Group #27141732 — subgroups

Subgroups — #27141732

Order 12416.

272 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 113 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	#20260759	C₂	1	2	yes	maximal
2	C₂×#15690729	C₂	1	2	yes	maximal
3	#20260504	C₂	1	2	yes	maximal
4	C₂×C₄×Q₈×C₉₇	C₂	1	1	yes	maximal, characteristic
5	C₂×#13095495	C₂²	1	4	yes	—
6	C₄⋊#11715895	—	2	2	no	—
7	C₄⋊#11715895	—	2	2	no	—
8	#15690729	—	2	4	no	—
9	C₂²×Q₈×C₉₇	C₂²	1	1	yes	characteristic
10	C₄×Q₈×C₉₇	—	2	2	no	—
11	C₄×Q₈×C₉₇	—	2	2	no	—
12	C₂×C₄⋊C₄×C₉₇	C₄	1	2	yes	—
13	C₂×C₄⋊C₄×C₉₇	C₂²	1	1	yes	characteristic
14	C₂×C₄²×C₉₇	C₄	1	2	yes	—
15	C₂×C₄²×C₉₇	C₂²	1	1	yes	characteristic
16	#13095495	—	2	4	no	—
17	#13095495	—	2	4	no	—
18	C₂×#11715895	—	2	4	no	—
19	C₂×Q₈×C₉₇	—	2	2	no	—
20	C₂×Q₈×C₉₇	—	2	2	no	—
21	C₂×Q₈×C₉₇	D₄	1	2	yes	—
22	C₂×Q₈×C₉₇	Q₈	1	2	yes	—
23	C₂²×C₄×C₉₇	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₄×C₉₇	C₂×C₄	1	2	yes	—
25	C₂²×C₄×C₉₇	C₂×C₄	1	2	yes	—
26	C₂²×C₄×C₉₇	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₄×C₉₇	C₂³	1	1	yes	characteristic
28	C₄⋊C₄×C₉₇	—	2	4	no	—
29	C₄⋊C₄×C₉₇	—	2	2	no	—
30	C₄²×C₉₇	—	2	2	no	—
31	C₄²×C₉₇	—	2	2	no	—
32	C₄²×C₉₇	D₄	1	2	yes	—
33	C₄²×C₉₇	D₄	1	2	yes	—
34	C₂³×C₉₇	C₂²×C₄	1	1	yes	characteristic
35	Q₈×C₉₇	—	2	4	no	—
36	Q₈×C₉₇	—	2	4	no	—
37	C₂×C₄×C₉₇	—	2	2	no	—
38	C₂×C₄×C₉₇	—	2	2	no	—
39	C₂×C₄×C₉₇	—	2	2	no	—
40	C₂×C₄×C₉₇	—	2	2	no	—
41	C₂×C₄×C₉₇	—	4	1	no	—
42	C₂×C₄×C₉₇	—	4	2	no	—
43	C₂×C₄×C₉₇	C₂²⋊C₄	1	1	yes	characteristic
44	C₂×C₄×C₉₇	C₂²⋊C₄	1	2	yes	—
45	C₂×C₄×C₉₇	C₂²⋊C₄	1	1	yes	characteristic
46	C₂×C₄×C₉₇	C₄⋊C₄	1	4	yes	—
47	C₂×C₄×C₉₇	C₂×D₄	1	2	yes	—
48	C₂×C₄×C₉₇	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₉₇	C₂×Q₈	1	1	yes	characteristic
50	C₂×C₄×C₉₇	#41	1	2	yes	—
51	#11715895	—	4	4	no	—
52	C₂²×C₉₇	—	2	1	no	—
53	C₂²×C₉₇	—	2	1	no	—
54	C₂²×C₉₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
55	C₂²×C₉₇	C₂×C₄⋊C₄	1	1	yes	characteristic
56	C₂²×C₉₇	#117	1	1	yes	characteristic
57	C₄×C₉₇	—	2	4	no	—
58	C₄×C₉₇	—	2	1	no	—
59	C₄×C₉₇	—	2	2	no	—
60	C₄×C₉₇	—	4	2	no	—
61	C₄×C₉₇	#101	1	2	yes	complemented
62	C₄×C₉₇	#121	1	2	yes	—
63	C₄×C₉₇	C₂²⋊Q₈	1	2	yes	—
64	C₂×C₉₇	—	2	1	no	—
65	C₂×C₉₇	—	2	1	no	—
66	C₂×C₉₇	#380	1	1	yes	characteristic
67	C₂×C₉₇	C₂×#101	1	1	yes	characteristic
68	C₂×C₉₇	#413	1	1	yes	characteristic
69	#1801	—	97	1	no	maximal, Sylow
70	C₉₇	#1801	1	1	yes	characteristic, complemented, Sylow
71	C₂×C₄×Q₈	D₉₇	1	1	yes	characteristic
72	#423	—	97	2	no	—
73	C₂×#101	—	97	2	no	—
74	#343	—	97	2	no	—
75	C₂²×Q₈	C₂×D₉₇	1	1	yes	characteristic
76	C₂×M₁₆	—	97	4	no	—
77	C₄×Q₈	—	2	2	no	—
78	C₄×Q₈	—	2	2	no	—
79	C₂×C₄⋊C₄	Dic₉₇	1	2	yes	—
80	C₂×C₄⋊C₄	C₂×D₉₇	1	1	yes	characteristic
81	C₂×C₄²	Dic₉₇	1	2	yes	—
82	C₂×C₄²	C₂×D₉₇	1	1	yes	characteristic
83	C₄⋊C₈	—	194	2	no	—
84	C₄⋊C₈	—	194	2	no	—
85	#101	—	194	4	no	—
86	C₂×Q₈	—	2	2	no	—
87	C₂×Q₈	—	2	2	no	—
88	C₂×Q₈	Dic₁₉₄	1	2	yes	—
89	C₂×Q₈	D₃₈₈	1	2	yes	—
90	C₂²×C₄	C₂×Dic₉₇	1	1	yes	characteristic
91	C₂²×C₄	C₂×Dic₉₇	1	2	yes	—
92	C₂²×C₄	C₂×Dic₉₇	1	2	yes	—
93	C₂²×C₄	C₂×Dic₉₇	1	1	yes	characteristic
94	C₂²×C₄	C₂²×D₉₇	1	1	yes	characteristic
95	M₁₆	—	194	4	no	—
96	M₁₆	—	194	4	no	—
97	C₂×C₈	—	194	4	no	—
98	C₄⋊C₄	—	2	4	no	—
99	C₄⋊C₄	—	2	2	no	—
100	C₄²	—	2	2	no	—
101	C₄²	—	2	2	no	—
102	C₄²	C₂²⋊D₉₇	1	2	yes	—
103	C₄²	C₂²⋊D₉₇	1	2	yes	—
104	C₂³	C₂²×Dic₉₇	1	1	yes	characteristic
105	Q₈	—	2	4	no	—
106	Q₈	—	2	4	no	—
107	C₂×C₄	—	2	2	no	—
108	C₂×C₄	—	2	2	no	—
109	C₂×C₄	—	2	2	no	—
110	C₂×C₄	—	2	2	no	—
111	C₂×C₄	—	4	1	no	—
112	C₂×C₄	—	4	2	no	—
113	C₂×C₄	C₂²⋊Dic₉₇	1	1	yes	characteristic
114	C₂×C₄	C₂²⋊Dic₉₇	1	2	yes	—
115	C₂×C₄	C₂²⋊Dic₉₇	1	1	yes	characteristic
116	C₂×C₄	C₄⋊Dic₉₇	1	4	yes	—
117	C₂×C₄	C₂×C₂²⋊D₉₇	1	2	yes	—
118	C₂×C₄	C₂×D₃₈₈	1	1	yes	characteristic
119	C₂×C₄	C₂×Dic₁₉₄	1	1	yes	characteristic
120	C₂×C₄	#13095511	1	2	yes	—
121	C₈	—	388	4	no	—
122	C₂²	—	2	1	no	—
123	C₂²	—	2	1	no	—
124	C₂²	C₂×C₂²⋊Dic₉₇	1	1	yes	characteristic
125	C₂²	C₂×C₄⋊Dic₉₇	1	1	yes	characteristic
126	C₂²	#15690769	1	1	yes	characteristic
127	C₄	—	2	1	no	—
128	C₄	—	2	2	no	—
129	C₄	—	2	4	no	—
130	C₄	—	4	2	no	—
131	C₄	#15690729	1	2	yes	complemented
132	C₄	#15690784	1	2	yes	—
133	C₄	#15690789	1	2	yes	—
134	C₂	—	2	1	no	—
135	C₂	—	2	1	no	—
136	C₂	#20260595	1	1	yes	characteristic
137	C₂	C₂×#15690729	1	1	yes	characteristic
138	C₂	#20260719	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20260759

C₂

yes

maximal

C₂×#15690729

C₂

yes

maximal

#20260504

C₂

yes

maximal

C₂×C₄×Q₈×C₉₇

C₂

yes

maximal, characteristic

C₂×#13095495

C₂²

yes

—

C₄⋊#11715895

—

C₄⋊#11715895

—

#15690729

—

C₂²×Q₈×C₉₇

C₂²

yes

characteristic

C₄×Q₈×C₉₇

—

C₄×Q₈×C₉₇

—

C₂×C₄⋊C₄×C₉₇

C₄

yes

—

C₂×C₄⋊C₄×C₉₇

C₂²

yes

characteristic

C₂×C₄²×C₉₇

C₄

yes

—

C₂×C₄²×C₉₇

C₂²

yes

—

—

—

C₂×Q₈×C₉₇

—

C₂×Q₈×C₉₇

—

C₂×Q₈×C₉₇

D₄

yes

—

C₂×Q₈×C₉₇

Q₈

yes

—

C₂²×C₄×C₉₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₉₇

C₂×C₄

yes

—

C₂²×C₄×C₉₇

C₂×C₄

yes

—

C₂²×C₄×C₉₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₉₇

C₂³

yes

characteristic

C₄⋊C₄×C₉₇

—

C₄⋊C₄×C₉₇

—

C₄²×C₉₇

—

C₄²×C₉₇

—

C₄²×C₉₇

D₄

yes

—

C₄²×C₉₇

D₄

yes

—

C₂³×C₉₇

C₂²×C₄

yes

characteristic

Q₈×C₉₇

—

Q₈×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₉₇

C₂²⋊C₄

yes

—

C₂×C₄×C₉₇

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₉₇

C₄⋊C₄

yes

—

C₂×C₄×C₉₇

C₂×D₄

yes

—

C₂×C₄×C₉₇

C₂×D₄

yes

characteristic

C₂×C₄×C₉₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₉₇

#41

yes

—

#11715895

—

C₂²×C₉₇

—

C₂²×C₉₇

—

C₂²×C₉₇

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₉₇

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₉₇

#117

yes

characteristic

C₄×C₉₇

—

C₄×C₉₇

—

C₄×C₉₇

—

C₄×C₉₇

—

C₄×C₉₇

#101

yes

complemented

C₄×C₉₇

#121

yes

—

C₄×C₉₇

C₂²⋊Q₈

yes

—

C₂×C₉₇

—

C₂×C₉₇

—

C₂×C₉₇

yes

C₂×C₉₇

yes

C₂×C₉₇

yes

—

C₉₇

yes

characteristic, complemented, Sylow

C₂×C₄×Q₈

D₉₇

yes

—

—

—

C₂²×Q₈

C₂×D₉₇

yes

characteristic

C₂×M₁₆

—

C₄×Q₈

—

C₄×Q₈

—

C₂×C₄⋊C₄

Dic₉₇

yes

—

C₂×C₄⋊C₄

C₂×D₉₇

yes

characteristic

C₂×C₄²

Dic₉₇

yes

—

C₂×C₄²

C₂×D₉₇

yes

characteristic

C₄⋊C₈

—

194

—

C₄⋊C₈

—

194

—

#101

—

194

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

Dic₁₉₄

yes

—

C₂×Q₈

D₃₈₈

yes

—

C₂²×C₄

C₂×Dic₉₇

yes

characteristic

C₂²×C₄

C₂×Dic₉₇

yes

—

C₂²×C₄

C₂×Dic₉₇

yes

—

C₂²×C₄

C₂×Dic₉₇

yes

characteristic

C₂²×C₄

C₂²×D₉₇

yes

characteristic

M₁₆

—

194

—

M₁₆

—

194

—

C₂×C₈

—

194

—

C₄⋊C₄

—

C₄⋊C₄

—

100

C₄²

—

101

C₄²

—

102

C₄²

C₂²⋊D₉₇

yes

—

103

C₄²

C₂²⋊D₉₇

yes

—

104

C₂³

C₂²×Dic₉₇

yes

characteristic

105

Q₈

—

106

Q₈

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

C₂²⋊Dic₉₇

yes

characteristic

114

C₂×C₄

C₂²⋊Dic₉₇

yes

—

115

C₂×C₄

C₂²⋊Dic₉₇

yes

characteristic

116

C₂×C₄

C₄⋊Dic₉₇

yes

—

117

C₂×C₄

C₂×C₂²⋊D₉₇

yes

—

118

C₂×C₄

C₂×D₃₈₈

yes

characteristic

119

C₂×C₄

C₂×Dic₁₉₄

yes

characteristic

120

C₂×C₄

#13095511

yes

—

121

C₈

—

388

—

122

C₂²

—

123

C₂²

—

124

C₂²

C₂×C₂²⋊Dic₉₇

yes

characteristic

125

C₂²

C₂×C₄⋊Dic₉₇

yes

characteristic

126

C₂²

#15690769

yes

characteristic

127

C₄

—

128

C₄

—

129

C₄

—

130

C₄

—

131

C₄

#15690729

yes

complemented

132

C₄

#15690784

yes

—

133

C₄

#15690789

yes

—

134

C₂

—

135

C₂

—

136

C₂

#20260595

yes

characteristic

137

C₂

C₂×#15690729

yes

characteristic

138

C₂

#20260719

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊Dic₉₇
commutator, prime radical	C₂³×C₉₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₉₇
Fitting	C₂×C₄×Q₈×C₉₇	C₂
socle	C₂²×C₉₇	C₂×C₂²⋊C₄
cosocle	C₂²×C₄×C₉₇	C₂³
hypercenter	C₂×C₄×Q₈	D₉₇
hypocenter	C₉₇	#1801
LCS	G ▷ C₂³×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×Q₈

subgroup

G/H

center

C₂²

C₂×C₂²⋊Dic₉₇

commutator, prime radical

C₂³×C₉₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₉₇

Fitting

C₂×C₄×Q₈×C₉₇

C₂

socle

C₂²×C₉₇

C₂×C₂²⋊C₄

cosocle

C₂²×C₄×C₉₇

C₂³

hypercenter

C₂×C₄×Q₈

D₉₇

hypocenter

C₉₇

#1801

LCS

G ▷ C₂³×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×Q₈

N	H ≅ G/N	type
61. C₄×C₉₇	#101	semidirect
70. C₉₇	#1801	semidirect
131. C₄	#15690729	semidirect

H ≅ G/N

type

61. C₄×C₉₇

#101

semidirect

70. C₉₇

#1801

semidirect

131. C₄

#15690729

semidirect

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

subgroup	index
#20260759	2
C₂×#15690729	2
#20260504	2
C₂×C₄×Q₈×C₉₇	2
#1801	97

subgroup

index

C₂×C₄×Q₈×C₉₇

Subgroups — #27141732

Conjugacy classes of subgroups

Complements

Maximal subgroups