Group #27157452 — subgroups

Subgroups — #27157452

Order 12416.

272 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 98 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20261694	C₂	1	1	yes	maximal, characteristic
2	C₄²×#87604	C₂	1	1	yes	maximal, characteristic
3	#20261600	C₂	1	4	yes	maximal
4	#97×D₉₇	C₂	1	1	yes	maximal, characteristic
5	C₂×#13095519	C₂²	1	2	yes	—
6	C₂×C₄×#87604	C₄	1	4	yes	—
7	C₂×C₄×#87604	C₂²	1	2	yes	—
8	#15690865	C₄	1	4	yes	complemented
9	C₂×C₈×D₉₇	C₂²	1	2	yes	—
10	C₄²×D₉₇	C₂²	1	1	yes	characteristic
11	#15690721	C₄	1	1	yes	characteristic, complemented
12	#15690720	C₄	1	2	yes	complemented
13	#97×C₉₇	C₄	1	1	yes	characteristic, complemented
14	C₂²×#87604	C₂×C₄	1	4	yes	—
15	#13095519	—	2	4	no	—
16	C₂×#11715897	C₂×C₄	1	4	yes	—
17	C₄×#87604	—	2	8	no	—
18	C₄×#87604	—	2	4	no	—
19	C₂×C₄×D₉₇	C₂×C₄	1	2	yes	—
20	C₂×C₄×D₉₇	C₂³	1	1	yes	characteristic
21	C₂×#11715895	C₂×C₄	1	2	yes	—
22	C₈×D₉₇	—	2	4	no	—
23	C₄×Dic₉₇	C₂×C₄	1	1	yes	characteristic
24	C₄×Dic₉₇	C₂×C₄	1	2	yes	—
25	C₂×C₈×C₉₇	C₂×C₄	1	2	yes	—
26	C₄²×C₉₇	C₂×C₄	1	1	yes	characteristic
27	C₂²×D₉₇	C₂²×C₄	1	1	yes	characteristic
28	C₂×#87604	—	2	4	no	—
29	C₂×#87604	—	2	4	no	—
30	C₂×#87604	M₁₆	1	8	yes	—
31	C₂×C₄×C₉₇	C₄²	1	2	yes	—
32	C₂×C₄×C₉₇	C₂²×C₄	1	1	yes	characteristic
33	C₂×Dic₉₇	C₄²	1	2	yes	—
34	C₂×Dic₉₇	C₂²×C₄	1	1	yes	characteristic
35	C₄×D₉₇	—	2	4	no	—
36	C₄×D₉₇	#41	1	4	yes	—
37	#11715897	—	2	4	no	—
38	C₈×C₉₇	—	2	2	no	—
39	#11715895	—	2	2	no	—
40	C₂²×C₉₇	C₂×C₄²	1	1	yes	characteristic
41	C₂×D₉₇	#117	1	2	yes	—
42	C₂×D₉₇	C₂×M₁₆	1	4	yes	—
43	#87604	—	2	8	no	—
44	C₄×C₉₇	—	2	2	no	—
45	C₄×C₉₇	#117	1	2	yes	—
46	Dic₉₇	—	2	2	no	—
47	Dic₉₇	#117	1	2	yes	—
48	C₂×C₉₇	#376	1	1	yes	characteristic
49	C₂×C₉₇	C₄×M₁₆	1	2	yes	—
50	D₉₇	#432	1	4	yes	—
51	#1745	—	97	1	no	maximal, Sylow
52	C₉₇	#1745	1	1	yes	characteristic, complemented, Sylow
53	C₂×#97	—	97	1	no	—
54	C₂×#97	—	97	1	no	—
55	C₄³	—	97	1	no	—
56	#336	—	97	4	no	—
57	C₂²×C₈	—	97	2	no	—
58	C₂²×C₈	—	97	2	no	—
59	C₂×C₄²	—	97	1	no	—
60	C₂×C₄²	—	97	2	no	—
61	C₂×C₄²	—	97	4	no	—
62	#97	—	97	4	no	—
63	#97	—	97	2	no	—
64	#97	—	97	1	no	—
65	#97	#87604	1	1	yes	characteristic, complemented
66	C₂²×C₄	—	97	1	no	—
67	C₂²×C₄	—	97	4	no	—
68	C₂²×C₄	—	97	2	no	—
69	C₂×C₈	—	97	4	no	—
70	C₂×C₈	—	97	2	no	—
71	C₂×C₈	—	194	4	no	—
72	C₂×C₈	—	194	4	no	—
73	C₂×C₈	C₂×#87604	1	2	yes	—
74	C₄²	—	97	2	no	—
75	C₄²	—	97	1	no	—
76	C₄²	—	194	4	no	—
77	C₄²	—	194	8	no	—
78	C₄²	C₂×#87604	1	1	yes	characteristic
79	C₂³	—	97	1	no	—
80	C₂×C₄	—	97	1	no	—
81	C₂×C₄	—	97	4	no	—
82	C₂×C₄	—	97	8	no	—
83	C₂×C₄	—	97	2	no	—
84	C₂×C₄	—	194	4	no	—
85	C₂×C₄	—	194	4	no	—
86	C₂×C₄	—	194	4	no	—
87	C₂×C₄	C₄×#87604	1	2	yes	—
88	C₂×C₄	C₂²×#87604	1	1	yes	characteristic
89	C₈	—	2	2	no	—
90	C₈	—	194	4	no	—
91	C₈	—	194	2	no	—
92	C₂²	—	97	2	no	—
93	C₂²	—	97	4	no	—
94	C₂²	C₂×C₄×#87604	1	1	yes	characteristic
95	C₄	—	2	2	no	—
96	C₄	—	97	2	no	—
97	C₄	—	194	8	no	—
98	C₄	—	194	2	no	—
99	C₄	#15690889	1	2	yes	—
100	C₂	—	97	4	no	—
101	C₂	#20261671	1	1	yes	characteristic
102	C₂	M₁₆×#87604	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20261694

C₂

yes

maximal, characteristic

C₄²×#87604

C₂

yes

maximal, characteristic

#20261600

C₂

yes

maximal

#97×D₉₇

C₂

yes

maximal, characteristic

C₂×#13095519

C₂²

yes

—

C₂×C₄×#87604

C₄

yes

—

C₂×C₄×#87604

C₂²

yes

—

#15690865

C₄

yes

complemented

C₂×C₈×D₉₇

C₂²

yes

—

C₄²×D₉₇

C₂²

yes

characteristic

#15690721

C₄

yes

characteristic, complemented

#15690720

C₄

yes

complemented

#97×C₉₇

C₄

yes

characteristic, complemented

C₂²×#87604

C₂×C₄

yes

—

#13095519

—

C₂×#11715897

C₂×C₄

yes

—

C₄×#87604

—

C₄×#87604

—

C₂×C₄×D₉₇

C₂×C₄

yes

—

C₂×C₄×D₉₇

C₂³

yes

characteristic

C₂×#11715895

C₂×C₄

yes

—

C₈×D₉₇

—

C₄×Dic₉₇

C₂×C₄

yes

characteristic

C₄×Dic₉₇

C₂×C₄

yes

—

C₂×C₈×C₉₇

C₂×C₄

yes

—

C₄²×C₉₇

C₂×C₄

yes

characteristic

C₂²×D₉₇

C₂²×C₄

yes

—

—

M₁₆

yes

—

C₂×C₄×C₉₇

C₄²

yes

—

C₂×C₄×C₉₇

C₂²×C₄

yes

characteristic

C₂×Dic₉₇

C₄²

yes

—

C₂×Dic₉₇

C₂²×C₄

yes

characteristic

C₄×D₉₇

—

C₄×D₉₇

#41

yes

—

#11715897

—

C₈×C₉₇

—

#11715895

—

C₂²×C₉₇

C₂×C₄²

yes

characteristic

C₂×D₉₇

#117

yes

—

C₂×D₉₇

C₂×M₁₆

yes

—

#87604

—

C₄×C₉₇

—

C₄×C₉₇

#117

yes

—

Dic₉₇

—

Dic₉₇

#117

yes

—

C₂×C₉₇

#376

yes

characteristic

C₂×C₉₇

C₄×M₁₆

yes

—

D₉₇

#432

yes

—

#1745

—

maximal, Sylow

C₉₇

#1745

yes

characteristic, complemented, Sylow

C₂×#97

—

C₂×#97

—

C₄³

—

#336

—

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

—

—

—

—

yes

characteristic, complemented

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

194

—

C₂×C₈

—

194

—

C₂×C₈

C₂×#87604

yes

—

C₄²

—

C₄²

—

C₄²

—

194

—

C₄²

—

194

—

C₄²

C₂×#87604

yes

characteristic

C₂³

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

194

—

C₂×C₄

—

194

—

C₂×C₄

—

194

—

C₂×C₄

C₄×#87604

yes

—

C₂×C₄

C₂²×#87604

yes

characteristic

C₈

—

C₈

—

194

—

C₈

—

194

—

C₂²

—

C₂²

—

C₂²

C₂×C₄×#87604

yes

characteristic

C₄

—

C₄

—

C₄

—

194

—

C₄

—

194

—

C₄

#15690889

yes

—

100

C₂

—

101

C₂

#20261671

yes

characteristic

102

C₂

M₁₆×#87604

yes

—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×#87604
commutator, socle	C₂²×C₉₇	C₂×C₄²
Fitting	#97×C₉₇	C₄
prime radical	C₂²×D₉₇	C₂²×C₄
cosocle	C₂×C₄×D₉₇	C₂³
hypercenter	#97	#87604
hypocenter	C₉₇	#1745
LCS	G ▷ C₂²×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂×C₄ ◁ #97

subgroup

G/H

center, Frattini

C₂×C₄

C₂²×#87604

commutator, socle

C₂²×C₉₇

C₂×C₄²

Fitting

#97×C₉₇

C₄

prime radical

C₂²×D₉₇

C₂²×C₄

cosocle

C₂×C₄×D₉₇

C₂³

C₉₇

G ▷ C₂²×C₉₇ ▷ C₉₇

C₁ ◁ C₂×C₄ ◁ #97

N	H ≅ G/N	type
8. #15690865	C₄	semidirect
11. #15690721	C₄	semidirect
12. #15690720	C₄	semidirect
13. #97×C₉₇	C₄	semidirect
52. C₉₇	#1745	semidirect
65. #97	#87604	semidirect

H ≅ G/N

type

8. #15690865

C₄

semidirect

11. #15690721

C₄

semidirect

12. #15690720

C₄

semidirect

13. #97×C₉₇

C₄

semidirect

52. C₉₇

#1745

semidirect

65. #97

#87604

semidirect

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

subgroup	index
#20261694	2
C₄²×#87604	2
#20261600	2
#97×D₉₇	2
#1745	97

subgroup

index

#20261694

C₄²×#87604

#20261600

#97×D₉₇

#1745

Subgroups — #27157452

Conjugacy classes of subgroups

Complements

Maximal subgroups