Subgroups — #27143978

Order 12416.

152 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20261137	C₂	1	1	yes	maximal, characteristic, complemented
2	#20261112	C₂	1	1	yes	maximal, characteristic
3	#20260537	C₂	1	4	yes	maximal, complemented
4	#346×C₉₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×D₇₇₆	—	2	1	no	—
6	C₂×D₇₇₆	C₂²	1	2	yes	—
7	#15690807	—	2	1	no	—
8	#15690802	—	2	1	no	—
9	#15690744	C₂²	1	2	yes	—
10	#15690742	—	2	1	no	—
11	C₂×C₁₆×C₉₇	C₂²	1	2	yes	—
12	C₄×C₈×C₉₇	C₂²	1	1	yes	characteristic
13	C₂×D₃₈₈	—	2	2	no	—
14	C₂×D₃₈₈	—	4	2	no	—
15	D₇₇₆	—	2	4	no	—
16	D₇₇₆	—	4	2	no	—
17	C₄⋊Dic₉₇	—	2	2	no	—
18	#13095490	—	4	2	no	—
19	C₂×C₈×C₉₇	D₄	1	1	yes	characteristic
20	C₂×C₈×C₉₇	C₂³	1	1	yes	characteristic
21	C₄²×C₉₇	D₄	1	1	yes	characteristic
22	C₁₆×C₉₇	—	2	2	no	—
23	C₂²×D₉₇	—	4	2	no	—
24	C₂×C₄×C₉₇	D₈	1	2	yes	—
25	C₂×C₄×C₉₇	C₂×D₄	1	1	yes	characteristic
26	C₂×Dic₉₇	—	4	2	no	—
27	D₃₈₈	—	4	2	no	—
28	D₃₈₈	—	4	2	no	—
29	D₃₈₈	—	8	4	no	—
30	C₈×C₉₇	SD₁₆	1	2	yes	—
31	C₈×C₉₇	#41	1	2	yes	—
32	C₂²×C₉₇	C₂×D₈	1	1	yes	characteristic
33	C₂×D₉₇	—	8	2	no	—
34	C₂×D₉₇	—	8	4	no	—
35	C₄×C₉₇	—	2	2	no	—
36	C₄×C₉₇	#124	1	1	yes	characteristic
37	C₄×C₉₇	C₂×SD₁₆	1	1	yes	characteristic
38	Dic₉₇	—	8	2	no	—
39	C₂×C₉₇	#486	1	1	yes	characteristic
40	C₂×C₉₇	#509	1	2	yes	—
41	D₉₇	—	16	2	no	—
42	#2244	—	97	1	no	maximal, Sylow
43	C₉₇	#2244	1	1	yes	characteristic, complemented, Sylow
44	#499	—	97	1	no	—
45	#493	—	97	1	no	—
46	#357	—	97	4	no	—
47	#346	D₉₇	1	1	yes	characteristic, complemented
48	C₂×D₈	—	97	2	no	—
49	C₂×D₈	—	194	1	no	—
50	#127	—	194	1	no	—
51	#125	—	194	1	no	—
52	C₂×C₁₆	C₂×D₉₇	1	2	yes	—
53	#107	—	97	2	no	—
54	#106	—	194	1	no	—
55	C₄×C₈	C₂×D₉₇	1	1	yes	characteristic
56	C₂×D₄	—	388	2	no	—
57	C₂×D₄	—	194	2	no	—
58	D₈	—	388	2	no	—
59	D₈	—	194	4	no	—
60	C₂×C₈	D₃₈₈	1	1	yes	characteristic
61	C₂×C₈	C₂²×D₉₇	1	1	yes	characteristic
62	C₄⋊C₄	—	388	2	no	—
63	C₄⋊C₄	—	194	2	no	—
64	C₄²	D₃₈₈	1	1	yes	characteristic
65	C₁₆	—	2	2	no	—
66	C₂³	—	388	2	no	—
67	D₄	—	776	4	no	—
68	D₄	—	388	2	no	—
69	D₄	—	388	2	no	—
70	C₂×C₄	—	388	2	no	—
71	C₂×C₄	D₇₇₆	1	2	yes	—
72	C₂×C₄	C₂×D₃₈₈	1	1	yes	characteristic
73	C₈	#13095500	1	2	yes	—
74	C₈	#13095511	1	2	yes	—
75	C₂²	—	776	4	no	—
76	C₂²	—	776	2	no	—
77	C₂²	C₂×D₇₇₆	1	1	yes	characteristic
78	C₄	—	2	2	no	—
79	C₄	—	776	2	no	—
80	C₄	#15690798	1	1	yes	characteristic
81	C₄	C₂×#13095500	1	1	yes	characteristic
82	C₂	—	1552	2	no	—
83	C₂	#20261082	1	1	yes	characteristic
84	C₂	#20261173	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₇₇₆
commutator, cosocle	C₂×C₈×C₉₇	C₂³
Frattini	C₂×C₈	C₂²×D₉₇
Fitting	#346×C₉₇	C₂
socle	C₂²×C₉₇	C₂×D₈
prime radical	#20261112	C₂
hypercenter	#346	D₉₇
hypocenter	C₉₇	#2244
LCS	G ▷ C₂×C₈×C₉₇ ▷ C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #346

Complements

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

N	H ≅ G/N	type
1. #20261137	C₂	semidirect
3. #20260537	C₂	semidirect
4. #346×C₉₇	C₂	semidirect
43. C₉₇	#2244	semidirect
47. #346	D₉₇	semidirect

Maximal subgroups

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

subgroup	index
#20261137	2
#20261112	2
#20260537	2
#346×C₉₇	2
#2244	97