Group #38769972 — subgroups

Subgroups — #38769972

Order 25472.

342 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 75 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×#20263988	C₂	1	2	yes	maximal
2	C₂²×Dic₇₉₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#20263873	C₂	1	2	yes	maximal, complemented
4	#27163984	C₂	1	1	yes	maximal, characteristic
5	#341×C₁₉₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#15691886	C₂²	1	1	yes	characteristic
7	C₂²×Dic₃₉₈	C₂²	1	2	yes	—
8	#20263988	—	2	4	no	—
9	#20263988	—	2	4	no	—
10	C₂×Dic₇₉₆	—	2	4	no	—
11	C₂×Dic₇₉₆	—	2	2	no	—
12	C₂×Dic₇₉₆	—	2	2	no	—
13	C₂×#15691875	—	2	2	no	—
14	C₂×#15691870	C₂²	1	1	yes	characteristic
15	#20263873	—	2	4	no	—
16	(C₂×D₁₉₉)⋊C₈	—	2	2	no	—
17	C₂×M₁₆×C₁₉₉	C₂²	1	2	yes	—
18	C₂²×C₈×C₁₉₉	C₂²	1	1	yes	characteristic
19	#15691886	—	2	2	no	—
20	#15691886	—	2	2	no	—
21	C₂×Dic₃₉₈	—	2	2	no	—
22	C₂×Dic₃₉₈	—	2	2	no	—
23	C₂×Dic₃₉₈	—	2	2	no	—
24	C₂×Dic₃₉₈	—	2	2	no	—
25	C₂×Dic₃₉₈	—	2	2	no	—
26	C₂×Dic₃₉₈	D₄	1	1	yes	characteristic
27	C₂×Dic₃₉₈	D₄	1	4	yes	—
28	C₂×D₇₉₆	D₄	1	1	yes	characteristic
29	C₂×C₂²⋊D₁₉₉	—	2	1	no	—
30	C₂²×Dic₁₉₉	—	2	2	no	—
31	C₂×C₄×D₁₉₉	—	2	1	no	—
32	Dic₇₉₆	—	2	8	no	—
33	Dic₇₉₆	—	2	8	no	—
34	#15691875	—	2	8	no	—
35	#15691870	—	2	2	no	—
36	C₂×C₁₉₉⋊C₈	—	2	1	no	—
37	C₂²×C₄×C₁₉₉	C₂³	1	1	yes	characteristic
38	M₁₆×C₁₉₉	—	2	2	no	—
39	M₁₆×C₁₉₉	D₄	1	4	yes	—
40	C₂×C₈×C₁₉₉	—	2	2	no	—
41	C₂×C₈×C₁₉₉	—	2	2	no	—
42	C₂×C₈×C₁₉₉	D₄	1	2	yes	—
43	C₂²×D₁₉₉	—	2	1	no	—
44	Dic₃₉₈	—	2	4	no	—
45	Dic₃₉₈	—	2	4	no	—
46	Dic₃₉₈	—	2	2	no	—
47	Dic₃₉₈	—	4	2	no	—
48	Dic₃₉₈	—	4	2	no	—
49	D₇₉₆	—	2	2	no	—
50	C₂²⋊D₁₉₉	—	4	2	no	—
51	C₂×Dic₁₉₉	—	2	1	no	—
52	C₂×Dic₁₉₉	—	2	4	no	—
53	C₂×Dic₁₉₉	—	4	2	no	—
54	C₂×Dic₁₉₉	—	4	2	no	—
55	C₄×D₁₉₉	—	4	1	no	—
56	C₄×D₁₉₉	—	4	1	no	—
57	C₁₉₉⋊C₈	—	4	1	no	—
58	C₂³×C₁₉₉	#41	1	1	yes	characteristic
59	C₂×C₄×C₁₉₉	C₂×D₄	1	1	yes	characteristic
60	C₂×C₄×C₁₉₉	C₂×D₄	1	2	yes	—
61	C₂×C₄×C₁₉₉	C₂×D₄	1	1	yes	characteristic
62	C₂×C₄×C₁₉₉	C₂×D₄	1	2	yes	—
63	C₈×C₁₉₉	—	2	2	no	—
64	C₈×C₁₉₉	—	2	4	no	—
65	C₂²×C₁₉₉	—	2	2	no	—
66	C₂²×C₁₉₉	#121	1	2	yes	—
67	C₂²×C₁₉₉	#121	1	1	yes	characteristic
68	C₂×D₁₉₉	—	4	1	no	—
69	C₂×D₁₉₉	—	4	2	no	—
70	C₄×C₁₉₉	#120	1	1	yes	characteristic
71	C₄×C₁₉₉	#120	1	2	yes	—
72	C₄×C₁₉₉	#127	1	1	yes	characteristic
73	Dic₁₉₉	—	4	1	no	—
74	Dic₁₉₉	—	4	4	no	—
75	C₂×C₁₉₉	—	2	2	no	—
76	C₂×C₁₉₉	#392	1	1	yes	characteristic
77	C₂×C₁₉₉	#473	1	2	yes	—
78	D₁₉₉	—	8	1	no	—
79	C₁₉₉	#2020	1	1	yes	characteristic, complemented, Sylow
80	#2020	—	199	1	no	maximal, Sylow
81	C₂×#137	—	199	2	no	—
82	C₂²×Q₁₆	—	199	1	no	—
83	C₂×#101	—	199	2	no	—
84	#408	—	199	1	no	—
85	#341	D₁₉₉	1	1	yes	characteristic, complemented
86	C₂×#41	—	199	1	no	—
87	C₂²×Q₈	—	199	2	no	—
88	#137	—	398	4	no	—
89	#137	—	398	4	no	—
90	C₂×Q₁₆	—	398	4	no	—
91	C₂×Q₁₆	—	398	2	no	—
92	C₂×Q₁₆	—	398	2	no	—
93	C₂×SD₁₆	—	398	2	no	—
94	C₂×M₁₆	—	199	1	no	—
95	C₂×M₁₆	C₂×D₁₉₉	1	2	yes	—
96	C₂²×C₈	C₂×D₁₉₉	1	1	yes	characteristic
97	#101	—	398	4	no	—
98	C₂²⋊C₈	—	398	2	no	—
99	#41	—	398	2	no	—
100	#41	—	398	2	no	—
101	C₂×Q₈	—	398	2	no	—
102	C₂×Q₈	—	398	2	no	—
103	C₂×Q₈	—	398	2	no	—
104	C₂×Q₈	—	398	2	no	—
105	C₂×Q₈	—	398	2	no	—
106	C₂×Q₈	—	199	1	no	—
107	C₂×Q₈	—	199	4	no	—
108	C₂×D₄	—	398	1	no	—
109	C₂×D₄	—	199	1	no	—
110	C₂²×C₄	—	398	2	no	—
111	C₂²×C₄	—	398	1	no	—
112	C₂²×C₄	C₂²×D₁₉₉	1	1	yes	characteristic
113	Q₁₆	—	398	8	no	—
114	Q₁₆	—	398	8	no	—
115	SD₁₆	—	398	8	no	—
116	M₁₆	—	2	2	no	—
117	M₁₆	—	398	2	no	—
118	M₁₆	D₇₉₆	1	4	yes	—
119	C₂×C₈	—	2	2	no	—
120	C₂×C₈	—	2	2	no	—
121	C₂×C₈	—	398	1	no	—
122	C₂×C₈	D₇₉₆	1	2	yes	—
123	C₂³	—	398	1	no	—
124	C₂³	#15691887	1	1	yes	characteristic
125	Q₈	—	796	2	no	—
126	Q₈	—	796	2	no	—
127	Q₈	—	398	4	no	—
128	Q₈	—	398	4	no	—
129	Q₈	—	398	2	no	—
130	D₄	—	796	2	no	—
131	D₄	—	398	2	no	—
132	C₂×C₄	—	796	2	no	—
133	C₂×C₄	—	796	2	no	—
134	C₂×C₄	—	796	1	no	—
135	C₂×C₄	—	796	1	no	—
136	C₂×C₄	—	398	4	no	—
137	C₂×C₄	—	398	1	no	—
138	C₂×C₄	D₄×D₁₉₉	1	2	yes	—
139	C₂×C₄	D₄×D₁₉₉	1	1	yes	characteristic
140	C₂×C₄	C₂×D₇₉₆	1	2	yes	—
141	C₂×C₄	C₂×D₇₉₆	1	1	yes	characteristic
142	C₈	—	2	2	no	—
143	C₈	—	2	4	no	—
144	C₈	—	796	1	no	—
145	C₂²	—	2	2	no	—
146	C₂²	—	796	1	no	—
147	C₂²	—	796	2	no	—
148	C₂²	#20263930	1	2	yes	—
149	C₂²	#20263930	1	1	yes	characteristic
150	C₄	—	796	1	no	—
151	C₄	—	796	4	no	—
152	C₄	#20263925	1	1	yes	characteristic
153	C₄	#20263925	1	2	yes	—
154	C₄	#20263952	1	1	yes	characteristic
155	C₂	—	2	2	no	—
156	C₂	—	1592	1	no	—
157	C₂	#27163928	1	1	yes	characteristic
158	C₂	#27164294	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#20263988

C₂

yes

maximal

C₂²×Dic₇₉₆

C₂

yes

maximal, characteristic, complemented

C₂×#20263873

C₂

yes

maximal, complemented

#27163984

C₂

yes

maximal, characteristic

#341×C₁₉₉

C₂

yes

maximal, characteristic, complemented

C₂×#15691886

C₂²

yes

characteristic

C₂²×Dic₃₉₈

C₂²

yes

—

#20263988

—

#20263988

—

C₂×Dic₇₉₆

—

C₂×Dic₇₉₆

—

C₂×Dic₇₉₆

—

C₂×#15691875

—

C₂×#15691870

C₂²

yes

characteristic

#20263873

—

(C₂×D₁₉₉)⋊C₈

—

C₂×M₁₆×C₁₉₉

C₂²

yes

—

C₂²×C₈×C₁₉₉

C₂²

yes

characteristic

#15691886

—

#15691886

—

C₂×Dic₃₉₈

—

C₂×Dic₃₉₈

—

C₂×Dic₃₉₈

—

C₂×Dic₃₉₈

—

C₂×Dic₃₉₈

—

C₂×Dic₃₉₈

D₄

yes

characteristic

C₂×Dic₃₉₈

D₄

yes

—

C₂×D₇₉₆

D₄

yes

characteristic

C₂×C₂²⋊D₁₉₉

—

C₂²×Dic₁₉₉

—

C₂×C₄×D₁₉₉

—

Dic₇₉₆

—

Dic₇₉₆

—

#15691875

—

#15691870

—

C₂×C₁₉₉⋊C₈

—

C₂²×C₄×C₁₉₉

C₂³

yes

characteristic

M₁₆×C₁₉₉

—

M₁₆×C₁₉₉

D₄

yes

—

C₂×C₈×C₁₉₉

—

C₂×C₈×C₁₉₉

—

C₂×C₈×C₁₉₉

D₄

yes

—

C₂²×D₁₉₉

—

Dic₃₉₈

—

Dic₃₉₈

—

Dic₃₉₈

—

Dic₃₉₈

—

Dic₃₉₈

—

D₇₉₆

—

C₂²⋊D₁₉₉

—

C₂×Dic₁₉₉

—

C₂×Dic₁₉₉

—

C₂×Dic₁₉₉

—

C₂×Dic₁₉₉

—

C₄×D₁₉₉

—

C₄×D₁₉₉

—

C₁₉₉⋊C₈

—

C₂³×C₁₉₉

#41

yes

characteristic

C₂×C₄×C₁₉₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₉₉

C₂×D₄

yes

—

C₂×C₄×C₁₉₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₉₉

C₂×D₄

yes

—

C₈×C₁₉₉

—

C₈×C₁₉₉

—

C₂²×C₁₉₉

—

C₂²×C₁₉₉

#121

yes

—

C₂²×C₁₉₉

#121

yes

characteristic

C₂×D₁₉₉

—

C₂×D₁₉₉

—

C₄×C₁₉₉

#120

yes

characteristic

C₄×C₁₉₉

#120

yes

—

C₄×C₁₉₉

#127

yes

characteristic

Dic₁₉₉

—

Dic₁₉₉

—

C₂×C₁₉₉

—

C₂×C₁₉₉

#392

yes

characteristic

C₂×C₁₉₉

#473

yes

—

D₁₉₉

—

C₁₉₉

#2020

yes

characteristic, complemented, Sylow

#2020

—

199

maximal, Sylow

C₂×#137

—

199

—

C₂²×Q₁₆

—

199

—

C₂×#101

—

199

—

#408

—

199

—

#341

D₁₉₉

yes

characteristic, complemented

C₂×#41

—

199

—

C₂²×Q₈

—

199

—

#137

—

398

—

#137

—

398

—

C₂×Q₁₆

—

398

—

C₂×Q₁₆

—

398

—

C₂×Q₁₆

—

398

—

C₂×SD₁₆

—

398

—

C₂×M₁₆

—

199

—

C₂×M₁₆

C₂×D₁₉₉

yes

—

C₂²×C₈

C₂×D₁₉₉

yes

characteristic

#101

—

398

—

C₂²⋊C₈

—

398

—

#41

—

398

—

100

#41

—

398

—

101

C₂×Q₈

—

398

—

102

C₂×Q₈

—

398

—

103

C₂×Q₈

—

398

—

104

C₂×Q₈

—

398

—

105

C₂×Q₈

—

398

—

106

C₂×Q₈

—

199

—

107

C₂×Q₈

—

199

—

108

C₂×D₄

—

398

—

109

C₂×D₄

—

199

—

110

C₂²×C₄

—

398

—

111

C₂²×C₄

—

398

—

112

C₂²×C₄

C₂²×D₁₉₉

yes

characteristic

113

Q₁₆

—

398

—

114

Q₁₆

—

398

—

115

SD₁₆

—

398

—

116

M₁₆

—

117

M₁₆

—

398

—

118

M₁₆

D₇₉₆

yes

—

119

C₂×C₈

—

120

C₂×C₈

—

121

C₂×C₈

—

398

—

122

C₂×C₈

D₇₉₆

yes

—

123

C₂³

—

398

—

124

C₂³

#15691887

yes

characteristic

125

Q₈

—

796

—

126

Q₈

—

796

—

127

Q₈

—

398

—

128

Q₈

—

398

—

129

Q₈

—

398

—

130

D₄

—

796

—

131

D₄

—

398

—

132

C₂×C₄

—

796

—

133

C₂×C₄

—

796

—

134

C₂×C₄

—

796

—

135

C₂×C₄

—

796

—

136

C₂×C₄

—

398

—

137

C₂×C₄

—

398

—

138

C₂×C₄

D₄×D₁₉₉

yes

—

139

C₂×C₄

D₄×D₁₉₉

yes

characteristic

140

C₂×C₄

C₂×D₇₉₆

yes

—

141

C₂×C₄

C₂×D₇₉₆

yes

characteristic

142

C₈

—

143

C₈

—

144

C₈

—

796

—

145

C₂²

—

146

C₂²

—

796

—

147

C₂²

—

796

—

148

C₂²

#20263930

yes

—

149

C₂²

#20263930

yes

characteristic

150

C₄

—

796

—

151

C₄

—

796

—

152

C₄

#20263925

yes

characteristic

153

C₄

#20263925

yes

—

154

C₄

#20263952

yes

characteristic

155

C₂

—

156

C₂

—

1592

—

157

C₂

#27163928

yes

characteristic

158

C₂

#27164294

yes

—

	subgroup	G/H
center	C₂²	#20263930
commutator, cosocle	C₂²×C₄×C₁₉₉	C₂³
Frattini	C₂²×C₄	C₂²×D₁₉₉
Fitting	#341×C₁₉₉	C₂
socle	C₂²×C₁₉₉	#121
prime radical	C₂×#15691886	C₂²
hypercenter	#341	D₁₉₉
hypocenter	C₁₉₉	#2020
LCS	G ▷ C₂²×C₄×C₁₉₉ ▷ C₂×C₁₉₉ ▷ C₁₉₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #341

subgroup

G/H

center

C₂²

#20263930

commutator, cosocle

C₂²×C₄×C₁₉₉

C₂³

Frattini

C₂²×C₄

C₂²×D₁₉₉

Fitting

#341×C₁₉₉

C₂

C₂²×C₁₉₉

C₂²

D₁₉₉

C₁₉₉

G ▷ C₂²×C₄×C₁₉₉ ▷ C₂×C₁₉₉ ▷ C₁₉₉

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #341

N	H ≅ G/N	type
2. C₂²×Dic₇₉₆	C₂	semidirect
3. C₂×#20263873	C₂	semidirect
5. #341×C₁₉₉	C₂	semidirect
79. C₁₉₉	#2020	semidirect
85. #341	D₁₉₉	semidirect

H ≅ G/N

type

2. C₂²×Dic₇₉₆

C₂

semidirect

3. C₂×#20263873

C₂

semidirect

5. #341×C₁₉₉

C₂

semidirect

79. C₁₉₉

#2020

semidirect

85. #341

D₁₉₉

semidirect

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

subgroup	index
C₂×#20263988	2
C₂²×Dic₇₉₆	2
C₂×#20263873	2
#27163984	2
#341×C₁₉₉	2
#2020	199

subgroup

index

C₂²×Dic₇₉₆

#341×C₁₉₉

199

Subgroups — #38769972

Conjugacy classes of subgroups

Complements

Maximal subgroups