Subgroups — #27156698

Order 12416.

100 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20261657	C₂	1	1	yes	maximal, characteristic, complemented
2	C₈×#11715897	C₂	1	1	yes	maximal, characteristic, complemented
3	#20260852	C₂	1	1	yes	maximal, characteristic
4	#15690884	—	2	1	no	—
5	C₄×#11715897	—	2	1	no	—
6	#15690817	—	2	1	no	—
7	#15690746	C₄	1	1	yes	characteristic
8	#15690737	—	2	1	no	—
9	C₈×Dic₉₇	C₂²	1	1	yes	characteristic
10	#135×C₉₇	C₄	1	1	yes	characteristic
11	C₂×#11715897	—	2	2	no	—
12	#13095514	—	4	1	no	—
13	#13095495	—	2	1	no	—
14	#13095495	—	4	1	no	—
15	C₂×#11715895	—	4	1	no	—
16	C₂×#11715895	D₄	1	1	yes	characteristic
17	C₄×Dic₉₇	D₄	1	1	yes	characteristic
18	#41×C₉₇	—	2	1	no	—
19	Q₁₆×C₉₇	C₈	1	1	yes	characteristic, complemented
20	SD₁₆×C₉₇	—	2	1	no	—
21	D₈×C₉₇	C₈	1	1	yes	characteristic, complemented
22	C₂×C₈×C₉₇	C₂×C₄	1	1	yes	characteristic
23	Q₈×C₉₇	—	2	1	no	—
24	D₄×C₉₇	—	2	1	no	—
25	D₄×C₉₇	—	4	1	no	—
26	C₂×C₄×C₉₇	—	4	1	no	—
27	C₂×C₄×C₉₇	C₂²⋊C₄	1	1	yes	characteristic
28	C₂×Dic₉₇	—	2	1	no	—
29	#11715897	—	2	4	no	—
30	C₈×C₉₇	C₂×C₈	1	1	yes	characteristic
31	C₈×C₉₇	M₁₆	1	1	yes	characteristic
32	#11715895	—	4	1	no	—
33	#11715895	D₈	1	1	yes	characteristic
34	#11715895	SD₁₆	1	1	yes	characteristic
35	C₂²×C₉₇	—	4	1	no	—
36	C₂²×C₉₇	#104	1	1	yes	characteristic
37	C₄×C₉₇	—	4	1	no	—
38	C₄×C₉₇	C₂²⋊C₈	1	1	yes	characteristic
39	C₄×C₉₇	#102	1	1	yes	characteristic
40	Dic₉₇	—	2	2	no	—
41	C₂×C₉₇	—	2	1	no	—
42	C₂×C₉₇	—	8	1	no	—
43	C₂×C₉₇	D₄⋊C₈	1	1	yes	characteristic
44	#1335	—	97	1	no	maximal, Sylow
45	C₉₇	#1335	1	1	yes	characteristic, complemented, Sylow
46	#443	—	97	1	no	—
47	#364	—	97	1	no	—
48	C₈²	—	97	1	no	—
49	#135	#87604	1	1	yes	characteristic
50	#131	—	194	1	no	—
51	M₃₂	—	194	1	no	—
52	#108	—	97	1	no	—
53	#104	—	194	1	no	—
54	C₄×C₈	—	97	1	no	—
55	C₄×C₈	—	194	1	no	—
56	#41	—	2	1	no	—
57	Q₁₆	#11715897	1	1	yes	characteristic, complemented
58	SD₁₆	—	2	1	no	—
59	D₈	#11715897	1	1	yes	characteristic, complemented
60	M₁₆	—	388	1	no	—
61	M₁₆	—	194	1	no	—
62	C₂×C₈	—	97	1	no	—
63	C₂×C₈	—	388	1	no	—
64	C₂×C₈	—	194	2	no	—
65	C₂×C₈	C₂×#87604	1	1	yes	characteristic
66	C₄²	—	97	1	no	—
67	C₁₆	—	388	1	no	—
68	Q₈	—	2	1	no	—
69	D₄	—	2	1	no	—
70	D₄	—	4	1	no	—
71	C₂×C₄	—	4	1	no	—
72	C₂×C₄	—	194	1	no	—
73	C₂×C₄	C₂²⋊#87604	1	1	yes	characteristic
74	C₈	—	97	1	no	—
75	C₈	—	97	1	no	—
76	C₈	—	388	1	no	—
77	C₈	—	194	4	no	—
78	C₈	C₂×#11715897	1	1	yes	characteristic
79	C₈	#13095520	1	1	yes	characteristic
80	C₂²	—	4	1	no	—
81	C₂²	#15690874	1	1	yes	characteristic
82	C₄	—	4	1	no	—
83	C₄	—	194	2	no	—
84	C₄	C₂²⋊#11715897	1	1	yes	characteristic
85	C₄	#15690872	1	1	yes	characteristic
86	C₂	—	2	1	no	—
87	C₂	—	8	1	no	—
88	C₂	D₄⋊#11715897	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#15690872
commutator	C₈×C₉₇	C₂×C₈
Frattini	C₂×C₈	C₂×#87604
Fitting, prime radical	#135×C₉₇	C₄
socle	C₂×C₉₇	D₄⋊C₈
cosocle	C₈×Dic₉₇	C₂²
hypercenter	#135	#87604
hypocenter	C₉₇	#1335
LCS	G ▷ C₈×C₉₇ ▷ C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #135

Complements

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

N	H ≅ G/N	type
1. #20261657	C₂	semidirect
2. C₈×#11715897	C₂	semidirect
19. Q₁₆×C₉₇	C₈	semidirect
21. D₈×C₉₇	C₈	semidirect
45. C₉₇	#1335	semidirect
57. Q₁₆	#11715897	semidirect
59. D₈	#11715897	semidirect

Maximal subgroups

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

subgroup	index
#20261657	2
C₈×#11715897	2
#20260852	2
#1335	97