Subgroups — #13243161

Order 1792.

526 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄²×C₇×C₈	C₂	1	1	yes	maximal, characteristic
2	C₂×#89249	C₂	1	4	yes	maximal
3	#11727142	C₂	1	2	yes	maximal
4	C₂×C₄×C₇×C₈	C₄	1	4	yes	—
5	C₂×C₄×C₇×C₈	C₂²	1	2	yes	—
6	C₄³×C₇	C₂²	1	1	yes	characteristic
7	C₂×C₄×C₇⋊C₈	C₂²	1	4	yes	—
8	#89319	C₄	1	8	yes	—
9	#89249	C₄	1	16	yes	complemented
10	#7843	—	7	1	no	maximal, Sylow
11	C₂²×C₇×C₈	C₂×C₄	1	4	yes	—
12	C₂×C₄²×C₇	C₂×C₄	1	4	yes	—
13	C₂×C₄²×C₇	C₂×C₄	1	2	yes	—
14	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
15	C₂²×C₇⋊C₈	C₂×C₄	1	8	yes	—
16	C₄×C₇×C₈	—	2	8	no	—
17	C₄×C₇×C₈	C₂×C₄	1	8	yes	—
18	C₄×C₇⋊C₈	C₂×C₄	1	16	yes	—
19	#1744	—	7	2	no	—
20	C₄²×C₈	D₇	1	1	yes	characteristic
21	C₂×#322	—	7	4	no	—
22	C₂²×C₄×C₇	C₄²	1	4	yes	—
23	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
24	C₂²×C₄×C₇	C₂²×C₄	1	2	yes	—
25	C₂×C₇×C₈	—	2	8	no	—
26	C₂×C₇×C₈	C₄²	1	8	yes	—
27	C₄²×C₇	—	2	8	no	—
28	C₄²×C₇	M₁₆	1	8	yes	—
29	C₄²×C₇	C₂²×C₄	1	4	yes	—
30	C₂×C₇⋊C₈	—	2	16	no	—
31	C₂×C₇⋊C₈	C₄²	1	16	yes	—
32	C₂×C₄×C₈	—	7	4	no	—
33	C₂×C₄×C₈	Dic₇	1	4	yes	—
34	C₂×C₄×C₈	C₂×D₇	1	2	yes	—
35	C₄³	C₂×D₇	1	1	yes	characteristic
36	#336	—	7	8	no	—
37	#322	—	7	16	no	—
38	C₂³×C₇	C₂×C₄²	1	1	yes	characteristic
39	C₂×C₄×C₇	—	2	8	no	—
40	C₂×C₄×C₇	#97	1	8	yes	—
41	C₂×C₄×C₇	C₂×C₄²	1	4	yes	—
42	C₂×C₄×C₇	C₂×C₄²	1	2	yes	—
43	C₂×C₄×C₇	C₂×M₁₆	1	4	yes	—
44	C₂×C₄×C₇	#131	1	8	yes	—
45	C₇×C₈	—	2	8	no	—
46	C₇⋊C₈	—	2	16	no	—
47	C₂²×C₈	—	7	8	no	—
48	C₂²×C₈	C₂×Dic₇	1	4	yes	—
49	C₂×C₄²	C₄×D₇	1	4	yes	—
50	C₂×C₄²	C₂×Dic₇	1	2	yes	—
51	C₂×C₄²	C₂²×D₇	1	1	yes	characteristic
52	C₄×C₈	—	2	8	no	—
53	C₄×C₈	—	7	16	no	—
54	C₄×C₈	C₄×D₇	1	8	yes	—
55	C₂²×C₇	C₄³	1	1	yes	characteristic
56	C₂²×C₇	C₂×#97	1	2	yes	—
57	C₂²×C₇	#405	1	4	yes	—
58	C₄×C₇	—	2	8	no	—
59	C₄×C₇	C₄×M₁₆	1	4	yes	—
60	C₄×C₇	#405	1	8	yes	—
61	C₂²×C₄	C₄×Dic₇	1	4	yes	—
62	C₂²×C₄	C₂×C₄×D₇	1	2	yes	—
63	C₂²×C₄	C₂²×Dic₇	1	1	yes	characteristic
64	C₂×C₈	—	2	8	no	—
65	C₂×C₈	—	7	16	no	—
66	C₂×C₈	—	14	16	no	—
67	C₂×C₈	C₄×Dic₇	1	8	yes	—
68	C₄²	—	2	8	no	—
69	C₄²	#1136	1	8	yes	—
70	C₄²	C₂×C₄×D₇	1	4	yes	—
71	C₂×C₇	#1452	1	4	yes	—
72	C₂×C₇	C₄×#97	1	1	yes	characteristic
73	C₂×C₇	#1726	1	2	yes	—
74	C₂³	C₂×C₄×Dic₇	1	1	yes	characteristic
75	C₂×C₄	—	2	8	no	—
76	C₂×C₄	#6377	1	8	yes	—
77	C₂×C₄	C₄²×D₇	1	4	yes	—
78	C₂×C₄	#6464	1	8	yes	—
79	C₂×C₄	C₂×#1136	1	4	yes	—
80	C₂×C₄	C₂×C₄×Dic₇	1	2	yes	—
81	C₈	—	2	8	no	—
82	C₈	—	14	16	no	—
83	C₇	#7843	1	1	yes	characteristic, complemented, Sylow
84	C₂²	C₂×#6377	1	2	yes	—
85	C₂²	C₄²×Dic₇	1	1	yes	characteristic
86	C₂²	#89873	1	4	yes	—
87	C₄	—	2	8	no	—
88	C₄	#89461	1	8	yes	—
89	C₄	C₄×#1136	1	4	yes	—
90	C₂	C₄×#6377	1	1	yes	characteristic
91	C₂	#11727480	1	4	yes	—
92	C₂	#11727503	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄²	C₂²×D₇
commutator	C₂²×C₇	C₄³
Fitting	C₄²×C₇×C₈	C₂
socle, prime radical	C₂³×C₇	C₂×C₄²
cosocle	C₂×C₄²×C₇	C₂³
hypercenter	C₄²×C₈	D₇
hypocenter	C₇	#7843
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₄² ◁ C₄²×C₈

Complements

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

N	H ≅ G/N	type
9. #89249	C₄	semidirect
83. C₇	#7843	semidirect

Maximal subgroups

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

subgroup	index
C₄²×C₇×C₈	2
C₂×#89249	2
#11727142	2
#7843	7