Group #27157331 — subgroups

Subgroups — #27157331

Order 12416.

160 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 160 conjugacy classes of subgroups — too many to draw. Only its 64 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	#20261784	C₂	1	2	yes	maximal
2	#20261596	C₂	1	1	yes	maximal, characteristic
3	#20261585	C₂	1	2	yes	maximal
4	#20261576	C₂	1	1	yes	maximal, characteristic
5	#20260829	C₂	1	1	yes	maximal, characteristic
6	#15690877	—	2	2	no	—
7	C₄⋊#11715897	C₂²	1	2	yes	—
8	C₄×#11715897	—	2	2	no	—
9	C₄×#11715897	C₂²	1	1	yes	characteristic
10	C₄×#11715897	C₂²	1	1	yes	characteristic
11	C₂×Dic₃₈₈	C₄	1	1	yes	characteristic
12	#15690775	C₂²	1	2	yes	—
13	#15690733	C₄	1	2	yes	—
14	C₈×Dic₉₇	C₂²	1	1	yes	characteristic
15	#107×C₉₇	C₄	1	1	yes	characteristic
16	C₂×#11715897	—	2	2	no	—
17	C₂×#11715897	—	2	1	no	—
18	C₂×#11715897	D₄	1	2	yes	—
19	C₂×Dic₁₉₄	C₂×C₄	1	2	yes	—
20	Dic₃₈₈	C₈	1	4	yes	complemented
21	C₂×#11715895	C₂×C₄	1	1	yes	characteristic
22	#13095490	—	2	2	no	—
23	C₄×Dic₉₇	—	2	2	no	—
24	C₄×Dic₉₇	C₂³	1	1	yes	characteristic
25	C₂×C₈×C₉₇	C₂×C₄	1	1	yes	characteristic
26	C₄⋊C₄×C₉₇	C₂×C₄	1	2	yes	—
27	C₂×C₄×C₉₇	—	2	2	no	—
28	C₂×C₄×C₉₇	C₂²×C₄	1	1	yes	characteristic
29	C₂×Dic₉₇	—	2	2	no	—
30	C₂×Dic₉₇	C₂×D₄	1	1	yes	characteristic
31	C₂×Dic₉₇	#41	1	1	yes	characteristic
32	Dic₁₉₄	—	2	2	no	—
33	Dic₁₉₄	C₂×C₈	1	4	yes	—
34	#11715897	—	2	2	no	—
35	#11715897	—	4	1	no	—
36	#11715897	—	4	2	no	—
37	C₈×C₉₇	C₂×C₈	1	2	yes	—
38	#11715895	—	2	1	no	—
39	C₂²×C₉₇	C₄×D₄	1	1	yes	characteristic
40	C₄×C₉₇	—	4	2	no	—
41	C₄×C₉₇	C₂²×C₈	1	1	yes	characteristic
42	C₄×C₉₇	#131	1	1	yes	characteristic
43	Dic₉₇	—	2	4	no	—
44	Dic₉₇	—	2	1	no	—
45	Dic₉₇	#137	1	2	yes	—
46	C₂×C₉₇	C₈×D₄	1	1	yes	characteristic
47	C₂×C₉₇	#441	1	1	yes	characteristic
48	C₂×C₉₇	#444	1	1	yes	characteristic
49	#1579	—	97	1	no	maximal, Sylow
50	C₉₇	#1579	1	1	yes	characteristic, complemented, Sylow
51	C₈×Q₈	—	97	2	no	—
52	C₄×Q₁₆	—	97	1	no	—
53	#334	—	97	1	no	—
54	Q₈⋊C₈	—	97	2	no	—
55	#322	—	97	1	no	—
56	C₂×Q₁₆	—	97	1	no	—
57	C₄×Q₈	—	97	2	no	—
58	#107	#87604	1	1	yes	characteristic
59	C₄⋊C₈	—	97	2	no	—
60	C₄⋊C₈	—	194	2	no	—
61	#103	—	97	2	no	—
62	C₄×C₈	—	97	1	no	—
63	C₄×C₈	—	97	1	no	—
64	C₄×C₈	—	97	1	no	—
65	C₄×C₈	—	194	2	no	—
66	C₂×Q₈	—	97	2	no	—
67	Q₁₆	—	97	4	no	—
68	C₂×C₈	—	97	2	no	—
69	C₂×C₈	—	97	1	no	—
70	C₂×C₈	—	194	1	no	—
71	C₂×C₈	—	194	2	no	—
72	C₂×C₈	C₂×#87604	1	1	yes	characteristic
73	C₄⋊C₄	—	194	2	no	—
74	C₄⋊C₄	C₂×#87604	1	2	yes	—
75	C₄²	—	97	1	no	—
76	C₄²	—	194	2	no	—
77	Q₈	—	97	4	no	—
78	Q₈	—	194	2	no	—
79	C₂×C₄	—	2	2	no	—
80	C₂×C₄	—	97	1	no	—
81	C₂×C₄	—	97	1	no	—
82	C₂×C₄	—	194	2	no	—
83	C₂×C₄	C₂²×#87604	1	1	yes	characteristic
84	C₈	—	388	2	no	—
85	C₈	—	388	1	no	—
86	C₈	—	194	2	no	—
87	C₈	—	194	1	no	—
88	C₈	#13095519	1	2	yes	—
89	C₂²	D₄×#87604	1	1	yes	characteristic
90	C₄	—	4	2	no	—
91	C₄	—	97	2	no	—
92	C₄	—	194	4	no	—
93	C₄	—	194	1	no	—
94	C₄	C₂×#13095519	1	1	yes	characteristic
95	C₄	#15690898	1	1	yes	characteristic
96	C₂	#20261763	1	1	yes	characteristic
97	C₂	#20261775	1	1	yes	characteristic
98	C₂	#20261782	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20261784

C₂

yes

maximal

#20261596

C₂

yes

maximal, characteristic

#20261585

C₂

yes

maximal

#20261576

C₂

yes

maximal, characteristic

#20260829

C₂

yes

maximal, characteristic

#15690877

—

C₄⋊#11715897

C₂²

yes

—

C₄×#11715897

—

C₄×#11715897

C₂²

yes

characteristic

C₄×#11715897

C₂²

yes

characteristic

C₂×Dic₃₈₈

C₄

yes

characteristic

#15690775

C₂²

yes

—

#15690733

C₄

yes

—

C₈×Dic₉₇

C₂²

yes

characteristic

#107×C₉₇

C₄

yes

—

—

D₄

yes

—

C₂×Dic₁₉₄

C₂×C₄

yes

—

Dic₃₈₈

C₈

yes

complemented

C₂×#11715895

C₂×C₄

yes

characteristic

#13095490

—

C₄×Dic₉₇

—

C₄×Dic₉₇

C₂³

yes

characteristic

C₂×C₈×C₉₇

C₂×C₄

yes

characteristic

C₄⋊C₄×C₉₇

C₂×C₄

yes

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

C₂²×C₄

yes

characteristic

C₂×Dic₉₇

—

C₂×Dic₉₇

C₂×D₄

yes

characteristic

C₂×Dic₉₇

#41

yes

characteristic

Dic₁₉₄

—

Dic₁₉₄

C₂×C₈

yes

—

#11715897

—

#11715897

—

#11715897

—

C₈×C₉₇

C₂×C₈

yes

—

#11715895

—

C₂²×C₉₇

C₄×D₄

yes

characteristic

C₄×C₉₇

—

C₄×C₉₇

C₂²×C₈

yes

characteristic

C₄×C₉₇

#131

yes

characteristic

Dic₉₇

—

Dic₉₇

—

Dic₉₇

#137

yes

—

C₂×C₉₇

C₈×D₄

yes

C₂×C₉₇

yes

C₂×C₉₇

yes

—

C₉₇

yes

characteristic, complemented, Sylow

C₈×Q₈

—

C₄×Q₁₆

—

#334

—

Q₈⋊C₈

—

#322

—

C₂×Q₁₆

—

C₄×Q₈

—

#107

#87604

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

194

—

#103

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

194

—

C₂×Q₈

—

Q₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

194

—

C₂×C₈

—

194

—

C₂×C₈

C₂×#87604

yes

characteristic

C₄⋊C₄

—

194

—

C₄⋊C₄

C₂×#87604

yes

—

C₄²

—

C₄²

—

194

—

Q₈

—

Q₈

—

194

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

194

—

C₂×C₄

C₂²×#87604

yes

characteristic

C₈

—

388

—

C₈

—

388

—

C₈

—

194

—

C₈

—

194

—

C₈

#13095519

yes

—

C₂²

D₄×#87604

yes

characteristic

C₄

—

C₄

—

C₄

—

194

—

C₄

—

194

—

C₄

yes

C₄

yes

C₂

yes

C₂

yes

C₂

yes

	subgroup	G/H
center	C₂²	D₄×#87604
commutator	C₄×C₉₇	C₂²×C₈
Frattini	C₂×C₄	C₂²×#87604
Fitting	#107×C₉₇	C₄
socle, prime radical	C₂²×C₉₇	C₄×D₄
cosocle	C₄×Dic₉₇	C₂³
hypercenter	#107	#87604
hypocenter	C₉₇	#1579
LCS	G ▷ C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #107

subgroup

G/H

center

C₂²

D₄×#87604

commutator

C₄×C₉₇

C₂²×C₈

Frattini

C₂×C₄

C₂²×#87604

Fitting

#107×C₉₇

C₄

socle, prime radical

C₂²×C₉₇

C₄×D₄

cosocle

C₄×Dic₉₇

C₂³

C₉₇

G ▷ C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ #107

N	H ≅ G/N	type
20. Dic₃₈₈	C₈	semidirect
50. C₉₇	#1579	semidirect

H ≅ G/N

type

20. Dic₃₈₈

C₈

semidirect

50. C₉₇

#1579

semidirect

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

subgroup	index
#20261784	2
#20261596	2
#20261585	2
#20261576	2
#20260829	2
#1579	97

subgroup

index

Subgroups — #27157331

Conjugacy classes of subgroups

Complements

Maximal subgroups