Group #38753092 — subgroups

Subgroups — #38753092

Order 25216.

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	#27161636	C₂	1	1	yes	maximal, characteristic
2	#27161447	C₂	1	4	yes	maximal, complemented
3	#27161429	C₂	1	4	yes	maximal, complemented
4	#27161088	C₂	1	4	yes	maximal
5	C₄×#15691342	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊Q₈×C₁₉₇	C₂	1	1	yes	maximal, characteristic
7	C₂×#15691342	C₂²	1	2	yes	—
8	#20263326	C₂²	1	2	yes	—
9	#20263318	C₂²	1	2	yes	—
10	#20263305	—	2	2	no	—
11	C₄×Dic₃₉₄	—	2	2	no	—
12	#20263285	C₂²	1	2	yes	—
13	#20263250	C₂²	1	16	yes	—
14	#20263237	C₂²	1	2	yes	—
15	C₄×#13096802	C₂²	1	2	yes	—
16	C₂²×Q₈×C₁₉₇	C₂²	1	2	yes	—
17	C₄⋊Q₈×C₁₉₇	—	2	2	no	—
18	C₄⋊Q₈×C₁₉₇	C₂²	1	2	yes	—
19	C₄⋊Q₈×C₁₉₇	C₂²	1	2	yes	—
20	C₂×C₄⋊C₄×C₁₉₇	—	2	2	no	—
21	C₂×C₄²×C₁₉₇	C₂²	1	1	yes	characteristic
22	C₂×Dic₃₉₄	—	2	2	no	—
23	#15691342	—	2	4	no	—
24	C₂×#13096802	C₂³	1	4	yes	—
25	C₄⋊Dic₁₉₇	—	2	2	no	—
26	C₄⋊Dic₁₉₇	C₂³	1	4	yes	—
27	#15691337	—	2	4	no	—
28	C₂²⋊Dic₁₉₇	—	2	2	no	—
29	C₄×Dic₁₉₇	—	2	2	no	—
30	C₂×Q₈×C₁₉₇	—	2	2	no	—
31	C₂×Q₈×C₁₉₇	—	2	8	no	—
32	C₂×Q₈×C₁₉₇	C₂³	1	4	yes	—
33	C₂²×C₄×C₁₉₇	—	2	2	no	—
34	C₂²×C₄×C₁₉₇	D₄	1	2	yes	—
35	C₂²×C₄×C₁₉₇	C₂³	1	1	yes	characteristic
36	C₄⋊C₄×C₁₉₇	—	2	4	no	—
37	C₄⋊C₄×C₁₉₇	—	2	4	no	—
38	C₄²×C₁₉₇	D₄	1	2	yes	—
39	C₄²×C₁₉₇	C₂³	1	1	yes	characteristic
40	C₄²×C₁₉₇	C₂³	1	1	yes	characteristic
41	Dic₃₉₄	—	4	4	no	—
42	C₂×Dic₁₉₇	—	2	4	no	—
43	#13096802	—	2	4	no	—
44	C₂³×C₁₉₇	C₂×D₄	1	1	yes	characteristic
45	Q₈×C₁₉₇	—	2	8	no	—
46	Q₈×C₁₉₇	—	4	4	no	—
47	C₂×C₄×C₁₉₇	—	2	4	no	—
48	C₂×C₄×C₁₉₇	—	2	4	no	—
49	C₂×C₄×C₁₉₇	—	4	4	no	—
50	C₂×C₄×C₁₉₇	C₂×D₄	1	2	yes	—
51	C₂×C₄×C₁₉₇	C₂×D₄	1	1	yes	characteristic
52	C₂×C₄×C₁₉₇	C₂×D₄	1	2	yes	—
53	C₂×C₄×C₁₉₇	#41	1	4	yes	—
54	C₂×C₄×C₁₉₇	C₂⁴	1	1	yes	characteristic
55	C₂²×C₁₉₇	—	2	2	no	—
56	C₂²×C₁₉₇	#124	1	2	yes	—
57	C₂²×C₁₉₇	C₂²×D₄	1	1	yes	characteristic
58	C₄×C₁₉₇	—	2	2	no	—
59	C₄×C₁₉₇	—	4	4	no	—
60	C₄×C₁₉₇	#124	1	2	yes	—
61	C₄×C₁₉₇	#137	1	4	yes	—
62	C₄×C₁₉₇	C₂×#41	1	2	yes	—
63	Dic₁₉₇	—	4	4	no	—
64	C₂×C₁₉₇	—	2	2	no	—
65	C₂×C₁₉₇	C₂×#124	1	1	yes	characteristic
66	C₂×C₁₉₇	C₂×#137	1	2	yes	—
67	C₁₉₇	#3139	1	1	yes	characteristic, complemented, Sylow
68	#3139	—	197	1	no	maximal, Sylow
69	#533	—	197	1	no	—
70	C₂×C₄⋊Q₈	D₁₉₇	1	1	yes	characteristic
71	#491	—	197	4	no	—
72	#487	—	197	4	no	—
73	#419	—	197	4	no	—
74	C₄×M₁₆	—	197	1	no	—
75	C₂²×Q₈	C₂×D₁₉₇	1	2	yes	—
76	C₂×M₁₆	—	197	2	no	—
77	C₄⋊Q₈	—	2	2	no	—
78	C₄⋊Q₈	—	197	2	no	—
79	C₄⋊Q₈	C₂×D₁₉₇	1	2	yes	—
80	C₄⋊Q₈	C₂×D₁₉₇	1	2	yes	—
81	#125	—	197	2	no	—
82	C₂²⋊Q₈	—	394	2	no	—
83	C₄×Q₈	—	394	2	no	—
84	#117	—	197	2	no	—
85	C₂×C₄⋊C₄	—	2	2	no	—
86	C₂×C₄²	C₂×D₁₉₇	1	1	yes	characteristic
87	#103	—	197	16	no	—
88	#97	—	197	2	no	—
89	C₄×C₈	—	197	2	no	—
90	C₂×Q₈	—	2	2	no	—
91	C₂×Q₈	—	2	8	no	—
92	C₂×Q₈	—	394	2	no	—
93	C₂×Q₈	C₂²×D₁₉₇	1	4	yes	—
94	C₂²×C₄	—	2	2	no	—
95	C₂²×C₄	C₂²⋊D₁₉₇	1	2	yes	—
96	C₂²×C₄	C₂²×D₁₉₇	1	1	yes	characteristic
97	M₁₆	—	394	4	no	—
98	C₂×C₈	—	197	4	no	—
99	C₄⋊C₄	—	2	4	no	—
100	C₄⋊C₄	—	2	4	no	—
101	C₄⋊C₄	—	394	4	no	—
102	C₄⋊C₄	—	394	2	no	—
103	C₄⋊C₄	—	197	4	no	—
104	C₂²⋊C₄	—	394	2	no	—
105	C₄²	—	394	2	no	—
106	C₄²	C₂²⋊D₁₉₇	1	2	yes	—
107	C₄²	C₂²×D₁₉₇	1	1	yes	characteristic
108	C₄²	C₂²×D₁₉₇	1	1	yes	characteristic
109	C₂³	C₂×C₂²⋊D₁₉₇	1	1	yes	characteristic
110	Q₈	—	2	8	no	—
111	Q₈	—	4	4	no	—
112	Q₈	—	788	4	no	—
113	C₂×C₄	—	2	4	no	—
114	C₂×C₄	—	2	4	no	—
115	C₂×C₄	—	4	4	no	—
116	C₂×C₄	—	394	4	no	—
117	C₂×C₄	C₂×C₂²⋊D₁₉₇	1	2	yes	—
118	C₂×C₄	C₂×C₂²⋊D₁₉₇	1	1	yes	characteristic
119	C₂×C₄	C₂×C₂²⋊D₁₉₇	1	2	yes	—
120	C₂×C₄	#15691357	1	4	yes	—
121	C₂×C₄	C₂³×D₁₉₇	1	1	yes	characteristic
122	C₈	—	394	4	no	—
123	C₂²	—	2	2	no	—
124	C₂²	#20263313	1	2	yes	—
125	C₂²	C₂²×C₂²⋊D₁₉₇	1	1	yes	characteristic
126	C₄	—	2	2	no	—
127	C₄	—	4	4	no	—
128	C₄	—	788	4	no	—
129	C₄	#20263313	1	2	yes	—
130	C₄	#20263357	1	4	yes	—
131	C₄	C₂×#15691357	1	2	yes	—
132	C₂	—	2	2	no	—
133	C₂	C₂×#20263313	1	1	yes	characteristic
134	C₂	C₂×#20263357	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#27161636

C₂

yes

maximal, characteristic

#27161447

C₂

yes

maximal, complemented

#27161429

C₂

yes

maximal, complemented

#27161088

C₂

yes

maximal

C₄×#15691342

C₂

yes

maximal, characteristic

C₂×C₄⋊Q₈×C₁₉₇

C₂

yes

maximal, characteristic

C₂×#15691342

C₂²

yes

—

#20263326

C₂²

yes

—

#20263318

C₂²

yes

—

#20263305

—

C₄×Dic₃₉₄

—

#20263285

C₂²

yes

—

#20263250

C₂²

yes

—

#20263237

C₂²

yes

—

C₄×#13096802

C₂²

yes

—

C₂²×Q₈×C₁₉₇

C₂²

yes

—

C₄⋊Q₈×C₁₉₇

—

C₄⋊Q₈×C₁₉₇

C₂²

yes

—

C₄⋊Q₈×C₁₉₇

C₂²

yes

—

C₂×C₄⋊C₄×C₁₉₇

—

C₂×C₄²×C₁₉₇

C₂²

yes

characteristic

C₂×Dic₃₉₄

—

#15691342

—

C₂×#13096802

C₂³

yes

—

C₄⋊Dic₁₉₇

—

C₄⋊Dic₁₉₇

C₂³

yes

—

#15691337

—

C₂²⋊Dic₁₉₇

—

C₄×Dic₁₉₇

—

C₂×Q₈×C₁₉₇

—

C₂×Q₈×C₁₉₇

—

C₂×Q₈×C₁₉₇

C₂³

yes

—

C₂²×C₄×C₁₉₇

—

C₂²×C₄×C₁₉₇

D₄

yes

—

C₂²×C₄×C₁₉₇

C₂³

yes

characteristic

C₄⋊C₄×C₁₉₇

—

C₄⋊C₄×C₁₉₇

—

C₄²×C₁₉₇

D₄

yes

—

C₄²×C₁₉₇

C₂³

yes

characteristic

C₄²×C₁₉₇

C₂³

yes

characteristic

Dic₃₉₄

—

C₂×Dic₁₉₇

—

#13096802

—

C₂³×C₁₉₇

C₂×D₄

yes

characteristic

Q₈×C₁₉₇

—

Q₈×C₁₉₇

—

C₂×C₄×C₁₉₇

—

C₂×C₄×C₁₉₇

—

C₂×C₄×C₁₉₇

—

C₂×C₄×C₁₉₇

C₂×D₄

yes

—

C₂×C₄×C₁₉₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₉₇

C₂×D₄

yes

—

C₂×C₄×C₁₉₇

#41

yes

—

C₂×C₄×C₁₉₇

C₂⁴

yes

characteristic

C₂²×C₁₉₇

—

C₂²×C₁₉₇

#124

yes

—

C₂²×C₁₉₇

C₂²×D₄

yes

characteristic

C₄×C₁₉₇

—

C₄×C₁₉₇

—

C₄×C₁₉₇

#124

yes

—

C₄×C₁₉₇

#137

yes

—

C₄×C₁₉₇

C₂×#41

yes

—

Dic₁₉₇

—

C₂×C₁₉₇

—

C₂×C₁₉₇

C₂×#124

yes

characteristic

C₂×C₁₉₇

C₂×#137

yes

—

C₁₉₇

#3139

yes

characteristic, complemented, Sylow

#3139

—

197

maximal, Sylow

#533

—

197

—

C₂×C₄⋊Q₈

D₁₉₇

yes

characteristic

#491

—

197

—

#487

—

197

—

#419

—

197

—

C₄×M₁₆

—

197

—

C₂²×Q₈

C₂×D₁₉₇

yes

—

C₂×M₁₆

—

197

—

C₄⋊Q₈

—

C₄⋊Q₈

—

197

—

C₄⋊Q₈

C₂×D₁₉₇

yes

—

C₄⋊Q₈

C₂×D₁₉₇

yes

—

#125

—

197

—

C₂²⋊Q₈

—

394

—

C₄×Q₈

—

394

—

#117

—

197

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×D₁₉₇

yes

characteristic

#103

—

197

—

#97

—

197

—

C₄×C₈

—

197

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

394

—

C₂×Q₈

C₂²×D₁₉₇

yes

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊D₁₉₇

yes

—

C₂²×C₄

C₂²×D₁₉₇

yes

characteristic

M₁₆

—

394

—

C₂×C₈

—

197

—

C₄⋊C₄

—

100

C₄⋊C₄

—

101

C₄⋊C₄

—

394

—

102

C₄⋊C₄

—

394

—

103

C₄⋊C₄

—

197

—

104

C₂²⋊C₄

—

394

—

105

C₄²

—

394

—

106

C₄²

C₂²⋊D₁₉₇

yes

—

107

C₄²

C₂²×D₁₉₇

yes

characteristic

108

C₄²

C₂²×D₁₉₇

yes

characteristic

109

C₂³

C₂×C₂²⋊D₁₉₇

yes

characteristic

110

Q₈

—

111

Q₈

—

112

Q₈

—

788

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

394

—

117

C₂×C₄

C₂×C₂²⋊D₁₉₇

yes

—

118

C₂×C₄

C₂×C₂²⋊D₁₉₇

yes

characteristic

119

C₂×C₄

C₂×C₂²⋊D₁₉₇

yes

—

120

C₂×C₄

#15691357

yes

—

121

C₂×C₄

C₂³×D₁₉₇

yes

characteristic

122

C₈

—

394

—

123

C₂²

—

124

C₂²

#20263313

yes

—

125

C₂²

C₂²×C₂²⋊D₁₉₇

yes

characteristic

126

C₄

—

127

C₄

—

128

C₄

—

788

—

129

C₄

#20263313

yes

—

130

C₄

#20263357

yes

—

131

C₄

C₂×#15691357

yes

—

132

C₂

—

133

C₂

C₂×#20263313

yes

characteristic

134

C₂

C₂×#20263357

yes

—

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₁₉₇
commutator, cosocle	C₂×C₄×C₁₉₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₉₇
Fitting	C₂×C₄⋊Q₈×C₁₉₇	C₂
socle	C₂²×C₁₉₇	C₂²×D₄
prime radical	C₂³×C₁₉₇	C₂×D₄
hypercenter	C₂×C₄⋊Q₈	D₁₉₇
hypocenter	C₁₉₇	#3139
LCS	G ▷ C₂×C₄×C₁₉₇ ▷ C₂×C₁₉₇ ▷ C₁₉₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

subgroup

G/H

center

C₂²

C₂²×C₂²⋊D₁₉₇

commutator, cosocle

C₂×C₄×C₁₉₇

C₂⁴

Frattini

C₂×C₄

C₂³×D₁₉₇

Fitting

C₂×C₄⋊Q₈×C₁₉₇

C₂

socle

C₂²×C₁₉₇

C₂²×D₄

prime radical

C₂³×C₁₉₇

C₂×D₄

hypercenter

C₂×C₄⋊Q₈

D₁₉₇

hypocenter

C₁₉₇

#3139

LCS

G ▷ C₂×C₄×C₁₉₇ ▷ C₂×C₁₉₇ ▷ C₁₉₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

N	H ≅ G/N	type
2. #27161447	C₂	semidirect
3. #27161429	C₂	semidirect
67. C₁₉₇	#3139	semidirect

H ≅ G/N

type

2. #27161447

C₂

semidirect

3. #27161429

C₂

semidirect

67. C₁₉₇

#3139

semidirect

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

subgroup	index
#27161636	2
#27161447	2
#27161429	2
#27161088	2
C₄×#15691342	2
C₂×C₄⋊Q₈×C₁₉₇	2
#3139	197

subgroup

index

C₂×C₄⋊Q₈×C₁₉₇

197

Subgroups — #38753092

Conjugacy classes of subgroups

Complements

Maximal subgroups