Group #38768940 — subgroups

Subgroups — #38768940

Order 25472.

344 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 344 conjugacy classes of subgroups — too many to draw. Only its 125 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×C₂²⋊(C₁₉₉⋊C₈)	C₂	1	1	yes	maximal, characteristic
2	#27163878	C₂	1	1	yes	maximal, characteristic
3	#27163780	C₂	1	4	yes	maximal, complemented
4	C₂²×C₄⋊C₄×C₁₉₉	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₁₉₉⋊C₈	C₂²	1	2	yes	—
6	C₂×C₄⋊Dic₁₉₉	C₂²	1	2	yes	—
7	C₂×C₂²⋊Dic₁₉₉	—	2	1	no	—
8	C₂²⋊(C₁₉₉⋊C₈)	—	2	2	no	—
9	C₁₉₉⋊#95	—	2	1	no	—
10	C₂³×C₄×C₁₉₉	—	2	1	no	—
11	C₂³×C₄×C₁₉₉	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊C₄×C₁₉₉	—	2	1	no	—
13	C₂×C₄⋊C₄×C₁₉₉	—	2	2	no	—
14	C₂×C₄⋊C₄×C₁₉₉	C₄	1	4	yes	complemented
15	C₂×C₄⋊C₄×C₁₉₉	C₂²	1	2	yes	—
16	C₂²×Dic₁₉₉	—	2	2	no	—
17	C₂×C₁₉₉⋊C₈	—	2	2	no	—
18	C₂×C₁₉₉⋊C₈	—	2	4	no	—
19	C₄⋊Dic₁₉₉	—	2	4	no	—
20	C₂²⋊Dic₁₉₉	—	4	2	no	—
21	C₂⁴×C₁₉₉	D₄	1	1	yes	characteristic
22	C₂²×C₄×C₁₉₉	—	2	4	no	—
23	C₂²×C₄×C₁₉₉	—	2	4	no	—
24	C₂²×C₄×C₁₉₉	—	2	2	no	—
25	C₂²×C₄×C₁₉₉	—	4	4	no	—
26	C₂²×C₄×C₁₉₉	C₂×C₄	1	2	yes	—
27	C₂²×C₄×C₁₉₉	D₄	1	1	yes	characteristic
28	C₂²×C₄×C₁₉₉	D₄	1	2	yes	—
29	C₂²×C₄×C₁₉₉	C₂³	1	1	yes	characteristic
30	C₄⋊C₄×C₁₉₉	—	2	4	no	—
31	C₄⋊C₄×C₁₉₉	—	2	2	no	—
32	C₄⋊C₄×C₁₉₉	C₂×C₄	1	4	yes	—
33	C₂×Dic₁₉₉	—	4	2	no	—
34	C₂×Dic₁₉₉	—	4	2	no	—
35	C₂×Dic₁₉₉	—	4	2	no	—
36	C₁₉₉⋊C₈	—	4	2	no	—
37	C₂³×C₁₉₉	—	2	2	no	—
38	C₂³×C₁₉₉	—	2	2	no	—
39	C₂³×C₁₉₉	C₂²⋊C₄	1	2	yes	—
40	C₂³×C₁₉₉	SD₁₆	1	2	yes	—
41	C₂³×C₁₉₉	Q₁₆	1	2	yes	—
42	C₂³×C₁₉₉	C₂×D₄	1	1	yes	characteristic
43	C₂×C₄×C₁₉₉	—	2	2	no	—
44	C₂×C₄×C₁₉₉	—	2	4	no	—
45	C₂×C₄×C₁₉₉	—	2	4	no	—
46	C₂×C₄×C₁₉₉	—	2	4	no	—
47	C₂×C₄×C₁₉₉	—	4	8	no	—
48	C₂×C₄×C₁₉₉	—	4	4	no	—
49	C₂×C₄×C₁₉₉	C₂²⋊C₄	1	2	yes	—
50	C₂×C₄×C₁₉₉	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₄×C₁₉₉	C₂×D₄	1	1	yes	characteristic
52	C₂×C₄×C₁₉₉	#41	1	4	yes	—
53	C₂²×C₁₉₉	—	2	2	no	—
54	C₂²×C₁₉₉	—	2	2	no	—
55	C₂²×C₁₉₉	—	2	4	no	—
56	C₂²×C₁₉₉	—	2	4	no	—
57	C₂²×C₁₉₉	#103	1	4	yes	—
58	C₂²×C₁₉₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
59	C₂²×C₁₉₉	#123	1	2	yes	—
60	C₂²×C₁₉₉	C₂×SD₁₆	1	1	yes	characteristic
61	C₂²×C₁₉₉	C₂×Q₁₆	1	1	yes	characteristic
62	C₂²×C₁₉₉	AGL(1,ℤ/8)	1	2	yes	—
63	C₄×C₁₉₉	—	2	2	no	—
64	C₄×C₁₉₉	—	4	4	no	—
65	C₄×C₁₉₉	#117	1	2	yes	—
66	C₄×C₁₉₉	#123	1	2	yes	—
67	Dic₁₉₉	—	8	2	no	—
68	C₂×C₁₉₉	—	2	4	no	—
69	C₂×C₁₉₉	#381	1	1	yes	characteristic
70	C₂×C₁₉₉	C₂×#103	1	1	yes	characteristic
71	C₂×C₁₉₉	#418	1	1	yes	characteristic
72	C₂×C₁₉₉	#481	1	2	yes	—
73	C₂×C₁₉₉	#484	1	2	yes	—
74	C₁₉₉	#1824	1	1	yes	characteristic, complemented, Sylow
75	#1824	—	199	1	no	maximal, Sylow
76	C₂²×C₄⋊C₄	D₁₉₉	1	1	yes	characteristic
77	C₂×C₂²⋊C₈	—	199	1	no	—
78	#380	—	199	1	no	—
79	#340	—	199	4	no	—
80	C₂³×C₄	—	2	1	no	—
81	C₂³×C₄	C₂×D₁₉₉	1	1	yes	characteristic
82	C₂²×C₈	—	199	2	no	—
83	C₂×C₄⋊C₄	—	2	1	no	—
84	C₂×C₄⋊C₄	—	2	2	no	—
85	C₂×C₄⋊C₄	—	199	2	no	—
86	C₂×C₄⋊C₄	Dic₁₉₉	1	4	yes	complemented
87	C₂×C₄⋊C₄	C₂×D₁₉₉	1	2	yes	—
88	C₂×C₂²⋊C₄	—	398	1	no	—
89	C₂²⋊C₈	—	398	2	no	—
90	#95	—	398	1	no	—
91	C₂⁴	C₂²⋊D₁₉₉	1	1	yes	characteristic
92	C₂²×C₄	—	2	4	no	—
93	C₂²×C₄	—	2	4	no	—
94	C₂²×C₄	—	2	2	no	—
95	C₂²×C₄	—	4	4	no	—
96	C₂²×C₄	—	398	2	no	—
97	C₂²×C₄	C₂×Dic₁₉₉	1	2	yes	—
98	C₂²×C₄	C₂²⋊D₁₉₉	1	1	yes	characteristic
99	C₂²×C₄	C₂²⋊D₁₉₉	1	2	yes	—
100	C₂²×C₄	C₂²×D₁₉₉	1	1	yes	characteristic
101	C₂×C₈	—	398	2	no	—
102	C₂×C₈	—	398	4	no	—
103	C₄⋊C₄	—	2	4	no	—
104	C₄⋊C₄	—	2	2	no	—
105	C₄⋊C₄	—	398	4	no	—
106	C₄⋊C₄	C₂×Dic₁₉₉	1	4	yes	—
107	C₂²⋊C₄	—	796	2	no	—
108	C₂³	—	2	2	no	—
109	C₂³	—	2	2	no	—
110	C₂³	C₂²⋊Dic₁₉₉	1	2	yes	—
111	C₂³	#15691873	1	2	yes	—
112	C₂³	#15691876	1	2	yes	—
113	C₂³	C₂×C₂²⋊D₁₉₉	1	1	yes	characteristic
114	C₂×C₄	—	2	2	no	—
115	C₂×C₄	—	2	4	no	—
116	C₂×C₄	—	2	4	no	—
117	C₂×C₄	—	2	4	no	—
118	C₂×C₄	—	4	8	no	—
119	C₂×C₄	—	4	4	no	—
120	C₂×C₄	—	796	2	no	—
121	C₂×C₄	—	796	2	no	—
122	C₂×C₄	—	796	2	no	—
123	C₂×C₄	C₂²⋊Dic₁₉₉	1	2	yes	—
124	C₂×C₄	C₂²×Dic₁₉₉	1	1	yes	characteristic
125	C₂×C₄	C₂×C₂²⋊D₁₉₉	1	1	yes	characteristic
126	C₂×C₄	#15691886	1	4	yes	—
127	C₈	—	796	2	no	—
128	C₂²	—	2	2	no	—
129	C₂²	—	2	4	no	—
130	C₂²	—	2	4	no	—
131	C₂²	—	2	2	no	—
132	C₂²	#20263879	1	4	yes	—
133	C₂²	C₂×C₂²⋊Dic₁₉₉	1	1	yes	characteristic
134	C₂²	#20263938	1	2	yes	—
135	C₂²	C₂×#15691873	1	1	yes	characteristic
136	C₂²	C₂×#15691876	1	1	yes	characteristic
137	C₂²	#20263981	1	2	yes	—
138	C₄	—	2	2	no	—
139	C₄	—	4	4	no	—
140	C₄	—	1592	2	no	—
141	C₄	#20263914	1	2	yes	—
142	C₄	#20263938	1	2	yes	—
143	C₂	—	2	4	no	—
144	C₂	#27163881	1	1	yes	characteristic
145	C₂	C₂×#20263879	1	1	yes	characteristic
146	C₂	#27164023	1	1	yes	characteristic
147	C₂	#27164336	1	2	yes	—
148	C₂	#27164353	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₂²⋊(C₁₉₉⋊C₈)

C₂

yes

maximal, characteristic

#27163878

C₂

yes

maximal, characteristic

#27163780

C₂

yes

maximal, complemented

C₂²×C₄⋊C₄×C₁₉₉

C₂

yes

maximal, characteristic

C₂²×C₁₉₉⋊C₈

C₂²

yes

—

C₂×C₄⋊Dic₁₉₉

C₂²

yes

—

C₂×C₂²⋊Dic₁₉₉

—

C₂²⋊(C₁₉₉⋊C₈)

—

C₁₉₉⋊#95

—

C₂³×C₄×C₁₉₉

—

C₂³×C₄×C₁₉₉

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₁₉₉

—

C₂×C₄⋊C₄×C₁₉₉

—

C₂×C₄⋊C₄×C₁₉₉

C₄

yes

complemented

C₂×C₄⋊C₄×C₁₉₉

C₂²

yes

—

C₂²×Dic₁₉₉

—

C₂×C₁₉₉⋊C₈

—

C₂×C₁₉₉⋊C₈

—

C₄⋊Dic₁₉₉

—

C₂²⋊Dic₁₉₉

—

C₂⁴×C₁₉₉

D₄

yes

characteristic

C₂²×C₄×C₁₉₉

—

C₂²×C₄×C₁₉₉

—

C₂²×C₄×C₁₉₉

—

C₂²×C₄×C₁₉₉

—

C₂²×C₄×C₁₉₉

C₂×C₄

yes

—

C₂²×C₄×C₁₉₉

D₄

yes

characteristic

C₂²×C₄×C₁₉₉

D₄

yes

—

C₂²×C₄×C₁₉₉

C₂³

yes

characteristic

C₄⋊C₄×C₁₉₉

—

C₄⋊C₄×C₁₉₉

—

C₄⋊C₄×C₁₉₉

C₂×C₄

yes

—

C₂×Dic₁₉₉

—

C₂×Dic₁₉₉

—

C₂×Dic₁₉₉

—

C₁₉₉⋊C₈

—

C₂³×C₁₉₉

—

C₂³×C₁₉₉

—

C₂³×C₁₉₉

C₂²⋊C₄

yes

—

C₂³×C₁₉₉

SD₁₆

yes

—

C₂³×C₁₉₉

Q₁₆

yes

—

C₂³×C₁₉₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₉₉

—

C₂×C₄×C₁₉₉

—

C₂×C₄×C₁₉₉

—

C₂×C₄×C₁₉₉

—

C₂×C₄×C₁₉₉

—

C₂×C₄×C₁₉₉

—

C₂×C₄×C₁₉₉

C₂²⋊C₄

yes

—

C₂×C₄×C₁₉₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₉₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₉₉

#41

yes

—

C₂²×C₁₉₉

—

C₂²×C₁₉₉

—

C₂²×C₁₉₉

—

C₂²×C₁₉₉

—

C₂²×C₁₉₉

#103

yes

—

C₂²×C₁₉₉

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₁₉₉

#123

yes

—

C₂²×C₁₉₉

C₂×SD₁₆

yes

characteristic

C₂²×C₁₉₉

C₂×Q₁₆

yes

characteristic

C₂²×C₁₉₉

AGL(1,ℤ/8)

yes

—

C₄×C₁₉₉

—

C₄×C₁₉₉

—

C₄×C₁₉₉

#117

yes

—

C₄×C₁₉₉

#123

yes

—

Dic₁₉₉

—

C₂×C₁₉₉

—

C₂×C₁₉₉

yes

C₂×C₁₉₉

yes

C₂×C₁₉₉

yes

C₂×C₁₉₉

yes

—

C₂×C₁₉₉

#484

yes

—

C₁₉₉

#1824

yes

characteristic, complemented, Sylow

#1824

—

199

maximal, Sylow

C₂²×C₄⋊C₄

D₁₉₉

yes

characteristic

C₂×C₂²⋊C₈

—

199

—

#380

—

199

—

#340

—

199

—

C₂³×C₄

—

C₂³×C₄

C₂×D₁₉₉

yes

characteristic

C₂²×C₈

—

199

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

199

—

C₂×C₄⋊C₄

Dic₁₉₉

yes

complemented

C₂×C₄⋊C₄

C₂×D₁₉₉

yes

—

C₂×C₂²⋊C₄

—

398

—

C₂²⋊C₈

—

398

—

#95

—

398

—

C₂⁴

C₂²⋊D₁₉₉

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

398

—

C₂²×C₄

C₂×Dic₁₉₉

yes

—

C₂²×C₄

C₂²⋊D₁₉₉

yes

characteristic

C₂²×C₄

C₂²⋊D₁₉₉

yes

—

100

C₂²×C₄

C₂²×D₁₉₉

yes

characteristic

101

C₂×C₈

—

398

—

102

C₂×C₈

—

398

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

398

—

106

C₄⋊C₄

C₂×Dic₁₉₉

yes

—

107

C₂²⋊C₄

—

796

—

108

C₂³

—

109

C₂³

—

110

C₂³

C₂²⋊Dic₁₉₉

yes

—

111

C₂³

#15691873

yes

—

112

C₂³

#15691876

yes

—

113

C₂³

C₂×C₂²⋊D₁₉₉

yes

characteristic

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

796

—

121

C₂×C₄

—

796

—

122

C₂×C₄

—

796

—

123

C₂×C₄

C₂²⋊Dic₁₉₉

yes

—

124

C₂×C₄

C₂²×Dic₁₉₉

yes

characteristic

125

C₂×C₄

C₂×C₂²⋊D₁₉₉

yes

characteristic

126

C₂×C₄

#15691886

yes

—

127

C₈

—

796

—

128

C₂²

—

129

C₂²

—

130

C₂²

—

131

C₂²

—

132

C₂²

#20263879

yes

—

133

C₂²

C₂×C₂²⋊Dic₁₉₉

yes

characteristic

134

C₂²

#20263938

yes

—

135

C₂²

C₂×#15691873

yes

characteristic

136

C₂²

C₂×#15691876

yes

characteristic

137

C₂²

#20263981

yes

—

138

C₄

—

139

C₄

—

140

C₄

—

1592

—

141

C₄

#20263914

yes

—

142

C₄

#20263938

yes

—

143

C₂

—

144

C₂

#27163881

yes

characteristic

145

C₂

C₂×#20263879

yes

characteristic

146

C₂

#27164023

yes

characteristic

147

C₂

#27164336

yes

—

148

C₂

#27164353

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₁₉₉
commutator	C₂×C₄×C₁₉₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₉₉
Fitting	C₂²×C₄⋊C₄×C₁₉₉	C₂
socle	C₂³×C₁₉₉	C₂×D₄
prime radical	C₂⁴×C₁₉₉	D₄
cosocle	C₂²×C₄×C₁₉₉	C₂³
hypercenter	C₂²×C₄⋊C₄	D₁₉₉
hypocenter	C₁₉₉	#1824
LCS	G ▷ C₂×C₄×C₁₉₉ ▷ C₂×C₁₉₉ ▷ C₁₉₉
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂²×C₄⋊C₄

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₁₉₉

commutator

C₂×C₄×C₁₉₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₉₉

Fitting

C₂²×C₄⋊C₄×C₁₉₉

C₂

socle

C₂³×C₁₉₉

C₂×D₄

prime radical

C₂⁴×C₁₉₉

D₄

cosocle

C₂²×C₄×C₁₉₉

C₂³

hypercenter

C₂²×C₄⋊C₄

D₁₉₉

hypocenter

C₁₉₉

#1824

LCS

G ▷ C₂×C₄×C₁₉₉ ▷ C₂×C₁₉₉ ▷ C₁₉₉

UCS

C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂²×C₄⋊C₄

N	H ≅ G/N	type
3. #27163780	C₂	semidirect
14. C₂×C₄⋊C₄×C₁₉₉	C₄	semidirect
74. C₁₉₉	#1824	semidirect
86. C₂×C₄⋊C₄	Dic₁₉₉	semidirect

H ≅ G/N

type

3. #27163780

C₂

semidirect

14. C₂×C₄⋊C₄×C₁₉₉

C₄

semidirect

74. C₁₉₉

#1824

semidirect

86. C₂×C₄⋊C₄

Dic₁₉₉

semidirect

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

subgroup	index
C₂×C₂²⋊(C₁₉₉⋊C₈)	2
#27163878	2
#27163780	2
C₂²×C₄⋊C₄×C₁₉₉	2
#1824	199

subgroup

index

C₂×C₂²⋊(C₁₉₉⋊C₈)

#27163878

#27163780

C₂²×C₄⋊C₄×C₁₉₉

#1824

199

Subgroups — #38768940

Conjugacy classes of subgroups

Complements

Maximal subgroups