Group #27157270 — subgroups

Subgroups — #27157270

Order 12416.

228 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 60 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	#20261758	C₂	1	2	yes	maximal
2	#20261595	C₂	1	4	yes	maximal, complemented
3	C₄⋊Q₈×D₉₇	C₂	1	1	yes	maximal, characteristic
4	#15690877	C₂²	1	4	yes	—
5	#15690867	—	2	2	no	—
6	#15690865	—	2	2	no	—
7	C₂×Q₈×D₉₇	C₄	1	2	yes	—
8	#15690810	C₄	1	1	yes	characteristic
9	#15690809	—	2	2	no	—
10	#15690808	C₂²	1	2	yes	—
11	C₄⋊C₄×D₉₇	—	2	2	no	—
12	C₄²×D₉₇	C₂²	1	1	yes	characteristic
13	C₄⋊Q₈×C₉₇	C₄	1	1	yes	characteristic
14	C₂×#11715897	—	2	4	no	—
15	C₂×Dic₁₉₄	—	2	2	no	—
16	C₂×Dic₁₉₄	C₂×C₄	1	2	yes	—
17	Q₈×D₉₇	—	4	4	no	—
18	C₂×C₄×D₉₇	—	2	2	no	—
19	C₂×C₄×D₉₇	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄×D₉₇	D₄	1	2	yes	—
21	C₄⋊Dic₉₇	—	2	2	no	—
22	#13095490	—	2	4	no	—
23	C₄×Dic₉₇	D₄	1	2	yes	—
24	C₄×Dic₉₇	C₂³	1	1	yes	characteristic
25	C₂×Q₈×C₉₇	C₂×C₄	1	2	yes	—
26	C₄⋊C₄×C₉₇	—	2	2	no	—
27	C₄²×C₉₇	C₂×C₄	1	1	yes	characteristic
28	C₂²×D₉₇	C₂²⋊C₄	1	1	yes	characteristic
29	Q₈×C₉₇	—	4	2	no	—
30	C₂×C₄×C₉₇	—	2	2	no	—
31	C₂×C₄×C₉₇	C₂²⋊C₄	1	2	yes	—
32	C₂×C₄×C₉₇	C₂²×C₄	1	1	yes	characteristic
33	C₂×Dic₉₇	—	2	2	no	—
34	C₂×Dic₉₇	C₂²⋊C₄	1	1	yes	characteristic
35	C₂×Dic₉₇	C₂×D₄	1	2	yes	—
36	C₄×D₉₇	—	2	2	no	—
37	C₄×D₉₇	—	2	4	no	—
38	C₄×D₉₇	—	4	4	no	—
39	Dic₁₉₄	—	4	4	no	—
40	Dic₁₉₄	—	4	2	no	—
41	#11715897	—	4	4	no	—
42	C₂²×C₉₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
43	C₂×D₉₇	—	2	2	no	—
44	C₂×D₉₇	#100	1	2	yes	—
45	C₄×C₉₇	—	2	1	no	—
46	C₄×C₉₇	—	2	2	no	—
47	C₄×C₉₇	—	4	2	no	—
48	Dic₉₇	—	2	1	no	—
49	Dic₉₇	—	4	2	no	—
50	Dic₉₇	#137	1	4	yes	—
51	C₂×C₉₇	C₂×#100	1	1	yes	characteristic
52	C₂×C₉₇	#419	1	2	yes	—
53	D₉₇	—	2	2	no	—
54	#1558	—	97	1	no	maximal, Sylow
55	C₉₇	#1558	1	1	yes	characteristic, complemented, Sylow
56	C₂×C₄⋊Q₈	—	97	1	no	—
57	#432	—	97	2	no	—
58	#333	—	97	4	no	—
59	C₂²×Q₈	—	97	2	no	—
60	C₄⋊Q₈	—	97	2	no	—
61	C₄⋊Q₈	—	97	1	no	—
62	C₄⋊Q₈	—	194	2	no	—
63	C₄⋊Q₈	#87604	1	1	yes	characteristic
64	C₂×C₄⋊C₄	—	194	2	no	—
65	C₂×C₄²	—	97	1	no	—
66	C₄⋊C₈	—	97	4	no	—
67	C₂²⋊C₈	—	194	2	no	—
68	#97	—	194	2	no	—
69	C₂×Q₈	—	97	2	no	—
70	C₂×Q₈	—	388	4	no	—
71	C₂×Q₈	—	194	2	no	—
72	C₂×Q₈	C₂×#87604	1	2	yes	—
73	C₂²×C₄	—	97	2	no	—
74	C₂²×C₄	—	97	1	no	—
75	C₂²×C₄	—	194	2	no	—
76	C₂×C₈	—	194	4	no	—
77	C₄⋊C₄	—	2	2	no	—
78	C₄⋊C₄	—	194	4	no	—
79	C₄⋊C₄	—	194	2	no	—
80	C₄²	—	97	2	no	—
81	C₄²	—	97	1	no	—
82	C₄²	C₂×#87604	1	1	yes	characteristic
83	C₂³	—	97	1	no	—
84	Q₈	—	4	2	no	—
85	Q₈	—	388	2	no	—
86	Q₈	—	388	4	no	—
87	C₂×C₄	—	2	2	no	—
88	C₂×C₄	—	97	2	no	—
89	C₂×C₄	—	97	1	no	—
90	C₂×C₄	—	388	4	no	—
91	C₂×C₄	—	194	4	no	—
92	C₂×C₄	—	194	2	no	—
93	C₂×C₄	—	194	2	no	—
94	C₂×C₄	C₂²⋊#87604	1	2	yes	—
95	C₂×C₄	C₂²×#87604	1	1	yes	characteristic
96	C₈	—	388	4	no	—
97	C₂²	—	97	2	no	—
98	C₂²	—	194	2	no	—
99	C₂²	C₂×C₂²⋊#87604	1	1	yes	characteristic
100	C₄	—	2	1	no	—
101	C₄	—	2	2	no	—
102	C₄	—	4	2	no	—
103	C₄	—	97	4	no	—
104	C₄	—	388	2	no	—
105	C₄	—	194	1	no	—
106	C₂	—	194	2	no	—
107	C₂	D₉₇⋊#100	1	1	yes	characteristic
108	C₂	#20261728	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20261758

C₂

yes

maximal

#20261595

C₂

yes

maximal, complemented

C₄⋊Q₈×D₉₇

C₂

yes

maximal, characteristic

#15690877

C₂²

yes

—

#15690867

—

#15690865

—

C₂×Q₈×D₉₇

C₄

yes

—

#15690810

C₄

yes

characteristic

#15690809

—

#15690808

C₂²

yes

—

C₄⋊C₄×D₉₇

—

C₄²×D₉₇

C₂²

yes

characteristic

C₄⋊Q₈×C₉₇

C₄

yes

characteristic

C₂×#11715897

—

C₂×Dic₁₉₄

—

C₂×Dic₁₉₄

C₂×C₄

yes

—

Q₈×D₉₇

—

C₂×C₄×D₉₇

—

C₂×C₄×D₉₇

C₂×C₄

yes

characteristic

C₂×C₄×D₉₇

D₄

yes

—

C₄⋊Dic₉₇

—

#13095490

—

C₄×Dic₉₇

D₄

yes

—

C₄×Dic₉₇

C₂³

yes

characteristic

C₂×Q₈×C₉₇

C₂×C₄

yes

—

C₄⋊C₄×C₉₇

—

C₄²×C₉₇

C₂×C₄

yes

characteristic

C₂²×D₉₇

C₂²⋊C₄

yes

characteristic

Q₈×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

C₂²⋊C₄

yes

—

C₂×C₄×C₉₇

C₂²×C₄

yes

characteristic

C₂×Dic₉₇

—

C₂×Dic₉₇

C₂²⋊C₄

yes

characteristic

C₂×Dic₉₇

C₂×D₄

yes

—

C₄×D₉₇

—

C₄×D₉₇

—

C₄×D₉₇

—

Dic₁₉₄

—

Dic₁₉₄

—

#11715897

—

C₂²×C₉₇

C₂×C₂²⋊C₄

yes

characteristic

C₂×D₉₇

—

C₂×D₉₇

#100

yes

—

C₄×C₉₇

—

C₄×C₉₇

—

C₄×C₉₇

—

Dic₉₇

—

Dic₉₇

—

Dic₉₇

#137

yes

—

C₂×C₉₇

C₂×#100

yes

characteristic

C₂×C₉₇

#419

yes

—

D₉₇

—

#1558

—

maximal, Sylow

C₉₇

#1558

yes

characteristic, complemented, Sylow

C₂×C₄⋊Q₈

—

#432

—

#333

—

C₂²×Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

194

—

C₄⋊Q₈

#87604

yes

characteristic

C₂×C₄⋊C₄

—

194

—

C₂×C₄²

—

C₄⋊C₈

—

C₂²⋊C₈

—

194

—

#97

—

194

—

C₂×Q₈

—

C₂×Q₈

—

388

—

C₂×Q₈

—

194

—

C₂×Q₈

C₂×#87604

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

194

—

C₂×C₈

—

194

—

C₄⋊C₄

—

C₄⋊C₄

—

194

—

C₄⋊C₄

—

194

—

C₄²

—

C₄²

—

C₄²

C₂×#87604

yes

characteristic

C₂³

—

Q₈

—

Q₈

—

388

—

Q₈

—

388

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

388

—

C₂×C₄

—

194

—

C₂×C₄

—

194

—

C₂×C₄

—

194

—

C₂×C₄

C₂²⋊#87604

yes

—

C₂×C₄

C₂²×#87604

yes

characteristic

C₈

—

388

—

C₂²

—

C₂²

—

194

—

C₂²

C₂×C₂²⋊#87604

yes

characteristic

100

C₄

—

101

C₄

—

102

C₄

—

103

C₄

—

104

C₄

—

388

—

105

C₄

—

194

—

106

C₂

—

194

—

107

C₂

D₉₇⋊#100

yes

characteristic

108

C₂

#20261728

yes

—

	subgroup	G/H
center	C₂²	C₂×C₂²⋊#87604
commutator	C₂×C₄×C₉₇	C₂²×C₄
Frattini	C₂×C₄	C₂²×#87604
Fitting	C₄⋊Q₈×C₉₇	C₄
socle	C₂²×C₉₇	C₂×C₂²⋊C₄
prime radical	C₂²×D₉₇	C₂²⋊C₄
cosocle	C₄×Dic₉₇	C₂³
hypercenter	C₄⋊Q₈	#87604
hypocenter	C₉₇	#1558
LCS	G ▷ C₂×C₄×C₉₇ ▷ C₂²×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

subgroup

G/H

center

C₂²

C₂×C₂²⋊#87604

commutator

C₂×C₄×C₉₇

C₂²×C₄

Frattini

C₂×C₄

C₂²×#87604

Fitting

C₄⋊Q₈×C₉₇

C₄

socle

C₂²×C₉₇

C₂×C₂²⋊C₄

prime radical

C₂²×D₉₇

C₂²⋊C₄

cosocle

C₄×Dic₉₇

C₂³

C₄⋊Q₈

C₉₇

G ▷ C₂×C₄×C₉₇ ▷ C₂²×C₉₇ ▷ C₉₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

N	H ≅ G/N	type
2. #20261595	C₂	semidirect
55. C₉₇	#1558	semidirect

H ≅ G/N

type

2. #20261595

C₂

semidirect

55. C₉₇

#1558

semidirect

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

subgroup	index
#20261758	2
#20261595	2
C₄⋊Q₈×D₉₇	2
#1558	97

subgroup

index

#20261758

#20261595

C₄⋊Q₈×D₉₇

#1558

Subgroups — #27157270

Conjugacy classes of subgroups

Complements

Maximal subgroups