Subgroups — #13507308

Order 1792.

530 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈×Q₈×D₇	C₂	1	1	yes	maximal, characteristic
2	C₇×#2183	C₂	1	1	yes	maximal, characteristic, complemented
3	#11728703	C₂	1	1	yes	maximal, characteristic, complemented
4	#11727038	C₂	1	1	yes	maximal, characteristic
5	#11727038	C₂	1	2	yes	maximal
6	#11727036	C₂	1	2	yes	maximal, complemented
7	#11727034	C₂	1	1	yes	maximal, characteristic, complemented
8	#11726825	C₂	1	2	yes	maximal
9	#11726822	C₂	1	2	yes	maximal, complemented
10	C₄×#6372	C₂	1	1	yes	maximal, characteristic
11	C₁₆×Dic₁₄	C₂	1	1	yes	maximal, characteristic, complemented
12	C₄×Q₈×D₇	C₄	1	1	yes	characteristic
13	C₇×C₈×Q₈	C₂²	1	1	yes	characteristic
14	Q₈×C₇⋊C₈	C₄	1	1	yes	characteristic
15	C₂×#6372	C₂²	1	1	yes	characteristic
16	C₂×#6372	C₂²	1	2	yes	—
17	C₂×#6372	C₂²	1	1	yes	characteristic
18	D₇×C₄⋊C₈	C₄	1	1	yes	characteristic
19	D₇×C₄⋊C₈	C₄	1	2	yes	—
20	#89602	C₄	1	2	yes	—
21	#89602	C₄	1	1	yes	characteristic
22	C₄×C₈×D₇	C₂²	1	1	yes	characteristic
23	C₄×C₈×D₇	C₂²	1	2	yes	—
24	C₈×Dic₁₄	C₂²	1	1	yes	characteristic
25	C₈×Dic₁₄	C₂²	1	2	yes	—
26	C₇×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
27	C₇×C₄⋊C₁₆	C₂²	1	2	yes	—
28	C₇×#346	C₂²	1	2	yes	—
29	C₄×C₇×C₁₆	C₂²	1	1	yes	characteristic
30	#89297	C₂²	1	1	yes	characteristic
31	#89297	C₂²	1	1	yes	characteristic
32	Dic₇⋊C₁₆	C₂²	1	2	yes	—
33	Dic₇⋊C₁₆	C₂²	1	2	yes	—
34	Dic₇⋊C₁₆	C₂²	1	2	yes	—
35	C₁₆×Dic₇	C₂²	1	2	yes	—
36	C₁₆×Dic₇	C₂²	1	2	yes	—
37	C₄⋊(C₇⋊C₁₆)	C₂²	1	1	yes	characteristic
38	C₄⋊(C₇⋊C₁₆)	C₂²	1	2	yes	—
39	#89256	C₂²	1	2	yes	—
40	C₄×C₇⋊C₁₆	C₂²	1	1	yes	characteristic
41	Q₈×M₃₂	—	7	1	no	maximal, Sylow
42	C₂×Q₈×D₇	C₈	1	1	yes	characteristic
43	C₄×C₇×Q₈	C₂×C₄	1	1	yes	characteristic
44	Q₈×Dic₇	C₈	1	1	yes	characteristic
45	C₂×C₈×D₇	C₂×C₄	1	1	yes	characteristic
46	C₂×C₈×D₇	C₂×C₄	1	2	yes	—
47	C₂×C₈×D₇	C₂³	1	1	yes	characteristic
48	D₇×C₄⋊C₄	C₈	1	1	yes	characteristic
49	D₇×C₄⋊C₄	C₈	1	2	yes	—
50	#6450	C₈	1	2	yes	—
51	#6450	C₈	1	1	yes	characteristic
52	C₄²×D₇	C₂×C₄	1	1	yes	characteristic
53	C₄²×D₇	C₂×C₄	1	2	yes	—
54	C₄×Dic₁₄	C₂×C₄	1	2	yes	—
55	C₄×Dic₁₄	C₂×C₄	1	1	yes	characteristic
56	C₂×C₇×C₁₆	C₂³	1	1	yes	characteristic
57	C₂×C₇×C₁₆	C₂³	1	2	yes	—
58	C₂×C₇×C₁₆	C₂³	1	1	yes	characteristic
59	C₇×C₄⋊C₈	C₂×C₄	1	2	yes	—
60	C₇×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
61	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
62	C₄×C₇×C₈	C₂³	1	2	yes	—
63	Dic₇⋊C₈	C₂×C₄	1	2	yes	—
64	Dic₇⋊C₈	C₂×C₄	1	2	yes	—
65	Dic₇⋊C₈	C₂×C₄	1	2	yes	—
66	C₈×Dic₇	C₂×C₄	1	1	yes	characteristic
67	C₈×Dic₇	C₂×C₄	1	2	yes	—
68	C₈×Dic₇	C₂³	1	2	yes	—
69	C₈×Dic₇	C₂³	1	1	yes	characteristic
70	C₂×C₇⋊C₁₆	C₂³	1	1	yes	characteristic
71	C₂×C₇⋊C₁₆	C₂³	1	2	yes	—
72	C₂×C₇⋊C₁₆	C₂³	1	1	yes	characteristic
73	C₄⋊(C₇⋊C₈)	C₂×C₄	1	1	yes	characteristic
74	C₄⋊(C₇⋊C₈)	C₂×C₄	1	2	yes	—
75	C₄×C₇⋊C₈	C₂×C₄	1	1	yes	characteristic
76	C₄×C₇⋊C₈	C₂×C₄	1	2	yes	—
77	#6372	—	2	2	no	—
78	#6372	—	2	4	no	—
79	#6372	Q₈	1	4	yes	complemented
80	C₂×C₈×Q₈	—	7	1	no	—
81	#2183	—	7	1	no	—
82	#2183	—	7	2	no	—
83	#2183	—	7	2	no	—
84	#2183	D₇	1	1	yes	characteristic, complemented
85	Q₈×C₁₆	—	7	1	no	—
86	Q₈×C₁₆	—	7	1	no	—
87	#2150	—	7	2	no	—
88	#2150	—	7	1	no	—
89	C₄×M₃₂	—	7	2	no	—
90	C₄×M₃₂	—	7	1	no	—
91	C₂×C₇×Q₈	C₂×C₈	1	1	yes	characteristic
92	Q₈×D₇	—	2	4	no	—
93	C₂×C₄×D₇	C₂×C₈	1	1	yes	characteristic
94	C₂×C₄×D₇	C₂×C₈	1	2	yes	—
95	C₂×C₄×D₇	C₂×C₈	1	1	yes	characteristic
96	C₂×C₄×D₇	C₂×C₈	1	2	yes	—
97	C₂×C₄×D₇	C₂²×C₄	1	1	yes	characteristic
98	C₂×Dic₁₄	C₂×C₈	1	2	yes	—
99	C₂×Dic₁₄	C₂×C₈	1	1	yes	characteristic
100	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
101	C₂×C₇×C₈	C₂²×C₄	1	2	yes	—
102	C₂×C₇×C₈	C₂⁴	1	1	yes	characteristic
103	C₇×C₄⋊C₄	C₂×C₈	1	2	yes	—
104	C₇×C₄⋊C₄	C₂×C₈	1	1	yes	characteristic
105	C₄²×C₇	C₂²×C₄	1	2	yes	—
106	C₄²×C₇	C₂²×C₄	1	1	yes	characteristic
107	C₄⋊Dic₇	C₂×C₈	1	2	yes	—
108	C₄⋊Dic₇	C₂×C₈	1	1	yes	characteristic
109	Dic₇⋊C₄	C₂×C₈	1	2	yes	—
110	Dic₇⋊C₄	C₂×C₈	1	2	yes	—
111	Dic₇⋊C₄	C₂×C₈	1	2	yes	—
112	C₄×Dic₇	C₂×C₈	1	1	yes	characteristic
113	C₄×Dic₇	C₂×C₈	1	2	yes	—
114	C₄×Dic₇	C₂×C₈	1	2	yes	—
115	C₄×Dic₇	C₂×C₈	1	1	yes	characteristic
116	C₄×Dic₇	C₂²×C₄	1	1	yes	characteristic
117	C₄×Dic₇	C₂²×C₄	1	2	yes	—
118	C₂×C₇⋊C₈	C₂²×C₄	1	2	yes	—
119	C₂×C₇⋊C₈	C₂²×C₄	1	1	yes	characteristic
120	C₂×C₇⋊C₈	C₂²×C₄	1	1	yes	characteristic
121	C₈×D₇	—	2	2	no	—
122	C₈×D₇	—	2	4	no	—
123	C₈×D₇	C₂×Q₈	1	2	yes	—
124	C₈×D₇	#41	1	2	yes	—
125	C₇×C₁₆	—	2	1	no	—
126	C₇×C₁₆	—	2	2	no	—
127	C₇×C₁₆	C₂×Q₈	1	2	yes	complemented
128	C₇⋊C₁₆	—	2	1	no	—
129	C₇⋊C₁₆	—	2	2	no	—
130	C₇⋊C₁₆	C₂×Q₈	1	2	yes	complemented
131	C₂×C₄×Q₈	—	7	1	no	—
132	C₂×M₃₂	—	7	1	no	—
133	C₂×M₃₂	—	7	2	no	—
134	C₂×M₃₂	—	7	1	no	—
135	C₈×Q₈	—	7	1	no	—
136	C₈×Q₈	—	7	2	no	—
137	C₈×Q₈	—	7	1	no	—
138	C₈×Q₈	—	7	1	no	—
139	C₈×Q₈	—	7	2	no	—
140	C₈×Q₈	C₂×D₇	1	1	yes	characteristic
141	C₂×C₄⋊C₈	—	7	2	no	—
142	C₂×C₄⋊C₈	—	7	1	no	—
143	C₂×C₄×C₈	—	7	2	no	—
144	C₂×C₄×C₈	—	7	1	no	—
145	C₄⋊C₁₆	—	7	2	no	—
146	C₄⋊C₁₆	—	7	2	no	—
147	C₄⋊C₁₆	—	7	2	no	—
148	C₄⋊C₁₆	—	7	1	no	—
149	C₄⋊C₁₆	—	7	2	no	—
150	C₄⋊C₁₆	C₂×D₇	1	1	yes	characteristic
151	C₄⋊C₁₆	C₂×D₇	1	2	yes	—
152	#346	—	7	1	no	—
153	#346	—	7	1	no	—
154	#346	—	7	2	no	—
155	#346	C₂×D₇	1	2	yes	—
156	C₄×C₁₆	—	7	2	no	—
157	C₄×C₁₆	—	7	2	no	—
158	C₄×C₁₆	—	7	1	no	—
159	C₄×C₁₆	C₂×D₇	1	1	yes	characteristic
160	C₂²×D₇	C₂²×C₈	1	1	yes	characteristic
161	C₇×Q₈	—	2	2	no	—
162	C₂×C₄×C₇	C₂²×C₈	1	1	yes	characteristic
163	C₂×C₄×C₇	C₂²×C₈	1	2	yes	—
164	C₂×C₄×C₇	C₂²×C₈	1	2	yes	—
165	C₂×C₄×C₇	C₂²×C₈	1	1	yes	characteristic
166	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
167	C₂×Dic₇	C₂²×C₈	1	2	yes	—
168	C₂×Dic₇	C₂²×C₈	1	1	yes	characteristic
169	C₂×Dic₇	C₂²×C₈	1	1	yes	characteristic
170	C₂×Dic₇	C₂²×C₈	1	1	yes	characteristic
171	C₂×Dic₇	C₂²×C₈	1	2	yes	—
172	C₄×D₇	—	2	2	no	—
173	C₄×D₇	—	2	4	no	—
174	C₄×D₇	—	4	1	no	—
175	C₄×D₇	—	4	2	no	—
176	C₄×D₇	C₄×Q₈	1	2	yes	—
177	C₄×D₇	#131	1	2	yes	—
178	Dic₁₄	—	2	4	no	—
179	Dic₁₄	M₃₂	1	4	yes	complemented
180	C₇×C₈	—	2	1	no	—
181	C₇×C₈	—	2	2	no	—
182	C₇×C₈	C₂²×Q₈	1	1	yes	characteristic
183	C₇×C₈	C₂×#41	1	1	yes	characteristic
184	C₇⋊C₈	—	2	2	no	—
185	C₇⋊C₈	—	2	1	no	—
186	C₇⋊C₈	C₄×Q₈	1	2	yes	—
187	C₂²×Q₈	—	7	1	no	—
188	C₂²×C₈	—	7	1	no	—
189	C₂²×C₈	—	7	2	no	—
190	C₂²×C₈	—	7	1	no	—
191	C₄×Q₈	—	7	1	no	—
192	C₄×Q₈	—	7	1	no	—
193	C₄×Q₈	—	7	2	no	—
194	C₄×Q₈	—	7	1	no	—
195	C₄×Q₈	—	7	2	no	—
196	C₄×Q₈	C₄×D₇	1	1	yes	characteristic
197	C₂×C₄⋊C₄	—	7	2	no	—
198	C₂×C₄⋊C₄	—	7	1	no	—
199	C₂×C₄²	—	7	2	no	—
200	C₂×C₄²	—	7	1	no	—
201	M₃₂	—	7	4	no	—
202	M₃₂	—	14	4	no	—
203	M₃₂	—	14	2	no	—
204	C₂×C₁₆	—	7	1	no	—
205	C₂×C₁₆	—	7	2	no	—
206	C₂×C₁₆	—	7	1	no	—
207	C₂×C₁₆	C₂²×D₇	1	2	yes	—
208	C₂×C₁₆	C₂²×D₇	1	1	yes	characteristic
209	C₂×C₁₆	C₂²×D₇	1	1	yes	characteristic
210	C₄⋊C₈	—	7	2	no	—
211	C₄⋊C₈	—	7	2	no	—
212	C₄⋊C₈	—	7	2	no	—
213	C₄⋊C₈	—	7	1	no	—
214	C₄⋊C₈	—	7	2	no	—
215	C₄⋊C₈	C₄×D₇	1	2	yes	—
216	C₄⋊C₈	C₄×D₇	1	1	yes	characteristic
217	C₄×C₈	—	7	2	no	—
218	C₄×C₈	—	7	1	no	—
219	C₄×C₈	—	7	2	no	—
220	C₄×C₈	—	7	1	no	—
221	C₄×C₈	—	7	2	no	—
222	C₄×C₈	—	7	1	no	—
223	C₄×C₈	C₂²×D₇	1	1	yes	characteristic
224	C₄×C₈	C₂²×D₇	1	2	yes	—
225	C₂²×C₇	C₂³×C₈	1	1	yes	characteristic
226	C₂×D₇	—	2	1	no	—
227	C₂×D₇	—	2	1	no	—
228	C₂×D₇	C₈×Q₈	1	2	yes	—
229	C₄×C₇	—	2	2	no	—
230	C₄×C₇	—	2	1	no	—
231	C₄×C₇	—	2	2	no	—
232	C₄×C₇	C₂×M₃₂	1	2	yes	—
233	C₄×C₇	C₂×C₄×Q₈	1	1	yes	characteristic
234	C₄×C₇	C₂×#131	1	1	yes	characteristic
235	Dic₇	—	2	1	no	—
236	Dic₇	—	2	2	no	—
237	Dic₇	—	2	1	no	—
238	Dic₇	C₈×Q₈	1	2	yes	—
239	Dic₇	C₂×M₃₂	1	4	yes	—
240	C₂×Q₈	—	7	1	no	—
241	C₂×Q₈	—	7	2	no	—
242	C₂×Q₈	—	14	4	no	—
243	C₂×Q₈	C₈×D₇	1	1	yes	characteristic
244	C₂²×C₄	—	7	1	no	—
245	C₂²×C₄	—	7	2	no	—
246	C₂²×C₄	—	7	1	no	—
247	C₂²×C₄	—	7	2	no	—
248	C₂²×C₄	—	7	1	no	—
249	C₂×C₈	—	7	1	no	—
250	C₂×C₈	—	7	2	no	—
251	C₂×C₈	—	7	2	no	—
252	C₂×C₈	—	7	2	no	—
253	C₂×C₈	—	7	1	no	—
254	C₂×C₈	—	14	2	no	—
255	C₂×C₈	—	14	4	no	—
256	C₂×C₈	C₂×C₄×D₇	1	1	yes	characteristic
257	C₂×C₈	C₂×C₄×D₇	1	2	yes	—
258	C₂×C₈	C₂³×D₇	1	1	yes	characteristic
259	C₄⋊C₄	—	7	2	no	—
260	C₄⋊C₄	—	7	2	no	—
261	C₄⋊C₄	—	7	2	no	—
262	C₄⋊C₄	—	7	2	no	—
263	C₄⋊C₄	—	7	1	no	—
264	C₄⋊C₄	C₈×D₇	1	2	yes	—
265	C₄⋊C₄	C₈×D₇	1	1	yes	characteristic
266	C₄²	—	7	2	no	—
267	C₄²	—	7	2	no	—
268	C₄²	—	7	1	no	—
269	C₄²	—	7	1	no	—
270	C₄²	—	7	2	no	—
271	C₄²	—	7	1	no	—
272	C₄²	C₂×C₄×D₇	1	2	yes	—
273	C₄²	C₂×C₄×D₇	1	1	yes	characteristic
274	C₁₆	—	2	2	no	—
275	C₁₆	—	2	1	no	—
276	C₁₆	—	7	2	no	—
277	C₁₆	—	14	1	no	—
278	C₁₆	—	14	2	no	—
279	C₁₆	Q₈×D₇	1	2	yes	complemented
280	C₂×C₇	C₂×C₈×Q₈	1	1	yes	characteristic
281	C₂×C₇	C₂²×M₃₂	1	1	yes	characteristic
282	C₂×C₇	#3407	1	1	yes	characteristic
283	D₇	—	2	2	no	—
284	C₂³	—	7	1	no	—
285	Q₈	—	2	2	no	—
286	Q₈	—	7	4	no	—
287	Q₈	—	14	4	no	—
288	C₂×C₄	—	7	1	no	—
289	C₂×C₄	—	7	2	no	—
290	C₂×C₄	—	7	2	no	—
291	C₂×C₄	—	7	2	no	—
292	C₂×C₄	—	7	2	no	—
293	C₂×C₄	—	7	1	no	—
294	C₂×C₄	—	7	1	no	—
295	C₂×C₄	—	14	4	no	—
296	C₂×C₄	—	14	2	no	—
297	C₂×C₄	—	28	2	no	—
298	C₂×C₄	—	28	1	no	—
299	C₂×C₄	C₂×C₈×D₇	1	2	yes	—
300	C₂×C₄	C₂×C₈×D₇	1	1	yes	characteristic
301	C₂×C₄	C₂×C₈×D₇	1	2	yes	—
302	C₂×C₄	C₂×C₈×D₇	1	1	yes	characteristic
303	C₂×C₄	C₂²×C₄×D₇	1	1	yes	characteristic
304	C₈	—	2	1	no	—
305	C₈	—	2	2	no	—
306	C₈	—	7	2	no	—
307	C₈	—	14	2	no	—
308	C₈	—	14	1	no	—
309	C₈	C₂×Q₈×D₇	1	1	yes	characteristic
310	C₈	D₇×#41	1	1	yes	characteristic
311	C₇	Q₈×M₃₂	1	1	yes	characteristic, complemented, Sylow
312	C₂²	—	7	2	no	—
313	C₂²	—	14	1	no	—
314	C₂²	—	14	1	no	—
315	C₂²	C₂²×C₈×D₇	1	1	yes	characteristic
316	C₄	—	2	2	no	—
317	C₄	—	2	2	no	—
318	C₄	—	2	1	no	—
319	C₄	—	7	2	no	—
320	C₄	—	7	4	no	—
321	C₄	—	14	1	no	—
322	C₄	—	14	2	no	—
323	C₄	—	14	1	no	—
324	C₄	D₇×M₃₂	1	2	yes	—
325	C₄	C₄×Q₈×D₇	1	1	yes	characteristic
326	C₄	D₇×#131	1	1	yes	characteristic
327	C₂	—	14	2	no	—
328	C₂	C₈×Q₈×D₇	1	1	yes	characteristic
329	C₂	C₂×D₇×M₃₂	1	1	yes	characteristic
330	C₂	#11739457	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₇
commutator, socle	C₂²×C₇	C₂³×C₈
Fitting	C₇×#2183	C₂
prime radical	C₂²×D₇	C₂²×C₈
cosocle	C₂×C₇×C₈	C₂⁴
hypercenter	#2183	D₇
hypocenter	C₇	Q₈×M₃₂
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₈ ◁ #2183

Complements

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

N	H ≅ G/N	type
2. C₇×#2183	C₂	semidirect
3. #11728703	C₂	semidirect
6. #11727036	C₂	semidirect
7. #11727034	C₂	semidirect
9. #11726822	C₂	semidirect
11. C₁₆×Dic₁₄	C₂	semidirect
79. #6372	Q₈	semidirect
84. #2183	D₇	semidirect
127. C₇×C₁₆	C₂×Q₈	semidirect
130. C₇⋊C₁₆	C₂×Q₈	semidirect
179. Dic₁₄	M₃₂	semidirect
279. C₁₆	Q₈×D₇	semidirect
311. C₇	Q₈×M₃₂	semidirect

Maximal subgroups

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

subgroup	index
C₈×Q₈×D₇	2
C₇×#2183	2
#11728703	2
#11727038	2
#11727036	2
#11727034	2
#11726825	2
#11726822	2
C₄×#6372	2
C₁₆×Dic₁₄	2
Q₈×M₃₂	7