Group #27143280 — subgroups

Subgroups — #27143280

Order 12416.

254 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 254 conjugacy classes of subgroups — too many to draw. Only its 91 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×#15690741	C₂	1	2	yes	maximal
2	C₂×#15690733	C₂	1	2	yes	maximal
3	#20260641	C₂	1	1	yes	maximal, characteristic
4	#20260490	C₂	1	1	yes	maximal, characteristic
5	#384×C₉₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₁₉₄	C₂²	1	1	yes	characteristic
7	C₂²×#11715895	C₂²	1	2	yes	—
8	C₂×C₄⋊Dic₉₇	C₂²	1	1	yes	characteristic
9	#15690741	—	2	4	no	—
10	#15690733	—	2	4	no	—
11	#15690717	—	2	2	no	—
12	C₂×C₄⋊C₄×C₉₇	—	2	2	no	—
13	C₂×C₄⋊C₄×C₉₇	C₂²	1	2	yes	—
14	C₂×C₄²×C₉₇	C₂²	1	1	yes	characteristic
15	C₂×Dic₁₉₄	—	2	2	no	—
16	C₂×Dic₁₉₄	—	2	2	no	—
17	C₂×Dic₁₉₄	—	2	1	no	—
18	C₂×Dic₁₉₄	Q₈	1	2	yes	—
19	C₂²×Dic₉₇	—	2	1	no	—
20	C₂²×Dic₉₇	—	2	1	no	—
21	C₂×#11715895	—	2	2	no	—
22	C₂×#11715895	—	2	2	no	—
23	C₂×#11715895	D₄	1	4	yes	—
24	C₄⋊Dic₉₇	—	2	2	no	—
25	C₂²×C₄×C₉₇	—	2	2	no	—
26	C₂²×C₄×C₉₇	D₄	1	2	yes	—
27	C₂²×C₄×C₉₇	C₂³	1	1	yes	characteristic
28	C₄⋊C₄×C₉₇	—	2	2	no	—
29	C₄⋊C₄×C₉₇	—	2	2	no	—
30	C₄⋊C₄×C₉₇	—	4	4	no	—
31	C₄²×C₉₇	—	2	2	no	—
32	C₂³×C₉₇	C₂×D₄	1	1	yes	characteristic
33	C₂×C₄×C₉₇	—	2	2	no	—
34	C₂×C₄×C₉₇	—	2	2	no	—
35	C₂×C₄×C₉₇	—	4	2	no	—
36	C₂×C₄×C₉₇	—	4	4	no	—
37	C₂×C₄×C₉₇	SD₁₆	1	4	yes	—
38	C₂×C₄×C₉₇	C₂×D₄	1	2	yes	—
39	C₂×C₄×C₉₇	C₂×Q₈	1	1	yes	characteristic
40	C₂×C₄×C₉₇	#41	1	2	yes	—
41	C₂×C₄×C₉₇	#41	1	1	yes	characteristic
42	C₂×Dic₉₇	—	2	2	no	—
43	C₂×Dic₉₇	—	4	1	no	—
44	C₂×Dic₉₇	—	4	1	no	—
45	C₂×Dic₉₇	—	4	1	no	—
46	C₂×Dic₉₇	—	4	2	no	—
47	Dic₁₉₄	—	2	4	no	—
48	Dic₁₉₄	—	4	2	no	—
49	#11715895	—	2	4	no	—
50	C₂²×C₉₇	C₂²⋊Q₈	1	1	yes	characteristic
51	C₂²×C₉₇	#123	1	1	yes	characteristic
52	C₂²×C₉₇	#127	1	1	yes	characteristic
53	C₂²×C₉₇	C₂×SD₁₆	1	2	yes	—
54	C₂²×C₉₇	#137	1	2	yes	—
55	C₄×C₉₇	—	2	4	no	—
56	C₄×C₉₇	—	8	2	no	—
57	C₄×C₉₇	C₂²⋊Q₈	1	2	yes	—
58	C₄×C₉₇	#123	1	2	yes	—
59	Dic₉₇	—	4	2	no	—
60	Dic₉₇	—	8	1	no	—
61	C₂×C₉₇	#399	1	1	yes	characteristic
62	C₂×C₉₇	#475	1	2	yes	—
63	C₂×C₉₇	#483	1	2	yes	—
64	C₂×C₉₇	#492	1	1	yes	characteristic
65	C₂×C₉₇	#497	1	1	yes	characteristic
66	#2096	—	97	1	no	maximal, Sylow
67	C₉₇	#2096	1	1	yes	characteristic, complemented, Sylow
68	C₂×#106	—	97	2	no	—
69	C₂×#103	—	97	2	no	—
70	#391	—	97	1	no	—
71	#384	D₉₇	1	1	yes	characteristic
72	#336	—	97	1	no	—
73	C₂²×Q₈	—	97	1	no	—
74	C₂²×C₈	—	97	2	no	—
75	C₂×C₄⋊C₄	—	2	2	no	—
76	C₂×C₄⋊C₄	—	97	1	no	—
77	C₂×C₄⋊C₄	C₂×D₉₇	1	2	yes	—
78	C₂×C₄²	C₂×D₉₇	1	1	yes	characteristic
79	#106	—	194	4	no	—
80	#103	—	194	4	no	—
81	#95	—	194	2	no	—
82	C₂×Q₈	—	97	2	no	—
83	C₂×Q₈	—	194	2	no	—
84	C₂×Q₈	—	194	2	no	—
85	C₂×Q₈	—	194	1	no	—
86	C₂²×C₄	—	2	2	no	—
87	C₂²×C₄	—	194	1	no	—
88	C₂²×C₄	—	194	1	no	—
89	C₂²×C₄	C₂²⋊D₉₇	1	2	yes	—
90	C₂²×C₄	C₂²×D₉₇	1	1	yes	characteristic
91	C₂×C₈	—	97	4	no	—
92	C₂×C₈	—	194	2	no	—
93	C₂×C₈	—	194	2	no	—
94	C₄⋊C₄	—	2	2	no	—
95	C₄⋊C₄	—	2	2	no	—
96	C₄⋊C₄	—	4	4	no	—
97	C₄⋊C₄	—	194	2	no	—
98	C₄²	—	2	2	no	—
99	C₂³	C₂×C₂²⋊D₉₇	1	1	yes	characteristic
100	Q₈	—	388	2	no	—
101	Q₈	—	194	4	no	—
102	C₂×C₄	—	2	2	no	—
103	C₂×C₄	—	2	2	no	—
104	C₂×C₄	—	4	2	no	—
105	C₂×C₄	—	4	4	no	—
106	C₂×C₄	—	388	1	no	—
107	C₂×C₄	—	388	2	no	—
108	C₂×C₄	—	388	1	no	—
109	C₂×C₄	—	388	1	no	—
110	C₂×C₄	—	194	2	no	—
111	C₂×C₄	#13095499	1	4	yes	—
112	C₂×C₄	D₄×D₉₇	1	2	yes	—
113	C₂×C₄	Q₈×D₉₇	1	1	yes	characteristic
114	C₂×C₄	#13095511	1	2	yes	—
115	C₂×C₄	#13095512	1	1	yes	characteristic
116	C₈	—	194	4	no	—
117	C₂²	#15690786	1	1	yes	characteristic
118	C₂²	#15690793	1	1	yes	characteristic
119	C₂²	#15690806	1	1	yes	characteristic
120	C₂²	C₂×#13095499	1	2	yes	—
121	C₂²	#15690841	1	2	yes	—
122	C₄	—	2	4	no	—
123	C₄	—	8	2	no	—
124	C₄	—	776	1	no	—
125	C₄	—	388	2	no	—
126	C₄	#15690788	1	2	yes	—
127	C₄	#15690795	1	2	yes	—
128	C₂	#20260670	1	1	yes	characteristic
129	C₂	#20261017	1	2	yes	—
130	C₂	#20261065	1	2	yes	—
131	C₂	#20261107	1	1	yes	characteristic
132	C₂	#20261127	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15690741

C₂

yes

maximal

C₂×#15690733

C₂

yes

maximal

#20260641

C₂

yes

maximal, characteristic

#20260490

C₂

yes

maximal, characteristic

#384×C₉₇

C₂

yes

maximal, characteristic

C₂²×Dic₁₉₄

C₂²

yes

characteristic

C₂²×#11715895

C₂²

yes

—

C₂×C₄⋊Dic₉₇

C₂²

yes

—

—

—

C₂×C₄⋊C₄×C₉₇

—

C₂×C₄⋊C₄×C₉₇

C₂²

yes

—

C₂×C₄²×C₉₇

C₂²

yes

characteristic

C₂×Dic₁₉₄

—

C₂×Dic₁₉₄

—

C₂×Dic₁₉₄

—

C₂×Dic₁₉₄

Q₈

yes

—

C₂²×Dic₉₇

—

C₂²×Dic₉₇

—

C₂×#11715895

—

C₂×#11715895

—

C₂×#11715895

D₄

yes

—

C₄⋊Dic₉₇

—

C₂²×C₄×C₉₇

—

C₂²×C₄×C₉₇

D₄

yes

—

C₂²×C₄×C₉₇

C₂³

yes

characteristic

C₄⋊C₄×C₉₇

—

C₄⋊C₄×C₉₇

—

C₄⋊C₄×C₉₇

—

C₄²×C₉₇

—

C₂³×C₉₇

C₂×D₄

yes

characteristic

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

SD₁₆

yes

—

C₂×C₄×C₉₇

C₂×D₄

yes

—

C₂×C₄×C₉₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₉₇

#41

yes

—

C₂×C₄×C₉₇

#41

yes

characteristic

C₂×Dic₉₇

—

C₂×Dic₉₇

—

C₂×Dic₉₇

—

C₂×Dic₉₇

—

C₂×Dic₉₇

—

Dic₁₉₄

—

Dic₁₉₄

—

#11715895

—

C₂²×C₉₇

C₂²⋊Q₈

yes

C₂²×C₉₇

yes

C₂²×C₉₇

yes

C₂²×C₉₇

C₂×SD₁₆

yes

—

C₂²×C₉₇

#137

yes

—

C₄×C₉₇

—

C₄×C₉₇

—

C₄×C₉₇

C₂²⋊Q₈

yes

—

C₄×C₉₇

#123

yes

—

Dic₉₇

—

Dic₉₇

—

C₂×C₉₇

#399

yes

characteristic

C₂×C₉₇

#475

yes

—

C₂×C₉₇

#483

yes

—

C₂×C₉₇

yes

C₂×C₉₇

yes

—

C₉₇

yes

characteristic, complemented, Sylow

—

—

—

D₉₇

yes

characteristic

#336

—

C₂²×Q₈

—

C₂²×C₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₉₇

yes

—

C₂×C₄²

C₂×D₉₇

yes

characteristic

#106

—

194

—

#103

—

194

—

#95

—

194

—

C₂×Q₈

—

C₂×Q₈

—

194

—

C₂×Q₈

—

194

—

C₂×Q₈

—

194

—

C₂²×C₄

—

C₂²×C₄

—

194

—

C₂²×C₄

—

194

—

C₂²×C₄

C₂²⋊D₉₇

yes

—

C₂²×C₄

C₂²×D₉₇

yes

characteristic

C₂×C₈

—

C₂×C₈

—

194

—

C₂×C₈

—

194

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

194

—

C₄²

—

C₂³

C₂×C₂²⋊D₉₇

yes

characteristic

100

Q₈

—

388

—

101

Q₈

—

194

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

388

—

107

C₂×C₄

—

388

—

108

C₂×C₄

—

388

—

109

C₂×C₄

—

388

—

110

C₂×C₄

—

194

—

111

C₂×C₄

#13095499

yes

—

112

C₂×C₄

D₄×D₉₇

yes

—

113

C₂×C₄

Q₈×D₉₇

yes

characteristic

114

C₂×C₄

#13095511

yes

—

115

C₂×C₄

#13095512

yes

characteristic

116

C₈

—

194

—

117

C₂²

#15690786

yes

characteristic

118

C₂²

#15690793

yes

characteristic

119

C₂²

#15690806

yes

characteristic

120

C₂²

C₂×#13095499

yes

—

121

C₂²

#15690841

yes

—

122

C₄

—

123

C₄

—

124

C₄

—

776

—

125

C₄

—

388

—

126

C₄

#15690788

yes

—

127

C₄

#15690795

yes

—

128

C₂

#20260670

yes

characteristic

129

C₂

#20261017

yes

—

130

C₂

#20261065

yes

—

131

C₂

#20261107

yes

characteristic

132

C₂

#20261127

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₉₇
commutator, cosocle	C₂²×C₄×C₉₇	C₂³
Frattini	C₂²×C₄	C₂²×D₉₇
Fitting	#384×C₉₇	C₂
socle, prime radical	C₂³×C₉₇	C₂×D₄
hypercenter	#384	D₉₇
hypocenter	C₉₇	#2096
LCS	G ▷ C₂²×C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #384

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₉₇

commutator, cosocle

C₂²×C₄×C₉₇

C₂³

Frattini

C₂²×C₄

C₂²×D₉₇

Fitting

#384×C₉₇

C₂

socle, prime radical

C₂³×C₉₇

C₂×D₄

D₉₇

C₉₇

G ▷ C₂²×C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #384

N	H ≅ G/N	type
67. C₉₇	#2096	semidirect

H ≅ G/N

type

67. C₉₇

#2096

semidirect

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

subgroup	index
C₂×#15690741	2
C₂×#15690733	2
#20260641	2
#20260490	2
#384×C₉₇	2
#2096	97

subgroup

index

#384×C₉₇

Subgroups — #27143280

Conjugacy classes of subgroups

Complements

Maximal subgroups