Group #27143787 — subgroups

Subgroups — #27143787

Order 12416.

154 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 154 conjugacy classes of subgroups — too many to draw. Only its 55 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×Dic₇₇₆	C₂	1	2	yes	maximal
2	#20261115	C₂	1	1	yes	maximal, characteristic
3	C₄×Dic₃₈₈	C₂	1	1	yes	maximal, characteristic
4	#20260540	C₂	1	2	yes	maximal
5	C₄⋊C₁₆×C₉₇	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₃₈₈	—	2	1	no	—
7	C₂×Dic₃₈₈	C₂²	1	2	yes	—
8	C₂×Dic₃₈₈	C₂²	1	1	yes	characteristic
9	#15690810	—	2	1	no	—
10	C₄×Dic₁₉₄	—	2	1	no	—
11	Dic₇₇₆	—	2	4	no	—
12	#15690744	C₂²	1	1	yes	characteristic
13	#15690735	—	2	1	no	—
14	C₂×C₁₆×C₉₇	C₂²	1	2	yes	—
15	C₄×C₈×C₉₇	C₂²	1	1	yes	characteristic
16	C₂×Dic₁₉₄	—	2	1	no	—
17	C₂×Dic₁₉₄	—	2	2	no	—
18	Dic₃₈₈	—	2	2	no	—
19	Dic₃₈₈	—	2	1	no	—
20	Dic₃₈₈	—	2	2	no	—
21	Dic₃₈₈	—	4	2	no	—
22	Dic₃₈₈	D₄	1	2	yes	—
23	C₄⋊Dic₉₇	—	2	1	no	—
24	C₄⋊Dic₉₇	—	4	1	no	—
25	C₄⋊Dic₉₇	—	4	1	no	—
26	#13095490	—	4	1	no	—
27	C₄×Dic₉₇	—	4	1	no	—
28	C₂×C₈×C₉₇	D₄	1	1	yes	characteristic
29	C₂×C₈×C₉₇	C₂³	1	1	yes	characteristic
30	C₄²×C₉₇	D₄	1	1	yes	characteristic
31	C₁₆×C₉₇	—	2	2	no	—
32	C₂×C₄×C₉₇	D₈	1	1	yes	characteristic
33	C₂×C₄×C₉₇	D₈	1	1	yes	characteristic
34	C₂×C₄×C₉₇	C₂×D₄	1	1	yes	characteristic
35	C₂×Dic₉₇	—	4	1	no	—
36	C₂×Dic₉₇	—	4	1	no	—
37	C₂×Dic₉₇	—	4	2	no	—
38	Dic₁₉₄	—	2	2	no	—
39	Dic₁₉₄	—	4	1	no	—
40	Dic₁₉₄	—	4	2	no	—
41	Dic₁₉₄	—	4	2	no	—
42	C₈×C₉₇	—	2	1	no	—
43	C₈×C₉₇	C₂×D₄	1	1	yes	characteristic
44	C₈×C₉₇	#41	1	1	yes	characteristic
45	C₂²×C₉₇	C₂×D₈	1	1	yes	characteristic
46	C₄×C₉₇	—	2	1	no	—
47	C₄×C₉₇	Q₃₂	1	2	yes	complemented
48	C₄×C₉₇	#121	1	1	yes	characteristic
49	C₄×C₉₇	AGL(1,ℤ/8)	1	1	yes	characteristic
50	Dic₉₇	—	4	2	no	—
51	Dic₉₇	—	8	2	no	—
52	Dic₉₇	—	8	1	no	—
53	C₂×C₉₇	#459	1	1	yes	characteristic
54	C₂×C₉₇	C₂×Q₃₂	1	1	yes	characteristic
55	C₂×C₉₇	#510	1	1	yes	characteristic
56	#2209	—	97	1	no	maximal, Sylow
57	C₉₇	#2209	1	1	yes	characteristic, complemented, Sylow
58	C₂×Q₃₂	—	97	2	no	—
59	#494	—	97	1	no	—
60	C₄×Q₁₆	—	97	1	no	—
61	C₄⋊C₁₆	D₉₇	1	1	yes	characteristic
62	#358	—	97	2	no	—
63	C₂×Q₁₆	—	97	1	no	—
64	C₂×Q₁₆	—	97	2	no	—
65	C₂×Q₁₆	—	194	1	no	—
66	C₄⋊Q₈	—	194	1	no	—
67	C₄×Q₈	—	194	1	no	—
68	Q₃₂	—	194	4	no	—
69	C₂×C₁₆	C₂×D₉₇	1	2	yes	—
70	#107	—	97	1	no	—
71	#103	—	194	1	no	—
72	C₄×C₈	C₂×D₉₇	1	1	yes	characteristic
73	C₂×Q₈	—	194	2	no	—
74	C₂×Q₈	—	194	1	no	—
75	Q₁₆	—	97	2	no	—
76	Q₁₆	—	388	2	no	—
77	Q₁₆	—	194	2	no	—
78	Q₁₆	—	194	2	no	—
79	Q₁₆	—	194	1	no	—
80	C₂×C₈	D₃₈₈	1	1	yes	characteristic
81	C₂×C₈	C₂²×D₉₇	1	1	yes	characteristic
82	C₄⋊C₄	—	388	1	no	—
83	C₄⋊C₄	—	388	1	no	—
84	C₄⋊C₄	—	388	1	no	—
85	C₄⋊C₄	—	194	1	no	—
86	C₄²	—	388	1	no	—
87	C₄²	D₃₈₈	1	1	yes	characteristic
88	C₁₆	—	2	2	no	—
89	Q₈	—	388	2	no	—
90	Q₈	—	388	2	no	—
91	Q₈	—	388	1	no	—
92	Q₈	—	194	2	no	—
93	C₂×C₄	—	388	1	no	—
94	C₂×C₄	—	388	2	no	—
95	C₂×C₄	—	388	1	no	—
96	C₂×C₄	D₇₇₆	1	1	yes	characteristic
97	C₂×C₄	D₇₇₆	1	1	yes	characteristic
98	C₂×C₄	C₂×D₃₈₈	1	1	yes	characteristic
99	C₈	—	2	1	no	—
100	C₈	D₄×D₉₇	1	1	yes	characteristic
101	C₈	#13095512	1	1	yes	characteristic
102	C₂²	C₂×D₇₇₆	1	1	yes	characteristic
103	C₄	—	2	1	no	—
104	C₄	—	776	1	no	—
105	C₄	—	776	2	no	—
106	C₄	—	388	2	no	—
107	C₄	Dic₇₇₆	1	2	yes	complemented
108	C₄	#15690785	1	1	yes	characteristic
109	C₄	#15690838	1	1	yes	characteristic
110	C₂	#20260931	1	1	yes	characteristic
111	C₂	C₂×Dic₇₇₆	1	1	yes	characteristic
112	C₂	#20261178	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×Dic₇₇₆

C₂

yes

maximal

#20261115

C₂

yes

maximal, characteristic

C₄×Dic₃₈₈

C₂

yes

maximal, characteristic

#20260540

C₂

yes

maximal

C₄⋊C₁₆×C₉₇

C₂

yes

maximal, characteristic

C₂×Dic₃₈₈

—

C₂×Dic₃₈₈

C₂²

yes

—

C₂×Dic₃₈₈

C₂²

yes

characteristic

#15690810

—

C₄×Dic₁₉₄

—

Dic₇₇₆

—

#15690744

C₂²

yes

characteristic

#15690735

—

C₂×C₁₆×C₉₇

C₂²

yes

—

C₄×C₈×C₉₇

C₂²

yes

characteristic

C₂×Dic₁₉₄

—

C₂×Dic₁₉₄

—

Dic₃₈₈

—

Dic₃₈₈

—

Dic₃₈₈

—

Dic₃₈₈

—

Dic₃₈₈

D₄

yes

—

C₄⋊Dic₉₇

—

C₄⋊Dic₉₇

—

C₄⋊Dic₉₇

—

#13095490

—

C₄×Dic₉₇

—

C₂×C₈×C₉₇

D₄

yes

characteristic

C₂×C₈×C₉₇

C₂³

yes

characteristic

C₄²×C₉₇

D₄

yes

characteristic

C₁₆×C₉₇

—

C₂×C₄×C₉₇

D₈

yes

characteristic

C₂×C₄×C₉₇

D₈

yes

characteristic

C₂×C₄×C₉₇

C₂×D₄

yes

characteristic

C₂×Dic₉₇

—

C₂×Dic₉₇

—

C₂×Dic₉₇

—

Dic₁₉₄

—

Dic₁₉₄

—

Dic₁₉₄

—

Dic₁₉₄

—

C₈×C₉₇

—

C₈×C₉₇

C₂×D₄

yes

characteristic

C₈×C₉₇

#41

yes

characteristic

C₂²×C₉₇

C₂×D₈

yes

characteristic

C₄×C₉₇

—

C₄×C₉₇

Q₃₂

yes

C₄×C₉₇

yes

C₄×C₉₇

yes

Dic₉₇

—

Dic₉₇

—

Dic₉₇

—

C₂×C₉₇

#459

yes

characteristic

C₂×C₉₇

C₂×Q₃₂

yes

C₂×C₉₇

yes

—

C₉₇

yes

characteristic, complemented, Sylow

C₂×Q₃₂

—

#494

—

C₄×Q₁₆

—

C₄⋊C₁₆

D₉₇

yes

characteristic

#358

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

194

—

C₄⋊Q₈

—

194

—

C₄×Q₈

—

194

—

Q₃₂

—

194

—

C₂×C₁₆

C₂×D₉₇

yes

—

#107

—

#103

—

194

—

C₄×C₈

C₂×D₉₇

yes

characteristic

C₂×Q₈

—

194

—

C₂×Q₈

—

194

—

Q₁₆

—

Q₁₆

—

388

—

Q₁₆

—

194

—

Q₁₆

—

194

—

Q₁₆

—

194

—

C₂×C₈

D₃₈₈

yes

characteristic

C₂×C₈

C₂²×D₉₇

yes

characteristic

C₄⋊C₄

—

388

—

C₄⋊C₄

—

388

—

C₄⋊C₄

—

388

—

C₄⋊C₄

—

194

—

C₄²

—

388

—

C₄²

D₃₈₈

yes

characteristic

C₁₆

—

Q₈

—

388

—

Q₈

—

388

—

Q₈

—

388

—

Q₈

—

194

—

C₂×C₄

—

388

—

C₂×C₄

—

388

—

C₂×C₄

—

388

—

C₂×C₄

D₇₇₆

yes

characteristic

C₂×C₄

D₇₇₆

yes

characteristic

C₂×C₄

C₂×D₃₈₈

yes

characteristic

C₈

—

100

C₈

D₄×D₉₇

yes

characteristic

101

C₈

#13095512

yes

characteristic

102

C₂²

C₂×D₇₇₆

yes

characteristic

103

C₄

—

104

C₄

—

776

—

105

C₄

—

776

—

106

C₄

—

388

—

107

C₄

Dic₇₇₆

yes

complemented

108

C₄

#15690785

yes

characteristic

109

C₄

#15690838

yes

characteristic

110

C₂

#20260931

yes

characteristic

111

C₂

C₂×Dic₇₇₆

yes

characteristic

112

C₂

#20261178

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×D₇₇₆
commutator, cosocle	C₂×C₈×C₉₇	C₂³
Frattini	C₂×C₈	C₂²×D₉₇
Fitting	C₄⋊C₁₆×C₉₇	C₂
socle, prime radical	C₂²×C₉₇	C₂×D₈
hypercenter	C₄⋊C₁₆	D₉₇
hypocenter	C₉₇	#2209
LCS	G ▷ C₂×C₈×C₉₇ ▷ C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄⋊C₁₆

subgroup

G/H

center

C₂²

C₂×D₇₇₆

commutator, cosocle

C₂×C₈×C₉₇

C₂³

Frattini

C₂×C₈

C₂²×D₉₇

Fitting

C₄⋊C₁₆×C₉₇

C₂

socle, prime radical

C₂²×C₉₇

C₂×D₈

hypercenter

C₄⋊C₁₆

D₉₇

hypocenter

C₉₇

#2209

LCS

G ▷ C₂×C₈×C₉₇ ▷ C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄⋊C₁₆

N	H ≅ G/N	type
47. C₄×C₉₇	Q₃₂	semidirect
57. C₉₇	#2209	semidirect
107. C₄	Dic₇₇₆	semidirect

H ≅ G/N

type

47. C₄×C₉₇

Q₃₂

semidirect

57. C₉₇

#2209

semidirect

107. C₄

Dic₇₇₆

semidirect

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

subgroup	index
C₂×Dic₇₇₆	2
#20261115	2
C₄×Dic₃₈₈	2
#20260540	2
C₄⋊C₁₆×C₉₇	2
#2209	97

subgroup

index

C₂×Dic₇₇₆

#20261115

C₄×Dic₃₈₈

#20260540

C₄⋊C₁₆×C₉₇

#2209

Subgroups — #27143787

Conjugacy classes of subgroups

Complements

Maximal subgroups