Subgroups — C₇×#34484 #13211700

Order 1792.

2132 conjugacy classes of subgroups.

The subgroup lattice has 676 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×#554	C₂	1	1	yes	maximal, characteristic
2	C₇×#2378	C₂	1	4	yes	maximal, complemented
3	C₇×#2370	C₂	1	2	yes	maximal, complemented
4	C₇×#2369	C₂	1	1	yes	maximal, characteristic, complemented
5	C₇×#2362	C₂	1	2	yes	maximal, complemented
6	C₇×#2354	C₂	1	1	yes	maximal, characteristic
7	C₇×#2354	C₂	1	2	yes	maximal
8	C₇×Q₈×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
9	C₇×#2344	C₂	1	2	yes	maximal, complemented
10	C₇×#2330	C₂	1	2	yes	maximal, complemented
11	C₇×#2323	C₂	1	2	yes	maximal, complemented
12	C₇×#2323	C₂	1	4	yes	maximal, complemented
13	C₇×#2319	C₂	1	2	yes	maximal
14	C₇×#2316	C₂	1	2	yes	maximal, complemented
15	C₄×C₇×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
16	C₇×#2295	C₂	1	1	yes	maximal, characteristic, complemented
17	C₇×#2293	C₂	1	1	yes	maximal, characteristic
18	C₇×#554	C₄	1	16	yes	complemented
19	C₂×C₇×C₄⋊Q₈	C₂²	1	1	yes	characteristic
20	C₂×C₇×#125	C₂²	1	2	yes	—
21	C₂×C₇×C₂²⋊Q₈	C₂²	1	2	yes	—
22	C₂×C₇×C₂²⋊Q₈	C₂²	1	4	yes	—
23	C₂×C₄×C₇×Q₈	C₂²	1	2	yes	—
24	C₂×C₄×C₇×Q₈	C₂²	1	2	yes	—
25	C₂×C₄×C₇×Q₈	C₂²	1	1	yes	characteristic
26	C₂×C₄×C₇×D₄	C₂²	1	1	yes	characteristic
27	C₂×C₄×C₇×D₄	C₂²	1	2	yes	—
28	C₂×C₇×#117	C₂²	1	2	yes	—
29	C₂×C₇×#117	C₂²	1	2	yes	—
30	C₂×C₇×#117	C₂²	1	4	yes	—
31	C₂²×C₇×C₄⋊C₄	C₂²	1	2	yes	—
32	C₇×#391	C₂²	1	1	yes	characteristic
33	C₇×#391	C₂²	1	1	yes	characteristic
34	C₇×#391	C₂²	1	2	yes	—
35	C₇×#391	C₂²	1	4	yes	—
36	C₇×#390	C₂²	1	1	yes	characteristic
37	C₇×#390	C₂²	1	1	yes	characteristic
38	C₇×#390	C₂²	1	2	yes	—
39	C₇×#389	C₂²	1	1	yes	characteristic
40	C₇×#389	C₂²	1	1	yes	characteristic
41	C₇×#389	C₂²	1	1	yes	characteristic
42	C₇×#389	C₂²	1	1	yes	characteristic
43	C₇×#389	C₂²	1	2	yes	—
44	C₇×#389	C₂²	1	2	yes	—
45	C₇×#389	C₂²	1	4	yes	—
46	C₇×#389	C₂²	1	2	yes	—
47	C₇×#389	C₂²	1	2	yes	—
48	C₇×#389	C₂²	1	4	yes	—
49	C₇×#389	C₂²	1	2	yes	—
50	C₇×#389	C₂²	1	4	yes	—
51	C₇×#389	C₂²	1	4	yes	—
52	C₇×#388	C₂²	1	2	yes	—
53	C₇×#388	C₂²	1	2	yes	—
54	C₇×#388	C₂²	1	4	yes	—
55	C₇×#388	C₂²	1	4	yes	—
56	C₇×#388	C₂²	1	4	yes	—
57	C₇×#388	C₂²	1	4	yes	—
58	C₇×#388	C₂²	1	4	yes	—
59	C₇×#387	C₂²	1	2	yes	—
60	C₇×#387	C₂²	1	2	yes	—
61	C₇×#387	C₂²	1	4	yes	—
62	C₇×#387	C₂²	1	4	yes	—
63	C₇×#387	C₂²	1	4	yes	—
64	C₇×#387	C₂²	1	4	yes	—
65	C₇×#384	C₂²	1	1	yes	characteristic
66	C₇×#384	C₂²	1	2	yes	—
67	C₇×#381	C₂²	1	4	yes	—
68	C₇×#380	C₂²	1	4	yes	—
69	C₇×#380	C₂²	1	4	yes	—
70	C₇×#380	C₂²	1	2	yes	—
71	C₄×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
72	C₄×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
73	C₄×C₇×C₄⋊C₄	C₂²	1	2	yes	—
74	C₄×C₇×C₄⋊C₄	C₂²	1	2	yes	—
75	C₄×C₇×C₄⋊C₄	C₂²	1	2	yes	—
76	C₄×C₇×C₄⋊C₄	C₂²	1	2	yes	—
77	C₄×C₇×C₄⋊C₄	C₂²	1	2	yes	—
78	C₄×C₇×C₄⋊C₄	C₂²	1	4	yes	—
79	C₄×C₇×C₄⋊C₄	C₂²	1	2	yes	—
80	C₄×C₇×C₄⋊C₄	C₂²	1	2	yes	—
81	C₄×C₇×C₂²⋊C₄	C₂²	1	2	yes	—
82	C₇×#376	C₂²	1	2	yes	—
83	C₄³×C₇	C₂²	1	1	yes	characteristic
84	#34484	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
85	C₂²×C₇×Q₈	C₂³	1	1	yes	characteristic
86	C₂²×C₇×Q₈	C₂³	1	2	yes	—
87	C₂²×C₇×D₄	C₂³	1	1	yes	characteristic
88	C₂³×C₄×C₇	C₂³	1	2	yes	—
89	C₂³×C₄×C₇	C₂³	1	4	yes	—
90	C₇×C₄⋊Q₈	C₂×C₄	1	8	yes	—
91	C₇×#125	C₂×C₄	1	16	yes	—
92	C₇×C₂²⋊Q₈	C₂×C₄	1	32	yes	—
93	C₇×C₂²⋊Q₈	C₂×C₄	1	16	yes	—
94	C₄×C₇×Q₈	—	4	4	no	—
95	C₄×C₇×Q₈	—	4	4	no	—
96	C₄×C₇×Q₈	C₂×C₄	1	8	yes	—
97	C₄×C₇×D₄	C₂×C₄	1	16	yes	—
98	C₄×C₇×D₄	C₂×C₄	1	8	yes	—
99	C₇×#117	—	4	4	no	—
100	C₇×#117	—	4	8	no	—
101	C₇×#117	—	4	4	no	—
102	C₂×C₇×C₄⋊C₄	C₂×C₄	1	16	yes	—
103	C₂×C₇×C₄⋊C₄	C₂³	1	2	yes	—
104	C₂×C₇×C₄⋊C₄	C₂³	1	1	yes	characteristic
105	C₂×C₇×C₄⋊C₄	C₂³	1	2	yes	—
106	C₂×C₇×C₄⋊C₄	C₂³	1	1	yes	characteristic
107	C₂×C₇×C₄⋊C₄	C₂³	1	1	yes	characteristic
108	C₂×C₇×C₄⋊C₄	C₂³	1	2	yes	—
109	C₂×C₇×C₄⋊C₄	C₂³	1	1	yes	characteristic
110	C₂×C₇×C₄⋊C₄	C₂³	1	4	yes	—
111	C₂×C₇×C₄⋊C₄	C₂³	1	2	yes	—
112	C₂×C₇×C₄⋊C₄	C₂³	1	2	yes	—
113	C₂×C₇×C₄⋊C₄	C₂³	1	2	yes	—
114	C₂×C₇×C₄⋊C₄	C₂³	1	4	yes	—
115	C₂×C₇×C₄⋊C₄	C₂³	1	4	yes	—
116	C₂×C₇×C₄⋊C₄	C₂³	1	4	yes	—
117	C₂×C₇×C₄⋊C₄	C₂³	1	2	yes	—
118	C₂×C₇×C₄⋊C₄	C₂³	1	1	yes	characteristic
119	C₂×C₇×C₄⋊C₄	C₂³	1	2	yes	—
120	C₂×C₇×C₄⋊C₄	C₂³	1	4	yes	—
121	C₂×C₇×C₄⋊C₄	C₂³	1	1	yes	characteristic
122	C₂×C₇×C₄⋊C₄	C₂³	1	2	yes	—
123	C₂×C₇×C₄⋊C₄	C₂³	1	4	yes	—
124	C₂×C₇×C₄⋊C₄	C₂³	1	2	yes	—
125	C₂×C₇×C₂²⋊C₄	C₂³	1	2	yes	—
126	C₂×C₇×C₂²⋊C₄	C₂³	1	2	yes	—
127	C₂×C₇×C₂²⋊C₄	C₂³	1	2	yes	—
128	C₂×C₇×C₂²⋊C₄	C₂³	1	2	yes	—
129	C₂×C₇×C₂²⋊C₄	C₂³	1	4	yes	—
130	C₂×C₇×C₂²⋊C₄	C₂³	1	4	yes	—
131	C₂×C₇×C₂²⋊C₄	C₂³	1	4	yes	—
132	C₂×C₇×C₂²⋊C₄	C₂³	1	2	yes	—
133	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
134	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
135	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
136	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
137	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
138	C₂×C₄²×C₇	C₂³	1	2	yes	—
139	C₂×C₄²×C₇	C₂³	1	2	yes	—
140	C₂×C₄²×C₇	C₂³	1	4	yes	—
141	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
142	C₂×C₄²×C₇	C₂³	1	4	yes	—
143	C₂×C₄²×C₇	C₂³	1	4	yes	—
144	C₂×C₄²×C₇	C₂³	1	4	yes	—
145	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
146	C₂×C₄²×C₇	C₂³	1	2	yes	—
147	C₇×#95	C₂³	1	2	yes	—
148	C₇×#95	C₂³	1	2	yes	—
149	C₇×#95	C₂³	1	2	yes	—
150	C₇×#95	C₂³	1	2	yes	—
151	C₇×#95	C₂³	1	2	yes	—
152	C₇×#95	C₂³	1	2	yes	—
153	C₇×#95	C₂³	1	2	yes	—
154	C₇×#95	C₂³	1	4	yes	—
155	C₇×#95	C₂³	1	4	yes	—
156	C₇×#95	C₂³	1	4	yes	—
157	C₇×#95	C₂³	1	4	yes	—
158	C₇×#95	C₂³	1	4	yes	—
159	C₇×#95	C₂³	1	4	yes	—
160	C₇×#95	C₂³	1	2	yes	—
161	C₇×#95	C₂³	1	2	yes	—
162	C₇×#95	C₂³	1	2	yes	—
163	C₂×#554	C₂×C₇	1	1	yes	characteristic
164	#2378	C₂×C₇	1	4	yes	complemented
165	#2370	C₂×C₇	1	2	yes	complemented
166	#2369	C₂×C₇	1	1	yes	characteristic, complemented
167	#2362	C₂×C₇	1	2	yes	complemented
168	#2354	C₂×C₇	1	1	yes	characteristic
169	#2354	C₂×C₇	1	2	yes	—
170	Q₈×C₄⋊C₄	C₂×C₇	1	1	yes	characteristic, complemented
171	#2344	C₂×C₇	1	2	yes	complemented
172	#2330	C₂×C₇	1	2	yes	complemented
173	#2323	C₂×C₇	1	2	yes	complemented
174	#2323	C₂×C₇	1	4	yes	complemented
175	#2319	C₂×C₇	1	2	yes	—
176	#2316	C₂×C₇	1	2	yes	complemented
177	C₄×C₄⋊Q₈	C₂×C₇	1	1	yes	characteristic, complemented
178	#2295	C₂×C₇	1	1	yes	characteristic, complemented
179	#2293	C₂×C₇	1	1	yes	characteristic
180	C₂⁴×C₇	C₂⁴	1	2	yes	—
181	C₂×C₇×Q₈	—	2	2	no	—
182	C₂×C₇×Q₈	—	2	2	no	—
183	C₂×C₇×Q₈	—	4	2	no	—
184	C₂×C₇×Q₈	—	4	2	no	—
185	C₂×C₇×Q₈	C₂²×C₄	1	4	yes	—
186	C₂×C₇×Q₈	C₂²×C₄	1	8	yes	—
187	C₂×C₇×D₄	—	2	2	no	—
188	C₂×C₇×D₄	—	2	2	no	—
189	C₂×C₇×D₄	C₂²×C₄	1	4	yes	—
190	C₂²×C₄×C₇	—	2	4	no	—
191	C₂²×C₄×C₇	—	2	4	no	—
192	C₂²×C₄×C₇	—	4	4	no	—
193	C₂²×C₄×C₇	—	4	4	no	—
194	C₂²×C₄×C₇	C₂²×C₄	1	8	yes	—
195	C₂²×C₄×C₇	C₂²×C₄	1	16	yes	—
196	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
197	C₂²×C₄×C₇	C₂⁴	1	2	yes	—
198	C₂²×C₄×C₇	C₂⁴	1	2	yes	—
199	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
200	C₂²×C₄×C₇	C₂⁴	1	2	yes	—
201	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
202	C₂²×C₄×C₇	C₂⁴	1	2	yes	—
203	C₂²×C₄×C₇	C₂⁴	1	4	yes	—
204	C₂²×C₄×C₇	C₂⁴	1	2	yes	—
205	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
206	C₂²×C₄×C₇	C₂⁴	1	2	yes	—
207	C₂²×C₄×C₇	C₂⁴	1	4	yes	—
208	C₂²×C₄×C₇	C₂⁴	1	2	yes	—
209	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
210	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
211	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
212	C₇×C₄⋊C₄	—	2	4	no	—
213	C₇×C₄⋊C₄	—	2	2	no	—
214	C₇×C₄⋊C₄	—	2	4	no	—
215	C₇×C₄⋊C₄	—	2	2	no	—
216	C₇×C₄⋊C₄	—	2	2	no	—
217	C₇×C₄⋊C₄	—	2	4	no	—
218	C₇×C₄⋊C₄	—	2	2	no	—
219	C₇×C₄⋊C₄	—	4	2	no	—
220	C₇×C₄⋊C₄	—	4	4	no	—
221	C₇×C₄⋊C₄	—	4	2	no	—
222	C₇×C₄⋊C₄	—	4	2	no	—
223	C₇×C₄⋊C₄	—	4	4	no	—
224	C₇×C₄⋊C₄	—	4	4	no	—
225	C₇×C₄⋊C₄	—	4	2	no	—
226	C₇×C₄⋊C₄	—	4	4	no	—
227	C₇×C₄⋊C₄	C₂²×C₄	1	4	yes	—
228	C₇×C₄⋊C₄	C₂²×C₄	1	16	yes	—
229	C₇×C₄⋊C₄	C₂²×C₄	1	8	yes	—
230	C₇×C₄⋊C₄	C₂²×C₄	1	4	yes	—
231	C₇×C₄⋊C₄	C₂²×C₄	1	8	yes	—
232	C₇×C₄⋊C₄	C₂²×C₄	1	8	yes	—
233	C₇×C₄⋊C₄	C₂²×C₄	1	16	yes	—
234	C₇×C₂²⋊C₄	—	2	4	no	—
235	C₇×C₂²⋊C₄	—	4	2	no	—
236	C₇×C₂²⋊C₄	—	4	2	no	—
237	C₇×C₂²⋊C₄	—	4	2	no	—
238	C₇×C₂²⋊C₄	—	4	4	no	—
239	C₇×C₂²⋊C₄	—	4	4	no	—
240	C₇×C₂²⋊C₄	C₂²×C₄	1	16	yes	—
241	C₇×C₂²⋊C₄	C₂²×C₄	1	8	yes	—
242	C₄²×C₇	—	2	2	no	—
243	C₄²×C₇	—	2	8	no	—
244	C₄²×C₇	—	2	4	no	—
245	C₄²×C₇	—	2	2	no	—
246	C₄²×C₇	—	2	2	no	—
247	C₄²×C₇	—	2	2	no	—
248	C₄²×C₇	—	2	4	no	—
249	C₄²×C₇	—	2	2	no	—
250	C₄²×C₇	—	4	1	no	—
251	C₄²×C₇	—	4	4	no	—
252	C₄²×C₇	—	4	4	no	—
253	C₄²×C₇	—	4	4	no	—
254	C₄²×C₇	C₂²×C₄	1	8	yes	—
255	C₄²×C₇	C₂²×C₄	1	4	yes	—
256	#554	C₄×C₇	1	16	yes	complemented
257	C₂×C₄⋊Q₈	C₂²×C₇	1	1	yes	characteristic
258	C₂×#125	C₂²×C₇	1	2	yes	—
259	C₂×C₂²⋊Q₈	C₂²×C₇	1	2	yes	—
260	C₂×C₂²⋊Q₈	C₂²×C₇	1	4	yes	—
261	C₂×C₄×Q₈	C₂²×C₇	1	2	yes	—
262	C₂×C₄×Q₈	C₂²×C₇	1	2	yes	—
263	C₂×C₄×Q₈	C₂²×C₇	1	1	yes	characteristic
264	C₂×C₄×D₄	C₂²×C₇	1	2	yes	—
265	C₂×C₄×D₄	C₂²×C₇	1	1	yes	characteristic
266	C₂×#117	C₂²×C₇	1	2	yes	—
267	C₂×#117	C₂²×C₇	1	2	yes	—
268	C₂×#117	C₂²×C₇	1	4	yes	—
269	C₂²×C₄⋊C₄	C₂²×C₇	1	2	yes	—
270	#391	C₂²×C₇	1	2	yes	—
271	#391	C₂²×C₇	1	1	yes	characteristic
272	#391	C₂²×C₇	1	4	yes	—
273	#391	C₂²×C₇	1	1	yes	characteristic
274	#390	C₂²×C₇	1	1	yes	characteristic
275	#390	C₂²×C₇	1	2	yes	—
276	#390	C₂²×C₇	1	1	yes	characteristic
277	#389	C₂²×C₇	1	2	yes	—
278	#389	C₂²×C₇	1	2	yes	—
279	#389	C₂²×C₇	1	1	yes	characteristic
280	#389	C₂²×C₇	1	4	yes	—
281	#389	C₂²×C₇	1	2	yes	—
282	#389	C₂²×C₇	1	2	yes	—
283	#389	C₂²×C₇	1	4	yes	—
284	#389	C₂²×C₇	1	2	yes	—
285	#389	C₂²×C₇	1	4	yes	—
286	#389	C₂²×C₇	1	4	yes	—
287	#389	C₂²×C₇	1	1	yes	characteristic
288	#389	C₂²×C₇	1	1	yes	characteristic
289	#389	C₂²×C₇	1	1	yes	characteristic
290	#388	C₂²×C₇	1	4	yes	—
291	#388	C₂²×C₇	1	4	yes	—
292	#388	C₂²×C₇	1	2	yes	—
293	#388	C₂²×C₇	1	4	yes	—
294	#388	C₂²×C₇	1	4	yes	—
295	#388	C₂²×C₇	1	4	yes	—
296	#388	C₂²×C₇	1	2	yes	—
297	#387	C₂²×C₇	1	2	yes	—
298	#387	C₂²×C₇	1	4	yes	—
299	#387	C₂²×C₇	1	4	yes	—
300	#387	C₂²×C₇	1	4	yes	—
301	#387	C₂²×C₇	1	4	yes	—
302	#387	C₂²×C₇	1	2	yes	—
303	#384	C₂²×C₇	1	1	yes	characteristic
304	#384	C₂²×C₇	1	2	yes	—
305	#381	C₂²×C₇	1	4	yes	—
306	#380	C₂²×C₇	1	4	yes	—
307	#380	C₂²×C₇	1	4	yes	—
308	#380	C₂²×C₇	1	2	yes	—
309	C₄×C₄⋊C₄	C₂²×C₇	1	2	yes	—
310	C₄×C₄⋊C₄	C₂²×C₇	1	2	yes	—
311	C₄×C₄⋊C₄	C₂²×C₇	1	2	yes	—
312	C₄×C₄⋊C₄	C₂²×C₇	1	2	yes	—
313	C₄×C₄⋊C₄	C₂²×C₇	1	4	yes	—
314	C₄×C₄⋊C₄	C₂²×C₇	1	2	yes	—
315	C₄×C₄⋊C₄	C₂²×C₇	1	2	yes	—
316	C₄×C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
317	C₄×C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
318	C₄×C₄⋊C₄	C₂²×C₇	1	2	yes	—
319	C₄×C₂²⋊C₄	C₂²×C₇	1	2	yes	—
320	#376	C₂²×C₇	1	2	yes	—
321	C₄³	C₂²×C₇	1	1	yes	characteristic
322	C₂³×C₇	—	2	2	no	—
323	C₂³×C₇	—	2	2	no	—
324	C₂³×C₇	—	2	2	no	—
325	C₂³×C₇	—	2	2	no	—
326	C₂³×C₇	—	2	2	no	—
327	C₂³×C₇	—	2	2	no	—
328	C₂³×C₇	C₂³×C₄	1	4	yes	—
329	C₂³×C₇	C₂⁵	1	1	yes	characteristic
330	C₇×Q₈	—	2	8	no	—
331	C₇×Q₈	—	4	8	no	—
332	C₇×D₄	—	2	8	no	—
333	C₂×C₄×C₇	—	2	4	no	—
334	C₂×C₄×C₇	—	2	1	no	—
335	C₂×C₄×C₇	—	2	2	no	—
336	C₂×C₄×C₇	—	2	2	no	—
337	C₂×C₄×C₇	—	2	2	no	—
338	C₂×C₄×C₇	—	2	1	no	—
339	C₂×C₄×C₇	—	2	1	no	—
340	C₂×C₄×C₇	—	2	2	no	—
341	C₂×C₄×C₇	—	2	2	no	—
342	C₂×C₄×C₇	—	2	1	no	—
343	C₂×C₄×C₇	—	2	2	no	—
344	C₂×C₄×C₇	—	2	4	no	—
345	C₂×C₄×C₇	—	2	16	no	—
346	C₂×C₄×C₇	—	2	2	no	—
347	C₂×C₄×C₇	—	2	1	no	—
348	C₂×C₄×C₇	—	2	1	no	—
349	C₂×C₄×C₇	—	2	2	no	—
350	C₂×C₄×C₇	—	2	4	no	—
351	C₂×C₄×C₇	—	2	1	no	—
352	C₂×C₄×C₇	—	2	2	no	—
353	C₂×C₄×C₇	—	2	2	no	—
354	C₂×C₄×C₇	—	2	2	no	—
355	C₂×C₄×C₇	—	2	4	no	—
356	C₂×C₄×C₇	—	2	2	no	—
357	C₂×C₄×C₇	—	2	1	no	—
358	C₂×C₄×C₇	—	2	2	no	—
359	C₂×C₄×C₇	—	2	1	no	—
360	C₂×C₄×C₇	—	2	1	no	—
361	C₂×C₄×C₇	—	2	1	no	—
362	C₂×C₄×C₇	—	2	2	no	—
363	C₂×C₄×C₇	—	2	2	no	—
364	C₂×C₄×C₇	—	2	2	no	—
365	C₂×C₄×C₇	—	2	4	no	—
366	C₂×C₄×C₇	—	2	1	no	—
367	C₂×C₄×C₇	—	2	1	no	—
368	C₂×C₄×C₇	—	2	2	no	—
369	C₂×C₄×C₇	—	2	2	no	—
370	C₂×C₄×C₇	—	4	16	no	—
371	C₂×C₄×C₇	C₂³×C₄	1	4	yes	—
372	C₂×C₄×C₇	C₂³×C₄	1	2	yes	—
373	C₂×C₄×C₇	C₂³×C₄	1	4	yes	—
374	C₂×C₄×C₇	C₂³×C₄	1	4	yes	—
375	C₂×C₄×C₇	C₂³×C₄	1	8	yes	—
376	C₂×C₄×C₇	C₂³×C₄	1	2	yes	—
377	C₂×C₄×C₇	C₂³×C₄	1	2	yes	—
378	C₂×C₄×C₇	H(5,𝔽₂)	1	2	yes	—
379	C₂×C₄×C₇	H(5,𝔽₂)	1	2	yes	—
380	C₂×C₄×C₇	2₋¹⁺⁴	1	2	yes	—
381	C₂×C₄×C₇	2₋¹⁺⁴	1	2	yes	—
382	C₂×C₄×C₇	2₋¹⁺⁴	1	2	yes	—
383	C₂²×Q₈	C₂³×C₇	1	1	yes	characteristic
384	C₂²×Q₈	C₂³×C₇	1	2	yes	—
385	C₂²×D₄	C₂³×C₇	1	1	yes	characteristic
386	C₂³×C₄	C₂³×C₇	1	2	yes	—
387	C₂³×C₄	C₂³×C₇	1	4	yes	—
388	C₄⋊Q₈	C₂×C₄×C₇	1	8	yes	—
389	#125	C₂×C₄×C₇	1	16	yes	—
390	C₂²⋊Q₈	C₂×C₄×C₇	1	32	yes	—
391	C₂²⋊Q₈	C₂×C₄×C₇	1	16	yes	—
392	C₄×Q₈	—	4	4	no	—
393	C₄×Q₈	—	4	4	no	—
394	C₄×Q₈	C₂×C₄×C₇	1	8	yes	—
395	C₄×D₄	C₂×C₄×C₇	1	16	yes	—
396	C₄×D₄	C₂×C₄×C₇	1	8	yes	—
397	#117	—	4	4	no	—
398	#117	—	4	8	no	—
399	#117	—	4	4	no	—
400	C₂×C₄⋊C₄	C₂×C₄×C₇	1	16	yes	—
401	C₂×C₄⋊C₄	C₂³×C₇	1	2	yes	—
402	C₂×C₄⋊C₄	C₂³×C₇	1	1	yes	characteristic
403	C₂×C₄⋊C₄	C₂³×C₇	1	2	yes	—
404	C₂×C₄⋊C₄	C₂³×C₇	1	1	yes	characteristic
405	C₂×C₄⋊C₄	C₂³×C₇	1	2	yes	—
406	C₂×C₄⋊C₄	C₂³×C₇	1	1	yes	characteristic
407	C₂×C₄⋊C₄	C₂³×C₇	1	1	yes	characteristic
408	C₂×C₄⋊C₄	C₂³×C₇	1	4	yes	—
409	C₂×C₄⋊C₄	C₂³×C₇	1	2	yes	—
410	C₂×C₄⋊C₄	C₂³×C₇	1	2	yes	—
411	C₂×C₄⋊C₄	C₂³×C₇	1	4	yes	—
412	C₂×C₄⋊C₄	C₂³×C₇	1	4	yes	—
413	C₂×C₄⋊C₄	C₂³×C₇	1	2	yes	—
414	C₂×C₄⋊C₄	C₂³×C₇	1	2	yes	—
415	C₂×C₄⋊C₄	C₂³×C₇	1	4	yes	—
416	C₂×C₄⋊C₄	C₂³×C₇	1	1	yes	characteristic
417	C₂×C₄⋊C₄	C₂³×C₇	1	4	yes	—
418	C₂×C₄⋊C₄	C₂³×C₇	1	2	yes	—
419	C₂×C₄⋊C₄	C₂³×C₇	1	1	yes	characteristic
420	C₂×C₄⋊C₄	C₂³×C₇	1	2	yes	—
421	C₂×C₄⋊C₄	C₂³×C₇	1	2	yes	—
422	C₂×C₄⋊C₄	C₂³×C₇	1	4	yes	—
423	C₂×C₂²⋊C₄	C₂³×C₇	1	2	yes	—
424	C₂×C₂²⋊C₄	C₂³×C₇	1	2	yes	—
425	C₂×C₂²⋊C₄	C₂³×C₇	1	2	yes	—
426	C₂×C₂²⋊C₄	C₂³×C₇	1	4	yes	—
427	C₂×C₂²⋊C₄	C₂³×C₇	1	4	yes	—
428	C₂×C₂²⋊C₄	C₂³×C₇	1	2	yes	—
429	C₂×C₂²⋊C₄	C₂³×C₇	1	4	yes	—
430	C₂×C₂²⋊C₄	C₂³×C₇	1	2	yes	—
431	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
432	C₂×C₄²	C₂³×C₇	1	2	yes	—
433	C₂×C₄²	C₂³×C₇	1	4	yes	—
434	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
435	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
436	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
437	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
438	C₂×C₄²	C₂³×C₇	1	2	yes	—
439	C₂×C₄²	C₂³×C₇	1	4	yes	—
440	C₂×C₄²	C₂³×C₇	1	4	yes	—
441	C₂×C₄²	C₂³×C₇	1	4	yes	—
442	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
443	C₂×C₄²	C₂³×C₇	1	2	yes	—
444	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
445	#95	C₂³×C₇	1	2	yes	—
446	#95	C₂³×C₇	1	2	yes	—
447	#95	C₂³×C₇	1	2	yes	—
448	#95	C₂³×C₇	1	2	yes	—
449	#95	C₂³×C₇	1	2	yes	—
450	#95	C₂³×C₇	1	2	yes	—
451	#95	C₂³×C₇	1	2	yes	—
452	#95	C₂³×C₇	1	4	yes	—
453	#95	C₂³×C₇	1	4	yes	—
454	#95	C₂³×C₇	1	4	yes	—
455	#95	C₂³×C₇	1	4	yes	—
456	#95	C₂³×C₇	1	4	yes	—
457	#95	C₂³×C₇	1	4	yes	—
458	#95	C₂³×C₇	1	2	yes	—
459	#95	C₂³×C₇	1	2	yes	—
460	#95	C₂³×C₇	1	2	yes	—
461	C₂²×C₇	—	2	4	no	—
462	C₂²×C₇	—	2	4	no	—
463	C₂²×C₇	—	2	4	no	—
464	C₂²×C₇	—	4	2	no	—
465	C₂²×C₇	—	4	2	no	—
466	C₂²×C₇	—	4	2	no	—
467	C₂²×C₇	—	4	2	no	—
468	C₂²×C₇	C₂⁴×C₄	1	1	yes	characteristic
469	C₂²×C₇	C₂×H(5,𝔽₂)	1	1	yes	characteristic
470	C₂²×C₇	C₂×H(5,𝔽₂)	1	1	yes	characteristic
471	C₂²×C₇	C₂×2₋¹⁺⁴	1	1	yes	characteristic
472	C₂²×C₇	C₂×2₋¹⁺⁴	1	1	yes	characteristic
473	C₂²×C₇	C₂×2₋¹⁺⁴	1	1	yes	characteristic
474	C₂²×C₇	#585	1	1	yes	characteristic
475	C₄×C₇	—	2	4	no	—
476	C₄×C₇	—	2	4	no	—
477	C₄×C₇	—	2	4	no	—
478	C₄×C₇	—	2	8	no	—
479	C₄×C₇	—	2	2	no	—
480	C₄×C₇	—	2	2	no	—
481	C₄×C₇	—	4	2	no	—
482	C₄×C₇	—	4	2	no	—
483	C₄×C₇	—	4	1	no	—
484	C₄×C₇	—	4	2	no	—
485	C₄×C₇	—	4	1	no	—
486	C₄×C₇	—	4	2	no	—
487	C₄×C₇	—	4	4	no	—
488	C₄×C₇	—	4	1	no	—
489	C₄×C₇	#519	1	4	yes	—
490	C₄×C₇	#520	1	4	yes	—
491	C₂⁴	C₂⁴×C₇	1	2	yes	—
492	C₂×Q₈	—	2	2	no	—
493	C₂×Q₈	—	2	2	no	—
494	C₂×Q₈	—	4	2	no	—
495	C₂×Q₈	—	4	2	no	—
496	C₂×Q₈	C₂²×C₄×C₇	1	4	yes	—
497	C₂×Q₈	C₂²×C₄×C₇	1	8	yes	—
498	C₂×D₄	—	2	2	no	—
499	C₂×D₄	—	2	2	no	—
500	C₂×D₄	C₂²×C₄×C₇	1	4	yes	—
501	C₂²×C₄	—	2	4	no	—
502	C₂²×C₄	—	2	4	no	—
503	C₂²×C₄	—	4	4	no	—
504	C₂²×C₄	—	4	4	no	—
505	C₂²×C₄	C₂²×C₄×C₇	1	8	yes	—
506	C₂²×C₄	C₂²×C₄×C₇	1	16	yes	—
507	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
508	C₂²×C₄	C₂⁴×C₇	1	2	yes	—
509	C₂²×C₄	C₂⁴×C₇	1	2	yes	—
510	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
511	C₂²×C₄	C₂⁴×C₇	1	2	yes	—
512	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
513	C₂²×C₄	C₂⁴×C₇	1	2	yes	—
514	C₂²×C₄	C₂⁴×C₇	1	4	yes	—
515	C₂²×C₄	C₂⁴×C₇	1	2	yes	—
516	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
517	C₂²×C₄	C₂⁴×C₇	1	2	yes	—
518	C₂²×C₄	C₂⁴×C₇	1	4	yes	—
519	C₂²×C₄	C₂⁴×C₇	1	2	yes	—
520	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
521	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
522	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
523	C₄⋊C₄	—	2	4	no	—
524	C₄⋊C₄	—	2	2	no	—
525	C₄⋊C₄	—	2	4	no	—
526	C₄⋊C₄	—	2	2	no	—
527	C₄⋊C₄	—	2	4	no	—
528	C₄⋊C₄	—	2	2	no	—
529	C₄⋊C₄	—	2	2	no	—
530	C₄⋊C₄	—	4	2	no	—
531	C₄⋊C₄	—	4	2	no	—
532	C₄⋊C₄	—	4	4	no	—
533	C₄⋊C₄	—	4	4	no	—
534	C₄⋊C₄	—	4	2	no	—
535	C₄⋊C₄	—	4	4	no	—
536	C₄⋊C₄	—	4	2	no	—
537	C₄⋊C₄	—	4	4	no	—
538	C₄⋊C₄	C₂²×C₄×C₇	1	4	yes	—
539	C₄⋊C₄	C₂²×C₄×C₇	1	16	yes	—
540	C₄⋊C₄	C₂²×C₄×C₇	1	8	yes	—
541	C₄⋊C₄	C₂²×C₄×C₇	1	4	yes	—
542	C₄⋊C₄	C₂²×C₄×C₇	1	8	yes	—
543	C₄⋊C₄	C₂²×C₄×C₇	1	8	yes	—
544	C₄⋊C₄	C₂²×C₄×C₇	1	16	yes	—
545	C₂²⋊C₄	—	2	4	no	—
546	C₂²⋊C₄	—	4	2	no	—
547	C₂²⋊C₄	—	4	2	no	—
548	C₂²⋊C₄	—	4	4	no	—
549	C₂²⋊C₄	—	4	4	no	—
550	C₂²⋊C₄	—	4	2	no	—
551	C₂²⋊C₄	C₂²×C₄×C₇	1	16	yes	—
552	C₂²⋊C₄	C₂²×C₄×C₇	1	8	yes	—
553	C₄²	—	2	2	no	—
554	C₄²	—	2	8	no	—
555	C₄²	—	2	4	no	—
556	C₄²	—	2	2	no	—
557	C₄²	—	2	2	no	—
558	C₄²	—	2	2	no	—
559	C₄²	—	2	4	no	—
560	C₄²	—	2	2	no	—
561	C₄²	—	4	4	no	—
562	C₄²	—	4	4	no	—
563	C₄²	—	4	4	no	—
564	C₄²	—	4	1	no	—
565	C₄²	C₂²×C₄×C₇	1	8	yes	—
566	C₄²	C₂²×C₄×C₇	1	4	yes	—
567	C₂×C₇	—	4	4	no	—
568	C₂×C₇	C₂×#519	1	1	yes	characteristic
569	C₂×C₇	C₂×#520	1	1	yes	characteristic
570	C₂×C₇	C₄×H(5,𝔽₂)	1	1	yes	characteristic
571	C₂×C₇	#3544	1	1	yes	characteristic
572	C₂×C₇	#3547	1	1	yes	characteristic
573	C₂×C₇	#3548	1	1	yes	characteristic
574	C₂×C₇	#3555	1	1	yes	characteristic
575	C₂³	—	2	2	no	—
576	C₂³	—	2	2	no	—
577	C₂³	—	2	2	no	—
578	C₂³	—	2	2	no	—
579	C₂³	—	2	2	no	—
580	C₂³	—	2	2	no	—
581	C₂³	C₂³×C₄×C₇	1	4	yes	—
582	C₂³	C₂⁵×C₇	1	1	yes	characteristic
583	Q₈	—	2	8	no	—
584	Q₈	—	4	8	no	—
585	D₄	—	2	8	no	—
586	C₂×C₄	—	2	4	no	—
587	C₂×C₄	—	2	1	no	—
588	C₂×C₄	—	2	2	no	—
589	C₂×C₄	—	2	2	no	—
590	C₂×C₄	—	2	2	no	—
591	C₂×C₄	—	2	1	no	—
592	C₂×C₄	—	2	1	no	—
593	C₂×C₄	—	2	2	no	—
594	C₂×C₄	—	2	2	no	—
595	C₂×C₄	—	2	1	no	—
596	C₂×C₄	—	2	2	no	—
597	C₂×C₄	—	2	4	no	—
598	C₂×C₄	—	2	16	no	—
599	C₂×C₄	—	2	2	no	—
600	C₂×C₄	—	2	1	no	—
601	C₂×C₄	—	2	1	no	—
602	C₂×C₄	—	2	2	no	—
603	C₂×C₄	—	2	4	no	—
604	C₂×C₄	—	2	1	no	—
605	C₂×C₄	—	2	2	no	—
606	C₂×C₄	—	2	2	no	—
607	C₂×C₄	—	2	2	no	—
608	C₂×C₄	—	2	4	no	—
609	C₂×C₄	—	2	2	no	—
610	C₂×C₄	—	2	1	no	—
611	C₂×C₄	—	2	2	no	—
612	C₂×C₄	—	2	1	no	—
613	C₂×C₄	—	2	1	no	—
614	C₂×C₄	—	2	1	no	—
615	C₂×C₄	—	2	2	no	—
616	C₂×C₄	—	2	2	no	—
617	C₂×C₄	—	2	2	no	—
618	C₂×C₄	—	2	4	no	—
619	C₂×C₄	—	2	1	no	—
620	C₂×C₄	—	2	1	no	—
621	C₂×C₄	—	2	2	no	—
622	C₂×C₄	—	2	2	no	—
623	C₂×C₄	—	4	16	no	—
624	C₂×C₄	C₂³×C₄×C₇	1	4	yes	—
625	C₂×C₄	C₂³×C₄×C₇	1	2	yes	—
626	C₂×C₄	C₂³×C₄×C₇	1	4	yes	—
627	C₂×C₄	C₂³×C₄×C₇	1	4	yes	—
628	C₂×C₄	C₂³×C₄×C₇	1	8	yes	—
629	C₂×C₄	C₂³×C₄×C₇	1	2	yes	—
630	C₂×C₄	C₂³×C₄×C₇	1	2	yes	—
631	C₂×C₄	C₇×H(5,𝔽₂)	1	2	yes	—
632	C₂×C₄	C₇×H(5,𝔽₂)	1	2	yes	—
633	C₂×C₄	C₇×2₋¹⁺⁴	1	2	yes	—
634	C₂×C₄	C₇×2₋¹⁺⁴	1	2	yes	—
635	C₂×C₄	C₇×2₋¹⁺⁴	1	2	yes	—
636	C₇	#34484	1	1	yes	characteristic, direct product factor, Sylow
637	C₂²	—	2	4	no	—
638	C₂²	—	2	4	no	—
639	C₂²	—	2	4	no	—
640	C₂²	—	4	2	no	—
641	C₂²	—	4	2	no	—
642	C₂²	—	4	2	no	—
643	C₂²	—	4	2	no	—
644	C₂²	C₂⁴×C₄×C₇	1	1	yes	characteristic
645	C₂²	C₂×C₇×H(5,𝔽₂)	1	1	yes	characteristic
646	C₂²	C₂×C₇×H(5,𝔽₂)	1	1	yes	characteristic
647	C₂²	C₂×C₇×2₋¹⁺⁴	1	1	yes	characteristic
648	C₂²	C₂×C₇×2₋¹⁺⁴	1	1	yes	characteristic
649	C₂²	C₂×C₇×2₋¹⁺⁴	1	1	yes	characteristic
650	C₂²	C₇×#585	1	1	yes	characteristic
651	C₄	—	2	4	no	—
652	C₄	—	2	4	no	—
653	C₄	—	2	4	no	—
654	C₄	—	2	8	no	—
655	C₄	—	2	2	no	—
656	C₄	—	2	2	no	—
657	C₄	—	4	2	no	—
658	C₄	—	4	2	no	—
659	C₄	—	4	1	no	—
660	C₄	—	4	2	no	—
661	C₄	—	4	1	no	—
662	C₄	—	4	2	no	—
663	C₄	—	4	4	no	—
664	C₄	—	4	1	no	—
665	C₄	C₇×#519	1	4	yes	—
666	C₄	C₇×#520	1	4	yes	—
667	C₂	—	4	4	no	—
668	C₂	C₂×C₇×#519	1	1	yes	characteristic
669	C₂	C₂×C₇×#520	1	1	yes	characteristic
670	C₂	C₄×C₇×H(5,𝔽₂)	1	1	yes	characteristic
671	C₂	C₇×#3544	1	1	yes	characteristic
672	C₂	C₇×#3547	1	1	yes	characteristic
673	C₂	C₇×#3548	1	1	yes	characteristic
674	C₂	C₇×#3555	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₇	C₂⁵
commutator	C₂²	C₂⁴×C₄×C₇
Frattini, cosocle	C₂³	C₂⁵×C₇
prime radical	C₂²×C₇×D₄	C₂³
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₇×#2378	C₂	semidirect
3. C₇×#2370	C₂	semidirect
4. C₇×#2369	C₂	semidirect
5. C₇×#2362	C₂	semidirect
8. C₇×Q₈×C₄⋊C₄	C₂	semidirect
9. C₇×#2344	C₂	semidirect
10. C₇×#2330	C₂	semidirect
11. C₇×#2323	C₂	semidirect
12. C₇×#2323	C₂	semidirect
14. C₇×#2316	C₂	semidirect
15. C₄×C₇×C₄⋊Q₈	C₂	semidirect
16. C₇×#2295	C₂	semidirect
18. C₇×#554	C₄	semidirect
84. #34484	C₇	direct (N × H)
164. #2378	C₂×C₇	semidirect
165. #2370	C₂×C₇	semidirect
166. #2369	C₂×C₇	semidirect
167. #2362	C₂×C₇	semidirect
170. Q₈×C₄⋊C₄	C₂×C₇	semidirect
171. #2344	C₂×C₇	semidirect
172. #2330	C₂×C₇	semidirect
173. #2323	C₂×C₇	semidirect
174. #2323	C₂×C₇	semidirect
176. #2316	C₂×C₇	semidirect
177. C₄×C₄⋊Q₈	C₂×C₇	semidirect
178. #2295	C₂×C₇	semidirect
256. #554	C₄×C₇	semidirect
636. C₇	#34484	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×#554	2
C₇×#2378	2
C₇×#2370	2
C₇×#2369	2
C₇×#2362	2
C₇×#2354	2
C₇×Q₈×C₄⋊C₄	2
C₇×#2344	2
C₇×#2330	2
C₇×#2323	2
C₇×#2319	2
C₇×#2316	2
C₄×C₇×C₄⋊Q₈	2
C₇×#2295	2
C₇×#2293	2
#34484	7

Subgroups — C7×#34484 #13211700

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#34484 #13211700