Subgroups — #27156863

Order 12416.

108 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20261644	C₂	1	1	yes	maximal, characteristic, complemented
2	#20261642	C₂	1	1	yes	maximal, characteristic, complemented
3	#20261122	C₂	1	1	yes	maximal, characteristic
4	#15690871	—	2	1	no	—
5	#15690869	—	2	1	no	—
6	C₂×#13095499	—	2	1	no	—
7	C₂×#13095496	—	2	1	no	—
8	#15690798	C₂²	1	1	yes	characteristic
9	#15690721	C₄	1	1	yes	characteristic, complemented
10	#127×C₉₇	C₄	1	1	yes	characteristic, complemented
11	#13095520	—	4	1	no	—
12	C₂²⋊#87604	—	4	1	no	—
13	C₂×Dic₁₉₄	D₄	1	1	yes	characteristic
14	C₂×D₃₈₈	D₄	1	1	yes	characteristic
15	#13095499	—	4	2	no	—
16	#13095496	—	4	2	no	—
17	C₂×#11715895	—	2	1	no	—
18	#13095488	—	4	1	no	—
19	C₂×D₄×C₉₇	—	2	1	no	—
20	C₂×D₄×C₉₇	—	4	1	no	—
21	C₄²×C₉₇	C₂×C₄	1	1	yes	characteristic
22	C₂³×C₉₇	—	4	1	no	—
23	C₂²×D₉₇	—	2	1	no	—
24	C₂×#87604	—	8	1	no	—
25	D₄×C₉₇	—	4	1	no	—
26	D₄×C₉₇	—	4	1	no	—
27	D₄×C₉₇	—	8	2	no	—
28	C₂×C₄×C₉₇	—	2	1	no	—
29	C₂×C₄×C₉₇	C₂²⋊C₄	1	1	yes	characteristic
30	C₂×Dic₉₇	—	2	1	no	—
31	D₃₈₈	—	4	1	no	—
32	Dic₁₉₄	—	4	1	no	—
33	#11715897	—	8	1	no	—
34	#11715895	—	2	2	no	—
35	C₂²×C₉₇	—	8	2	no	—
36	C₂²×C₉₇	—	8	1	no	—
37	C₂²×C₉₇	#99	1	1	yes	characteristic
38	C₂×D₉₇	—	4	1	no	—
39	C₂×D₉₇	—	8	1	no	—
40	#87604	—	16	1	no	—
41	C₄×C₉₇	—	2	1	no	—
42	C₄×C₉₇	—	4	1	no	—
43	Dic₉₇	—	4	1	no	—
44	C₂×C₉₇	—	2	1	no	—
45	C₂×C₉₇	—	16	1	no	—
46	C₂×C₉₇	#353	1	1	yes	characteristic
47	D₉₇	—	8	1	no	—
48	#1406	—	97	1	no	maximal, Sylow
49	C₉₇	#1406	1	1	yes	characteristic, complemented, Sylow
50	#496	—	97	1	no	—
51	#355	—	97	1	no	—
52	#354	—	97	1	no	—
53	C₂×SD₁₆	—	194	1	no	—
54	C₂×D₈	—	194	1	no	—
55	#127	#87604	1	1	yes	characteristic, complemented
56	#124	—	97	1	no	—
57	#101	—	194	1	no	—
58	#99	—	194	1	no	—
59	#97	—	97	1	no	—
60	C₂×Q₈	—	97	1	no	—
61	C₂×D₄	—	2	1	no	—
62	C₂×D₄	—	4	1	no	—
63	C₂×D₄	—	97	1	no	—
64	SD₁₆	—	388	2	no	—
65	D₈	—	388	2	no	—
66	M₁₆	—	388	1	no	—
67	C₂×C₈	—	194	1	no	—
68	C₂²⋊C₄	—	388	1	no	—
69	C₂²⋊C₄	—	388	1	no	—
70	C₄²	C₂×#87604	1	1	yes	characteristic
71	C₂³	—	4	1	no	—
72	C₂³	—	194	1	no	—
73	Q₈	—	388	1	no	—
74	D₄	—	4	1	no	—
75	D₄	—	4	1	no	—
76	D₄	—	8	2	no	—
77	D₄	—	388	1	no	—
78	C₂×C₄	—	2	1	no	—
79	C₂×C₄	—	776	1	no	—
80	C₂×C₄	—	194	1	no	—
81	C₂×C₄	C₂²⋊#87604	1	1	yes	characteristic
82	C₈	—	776	1	no	—
83	C₈	—	194	2	no	—
84	C₂²	—	8	2	no	—
85	C₂²	—	8	1	no	—
86	C₂²	—	776	1	no	—
87	C₂²	—	388	1	no	—
88	C₂²	#15690868	1	1	yes	characteristic
89	C₄	—	2	1	no	—
90	C₄	—	4	1	no	—
91	C₄	—	1552	1	no	—
92	C₄	—	388	1	no	—
93	C₂	—	2	1	no	—
94	C₂	—	776	1	no	—
95	C₂	—	16	1	no	—
96	C₂	#20261639	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#20261639
commutator	C₄²×C₉₇	C₂×C₄
Frattini	C₄²	C₂×#87604
Fitting	#127×C₉₇	C₄
socle	C₂×C₉₇	#353
prime radical	#20261122	C₂
cosocle	#15690798	C₂²
hypercenter	#127	#87604
hypocenter	C₉₇	#1406
LCS	G ▷ C₄²×C₉₇ ▷ C₂×C₄×C₉₇ ▷ C₂²×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×C₄ ◁ C₄² ◁ #127

Complements

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

N	H ≅ G/N	type
1. #20261644	C₂	semidirect
2. #20261642	C₂	semidirect
9. #15690721	C₄	semidirect
10. #127×C₉₇	C₄	semidirect
49. C₉₇	#1406	semidirect
55. #127	#87604	semidirect

Maximal subgroups

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

subgroup	index
#20261644	2
#20261642	2
#20261122	2
#1406	97