Group #38753022 — subgroups

Subgroups — C₂×#27161429 #38753022

Order 25216.

424 conjugacy classes of subgroups; only the 181 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 181 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×#20263326	C₂	1	1	yes	maximal, characteristic
2	#27161429	C₂	1	8	yes	maximal, direct product factor
3	C₂×#20263250	C₂	1	4	yes	maximal
4	C₂×C₄×#13096802	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊Q₈×C₁₉₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₃₉₄	—	2	1	no	—
7	C₂²×#13096802	C₂²	1	2	yes	—
8	#20263326	—	2	2	no	—
9	#20263326	C₂²	1	4	yes	—
10	C₂×C₄⋊Dic₁₉₇	—	2	1	no	—
11	C₂×C₄⋊Dic₁₉₇	C₂²	1	2	yes	—
12	#20263250	C₂²	1	16	yes	—
13	C₄×#13096802	C₂²	1	4	yes	—
14	C₂²×Q₈×C₁₉₇	C₂²	1	2	yes	—
15	C₄⋊Q₈×C₁₉₇	—	2	2	no	—
16	C₄⋊Q₈×C₁₉₇	C₂²	1	4	yes	—
17	C₂×C₄⋊C₄×C₁₉₇	—	2	1	no	—
18	C₂×C₄⋊C₄×C₁₉₇	—	2	1	no	—
19	C₂×C₄²×C₁₉₇	C₂²	1	1	yes	characteristic
20	C₂×Dic₃₉₄	—	2	4	no	—
21	C₂×Dic₃₉₄	—	4	4	no	—
22	C₂²×Dic₁₉₇	—	2	2	no	—
23	C₂×#13096802	—	2	4	no	—
24	C₂×#13096802	C₂³	1	4	yes	—
25	C₄⋊Dic₁₉₇	—	2	4	no	—
26	C₄⋊Dic₁₉₇	—	2	2	no	—
27	C₄⋊Dic₁₉₇	C₂³	1	4	yes	—
28	C₂×Q₈×C₁₉₇	—	2	2	no	—
29	C₂×Q₈×C₁₉₇	—	2	8	no	—
30	C₂×Q₈×C₁₉₇	C₂³	1	4	yes	—
31	C₂²×C₄×C₁₉₇	—	2	2	no	—
32	C₂²×C₄×C₁₉₇	D₄	1	1	yes	characteristic
33	C₂²×C₄×C₁₉₇	D₄	1	1	yes	characteristic
34	C₂²×C₄×C₁₉₇	C₂³	1	1	yes	characteristic
35	C₄⋊C₄×C₁₉₇	—	2	4	no	—
36	C₄⋊C₄×C₁₉₇	—	2	4	no	—
37	C₄²×C₁₉₇	D₄	1	2	yes	—
38	C₄²×C₁₉₇	C₂³	1	2	yes	—
39	Dic₃₉₄	—	4	8	no	—
40	C₂×Dic₁₉₇	—	2	4	no	—
41	C₂×Dic₁₉₇	—	4	4	no	—
42	#13096802	—	2	4	no	—
43	C₂³×C₁₉₇	C₂×D₄	1	1	yes	characteristic
44	Q₈×C₁₉₇	—	2	8	no	—
45	Q₈×C₁₉₇	—	4	4	no	—
46	C₂×C₄×C₁₉₇	—	2	4	no	—
47	C₂×C₄×C₁₉₇	—	4	4	no	—
48	C₂×C₄×C₁₉₇	SD₁₆	1	4	yes	—
49	C₂×C₄×C₁₉₇	Q₁₆	1	4	yes	—
50	C₂×C₄×C₁₉₇	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₁₉₇	C₂×D₄	1	2	yes	—
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	#124	1	2	yes	—
56	C₂²×C₁₉₇	C₂×SD₁₆	1	2	yes	—
57	C₂²×C₁₉₇	C₂×Q₁₆	1	2	yes	—
58	C₂²×C₁₉₇	C₂²×D₄	1	1	yes	characteristic
59	C₄×C₁₉₇	—	4	4	no	—
60	C₄×C₁₉₇	#124	1	2	yes	—
61	C₄×C₁₉₇	C₂×SD₁₆	1	4	yes	—
62	C₄×C₁₉₇	C₂×Q₁₆	1	4	yes	—
63	C₄×C₁₉₇	C₂×#41	1	2	yes	—
64	Dic₁₉₇	—	4	4	no	—
65	C₂×C₁₉₇	#487	1	4	yes	complemented
66	C₂×C₁₉₇	C₂×#124	1	1	yes	characteristic
67	C₂×C₁₉₇	C₂²×SD₁₆	1	1	yes	characteristic
68	C₂×C₁₉₇	C₂²×Q₁₆	1	1	yes	characteristic
69	C₁₉₇	C₂×#487	1	1	yes	characteristic, complemented, Sylow
70	C₂×#487	—	197	1	no	maximal, Sylow
71	C₂×C₄⋊Q₈	—	197	1	no	—
72	C₂×C₄⋊Q₈	D₁₉₇	1	1	yes	characteristic
73	#487	—	197	8	no	—
74	C₂×#103	—	197	4	no	—
75	C₂×C₄×C₈	—	197	1	no	—
76	C₂²×Q₈	—	394	1	no	—
77	C₂²×Q₈	C₂×D₁₉₇	1	2	yes	—
78	C₂²×C₈	—	197	2	no	—
79	C₄⋊Q₈	—	2	2	no	—
80	C₄⋊Q₈	—	394	2	no	—
81	C₄⋊Q₈	—	197	4	no	—
82	C₄⋊Q₈	C₂×D₁₉₇	1	4	yes	—
83	C₂×C₄⋊C₄	—	2	1	no	—
84	C₂×C₄⋊C₄	—	2	1	no	—
85	C₂×C₄⋊C₄	—	394	1	no	—
86	C₂×C₄⋊C₄	—	197	2	no	—
87	C₂×C₄²	C₂×D₁₉₇	1	1	yes	characteristic
88	#103	—	197	16	no	—
89	C₄×C₈	—	197	4	no	—
90	C₂×Q₈	—	2	2	no	—
91	C₂×Q₈	—	2	8	no	—
92	C₂×Q₈	—	788	4	no	—
93	C₂×Q₈	—	394	4	no	—
94	C₂×Q₈	C₂²×D₁₉₇	1	4	yes	—
95	C₂²×C₄	—	2	2	no	—
96	C₂²×C₄	—	394	2	no	—
97	C₂²×C₄	C₂²⋊D₁₉₇	1	1	yes	characteristic
98	C₂²×C₄	C₂²⋊D₁₉₇	1	1	yes	characteristic
99	C₂²×C₄	C₂²×D₁₉₇	1	1	yes	characteristic
100	C₂×C₈	—	394	4	no	—
101	C₂×C₈	—	197	4	no	—
102	C₄⋊C₄	—	2	4	no	—
103	C₄⋊C₄	—	2	4	no	—
104	C₄⋊C₄	—	394	4	no	—
105	C₄⋊C₄	—	394	2	no	—
106	C₄⋊C₄	—	197	4	no	—
107	C₄²	C₂²⋊D₁₉₇	1	2	yes	—
108	C₄²	C₂²×D₁₉₇	1	2	yes	—
109	C₂³	C₂×C₂²⋊D₁₉₇	1	1	yes	characteristic
110	Q₈	—	2	8	no	—
111	Q₈	—	4	4	no	—
112	Q₈	—	788	8	no	—
113	C₂×C₄	—	2	4	no	—
114	C₂×C₄	—	4	4	no	—
115	C₂×C₄	—	788	4	no	—
116	C₂×C₄	—	394	4	no	—
117	C₂×C₄	#15691345	1	4	yes	—
118	C₂×C₄	#15691348	1	4	yes	—
119	C₂×C₄	C₂×C₂²⋊D₁₉₇	1	1	yes	characteristic
120	C₂×C₄	C₂×C₂²⋊D₁₉₇	1	2	yes	—
121	C₂×C₄	C₂×C₂²⋊D₁₉₇	1	2	yes	—
122	C₂×C₄	#15691357	1	4	yes	—
123	C₂×C₄	C₂³×D₁₉₇	1	1	yes	characteristic
124	C₈	—	394	4	no	—
125	C₂²	#20263313	1	2	yes	—
126	C₂²	C₂×#15691345	1	2	yes	—
127	C₂²	C₂×#15691348	1	2	yes	—
128	C₂²	C₂²×C₂²⋊D₁₉₇	1	1	yes	characteristic
129	C₄	—	4	4	no	—
130	C₄	—	788	4	no	—
131	C₄	#20263313	1	2	yes	—
132	C₄	C₂×#15691345	1	4	yes	—
133	C₄	C₂×#15691348	1	4	yes	—
134	C₄	C₂×#15691357	1	2	yes	—
135	C₂	#27161429	1	4	yes	direct product factor
136	C₂	C₂×#20263313	1	1	yes	characteristic
137	C₂	C₂²×#15691345	1	1	yes	characteristic
138	C₂	C₂²×#15691348	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#20263326

C₂

yes

maximal, characteristic

#27161429

C₂

yes

maximal, direct product factor

C₂×#20263250

C₂

yes

maximal

C₂×C₄×#13096802

C₂

yes

maximal, characteristic

C₂×C₄⋊Q₈×C₁₉₇

C₂

yes

maximal, characteristic

C₂²×Dic₃₉₄

—

C₂²×#13096802

C₂²

yes

—

#20263326

—

#20263326

C₂²

yes

—

C₂×C₄⋊Dic₁₉₇

—

C₂×C₄⋊Dic₁₉₇

C₂²

yes

—

#20263250

C₂²

yes

—

C₄×#13096802

C₂²

yes

—

C₂²×Q₈×C₁₉₇

C₂²

yes

—

C₄⋊Q₈×C₁₉₇

—

C₄⋊Q₈×C₁₉₇

C₂²

yes

—

C₂×C₄⋊C₄×C₁₉₇

—

C₂×C₄⋊C₄×C₁₉₇

—

C₂×C₄²×C₁₉₇

C₂²

yes

characteristic

C₂×Dic₃₉₄

—

C₂×Dic₃₉₄

—

C₂²×Dic₁₉₇

—

C₂×#13096802

—

C₂×#13096802

C₂³

yes

—

C₄⋊Dic₁₉₇

—

C₄⋊Dic₁₉₇

—

C₄⋊Dic₁₉₇

C₂³

yes

—

C₂×Q₈×C₁₉₇

—

C₂×Q₈×C₁₉₇

—

C₂×Q₈×C₁₉₇

C₂³

yes

—

C₂²×C₄×C₁₉₇

—

C₂²×C₄×C₁₉₇

D₄

yes

characteristic

C₂²×C₄×C₁₉₇

D₄

yes

characteristic

C₂²×C₄×C₁₉₇

C₂³

yes

characteristic

C₄⋊C₄×C₁₉₇

—

C₄⋊C₄×C₁₉₇

—

C₄²×C₁₉₇

D₄

yes

—

C₄²×C₁₉₇

C₂³

yes

—

Dic₃₉₄

—

C₂×Dic₁₉₇

—

C₂×Dic₁₉₇

—

#13096802

—

C₂³×C₁₉₇

C₂×D₄

yes

characteristic

Q₈×C₁₉₇

—

Q₈×C₁₉₇

—

C₂×C₄×C₁₉₇

—

C₂×C₄×C₁₉₇

—

C₂×C₄×C₁₉₇

SD₁₆

yes

—

C₂×C₄×C₁₉₇

Q₁₆

yes

—

C₂×C₄×C₁₉₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₉₇

C₂×D₄

yes

—

C₂×C₄×C₁₉₇

C₂×D₄

yes

—

C₂×C₄×C₁₉₇

#41

yes

—

C₂×C₄×C₁₉₇

C₂⁴

yes

characteristic

C₂²×C₁₉₇

#124

yes

—

C₂²×C₁₉₇

C₂×SD₁₆

yes

—

C₂²×C₁₉₇

C₂×Q₁₆

yes

—

C₂²×C₁₉₇

C₂²×D₄

yes

characteristic

C₄×C₁₉₇

—

C₄×C₁₉₇

#124

yes

—

C₄×C₁₉₇

C₂×SD₁₆

yes

—

C₄×C₁₉₇

C₂×Q₁₆

yes

—

C₄×C₁₉₇

C₂×#41

yes

—

Dic₁₉₇

—

C₂×C₁₉₇

yes

C₂×C₁₉₇

yes

C₂×C₁₉₇

C₂²×SD₁₆

yes

characteristic

C₂×C₁₉₇

C₂²×Q₁₆

yes

characteristic

C₁₉₇

C₂×#487

yes

characteristic, complemented, Sylow

C₂×#487

—

197

maximal, Sylow

C₂×C₄⋊Q₈

—

197

—

C₂×C₄⋊Q₈

D₁₉₇

yes

characteristic

#487

—

197

—

C₂×#103

—

197

—

C₂×C₄×C₈

—

197

—

C₂²×Q₈

—

394

—

C₂²×Q₈

C₂×D₁₉₇

yes

—

C₂²×C₈

—

197

—

C₄⋊Q₈

—

C₄⋊Q₈

—

394

—

C₄⋊Q₈

—

197

—

C₄⋊Q₈

C₂×D₁₉₇

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

394

—

C₂×C₄⋊C₄

—

197

—

C₂×C₄²

C₂×D₁₉₇

yes

characteristic

#103

—

197

—

C₄×C₈

—

197

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

788

—

C₂×Q₈

—

394

—

C₂×Q₈

C₂²×D₁₉₇

yes

—

C₂²×C₄

—

C₂²×C₄

—

394

—

C₂²×C₄

C₂²⋊D₁₉₇

yes

characteristic

C₂²×C₄

C₂²⋊D₁₉₇

yes

characteristic

C₂²×C₄

C₂²×D₁₉₇

yes

characteristic

100

C₂×C₈

—

394

—

101

C₂×C₈

—

197

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

394

—

105

C₄⋊C₄

—

394

—

106

C₄⋊C₄

—

197

—

107

C₄²

C₂²⋊D₁₉₇

yes

—

108

C₄²

C₂²×D₁₉₇

yes

—

109

C₂³

C₂×C₂²⋊D₁₉₇

yes

characteristic

110

Q₈

—

111

Q₈

—

112

Q₈

—

788

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

788

—

116

C₂×C₄

—

394

—

117

C₂×C₄

#15691345

yes

—

118

C₂×C₄

#15691348

yes

—

119

C₂×C₄

C₂×C₂²⋊D₁₉₇

yes

characteristic

120

C₂×C₄

C₂×C₂²⋊D₁₉₇

yes

—

121

C₂×C₄

C₂×C₂²⋊D₁₉₇

yes

—

122

C₂×C₄

#15691357

yes

—

123

C₂×C₄

C₂³×D₁₉₇

yes

characteristic

124

C₈

—

394

—

125

C₂²

#20263313

yes

—

126

C₂²

C₂×#15691345

yes

—

127

C₂²

C₂×#15691348

yes

—

128

C₂²

C₂²×C₂²⋊D₁₉₇

yes

characteristic

129

C₄

—

130

C₄

—

788

—

131

C₄

#20263313

yes

—

132

C₄

C₂×#15691345

yes

—

133

C₄

C₂×#15691348

yes

—

134

C₄

C₂×#15691357

yes

—

135

C₂

#27161429

yes

direct product factor

136

C₂

C₂×#20263313

yes

characteristic

137

C₂

C₂²×#15691345

yes

characteristic

138

C₂

C₂²×#15691348

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₁₉₇
commutator, cosocle	C₂×C₄×C₁₉₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₉₇
Fitting	C₂×C₄⋊Q₈×C₁₉₇	C₂
socle, prime radical	C₂³×C₁₉₇	C₂×D₄
hypercenter	C₂×C₄⋊Q₈	D₁₉₇
hypocenter	C₁₉₇	C₂×#487
LCS	G ▷ C₂×C₄×C₁₉₇ ▷ C₂×C₁₉₇ ▷ C₁₉₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₁₉₇

commutator, cosocle

C₂×C₄×C₁₉₇

C₂⁴

Frattini

C₂×C₄

C₂³×D₁₉₇

Fitting

C₂×C₄⋊Q₈×C₁₉₇

C₂

socle, prime radical

C₂³×C₁₉₇

C₂×D₄

hypercenter

C₂×C₄⋊Q₈

D₁₉₇

hypocenter

C₁₉₇

C₂×#487

LCS

G ▷ C₂×C₄×C₁₉₇ ▷ C₂×C₁₉₇ ▷ C₁₉₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

N	H ≅ G/N	type
2. #27161429	C₂	direct (N × H)
65. C₂×C₁₉₇	#487	semidirect
69. C₁₉₇	C₂×#487	semidirect
135. C₂	#27161429	direct (N × H)

H ≅ G/N

type

2. #27161429

C₂

direct (N × H)

65. C₂×C₁₉₇

#487

semidirect

69. C₁₉₇

C₂×#487

semidirect

135. C₂

#27161429

direct (N × H)

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

subgroup	index
C₂×#20263326	2
#27161429	2
C₂×#20263250	2
C₂×C₄×#13096802	2
C₂×C₄⋊Q₈×C₁₉₇	2
C₂×#487	197

subgroup

index

C₂×C₄⋊Q₈×C₁₉₇

197

Subgroups — C₂×#27161429 #38753022

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×#27161429 #38753022

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#27161429 #38753022