Subgroups — #13262806

Order 1792.

282 conjugacy classes of subgroups.

The subgroup lattice has 136 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×#346	C₂	1	1	yes	maximal, characteristic
2	C₄×#6386	C₂	1	1	yes	maximal, characteristic
3	#11727478	C₂	1	1	yes	maximal, characteristic
4	#11725318	C₂	1	4	yes	maximal, complemented
5	C₂²×C₇×C₁₆	C₄	1	2	yes	—
6	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
7	C₂×#6386	C₄	1	2	yes	—
8	#89697	C₄	1	2	yes	—
9	C₇×#346	C₄	1	2	yes	—
10	C₇×#346	C₂²	1	2	yes	—
11	#89256	C₂²	1	2	yes	—
12	C₄×C₇⋊C₁₆	C₄	1	2	yes	—
13	#89250	C₂²	1	2	yes	—
14	C₈×C₇⋊C₈	C₄	1	2	yes	—
15	#11406	—	7	1	no	maximal, Sylow
16	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
17	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
18	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
19	C₂×C₇×C₁₆	—	2	4	no	—
20	C₂×C₇×C₁₆	C₂×C₄	1	4	yes	—
21	C₄×C₇×C₈	C₂×C₄	1	1	yes	characteristic
22	C₄×C₇×C₈	C₂×C₄	1	1	yes	characteristic
23	C₄×C₇×C₈	C₂×C₄	1	1	yes	characteristic
24	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
25	C₂²⋊(C₇⋊C₈)	—	2	2	no	—
26	#6386	C₈	1	8	yes	complemented
27	C₂×C₇⋊C₁₆	C₂×C₄	1	4	yes	—
28	C₄⋊(C₇⋊C₈)	—	2	2	no	—
29	C₄×C₇⋊C₈	C₂×C₄	1	4	yes	—
30	C₄×M₃₂	—	7	1	no	—
31	C₂×#346	D₇	1	1	yes	characteristic
32	#1450	—	7	1	no	—
33	#1313	—	7	4	no	—
34	C₂²×C₄×C₇	C₄²	1	1	yes	characteristic
35	C₂²×C₄×C₇	M₁₆	1	2	yes	—
36	C₂×C₇×C₈	C₄²	1	1	yes	characteristic
37	C₂×C₇×C₈	C₄²	1	1	yes	characteristic
38	C₂×C₇×C₈	C₂×C₈	1	4	yes	—
39	C₂×C₇×C₈	M₁₆	1	4	yes	—
40	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
42	C₄²×C₇	C₄²	1	1	yes	characteristic
43	C₄²×C₇	M₁₆	1	2	yes	—
44	C₄²×C₇	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₇⋊C₈	—	2	4	no	—
46	C₇×C₁₆	—	2	4	no	—
47	C₇⋊C₁₆	C₂×C₈	1	8	yes	—
48	C₂×M₃₂	—	7	2	no	—
49	C₂²×C₁₆	Dic₇	1	2	yes	—
50	#431	—	7	2	no	—
51	C₂×C₄×C₈	C₂×D₇	1	1	yes	characteristic
52	#346	—	7	2	no	—
53	#346	Dic₇	1	2	yes	—
54	#346	C₂×D₇	1	2	yes	—
55	C₄×C₁₆	—	7	2	no	—
56	#322	—	7	2	no	—
57	C₈²	—	7	2	no	—
58	C₂³×C₇	#97	1	1	yes	characteristic
59	C₂×C₄×C₇	—	4	2	no	—
60	C₂×C₄×C₇	C₄×C₈	1	2	yes	—
61	C₂×C₄×C₇	#97	1	2	yes	—
62	C₂×C₄×C₇	#97	1	2	yes	—
63	C₂×C₄×C₇	#97	1	1	yes	characteristic
64	C₂×C₄×C₇	C₂×C₄²	1	1	yes	characteristic
65	C₂×C₄×C₇	C₂×M₁₆	1	2	yes	—
66	C₇×C₈	C₄×C₈	1	2	yes	—
67	C₇×C₈	#97	1	2	yes	—
68	C₇×C₈	C₂²×C₈	1	2	yes	—
69	C₇×C₈	C₂×M₁₆	1	2	yes	—
70	C₇⋊C₈	—	4	4	no	—
71	C₂²×C₈	C₄×D₇	1	1	yes	characteristic
72	C₂²×C₈	C₂×Dic₇	1	1	yes	characteristic
73	C₂×C₄²	C₄×D₇	1	1	yes	characteristic
74	M₃₂	—	7	8	no	—
75	C₂×C₁₆	—	2	4	no	—
76	C₂×C₁₆	—	7	4	no	—
77	C₂×C₁₆	C₂×Dic₇	1	4	yes	—
78	C₄⋊C₈	—	14	2	no	—
79	C₂²⋊C₈	—	14	2	no	—
80	C₄×C₈	—	7	4	no	—
81	C₄×C₈	C₄×D₇	1	1	yes	characteristic
82	C₄×C₈	C₄×D₇	1	1	yes	characteristic
83	C₄×C₈	C₂×Dic₇	1	1	yes	characteristic
84	C₄×C₈	C₂²×D₇	1	1	yes	characteristic
85	C₂²×C₇	—	2	2	no	—
86	C₂²×C₇	#322	1	2	yes	—
87	C₂²×C₇	C₂×#97	1	1	yes	characteristic
88	C₄×C₇	—	2	2	no	—
89	C₄×C₇	—	2	2	no	—
90	C₄×C₇	#322	1	2	yes	—
91	C₄×C₇	C₂×C₄×C₈	1	1	yes	characteristic
92	C₄×C₇	C₂×#97	1	1	yes	characteristic
93	C₂²×C₄	#1131	1	2	yes	—
94	C₂²×C₄	C₄×Dic₇	1	1	yes	characteristic
95	C₂×C₈	—	14	4	no	—
96	C₂×C₈	C₈×D₇	1	4	yes	—
97	C₂×C₈	#1136	1	4	yes	—
98	C₂×C₈	C₄×Dic₇	1	1	yes	characteristic
99	C₂×C₈	C₄×Dic₇	1	1	yes	characteristic
100	C₂×C₈	C₂×C₄×D₇	1	1	yes	characteristic
101	C₂×C₈	C₂²×Dic₇	1	1	yes	characteristic
102	C₄²	#1131	1	2	yes	—
103	C₄²	C₄×Dic₇	1	1	yes	characteristic
104	C₄²	C₂×C₄×D₇	1	1	yes	characteristic
105	C₁₆	—	2	4	no	—
106	C₁₆	—	7	8	no	—
107	C₂×C₇	—	2	2	no	—
108	C₂×C₇	C₂×#322	1	1	yes	characteristic
109	C₂×C₇	#2110	1	2	yes	—
110	C₂³	#6389	1	1	yes	characteristic
111	C₂×C₄	—	4	2	no	—
112	C₂×C₄	#6377	1	2	yes	—
113	C₂×C₄	C₈×Dic₇	1	2	yes	—
114	C₂×C₄	#6389	1	2	yes	—
115	C₂×C₄	#6389	1	1	yes	characteristic
116	C₂×C₄	C₂×#1131	1	2	yes	—
117	C₂×C₄	C₂×C₄×Dic₇	1	1	yes	characteristic
118	C₈	—	28	4	no	—
119	C₈	#6377	1	2	yes	—
120	C₈	C₈×Dic₇	1	2	yes	—
121	C₈	C₂×C₈×D₇	1	2	yes	—
122	C₈	C₂×#1136	1	2	yes	—
123	C₇	#11406	1	1	yes	characteristic, complemented, Sylow
124	C₂²	—	2	2	no	—
125	C₂²	#89249	1	2	yes	—
126	C₂²	C₂×#6389	1	1	yes	characteristic
127	C₄	—	2	2	no	—
128	C₄	—	2	2	no	—
129	C₄	#89249	1	2	yes	—
130	C₄	C₂×#6377	1	1	yes	characteristic
131	C₄	C₂×C₈×Dic₇	1	1	yes	characteristic
132	C₂	—	2	2	no	—
133	C₂	C₂×#89249	1	1	yes	characteristic
134	C₂	#11729179	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	C₄×Dic₇
commutator	C₄×C₇	C₂×C₄×C₈
Frattini	C₄×C₈	C₂²×D₇
Fitting	C₂×C₇×#346	C₂
socle	C₂²×C₇	C₂×#97
prime radical	C₂³×C₇	#97
cosocle	C₄×C₇×C₈	C₂³
hypercenter	C₂×#346	D₇
hypocenter	C₇	#11406
LCS	G ▷ C₄×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₈ ◁ C₂×#346

Complements

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

N	H ≅ G/N	type
4. #11725318	C₂	semidirect
26. #6386	C₈	semidirect
123. C₇	#11406	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₇×#346	2
C₄×#6386	2
#11727478	2
#11725318	2
#11406	7