Group #27142726 — subgroups

Subgroups — #27142726

Order 12416.

260 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 105 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	1	yes	maximal, characteristic
2	C₄×#13095495	C₂	1	1	yes	maximal, characteristic
3	#20260504	C₂	1	4	yes	maximal
4	C₂×C₄⋊Q₈×C₉₇	C₂	1	1	yes	maximal, characteristic
5	C₂×#13095495	C₂²	1	2	yes	—
6	C₂×#13095495	C₂²	1	2	yes	—
7	C₄⋊#11715895	—	2	1	no	—
8	C₄⋊#11715895	—	2	1	no	—
9	#15690721	—	2	1	no	—
10	C₄×#11715895	—	2	1	no	—
11	C₂²×Q₈×C₉₇	C₄	1	2	yes	—
12	C₄⋊Q₈×C₉₇	—	2	1	no	—
13	C₄⋊Q₈×C₉₇	—	2	1	no	—
14	C₄⋊Q₈×C₉₇	—	2	1	no	—
15	C₄⋊Q₈×C₉₇	—	2	1	no	—
16	C₂×C₄⋊C₄×C₉₇	C₄	1	2	yes	—
17	C₂×C₄⋊C₄×C₉₇	C₂²	1	2	yes	—
18	C₂×C₄²×C₉₇	C₂²	1	1	yes	characteristic
19	#13095495	—	2	2	no	—
20	#13095495	—	2	4	no	—
21	#13095495	Q₈	1	4	yes	complemented
22	C₂×#11715895	—	2	2	no	—
23	C₂×#11715895	—	2	2	no	—
24	C₂×Q₈×C₉₇	—	2	2	no	—
25	C₂×Q₈×C₉₇	—	2	2	no	—
26	C₂×Q₈×C₉₇	—	4	4	no	—
27	C₂²×C₄×C₉₇	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₄×C₉₇	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₄×C₉₇	C₂×C₄	1	4	yes	—
30	C₂²×C₄×C₉₇	C₂³	1	1	yes	characteristic
31	C₄⋊C₄×C₉₇	—	2	2	no	—
32	C₄⋊C₄×C₉₇	—	2	2	no	—
33	C₄⋊C₄×C₉₇	—	2	2	no	—
34	C₄⋊C₄×C₉₇	—	2	2	no	—
35	C₄²×C₉₇	D₄	1	1	yes	characteristic
36	C₄²×C₉₇	D₄	1	1	yes	characteristic
37	C₄²×C₉₇	D₄	1	1	yes	characteristic
38	C₄²×C₉₇	D₄	1	1	yes	characteristic
39	C₂³×C₉₇	C₂²×C₄	1	1	yes	characteristic
40	Q₈×C₉₇	—	4	4	no	—
41	Q₈×C₉₇	—	4	4	no	—
42	C₂×C₄×C₉₇	—	2	2	no	—
43	C₂×C₄×C₉₇	—	2	2	no	—
44	C₂×C₄×C₉₇	—	2	4	no	—
45	C₂×C₄×C₉₇	—	4	4	no	—
46	C₂×C₄×C₉₇	C₂²⋊C₄	1	1	yes	characteristic
47	C₂×C₄×C₉₇	C₂²⋊C₄	1	1	yes	characteristic
48	C₂×C₄×C₉₇	C₂²⋊C₄	1	1	yes	characteristic
49	C₂×C₄×C₉₇	C₂²⋊C₄	1	1	yes	characteristic
50	C₂×C₄×C₉₇	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₉₇	C₂×D₄	1	1	yes	characteristic
52	C₂×C₄×C₉₇	C₂×Q₈	1	2	yes	—
53	C₂×C₄×C₉₇	#41	1	2	yes	—
54	#11715895	—	2	4	no	—
55	#11715895	—	4	2	no	—
56	C₂²×C₉₇	—	2	1	no	—
57	C₂²×C₉₇	—	2	1	no	—
58	C₂²×C₉₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
59	C₂²×C₉₇	C₄×Q₈	1	2	yes	—
60	C₄×C₉₇	—	2	1	no	—
61	C₄×C₉₇	—	2	1	no	—
62	C₄×C₉₇	—	4	4	no	—
63	C₄×C₉₇	#100	1	2	yes	—
64	C₄×C₉₇	#101	1	2	yes	—
65	C₄×C₉₇	C₂²⋊Q₈	1	2	yes	—
66	C₄×C₉₇	#124	1	1	yes	characteristic
67	C₄×C₉₇	C₄⋊Q₈	1	1	yes	characteristic
68	C₂×C₉₇	—	2	2	no	—
69	C₂×C₉₇	#391	1	1	yes	characteristic
70	C₂×C₉₇	C₂×#100	1	1	yes	characteristic
71	C₂×C₉₇	C₂×#101	1	1	yes	characteristic
72	#1997	—	97	1	no	maximal, Sylow
73	C₉₇	#1997	1	1	yes	characteristic, complemented, Sylow
74	C₂×C₄⋊Q₈	D₉₇	1	1	yes	characteristic
75	#423	—	97	1	no	—
76	C₄×M₁₆	—	97	1	no	—
77	#343	—	97	4	no	—
78	C₂²×Q₈	Dic₉₇	1	2	yes	—
79	C₂×M₁₆	—	97	2	no	—
80	C₂×M₁₆	—	97	2	no	—
81	C₄⋊Q₈	—	2	1	no	—
82	C₄⋊Q₈	—	2	1	no	—
83	C₄⋊Q₈	—	2	1	no	—
84	C₄⋊Q₈	—	2	1	no	—
85	C₂×C₄⋊C₄	Dic₉₇	1	2	yes	—
86	C₂×C₄⋊C₄	C₂×D₉₇	1	2	yes	—
87	C₂×C₄²	C₂×D₉₇	1	1	yes	characteristic
88	C₄⋊C₈	—	194	1	no	—
89	C₄⋊C₈	—	194	1	no	—
90	#97	—	194	1	no	—
91	C₄×C₈	—	194	1	no	—
92	C₂×Q₈	—	2	2	no	—
93	C₂×Q₈	—	2	2	no	—
94	C₂×Q₈	—	4	4	no	—
95	C₂²×C₄	C₂×Dic₉₇	1	1	yes	characteristic
96	C₂²×C₄	C₂×Dic₉₇	1	1	yes	characteristic
97	C₂²×C₄	C₂×Dic₉₇	1	4	yes	—
98	C₂²×C₄	C₂²×D₉₇	1	1	yes	characteristic
99	M₁₆	—	97	4	no	—
100	M₁₆	—	194	4	no	—
101	M₁₆	—	194	2	no	—
102	C₂×C₈	—	194	2	no	—
103	C₂×C₈	—	194	2	no	—
104	C₄⋊C₄	—	2	2	no	—
105	C₄⋊C₄	—	2	2	no	—
106	C₄⋊C₄	—	2	2	no	—
107	C₄⋊C₄	—	2	2	no	—
108	C₄²	C₂²⋊D₉₇	1	1	yes	characteristic
109	C₄²	C₂²⋊D₉₇	1	1	yes	characteristic
110	C₄²	C₂²⋊D₉₇	1	1	yes	characteristic
111	C₄²	C₂²⋊D₉₇	1	1	yes	characteristic
112	C₂³	C₂²×Dic₉₇	1	1	yes	characteristic
113	Q₈	—	4	4	no	—
114	Q₈	—	4	4	no	—
115	C₂×C₄	—	2	2	no	—
116	C₂×C₄	—	2	2	no	—
117	C₂×C₄	—	2	4	no	—
118	C₂×C₄	—	4	4	no	—
119	C₂×C₄	C₂²⋊Dic₉₇	1	1	yes	characteristic
120	C₂×C₄	C₂²⋊Dic₉₇	1	1	yes	characteristic
121	C₂×C₄	C₂²⋊Dic₉₇	1	1	yes	characteristic
122	C₂×C₄	C₂²⋊Dic₉₇	1	1	yes	characteristic
123	C₂×C₄	C₂×C₂²⋊D₉₇	1	1	yes	characteristic
124	C₂×C₄	C₂×C₂²⋊D₉₇	1	1	yes	characteristic
125	C₂×C₄	Q₈×D₉₇	1	2	yes	—
126	C₂×C₄	#13095512	1	2	yes	—
127	C₈	—	388	2	no	—
128	C₈	—	194	4	no	—
129	C₂²	—	2	1	no	—
130	C₂²	—	2	1	no	—
131	C₂²	C₂×C₂²⋊Dic₉₇	1	1	yes	characteristic
132	C₂²	Q₈×Dic₉₇	1	2	yes	—
133	C₄	—	2	1	no	—
134	C₄	—	2	1	no	—
135	C₄	—	4	4	no	—
136	C₄	#15690727	1	2	yes	—
137	C₄	#15690729	1	2	yes	—
138	C₄	#15690786	1	2	yes	—
139	C₄	#15690799	1	1	yes	characteristic
140	C₄	#15690808	1	1	yes	characteristic
141	C₂	—	2	2	no	—
142	C₂	#20260642	1	1	yes	characteristic
143	C₂	C₂×#15690727	1	1	yes	characteristic
144	C₂	C₂×#15690729	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20260759

C₂

yes

maximal, characteristic

C₄×#13095495

C₂

yes

maximal, characteristic

#20260504

C₂

yes

maximal

C₂×C₄⋊Q₈×C₉₇

C₂

yes

maximal, characteristic

C₂×#13095495

C₂²

yes

—

C₂×#13095495

C₂²

yes

—

—

—

—

—

C₂²×Q₈×C₉₇

C₄

yes

—

C₄⋊Q₈×C₉₇

—

C₄⋊Q₈×C₉₇

—

C₄⋊Q₈×C₉₇

—

C₄⋊Q₈×C₉₇

—

C₂×C₄⋊C₄×C₉₇

C₄

yes

—

C₂×C₄⋊C₄×C₉₇

C₂²

yes

—

C₂×C₄²×C₉₇

C₂²

yes

—

—

Q₈

yes

complemented

C₂×#11715895

—

C₂×#11715895

—

C₂×Q₈×C₉₇

—

C₂×Q₈×C₉₇

—

C₂×Q₈×C₉₇

—

C₂²×C₄×C₉₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₉₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₉₇

C₂×C₄

yes

—

C₂²×C₄×C₉₇

C₂³

yes

characteristic

C₄⋊C₄×C₉₇

—

C₄⋊C₄×C₉₇

—

C₄⋊C₄×C₉₇

—

C₄⋊C₄×C₉₇

—

C₄²×C₉₇

D₄

yes

characteristic

C₄²×C₉₇

D₄

yes

characteristic

C₄²×C₉₇

D₄

yes

characteristic

C₄²×C₉₇

D₄

yes

characteristic

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

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₉₇

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₉₇

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₉₇

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₉₇

C₂×D₄

yes

characteristic

C₂×C₄×C₉₇

C₂×D₄

yes

characteristic

C₂×C₄×C₉₇

C₂×Q₈

yes

—

C₂×C₄×C₉₇

#41

yes

—

#11715895

—

#11715895

—

C₂²×C₉₇

—

C₂²×C₉₇

—

C₂²×C₉₇

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₉₇

C₄×Q₈

yes

—

C₄×C₉₇

—

C₄×C₉₇

—

C₄×C₉₇

—

C₄×C₉₇

#100

yes

—

C₄×C₉₇

#101

yes

—

C₄×C₉₇

C₂²⋊Q₈

yes

—

C₄×C₉₇

#124

yes

characteristic

C₄×C₉₇

C₄⋊Q₈

yes

characteristic

C₂×C₉₇

—

C₂×C₉₇

yes

C₂×C₉₇

yes

C₂×C₉₇

yes

—

C₉₇

yes

characteristic, complemented, Sylow

C₂×C₄⋊Q₈

D₉₇

yes

characteristic

#423

—

C₄×M₁₆

—

#343

—

C₂²×Q₈

Dic₉₇

yes

—

C₂×M₁₆

—

C₂×M₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₂×C₄⋊C₄

Dic₉₇

yes

—

C₂×C₄⋊C₄

C₂×D₉₇

yes

—

C₂×C₄²

C₂×D₉₇

yes

characteristic

C₄⋊C₈

—

194

—

C₄⋊C₈

—

194

—

#97

—

194

—

C₄×C₈

—

194

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

C₂×Dic₉₇

yes

characteristic

C₂²×C₄

C₂×Dic₉₇

yes

characteristic

C₂²×C₄

C₂×Dic₉₇

yes

—

C₂²×C₄

C₂²×D₉₇

yes

characteristic

M₁₆

—

100

M₁₆

—

194

—

101

M₁₆

—

194

—

102

C₂×C₈

—

194

—

103

C₂×C₈

—

194

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

106

C₄⋊C₄

—

107

C₄⋊C₄

—

108

C₄²

C₂²⋊D₉₇

yes

characteristic

109

C₄²

C₂²⋊D₉₇

yes

characteristic

110

C₄²

C₂²⋊D₉₇

yes

characteristic

111

C₄²

C₂²⋊D₉₇

yes

characteristic

112

C₂³

C₂²×Dic₉₇

yes

characteristic

113

Q₈

—

114

Q₈

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

C₂²⋊Dic₉₇

yes

characteristic

120

C₂×C₄

C₂²⋊Dic₉₇

yes

characteristic

121

C₂×C₄

C₂²⋊Dic₉₇

yes

characteristic

122

C₂×C₄

C₂²⋊Dic₉₇

yes

characteristic

123

C₂×C₄

C₂×C₂²⋊D₉₇

yes

characteristic

124

C₂×C₄

C₂×C₂²⋊D₉₇

yes

characteristic

125

C₂×C₄

Q₈×D₉₇

yes

—

126

C₂×C₄

#13095512

yes

—

127

C₈

—

388

—

128

C₈

—

194

—

129

C₂²

—

130

C₂²

—

131

C₂²

C₂×C₂²⋊Dic₉₇

yes

characteristic

132

C₂²

Q₈×Dic₉₇

yes

—

133

C₄

—

134

C₄

—

135

C₄

—

136

C₄

#15690727

yes

—

137

C₄

#15690729

yes

—

138

C₄

#15690786

yes

—

139

C₄

#15690799

yes

characteristic

140

C₄

#15690808

yes

characteristic

141

C₂

—

142

C₂

#20260642

yes

characteristic

143

C₂

C₂×#15690727

yes

characteristic

144

C₂

C₂×#15690729

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	#1997
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

#1997

LCS

G ▷ C₂³×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

N	H ≅ G/N	type
21. #13095495	Q₈	semidirect
73. C₉₇	#1997	semidirect

H ≅ G/N

type

21. #13095495

Q₈

semidirect

73. C₉₇

#1997

semidirect

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

subgroup	index
#20260759	2
C₄×#13095495	2
#20260504	2
C₂×C₄⋊Q₈×C₉₇	2
#1997	97

subgroup

index

C₂×C₄⋊Q₈×C₉₇

Subgroups — #27142726

Conjugacy classes of subgroups

Complements

Maximal subgroups