Subgroups — #27144017

Order 12416.

128 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20261139	C₂	1	2	yes	maximal
2	#20260563	C₂	1	1	yes	maximal, characteristic
3	#20260562	C₂	1	2	yes	maximal
4	C₁₆×Dic₉₇	C₂	1	1	yes	maximal, characteristic
5	#367×C₉₇	C₂	1	1	yes	maximal, characteristic
6	#15690809	—	2	2	no	—
7	C₂×#13095486	C₂²	1	1	yes	characteristic
8	#15690744	C₂²	1	2	yes	—
9	#15690743	—	2	2	no	—
10	C₈×Dic₉₇	C₂²	1	1	yes	characteristic
11	C₂×C₁₆×C₉₇	C₂²	1	1	yes	characteristic
12	#107×C₉₇	C₂²	1	2	yes	—
13	C₂×Dic₁₉₄	—	4	2	no	—
14	C₂×#11715895	D₄	1	1	yes	characteristic
15	C₄⋊Dic₉₇	—	2	2	no	—
16	#13095490	—	4	2	no	—
17	C₄×Dic₉₇	D₄	1	1	yes	characteristic
18	#13095486	Q₈	1	2	yes	complemented
19	C₂×C₈×C₉₇	C₂³	1	1	yes	characteristic
20	C₄⋊C₄×C₉₇	—	2	2	no	—
21	C₁₆×C₉₇	Q₈	1	2	yes	complemented
22	C₂×C₄×C₉₇	—	4	2	no	—
23	C₂×C₄×C₉₇	C₂×D₄	1	1	yes	characteristic
24	C₂×Dic₉₇	—	4	2	no	—
25	C₂×Dic₉₇	D₈	1	2	yes	—
26	Dic₁₉₄	—	8	4	no	—
27	C₈×C₉₇	C₂×Q₈	1	1	yes	characteristic
28	C₈×C₉₇	C₂×Q₈	1	1	yes	characteristic
29	#11715895	Q₁₆	1	2	yes	—
30	C₂²×C₉₇	C₂×D₈	1	1	yes	characteristic
31	C₄×C₉₇	—	8	2	no	—
32	C₄×C₉₇	C₄⋊Q₈	1	1	yes	characteristic
33	C₄×C₉₇	C₂×Q₁₆	1	1	yes	characteristic
34	Dic₉₇	—	8	2	no	—
35	Dic₉₇	SD₃₂	1	4	yes	—
36	C₂×C₉₇	#500	1	1	yes	characteristic
37	C₂×C₉₇	C₂×SD₃₂	1	2	yes	—
38	#2254	—	97	1	no	maximal, Sylow
39	C₉₇	#2254	1	1	yes	characteristic, complemented, Sylow
40	#500	—	97	2	no	—
41	#367	—	97	1	no	—
42	#367	—	97	2	no	—
43	#367	D₉₇	1	1	yes	characteristic
44	C₄×C₁₆	—	97	1	no	—
45	C₄⋊Q₈	—	194	2	no	—
46	C₂×C₁₆	—	97	1	no	—
47	C₂×C₁₆	C₂×D₉₇	1	1	yes	characteristic
48	#107	—	97	2	no	—
49	#107	—	194	2	no	—
50	#107	C₂×D₉₇	1	2	yes	—
51	C₄×C₈	—	97	1	no	—
52	C₂×Q₈	—	388	2	no	—
53	C₂×C₈	—	97	1	no	—
54	C₂×C₈	C₂²×D₉₇	1	1	yes	characteristic
55	C₄⋊C₄	—	2	2	no	—
56	C₄⋊C₄	—	388	2	no	—
57	C₄⋊C₄	—	194	2	no	—
58	C₄²	—	97	1	no	—
59	C₁₆	—	97	2	no	—
60	C₁₆	Dic₁₉₄	1	2	yes	complemented
61	Q₈	—	776	4	no	—
62	C₂×C₄	—	4	2	no	—
63	C₂×C₄	—	97	2	no	—
64	C₂×C₄	—	388	2	no	—
65	C₂×C₄	D₄×D₉₇	1	1	yes	characteristic
66	C₈	—	97	2	no	—
67	C₈	Q₈×D₉₇	1	1	yes	characteristic
68	C₈	C₂×Dic₁₉₄	1	1	yes	characteristic
69	C₂²	D₈×D₉₇	1	1	yes	characteristic
70	C₄	—	8	2	no	—
71	C₄	—	776	2	no	—
72	C₄	—	97	4	no	—
73	C₄	#15690809	1	1	yes	characteristic
74	C₄	Q₁₆×D₉₇	1	1	yes	characteristic
75	C₂	#20261139	1	1	yes	characteristic
76	C₂	SD₃₂×D₉₇	1	2	yes	—

	subgroup	G/H
center	C₂²	D₈×D₉₇
commutator, cosocle	C₂×C₈×C₉₇	C₂³
Frattini	C₂×C₈	C₂²×D₉₇
Fitting	#367×C₉₇	C₂
socle, prime radical	C₂²×C₉₇	C₂×D₈
hypercenter	#367	D₉₇
hypocenter	C₉₇	#2254
LCS	G ▷ C₂×C₈×C₉₇ ▷ C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #367

Complements

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

N	H ≅ G/N	type
18. #13095486	Q₈	semidirect
21. C₁₆×C₉₇	Q₈	semidirect
39. C₉₇	#2254	semidirect
60. C₁₆	Dic₁₉₄	semidirect

Maximal subgroups

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

subgroup	index
#20261139	2
#20260563	2
#20260562	2
C₁₆×Dic₉₇	2
#367×C₉₇	2
#2254	97