Subgroups — #13507332

Order 1792.

526 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#2966	C₂	1	1	yes	maximal, characteristic
2	#11728703	C₂	1	1	yes	maximal, characteristic, complemented
3	Q₈×C₇⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
4	#11728612	C₂	1	1	yes	maximal, characteristic, complemented
5	#11728611	C₂	1	2	yes	maximal, complemented
6	D₄×C₇⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
7	D₄×C₇⋊C₁₆	C₂	1	2	yes	maximal, complemented
8	#11727637	C₂	1	2	yes	maximal, complemented
9	#11727635	C₂	1	2	yes	maximal, complemented
10	#11727499	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₄×C₇⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
12	C₇×#529	C₄	1	1	yes	characteristic
13	C₇×#446	C₄	1	1	yes	characteristic
14	C₇×C₈×Q₈	C₂²	1	1	yes	characteristic
15	C₇×#436	C₄	1	1	yes	characteristic
16	C₇×#435	C₄	1	2	yes	—
17	C₇×C₈×D₄	C₂²	1	2	yes	—
18	C₇×C₈×D₄	C₂²	1	1	yes	characteristic
19	C₇×#433	C₄	1	2	yes	—
20	C₇×#432	C₄	1	1	yes	characteristic
21	C₇×#405	C₂²	1	2	yes	—
22	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
23	C₂×#6386	C₂²	1	2	yes	—
24	C₂²×C₇⋊C₁₆	C₂²	1	2	yes	—
25	C₂²×C₇⋊C₁₆	C₂²	1	1	yes	characteristic
26	C₂²×C₇⋊C₁₆	C₂²	1	1	yes	characteristic
27	C₂²⋊(C₇⋊C₁₆)	C₂²	1	2	yes	—
28	C₂²⋊(C₇⋊C₁₆)	C₂²	1	2	yes	—
29	C₂²⋊(C₇⋊C₁₆)	C₂²	1	2	yes	—
30	C₄⋊(C₇⋊C₁₆)	C₂²	1	2	yes	—
31	C₄⋊(C₇⋊C₁₆)	C₂²	1	2	yes	—
32	C₄⋊(C₇⋊C₁₆)	C₂²	1	1	yes	characteristic
33	C₄⋊(C₇⋊C₁₆)	C₂²	1	1	yes	characteristic
34	#89256	C₂²	1	2	yes	—
35	#89256	C₂²	1	2	yes	—
36	C₄×C₇⋊C₁₆	C₂²	1	1	yes	characteristic
37	C₄×C₇⋊C₁₆	C₂²	1	2	yes	—
38	C₄×C₇⋊C₁₆	C₂²	1	1	yes	characteristic
39	C₄×C₇⋊C₁₆	C₂²	1	1	yes	characteristic
40	C₄×C₇⋊C₁₆	C₂²	1	1	yes	characteristic
41	#33480	—	7	1	no	maximal, Sylow
42	C₂×C₇×M₁₆	C₂×C₄	1	2	yes	—
43	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
44	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
45	C₂²×C₇×C₈	C₂³	1	2	yes	—
46	C₇×#126	C₈	1	2	yes	—
47	C₇×#125	C₈	1	1	yes	characteristic
48	C₇×#124	C₈	1	1	yes	characteristic
49	C₇×#123	C₈	1	2	yes	—
50	C₇×C₂²⋊Q₈	C₈	1	1	yes	characteristic
51	C₇×#121	C₈	1	1	yes	characteristic
52	C₄×C₇×Q₈	C₂×C₄	1	1	yes	characteristic
53	C₄×C₇×D₄	C₂×C₄	1	2	yes	—
54	C₄×C₇×D₄	C₂×C₄	1	1	yes	characteristic
55	C₇×#117	C₂×C₄	1	2	yes	—
56	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
57	C₇×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
58	C₇×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
59	C₇×C₄⋊C₈	C₂×C₄	1	2	yes	—
60	C₇×C₄⋊C₈	C₂×C₄	1	2	yes	—
61	C₇×C₂²⋊C₈	C₂×C₄	1	2	yes	—
62	C₇×C₂²⋊C₈	C₂×C₄	1	2	yes	—
63	C₇×C₂²⋊C₈	C₂×C₄	1	2	yes	—
64	C₇×#97	C₂×C₄	1	2	yes	—
65	C₇×#97	C₂×C₄	1	2	yes	—
66	C₄×C₇×C₈	C₂×C₄	1	1	yes	characteristic
67	C₄×C₇×C₈	C₂×C₄	1	1	yes	characteristic
68	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
69	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
70	C₄×C₇×C₈	C₂³	1	2	yes	—
71	#6386	—	2	4	no	—
72	C₂×C₇⋊C₁₆	—	2	4	no	—
73	C₂×C₇⋊C₁₆	—	2	2	no	—
74	C₂×C₇⋊C₁₆	—	2	2	no	—
75	C₂×C₇⋊C₁₆	C₂³	1	1	yes	characteristic
76	C₂×C₇⋊C₁₆	C₂³	1	1	yes	characteristic
77	C₂×C₇⋊C₁₆	C₂³	1	2	yes	—
78	C₂×C₇⋊C₁₆	C₂³	1	2	yes	—
79	C₂×C₇⋊C₁₆	C₂³	1	1	yes	characteristic
80	C₂×C₇⋊C₁₆	C₂³	1	1	yes	characteristic
81	#2966	D₇	1	1	yes	characteristic
82	#2183	—	7	1	no	—
83	Q₈×C₁₆	—	7	1	no	—
84	#2169	—	7	1	no	—
85	#2168	—	7	2	no	—
86	D₄×C₁₆	—	7	1	no	—
87	D₄×C₁₆	—	7	2	no	—
88	#2164	—	7	2	no	—
89	#2163	—	7	1	no	—
90	#2108	—	7	2	no	—
91	C₂×C₄×C₁₆	—	7	1	no	—
92	C₂×C₇×Q₈	C₂×C₈	1	1	yes	characteristic
93	C₂×C₇×D₄	C₂×C₈	1	2	yes	—
94	C₂×C₇×D₄	C₂×C₈	1	1	yes	characteristic
95	C₂²×C₄×C₇	C₂×C₈	1	1	yes	characteristic
96	C₂²×C₄×C₇	C₂×C₈	1	1	yes	characteristic
97	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
98	C₂²×C₄×C₇	C₂²×C₄	1	2	yes	—
99	C₇×M₁₆	—	2	4	no	—
100	C₂×C₇×C₈	—	2	2	no	—
101	C₂×C₇×C₈	—	2	1	no	—
102	C₂×C₇×C₈	—	2	1	no	—
103	C₂×C₇×C₈	—	2	2	no	—
104	C₂×C₇×C₈	—	2	2	no	—
105	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
106	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
107	C₂×C₇×C₈	C₂²×C₄	1	2	yes	—
108	C₂×C₇×C₈	C₂²×C₄	1	2	yes	—
109	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
110	C₂×C₇×C₈	C₂⁴	1	1	yes	characteristic
111	C₇×C₄⋊C₄	C₂×C₈	1	2	yes	—
112	C₇×C₄⋊C₄	C₂×C₈	1	2	yes	—
113	C₇×C₄⋊C₄	C₂×C₈	1	1	yes	characteristic
114	C₇×C₄⋊C₄	C₂×C₈	1	2	yes	—
115	C₇×C₄⋊C₄	C₂×C₈	1	1	yes	characteristic
116	C₇×C₄⋊C₄	C₂×C₈	1	2	yes	—
117	C₇×C₂²⋊C₄	C₂×C₈	1	2	yes	—
118	C₇×C₂²⋊C₄	C₂×C₈	1	2	yes	—
119	C₇×C₂²⋊C₄	C₂×C₈	1	2	yes	—
120	C₇×C₂²⋊C₄	C₂×C₈	1	2	yes	—
121	C₇×C₂²⋊C₄	C₂×C₈	1	2	yes	—
122	C₄²×C₇	C₂×C₈	1	1	yes	characteristic
123	C₄²×C₇	C₂×C₈	1	1	yes	characteristic
124	C₄²×C₇	C₂²×C₄	1	1	yes	characteristic
125	C₄²×C₇	C₂²×C₄	1	1	yes	characteristic
126	C₄²×C₇	C₂²×C₄	1	2	yes	—
127	C₇⋊C₁₆	—	2	1	no	—
128	C₇⋊C₁₆	—	2	1	no	—
129	C₇⋊C₁₆	—	2	2	no	—
130	C₇⋊C₁₆	—	2	2	no	—
131	C₇⋊C₁₆	#41	1	2	yes	—
132	C₇⋊C₁₆	#41	1	2	yes	—
133	#529	Dic₇	1	1	yes	characteristic
134	C₂×M₃₂	—	7	2	no	—
135	C₂²×C₁₆	—	7	1	no	—
136	C₂²×C₁₆	—	7	1	no	—
137	C₂²×C₁₆	—	7	2	no	—
138	#446	Dic₇	1	1	yes	characteristic
139	C₈×Q₈	C₂×D₇	1	1	yes	characteristic
140	#436	Dic₇	1	1	yes	characteristic
141	#435	Dic₇	1	2	yes	—
142	C₈×D₄	C₂×D₇	1	2	yes	—
143	C₈×D₄	C₂×D₇	1	1	yes	characteristic
144	#433	Dic₇	1	2	yes	—
145	#432	Dic₇	1	1	yes	characteristic
146	#405	C₂×D₇	1	2	yes	—
147	C₂×C₄×C₈	C₂×D₇	1	1	yes	characteristic
148	C₄⋊C₁₆	—	7	1	no	—
149	C₄⋊C₁₆	—	7	1	no	—
150	C₄⋊C₁₆	—	7	2	no	—
151	C₄⋊C₁₆	—	7	2	no	—
152	C₂²⋊C₁₆	—	7	2	no	—
153	C₂²⋊C₁₆	—	7	2	no	—
154	C₂²⋊C₁₆	—	7	2	no	—
155	#346	—	7	2	no	—
156	#346	—	7	2	no	—
157	C₄×C₁₆	—	7	1	no	—
158	C₄×C₁₆	—	7	1	no	—
159	C₄×C₁₆	—	7	1	no	—
160	C₄×C₁₆	—	7	1	no	—
161	C₄×C₁₆	—	7	2	no	—
162	C₂³×C₇	C₂²×C₈	1	2	yes	—
163	C₂³×C₇	C₂²×C₈	1	1	yes	characteristic
164	C₇×Q₈	—	2	2	no	—
165	C₇×D₄	—	2	4	no	—
166	C₇×D₄	—	2	2	no	—
167	C₂×C₄×C₇	—	2	2	no	—
168	C₂×C₄×C₇	—	2	1	no	—
169	C₂×C₄×C₇	—	2	2	no	—
170	C₂×C₄×C₇	—	2	1	no	—
171	C₂×C₄×C₇	—	2	2	no	—
172	C₂×C₄×C₇	—	2	2	no	—
173	C₂×C₄×C₇	C₂²×C₈	1	1	yes	characteristic
174	C₂×C₄×C₇	C₂²×C₈	1	1	yes	characteristic
175	C₂×C₄×C₇	C₂²×C₈	1	1	yes	characteristic
176	C₂×C₄×C₇	C₂²×C₈	1	1	yes	characteristic
177	C₂×C₄×C₇	C₂²×C₈	1	2	yes	—
178	C₂×C₄×C₇	C₂²×C₈	1	2	yes	—
179	C₂×C₄×C₇	C₂²×C₈	1	2	yes	—
180	C₂×C₄×C₇	C₂²×C₈	1	1	yes	characteristic
181	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
182	C₇×C₈	—	2	1	no	—
183	C₇×C₈	—	2	2	no	—
184	C₇×C₈	—	2	1	no	—
185	C₇×C₈	—	2	2	no	—
186	C₇×C₈	#131	1	2	yes	—
187	C₇×C₈	C₂×#41	1	1	yes	characteristic
188	C₇×C₈	C₂×#41	1	1	yes	characteristic
189	C₂×M₁₆	C₂×Dic₇	1	2	yes	—
190	C₂²×C₈	C₂×Dic₇	1	1	yes	characteristic
191	C₂²×C₈	C₂²×D₇	1	1	yes	characteristic
192	C₂²×C₈	C₂²×D₇	1	2	yes	—
193	#126	C₇⋊C₈	1	2	yes	—
194	#125	C₇⋊C₈	1	1	yes	characteristic
195	#124	C₇⋊C₈	1	1	yes	characteristic
196	#123	C₇⋊C₈	1	2	yes	—
197	C₂²⋊Q₈	C₇⋊C₈	1	1	yes	characteristic
198	#121	C₇⋊C₈	1	1	yes	characteristic
199	C₄×Q₈	C₂×Dic₇	1	1	yes	characteristic
200	C₄×D₄	C₂×Dic₇	1	2	yes	—
201	C₄×D₄	C₂×Dic₇	1	1	yes	characteristic
202	#117	C₂×Dic₇	1	2	yes	—
203	C₂×C₄²	C₂×Dic₇	1	1	yes	characteristic
204	M₃₂	—	14	4	no	—
205	C₂×C₁₆	—	7	1	no	—
206	C₂×C₁₆	—	7	1	no	—
207	C₂×C₁₆	—	7	1	no	—
208	C₂×C₁₆	—	7	1	no	—
209	C₂×C₁₆	—	7	2	no	—
210	C₂×C₁₆	—	7	2	no	—
211	C₂×C₁₆	—	14	2	no	—
212	C₂×C₁₆	—	14	4	no	—
213	C₂×C₁₆	—	14	2	no	—
214	C₄⋊C₈	C₂×Dic₇	1	1	yes	characteristic
215	C₄⋊C₈	C₂×Dic₇	1	1	yes	characteristic
216	C₄⋊C₈	C₂×Dic₇	1	2	yes	—
217	C₄⋊C₈	C₂×Dic₇	1	2	yes	—
218	C₂²⋊C₈	C₂×Dic₇	1	2	yes	—
219	C₂²⋊C₈	C₂×Dic₇	1	2	yes	—
220	C₂²⋊C₈	C₂×Dic₇	1	2	yes	—
221	#97	C₂×Dic₇	1	2	yes	—
222	#97	C₂×Dic₇	1	2	yes	—
223	C₄×C₈	C₂×Dic₇	1	1	yes	characteristic
224	C₄×C₈	C₂×Dic₇	1	1	yes	characteristic
225	C₄×C₈	C₂²×D₇	1	1	yes	characteristic
226	C₄×C₈	C₂²×D₇	1	1	yes	characteristic
227	C₄×C₈	C₂²×D₇	1	2	yes	—
228	C₂²×C₇	—	2	1	no	—
229	C₂²×C₇	—	2	1	no	—
230	C₂²×C₇	—	2	2	no	—
231	C₂²×C₇	—	2	2	no	—
232	C₂²×C₇	—	2	2	no	—
233	C₂²×C₇	#504	1	2	yes	—
234	C₂²×C₇	C₂³×C₈	1	1	yes	characteristic
235	C₄×C₇	—	2	1	no	—
236	C₄×C₇	—	2	1	no	—
237	C₄×C₇	—	2	1	no	—
238	C₄×C₇	—	2	2	no	—
239	C₄×C₇	—	2	2	no	—
240	C₄×C₇	—	2	1	no	—
241	C₄×C₇	—	2	2	no	—
242	C₄×C₇	#504	1	2	yes	—
243	C₄×C₇	C₄×#41	1	1	yes	characteristic
244	C₄×C₇	C₂×#131	1	1	yes	characteristic
245	C₂×Q₈	C₂×C₇⋊C₈	1	1	yes	characteristic
246	C₂×D₄	C₂×C₇⋊C₈	1	2	yes	—
247	C₂×D₄	C₂×C₇⋊C₈	1	1	yes	characteristic
248	C₂²×C₄	C₂×C₇⋊C₈	1	1	yes	characteristic
249	C₂²×C₄	C₂×C₇⋊C₈	1	1	yes	characteristic
250	C₂²×C₄	C₂²×Dic₇	1	1	yes	characteristic
251	C₂²×C₄	C₂²×Dic₇	1	2	yes	—
252	M₁₆	—	2	4	no	—
253	C₂×C₈	—	2	2	no	—
254	C₂×C₈	—	2	1	no	—
255	C₂×C₈	—	2	1	no	—
256	C₂×C₈	—	2	2	no	—
257	C₂×C₈	—	2	2	no	—
258	C₂×C₈	C₂²×Dic₇	1	1	yes	characteristic
259	C₂×C₈	C₂²×Dic₇	1	1	yes	characteristic
260	C₂×C₈	C₂²×Dic₇	1	2	yes	—
261	C₂×C₈	C₂²×Dic₇	1	2	yes	—
262	C₂×C₈	C₂²×Dic₇	1	1	yes	characteristic
263	C₂×C₈	C₂³×D₇	1	1	yes	characteristic
264	C₄⋊C₄	C₂×C₇⋊C₈	1	2	yes	—
265	C₄⋊C₄	C₂×C₇⋊C₈	1	2	yes	—
266	C₄⋊C₄	C₂×C₇⋊C₈	1	1	yes	characteristic
267	C₄⋊C₄	C₂×C₇⋊C₈	1	2	yes	—
268	C₄⋊C₄	C₂×C₇⋊C₈	1	1	yes	characteristic
269	C₄⋊C₄	C₂×C₇⋊C₈	1	2	yes	—
270	C₂²⋊C₄	C₂×C₇⋊C₈	1	2	yes	—
271	C₂²⋊C₄	C₂×C₇⋊C₈	1	2	yes	—
272	C₂²⋊C₄	C₂×C₇⋊C₈	1	2	yes	—
273	C₂²⋊C₄	C₂×C₇⋊C₈	1	2	yes	—
274	C₂²⋊C₄	C₂×C₇⋊C₈	1	2	yes	—
275	C₄²	C₂×C₇⋊C₈	1	1	yes	characteristic
276	C₄²	C₂×C₇⋊C₈	1	1	yes	characteristic
277	C₄²	C₂²×Dic₇	1	1	yes	characteristic
278	C₄²	C₂²×Dic₇	1	1	yes	characteristic
279	C₄²	C₂²×Dic₇	1	2	yes	—
280	C₁₆	—	7	2	no	—
281	C₁₆	—	7	2	no	—
282	C₁₆	—	14	1	no	—
283	C₁₆	—	14	1	no	—
284	C₁₆	—	14	2	no	—
285	C₁₆	—	14	2	no	—
286	C₂×C₇	—	2	2	no	—
287	C₂×C₇	—	4	2	no	—
288	C₂×C₇	C₈×#41	1	1	yes	characteristic
289	C₂×C₇	C₂×#504	1	1	yes	characteristic
290	C₂×C₇	C₂×#504	1	1	yes	characteristic
291	C₂³	C₂²×C₇⋊C₈	1	2	yes	—
292	C₂³	C₂²×C₇⋊C₈	1	1	yes	characteristic
293	Q₈	—	2	2	no	—
294	D₄	—	2	4	no	—
295	D₄	—	2	2	no	—
296	C₂×C₄	—	2	2	no	—
297	C₂×C₄	—	2	2	no	—
298	C₂×C₄	—	2	1	no	—
299	C₂×C₄	—	2	2	no	—
300	C₂×C₄	—	2	1	no	—
301	C₂×C₄	—	2	2	no	—
302	C₂×C₄	C₂²×C₇⋊C₈	1	1	yes	characteristic
303	C₂×C₄	C₂²×C₇⋊C₈	1	1	yes	characteristic
304	C₂×C₄	C₂²×C₇⋊C₈	1	1	yes	characteristic
305	C₂×C₄	C₂²×C₇⋊C₈	1	1	yes	characteristic
306	C₂×C₄	C₂²×C₇⋊C₈	1	2	yes	—
307	C₂×C₄	C₂²×C₇⋊C₈	1	2	yes	—
308	C₂×C₄	C₂²×C₇⋊C₈	1	2	yes	—
309	C₂×C₄	C₂²×C₇⋊C₈	1	1	yes	characteristic
310	C₂×C₄	C₂³×Dic₇	1	1	yes	characteristic
311	C₈	—	2	1	no	—
312	C₈	—	2	2	no	—
313	C₈	—	2	1	no	—
314	C₈	—	2	2	no	—
315	C₈	#6511	1	2	yes	—
316	C₈	D₇×#41	1	1	yes	characteristic
317	C₈	D₇×#41	1	1	yes	characteristic
318	C₇	#33480	1	1	yes	characteristic, complemented, Sylow
319	C₂²	—	2	1	no	—
320	C₂²	—	2	1	no	—
321	C₂²	—	2	2	no	—
322	C₂²	—	2	2	no	—
323	C₂²	—	2	2	no	—
324	C₂²	#89912	1	2	yes	—
325	C₂²	C₂³×C₇⋊C₈	1	1	yes	characteristic
326	C₄	—	2	1	no	—
327	C₄	—	2	1	no	—
328	C₄	—	2	1	no	—
329	C₄	—	2	2	no	—
330	C₄	—	2	2	no	—
331	C₄	—	2	1	no	—
332	C₄	—	2	2	no	—
333	C₄	#89912	1	2	yes	—
334	C₄	C₂×#6511	1	1	yes	characteristic
335	C₄	#41×Dic₇	1	1	yes	characteristic
336	C₂	—	2	2	no	—
337	C₂	—	4	2	no	—
338	C₂	#41×C₇⋊C₈	1	1	yes	characteristic
339	C₂	C₂×#89912	1	1	yes	characteristic
340	C₂	C₂×#89912	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₇
commutator, socle	C₂²×C₇	C₂³×C₈
Fitting	C₇×#2966	C₂
prime radical	C₇×#121	C₈
cosocle	C₂×C₇×C₈	C₂⁴
hypercenter	#2966	D₇
hypocenter	C₇	#33480
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₈ ◁ #2966

Complements

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

N	H ≅ G/N	type
2. #11728703	C₂	semidirect
3. Q₈×C₇⋊C₁₆	C₂	semidirect
4. #11728612	C₂	semidirect
5. #11728611	C₂	semidirect
6. D₄×C₇⋊C₁₆	C₂	semidirect
7. D₄×C₇⋊C₁₆	C₂	semidirect
8. #11727637	C₂	semidirect
9. #11727635	C₂	semidirect
10. #11727499	C₂	semidirect
11. C₂×C₄×C₇⋊C₁₆	C₂	semidirect
318. C₇	#33480	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₇×#2966	2
#11728703	2
Q₈×C₇⋊C₁₆	2
#11728612	2
#11728611	2
D₄×C₇⋊C₁₆	2
#11727637	2
#11727635	2
#11727499	2
C₂×C₄×C₇⋊C₁₆	2
#33480	7