Subgroups — #27143982

Order 12416.

132 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20261137	C₂	1	1	yes	maximal, characteristic
2	#20261115	C₂	1	1	yes	maximal, characteristic
3	#20260540	C₂	1	4	yes	maximal
4	#346×C₉₇	C₂	1	1	yes	maximal, characteristic
5	C₂×Dic₃₈₈	—	2	1	no	—
6	C₂×Dic₃₈₈	C₂²	1	2	yes	—
7	#15690810	—	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×Dic₁₉₄	—	2	2	no	—
14	Dic₃₈₈	—	2	4	no	—
15	Dic₃₈₈	—	4	2	no	—
16	C₄⋊Dic₉₇	—	2	2	no	—
17	C₄⋊Dic₉₇	—	4	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	C₂×C₄×C₉₇	D₈	1	2	yes	—
24	C₂×C₄×C₉₇	C₂×D₄	1	1	yes	characteristic
25	C₂×Dic₉₇	—	4	2	no	—
26	C₂×Dic₉₇	—	4	2	no	—
27	Dic₁₉₄	—	4	2	no	—
28	Dic₁₉₄	—	4	2	no	—
29	C₈×C₉₇	SD₁₆	1	2	yes	—
30	C₈×C₉₇	#41	1	2	yes	—
31	C₂²×C₉₇	C₂×D₈	1	1	yes	characteristic
32	C₄×C₉₇	—	2	2	no	—
33	C₄×C₉₇	#124	1	1	yes	characteristic
34	C₄×C₉₇	C₂×SD₁₆	1	1	yes	characteristic
35	Dic₉₇	—	8	2	no	—
36	Dic₉₇	—	8	2	no	—
37	C₂×C₉₇	#486	1	1	yes	characteristic
38	C₂×C₉₇	#510	1	2	yes	—
39	#2245	—	97	1	no	maximal, Sylow
40	C₉₇	#2245	1	1	yes	characteristic, complemented, Sylow
41	#499	—	97	1	no	—
42	#494	—	97	1	no	—
43	#358	—	97	4	no	—
44	#346	D₉₇	1	1	yes	characteristic
45	C₂×Q₁₆	—	97	2	no	—
46	C₂×Q₁₆	—	194	1	no	—
47	C₄⋊Q₈	—	194	1	no	—
48	#125	—	194	1	no	—
49	C₂×C₁₆	C₂×D₉₇	1	2	yes	—
50	#107	—	97	2	no	—
51	#106	—	194	1	no	—
52	C₄×C₈	C₂×D₉₇	1	1	yes	characteristic
53	C₂×Q₈	—	194	2	no	—
54	Q₁₆	—	388	2	no	—
55	Q₁₆	—	194	4	no	—
56	C₂×C₈	D₃₈₈	1	1	yes	characteristic
57	C₂×C₈	C₂²×D₉₇	1	1	yes	characteristic
58	C₄⋊C₄	—	388	2	no	—
59	C₄⋊C₄	—	388	2	no	—
60	C₄⋊C₄	—	194	2	no	—
61	C₄²	D₃₈₈	1	1	yes	characteristic
62	C₁₆	—	2	2	no	—
63	Q₈	—	388	2	no	—
64	Q₈	—	388	2	no	—
65	C₂×C₄	—	388	2	no	—
66	C₂×C₄	—	388	2	no	—
67	C₂×C₄	D₇₇₆	1	2	yes	—
68	C₂×C₄	C₂×D₃₈₈	1	1	yes	characteristic
69	C₈	#13095500	1	2	yes	—
70	C₈	#13095511	1	2	yes	—
71	C₂²	C₂×D₇₇₆	1	1	yes	characteristic
72	C₄	—	2	2	no	—
73	C₄	—	776	2	no	—
74	C₄	—	776	2	no	—
75	C₄	#15690798	1	1	yes	characteristic
76	C₄	C₂×#13095500	1	1	yes	characteristic
77	C₂	#20261082	1	1	yes	characteristic
78	C₂	#20261178	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, prime radical	C₂²×C₉₇	C₂×D₈
hypercenter	#346	D₉₇
hypocenter	C₉₇	#2245
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
40. C₉₇	#2245	semidirect

Maximal subgroups

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

subgroup	index
#20261137	2
#20261115	2
#20260540	2
#346×C₉₇	2
#2245	97