Group #38768934 — subgroups

Subgroups — #38768934

Order 25472.

344 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 344 conjugacy classes of subgroups — too many to draw. Only its 117 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	C₂²×C₄⋊Dic₁₉₉	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂²⋊(C₁₉₉⋊C₈)	C₂	1	1	yes	maximal, characteristic
3	#27163780	C₂	1	4	yes	maximal, complemented
4	#380×C₁₉₉	C₂	1	1	yes	maximal, characteristic
5	C₂³×Dic₁₉₉	—	2	1	no	—
6	C₂²×C₁₉₉⋊C₈	C₂²	1	2	yes	—
7	C₂×C₄⋊Dic₁₉₉	—	2	1	no	—
8	C₂×C₄⋊Dic₁₉₉	—	2	2	no	—
9	C₂×C₄⋊Dic₁₉₉	C₄	1	4	yes	complemented
10	C₂×C₄⋊Dic₁₉₉	C₂²	1	2	yes	—
11	C₂²⋊(C₁₉₉⋊C₈)	—	2	2	no	—
12	C₂³×C₄×C₁₉₉	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊C₄×C₁₉₉	C₂²	1	2	yes	—
14	C₂×C₂²⋊C₄×C₁₉₉	—	2	1	no	—
15	#95×C₁₉₉	—	2	1	no	—
16	C₂²×Dic₁₉₉	—	2	4	no	—
17	C₂²×Dic₁₉₉	—	2	2	no	—
18	C₂²×Dic₁₉₉	—	4	2	no	—
19	C₂²×Dic₁₉₉	—	4	2	no	—
20	C₂×C₁₉₉⋊C₈	—	2	2	no	—
21	C₂×C₁₉₉⋊C₈	—	2	4	no	—
22	C₄⋊Dic₁₉₉	—	2	4	no	—
23	C₄⋊Dic₁₉₉	—	2	2	no	—
24	C₄⋊Dic₁₉₉	C₂×C₄	1	4	yes	—
25	C₂⁴×C₁₉₉	D₄	1	1	yes	characteristic
26	C₂²×C₄×C₁₉₉	—	2	4	no	—
27	C₂²×C₄×C₁₉₉	—	2	2	no	—
28	C₂²×C₄×C₁₉₉	C₂×C₄	1	2	yes	—
29	C₂²×C₄×C₁₉₉	D₄	1	1	yes	characteristic
30	C₂²×C₄×C₁₉₉	D₄	1	2	yes	—
31	C₂²×C₄×C₁₉₉	C₂³	1	1	yes	characteristic
32	C₄⋊C₄×C₁₉₉	—	2	4	no	—
33	C₂²⋊C₄×C₁₉₉	—	4	2	no	—
34	C₂×Dic₁₉₉	—	2	4	no	—
35	C₂×Dic₁₉₉	—	4	8	no	—
36	C₂×Dic₁₉₉	—	4	2	no	—
37	C₂×Dic₁₉₉	—	4	2	no	—
38	C₁₉₉⋊C₈	—	4	2	no	—
39	C₂³×C₁₉₉	—	2	2	no	—
40	C₂³×C₁₉₉	—	2	2	no	—
41	C₂³×C₁₉₉	C₂²⋊C₄	1	2	yes	—
42	C₂³×C₁₉₉	D₈	1	2	yes	—
43	C₂³×C₁₉₉	SD₁₆	1	2	yes	—
44	C₂³×C₁₉₉	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₁₉₉	—	2	2	no	—
46	C₂×C₄×C₁₉₉	—	2	4	no	—
47	C₂×C₄×C₁₉₉	—	2	4	no	—
48	C₂×C₄×C₁₉₉	—	4	2	no	—
49	C₂×C₄×C₁₉₉	—	4	2	no	—
50	C₂×C₄×C₁₉₉	—	4	2	no	—
51	C₂×C₄×C₁₉₉	C₂²⋊C₄	1	2	yes	—
52	C₂×C₄×C₁₉₉	C₂²×C₄	1	1	yes	characteristic
53	C₂×C₄×C₁₉₉	C₂×D₄	1	1	yes	characteristic
54	C₂×C₄×C₁₉₉	#41	1	4	yes	—
55	C₂²×C₁₉₉	—	2	2	no	—
56	C₂²×C₁₉₉	—	2	2	no	—
57	C₂²×C₁₉₉	—	2	4	no	—
58	C₂²×C₁₉₉	—	2	4	no	—
59	C₂²×C₁₉₉	#102	1	4	yes	—
60	C₂²×C₁₉₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
61	C₂²×C₁₉₉	#123	1	2	yes	—
62	C₂²×C₁₉₉	C₂×D₈	1	1	yes	characteristic
63	C₂²×C₁₉₉	C₂×SD₁₆	1	1	yes	characteristic
64	C₂²×C₁₉₉	#137	1	2	yes	—
65	C₄×C₁₉₉	—	2	2	no	—
66	C₄×C₁₉₉	—	8	2	no	—
67	C₄×C₁₉₉	#117	1	2	yes	—
68	C₄×C₁₉₉	#123	1	2	yes	—
69	Dic₁₉₉	—	4	4	no	—
70	C₂×C₁₉₉	—	2	4	no	—
71	C₂×C₁₉₉	#381	1	1	yes	characteristic
72	C₂×C₁₉₉	C₂×#102	1	1	yes	characteristic
73	C₂×C₁₉₉	#419	1	1	yes	characteristic
74	C₂×C₁₉₉	#480	1	2	yes	—
75	C₂×C₁₉₉	#483	1	2	yes	—
76	C₁₉₉	#1823	1	1	yes	characteristic, complemented, Sylow
77	#1823	—	199	1	no	maximal, Sylow
78	C₂²×C₄⋊C₄	—	199	1	no	—
79	C₂×C₂²⋊C₈	—	199	1	no	—
80	#380	D₁₉₉	1	1	yes	characteristic
81	#340	—	199	4	no	—
82	C₂³×C₄	—	398	1	no	—
83	C₂³×C₄	C₂×D₁₉₉	1	1	yes	characteristic
84	C₂²×C₈	—	199	2	no	—
85	C₂×C₄⋊C₄	—	398	1	no	—
86	C₂×C₄⋊C₄	—	398	2	no	—
87	C₂×C₄⋊C₄	—	199	4	no	—
88	C₂×C₄⋊C₄	—	199	2	no	—
89	C₂×C₄⋊C₄	C₂×D₁₉₉	1	2	yes	—
90	C₂×C₂²⋊C₄	—	2	1	no	—
91	C₂²⋊C₈	—	398	2	no	—
92	#95	—	2	1	no	—
93	C₂⁴	C₂²⋊D₁₉₉	1	1	yes	characteristic
94	C₂²×C₄	—	2	4	no	—
95	C₂²×C₄	—	2	2	no	—
96	C₂²×C₄	—	796	2	no	—
97	C₂²×C₄	—	796	2	no	—
98	C₂²×C₄	—	398	4	no	—
99	C₂²×C₄	—	398	2	no	—
100	C₂²×C₄	C₄×D₁₉₉	1	2	yes	—
101	C₂²×C₄	C₂²⋊D₁₉₉	1	1	yes	characteristic
102	C₂²×C₄	D₇₉₆	1	2	yes	—
103	C₂²×C₄	C₂²×D₁₉₉	1	1	yes	characteristic
104	C₂×C₈	—	398	2	no	—
105	C₂×C₈	—	398	4	no	—
106	C₄⋊C₄	—	2	4	no	—
107	C₄⋊C₄	—	398	4	no	—
108	C₄⋊C₄	—	398	2	no	—
109	C₄⋊C₄	—	199	4	no	—
110	C₂²⋊C₄	—	4	2	no	—
111	C₂³	—	2	2	no	—
112	C₂³	—	2	2	no	—
113	C₂³	(C₂×D₁₉₉)⋊C₄	1	2	yes	—
114	C₂³	#15691871	1	2	yes	—
115	C₂³	#15691873	1	2	yes	—
116	C₂³	C₂×C₂²⋊D₁₉₉	1	1	yes	characteristic
117	C₂×C₄	—	2	2	no	—
118	C₂×C₄	—	2	4	no	—
119	C₂×C₄	—	2	4	no	—
120	C₂×C₄	—	4	2	no	—
121	C₂×C₄	—	4	2	no	—
122	C₂×C₄	—	4	2	no	—
123	C₂×C₄	—	796	2	no	—
124	C₂×C₄	—	796	2	no	—
125	C₂×C₄	—	796	8	no	—
126	C₂×C₄	—	398	4	no	—
127	C₂×C₄	(C₂×D₁₉₉)⋊C₄	1	2	yes	—
128	C₂×C₄	C₂×C₄×D₁₉₉	1	1	yes	characteristic
129	C₂×C₄	C₂×D₇₉₆	1	1	yes	characteristic
130	C₂×C₄	#15691885	1	4	yes	—
131	C₈	—	796	2	no	—
132	C₂²	—	2	2	no	—
133	C₂²	—	2	4	no	—
134	C₂²	—	2	4	no	—
135	C₂²	—	2	2	no	—
136	C₂²	#20263875	1	4	yes	—
137	C₂²	C₂×(C₂×D₁₉₉)⋊C₄	1	1	yes	characteristic
138	C₂²	#20263936	1	2	yes	—
139	C₂²	C₂×#15691871	1	1	yes	characteristic
140	C₂²	C₂×#15691873	1	1	yes	characteristic
141	C₂²	#20263985	1	2	yes	—
142	C₄	—	2	2	no	—
143	C₄	—	8	2	no	—
144	C₄	—	796	4	no	—
145	C₄	#20263911	1	2	yes	—
146	C₄	#20263939	1	2	yes	—
147	C₂	—	2	4	no	—
148	C₂	#27163879	1	1	yes	characteristic
149	C₂	C₂×#20263875	1	1	yes	characteristic
150	C₂	#27164026	1	1	yes	characteristic
151	C₂	#27164329	1	2	yes	—
152	C₂	#27164346	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×C₄⋊Dic₁₉₉

C₂

yes

maximal, characteristic

C₂×C₂²⋊(C₁₉₉⋊C₈)

C₂

yes

maximal, characteristic

#27163780

C₂

yes

maximal, complemented

#380×C₁₉₉

C₂

yes

maximal, characteristic

C₂³×Dic₁₉₉

—

C₂²×C₁₉₉⋊C₈

C₂²

yes

—

C₂×C₄⋊Dic₁₉₉

—

C₂×C₄⋊Dic₁₉₉

—

C₂×C₄⋊Dic₁₉₉

C₄

yes

complemented

C₂×C₄⋊Dic₁₉₉

C₂²

yes

—

C₂²⋊(C₁₉₉⋊C₈)

—

C₂³×C₄×C₁₉₉

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₁₉₉

C₂²

yes

—

C₂×C₂²⋊C₄×C₁₉₉

—

#95×C₁₉₉

—

C₂²×Dic₁₉₉

—

C₂²×Dic₁₉₉

—

C₂²×Dic₁₉₉

—

C₂²×Dic₁₉₉

—

C₂×C₁₉₉⋊C₈

—

C₂×C₁₉₉⋊C₈

—

C₄⋊Dic₁₉₉

—

C₄⋊Dic₁₉₉

—

C₄⋊Dic₁₉₉

C₂×C₄

yes

—

C₂⁴×C₁₉₉

D₄

yes

characteristic

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

—

C₂²⋊C₄×C₁₉₉

—

C₂×Dic₁₉₉

—

C₂×Dic₁₉₉

—

C₂×Dic₁₉₉

—

C₂×Dic₁₉₉

—

C₁₉₉⋊C₈

—

C₂³×C₁₉₉

—

C₂³×C₁₉₉

—

C₂³×C₁₉₉

C₂²⋊C₄

yes

—

C₂³×C₁₉₉

D₈

yes

—

C₂³×C₁₉₉

SD₁₆

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

#102

yes

—

C₂²×C₁₉₉

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₁₉₉

#123

yes

—

C₂²×C₁₉₉

C₂×D₈

yes

characteristic

C₂²×C₁₉₉

C₂×SD₁₆

yes

characteristic

C₂²×C₁₉₉

#137

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

#483

yes

—

C₁₉₉

#1823

yes

characteristic, complemented, Sylow

#1823

—

199

maximal, Sylow

C₂²×C₄⋊C₄

—

199

—

C₂×C₂²⋊C₈

—

199

—

#380

D₁₉₉

yes

characteristic

#340

—

199

—

C₂³×C₄

—

398

—

C₂³×C₄

C₂×D₁₉₉

yes

characteristic

C₂²×C₈

—

199

—

C₂×C₄⋊C₄

—

398

—

C₂×C₄⋊C₄

—

398

—

C₂×C₄⋊C₄

—

199

—

C₂×C₄⋊C₄

—

199

—

C₂×C₄⋊C₄

C₂×D₁₉₉

yes

—

C₂×C₂²⋊C₄

—

C₂²⋊C₈

—

398

—

#95

—

C₂⁴

C₂²⋊D₁₉₉

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

796

—

C₂²×C₄

—

796

—

C₂²×C₄

—

398

—

C₂²×C₄

—

398

—

100

C₂²×C₄

C₄×D₁₉₉

yes

—

101

C₂²×C₄

C₂²⋊D₁₉₉

yes

characteristic

102

C₂²×C₄

D₇₉₆

yes

—

103

C₂²×C₄

C₂²×D₁₉₉

yes

characteristic

104

C₂×C₈

—

398

—

105

C₂×C₈

—

398

—

106

C₄⋊C₄

—

107

C₄⋊C₄

—

398

—

108

C₄⋊C₄

—

398

—

109

C₄⋊C₄

—

199

—

110

C₂²⋊C₄

—

111

C₂³

—

112

C₂³

—

113

C₂³

(C₂×D₁₉₉)⋊C₄

yes

—

114

C₂³

#15691871

yes

—

115

C₂³

#15691873

yes

—

116

C₂³

C₂×C₂²⋊D₁₉₉

yes

characteristic

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

796

—

124

C₂×C₄

—

796

—

125

C₂×C₄

—

796

—

126

C₂×C₄

—

398

—

127

C₂×C₄

(C₂×D₁₉₉)⋊C₄

yes

—

128

C₂×C₄

C₂×C₄×D₁₉₉

yes

characteristic

129

C₂×C₄

C₂×D₇₉₆

yes

characteristic

130

C₂×C₄

#15691885

yes

—

131

C₈

—

796

—

132

C₂²

—

133

C₂²

—

134

C₂²

—

135

C₂²

—

136

C₂²

#20263875

yes

—

137

C₂²

C₂×(C₂×D₁₉₉)⋊C₄

yes

characteristic

138

C₂²

#20263936

yes

—

139

C₂²

C₂×#15691871

yes

characteristic

140

C₂²

C₂×#15691873

yes

characteristic

141

C₂²

#20263985

yes

—

142

C₄

—

143

C₄

—

144

C₄

—

796

—

145

C₄

#20263911

yes

—

146

C₄

#20263939

yes

—

147

C₂

—

148

C₂

#27163879

yes

characteristic

149

C₂

C₂×#20263875

yes

characteristic

150

C₂

#27164026

yes

characteristic

151

C₂

#27164329

yes

—

152

C₂

#27164346

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₁₉₉
commutator	C₂×C₄×C₁₉₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₉₉
Fitting	#380×C₁₉₉	C₂
socle	C₂³×C₁₉₉	C₂×D₄
prime radical	C₂⁴×C₁₉₉	D₄
cosocle	C₂²×C₄×C₁₉₉	C₂³
hypercenter	#380	D₁₉₉
hypocenter	C₁₉₉	#1823
LCS	G ▷ C₂×C₄×C₁₉₉ ▷ C₂×C₁₉₉ ▷ C₁₉₉
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ #380

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₁₉₉

commutator

C₂×C₄×C₁₉₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₉₉

Fitting

#380×C₁₉₉

C₂

socle

C₂³×C₁₉₉

C₂×D₄

prime radical

C₂⁴×C₁₉₉

D₄

cosocle

C₂²×C₄×C₁₉₉

C₂³

D₁₉₉

C₁₉₉

G ▷ C₂×C₄×C₁₉₉ ▷ C₂×C₁₉₉ ▷ C₁₉₉

UCS

C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ #380

N	H ≅ G/N	type
3. #27163780	C₂	semidirect
9. C₂×C₄⋊Dic₁₉₉	C₄	semidirect
76. C₁₉₉	#1823	semidirect

H ≅ G/N

type

3. #27163780

C₂

semidirect

9. C₂×C₄⋊Dic₁₉₉

C₄

semidirect

76. C₁₉₉

#1823

semidirect

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

subgroup	index
C₂²×C₄⋊Dic₁₉₉	2
C₂×C₂²⋊(C₁₉₉⋊C₈)	2
#27163780	2
#380×C₁₉₉	2
#1823	199

subgroup

index

C₂²×C₄⋊Dic₁₉₉

C₂×C₂²⋊(C₁₉₉⋊C₈)

#27163780

#380×C₁₉₉

#1823

199

Subgroups — #38768934

Conjugacy classes of subgroups

Complements

Maximal subgroups