Group #27140857 — subgroups

Subgroups — #27140857

Order 12416.

218 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 218 conjugacy classes of subgroups — too many to draw. Only its 57 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	#20261107	C₂	1	1	yes	maximal, characteristic
2	#20261030	C₂	1	2	yes	maximal, complemented
3	#20260479	C₂	1	1	yes	maximal, characteristic
4	Q₈⋊#11715895	C₂	1	2	yes	maximal, complemented
5	#559×C₉₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#13095499	—	2	2	no	—
7	#15690810	C₂²	1	1	yes	characteristic
8	#15690743	—	2	2	no	—
9	C₄⋊#11715895	C₂²	1	2	yes	—
10	#15690734	—	2	2	no	—
11	C₄×#11715895	C₂²	1	1	yes	characteristic
12	#127×C₉₇	C₂²	1	1	yes	characteristic
13	#124×C₉₇	—	2	2	no	—
14	#123×C₉₇	—	2	2	no	—
15	C₄×Q₈×C₉₇	C₂²	1	2	yes	—
16	C₄×D₄×C₉₇	—	2	2	no	—
17	C₂×Dic₁₉₄	—	2	1	no	—
18	#13095499	—	4	4	no	—
19	C₂×#11715895	—	2	1	no	—
20	C₂×#11715895	—	2	2	no	—
21	C₄⋊Dic₉₇	—	2	2	no	—
22	C₂×Q₈×C₉₇	D₄	1	2	yes	—
23	C₂×D₄×C₉₇	—	2	2	no	—
24	C₂×D₄×C₉₇	D₄	1	2	yes	—
25	C₂²×C₄×C₉₇	—	2	2	no	—
26	C₄⋊C₄×C₉₇	—	2	2	no	—
27	C₄⋊C₄×C₉₇	D₄	1	2	yes	—
28	C₂²⋊C₄×C₉₇	—	2	4	no	—
29	C₂²⋊C₄×C₉₇	—	2	2	no	—
30	C₄²×C₉₇	—	2	2	no	—
31	C₄²×C₉₇	C₂³	1	1	yes	characteristic
32	C₂³×C₉₇	—	2	2	no	—
33	Q₈×C₉₇	—	2	2	no	—
34	Q₈×C₉₇	—	2	2	no	—
35	D₄×C₉₇	—	4	2	no	—
36	D₄×C₉₇	—	4	2	no	—
37	D₄×C₉₇	—	4	2	no	—
38	C₂×C₄×C₉₇	—	2	2	no	—
39	C₂×C₄×C₉₇	—	2	2	no	—
40	C₂×C₄×C₉₇	—	8	2	no	—
41	C₂×C₄×C₉₇	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₉₇	C₂×D₄	1	2	yes	—
43	C₂×Dic₉₇	—	4	1	no	—
44	Dic₁₉₄	—	4	2	no	—
45	#11715895	—	2	2	no	—
46	#11715895	—	4	2	no	—
47	C₂²×C₉₇	—	4	4	no	—
48	C₂²×C₉₇	—	4	2	no	—
49	C₂²×C₉₇	#120	1	1	yes	characteristic
50	C₄×C₉₇	—	2	1	no	—
51	C₄×C₉₇	—	4	2	no	—
52	C₄×C₉₇	—	4	2	no	—
53	C₄×C₉₇	#135	1	2	yes	—
54	C₄×C₉₇	#137	1	2	yes	—
55	Dic₉₇	—	8	1	no	—
56	C₂×C₉₇	—	8	2	no	—
57	C₂×C₉₇	#452	1	2	yes	—
58	C₂×C₉₇	#453	1	1	yes	characteristic
59	#1630	—	97	1	no	maximal, Sylow
60	C₉₇	#1630	1	1	yes	characteristic, complemented, Sylow
61	#559	D₉₇	1	1	yes	characteristic
62	#492	—	97	1	no	—
63	#477	—	97	2	no	—
64	#331	—	97	1	no	—
65	Q₈⋊C₈	—	97	2	no	—
66	C₂×SD₁₆	—	194	2	no	—
67	C₄⋊Q₈	—	97	1	no	—
68	#127	C₂×D₉₇	1	1	yes	characteristic
69	#124	—	2	2	no	—
70	#123	—	2	2	no	—
71	C₄×Q₈	C₂×D₉₇	1	2	yes	—
72	C₄×D₄	—	2	2	no	—
73	#107	—	194	2	no	—
74	C₄⋊C₈	—	97	2	no	—
75	#103	—	194	2	no	—
76	C₄×C₈	—	97	1	no	—
77	C₂×Q₈	—	194	1	no	—
78	C₂×Q₈	C₂²⋊D₉₇	1	2	yes	—
79	C₂×D₄	—	2	2	no	—
80	C₂×D₄	C₂²⋊D₉₇	1	2	yes	—
81	C₂²×C₄	—	2	2	no	—
82	SD₁₆	—	388	4	no	—
83	C₂×C₈	—	194	2	no	—
84	C₂×C₈	—	194	1	no	—
85	C₄⋊C₄	—	2	2	no	—
86	C₄⋊C₄	—	194	2	no	—
87	C₄⋊C₄	C₂²⋊D₉₇	1	2	yes	—
88	C₂²⋊C₄	—	2	4	no	—
89	C₂²⋊C₄	—	2	2	no	—
90	C₄²	—	2	2	no	—
91	C₄²	C₂²×D₉₇	1	1	yes	characteristic
92	C₂³	—	2	2	no	—
93	Q₈	—	2	2	no	—
94	Q₈	—	2	2	no	—
95	Q₈	—	388	2	no	—
96	D₄	—	4	2	no	—
97	D₄	—	4	2	no	—
98	D₄	—	4	2	no	—
99	C₂×C₄	—	2	2	no	—
100	C₂×C₄	—	2	2	no	—
101	C₂×C₄	—	8	2	no	—
102	C₂×C₄	—	388	1	no	—
103	C₂×C₄	C₂×C₂²⋊D₉₇	1	1	yes	characteristic
104	C₂×C₄	C₂×C₂²⋊D₉₇	1	2	yes	—
105	C₈	—	388	2	no	—
106	C₈	—	194	2	no	—
107	C₂²	—	4	4	no	—
108	C₂²	—	4	2	no	—
109	C₂²	#15690779	1	1	yes	characteristic
110	C₄	—	2	1	no	—
111	C₄	—	4	2	no	—
112	C₄	—	4	2	no	—
113	C₄	—	776	1	no	—
114	C₄	#15690830	1	2	yes	—
115	C₄	#15690840	1	2	yes	—
116	C₂	—	8	2	no	—
117	C₂	#20260897	1	2	yes	—
118	C₂	#20260903	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20261107

C₂

yes

maximal, characteristic

#20261030

C₂

yes

maximal, complemented

#20260479

C₂

yes

maximal, characteristic

Q₈⋊#11715895

C₂

yes

maximal, complemented

#559×C₉₇

C₂

yes

maximal, characteristic

C₂×#13095499

—

#15690810

C₂²

yes

characteristic

#15690743

—

C₄⋊#11715895

C₂²

yes

—

#15690734

—

C₄×#11715895

C₂²

yes

characteristic

#127×C₉₇

C₂²

yes

characteristic

#124×C₉₇

—

#123×C₉₇

—

C₄×Q₈×C₉₇

C₂²

yes

—

C₄×D₄×C₉₇

—

C₂×Dic₁₉₄

—

#13095499

—

C₂×#11715895

—

C₂×#11715895

—

C₄⋊Dic₉₇

—

C₂×Q₈×C₉₇

D₄

yes

—

C₂×D₄×C₉₇

—

C₂×D₄×C₉₇

D₄

yes

—

C₂²×C₄×C₉₇

—

C₄⋊C₄×C₉₇

—

C₄⋊C₄×C₉₇

D₄

yes

—

C₂²⋊C₄×C₉₇

—

C₂²⋊C₄×C₉₇

—

C₄²×C₉₇

—

C₄²×C₉₇

C₂³

yes

characteristic

C₂³×C₉₇

—

Q₈×C₉₇

—

Q₈×C₉₇

—

D₄×C₉₇

—

D₄×C₉₇

—

D₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

C₂×D₄

yes

characteristic

C₂×C₄×C₉₇

C₂×D₄

yes

—

C₂×Dic₉₇

—

Dic₁₉₄

—

#11715895

—

#11715895

—

C₂²×C₉₇

—

C₂²×C₉₇

—

C₂²×C₉₇

#120

yes

characteristic

C₄×C₉₇

—

C₄×C₉₇

—

C₄×C₉₇

—

C₄×C₉₇

#135

yes

—

C₄×C₉₇

#137

yes

—

Dic₉₇

—

C₂×C₉₇

—

C₂×C₉₇

#452

yes

—

C₂×C₉₇

yes

—

C₉₇

yes

characteristic, complemented, Sylow

#559

D₉₇

yes

—

—

—

Q₈⋊C₈

—

C₂×SD₁₆

—

194

—

C₄⋊Q₈

—

#127

C₂×D₉₇

yes

characteristic

#124

—

#123

—

C₄×Q₈

C₂×D₉₇

yes

—

C₄×D₄

—

#107

—

194

—

C₄⋊C₈

—

#103

—

194

—

C₄×C₈

—

C₂×Q₈

—

194

—

C₂×Q₈

C₂²⋊D₉₇

yes

—

C₂×D₄

—

C₂×D₄

C₂²⋊D₉₇

yes

—

C₂²×C₄

—

SD₁₆

—

388

—

C₂×C₈

—

194

—

C₂×C₈

—

194

—

C₄⋊C₄

—

C₄⋊C₄

—

194

—

C₄⋊C₄

C₂²⋊D₉₇

yes

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

C₂²×D₉₇

yes

characteristic

C₂³

—

Q₈

—

Q₈

—

Q₈

—

388

—

D₄

—

D₄

—

D₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

388

—

103

C₂×C₄

C₂×C₂²⋊D₉₇

yes

characteristic

104

C₂×C₄

C₂×C₂²⋊D₉₇

yes

—

105

C₈

—

388

—

106

C₈

—

194

—

107

C₂²

—

108

C₂²

—

109

C₂²

#15690779

yes

characteristic

110

C₄

—

111

C₄

—

112

C₄

—

113

C₄

—

776

—

114

C₄

#15690830

yes

—

115

C₄

#15690840

yes

—

116

C₂

—

117

C₂

#20260897

yes

—

118

C₂

#20260903

yes

characteristic

	subgroup	G/H
center	C₂²	#15690779
commutator, cosocle	C₄²×C₉₇	C₂³
Frattini	C₄²	C₂²×D₉₇
Fitting	#559×C₉₇	C₂
socle	C₂²×C₉₇	#120
prime radical	#127×C₉₇	C₂²
hypercenter	#559	D₉₇
hypocenter	C₉₇	#1630
LCS	G ▷ C₄²×C₉₇ ▷ C₂²×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #559

subgroup

G/H

center

C₂²

#15690779

commutator, cosocle

C₄²×C₉₇

C₂³

Frattini

C₄²

C₂²×D₉₇

Fitting

#559×C₉₇

C₂

socle

C₂²×C₉₇

#120

prime radical

#127×C₉₇

C₂²

D₉₇

C₉₇

G ▷ C₄²×C₉₇ ▷ C₂²×C₉₇ ▷ C₉₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #559

N	H ≅ G/N	type
2. #20261030	C₂	semidirect
4. Q₈⋊#11715895	C₂	semidirect
60. C₉₇	#1630	semidirect

H ≅ G/N

type

2. #20261030

C₂

semidirect

4. Q₈⋊#11715895

C₂

semidirect

60. C₉₇

#1630

semidirect

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

subgroup	index
#20261107	2
#20261030	2
#20260479	2
Q₈⋊#11715895	2
#559×C₉₇	2
#1630	97

subgroup

index

#559×C₉₇

Subgroups — #27140857

Conjugacy classes of subgroups

Complements

Maximal subgroups