Subgroups — #13507291

Order 1792.

506 conjugacy classes of subgroups.

The subgroup lattice has 194 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	Q₈×C₇⋊C₁₆	C₂	1	4	yes	maximal, complemented
4	#11727636	C₂	1	2	yes	maximal
5	#11727635	C₂	1	2	yes	maximal
6	#11727497	C₂	1	1	yes	maximal, characteristic
7	C₂×C₄×C₇⋊C₁₆	C₂	1	1	yes	maximal, characteristic
8	C₇×#533	C₄	1	1	yes	characteristic
9	C₇×#446	C₄	1	4	yes	—
10	C₇×C₈×Q₈	C₂²	1	4	yes	—
11	C₇×#424	C₄	1	2	yes	—
12	C₇×#423	C₄	1	1	yes	characteristic
13	C₇×#405	C₂²	1	2	yes	—
14	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
15	C₂×#6386	C₂²	1	2	yes	—
16	C₂²×C₇⋊C₁₆	C₂²	1	1	yes	characteristic
17	C₂²×C₇⋊C₁₆	C₂²	1	1	yes	characteristic
18	C₄⋊(C₇⋊C₁₆)	C₂²	1	8	yes	—
19	C₄⋊(C₇⋊C₁₆)	C₂²	1	4	yes	—
20	#89256	C₂²	1	4	yes	—
21	C₄×C₇⋊C₁₆	C₂²	1	4	yes	—
22	C₄×C₇⋊C₁₆	C₂²	1	2	yes	—
23	C₄×C₇⋊C₁₆	C₂²	1	2	yes	—
24	#33474	—	7	1	no	maximal, Sylow
25	C₂×C₇×M₁₆	C₂×C₄	1	2	yes	—
26	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
28	C₇×C₄⋊Q₈	C₈	1	2	yes	—
29	C₇×#125	C₈	1	2	yes	—
30	C₇×C₂²⋊Q₈	C₈	1	4	yes	—
31	C₄×C₇×Q₈	C₂×C₄	1	4	yes	—
32	C₇×#117	C₂×C₄	1	2	yes	—
33	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
34	C₇×C₄⋊C₈	C₂×C₄	1	4	yes	—
35	C₇×C₄⋊C₈	C₂×C₄	1	8	yes	—
36	C₇×#97	C₂×C₄	1	4	yes	—
37	C₄×C₇×C₈	C₂×C₄	1	2	yes	—
38	C₄×C₇×C₈	C₂³	1	2	yes	—
39	C₄×C₇×C₈	C₂³	1	4	yes	—
40	#6386	—	2	4	no	—
41	C₂×C₇⋊C₁₆	—	2	2	no	—
42	C₂×C₇⋊C₁₆	Q₈	1	4	yes	complemented
43	C₂×C₇⋊C₁₆	C₂³	1	2	yes	—
44	C₂×C₇⋊C₁₆	C₂³	1	4	yes	—
45	C₂×C₇⋊C₁₆	C₂³	1	1	yes	characteristic
46	C₂×C₇⋊C₁₆	C₂³	1	1	yes	characteristic
47	#2960	D₇	1	1	yes	characteristic
48	#2183	—	7	4	no	—
49	Q₈×C₁₆	—	7	4	no	—
50	#2151	—	7	2	no	—
51	#2150	—	7	1	no	—
52	#2108	—	7	2	no	—
53	C₂×C₄×C₁₆	—	7	1	no	—
54	C₂×C₇×Q₈	C₂×C₈	1	4	yes	—
55	C₂²×C₄×C₇	C₂×C₈	1	2	yes	—
56	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
57	C₇×M₁₆	—	2	4	no	—
58	C₂×C₇×C₈	—	2	2	no	—
59	C₂×C₇×C₈	C₂²×C₄	1	2	yes	—
60	C₂×C₇×C₈	C₂²×C₄	1	4	yes	—
61	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
62	C₂×C₇×C₈	C₂×Q₈	1	2	yes	—
63	C₂×C₇×C₈	#41	1	2	yes	—
64	C₂×C₇×C₈	C₂⁴	1	1	yes	characteristic
65	C₇×C₄⋊C₄	C₂×C₈	1	4	yes	—
66	C₇×C₄⋊C₄	C₂×C₈	1	8	yes	—
67	C₇×C₄⋊C₄	C₂×C₈	1	4	yes	—
68	C₇×C₂²⋊C₄	C₂×C₈	1	4	yes	—
69	C₄²×C₇	C₂×C₈	1	1	yes	characteristic
70	C₄²×C₇	C₂×C₈	1	1	yes	characteristic
71	C₄²×C₇	C₂²×C₄	1	2	yes	—
72	C₄²×C₇	C₂²×C₄	1	4	yes	—
73	C₇⋊C₁₆	—	2	2	no	—
74	C₇⋊C₁₆	—	2	4	no	—
75	C₇⋊C₁₆	C₂×Q₈	1	2	yes	—
76	C₇⋊C₁₆	C₂×Q₈	1	2	yes	—
77	#533	Dic₇	1	1	yes	characteristic
78	C₂×M₃₂	—	7	2	no	—
79	C₂²×C₁₆	—	7	1	no	—
80	C₂²×C₁₆	—	7	1	no	—
81	#446	Dic₇	1	4	yes	—
82	C₈×Q₈	C₂×D₇	1	4	yes	—
83	#424	Dic₇	1	2	yes	—
84	#423	Dic₇	1	1	yes	characteristic
85	#405	C₂×D₇	1	2	yes	—
86	C₂×C₄×C₈	C₂×D₇	1	1	yes	characteristic
87	C₄⋊C₁₆	—	7	4	no	—
88	C₄⋊C₁₆	—	7	8	no	—
89	#346	—	7	4	no	—
90	C₄×C₁₆	—	7	2	no	—
91	C₄×C₁₆	—	7	2	no	—
92	C₄×C₁₆	—	7	4	no	—
93	C₂³×C₇	C₂²×C₈	1	1	yes	characteristic
94	C₇×Q₈	—	2	8	no	—
95	C₂×C₄×C₇	—	2	4	no	—
96	C₂×C₄×C₇	C₄×Q₈	1	2	yes	—
97	C₂×C₄×C₇	C₂²×C₈	1	2	yes	—
98	C₂×C₄×C₇	C₂²×C₈	1	8	yes	—
99	C₂×C₄×C₇	C₂²×C₈	1	1	yes	characteristic
100	C₂×C₄×C₇	C₂²×C₈	1	2	yes	—
101	C₂×C₄×C₇	#131	1	2	yes	—
102	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
103	C₇×C₈	—	2	2	no	—
104	C₇×C₈	—	2	4	no	—
105	C₇×C₈	C₄×Q₈	1	2	yes	—
106	C₇×C₈	C₂²×Q₈	1	1	yes	characteristic
107	C₇×C₈	C₂×#41	1	1	yes	characteristic
108	C₂×M₁₆	C₂×Dic₇	1	2	yes	—
109	C₂²×C₈	C₂×Dic₇	1	1	yes	characteristic
110	C₂²×C₈	C₂²×D₇	1	1	yes	characteristic
111	C₄⋊Q₈	C₇⋊C₈	1	2	yes	—
112	#125	C₇⋊C₈	1	2	yes	—
113	C₂²⋊Q₈	C₇⋊C₈	1	4	yes	—
114	C₄×Q₈	C₂×Dic₇	1	4	yes	—
115	#117	C₂×Dic₇	1	2	yes	—
116	C₂×C₄²	C₂×Dic₇	1	1	yes	characteristic
117	M₃₂	—	14	4	no	—
118	C₂×C₁₆	—	7	4	no	—
119	C₂×C₁₆	—	7	1	no	—
120	C₂×C₁₆	—	7	1	no	—
121	C₂×C₁₆	—	7	2	no	—
122	C₂×C₁₆	—	7	4	no	—
123	C₂×C₁₆	—	14	2	no	—
124	C₄⋊C₈	C₂×Dic₇	1	4	yes	—
125	C₄⋊C₈	C₂×Dic₇	1	8	yes	—
126	#97	C₂×Dic₇	1	4	yes	—
127	C₄×C₈	C₂×Dic₇	1	2	yes	—
128	C₄×C₈	C₂²×D₇	1	2	yes	—
129	C₄×C₈	C₂²×D₇	1	4	yes	—
130	C₂²×C₇	—	2	2	no	—
131	C₂²×C₇	C₈×Q₈	1	2	yes	—
132	C₂²×C₇	C₂³×C₈	1	1	yes	characteristic
133	C₄×C₇	—	2	2	no	—
134	C₄×C₇	—	2	8	no	—
135	C₄×C₇	C₈×Q₈	1	2	yes	—
136	C₄×C₇	#504	1	4	yes	—
137	C₄×C₇	C₂×C₄×Q₈	1	1	yes	characteristic
138	C₄×C₇	C₂×#131	1	1	yes	characteristic
139	C₂×Q₈	C₂×C₇⋊C₈	1	4	yes	—
140	C₂²×C₄	C₂×C₇⋊C₈	1	2	yes	—
141	C₂²×C₄	C₂²×Dic₇	1	1	yes	characteristic
142	M₁₆	—	2	4	no	—
143	C₂×C₈	—	2	2	no	—
144	C₂×C₈	Q₈×D₇	1	2	yes	—
145	C₂×C₈	#1161	1	2	yes	—
146	C₂×C₈	C₂²×Dic₇	1	2	yes	—
147	C₂×C₈	C₂²×Dic₇	1	4	yes	—
148	C₂×C₈	C₂²×Dic₇	1	1	yes	characteristic
149	C₂×C₈	C₂³×D₇	1	1	yes	characteristic
150	C₄⋊C₄	C₂×C₇⋊C₈	1	4	yes	—
151	C₄⋊C₄	C₂×C₇⋊C₈	1	8	yes	—
152	C₄⋊C₄	C₂×C₇⋊C₈	1	4	yes	—
153	C₂²⋊C₄	C₂×C₇⋊C₈	1	4	yes	—
154	C₄²	C₂×C₇⋊C₈	1	1	yes	characteristic
155	C₄²	C₂×C₇⋊C₈	1	1	yes	characteristic
156	C₄²	C₂²×Dic₇	1	2	yes	—
157	C₄²	C₂²×Dic₇	1	4	yes	—
158	C₁₆	—	7	2	no	—
159	C₁₆	—	7	2	no	—
160	C₁₆	—	14	2	no	—
161	C₁₆	—	14	4	no	—
162	C₂×C₇	—	2	2	no	—
163	C₂×C₇	C₂×C₈×Q₈	1	1	yes	characteristic
164	C₂×C₇	C₂×#504	1	2	yes	—
165	C₂³	C₂²×C₇⋊C₈	1	1	yes	characteristic
166	Q₈	—	2	8	no	—
167	C₂×C₄	—	2	4	no	—
168	C₂×C₄	C₂²×C₇⋊C₈	1	2	yes	—
169	C₂×C₄	C₂²×C₇⋊C₈	1	8	yes	—
170	C₂×C₄	C₂²×C₇⋊C₈	1	2	yes	—
171	C₂×C₄	C₂²×C₇⋊C₈	1	1	yes	characteristic
172	C₂×C₄	Q₈×Dic₇	1	2	yes	—
173	C₂×C₄	#6511	1	2	yes	—
174	C₂×C₄	C₂³×Dic₇	1	1	yes	characteristic
175	C₈	—	2	2	no	—
176	C₈	—	2	4	no	—
177	C₈	Q₈×Dic₇	1	2	yes	—
178	C₈	C₂×Q₈×D₇	1	1	yes	characteristic
179	C₈	C₂×#1161	1	1	yes	characteristic
180	C₇	#33474	1	1	yes	characteristic, complemented, Sylow
181	C₂²	—	2	2	no	—
182	C₂²	Q₈×C₇⋊C₈	1	2	yes	—
183	C₂²	C₂³×C₇⋊C₈	1	1	yes	characteristic
184	C₄	—	2	2	no	—
185	C₄	—	2	8	no	—
186	C₄	Q₈×C₇⋊C₈	1	2	yes	—
187	C₄	#89912	1	4	yes	—
188	C₄	C₂×Q₈×Dic₇	1	1	yes	characteristic
189	C₄	C₂×#6511	1	1	yes	characteristic
190	C₂	—	2	2	no	—
191	C₂	C₂×Q₈×C₇⋊C₈	1	1	yes	characteristic
192	C₂	C₂×#89912	1	2	yes	—

	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	#33474
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. Q₈×C₇⋊C₁₆	C₂	semidirect
42. C₂×C₇⋊C₁₆	Q₈	semidirect
180. C₇	#33474	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
#11727497	2
C₂×C₄×C₇⋊C₁₆	2
#33474	7