Subgroups — #27150130

Order 12416.

1272 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15690807	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#15690806	C₂	1	12	yes	maximal, complemented
3	C₄²×Dic₉₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#127×C₉₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×D₃₈₈	C₂²	1	6	yes	—
6	C₂²×C₂²⋊D₉₇	C₂²	1	16	yes	—
7	#15690807	—	2	4	no	—
8	#15690806	—	2	48	no	—
9	C₂×C₄×Dic₉₇	C₂²	1	6	yes	—
10	C₂²×D₄×C₉₇	C₂²	1	6	yes	—
11	#127×C₉₇	—	2	4	no	—
12	C₂×C₄²×C₉₇	C₂²	1	1	yes	characteristic
13	C₂³×D₉₇	C₂³	1	4	yes	—
14	C₂×D₃₈₈	—	2	24	no	—
15	C₂×D₃₈₈	—	2	12	no	—
16	C₂×C₂²⋊D₉₇	—	2	96	no	—
17	C₂²×Dic₉₇	C₂³	1	4	yes	—
18	C₄×Dic₉₇	D₄	1	24	yes	complemented
19	C₂⁴×C₉₇	C₂³	1	4	yes	—
20	C₂×D₄×C₉₇	—	2	24	no	—
21	C₂×D₄×C₉₇	—	2	12	no	—
22	C₂²×C₄×C₉₇	C₂³	1	3	yes	—
23	C₄²×C₉₇	D₄	1	4	yes	complemented
24	C₂³×C₉₇	—	2	4	no	—
25	C₂³×C₉₇	—	2	24	no	—
26	C₂³×C₉₇	C₂⁴	1	1	yes	characteristic
27	C₂²×D₉₇	—	2	24	no	—
28	C₂²×D₉₇	—	2	4	no	—
29	D₄×C₉₇	—	4	24	no	—
30	C₂×C₄×C₉₇	C₂×D₄	1	12	yes	—
31	C₂×C₄×C₉₇	C₂×D₄	1	6	yes	—
32	C₂²⋊D₉₇	—	4	64	no	—
33	C₂×Dic₉₇	C₂×D₄	1	24	yes	—
34	D₃₈₈	—	4	24	no	—
35	C₂²×C₉₇	—	4	16	no	—
36	C₂²×C₉₇	—	4	12	no	—
37	C₂²×C₉₇	C₂²×D₄	1	6	yes	—
38	C₂²×C₉₇	C₂²×D₄	1	1	yes	characteristic
39	C₂×D₉₇	—	4	12	no	—
40	C₂×D₉₇	—	4	16	no	—
41	C₄×C₉₇	#127	1	12	yes	complemented
42	Dic₉₇	#127	1	16	yes	complemented
43	C₂×C₉₇	—	8	4	no	—
44	C₂×C₉₇	C₂×#127	1	4	yes	—
45	C₂×C₉₇	C₂×#127	1	3	yes	—
46	D₉₇	—	8	4	no	—
47	#2867	—	97	1	no	maximal, Sylow
48	C₉₇	#2867	1	1	yes	characteristic, complemented, Sylow
49	C₂×#127	—	97	12	no	—
50	C₂×#127	—	97	1	no	—
51	C₂×#127	D₉₇	1	1	yes	characteristic, complemented
52	C₄³	—	97	1	no	—
53	C₂²×D₄	—	97	16	no	—
54	C₂²×D₄	—	97	6	no	—
55	C₂²×D₄	C₂×D₉₇	1	6	yes	—
56	#127	—	2	4	no	—
57	#127	—	194	48	no	—
58	#127	—	194	4	no	—
59	C₂×C₄²	—	97	6	no	—
60	C₂×C₄²	C₂×D₉₇	1	1	yes	characteristic
61	C₂⁴	—	97	4	no	—
62	C₂⁴	C₂²×D₉₇	1	4	yes	—
63	C₂×D₄	—	2	24	no	—
64	C₂×D₄	—	2	12	no	—
65	C₂×D₄	—	194	96	no	—
66	C₂×D₄	—	194	24	no	—
67	C₂×D₄	—	194	12	no	—
68	C₂²×C₄	—	97	4	no	—
69	C₂²×C₄	C₂²×D₉₇	1	3	yes	—
70	C₄²	—	97	24	no	—
71	C₄²	C₂²⋊D₉₇	1	4	yes	complemented
72	C₂³	—	2	24	no	—
73	C₂³	—	2	4	no	—
74	C₂³	—	194	24	no	—
75	C₂³	—	194	4	no	—
76	C₂³	C₂³×D₉₇	1	1	yes	characteristic
77	D₄	—	4	24	no	—
78	D₄	—	388	64	no	—
79	D₄	—	388	24	no	—
80	C₂×C₄	—	97	24	no	—
81	C₂×C₄	D₄×D₉₇	1	12	yes	—
82	C₂×C₄	C₂×C₂²⋊D₉₇	1	6	yes	—
83	C₂²	—	4	16	no	—
84	C₂²	—	4	12	no	—
85	C₂²	—	388	16	no	—
86	C₂²	—	388	12	no	—
87	C₂²	C₂×D₄×D₉₇	1	6	yes	—
88	C₂²	C₂²×C₂²⋊D₉₇	1	1	yes	characteristic
89	C₄	—	97	16	no	—
90	C₄	#15690806	1	12	yes	complemented
91	C₂	—	8	4	no	—
92	C₂	—	776	4	no	—
93	C₂	#127×D₉₇	1	4	yes	—
94	C₂	C₂×#15690806	1	3	yes	—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₉₇
commutator, socle, cosocle	C₂³×C₉₇	C₂⁴
Fitting	C₂×#127×C₉₇	C₂
hypercenter	C₂×#127	D₉₇
hypocenter	C₉₇	#2867
LCS	G ▷ C₂³×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂³ ◁ C₂×#127

Complements

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

N	H ≅ G/N	type
1. C₂×#15690807	C₂	semidirect
2. C₂×#15690806	C₂	semidirect
3. C₄²×Dic₉₇	C₂	semidirect
4. C₂×#127×C₉₇	C₂	semidirect
18. C₄×Dic₉₇	D₄	semidirect
23. C₄²×C₉₇	D₄	semidirect
41. C₄×C₉₇	#127	semidirect
42. Dic₉₇	#127	semidirect
48. C₉₇	#2867	semidirect
51. C₂×#127	D₉₇	semidirect
71. C₄²	C₂²⋊D₉₇	semidirect
90. C₄	#15690806	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15690807	2
C₂×#15690806	2
C₄²×Dic₉₇	2
C₂×#127×C₉₇	2
#2867	97