Group #38769993 — subgroups

Subgroups — #38769993

Order 25472.

294 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 294 conjugacy classes of subgroups — too many to draw. Only its 87 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×#20263875	C₂	1	2	yes	maximal
2	#27163920	C₂	1	1	yes	maximal, characteristic
3	#27163780	C₂	1	2	yes	maximal, complemented
4	#27163773	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#125×C₁₉₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₇₉₆	C₂²	1	1	yes	characteristic
7	C₂²×C₁₉₉⋊C₈	C₂²	1	2	yes	—
8	C₂×C₄⋊Dic₁₉₉	C₂²	1	1	yes	characteristic
9	C₂×(C₂×D₁₉₉)⋊C₄	—	2	2	no	—
10	#20263875	—	2	4	no	—
11	#125×C₁₉₉	—	2	4	no	—
12	C₂×C₄⋊C₄×C₁₉₉	—	2	2	no	—
13	C₂×C₄⋊C₄×C₁₉₉	C₂²	1	2	yes	—
14	C₂×C₄²×C₁₉₉	C₂²	1	1	yes	characteristic
15	C₂³×D₁₉₉	—	2	1	no	—
16	C₂×D₇₉₆	—	2	2	no	—
17	C₂×D₇₉₆	—	2	2	no	—
18	C₂×D₇₉₆	—	2	1	no	—
19	C₂×D₇₉₆	Q₈	1	2	yes	—
20	C₂²×Dic₁₉₉	—	2	1	no	—
21	C₂×C₁₉₉⋊C₈	—	2	2	no	—
22	C₂×C₁₉₉⋊C₈	—	2	2	no	—
23	C₂×C₁₉₉⋊C₈	—	2	2	no	—
24	C₄⋊Dic₁₉₉	—	2	2	no	—
25	(C₂×D₁₉₉)⋊C₄	—	4	4	no	—
26	C₂²×C₄×C₁₉₉	—	2	2	no	—
27	C₂²×C₄×C₁₉₉	D₄	1	2	yes	—
28	C₂²×C₄×C₁₉₉	C₂³	1	1	yes	characteristic
29	C₄⋊C₄×C₁₉₉	—	2	8	no	—
30	C₄⋊C₄×C₁₉₉	—	2	2	no	—
31	C₄⋊C₄×C₁₉₉	D₄	1	4	yes	—
32	C₄²×C₁₉₉	—	2	2	no	—
33	C₂²×D₁₉₉	—	2	2	no	—
34	C₂²×D₁₉₉	—	4	1	no	—
35	C₂²×D₁₉₉	—	4	1	no	—
36	C₂²×D₁₉₉	—	4	2	no	—
37	C₂²×D₁₉₉	—	4	2	no	—
38	D₇₉₆	—	2	4	no	—
39	D₇₉₆	—	4	2	no	—
40	C₂×Dic₁₉₉	—	4	1	no	—
41	C₂×Dic₁₉₉	—	4	2	no	—
42	C₁₉₉⋊C₈	—	4	2	no	—
43	C₂³×C₁₉₉	C₂×D₄	1	1	yes	characteristic
44	C₂×C₄×C₁₉₉	—	2	2	no	—
45	C₂×C₄×C₁₉₉	—	2	4	no	—
46	C₂×C₄×C₁₉₉	—	2	2	no	—
47	C₂×C₄×C₁₉₉	—	2	2	no	—
48	C₂×C₄×C₁₉₉	—	4	2	no	—
49	C₂×C₄×C₁₉₉	—	4	2	no	—
50	C₂×C₄×C₁₉₉	C₂×D₄	1	2	yes	—
51	C₂×C₄×C₁₉₉	C₂×Q₈	1	1	yes	characteristic
52	C₂×C₄×C₁₉₉	#41	1	2	yes	—
53	C₂×C₄×C₁₉₉	#41	1	1	yes	characteristic
54	C₂²×C₁₉₉	#120	1	1	yes	characteristic
55	C₂²×C₁₉₉	C₂²⋊Q₈	1	1	yes	characteristic
56	C₂²×C₁₉₉	#124	1	1	yes	characteristic
57	C₂²×C₁₉₉	#135	1	2	yes	—
58	C₂²×C₁₉₉	AGL(1,ℤ/8)	1	2	yes	—
59	C₂×D₁₉₉	—	4	2	no	—
60	C₂×D₁₉₉	—	4	4	no	—
61	C₂×D₁₉₉	—	8	2	no	—
62	C₂×D₁₉₉	—	8	1	no	—
63	C₂×D₁₉₉	—	8	1	no	—
64	C₄×C₁₉₉	—	4	2	no	—
65	C₄×C₁₉₉	—	4	4	no	—
66	C₄×C₁₉₉	C₂²⋊Q₈	1	2	yes	—
67	C₄×C₁₉₉	#124	1	2	yes	—
68	Dic₁₉₉	—	8	1	no	—
69	C₂×C₁₉₉	#393	1	1	yes	characteristic
70	C₂×C₁₉₉	#449	1	2	yes	—
71	C₂×C₁₉₉	#478	1	2	yes	—
72	C₂×C₁₉₉	#488	1	1	yes	characteristic
73	C₂×C₁₉₉	#490	1	1	yes	characteristic
74	D₁₉₉	—	8	2	no	—
75	C₁₉₉	#2025	1	1	yes	characteristic, complemented, Sylow
76	#2025	—	199	1	no	maximal, Sylow
77	C₂×#125	D₁₉₉	1	1	yes	characteristic, complemented
78	C₂×#102	—	199	2	no	—
79	#390	—	199	1	no	—
80	#340	—	199	2	no	—
81	#336	—	199	1	no	—
82	C₂²×D₄	—	199	1	no	—
83	C₂²×C₈	—	199	2	no	—
84	#125	—	2	4	no	—
85	C₂×C₄⋊C₄	—	2	2	no	—
86	C₂×C₄⋊C₄	—	199	1	no	—
87	C₂×C₄⋊C₄	C₂×D₁₉₉	1	2	yes	—
88	C₂×C₂²⋊C₄	—	398	2	no	—
89	C₂×C₄²	C₂×D₁₉₉	1	1	yes	characteristic
90	#102	—	398	4	no	—
91	C₂⁴	—	398	1	no	—
92	C₂×D₄	—	398	2	no	—
93	C₂×D₄	—	398	1	no	—
94	C₂×D₄	—	398	2	no	—
95	C₂×D₄	—	199	2	no	—
96	C₂²×C₄	—	2	2	no	—
97	C₂²×C₄	—	398	1	no	—
98	C₂²×C₄	C₂²⋊D₁₉₉	1	2	yes	—
99	C₂²×C₄	C₂²×D₁₉₉	1	1	yes	characteristic
100	C₂×C₈	—	398	2	no	—
101	C₂×C₈	—	398	2	no	—
102	C₂×C₈	—	398	2	no	—
103	C₄⋊C₄	—	2	8	no	—
104	C₄⋊C₄	—	2	2	no	—
105	C₄⋊C₄	—	398	2	no	—
106	C₄⋊C₄	C₂²⋊D₁₉₉	1	4	yes	—
107	C₂²⋊C₄	—	796	4	no	—
108	C₄²	—	2	2	no	—
109	C₂³	—	796	1	no	—
110	C₂³	—	796	2	no	—
111	C₂³	—	796	2	no	—
112	C₂³	—	796	1	no	—
113	C₂³	—	398	2	no	—
114	C₂³	C₂×C₂²⋊D₁₉₉	1	1	yes	characteristic
115	D₄	—	796	2	no	—
116	D₄	—	398	4	no	—
117	C₂×C₄	—	2	2	no	—
118	C₂×C₄	—	2	2	no	—
119	C₂×C₄	—	2	2	no	—
120	C₂×C₄	—	2	4	no	—
121	C₂×C₄	—	4	2	no	—
122	C₂×C₄	—	4	2	no	—
123	C₂×C₄	—	796	1	no	—
124	C₂×C₄	—	796	2	no	—
125	C₂×C₄	C₂×C₂²⋊D₁₉₉	1	2	yes	—
126	C₂×C₄	Q₈×D₁₉₉	1	1	yes	characteristic
127	C₂×C₄	#15691887	1	1	yes	characteristic
128	C₂×C₄	#15691887	1	2	yes	—
129	C₈	—	796	2	no	—
130	C₂²	—	796	2	no	—
131	C₂²	—	796	4	no	—
132	C₂²	—	1592	2	no	—
133	C₂²	—	1592	1	no	—
134	C₂²	—	1592	1	no	—
135	C₂²	#20263924	1	1	yes	characteristic
136	C₂²	#20263931	1	1	yes	characteristic
137	C₂²	#20263944	1	1	yes	characteristic
138	C₂²	#20263975	1	2	yes	—
139	C₂²	#20263980	1	2	yes	—
140	C₄	—	4	2	no	—
141	C₄	—	4	4	no	—
142	C₄	—	1592	1	no	—
143	C₄	#20263931	1	2	yes	—
144	C₄	#20263944	1	2	yes	—
145	C₂	—	1592	2	no	—
146	C₂	#27163929	1	1	yes	characteristic
147	C₂	#27164163	1	2	yes	—
148	C₂	#27164315	1	2	yes	—
149	C₂	#27164375	1	1	yes	characteristic
150	C₂	#27164384	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#20263875

C₂

yes

maximal

#27163920

C₂

yes

maximal, characteristic

#27163780

C₂

yes

maximal, complemented

#27163773

C₂

yes

maximal, characteristic, complemented

C₂×#125×C₁₉₉

C₂

yes

maximal, characteristic, complemented

C₂²×D₇₉₆

C₂²

yes

characteristic

C₂²×C₁₉₉⋊C₈

C₂²

yes

—

C₂×C₄⋊Dic₁₉₉

C₂²

yes

characteristic

C₂×(C₂×D₁₉₉)⋊C₄

—

#20263875

—

#125×C₁₉₉

—

C₂×C₄⋊C₄×C₁₉₉

—

C₂×C₄⋊C₄×C₁₉₉

C₂²

yes

—

C₂×C₄²×C₁₉₉

C₂²

yes

characteristic

C₂³×D₁₉₉

—

C₂×D₇₉₆

—

C₂×D₇₉₆

—

C₂×D₇₉₆

—

C₂×D₇₉₆

Q₈

yes

—

C₂²×Dic₁₉₉

—

C₂×C₁₉₉⋊C₈

—

C₂×C₁₉₉⋊C₈

—

C₂×C₁₉₉⋊C₈

—

C₄⋊Dic₁₉₉

—

(C₂×D₁₉₉)⋊C₄

—

C₂²×C₄×C₁₉₉

—

C₂²×C₄×C₁₉₉

D₄

yes

—

C₂²×C₄×C₁₉₉

C₂³

yes

characteristic

C₄⋊C₄×C₁₉₉

—

C₄⋊C₄×C₁₉₉

—

C₄⋊C₄×C₁₉₉

D₄

yes

—

C₄²×C₁₉₉

—

C₂²×D₁₉₉

—

C₂²×D₁₉₉

—

C₂²×D₁₉₉

—

C₂²×D₁₉₉

—

C₂²×D₁₉₉

—

D₇₉₆

—

D₇₉₆

—

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×D₄

yes

—

C₂×C₄×C₁₉₉

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₉₉

#41

yes

—

C₂×C₄×C₁₉₉

yes

C₂²×C₁₉₉

yes

C₂²×C₁₉₉

C₂²⋊Q₈

yes

C₂²×C₁₉₉

yes

C₂²×C₁₉₉

yes

—

C₂²×C₁₉₉

AGL(1,ℤ/8)

yes

—

C₂×D₁₉₉

—

C₂×D₁₉₉

—

C₂×D₁₉₉

—

C₂×D₁₉₉

—

C₂×D₁₉₉

—

C₄×C₁₉₉

—

C₄×C₁₉₉

—

C₄×C₁₉₉

C₂²⋊Q₈

yes

—

C₄×C₁₉₉

#124

yes

—

Dic₁₉₉

—

C₂×C₁₉₉

#393

yes

characteristic

C₂×C₁₉₉

#449

yes

—

C₂×C₁₉₉

#478

yes

—

C₂×C₁₉₉

yes

C₂×C₁₉₉

yes

D₁₉₉

—

C₁₉₉

#2025

yes

characteristic, complemented, Sylow

#2025

—

199

maximal, Sylow

C₂×#125

D₁₉₉

yes

characteristic, complemented

C₂×#102

—

199

—

#390

—

199

—

#340

—

199

—

#336

—

199

—

C₂²×D₄

—

199

—

C₂²×C₈

—

199

—

#125

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

199

—

C₂×C₄⋊C₄

C₂×D₁₉₉

yes

—

C₂×C₂²⋊C₄

—

398

—

C₂×C₄²

C₂×D₁₉₉

yes

characteristic

#102

—

398

—

C₂⁴

—

398

—

C₂×D₄

—

398

—

C₂×D₄

—

398

—

C₂×D₄

—

398

—

C₂×D₄

—

199

—

C₂²×C₄

—

C₂²×C₄

—

398

—

C₂²×C₄

C₂²⋊D₁₉₉

yes

—

C₂²×C₄

C₂²×D₁₉₉

yes

characteristic

100

C₂×C₈

—

398

—

101

C₂×C₈

—

398

—

102

C₂×C₈

—

398

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

398

—

106

C₄⋊C₄

C₂²⋊D₁₉₉

yes

—

107

C₂²⋊C₄

—

796

—

108

C₄²

—

109

C₂³

—

796

—

110

C₂³

—

796

—

111

C₂³

—

796

—

112

C₂³

—

796

—

113

C₂³

—

398

—

114

C₂³

C₂×C₂²⋊D₁₉₉

yes

characteristic

115

D₄

—

796

—

116

D₄

—

398

—

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

C₂×C₂²⋊D₁₉₉

yes

—

126

C₂×C₄

Q₈×D₁₉₉

yes

characteristic

127

C₂×C₄

#15691887

yes

characteristic

128

C₂×C₄

#15691887

yes

—

129

C₈

—

796

—

130

C₂²

—

796

—

131

C₂²

—

796

—

132

C₂²

—

1592

—

133

C₂²

—

1592

—

134

C₂²

—

1592

—

135

C₂²

#20263924

yes

characteristic

136

C₂²

#20263931

yes

characteristic

137

C₂²

#20263944

yes

characteristic

138

C₂²

#20263975

yes

—

139

C₂²

#20263980

yes

—

140

C₄

—

141

C₄

—

142

C₄

—

1592

—

143

C₄

#20263931

yes

—

144

C₄

#20263944

yes

—

145

C₂

—

1592

—

146

C₂

#27163929

yes

characteristic

147

C₂

#27164163

yes

—

148

C₂

#27164315

yes

—

149

C₂

#27164375

yes

characteristic

150

C₂

#27164384

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₁₉₉
commutator, cosocle	C₂²×C₄×C₁₉₉	C₂³
Frattini	C₂²×C₄	C₂²×D₁₉₉
Fitting	C₂×#125×C₁₉₉	C₂
socle	C₂³×C₁₉₉	C₂×D₄
prime radical	C₂²×D₇₉₆	C₂²
hypercenter	C₂×#125	D₁₉₉
hypocenter	C₁₉₉	#2025
LCS	G ▷ C₂²×C₄×C₁₉₉ ▷ C₂×C₁₉₉ ▷ C₁₉₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#125

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₁₉₉

commutator, cosocle

C₂²×C₄×C₁₉₉

C₂³

Frattini

C₂²×C₄

C₂²×D₁₉₉

Fitting

C₂×#125×C₁₉₉

C₂

socle

C₂³×C₁₉₉

C₂×D₄

prime radical

C₂²×D₇₉₆

C₂²

D₁₉₉

C₁₉₉

G ▷ C₂²×C₄×C₁₉₉ ▷ C₂×C₁₉₉ ▷ C₁₉₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#125

N	H ≅ G/N	type
3. #27163780	C₂	semidirect
4. #27163773	C₂	semidirect
5. C₂×#125×C₁₉₉	C₂	semidirect
75. C₁₉₉	#2025	semidirect
77. C₂×#125	D₁₉₉	semidirect

H ≅ G/N

type

3. #27163780

C₂

semidirect

4. #27163773

C₂

semidirect

5. C₂×#125×C₁₉₉

C₂

semidirect

75. C₁₉₉

#2025

semidirect

77. C₂×#125

D₁₉₉

semidirect

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

subgroup	index
C₂×#20263875	2
#27163920	2
#27163780	2
#27163773	2
C₂×#125×C₁₉₉	2
#2025	199

subgroup

index

C₂×#125×C₁₉₉

199

Subgroups — #38769993

Conjugacy classes of subgroups

Complements

Maximal subgroups