Subgroups — #13507307

Order 1792.

530 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×Dic₁₄	C₂	1	1	yes	maximal, characteristic
2	C₇×#2150	C₂	1	1	yes	maximal, characteristic
3	#11729174	C₂	1	2	yes	maximal
4	Dic₇×M₃₂	C₂	1	2	yes	maximal
5	C₄×#6386	C₂	1	1	yes	maximal, characteristic
6	#11727036	C₂	1	4	yes	maximal, complemented
7	#11727035	C₂	1	2	yes	maximal, complemented
8	#11727034	C₂	1	2	yes	maximal, complemented
9	C₂×C₄×Dic₁₄	C₄	1	1	yes	characteristic
10	C₂×C₇×M₃₂	C₂²	1	2	yes	—
11	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
12	C₂×Dic₇⋊C₈	C₄	1	2	yes	—
13	C₂×C₈×Dic₇	C₂²	1	2	yes	—
14	C₂×#6386	C₂²	1	1	yes	characteristic
15	C₂×#6386	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊(C₇⋊C₈)	C₄	1	1	yes	characteristic
17	C₈×Dic₁₄	C₄	1	1	yes	characteristic
18	C₈×Dic₁₄	C₄	1	2	yes	—
19	C₈×Dic₁₄	C₄	1	1	yes	characteristic
20	C₈×Dic₁₄	C₂²	1	1	yes	characteristic
21	C₈×Dic₁₄	C₂²	1	1	yes	characteristic
22	C₈×Dic₁₄	C₂²	1	2	yes	—
23	C₇×C₄⋊C₁₆	C₂²	1	2	yes	—
24	C₇×C₄⋊C₁₆	C₂²	1	2	yes	—
25	#89297	C₂²	1	4	yes	—
26	Dic₇⋊C₁₆	C₂²	1	4	yes	—
27	Dic₇⋊C₁₆	C₂²	1	4	yes	—
28	C₁₆×Dic₇	C₂²	1	4	yes	—
29	#89256	C₂²	1	2	yes	—
30	C₄×C₇⋊C₁₆	C₂²	1	2	yes	—
31	Q₈×M₃₂	—	7	1	no	maximal, Sylow
32	C₂²×Dic₁₄	C₈	1	1	yes	characteristic
33	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
35	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄⋊Dic₇	C₈	1	1	yes	characteristic
37	C₂×Dic₇⋊C₄	C₈	1	2	yes	—
38	C₂×C₄×Dic₇	C₂×C₄	1	2	yes	—
39	C₂²×C₇⋊C₈	C₂×C₄	1	2	yes	—
40	C₄×Dic₁₄	C₈	1	1	yes	characteristic
41	C₄×Dic₁₄	C₈	1	2	yes	—
42	C₄×Dic₁₄	C₈	1	1	yes	characteristic
43	C₄×Dic₁₄	C₂×C₄	1	1	yes	characteristic
44	C₄×Dic₁₄	C₂×C₄	1	2	yes	—
45	C₄×Dic₁₄	C₂×C₄	1	1	yes	characteristic
46	C₇×M₃₂	—	2	4	no	—
47	C₂×C₇×C₁₆	C₂³	1	4	yes	—
48	C₄×C₇×C₈	C₂×C₄	1	1	yes	characteristic
49	C₄×C₇×C₈	C₂×C₄	1	1	yes	characteristic
50	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
51	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
52	Dic₇⋊C₈	C₂×C₄	1	2	yes	—
53	Dic₇⋊C₈	C₂×C₄	1	2	yes	—
54	Dic₇⋊C₈	C₂×C₄	1	2	yes	—
55	Dic₇⋊C₈	C₂×C₄	1	2	yes	—
56	C₈×Dic₇	C₂×C₄	1	2	yes	—
57	C₈×Dic₇	C₂×C₄	1	2	yes	—
58	C₈×Dic₇	C₂³	1	2	yes	—
59	C₈×Dic₇	C₂³	1	2	yes	—
60	#6386	—	2	2	no	—
61	#6386	Q₈	1	4	yes	complemented
62	C₂×C₇⋊C₁₆	C₂³	1	2	yes	—
63	C₂×C₇⋊C₁₆	C₂³	1	2	yes	—
64	C₄⋊(C₇⋊C₈)	C₂×C₄	1	1	yes	characteristic
65	C₄⋊(C₇⋊C₈)	C₂×C₄	1	1	yes	characteristic
66	C₄⋊(C₇⋊C₈)	C₂×C₄	1	2	yes	—
67	C₂×C₈×Q₈	—	7	1	no	—
68	#2183	—	7	2	no	—
69	#2183	—	7	4	no	—
70	Q₈×C₁₆	—	7	2	no	—
71	#2150	—	7	2	no	—
72	#2150	D₇	1	1	yes	characteristic
73	C₄×M₃₂	—	7	2	no	—
74	C₄×M₃₂	—	7	1	no	—
75	C₂²×C₄×C₇	C₂×C₈	1	1	yes	characteristic
76	C₂²×C₄×C₇	C₂×C₈	1	1	yes	characteristic
77	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
78	C₂²×Dic₇	C₂×C₈	1	2	yes	—
79	C₂²×Dic₇	C₂×C₈	1	2	yes	—
80	C₂×Dic₁₄	—	2	4	no	—
81	C₂×Dic₁₄	C₂×C₈	1	1	yes	characteristic
82	C₂×Dic₁₄	C₂×C₈	1	2	yes	—
83	C₂×Dic₁₄	C₂×C₈	1	1	yes	characteristic
84	C₂×C₇×C₈	—	2	2	no	—
85	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
86	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
87	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
88	C₂×C₇×C₈	C₂×Q₈	1	2	yes	—
89	C₂×C₇×C₈	#41	1	2	yes	—
90	C₂×C₇×C₈	C₂⁴	1	1	yes	characteristic
91	C₄²×C₇	C₂×C₈	1	1	yes	characteristic
92	C₄²×C₇	C₂×C₈	1	1	yes	characteristic
93	C₄²×C₇	C₂²×C₄	1	1	yes	characteristic
94	C₄²×C₇	C₂²×C₄	1	1	yes	characteristic
95	C₄⋊Dic₇	C₂×C₈	1	2	yes	—
96	C₄⋊Dic₇	C₂×C₈	1	1	yes	characteristic
97	C₄⋊Dic₇	C₂×C₈	1	1	yes	characteristic
98	Dic₇⋊C₄	C₂×C₈	1	2	yes	—
99	Dic₇⋊C₄	C₂×C₈	1	2	yes	—
100	Dic₇⋊C₄	C₂×C₈	1	2	yes	—
101	Dic₇⋊C₄	C₂×C₈	1	2	yes	—
102	C₄×Dic₇	C₂×C₈	1	2	yes	—
103	C₄×Dic₇	C₂×C₈	1	2	yes	—
104	C₄×Dic₇	C₂²×C₄	1	2	yes	—
105	C₄×Dic₇	C₂²×C₄	1	2	yes	—
106	C₂×C₇⋊C₈	—	2	4	no	—
107	C₂×C₇⋊C₈	C₂²×C₄	1	2	yes	—
108	C₂×C₇⋊C₈	C₂²×C₄	1	2	yes	—
109	C₇×C₁₆	—	2	4	no	—
110	C₇⋊C₁₆	—	2	2	no	—
111	C₇⋊C₁₆	C₂×Q₈	1	4	yes	complemented
112	C₂×C₄×Q₈	—	7	1	no	—
113	C₂×M₃₂	—	7	1	no	—
114	C₂×M₃₂	—	7	1	no	—
115	C₂×M₃₂	C₂×D₇	1	2	yes	—
116	C₈×Q₈	—	7	2	no	—
117	C₈×Q₈	—	7	1	no	—
118	C₈×Q₈	—	7	1	no	—
119	C₈×Q₈	—	7	1	no	—
120	C₈×Q₈	—	7	2	no	—
121	C₈×Q₈	—	7	1	no	—
122	C₂×C₄⋊C₈	—	7	2	no	—
123	C₂×C₄⋊C₈	—	7	1	no	—
124	C₂×C₄×C₈	—	7	2	no	—
125	C₂×C₄×C₈	C₂×D₇	1	1	yes	characteristic
126	C₄⋊C₁₆	—	7	4	no	—
127	C₄⋊C₁₆	—	7	4	no	—
128	C₄⋊C₁₆	C₂×D₇	1	2	yes	—
129	C₄⋊C₁₆	C₂×D₇	1	2	yes	—
130	#346	—	7	4	no	—
131	#346	—	7	2	no	—
132	C₄×C₁₆	—	7	4	no	—
133	C₄×C₁₆	—	7	2	no	—
134	C₂³×C₇	C₂²×C₈	1	1	yes	characteristic
135	C₂×C₄×C₇	—	2	2	no	—
136	C₂×C₄×C₇	—	4	1	no	—
137	C₂×C₄×C₇	C₄×Q₈	1	2	yes	—
138	C₂×C₄×C₇	C₂²×C₈	1	1	yes	characteristic
139	C₂×C₄×C₇	C₂²×C₈	1	1	yes	characteristic
140	C₂×C₄×C₇	C₂²×C₈	1	1	yes	characteristic
141	C₂×C₄×C₇	C₂²×C₈	1	1	yes	characteristic
142	C₂×C₄×C₇	C₂²×C₈	1	1	yes	characteristic
143	C₂×C₄×C₇	#131	1	2	yes	—
144	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
145	C₂×Dic₇	—	2	4	no	—
146	C₂×Dic₇	—	4	2	no	—
147	C₂×Dic₇	C₂²×C₈	1	2	yes	—
148	C₂×Dic₇	C₂²×C₈	1	2	yes	—
149	C₂×Dic₇	C₂²×C₈	1	2	yes	—
150	C₂×Dic₇	C₂²×C₈	1	2	yes	—
151	Dic₁₄	—	2	2	no	—
152	Dic₁₄	—	2	4	no	—
153	Dic₁₄	M₃₂	1	4	yes	complemented
154	C₇×C₈	—	2	1	no	—
155	C₇×C₈	—	2	1	no	—
156	C₇×C₈	C₄×Q₈	1	2	yes	—
157	C₇×C₈	C₂²×Q₈	1	1	yes	characteristic
158	C₇×C₈	C₂×#41	1	1	yes	characteristic
159	C₇⋊C₈	—	2	2	no	—
160	C₇⋊C₈	—	2	2	no	—
161	C₂²×Q₈	—	7	1	no	—
162	C₂²×C₈	—	7	2	no	—
163	C₂²×C₈	C₄×D₇	1	1	yes	characteristic
164	C₂²×C₈	C₂²×D₇	1	1	yes	characteristic
165	C₄×Q₈	—	7	1	no	—
166	C₄×Q₈	—	7	2	no	—
167	C₄×Q₈	—	7	1	no	—
168	C₄×Q₈	—	7	1	no	—
169	C₄×Q₈	—	7	2	no	—
170	C₄×Q₈	—	7	1	no	—
171	C₂×C₄⋊C₄	—	7	2	no	—
172	C₂×C₄⋊C₄	—	7	1	no	—
173	C₂×C₄²	—	7	2	no	—
174	C₂×C₄²	C₄×D₇	1	1	yes	characteristic
175	M₃₂	—	2	4	no	—
176	M₃₂	—	7	4	no	—
177	M₃₂	—	14	2	no	—
178	C₂×C₁₆	—	7	2	no	—
179	C₂×C₁₆	—	7	2	no	—
180	C₂×C₁₆	C₂²×D₇	1	4	yes	—
181	C₄⋊C₈	—	7	2	no	—
182	C₄⋊C₈	—	7	2	no	—
183	C₄⋊C₈	—	7	2	no	—
184	C₄⋊C₈	—	7	2	no	—
185	C₄⋊C₈	—	7	1	no	—
186	C₄⋊C₈	—	7	2	no	—
187	C₄⋊C₈	—	7	1	no	—
188	C₄×C₈	—	7	2	no	—
189	C₄×C₈	—	7	2	no	—
190	C₄×C₈	—	7	2	no	—
191	C₄×C₈	—	7	2	no	—
192	C₄×C₈	C₄×D₇	1	1	yes	characteristic
193	C₄×C₈	C₄×D₇	1	1	yes	characteristic
194	C₄×C₈	C₂²×D₇	1	1	yes	characteristic
195	C₄×C₈	C₂²×D₇	1	1	yes	characteristic
196	C₂²×C₇	—	2	1	no	—
197	C₂²×C₇	—	2	1	no	—
198	C₂²×C₇	C₈×Q₈	1	2	yes	—
199	C₂²×C₇	C₂³×C₈	1	1	yes	characteristic
200	C₄×C₇	—	2	1	no	—
201	C₄×C₇	—	2	1	no	—
202	C₄×C₇	—	2	1	no	—
203	C₄×C₇	C₈×Q₈	1	2	yes	—
204	C₄×C₇	C₂×M₃₂	1	2	yes	—
205	C₄×C₇	C₂×C₄×Q₈	1	1	yes	characteristic
206	C₄×C₇	C₂×#131	1	1	yes	characteristic
207	Dic₇	—	2	2	no	—
208	Dic₇	—	2	2	no	—
209	Dic₇	—	2	2	no	—
210	Dic₇	C₂×M₃₂	1	4	yes	—
211	C₂×Q₈	—	7	1	no	—
212	C₂×Q₈	—	7	2	no	—
213	C₂×Q₈	—	7	1	no	—
214	C₂×Q₈	—	14	4	no	—
215	C₂²×C₄	—	7	2	no	—
216	C₂²×C₄	—	7	2	no	—
217	C₂²×C₄	C₈×D₇	1	1	yes	characteristic
218	C₂²×C₄	C₈×D₇	1	1	yes	characteristic
219	C₂²×C₄	C₂×C₄×D₇	1	1	yes	characteristic
220	C₂×C₈	—	2	2	no	—
221	C₂×C₈	—	7	2	no	—
222	C₂×C₈	—	7	2	no	—
223	C₂×C₈	—	14	4	no	—
224	C₂×C₈	C₂×C₄×D₇	1	1	yes	characteristic
225	C₂×C₈	C₂×C₄×D₇	1	1	yes	characteristic
226	C₂×C₈	C₂×C₄×D₇	1	1	yes	characteristic
227	C₂×C₈	#1159	1	2	yes	—
228	C₂×C₈	Q₈×D₇	1	2	yes	—
229	C₂×C₈	C₂³×D₇	1	1	yes	characteristic
230	C₄⋊C₄	—	7	2	no	—
231	C₄⋊C₄	—	7	2	no	—
232	C₄⋊C₄	—	7	2	no	—
233	C₄⋊C₄	—	7	2	no	—
234	C₄⋊C₄	—	7	2	no	—
235	C₄⋊C₄	—	7	1	no	—
236	C₄⋊C₄	—	7	1	no	—
237	C₄²	—	7	2	no	—
238	C₄²	—	7	2	no	—
239	C₄²	—	7	2	no	—
240	C₄²	—	7	2	no	—
241	C₄²	C₈×D₇	1	1	yes	characteristic
242	C₄²	C₈×D₇	1	1	yes	characteristic
243	C₄²	C₂×C₄×D₇	1	1	yes	characteristic
244	C₄²	C₂×C₄×D₇	1	1	yes	characteristic
245	C₁₆	—	2	4	no	—
246	C₁₆	—	7	4	no	—
247	C₁₆	—	14	2	no	—
248	C₂×C₇	—	2	2	no	—
249	C₂×C₇	C₂×C₈×Q₈	1	1	yes	characteristic
250	C₂×C₇	C₂²×M₃₂	1	1	yes	characteristic
251	C₂×C₇	#3407	1	1	yes	characteristic
252	C₂³	C₂×C₈×D₇	1	1	yes	characteristic
253	Q₈	—	7	4	no	—
254	Q₈	—	14	4	no	—
255	Q₈	—	14	2	no	—
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	4	1	no	—
258	C₂×C₄	—	7	2	no	—
259	C₂×C₄	—	7	2	no	—
260	C₂×C₄	—	7	2	no	—
261	C₂×C₄	—	7	2	no	—
262	C₂×C₄	—	14	4	no	—
263	C₂×C₄	—	28	2	no	—
264	C₂×C₄	#6450	1	2	yes	—
265	C₂×C₄	C₂×C₈×D₇	1	1	yes	characteristic
266	C₂×C₄	C₂×C₈×D₇	1	1	yes	characteristic
267	C₂×C₄	C₂×C₈×D₇	1	1	yes	characteristic
268	C₂×C₄	C₂×C₈×D₇	1	1	yes	characteristic
269	C₂×C₄	C₂×C₈×D₇	1	1	yes	characteristic
270	C₂×C₄	#6470	1	2	yes	—
271	C₂×C₄	C₂²×C₄×D₇	1	1	yes	characteristic
272	C₈	—	2	1	no	—
273	C₈	—	2	1	no	—
274	C₈	—	14	2	no	—
275	C₈	—	14	2	no	—
276	C₈	#6450	1	2	yes	—
277	C₈	C₂×#1159	1	1	yes	characteristic
278	C₈	C₂×Q₈×D₇	1	1	yes	characteristic
279	C₇	Q₈×M₃₂	1	1	yes	characteristic, complemented, Sylow
280	C₂²	—	2	1	no	—
281	C₂²	—	2	1	no	—
282	C₂²	#89602	1	2	yes	—
283	C₂²	C₂²×C₈×D₇	1	1	yes	characteristic
284	C₄	—	2	1	no	—
285	C₄	—	2	1	no	—
286	C₄	—	2	1	no	—
287	C₄	—	7	4	no	—
288	C₄	—	14	2	no	—
289	C₄	—	14	2	no	—
290	C₄	—	14	2	no	—
291	C₄	#89602	1	2	yes	—
292	C₄	D₇×M₃₂	1	2	yes	—
293	C₄	C₂×#6450	1	1	yes	characteristic
294	C₄	C₂×#6470	1	1	yes	characteristic
295	C₂	—	2	2	no	—
296	C₂	C₂×#89602	1	1	yes	characteristic
297	C₂	C₂×D₇×M₃₂	1	1	yes	characteristic
298	C₂	#11739452	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₇
commutator, socle	C₂²×C₇	C₂³×C₈
Fitting	C₇×#2150	C₂
prime radical	C₂³×C₇	C₂²×C₈
cosocle	C₂×C₇×C₈	C₂⁴
hypercenter	#2150	D₇
hypocenter	C₇	Q₈×M₃₂
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₈ ◁ #2150

Complements

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

N	H ≅ G/N	type
6. #11727036	C₂	semidirect
7. #11727035	C₂	semidirect
8. #11727034	C₂	semidirect
61. #6386	Q₈	semidirect
111. C₇⋊C₁₆	C₂×Q₈	semidirect
153. Dic₁₄	M₃₂	semidirect
279. 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
C₂×C₈×Dic₁₄	2
C₇×#2150	2
#11729174	2
Dic₇×M₃₂	2
C₄×#6386	2
#11727036	2
#11727035	2
#11727034	2
Q₈×M₃₂	7