Subgroups — #13507305

Order 1792.

498 conjugacy classes of subgroups.

The subgroup lattice has 256 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#2960	C₂	1	1	yes	maximal, characteristic
2	#11728703	C₂	1	4	yes	maximal, complemented
3	#11728703	C₂	1	2	yes	maximal, complemented
4	Q₈×C₇⋊C₁₆	C₂	1	2	yes	maximal, complemented
5	#11727636	C₂	1	2	yes	maximal
6	#11727635	C₂	1	2	yes	maximal
7	C₂×C₄⋊(C₇⋊C₁₆)	C₂	1	1	yes	maximal, characteristic
8	C₄×#6386	C₂	1	1	yes	maximal, characteristic
9	C₇×#533	C₄	1	1	yes	characteristic
10	C₇×#446	C₄	1	2	yes	—
11	C₇×#446	C₄	1	2	yes	—
12	C₇×C₈×Q₈	C₂²	1	2	yes	—
13	C₇×C₈×Q₈	C₂²	1	2	yes	—
14	C₇×#424	C₄	1	2	yes	—
15	C₇×#423	C₄	1	1	yes	characteristic
16	C₇×#405	C₂²	1	2	yes	—
17	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
18	C₂×#6386	C₂²	1	1	yes	characteristic
19	C₂×#6386	C₂²	1	1	yes	characteristic
20	C₂²×C₇⋊C₁₆	C₂²	1	2	yes	—
21	C₄⋊(C₇⋊C₁₆)	C₂²	1	4	yes	—
22	C₄⋊(C₇⋊C₁₆)	C₂²	1	4	yes	—
23	C₄⋊(C₇⋊C₁₆)	C₂²	1	1	yes	characteristic
24	C₄⋊(C₇⋊C₁₆)	C₂²	1	1	yes	characteristic
25	C₄⋊(C₇⋊C₁₆)	C₂²	1	2	yes	—
26	#89256	C₂²	1	4	yes	—
27	#89256	C₂²	1	2	yes	—
28	C₄×C₇⋊C₁₆	C₂²	1	4	yes	—
29	C₄×C₇⋊C₁₆	C₂²	1	2	yes	—
30	#33476	—	7	1	no	maximal, Sylow
31	C₂×C₇×M₁₆	C₂×C₄	1	2	yes	—
32	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
34	C₇×C₄⋊Q₈	C₈	1	1	yes	characteristic
35	C₇×C₄⋊Q₈	C₈	1	1	yes	characteristic
36	C₇×#125	C₈	1	2	yes	—
37	C₇×C₂²⋊Q₈	C₈	1	2	yes	—
38	C₇×C₂²⋊Q₈	C₈	1	2	yes	—
39	C₄×C₇×Q₈	C₂×C₄	1	2	yes	—
40	C₄×C₇×Q₈	C₂×C₄	1	2	yes	—
41	C₇×#117	C₂×C₄	1	2	yes	—
42	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
43	C₇×C₄⋊C₈	C₂×C₄	1	2	yes	—
44	C₇×C₄⋊C₈	C₂×C₄	1	2	yes	—
45	C₇×C₄⋊C₈	C₂×C₄	1	4	yes	—
46	C₇×C₄⋊C₈	C₂×C₄	1	4	yes	—
47	C₇×#97	C₂×C₄	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	4	yes	—
52	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
53	#6386	—	2	2	no	—
54	#6386	Q₈	1	4	yes	complemented
55	C₂×C₇⋊C₁₆	—	2	4	no	—
56	C₂×C₇⋊C₁₆	C₂³	1	2	yes	—
57	C₂×C₇⋊C₁₆	C₂³	1	2	yes	—
58	C₂×C₇⋊C₁₆	C₂³	1	2	yes	—
59	C₂×C₇⋊C₁₆	C₂³	1	2	yes	—
60	#2960	D₇	1	1	yes	characteristic
61	#2183	—	7	2	no	—
62	#2183	—	7	4	no	—
63	Q₈×C₁₆	—	7	2	no	—
64	#2151	—	7	2	no	—
65	C₂×C₄⋊C₁₆	—	7	1	no	—
66	#2108	—	7	2	no	—
67	C₄×M₃₂	—	7	1	no	—
68	C₂×C₇×Q₈	C₂×C₈	1	2	yes	—
69	C₂×C₇×Q₈	C₂×C₈	1	2	yes	—
70	C₂²×C₄×C₇	C₂×C₈	1	1	yes	characteristic
71	C₂²×C₄×C₇	C₂×C₈	1	1	yes	characteristic
72	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
73	C₇×M₁₆	—	2	4	no	—
74	C₂×C₇×C₈	—	2	2	no	—
75	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
76	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₇×C₈	C₂²×C₄	1	4	yes	—
78	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₇×C₈	C₂×Q₈	1	2	yes	—
80	C₂×C₇×C₈	#41	1	2	yes	—
81	C₂×C₇×C₈	C₂⁴	1	1	yes	characteristic
82	C₇×C₄⋊C₄	C₂×C₈	1	2	yes	—
83	C₇×C₄⋊C₄	C₂×C₈	1	2	yes	—
84	C₇×C₄⋊C₄	C₂×C₈	1	4	yes	—
85	C₇×C₄⋊C₄	C₂×C₈	1	4	yes	—
86	C₇×C₄⋊C₄	C₂×C₈	1	2	yes	—
87	C₇×C₄⋊C₄	C₂×C₈	1	2	yes	—
88	C₇×C₂²⋊C₄	C₂×C₈	1	4	yes	—
89	C₄²×C₇	C₂×C₈	1	1	yes	characteristic
90	C₄²×C₇	C₂×C₈	1	1	yes	characteristic
91	C₄²×C₇	C₂²×C₄	1	1	yes	characteristic
92	C₄²×C₇	C₂²×C₄	1	1	yes	characteristic
93	C₄²×C₇	C₂²×C₄	1	4	yes	—
94	C₇⋊C₁₆	—	2	2	no	—
95	C₇⋊C₁₆	—	2	2	no	—
96	C₇⋊C₁₆	—	2	2	no	—
97	C₇⋊C₁₆	C₂×Q₈	1	4	yes	—
98	#533	Dic₇	1	1	yes	characteristic
99	C₂×M₃₂	—	7	1	no	—
100	C₂×M₃₂	—	7	1	no	—
101	C₂²×C₁₆	—	7	2	no	—
102	#446	Dic₇	1	2	yes	—
103	#446	Dic₇	1	2	yes	—
104	C₈×Q₈	C₂×D₇	1	2	yes	—
105	C₈×Q₈	C₂×D₇	1	2	yes	—
106	#424	Dic₇	1	2	yes	—
107	#423	Dic₇	1	1	yes	characteristic
108	#405	C₂×D₇	1	2	yes	—
109	C₂×C₄×C₈	C₂×D₇	1	1	yes	characteristic
110	C₄⋊C₁₆	—	7	1	no	—
111	C₄⋊C₁₆	—	7	1	no	—
112	C₄⋊C₁₆	—	7	2	no	—
113	C₄⋊C₁₆	—	7	4	no	—
114	C₄⋊C₁₆	—	7	4	no	—
115	#346	—	7	2	no	—
116	#346	—	7	4	no	—
117	C₄×C₁₆	—	7	2	no	—
118	C₄×C₁₆	—	7	4	no	—
119	C₂³×C₇	C₂²×C₈	1	1	yes	characteristic
120	C₇×Q₈	—	2	4	no	—
121	C₇×Q₈	—	4	2	no	—
122	C₂×C₄×C₇	—	2	2	no	—
123	C₂×C₄×C₇	—	4	1	no	—
124	C₂×C₄×C₇	C₄×Q₈	1	2	yes	—
125	C₂×C₄×C₇	C₂²×C₈	1	1	yes	characteristic
126	C₂×C₄×C₇	C₂²×C₈	1	1	yes	characteristic
127	C₂×C₄×C₇	C₂²×C₈	1	1	yes	characteristic
128	C₂×C₄×C₇	C₂²×C₈	1	4	yes	—
129	C₂×C₄×C₇	C₂²×C₈	1	4	yes	—
130	C₂×C₄×C₇	C₂²×C₈	1	1	yes	characteristic
131	C₂×C₄×C₇	C₂²×C₈	1	1	yes	characteristic
132	C₂×C₄×C₇	#131	1	2	yes	—
133	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
134	C₇×C₈	—	2	1	no	—
135	C₇×C₈	—	2	1	no	—
136	C₇×C₈	—	2	4	no	—
137	C₇×C₈	C₄×Q₈	1	2	yes	—
138	C₇×C₈	C₂²×Q₈	1	1	yes	characteristic
139	C₇×C₈	C₂×#41	1	1	yes	characteristic
140	C₂×M₁₆	C₂×Dic₇	1	2	yes	—
141	C₂²×C₈	C₂×Dic₇	1	1	yes	characteristic
142	C₂²×C₈	C₂²×D₇	1	1	yes	characteristic
143	C₄⋊Q₈	C₇⋊C₈	1	1	yes	characteristic
144	C₄⋊Q₈	C₇⋊C₈	1	1	yes	characteristic
145	#125	C₇⋊C₈	1	2	yes	—
146	C₂²⋊Q₈	C₇⋊C₈	1	2	yes	—
147	C₂²⋊Q₈	C₇⋊C₈	1	2	yes	—
148	C₄×Q₈	C₂×Dic₇	1	2	yes	—
149	C₄×Q₈	C₂×Dic₇	1	2	yes	—
150	#117	C₂×Dic₇	1	2	yes	—
151	C₂×C₄²	C₂×Dic₇	1	1	yes	characteristic
152	M₃₂	—	7	4	no	—
153	M₃₂	—	14	2	no	—
154	C₂×C₁₆	—	7	2	no	—
155	C₂×C₁₆	—	7	2	no	—
156	C₂×C₁₆	—	7	2	no	—
157	C₂×C₁₆	—	7	2	no	—
158	C₂×C₁₆	—	14	4	no	—
159	C₄⋊C₈	C₂×Dic₇	1	2	yes	—
160	C₄⋊C₈	C₂×Dic₇	1	2	yes	—
161	C₄⋊C₈	C₂×Dic₇	1	4	yes	—
162	C₄⋊C₈	C₂×Dic₇	1	4	yes	—
163	#97	C₂×Dic₇	1	4	yes	—
164	C₄×C₈	C₂×Dic₇	1	1	yes	characteristic
165	C₄×C₈	C₂×Dic₇	1	1	yes	characteristic
166	C₄×C₈	C₂²×D₇	1	1	yes	characteristic
167	C₄×C₈	C₂²×D₇	1	4	yes	—
168	C₄×C₈	C₂²×D₇	1	1	yes	characteristic
169	C₂²×C₇	—	2	1	no	—
170	C₂²×C₇	—	2	1	no	—
171	C₂²×C₇	C₈×Q₈	1	2	yes	—
172	C₂²×C₇	C₂³×C₈	1	1	yes	characteristic
173	C₄×C₇	—	2	1	no	—
174	C₄×C₇	—	2	4	no	—
175	C₄×C₇	—	2	1	no	—
176	C₄×C₇	—	2	1	no	—
177	C₄×C₇	—	2	4	no	—
178	C₄×C₇	C₈×Q₈	1	2	yes	—
179	C₄×C₇	#504	1	2	yes	—
180	C₄×C₇	C₂×C₄×Q₈	1	1	yes	characteristic
181	C₄×C₇	C₂×#131	1	1	yes	characteristic
182	C₂×Q₈	C₂×C₇⋊C₈	1	2	yes	—
183	C₂×Q₈	C₂×C₇⋊C₈	1	2	yes	—
184	C₂²×C₄	C₂×C₇⋊C₈	1	1	yes	characteristic
185	C₂²×C₄	C₂×C₇⋊C₈	1	1	yes	characteristic
186	C₂²×C₄	C₂²×Dic₇	1	1	yes	characteristic
187	M₁₆	—	2	4	no	—
188	C₂×C₈	—	2	2	no	—
189	C₂×C₈	Q₈×D₇	1	2	yes	—
190	C₂×C₈	#1161	1	2	yes	—
191	C₂×C₈	C₂²×Dic₇	1	1	yes	characteristic
192	C₂×C₈	C₂²×Dic₇	1	1	yes	characteristic
193	C₂×C₈	C₂²×Dic₇	1	4	yes	—
194	C₂×C₈	C₂²×Dic₇	1	1	yes	characteristic
195	C₂×C₈	C₂³×D₇	1	1	yes	characteristic
196	C₄⋊C₄	C₂×C₇⋊C₈	1	2	yes	—
197	C₄⋊C₄	C₂×C₇⋊C₈	1	2	yes	—
198	C₄⋊C₄	C₂×C₇⋊C₈	1	4	yes	—
199	C₄⋊C₄	C₂×C₇⋊C₈	1	4	yes	—
200	C₄⋊C₄	C₂×C₇⋊C₈	1	2	yes	—
201	C₄⋊C₄	C₂×C₇⋊C₈	1	2	yes	—
202	C₂²⋊C₄	C₂×C₇⋊C₈	1	4	yes	—
203	C₄²	C₂×C₇⋊C₈	1	1	yes	characteristic
204	C₄²	C₂×C₇⋊C₈	1	1	yes	characteristic
205	C₄²	C₂²×Dic₇	1	1	yes	characteristic
206	C₄²	C₂²×Dic₇	1	1	yes	characteristic
207	C₄²	C₂²×Dic₇	1	4	yes	—
208	C₁₆	—	7	4	no	—
209	C₁₆	—	14	2	no	—
210	C₁₆	—	14	2	no	—
211	C₁₆	—	14	2	no	—
212	C₂×C₇	—	2	2	no	—
213	C₂×C₇	C₂×C₈×Q₈	1	1	yes	characteristic
214	C₂×C₇	C₂×#504	1	1	yes	characteristic
215	C₂×C₇	#3407	1	1	yes	characteristic
216	C₂³	C₂²×C₇⋊C₈	1	1	yes	characteristic
217	Q₈	—	2	4	no	—
218	Q₈	—	4	2	no	—
219	C₂×C₄	—	2	2	no	—
220	C₂×C₄	—	4	1	no	—
221	C₂×C₄	C₂²×C₇⋊C₈	1	1	yes	characteristic
222	C₂×C₄	C₂²×C₇⋊C₈	1	1	yes	characteristic
223	C₂×C₄	C₂²×C₇⋊C₈	1	1	yes	characteristic
224	C₂×C₄	C₂²×C₇⋊C₈	1	4	yes	—
225	C₂×C₄	C₂²×C₇⋊C₈	1	4	yes	—
226	C₂×C₄	C₂²×C₇⋊C₈	1	1	yes	characteristic
227	C₂×C₄	C₂²×C₇⋊C₈	1	1	yes	characteristic
228	C₂×C₄	Q₈×Dic₇	1	2	yes	—
229	C₂×C₄	#6511	1	2	yes	—
230	C₂×C₄	C₂³×Dic₇	1	1	yes	characteristic
231	C₈	—	2	1	no	—
232	C₈	—	2	1	no	—
233	C₈	—	2	4	no	—
234	C₈	Q₈×Dic₇	1	2	yes	—
235	C₈	C₂×Q₈×D₇	1	1	yes	characteristic
236	C₈	C₂×#1161	1	1	yes	characteristic
237	C₇	#33476	1	1	yes	characteristic, complemented, Sylow
238	C₂²	—	2	1	no	—
239	C₂²	—	2	1	no	—
240	C₂²	Q₈×C₇⋊C₈	1	2	yes	—
241	C₂²	C₂³×C₇⋊C₈	1	1	yes	characteristic
242	C₄	—	2	1	no	—
243	C₄	—	2	4	no	—
244	C₄	—	2	1	no	—
245	C₄	—	2	1	no	—
246	C₄	—	2	4	no	—
247	C₄	Q₈×C₇⋊C₈	1	2	yes	—
248	C₄	#89912	1	2	yes	—
249	C₄	C₂×Q₈×Dic₇	1	1	yes	characteristic
250	C₄	C₂×#6511	1	1	yes	characteristic
251	C₂	—	2	2	no	—
252	C₂	C₂×Q₈×C₇⋊C₈	1	1	yes	characteristic
253	C₂	C₂×#89912	1	1	yes	characteristic
254	C₂	#11741225	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₇
commutator, socle	C₂²×C₇	C₂³×C₈
Fitting	C₇×#2960	C₂
prime radical	C₂³×C₇	C₂²×C₈
cosocle	C₂×C₇×C₈	C₂⁴
hypercenter	#2960	D₇
hypocenter	C₇	#33476
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₈ ◁ #2960

Complements

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

N	H ≅ G/N	type
2. #11728703	C₂	semidirect
3. #11728703	C₂	semidirect
4. Q₈×C₇⋊C₁₆	C₂	semidirect
54. #6386	Q₈	semidirect
237. C₇	#33476	semidirect

Maximal subgroups

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

subgroup	index
C₇×#2960	2
#11728703	2
Q₈×C₇⋊C₁₆	2
#11727636	2
#11727635	2
C₂×C₄⋊(C₇⋊C₁₆)	2
C₄×#6386	2
#33476	7