Subgroups — #27139709

Order 12416.

100 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20260556	C₂	1	1	yes	maximal, characteristic, complemented
2	C₈×#11715895	C₂	1	1	yes	maximal, characteristic, complemented
3	#443×C₉₇	C₂	1	1	yes	maximal, characteristic
4	#15690750	—	2	1	no	—
5	C₄×#11715895	—	2	1	no	—
6	#135×C₉₇	C₄	1	1	yes	characteristic
7	#131×C₉₇	—	2	1	no	—
8	#108×C₉₇	C₄	1	1	yes	characteristic
9	#104×C₉₇	—	2	1	no	—
10	C₄×C₈×C₉₇	C₂²	1	1	yes	characteristic
11	C₂×#11715895	—	2	2	no	—
12	#13095486	—	4	1	no	—
13	#41×C₉₇	—	2	1	no	—
14	Q₁₆×C₉₇	C₈	1	1	yes	characteristic, complemented
15	SD₁₆×C₉₇	—	2	1	no	—
16	D₈×C₉₇	C₈	1	1	yes	characteristic, complemented
17	M₁₆×C₉₇	—	2	1	no	—
18	M₁₆×C₉₇	—	4	1	no	—
19	C₂×C₈×C₉₇	—	4	1	no	—
20	C₂×C₈×C₉₇	C₂×C₄	1	1	yes	characteristic
21	C₂×C₈×C₉₇	D₄	1	1	yes	characteristic
22	C₄²×C₉₇	D₄	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	—	2	1	no	—
27	C₂×C₄×C₉₇	—	4	1	no	—
28	C₂×C₄×C₉₇	C₂²⋊C₄	1	1	yes	characteristic
29	C₈×C₉₇	—	4	1	no	—
30	C₈×C₉₇	C₂×C₈	1	1	yes	characteristic
31	C₈×C₉₇	M₁₆	1	1	yes	characteristic
32	C₈×C₉₇	D₈	1	1	yes	characteristic
33	C₈×C₉₇	SD₁₆	1	1	yes	characteristic
34	#11715895	—	2	4	no	—
35	C₂²×C₉₇	—	4	1	no	—
36	C₂²×C₉₇	#104	1	1	yes	characteristic
37	C₄×C₉₇	—	2	2	no	—
38	C₄×C₉₇	—	4	1	no	—
39	C₄×C₉₇	C₂²⋊C₈	1	1	yes	characteristic
40	C₄×C₉₇	#102	1	1	yes	characteristic
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	D₉₇	1	1	yes	characteristic
47	#364	—	97	1	no	—
48	C₈²	—	97	1	no	—
49	#135	Dic₉₇	1	1	yes	characteristic
50	#131	—	2	1	no	—
51	M₃₂	—	194	1	no	—
52	#108	Dic₉₇	1	1	yes	characteristic
53	#104	—	2	1	no	—
54	C₄×C₈	—	194	1	no	—
55	C₄×C₈	C₂×D₉₇	1	1	yes	characteristic
56	#41	—	2	1	no	—
57	Q₁₆	#11715895	1	1	yes	characteristic, complemented
58	SD₁₆	—	2	1	no	—
59	D₈	#11715895	1	1	yes	characteristic, complemented
60	M₁₆	—	2	1	no	—
61	M₁₆	—	4	1	no	—
62	C₂×C₈	—	4	1	no	—
63	C₂×C₈	—	194	2	no	—
64	C₂×C₈	C₂×Dic₉₇	1	1	yes	characteristic
65	C₂×C₈	C₂²⋊D₉₇	1	1	yes	characteristic
66	C₄²	C₂²⋊D₉₇	1	1	yes	characteristic
67	C₁₆	—	388	1	no	—
68	Q₈	—	2	1	no	—
69	D₄	—	2	1	no	—
70	D₄	—	4	1	no	—
71	C₂×C₄	—	2	1	no	—
72	C₂×C₄	—	4	1	no	—
73	C₂×C₄	C₂²⋊Dic₉₇	1	1	yes	characteristic
74	C₈	—	4	1	no	—
75	C₈	—	194	4	no	—
76	C₈	C₂×#11715895	1	1	yes	characteristic
77	C₈	#13095495	1	1	yes	characteristic
78	C₈	#13095496	1	1	yes	characteristic
79	C₈	#13095499	1	1	yes	characteristic
80	C₂²	—	4	1	no	—
81	C₂²	#15690737	1	1	yes	characteristic
82	C₄	—	2	2	no	—
83	C₄	—	4	1	no	—
84	C₄	C₂²⋊#11715895	1	1	yes	characteristic
85	C₄	#15690731	1	1	yes	characteristic
86	C₂	—	2	1	no	—
87	C₂	—	8	1	no	—
88	C₂	D₄⋊#11715895	1	1	yes	characteristic

	subgroup	G/H
center	C₈	#13095496
commutator	C₈×C₉₇	C₂×C₈
Frattini	C₄×C₈	C₂×D₉₇
Fitting	#443×C₉₇	C₂
socle	C₂×C₉₇	D₄⋊C₈
prime radical	#135×C₉₇	C₄
cosocle	C₄×C₈×C₉₇	C₂²
hypercenter	#443	D₉₇
hypocenter	C₉₇	#1335
LCS	G ▷ C₈×C₉₇ ▷ C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₈ ◁ C₂×C₈ ◁ C₄×C₈ ◁ #443

Complements

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

N	H ≅ G/N	type
1. #20260556	C₂	semidirect
2. C₈×#11715895	C₂	semidirect
14. Q₁₆×C₉₇	C₈	semidirect
16. D₈×C₉₇	C₈	semidirect
45. C₉₇	#1335	semidirect
57. Q₁₆	#11715895	semidirect
59. D₈	#11715895	semidirect

Maximal subgroups

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

subgroup	index
#20260556	2
C₈×#11715895	2
#443×C₉₇	2
#1335	97