Group #38770383 — subgroups

Subgroups — #38770383

Order 25472.

290 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 290 conjugacy classes of subgroups — too many to draw. Only its 93 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×#20263886	C₂	1	2	yes	maximal, complemented
2	C₂×#20263876	C₂	1	2	yes	maximal
3	#27163888	C₂	1	1	yes	maximal, characteristic, complemented
4	#27163773	C₂	1	1	yes	maximal, characteristic, complemented
5	#390×C₁₉₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₁₉₉⋊C₈	C₂²	1	2	yes	—
7	C₂×C₄⋊Dic₁₉₉	—	2	2	no	—
8	C₂×C₄⋊Dic₁₉₉	C₂²	1	2	yes	—
9	#20263886	—	2	4	no	—
10	#20263876	—	2	4	no	—
11	C₂²×D₄×C₁₉₉	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊C₄×C₁₉₉	C₂²	1	1	yes	characteristic
13	C₂×C₂²⋊C₄×C₁₉₉	—	2	2	no	—
14	C₂×C₄²×C₁₉₉	C₂²	1	1	yes	characteristic
15	C₂²×Dic₁₉₉	—	2	2	no	—
16	C₂×C₁₉₉⋊C₈	—	2	2	no	—
17	C₂×C₁₉₉⋊C₈	—	2	2	no	—
18	C₂×C₁₉₉⋊C₈	D₄	1	4	yes	—
19	C₄⋊Dic₁₉₉	—	2	2	no	—
20	C₄⋊Dic₁₉₉	—	2	2	no	—
21	C₄⋊Dic₁₉₉	—	4	4	no	—
22	C₂⁴×C₁₉₉	—	2	1	no	—
23	C₂×D₄×C₁₉₉	—	2	2	no	—
24	C₂×D₄×C₁₉₉	—	2	2	no	—
25	C₂×D₄×C₁₉₉	—	2	1	no	—
26	C₂×D₄×C₁₉₉	Q₈	1	2	yes	—
27	C₂²×C₄×C₁₉₉	—	2	1	no	—
28	C₂²×C₄×C₁₉₉	D₄	1	2	yes	—
29	C₂²×C₄×C₁₉₉	C₂³	1	1	yes	characteristic
30	C₄⋊C₄×C₁₉₉	—	2	2	no	—
31	C₂²⋊C₄×C₁₉₉	—	4	4	no	—
32	C₄²×C₁₉₉	—	2	2	no	—
33	C₂×Dic₁₉₉	—	4	2	no	—
34	C₂×Dic₁₉₉	—	4	4	no	—
35	C₁₉₉⋊C₈	—	2	4	no	—
36	C₂³×C₁₉₉	—	2	2	no	—
37	C₂³×C₁₉₉	—	4	1	no	—
38	C₂³×C₁₉₉	—	4	1	no	—
39	C₂³×C₁₉₉	—	4	2	no	—
40	C₂³×C₁₉₉	—	4	2	no	—
41	C₂³×C₁₉₉	C₂×D₄	1	1	yes	characteristic
42	D₄×C₁₉₉	—	2	4	no	—
43	D₄×C₁₉₉	—	4	2	no	—
44	C₂×C₄×C₁₉₉	—	2	2	no	—
45	C₂×C₄×C₁₉₉	—	2	2	no	—
46	C₂×C₄×C₁₉₉	—	4	1	no	—
47	C₂×C₄×C₁₉₉	—	4	2	no	—
48	C₂×C₄×C₁₉₉	SD₁₆	1	4	yes	—
49	C₂×C₄×C₁₉₉	C₂×D₄	1	2	yes	—
50	C₂×C₄×C₁₉₉	C₂×Q₈	1	1	yes	characteristic
51	C₂×C₄×C₁₉₉	#41	1	2	yes	—
52	C₂×C₄×C₁₉₉	#41	1	1	yes	characteristic
53	C₂²×C₁₉₉	—	4	2	no	—
54	C₂²×C₁₉₉	—	4	4	no	—
55	C₂²×C₁₉₉	—	8	2	no	—
56	C₂²×C₁₉₉	—	8	1	no	—
57	C₂²×C₁₉₉	—	8	1	no	—
58	C₂²×C₁₉₉	C₂²⋊Q₈	1	1	yes	characteristic
59	C₂²×C₁₉₉	#123	1	1	yes	characteristic
60	C₂²×C₁₉₉	#127	1	1	yes	characteristic
61	C₂²×C₁₉₉	C₂×SD₁₆	1	2	yes	—
62	C₂²×C₁₉₉	AGL(1,ℤ/8)	1	2	yes	—
63	C₄×C₁₉₉	—	2	4	no	—
64	C₄×C₁₉₉	—	8	1	no	—
65	C₄×C₁₉₉	C₂²⋊Q₈	1	2	yes	—
66	C₄×C₁₉₉	#123	1	2	yes	—
67	Dic₁₉₉	—	8	2	no	—
68	C₂×C₁₉₉	—	8	2	no	—
69	C₂×C₁₉₉	#399	1	1	yes	characteristic
70	C₂×C₁₉₉	#476	1	2	yes	—
71	C₂×C₁₉₉	#481	1	2	yes	—
72	C₂×C₁₉₉	#492	1	1	yes	characteristic
73	C₂×C₁₉₉	#496	1	1	yes	characteristic
74	C₁₉₉	#2094	1	1	yes	characteristic, complemented, Sylow
75	#2094	—	199	1	no	maximal, Sylow
76	C₂×#106	—	199	2	no	—
77	C₂×#102	—	199	2	no	—
78	#390	D₁₉₉	1	1	yes	characteristic
79	#384	—	199	1	no	—
80	#336	—	199	1	no	—
81	C₂²×D₄	C₂×D₁₉₉	1	1	yes	characteristic
82	C₂²×C₈	—	199	2	no	—
83	C₂×C₄⋊C₄	—	398	2	no	—
84	C₂×C₄⋊C₄	—	199	2	no	—
85	C₂×C₄⋊C₄	C₂×D₁₉₉	1	1	yes	characteristic
86	C₂×C₂²⋊C₄	—	2	2	no	—
87	C₂×C₄²	C₂×D₁₉₉	1	1	yes	characteristic
88	#106	—	398	4	no	—
89	#102	—	398	4	no	—
90	C₂⁴	—	2	1	no	—
91	C₂×D₄	—	2	2	no	—
92	C₂×D₄	—	2	2	no	—
93	C₂×D₄	—	2	1	no	—
94	C₂×D₄	Dic₃₉₈	1	2	yes	—
95	C₂²×C₄	—	2	1	no	—
96	C₂²×C₄	—	398	2	no	—
97	C₂²×C₄	C₂²⋊D₁₉₉	1	2	yes	—
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	—	796	4	no	—
104	C₄⋊C₄	—	398	2	no	—
105	C₄⋊C₄	—	398	2	no	—
106	C₂²⋊C₄	—	4	4	no	—
107	C₄²	—	2	2	no	—
108	C₂³	—	2	2	no	—
109	C₂³	—	4	1	no	—
110	C₂³	—	4	1	no	—
111	C₂³	—	4	2	no	—
112	C₂³	—	4	2	no	—
113	C₂³	C₂×C₂²⋊D₁₉₉	1	1	yes	characteristic
114	D₄	—	2	4	no	—
115	D₄	—	4	2	no	—
116	C₂×C₄	—	2	2	no	—
117	C₂×C₄	—	2	2	no	—
118	C₂×C₄	—	4	1	no	—
119	C₂×C₄	—	4	2	no	—
120	C₂×C₄	—	796	2	no	—
121	C₂×C₄	—	796	4	no	—
122	C₂×C₄	#15691874	1	4	yes	—
123	C₂×C₄	D₄×D₁₉₉	1	2	yes	—
124	C₂×C₄	C₂×Dic₃₉₈	1	1	yes	characteristic
125	C₂×C₄	#15691885	1	2	yes	—
126	C₂×C₄	#15691886	1	1	yes	characteristic
127	C₈	—	398	4	no	—
128	C₂²	—	4	2	no	—
129	C₂²	—	4	4	no	—
130	C₂²	—	8	2	no	—
131	C₂²	—	8	1	no	—
132	C₂²	—	8	1	no	—
133	C₂²	#20263934	1	1	yes	characteristic
134	C₂²	#20263936	1	1	yes	characteristic
135	C₂²	#20263951	1	1	yes	characteristic
136	C₂²	C₂×#15691874	1	2	yes	—
137	C₂²	#20263980	1	2	yes	—
138	C₄	—	2	4	no	—
139	C₄	—	8	1	no	—
140	C₄	—	1592	2	no	—
141	C₄	#20263932	1	2	yes	—
142	C₄	#20263937	1	2	yes	—
143	C₂	—	8	2	no	—
144	C₂	#27163954	1	1	yes	characteristic
145	C₂	#27164308	1	2	yes	—
146	C₂	#27164334	1	2	yes	—
147	C₂	#27164390	1	1	yes	characteristic
148	C₂	#27164408	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#20263886

C₂

yes

maximal, complemented

C₂×#20263876

C₂

yes

maximal

#27163888

C₂

yes

maximal, characteristic, complemented

#27163773

C₂

yes

maximal, characteristic, complemented

#390×C₁₉₉

C₂

yes

maximal, characteristic

C₂²×C₁₉₉⋊C₈

C₂²

yes

—

C₂×C₄⋊Dic₁₉₉

—

C₂×C₄⋊Dic₁₉₉

C₂²

yes

—

#20263886

—

#20263876

—

C₂²×D₄×C₁₉₉

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₁₉₉

C₂²

yes

characteristic

C₂×C₂²⋊C₄×C₁₉₉

—

C₂×C₄²×C₁₉₉

C₂²

yes

characteristic

C₂²×Dic₁₉₉

—

C₂×C₁₉₉⋊C₈

—

C₂×C₁₉₉⋊C₈

—

C₂×C₁₉₉⋊C₈

D₄

yes

—

C₄⋊Dic₁₉₉

—

C₄⋊Dic₁₉₉

—

C₄⋊Dic₁₉₉

—

C₂⁴×C₁₉₉

—

C₂×D₄×C₁₉₉

—

C₂×D₄×C₁₉₉

—

C₂×D₄×C₁₉₉

—

C₂×D₄×C₁₉₉

Q₈

yes

—

C₂²×C₄×C₁₉₉

—

C₂²×C₄×C₁₉₉

D₄

yes

—

C₂²×C₄×C₁₉₉

C₂³

yes

characteristic

C₄⋊C₄×C₁₉₉

—

C₂²⋊C₄×C₁₉₉

—

C₄²×C₁₉₉

—

C₂×Dic₁₉₉

—

C₂×Dic₁₉₉

—

C₁₉₉⋊C₈

—

C₂³×C₁₉₉

—

C₂³×C₁₉₉

—

C₂³×C₁₉₉

—

C₂³×C₁₉₉

—

C₂³×C₁₉₉

—

C₂³×C₁₉₉

C₂×D₄

yes

characteristic

D₄×C₁₉₉

—

D₄×C₁₉₉

—

C₂×C₄×C₁₉₉

—

C₂×C₄×C₁₉₉

—

C₂×C₄×C₁₉₉

—

C₂×C₄×C₁₉₉

—

C₂×C₄×C₁₉₉

SD₁₆

yes

—

C₂×C₄×C₁₉₉

C₂×D₄

yes

—

C₂×C₄×C₁₉₉

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₉₉

#41

yes

—

C₂×C₄×C₁₉₉

#41

yes

characteristic

C₂²×C₁₉₉

—

C₂²×C₁₉₉

—

C₂²×C₁₉₉

—

C₂²×C₁₉₉

—

C₂²×C₁₉₉

—

C₂²×C₁₉₉

C₂²⋊Q₈

yes

C₂²×C₁₉₉

yes

C₂²×C₁₉₉

yes

C₂²×C₁₉₉

C₂×SD₁₆

yes

—

C₂²×C₁₉₉

AGL(1,ℤ/8)

yes

—

C₄×C₁₉₉

—

C₄×C₁₉₉

—

C₄×C₁₉₉

C₂²⋊Q₈

yes

—

C₄×C₁₉₉

#123

yes

—

Dic₁₉₉

—

C₂×C₁₉₉

—

C₂×C₁₉₉

#399

yes

characteristic

C₂×C₁₉₉

#476

yes

—

C₂×C₁₉₉

#481

yes

—

C₂×C₁₉₉

yes

C₂×C₁₉₉

yes

C₁₉₉

yes

characteristic, complemented, Sylow

#2094

—

199

maximal, Sylow

C₂×#106

—

199

—

C₂×#102

—

199

—

#390

D₁₉₉

yes

characteristic

#384

—

199

—

#336

—

199

—

C₂²×D₄

C₂×D₁₉₉

yes

characteristic

C₂²×C₈

—

199

—

C₂×C₄⋊C₄

—

398

—

C₂×C₄⋊C₄

—

199

—

C₂×C₄⋊C₄

C₂×D₁₉₉

yes

characteristic

C₂×C₂²⋊C₄

—

C₂×C₄²

C₂×D₁₉₉

yes

characteristic

#106

—

398

—

#102

—

398

—

C₂⁴

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

Dic₃₉₈

yes

—

C₂²×C₄

—

C₂²×C₄

—

398

—

C₂²×C₄

C₂²⋊D₁₉₉

yes

—

C₂²×C₄

C₂²×D₁₉₉

yes

characteristic

C₂×C₈

—

398

—

100

C₂×C₈

—

398

—

101

C₂×C₈

—

199

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

796

—

104

C₄⋊C₄

—

398

—

105

C₄⋊C₄

—

398

—

106

C₂²⋊C₄

—

107

C₄²

—

108

C₂³

—

109

C₂³

—

110

C₂³

—

111

C₂³

—

112

C₂³

—

113

C₂³

C₂×C₂²⋊D₁₉₉

yes

characteristic

114

D₄

—

115

D₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

796

—

121

C₂×C₄

—

796

—

122

C₂×C₄

#15691874

yes

—

123

C₂×C₄

D₄×D₁₉₉

yes

—

124

C₂×C₄

C₂×Dic₃₉₈

yes

characteristic

125

C₂×C₄

#15691885

yes

—

126

C₂×C₄

#15691886

yes

characteristic

127

C₈

—

398

—

128

C₂²

—

129

C₂²

—

130

C₂²

—

131

C₂²

—

132

C₂²

—

133

C₂²

#20263934

yes

characteristic

134

C₂²

#20263936

yes

characteristic

135

C₂²

#20263951

yes

characteristic

136

C₂²

C₂×#15691874

yes

—

137

C₂²

#20263980

yes

—

138

C₄

—

139

C₄

—

140

C₄

—

1592

—

141

C₄

#20263932

yes

—

142

C₄

#20263937

yes

—

143

C₂

—

144

C₂

#27163954

yes

characteristic

145

C₂

#27164308

yes

—

146

C₂

#27164334

yes

—

147

C₂

#27164390

yes

characteristic

148

C₂

#27164408

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₁₉₉
commutator, cosocle	C₂²×C₄×C₁₉₉	C₂³
Frattini	C₂²×C₄	C₂²×D₁₉₉
Fitting	#390×C₁₉₉	C₂
socle	C₂³×C₁₉₉	C₂×D₄
prime radical	C₂²×D₄×C₁₉₉	C₂²
hypercenter	#390	D₁₉₉
hypocenter	C₁₉₉	#2094
LCS	G ▷ C₂²×C₄×C₁₉₉ ▷ C₂×C₁₉₉ ▷ C₁₉₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #390

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₁₉₉

commutator, cosocle

C₂²×C₄×C₁₉₉

C₂³

Frattini

C₂²×C₄

C₂²×D₁₉₉

Fitting

#390×C₁₉₉

C₂

socle

C₂³×C₁₉₉

C₂×D₄

prime radical

C₂²×D₄×C₁₉₉

C₂²

D₁₉₉

C₁₉₉

G ▷ C₂²×C₄×C₁₉₉ ▷ C₂×C₁₉₉ ▷ C₁₉₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #390

N	H ≅ G/N	type
1. C₂×#20263886	C₂	semidirect
3. #27163888	C₂	semidirect
4. #27163773	C₂	semidirect
74. C₁₉₉	#2094	semidirect

H ≅ G/N

type

1. C₂×#20263886

C₂

semidirect

3. #27163888

C₂

semidirect

4. #27163773

C₂

semidirect

74. C₁₉₉

#2094

semidirect

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

subgroup	index
C₂×#20263886	2
C₂×#20263876	2
#27163888	2
#27163773	2
#390×C₁₉₉	2
#2094	199

subgroup

index

#390×C₁₉₉

199

Subgroups — #38770383

Conjugacy classes of subgroups

Complements

Maximal subgroups