄	C₂²×C₄×C₇	1	8	yes	—
495	C₄²	C₂²×C₄×C₇	1	1	yes	characteristic
496	C₄²	C₂²×C₄×C₇	1	1	yes	characteristic
497	C₄²	C₂²×C₄×C₇	1	1	yes	characteristic
498	C₄²	C₂²×C₄×C₇	1	1	yes	characteristic
499	C₄²	C₂²×C₄×C₇	1	1	yes	characteristic
500	C₄²	C₂²×C₄×C₇	1	1	yes	characteristic
501	C₄²	C₂²×C₄×C₇	1	1	yes	characteristic
502	C₄²	C₂²×C₄×C₇	1	1	yes	characteristic
503	C₄²	C₂²×C₄×C₇	1	1	yes	characteristic
504	C₄²	C₂²×C₄×C₇	1	1	yes	characteristic
505	C₄²	C₂²×C₄×C₇	1	1	yes	characteristic
506	C₄²	C₂²×C₄×C₇	1	1	yes	characteristic
507	C₄²	C₂⁴×C₇	1	1	yes	characteristic
508	C₄²	C₂⁴×C₇	1	1	yes	characteristic
509	C₄²	C₂⁴×C₇	1	1	yes	characteristic
510	C₄²	C₂⁴×C₇	1	1	yes	characteristic
511	C₂×C₇	—	2	2	no	—
512	C₂×C₇	—	2	2	no	—
513	C₂×C₇	—	4	1	no	—
514	C₂×C₇	—	4	8	no	—
515	C₂×C₇	C₂×#518	1	1	yes	characteristic
516	C₂×C₇	C₂²×#131	1	1	yes	characteristic
517	C₂×C₇	C₂×#568	1	1	yes	characteristic
518	C₂³	—	2	2	no	—
519	C₂³	—	2	2	no	—
520	C₂³	—	2	8	no	—
521	C₂³	—	4	1	no	—
522	C₂³	—	4	4	no	—
523	C₂³	—	4	4	no	—
524	C₂³	C₂³×C₄×C₇	1	8	yes	—
525	C₂³	C₂³×C₄×C₇	1	1	yes	characteristic
526	C₂³	C₂³×C₄×C₇	1	1	yes	characteristic
527	C₂³	C₂³×C₄×C₇	1	1	yes	characteristic
528	Q₈	—	2	8	no	—
529	Q₈	—	4	4	no	—
530	D₄	—	2	16	no	—
531	D₄	—	2	8	no	—
532	D₄	—	2	8	no	—
533	D₄	—	2	8	no	—
534	D₄	—	2	8	no	—
535	D₄	—	4	4	no	—
536	D₄	—	4	4	no	—
537	D₄	—	4	8	no	—
538	C₂×C₄	—	2	2	no	—
539	C₂×C₄	—	2	2	no	—
540	C₂×C₄	—	2	16	no	—
541	C₂×C₄	—	2	2	no	—
542	C₂×C₄	—	2	2	no	—
543	C₂×C₄	—	2	2	no	—
544	C₂×C₄	—	2	2	no	—
545	C₂×C₄	—	2	2	no	—
546	C₂×C₄	—	2	2	no	—
547	C₂×C₄	—	2	2	no	—
548	C₂×C₄	—	2	2	no	—
549	C₂×C₄	—	2	2	no	—
550	C₂×C₄	—	2	2	no	—
551	C₂×C₄	—	2	2	no	—
552	C₂×C₄	—	4	1	no	—
553	C₂×C₄	—	4	1	no	—
554	C₂×C₄	—	4	1	no	—
555	C₂×C₄	—	4	4	no	—
556	C₂×C₄	—	4	8	no	—
557	C₂×C₄	C₇×#131	1	4	yes	complemented
558	C₂×C₄	C₂³×C₄×C₇	1	1	yes	characteristic
559	C₂×C₄	C₂³×C₄×C₇	1	1	yes	characteristic
560	C₂×C₄	C₂³×C₄×C₇	1	1	yes	characteristic
561	C₂×C₄	C₂³×C₄×C₇	1	1	yes	characteristic
562	C₂×C₄	C₂³×C₄×C₇	1	1	yes	characteristic
563	C₂×C₄	C₂³×C₄×C₇	1	1	yes	characteristic
564	C₂×C₄	C₂³×C₄×C₇	1	1	yes	characteristic
565	C₂×C₄	C₂³×C₄×C₇	1	8	yes	—
566	C₂×C₄	C₂³×C₄×C₇	1	1	yes	characteristic
567	C₂×C₄	C₂³×C₄×C₇	1	1	yes	characteristic
568	C₂×C₄	C₂³×C₄×C₇	1	1	yes	characteristic
569	C₂×C₄	C₂³×C₄×C₇	1	1	yes	characteristic
570	C₂×C₄	C₇×H(5,𝔽₂)	1	4	yes	—
571	C₂×C₄	C₂⁵×C₇	1	1	yes	characteristic
572	C₈	—	2	4	no	—
573	C₈	—	2	4	no	—
574	C₈	—	2	8	no	—
575	C₇	#60236	1	1	yes	characteristic, direct product factor, Sylow
576	C₂²	—	2	1	no	—
577	C₂²	—	2	1	no	—
578	C₂²	—	2	1	no	—
579	C₂²	—	2	1	no	—
580	C₂²	—	2	1	no	—
581	C₂²	—	2	8	no	—
582	C₂²	—	2	1	no	—
583	C₂²	—	2	8	no	—
584	C₂²	—	2	1	no	—
585	C₂²	—	2	8	no	—
586	C₂²	—	4	4	no	—
587	C₂²	—	8	8	no	—
588	C₂²	C₇×#518	1	2	yes	—
589	C₂²	C₂×C₇×#131	1	2	yes	—
590	C₂²	C₂⁴×C₄×C₇	1	1	yes	characteristic
591	C₄	—	2	1	no	—
592	C₄	—	2	1	no	—
593	C₄	—	2	1	no	—
594	C₄	—	2	1	no	—
595	C₄	—	2	1	no	—
596	C₄	—	2	1	no	—
597	C₄	—	2	1	no	—
598	C₄	—	2	8	no	—
599	C₄	C₇×#518	1	2	yes	—
600	C₄	C₂×C₇×#131	1	2	yes	—
601	C₄	C₂×C₇×#131	1	2	yes	—
602	C₄	C₇×#568	1	2	yes	—
603	C₄	C₂×C₇×H(5,𝔽₂)	1	2	yes	—
604	C₂	—	2	2	no	—
605	C₂	—	2	2	no	—
606	C₂	—	4	1	no	—
607	C₂	—	4	8	no	—
608	C₂	C₂×C₇×#518	1	1	yes	characteristic
609	C₂	C₂²×C₇×#131	1	1	yes	characteristic
610	C₂	C₂×C₇×#568	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₇	C₂⁵
commutator	C₂²	C₂⁴×C₄×C₇
Frattini, cosocle	C₂×C₄	C₂⁵×C₇
socle	C₂²×C₇	C₂⁴×C₄
prime radical	C₂×C₇×#532	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₇×#2981	C₂	semidirect
3. C₇×#2979	C₂	semidirect
4. C₇×#2975	C₂	semidirect
5. C₇×#2974	C₂	semidirect
6. C₇×#2972	C₂	semidirect
7. C₇×#2935	C₂	semidirect
8. C₇×#2931	C₂	semidirect
9. C₇×#2920	C₂	semidirect
10. C₂×C₇×#431	C₂	semidirect
11. C₇×#2880	C₂	semidirect
13. C₂×C₇×#131	C₂²	semidirect
30. C₇×#436	C₂²	semidirect
31. C₇×#436	C₂²	semidirect
32. C₇×#436	C₂²	semidirect
33. C₇×#435	C₂²	semidirect
34. C₇×C₈×D₄	C₂²	semidirect
35. C₇×C₈×D₄	C₂²	semidirect
36. C₇×C₈×D₄	C₂²	semidirect
38. C₇×#432	C₂²	semidirect
43. C₇×#431	C₂²	semidirect
44. C₇×#431	C₂²	semidirect
45. C₇×#431	C₂²	semidirect
46. C₇×#423	C₂²	semidirect
47. C₂×C₇×C₄⋊C₈	C₂²	semidirect
48. C₇×#408	C₂²	semidirect
49. C₇×#408	C₂²	semidirect
54. C₄×C₇×M₁₆	C₂²	semidirect
55. C₂×C₄×C₇×C₈	C₂²	semidirect
56. #60236	C₇	direct (N × H)
138. #2981	C₂×C₇	semidirect
139. #2979	C₂×C₇	semidirect
140. #2975	C₂×C₇	semidirect
141. #2974	C₂×C₇	semidirect
142. #2972	C₂×C₇	semidirect
143. #2935	C₂×C₇	semidirect
144. #2931	C₂×C₇	semidirect
145. #2920	C₂×C₇	semidirect
146. C₂×#431	C₂×C₇	semidirect
147. #2880	C₂×C₇	semidirect
226. C₂×#131	C₂²×C₇	semidirect
243. #436	C₂²×C₇	semidirect
244. #436	C₂²×C₇	semidirect
245. #436	C₂²×C₇	semidirect
246. #435	C₂²×C₇	semidirect
247. C₈×D₄	C₂²×C₇	semidirect
248. C₈×D₄	C₂²×C₇	semidirect
249. C₈×D₄	C₂²×C₇	semidirect
251. #432	C₂²×C₇	semidirect
256. #431	C₂²×C₇	semidirect
257. #431	C₂²×C₇	semidirect
258. #431	C₂²×C₇	semidirect
259. #423	C₂²×C₇	semidirect
260. C₂×C₄⋊C₈	C₂²×C₇	semidirect
261. #408	C₂²×C₇	semidirect
262. #408	C₂²×C₇	semidirect
267. C₄×M₁₆	C₂²×C₇	semidirect
268. C₂×C₄×C₈	C₂²×C₇	semidirect
308. C₂×C₄×C₇	#131	semidirect
557. C₂×C₄	C₇×#131	semidirect
575. C₇	#60236	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×#532	2
C₇×#2981	2
C₇×#2979	2
C₇×#2975	2
C₇×#2974	2
C₇×#2972	2
C₇×#2935	2
C₇×#2931	2
C₇×#2920	2
C₂×C₇×#431	2
C₇×#2880	2
#60236	7

Subgroups — C7×#60236 #13237452

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#60236 #13237452