Group #27152116 — subgroups

Subgroups — #27152116

Order 12416.

380 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 141 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	#20261292	C₂	1	1	yes	maximal, characteristic
2	#20261103	C₂	1	4	yes	maximal, complemented
3	#20261085	C₂	1	4	yes	maximal, complemented
4	#20260744	C₂	1	4	yes	maximal
5	C₄×#13095495	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊Q₈×C₉₇	C₂	1	1	yes	maximal, characteristic
7	C₂×#13095495	C₂²	1	2	yes	—
8	#15690810	C₂²	1	2	yes	—
9	#15690802	C₂²	1	2	yes	—
10	#15690789	—	2	2	no	—
11	C₄×Dic₁₉₄	—	2	2	no	—
12	#15690769	C₂²	1	2	yes	—
13	#15690734	C₂²	1	16	yes	—
14	#15690721	C₂²	1	2	yes	—
15	C₄×#11715895	C₂²	1	2	yes	—
16	C₂²×Q₈×C₉₇	C₂²	1	2	yes	—
17	C₄⋊Q₈×C₉₇	—	2	2	no	—
18	C₄⋊Q₈×C₉₇	C₂²	1	2	yes	—
19	C₄⋊Q₈×C₉₇	C₂²	1	2	yes	—
20	C₂×C₄⋊C₄×C₉₇	—	2	2	no	—
21	C₂×C₄²×C₉₇	C₂²	1	1	yes	characteristic
22	C₂×Dic₁₉₄	—	2	2	no	—
23	#13095495	—	2	4	no	—
24	C₂×#11715895	C₂³	1	4	yes	—
25	C₄⋊Dic₉₇	—	2	2	no	—
26	C₄⋊Dic₉₇	C₂³	1	4	yes	—
27	#13095490	—	2	4	no	—
28	C₂²⋊Dic₉₇	—	2	2	no	—
29	C₄×Dic₉₇	—	2	2	no	—
30	C₂×Q₈×C₉₇	—	2	2	no	—
31	C₂×Q₈×C₉₇	—	2	8	no	—
32	C₂×Q₈×C₉₇	C₂³	1	4	yes	—
33	C₂²×C₄×C₉₇	—	2	2	no	—
34	C₂²×C₄×C₉₇	D₄	1	2	yes	—
35	C₂²×C₄×C₉₇	C₂³	1	1	yes	characteristic
36	C₄⋊C₄×C₉₇	—	2	4	no	—
37	C₄⋊C₄×C₉₇	—	2	4	no	—
38	C₄²×C₉₇	D₄	1	2	yes	—
39	C₄²×C₉₇	C₂³	1	1	yes	characteristic
40	C₄²×C₉₇	C₂³	1	1	yes	characteristic
41	C₂³×C₉₇	C₂×D₄	1	1	yes	characteristic
42	Q₈×C₉₇	—	2	8	no	—
43	Q₈×C₉₇	—	4	4	no	—
44	C₂×C₄×C₉₇	—	2	4	no	—
45	C₂×C₄×C₉₇	—	2	4	no	—
46	C₂×C₄×C₉₇	—	4	4	no	—
47	C₂×C₄×C₉₇	C₂×D₄	1	2	yes	—
48	C₂×C₄×C₉₇	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₉₇	C₂×D₄	1	2	yes	—
50	C₂×C₄×C₉₇	#41	1	4	yes	—
51	C₂×C₄×C₉₇	C₂⁴	1	1	yes	characteristic
52	C₂×Dic₉₇	—	2	4	no	—
53	Dic₁₉₄	—	4	4	no	—
54	#11715895	—	2	4	no	—
55	C₂²×C₉₇	—	2	2	no	—
56	C₂²×C₉₇	#124	1	2	yes	—
57	C₂²×C₉₇	C₂²×D₄	1	1	yes	characteristic
58	C₄×C₉₇	—	2	2	no	—
59	C₄×C₉₇	—	4	4	no	—
60	C₄×C₉₇	#124	1	2	yes	—
61	C₄×C₉₇	#137	1	4	yes	—
62	C₄×C₉₇	C₂×#41	1	2	yes	—
63	Dic₉₇	—	4	4	no	—
64	C₂×C₉₇	—	2	2	no	—
65	C₂×C₉₇	C₂×#124	1	1	yes	characteristic
66	C₂×C₉₇	C₂×#137	1	2	yes	—
67	#3139	—	97	1	no	maximal, Sylow
68	C₉₇	#3139	1	1	yes	characteristic, complemented, Sylow
69	#533	—	97	1	no	—
70	C₂×C₄⋊Q₈	D₉₇	1	1	yes	characteristic
71	#491	—	97	4	no	—
72	#487	—	97	4	no	—
73	#419	—	97	4	no	—
74	C₄×M₁₆	—	97	1	no	—
75	C₂²×Q₈	C₂×D₉₇	1	2	yes	—
76	C₂×M₁₆	—	97	2	no	—
77	C₄⋊Q₈	—	2	2	no	—
78	C₄⋊Q₈	—	97	2	no	—
79	C₄⋊Q₈	C₂×D₉₇	1	2	yes	—
80	C₄⋊Q₈	C₂×D₉₇	1	2	yes	—
81	#125	—	97	2	no	—
82	C₂²⋊Q₈	—	194	2	no	—
83	C₄×Q₈	—	194	2	no	—
84	#117	—	97	2	no	—
85	C₂×C₄⋊C₄	—	2	2	no	—
86	C₂×C₄²	C₂×D₉₇	1	1	yes	characteristic
87	#103	—	97	16	no	—
88	#97	—	97	2	no	—
89	C₄×C₈	—	97	2	no	—
90	C₂×Q₈	—	2	2	no	—
91	C₂×Q₈	—	2	8	no	—
92	C₂×Q₈	—	194	2	no	—
93	C₂×Q₈	C₂²×D₉₇	1	4	yes	—
94	C₂²×C₄	—	2	2	no	—
95	C₂²×C₄	C₂²⋊D₉₇	1	2	yes	—
96	C₂²×C₄	C₂²×D₉₇	1	1	yes	characteristic
97	M₁₆	—	194	4	no	—
98	C₂×C₈	—	97	4	no	—
99	C₄⋊C₄	—	2	4	no	—
100	C₄⋊C₄	—	2	4	no	—
101	C₄⋊C₄	—	97	4	no	—
102	C₄⋊C₄	—	194	4	no	—
103	C₄⋊C₄	—	194	2	no	—
104	C₂²⋊C₄	—	194	2	no	—
105	C₄²	—	194	2	no	—
106	C₄²	C₂²⋊D₉₇	1	2	yes	—
107	C₄²	C₂²×D₉₇	1	1	yes	characteristic
108	C₄²	C₂²×D₉₇	1	1	yes	characteristic
109	C₂³	C₂×C₂²⋊D₉₇	1	1	yes	characteristic
110	Q₈	—	2	8	no	—
111	Q₈	—	4	4	no	—
112	Q₈	—	388	4	no	—
113	C₂×C₄	—	2	4	no	—
114	C₂×C₄	—	2	4	no	—
115	C₂×C₄	—	4	4	no	—
116	C₂×C₄	—	194	4	no	—
117	C₂×C₄	C₂×C₂²⋊D₉₇	1	2	yes	—
118	C₂×C₄	C₂×C₂²⋊D₉₇	1	1	yes	characteristic
119	C₂×C₄	C₂×C₂²⋊D₉₇	1	2	yes	—
120	C₂×C₄	#13095510	1	4	yes	—
121	C₂×C₄	C₂³×D₉₇	1	1	yes	characteristic
122	C₈	—	194	4	no	—
123	C₂²	—	2	2	no	—
124	C₂²	#15690797	1	2	yes	—
125	C₂²	C₂²×C₂²⋊D₉₇	1	1	yes	characteristic
126	C₄	—	2	2	no	—
127	C₄	—	4	4	no	—
128	C₄	—	388	4	no	—
129	C₄	#15690797	1	2	yes	—
130	C₄	#15690841	1	4	yes	—
131	C₄	C₂×#13095510	1	2	yes	—
132	C₂	—	2	2	no	—
133	C₂	C₂×#15690797	1	1	yes	characteristic
134	C₂	C₂×#15690841	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20261292

C₂

yes

maximal, characteristic

#20261103

C₂

yes

maximal, complemented

#20261085

C₂

yes

maximal, complemented

#20260744

C₂

yes

maximal

C₄×#13095495

C₂

yes

maximal, characteristic

C₂×C₄⋊Q₈×C₉₇

C₂

yes

maximal, characteristic

C₂×#13095495

C₂²

yes

—

#15690810

C₂²

yes

—

#15690802

C₂²

yes

—

#15690789

—

C₄×Dic₁₉₄

—

#15690769

C₂²

yes

—

#15690734

C₂²

yes

—

#15690721

C₂²

yes

—

C₄×#11715895

C₂²

yes

—

C₂²×Q₈×C₉₇

C₂²

yes

—

C₄⋊Q₈×C₉₇

—

C₄⋊Q₈×C₉₇

C₂²

yes

—

C₄⋊Q₈×C₉₇

C₂²

yes

—

C₂×C₄⋊C₄×C₉₇

—

C₂×C₄²×C₉₇

C₂²

yes

characteristic

C₂×Dic₁₉₄

—

#13095495

—

C₂×#11715895

C₂³

yes

—

C₄⋊Dic₉₇

—

C₄⋊Dic₉₇

C₂³

yes

—

#13095490

—

C₂²⋊Dic₉₇

—

C₄×Dic₉₇

—

C₂×Q₈×C₉₇

—

C₂×Q₈×C₉₇

—

C₂×Q₈×C₉₇

C₂³

yes

—

C₂²×C₄×C₉₇

—

C₂²×C₄×C₉₇

D₄

yes

—

C₂²×C₄×C₉₇

C₂³

yes

characteristic

C₄⋊C₄×C₉₇

—

C₄⋊C₄×C₉₇

—

C₄²×C₉₇

D₄

yes

—

C₄²×C₉₇

C₂³

yes

characteristic

C₄²×C₉₇

C₂³

yes

characteristic

C₂³×C₉₇

C₂×D₄

yes

characteristic

Q₈×C₉₇

—

Q₈×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

C₂×D₄

yes

—

C₂×C₄×C₉₇

C₂×D₄

yes

characteristic

C₂×C₄×C₉₇

C₂×D₄

yes

—

C₂×C₄×C₉₇

#41

yes

—

C₂×C₄×C₉₇

C₂⁴

yes

characteristic

C₂×Dic₉₇

—

Dic₁₉₄

—

#11715895

—

C₂²×C₉₇

—

C₂²×C₉₇

#124

yes

—

C₂²×C₉₇

C₂²×D₄

yes

characteristic

C₄×C₉₇

—

C₄×C₉₇

—

C₄×C₉₇

#124

yes

—

C₄×C₉₇

#137

yes

—

C₄×C₉₇

C₂×#41

yes

—

Dic₉₇

—

C₂×C₉₇

—

C₂×C₉₇

C₂×#124

yes

characteristic

C₂×C₉₇

C₂×#137

yes

—

#3139

—

maximal, Sylow

C₉₇

#3139

yes

characteristic, complemented, Sylow

#533

—

C₂×C₄⋊Q₈

D₉₇

yes

—

—

—

C₄×M₁₆

—

C₂²×Q₈

C₂×D₉₇

yes

—

C₂×M₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₉₇

yes

—

C₄⋊Q₈

C₂×D₉₇

yes

—

#125

—

C₂²⋊Q₈

—

194

—

C₄×Q₈

—

194

—

#117

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×D₉₇

yes

characteristic

#103

—

#97

—

C₄×C₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

194

—

C₂×Q₈

C₂²×D₉₇

yes

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊D₉₇

yes

—

C₂²×C₄

C₂²×D₉₇

yes

characteristic

M₁₆

—

194

—

C₂×C₈

—

C₄⋊C₄

—

100

C₄⋊C₄

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

194

—

103

C₄⋊C₄

—

194

—

104

C₂²⋊C₄

—

194

—

105

C₄²

—

194

—

106

C₄²

C₂²⋊D₉₇

yes

—

107

C₄²

C₂²×D₉₇

yes

characteristic

108

C₄²

C₂²×D₉₇

yes

characteristic

109

C₂³

C₂×C₂²⋊D₉₇

yes

characteristic

110

Q₈

—

111

Q₈

—

112

Q₈

—

388

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

194

—

117

C₂×C₄

C₂×C₂²⋊D₉₇

yes

—

118

C₂×C₄

C₂×C₂²⋊D₉₇

yes

characteristic

119

C₂×C₄

C₂×C₂²⋊D₉₇

yes

—

120

C₂×C₄

#13095510

yes

—

121

C₂×C₄

C₂³×D₉₇

yes

characteristic

122

C₈

—

194

—

123

C₂²

—

124

C₂²

#15690797

yes

—

125

C₂²

C₂²×C₂²⋊D₉₇

yes

characteristic

126

C₄

—

127

C₄

—

128

C₄

—

388

—

129

C₄

#15690797

yes

—

130

C₄

#15690841

yes

—

131

C₄

C₂×#13095510

yes

—

132

C₂

—

133

C₂

C₂×#15690797

yes

characteristic

134

C₂

C₂×#15690841

yes

—

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₉₇
commutator, cosocle	C₂×C₄×C₉₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₉₇
Fitting	C₂×C₄⋊Q₈×C₉₇	C₂
socle	C₂²×C₉₇	C₂²×D₄
prime radical	C₂³×C₉₇	C₂×D₄
hypercenter	C₂×C₄⋊Q₈	D₉₇
hypocenter	C₉₇	#3139
LCS	G ▷ C₂×C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

subgroup

G/H

center

C₂²

C₂²×C₂²⋊D₉₇

commutator, cosocle

C₂×C₄×C₉₇

C₂⁴

Frattini

C₂×C₄

C₂³×D₉₇

Fitting

C₂×C₄⋊Q₈×C₉₇

C₂

socle

C₂²×C₉₇

C₂²×D₄

prime radical

C₂³×C₉₇

C₂×D₄

hypercenter

C₂×C₄⋊Q₈

D₉₇

hypocenter

C₉₇

#3139

LCS

G ▷ C₂×C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

N	H ≅ G/N	type
2. #20261103	C₂	semidirect
3. #20261085	C₂	semidirect
68. C₉₇	#3139	semidirect

H ≅ G/N

type

2. #20261103

C₂

semidirect

3. #20261085

C₂

semidirect

68. C₉₇

#3139

semidirect

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

subgroup	index
#20261292	2
#20261103	2
#20261085	2
#20260744	2
C₄×#13095495	2
C₂×C₄⋊Q₈×C₉₇	2
#3139	97

subgroup

index

C₂×C₄⋊Q₈×C₉₇

Subgroups — #27152116

Conjugacy classes of subgroups

Complements

Maximal subgroups