Subgroups — #27139906

Order 12416.

94 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20261099	C₂	1	1	yes	maximal, characteristic
2	#20260528	C₂	1	1	yes	maximal, characteristic
3	#353×C₉₇	C₂	1	1	yes	maximal, characteristic
4	#15690802	C₄	1	1	yes	characteristic, complemented
5	#15690731	—	2	1	no	—
6	#15690731	—	2	1	no	—
7	#15690725	—	2	1	no	—
8	#15690721	C₄	1	1	yes	characteristic, complemented
9	#127×C₉₇	C₂²	1	1	yes	characteristic
10	#99×C₉₇	—	2	1	no	—
11	C₂×#11715895	—	2	1	no	—
12	C₄⋊Dic₉₇	—	2	1	no	—
13	#13095490	—	4	1	no	—
14	C₂²⋊Dic₉₇	—	4	1	no	—
15	C₂×D₄×C₉₇	—	4	1	no	—
16	C₂×D₄×C₉₇	D₄	1	1	yes	characteristic
17	C₂×D₄×C₉₇	D₄	1	1	yes	characteristic
18	C₂²⋊C₄×C₉₇	—	4	1	no	—
19	C₄²×C₉₇	C₂×C₄	1	1	yes	characteristic
20	C₂³×C₉₇	—	2	1	no	—
21	C₂³×C₉₇	—	2	1	no	—
22	D₄×C₉₇	—	4	1	no	—
23	D₄×C₉₇	—	4	1	no	—
24	D₄×C₉₇	—	8	2	no	—
25	C₂×C₄×C₉₇	—	2	1	no	—
26	C₂×C₄×C₉₇	—	8	1	no	—
27	C₂×C₄×C₉₇	C₂²⋊C₄	1	1	yes	characteristic
28	C₂×Dic₉₇	—	4	1	no	—
29	C₂×Dic₉₇	—	8	1	no	—
30	#11715895	—	4	1	no	—
31	C₂²×C₉₇	—	4	1	no	—
32	C₂²×C₉₇	—	4	1	no	—
33	C₂²×C₉₇	—	8	1	no	—
34	C₂²×C₉₇	—	8	1	no	—
35	C₂²×C₉₇	#99	1	1	yes	characteristic
36	C₄×C₉₇	—	2	1	no	—
37	C₄×C₉₇	—	4	1	no	—
38	C₄×C₉₇	—	16	1	no	—
39	Dic₉₇	—	8	1	no	—
40	Dic₉₇	—	16	1	no	—
41	C₂×C₉₇	—	2	1	no	—
42	C₂×C₉₇	—	8	1	no	—
43	C₂×C₉₇	—	8	1	no	—
44	C₂×C₉₇	#354	1	1	yes	characteristic
45	#1412	—	97	1	no	maximal, Sylow
46	C₉₇	#1412	1	1	yes	characteristic, complemented, Sylow
47	#490	—	97	1	no	—
48	#353	—	97	1	no	—
49	#353	D₉₇	1	1	yes	characteristic
50	#127	C₂×D₉₇	1	1	yes	characteristic
51	#125	—	97	1	no	—
52	#102	—	194	1	no	—
53	#102	—	194	1	no	—
54	#99	—	2	1	no	—
55	#99	—	194	1	no	—
56	#97	—	97	1	no	—
57	C₂×D₄	—	4	1	no	—
58	C₂×D₄	D₃₈₈	1	1	yes	characteristic
59	C₂×D₄	C₂²⋊D₉₇	1	1	yes	characteristic
60	C₂×C₈	—	194	1	no	—
61	C₄⋊C₄	—	388	1	no	—
62	C₄⋊C₄	—	194	1	no	—
63	C₂²⋊C₄	—	4	1	no	—
64	C₂²⋊C₄	—	388	1	no	—
65	C₄²	C₄×D₉₇	1	1	yes	characteristic
66	C₂³	—	2	1	no	—
67	C₂³	—	2	1	no	—
68	D₄	—	4	1	no	—
69	D₄	—	4	1	no	—
70	D₄	—	8	2	no	—
71	C₂×C₄	—	2	1	no	—
72	C₂×C₄	—	8	1	no	—
73	C₂×C₄	—	776	1	no	—
74	C₂×C₄	—	388	1	no	—
75	C₂×C₄	#13095488	1	1	yes	characteristic
76	C₈	—	388	1	no	—
77	C₂²	—	4	1	no	—
78	C₂²	—	4	1	no	—
79	C₂²	—	8	1	no	—
80	C₂²	—	8	1	no	—
81	C₂²	#15690724	1	1	yes	characteristic
82	C₄	—	2	1	no	—
83	C₄	—	4	1	no	—
84	C₄	—	776	1	no	—
85	C₄	—	16	1	no	—
86	C₄	—	1552	1	no	—
87	C₂	—	2	1	no	—
88	C₂	—	8	1	no	—
89	C₂	—	8	1	no	—
90	C₂	#20260529	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#20260529
commutator	C₄²×C₉₇	C₂×C₄
Frattini	#127	C₂×D₉₇
Fitting	#353×C₉₇	C₂
socle	C₂×C₉₇	#354
prime radical, cosocle	#127×C₉₇	C₂²
hypercenter	#353	D₉₇
hypocenter	C₉₇	#1412
LCS	G ▷ C₄²×C₉₇ ▷ C₂×C₄×C₉₇ ▷ C₂²×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×C₄ ◁ #127 ◁ #353

Complements

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

N	H ≅ G/N	type
4. #15690802	C₄	semidirect
8. #15690721	C₄	semidirect
46. C₉₇	#1412	semidirect

Maximal subgroups

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

subgroup	index
#20261099	2
#20260528	2
#353×C₉₇	2
#1412	97