Group #38770322 — subgroups

Subgroups — #38770322

Order 25472.

230 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 230 conjugacy classes of subgroups — too many to draw. Only its 91 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×#20263888	C₂	1	2	yes	maximal
2	#27163916	C₂	1	1	yes	maximal, characteristic
3	#27163780	C₂	1	2	yes	maximal
4	#27163773	C₂	1	1	yes	maximal, characteristic
5	#384×C₁₉₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₁₉₉⋊C₈	C₂²	1	2	yes	—
7	C₂×C₄⋊Dic₁₉₉	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊Dic₁₉₉	C₂²	1	1	yes	characteristic
9	C₂×Dic₁₉₉⋊C₄	—	2	1	no	—
10	#20263888	—	2	4	no	—
11	C₁₉₉⋊#95	—	2	1	no	—
12	C₂×C₄⋊C₄×C₁₉₉	—	2	1	no	—
13	C₂×C₄⋊C₄×C₁₉₉	—	2	1	no	—
14	C₂×C₄⋊C₄×C₁₉₉	C₂²	1	2	yes	—
15	C₂×C₄²×C₁₉₉	C₂²	1	1	yes	characteristic
16	C₂²×Dic₁₉₉	—	2	1	no	—
17	C₂²×Dic₁₉₉	—	2	1	no	—
18	C₂×C₁₉₉⋊C₈	—	2	2	no	—
19	C₂×C₁₉₉⋊C₈	—	2	2	no	—
20	C₂×C₁₉₉⋊C₈	Q₈	1	4	yes	—
21	C₄⋊Dic₁₉₉	—	2	1	no	—
22	C₄⋊Dic₁₉₉	—	2	2	no	—
23	C₄⋊Dic₁₉₉	D₄	1	2	yes	—
24	Dic₁₉₉⋊C₄	—	4	2	no	—
25	C₂²×C₄×C₁₉₉	—	2	2	no	—
26	C₂²×C₄×C₁₉₉	D₄	1	1	yes	characteristic
27	C₂²×C₄×C₁₉₉	D₄	1	1	yes	characteristic
28	C₂²×C₄×C₁₉₉	C₂³	1	1	yes	characteristic
29	C₄⋊C₄×C₁₉₉	—	2	2	no	—
30	C₄⋊C₄×C₁₉₉	—	2	2	no	—
31	C₄⋊C₄×C₁₉₉	—	4	2	no	—
32	C₄⋊C₄×C₁₉₉	—	4	2	no	—
33	C₄²×C₁₉₉	—	2	2	no	—
34	C₂×Dic₁₉₉	—	2	2	no	—
35	C₂×Dic₁₉₉	—	4	1	no	—
36	C₂×Dic₁₉₉	—	4	1	no	—
37	C₂×Dic₁₉₉	—	4	1	no	—
38	C₂×Dic₁₉₉	—	4	1	no	—
39	C₂×Dic₁₉₉	—	4	1	no	—
40	C₁₉₉⋊C₈	—	2	4	no	—
41	C₂³×C₁₉₉	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₁₉₉	—	2	1	no	—
43	C₂×C₄×C₁₉₉	—	2	1	no	—
44	C₂×C₄×C₁₉₉	—	2	1	no	—
45	C₂×C₄×C₁₉₉	—	2	1	no	—
46	C₂×C₄×C₁₉₉	—	4	2	no	—
47	C₂×C₄×C₁₉₉	—	4	2	no	—
48	C₂×C₄×C₁₉₉	—	4	2	no	—
49	C₂×C₄×C₁₉₉	D₈	1	2	yes	—
50	C₂×C₄×C₁₉₉	Q₁₆	1	2	yes	—
51	C₂×C₄×C₁₉₉	C₂×D₄	1	1	yes	characteristic
52	C₂×C₄×C₁₉₉	C₂×Q₈	1	2	yes	—
53	C₂×C₄×C₁₉₉	#41	1	1	yes	characteristic
54	C₂×C₄×C₁₉₉	#41	1	2	yes	—
55	C₂²×C₁₉₉	#121	1	1	yes	characteristic
56	C₂²×C₁₉₉	#123	1	1	yes	characteristic
57	C₂²×C₁₉₉	C₄⋊Q₈	1	1	yes	characteristic
58	C₂²×C₁₉₉	C₂×D₈	1	1	yes	characteristic
59	C₂²×C₁₉₉	C₂×Q₁₆	1	1	yes	characteristic
60	C₂²×C₁₉₉	AGL(1,ℤ/8)	1	1	yes	characteristic
61	C₂²×C₁₉₉	#137	1	1	yes	characteristic
62	C₄×C₁₉₉	—	2	2	no	—
63	C₄×C₁₉₉	—	2	2	no	—
64	C₄×C₁₉₉	—	8	2	no	—
65	C₄×C₁₉₉	C₂²⋊Q₈	1	2	yes	—
66	C₄×C₁₉₉	#125	1	2	yes	—
67	Dic₁₉₉	—	4	2	no	—
68	Dic₁₉₉	—	8	1	no	—
69	C₂×C₁₉₉	#398	1	1	yes	characteristic
70	C₂×C₁₉₉	#459	1	1	yes	characteristic
71	C₂×C₁₉₉	#462	1	1	yes	characteristic
72	C₂×C₁₉₉	#480	1	1	yes	characteristic
73	C₂×C₁₉₉	#484	1	1	yes	characteristic
74	C₂×C₁₉₉	#500	1	1	yes	characteristic
75	C₂×C₁₉₉	#501	1	1	yes	characteristic
76	C₁₉₉	#2083	1	1	yes	characteristic, complemented, Sylow
77	#2083	—	199	1	no	maximal, Sylow
78	C₂×#107	—	199	2	no	—
79	#389	—	199	1	no	—
80	#384	D₁₉₉	1	1	yes	characteristic
81	#340	—	199	2	no	—
82	#336	—	199	1	no	—
83	C₂²×C₈	—	199	2	no	—
84	C₂×C₄⋊C₄	—	2	1	no	—
85	C₂×C₄⋊C₄	—	2	1	no	—
86	C₂×C₄⋊C₄	—	398	1	no	—
87	C₂×C₄⋊C₄	—	199	1	no	—
88	C₂×C₄⋊C₄	—	199	1	no	—
89	C₂×C₄⋊C₄	C₂×D₁₉₉	1	2	yes	—
90	C₂×C₄²	C₂×D₁₉₉	1	1	yes	characteristic
91	#107	—	398	4	no	—
92	#95	—	398	1	no	—
93	C₂²×C₄	—	2	2	no	—
94	C₂²×C₄	—	398	1	no	—
95	C₂²×C₄	—	398	1	no	—
96	C₂²×C₄	C₂²⋊D₁₉₉	1	1	yes	characteristic
97	C₂²×C₄	C₂²⋊D₁₉₉	1	1	yes	characteristic
98	C₂²×C₄	C₂²×D₁₉₉	1	1	yes	characteristic
99	C₂×C₈	—	398	2	no	—
100	C₂×C₈	—	398	2	no	—
101	C₂×C₈	—	199	4	no	—
102	C₄⋊C₄	—	2	2	no	—
103	C₄⋊C₄	—	2	2	no	—
104	C₄⋊C₄	—	4	2	no	—
105	C₄⋊C₄	—	4	2	no	—
106	C₄⋊C₄	—	796	2	no	—
107	C₄⋊C₄	—	398	2	no	—
108	C₄⋊C₄	—	398	1	no	—
109	C₄⋊C₄	—	199	2	no	—
110	C₄²	—	2	2	no	—
111	C₂³	C₂×C₂²⋊D₁₉₉	1	1	yes	characteristic
112	C₂×C₄	—	2	1	no	—
113	C₂×C₄	—	2	1	no	—
114	C₂×C₄	—	2	1	no	—
115	C₂×C₄	—	2	1	no	—
116	C₂×C₄	—	4	2	no	—
117	C₂×C₄	—	4	2	no	—
118	C₂×C₄	—	4	2	no	—
119	C₂×C₄	—	796	1	no	—
120	C₂×C₄	—	796	1	no	—
121	C₂×C₄	—	796	1	no	—
122	C₂×C₄	—	796	1	no	—
123	C₂×C₄	—	796	1	no	—
124	C₂×C₄	—	398	2	no	—
125	C₂×C₄	#15691871	1	2	yes	—
126	C₂×C₄	#15691876	1	2	yes	—
127	C₂×C₄	D₄×D₁₉₉	1	1	yes	characteristic
128	C₂×C₄	Q₈×D₁₉₉	1	2	yes	—
129	C₂×C₄	#15691885	1	1	yes	characteristic
130	C₂×C₄	#15691886	1	2	yes	—
131	C₈	—	398	4	no	—
132	C₂²	#20263926	1	1	yes	characteristic
133	C₂²	#20263938	1	1	yes	characteristic
134	C₂²	#20263953	1	1	yes	characteristic
135	C₂²	C₂×#15691871	1	1	yes	characteristic
136	C₂²	C₂×#15691876	1	1	yes	characteristic
137	C₂²	#20263981	1	1	yes	characteristic
138	C₂²	#20263986	1	1	yes	characteristic
139	C₄	—	2	2	no	—
140	C₄	—	2	2	no	—
141	C₄	—	8	2	no	—
142	C₄	—	796	2	no	—
143	C₄	—	1592	1	no	—
144	C₄	#20263933	1	2	yes	—
145	C₄	#20263945	1	2	yes	—
146	C₂	#27163949	1	1	yes	characteristic
147	C₂	#27164210	1	1	yes	characteristic
148	C₂	#27164225	1	1	yes	characteristic
149	C₂	#27164331	1	1	yes	characteristic
150	C₂	#27164353	1	1	yes	characteristic
151	C₂	#27164421	1	1	yes	characteristic
152	C₂	#27164424	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#20263888

C₂

yes

maximal

#27163916

C₂

yes

maximal, characteristic

#27163780

C₂

yes

maximal

#27163773

C₂

yes

maximal, characteristic

#384×C₁₉₉

C₂

yes

maximal, characteristic

C₂²×C₁₉₉⋊C₈

C₂²

yes

—

C₂×C₄⋊Dic₁₉₉

C₂²

yes

characteristic

C₂×C₄⋊Dic₁₉₉

C₂²

yes

characteristic

C₂×Dic₁₉₉⋊C₄

—

#20263888

—

C₁₉₉⋊#95

—

C₂×C₄⋊C₄×C₁₉₉

—

C₂×C₄⋊C₄×C₁₉₉

—

C₂×C₄⋊C₄×C₁₉₉

C₂²

yes

—

C₂×C₄²×C₁₉₉

C₂²

yes

characteristic

C₂²×Dic₁₉₉

—

C₂²×Dic₁₉₉

—

C₂×C₁₉₉⋊C₈

—

C₂×C₁₉₉⋊C₈

—

C₂×C₁₉₉⋊C₈

Q₈

yes

—

C₄⋊Dic₁₉₉

—

C₄⋊Dic₁₉₉

—

C₄⋊Dic₁₉₉

D₄

yes

—

Dic₁₉₉⋊C₄

—

C₂²×C₄×C₁₉₉

—

C₂²×C₄×C₁₉₉

D₄

yes

characteristic

C₂²×C₄×C₁₉₉

D₄

yes

characteristic

C₂²×C₄×C₁₉₉

C₂³

yes

characteristic

C₄⋊C₄×C₁₉₉

—

C₄⋊C₄×C₁₉₉

—

C₄⋊C₄×C₁₉₉

—

C₄⋊C₄×C₁₉₉

—

C₄²×C₁₉₉

—

C₂×Dic₁₉₉

—

C₂×Dic₁₉₉

—

C₂×Dic₁₉₉

—

C₂×Dic₁₉₉

—

C₂×Dic₁₉₉

—

C₂×Dic₁₉₉

—

C₁₉₉⋊C₈

—

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

—

C₂×C₄×C₁₉₉

D₈

yes

—

C₂×C₄×C₁₉₉

Q₁₆

yes

—

C₂×C₄×C₁₉₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₉₉

C₂×Q₈

yes

—

C₂×C₄×C₁₉₉

#41

yes

characteristic

C₂×C₄×C₁₉₉

#41

yes

—

C₂²×C₁₉₉

yes

C₂²×C₁₉₉

yes

C₂²×C₁₉₉

C₄⋊Q₈

yes

characteristic

C₂²×C₁₉₉

C₂×D₈

yes

characteristic

C₂²×C₁₉₉

C₂×Q₁₆

yes

C₂²×C₁₉₉

yes

C₂²×C₁₉₉

yes

C₄×C₁₉₉

—

C₄×C₁₉₉

—

C₄×C₁₉₉

—

C₄×C₁₉₉

C₂²⋊Q₈

yes

—

C₄×C₁₉₉

#125

yes

—

Dic₁₉₉

—

Dic₁₉₉

—

C₂×C₁₉₉

yes

C₂×C₁₉₉

yes

C₂×C₁₉₉

yes

C₂×C₁₉₉

yes

C₂×C₁₉₉

yes

C₂×C₁₉₉

yes

C₂×C₁₉₉

yes

C₁₉₉

yes

characteristic, complemented, Sylow

#2083

—

199

maximal, Sylow

C₂×#107

—

199

—

#389

—

199

—

#384

D₁₉₉

yes

characteristic

#340

—

199

—

#336

—

199

—

C₂²×C₈

—

199

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

398

—

C₂×C₄⋊C₄

—

199

—

C₂×C₄⋊C₄

—

199

—

C₂×C₄⋊C₄

C₂×D₁₉₉

yes

—

C₂×C₄²

C₂×D₁₉₉

yes

characteristic

#107

—

398

—

#95

—

398

—

C₂²×C₄

—

C₂²×C₄

—

398

—

C₂²×C₄

—

398

—

C₂²×C₄

C₂²⋊D₁₉₉

yes

characteristic

C₂²×C₄

C₂²⋊D₁₉₉

yes

characteristic

C₂²×C₄

C₂²×D₁₉₉

yes

characteristic

C₂×C₈

—

398

—

100

C₂×C₈

—

398

—

101

C₂×C₈

—

199

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

106

C₄⋊C₄

—

796

—

107

C₄⋊C₄

—

398

—

108

C₄⋊C₄

—

398

—

109

C₄⋊C₄

—

199

—

110

C₄²

—

111

C₂³

C₂×C₂²⋊D₁₉₉

yes

characteristic

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

796

—

120

C₂×C₄

—

796

—

121

C₂×C₄

—

796

—

122

C₂×C₄

—

796

—

123

C₂×C₄

—

796

—

124

C₂×C₄

—

398

—

125

C₂×C₄

#15691871

yes

—

126

C₂×C₄

#15691876

yes

—

127

C₂×C₄

D₄×D₁₉₉

yes

characteristic

128

C₂×C₄

Q₈×D₁₉₉

yes

—

129

C₂×C₄

#15691885

yes

characteristic

130

C₂×C₄

#15691886

yes

—

131

C₈

—

398

—

132

C₂²

#20263926

yes

characteristic

133

C₂²

#20263938

yes

characteristic

134

C₂²

#20263953

yes

characteristic

135

C₂²

C₂×#15691871

yes

characteristic

136

C₂²

C₂×#15691876

yes

characteristic

137

C₂²

#20263981

yes

characteristic

138

C₂²

#20263986

yes

characteristic

139

C₄

—

140

C₄

—

141

C₄

—

142

C₄

—

796

—

143

C₄

—

1592

—

144

C₄

#20263933

yes

—

145

C₄

#20263945

yes

—

146

C₂

#27163949

yes

characteristic

147

C₂

#27164210

yes

characteristic

148

C₂

#27164225

yes

characteristic

149

C₂

#27164331

yes

characteristic

150

C₂

#27164353

yes

characteristic

151

C₂

#27164421

yes

characteristic

152

C₂

#27164424

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₁₉₉
commutator, cosocle	C₂²×C₄×C₁₉₉	C₂³
Frattini	C₂²×C₄	C₂²×D₁₉₉
Fitting	#384×C₁₉₉	C₂
socle, prime radical	C₂³×C₁₉₉	C₂×D₄
hypercenter	#384	D₁₉₉
hypocenter	C₁₉₉	#2083
LCS	G ▷ C₂²×C₄×C₁₉₉ ▷ C₂×C₁₉₉ ▷ C₁₉₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #384

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₁₉₉

commutator, cosocle

C₂²×C₄×C₁₉₉

C₂³

Frattini

C₂²×C₄

C₂²×D₁₉₉

Fitting

#384×C₁₉₉

C₂

socle, prime radical

C₂³×C₁₉₉

C₂×D₄

D₁₉₉

C₁₉₉

G ▷ C₂²×C₄×C₁₉₉ ▷ C₂×C₁₉₉ ▷ C₁₉₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #384

N	H ≅ G/N	type
76. C₁₉₉	#2083	semidirect

H ≅ G/N

type

76. C₁₉₉

#2083

semidirect

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

subgroup	index
C₂×#20263888	2
#27163916	2
#27163780	2
#27163773	2
#384×C₁₉₉	2
#2083	199

subgroup

index

#384×C₁₉₉

199

Subgroups — #38770322

Conjugacy classes of subgroups

Complements

Maximal subgroups