Group #27157653 — subgroups

Subgroups — #27157653

Order 12416.

260 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 114 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	C₂×C₄⋊#11715897	C₂	1	2	yes	maximal
2	#20261601	C₂	1	4	yes	maximal
3	C₄²×Dic₉₇	C₂	1	1	yes	maximal, characteristic
4	C₂²×#11715897	C₂²	1	4	yes	—
5	C₄⋊#11715897	—	2	4	no	—
6	C₂×C₄×Dic₉₇	C₄	1	2	yes	—
7	C₂×C₄×Dic₉₇	C₄	1	1	yes	characteristic
8	C₂×C₄×Dic₉₇	C₂²	1	2	yes	—
9	C₂×C₄×Dic₉₇	C₂²	1	1	yes	characteristic
10	C₂×C₄²×C₉₇	C₄	1	1	yes	characteristic
11	C₂×#11715897	—	2	4	no	—
12	C₂×#11715897	—	2	8	no	—
13	C₂²×Dic₉₇	C₂×C₄	1	1	yes	characteristic
14	C₂²×Dic₉₇	C₂×C₄	1	2	yes	—
15	C₂²×Dic₉₇	C₂³	1	1	yes	characteristic
16	C₄×Dic₉₇	—	2	4	no	—
17	C₄×Dic₉₇	—	2	4	no	—
18	C₄×Dic₉₇	C₈	1	4	yes	complemented
19	C₄×Dic₉₇	D₄	1	2	yes	—
20	C₄×Dic₉₇	Q₈	1	2	yes	—
21	C₂²×C₄×C₉₇	C₂×C₄	1	2	yes	—
22	C₂²×C₄×C₉₇	C₂×C₄	1	1	yes	characteristic
23	C₄²×C₉₇	—	2	2	no	—
24	C₂³×C₉₇	C₂²×C₄	1	1	yes	characteristic
25	C₂×C₄×C₉₇	—	2	2	no	—
26	C₂×C₄×C₉₇	—	2	2	no	—
27	C₂×C₄×C₉₇	C₄⋊C₄	1	2	yes	—
28	C₂×C₄×C₉₇	C₂×C₈	1	2	yes	—
29	C₂×C₄×C₉₇	M₁₆	1	2	yes	—
30	C₂×C₄×C₉₇	M₁₆	1	4	yes	—
31	C₂×Dic₉₇	—	2	2	no	—
32	C₂×Dic₉₇	—	2	2	no	—
33	C₂×Dic₉₇	—	2	2	no	—
34	C₂×Dic₉₇	C₄⋊C₄	1	2	yes	—
35	C₂×Dic₉₇	C₂×C₈	1	4	yes	—
36	C₂×Dic₉₇	C₂×D₄	1	1	yes	characteristic
37	C₂×Dic₉₇	C₂×Q₈	1	1	yes	characteristic
38	C₂×Dic₉₇	#41	1	4	yes	—
39	#11715897	—	4	4	no	—
40	C₂²×C₉₇	C₂×C₄⋊C₄	1	1	yes	characteristic
41	C₂²×C₉₇	#117	1	2	yes	—
42	C₂²×C₉₇	C₂²×C₈	1	1	yes	characteristic
43	C₂²×C₉₇	C₂×M₁₆	1	1	yes	characteristic
44	C₂²×C₉₇	C₂×M₁₆	1	2	yes	—
45	C₄×C₉₇	—	2	4	no	—
46	C₄×C₉₇	C₄⋊C₈	1	4	yes	—
47	Dic₉₇	—	2	2	no	—
48	Dic₉₇	—	2	4	no	—
49	Dic₉₇	#124	1	2	yes	—
50	Dic₉₇	#125	1	2	yes	—
51	C₂×C₉₇	#382	1	1	yes	characteristic
52	C₂×C₉₇	C₂×C₄⋊C₈	1	1	yes	characteristic
53	C₂×C₉₇	#423	1	1	yes	characteristic
54	C₂×C₉₇	#431	1	2	yes	—
55	C₂×C₉₇	#432	1	2	yes	—
56	#1831	—	97	1	no	maximal, Sylow
57	C₉₇	#1831	1	1	yes	characteristic, complemented, Sylow
58	C₂×C₄⋊C₈	—	97	2	no	—
59	C₄³	—	97	1	no	—
60	#336	—	97	4	no	—
61	C₂²×C₈	—	97	4	no	—
62	C₂×C₄²	—	97	1	no	—
63	C₂×C₄²	—	97	1	no	—
64	C₂×C₄²	—	97	2	no	—
65	C₂×C₄²	—	97	2	no	—
66	C₂×C₄²	#87604	1	1	yes	characteristic
67	C₄⋊C₈	—	194	4	no	—
68	C₂²×C₄	—	97	1	no	—
69	C₂²×C₄	—	97	1	no	—
70	C₂²×C₄	—	97	2	no	—
71	C₂²×C₄	C₂×#87604	1	2	yes	—
72	C₂²×C₄	C₂×#87604	1	1	yes	characteristic
73	C₂×C₈	—	194	4	no	—
74	C₂×C₈	—	194	8	no	—
75	C₄²	—	2	2	no	—
76	C₄²	—	97	2	no	—
77	C₄²	—	97	2	no	—
78	C₄²	—	97	4	no	—
79	C₄²	—	194	4	no	—
80	C₄²	—	194	4	no	—
81	C₂³	C₂²×#87604	1	1	yes	characteristic
82	C₂×C₄	—	2	2	no	—
83	C₂×C₄	—	2	2	no	—
84	C₂×C₄	—	97	1	no	—
85	C₂×C₄	—	97	1	no	—
86	C₂×C₄	—	97	4	no	—
87	C₂×C₄	—	97	2	no	—
88	C₂×C₄	—	97	4	no	—
89	C₂×C₄	—	194	2	no	—
90	C₂×C₄	—	194	2	no	—
91	C₂×C₄	—	194	2	no	—
92	C₂×C₄	C₄⋊#87604	1	2	yes	—
93	C₂×C₄	#13095519	1	2	yes	—
94	C₂×C₄	#13095520	1	2	yes	—
95	C₂×C₄	#13095521	1	4	yes	—
96	C₈	—	388	4	no	—
97	C₂²	C₂×C₄⋊#87604	1	1	yes	characteristic
98	C₂²	#15690889	1	2	yes	—
99	C₂²	C₂×#13095519	1	1	yes	characteristic
100	C₂²	C₂×#13095520	1	1	yes	characteristic
101	C₂²	C₂×#13095521	1	2	yes	—
102	C₄	—	2	4	no	—
103	C₄	—	97	2	no	—
104	C₄	—	97	2	no	—
105	C₄	—	194	2	no	—
106	C₄	—	194	4	no	—
107	C₄	#15690877	1	4	yes	—
108	C₂	#20261679	1	1	yes	characteristic
109	C₂	C₂×#15690877	1	1	yes	characteristic
110	C₂	#20261739	1	1	yes	characteristic
111	C₂	#20261754	1	2	yes	—
112	C₂	#20261756	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊#11715897

C₂

yes

maximal

#20261601

C₂

yes

maximal

C₄²×Dic₉₇

C₂

yes

maximal, characteristic

C₂²×#11715897

C₂²

yes

—

C₄⋊#11715897

—

C₂×C₄×Dic₉₇

C₄

yes

—

C₂×C₄×Dic₉₇

C₄

yes

characteristic

C₂×C₄×Dic₉₇

C₂²

yes

—

C₂×C₄×Dic₉₇

C₂²

yes

characteristic

C₂×C₄²×C₉₇

C₄

yes

characteristic

C₂×#11715897

—

C₂×#11715897

—

C₂²×Dic₉₇

C₂×C₄

yes

characteristic

C₂²×Dic₉₇

C₂×C₄

yes

—

C₂²×Dic₉₇

C₂³

yes

characteristic

C₄×Dic₉₇

—

C₄×Dic₉₇

—

C₄×Dic₉₇

C₈

yes

complemented

C₄×Dic₉₇

D₄

yes

—

C₄×Dic₉₇

Q₈

yes

—

C₂²×C₄×C₉₇

C₂×C₄

yes

—

C₂²×C₄×C₉₇

C₂×C₄

yes

characteristic

C₄²×C₉₇

—

C₂³×C₉₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

C₄⋊C₄

yes

—

C₂×C₄×C₉₇

C₂×C₈

yes

—

C₂×C₄×C₉₇

M₁₆

yes

—

C₂×C₄×C₉₇

M₁₆

yes

—

C₂×Dic₉₇

—

C₂×Dic₉₇

—

C₂×Dic₉₇

—

C₂×Dic₉₇

C₄⋊C₄

yes

—

C₂×Dic₉₇

C₂×C₈

yes

—

C₂×Dic₉₇

C₂×D₄

yes

characteristic

C₂×Dic₉₇

C₂×Q₈

yes

characteristic

C₂×Dic₉₇

#41

yes

—

#11715897

—

C₂²×C₉₇

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₉₇

#117

yes

—

C₂²×C₉₇

C₂²×C₈

yes

characteristic

C₂²×C₉₇

C₂×M₁₆

yes

characteristic

C₂²×C₉₇

C₂×M₁₆

yes

—

C₄×C₉₇

—

C₄×C₉₇

C₄⋊C₈

yes

—

Dic₉₇

—

Dic₉₇

—

Dic₉₇

#124

yes

—

Dic₉₇

#125

yes

—

C₂×C₉₇

#382

yes

characteristic

C₂×C₉₇

C₂×C₄⋊C₈

yes

C₂×C₉₇

yes

C₂×C₉₇

yes

—

C₂×C₉₇

#432

yes

—

#1831

—

maximal, Sylow

C₉₇

#1831

yes

characteristic, complemented, Sylow

C₂×C₄⋊C₈

—

C₄³

—

#336

—

C₂²×C₈

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

#87604

yes

characteristic

C₄⋊C₈

—

194

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×#87604

yes

—

C₂²×C₄

C₂×#87604

yes

characteristic

C₂×C₈

—

194

—

C₂×C₈

—

194

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

194

—

C₄²

—

194

—

C₂³

C₂²×#87604

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

194

—

C₂×C₄

—

194

—

C₂×C₄

—

194

—

C₂×C₄

C₄⋊#87604

yes

—

C₂×C₄

#13095519

yes

—

C₂×C₄

#13095520

yes

—

C₂×C₄

#13095521

yes

—

C₈

—

388

—

C₂²

C₂×C₄⋊#87604

yes

characteristic

C₂²

#15690889

yes

—

C₂²

C₂×#13095519

yes

characteristic

100

C₂²

C₂×#13095520

yes

characteristic

101

C₂²

C₂×#13095521

yes

—

102

C₄

—

103

C₄

—

104

C₄

—

105

C₄

—

194

—

106

C₄

—

194

—

107

C₄

#15690877

yes

—

108

C₂

#20261679

yes

characteristic

109

C₂

C₂×#15690877

yes

characteristic

110

C₂

#20261739

yes

characteristic

111

C₂

#20261754

yes

—

112

C₂

#20261756

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂²×#87604
commutator	C₂²×C₉₇	C₂²×C₈
Fitting	C₂×C₄²×C₉₇	C₄
socle, prime radical	C₂³×C₉₇	C₂²×C₄
cosocle	C₂²×Dic₉₇	C₂³
hypercenter	C₂×C₄²	#87604
hypocenter	C₉₇	#1831
LCS	G ▷ C₂²×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄²

subgroup

G/H

center, Frattini

C₂³

C₂²×#87604

commutator

C₂²×C₉₇

C₂²×C₈

Fitting

C₂×C₄²×C₉₇

C₄

socle, prime radical

C₂³×C₉₇

C₂²×C₄

cosocle

C₂²×Dic₉₇

C₂³

C₂×C₄²

C₉₇

G ▷ C₂²×C₉₇ ▷ C₉₇

C₁ ◁ C₂³ ◁ C₂×C₄²

N	H ≅ G/N	type
18. C₄×Dic₉₇	C₈	semidirect
57. C₉₇	#1831	semidirect

H ≅ G/N

type

18. C₄×Dic₉₇

C₈

semidirect

57. C₉₇

#1831

semidirect

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

subgroup	index
C₂×C₄⋊#11715897	2
#20261601	2
C₄²×Dic₉₇	2
#1831	97

subgroup

index

C₂×C₄⋊#11715897

#20261601

C₄²×Dic₉₇

#1831

Subgroups — #27157653

Conjugacy classes of subgroups

Complements

Maximal subgroups