Group #38767562 — subgroups

Subgroups — #38767562

Order 25472.

332 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 332 conjugacy classes of subgroups — too many to draw. Only its 67 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	#127×D₁₉₉	C₂	1	1	yes	maximal, characteristic
2	#27164074	C₂	1	2	yes	maximal, complemented
3	#27163762	C₂	1	1	yes	maximal, characteristic, complemented
4	#27163761	C₂	1	2	yes	maximal, complemented
5	#331×C₁₉₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₄×D₁₉₉	—	2	2	no	—
7	C₂×D₄×D₁₉₉	C₄	1	2	yes	—
8	#20263952	C₂²	1	1	yes	characteristic
9	#20263951	—	2	2	no	—
10	#20263951	C₄	1	2	yes	—
11	C₄²×D₁₉₉	C₂²	1	1	yes	characteristic
12	C₄⋊(C₁₉₉⋊C₈)	C₂²	1	2	yes	complemented
13	(C₂×D₁₉₉)⋊C₈	—	2	2	no	—
14	#20263865	—	2	2	no	—
15	#127×C₁₉₉	C₂²	1	1	yes	characteristic
16	C₄⋊C₈×C₁₉₉	C₂²	1	2	yes	complemented
17	C₂³×D₁₉₉	—	2	2	no	—
18	C₂×D₇₉₆	—	2	2	no	—
19	C₂×D₇₉₆	C₂×C₄	1	2	yes	—
20	C₂×C₂²⋊D₁₉₉	—	2	2	no	—
21	C₂×C₂²⋊D₁₉₉	—	2	4	no	—
22	D₄×D₁₉₉	—	4	4	no	—
23	D₄×D₁₉₉	—	4	4	no	—
24	D₄×D₁₉₉	—	4	4	no	—
25	C₂×C₄×D₁₉₉	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄×D₁₉₉	D₄	1	2	yes	—
27	C₂×C₁₉₉⋊C₈	—	2	2	no	—
28	C₄×Dic₁₉₉	C₂×C₄	1	1	yes	characteristic
29	C₄×Dic₁₉₉	D₄	1	2	yes	—
30	C₂×D₄×C₁₉₉	—	2	2	no	—
31	C₂×D₄×C₁₉₉	C₂×C₄	1	2	yes	—
32	C₂×C₈×C₁₉₉	—	2	2	no	—
33	C₄²×C₁₉₉	C₂³	1	1	yes	characteristic
34	C₂²×D₁₉₉	—	2	2	no	—
35	C₂²×D₁₉₉	—	4	8	no	—
36	C₂²×D₁₉₉	—	4	4	no	—
37	C₂²×D₁₉₉	C₂²⋊C₄	1	1	yes	characteristic
38	D₇₉₆	—	4	2	no	—
39	D₇₉₆	—	4	2	no	—
40	D₇₉₆	—	4	2	no	—
41	C₂²⋊D₁₉₉	—	4	4	no	—
42	C₂²⋊D₁₉₉	—	4	8	no	—
43	C₂×Dic₁₉₉	C₂²⋊C₄	1	2	yes	—
44	C₂×Dic₁₉₉	C₂²⋊C₄	1	1	yes	characteristic
45	C₄×D₁₉₉	—	2	2	no	—
46	C₄×D₁₉₉	—	2	2	no	—
47	C₄×D₁₉₉	—	2	2	no	—
48	C₁₉₉⋊C₈	—	4	2	no	—
49	C₂³×C₁₉₉	—	2	2	no	—
50	D₄×C₁₉₉	—	4	2	no	—
51	D₄×C₁₉₉	—	4	2	no	—
52	D₄×C₁₉₉	—	4	2	no	—
53	C₂×C₄×C₁₉₉	C₂²×C₄	1	1	yes	characteristic
54	C₂×C₄×C₁₉₉	C₂×D₄	1	2	yes	—
55	C₈×C₁₉₉	—	4	2	no	—
56	C₂²×C₁₉₉	—	4	2	no	—
57	C₂²×C₁₉₉	—	4	4	no	—
58	C₂²×C₁₉₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
59	C₂×D₁₉₉	—	2	2	no	—
60	C₂×D₁₉₉	—	4	2	no	—
61	C₂×D₁₉₉	—	4	4	no	—
62	C₂×D₁₉₉	—	8	8	no	—
63	C₂×D₁₉₉	#100	1	2	yes	—
64	C₄×C₁₉₉	—	2	1	no	—
65	C₄×C₁₉₉	AGL(1,ℤ/8)	1	2	yes	—
66	C₄×C₁₉₉	AGL(1,ℤ/8)	1	2	yes	—
67	Dic₁₉₉	—	2	1	no	—
68	Dic₁₉₉	—	2	2	no	—
69	C₂×C₁₉₉	—	8	2	no	—
70	C₂×C₁₉₉	C₂×#100	1	1	yes	characteristic
71	C₂×C₁₉₉	#418	1	2	yes	—
72	D₁₉₉	—	2	2	no	—
73	D₁₉₉	—	8	2	no	—
74	C₁₉₉	#1555	1	1	yes	characteristic, complemented, Sylow
75	#1555	—	199	1	no	maximal, Sylow
76	C₂×#127	—	199	1	no	—
77	#432	—	199	2	no	—
78	#331	—	199	1	no	—
79	#331	—	199	2	no	—
80	#331	D₁₉₉	1	1	yes	characteristic, complemented
81	C₂²×D₄	—	398	2	no	—
82	C₂²×D₄	—	199	2	no	—
83	#127	—	398	2	no	—
84	#127	—	199	2	no	—
85	#127	—	199	1	no	—
86	#127	C₂×D₁₉₉	1	1	yes	characteristic
87	C₂×C₄²	—	199	1	no	—
88	C₄⋊C₈	—	199	2	no	—
89	C₄⋊C₈	C₂×D₁₉₉	1	2	yes	complemented
90	C₂²⋊C₈	—	398	2	no	—
91	#97	—	398	2	no	—
92	C₂⁴	—	398	2	no	—
93	C₂×D₄	—	2	2	no	—
94	C₂×D₄	—	796	4	no	—
95	C₂×D₄	—	796	4	no	—
96	C₂×D₄	—	796	4	no	—
97	C₂×D₄	—	398	2	no	—
98	C₂×D₄	—	398	4	no	—
99	C₂×D₄	—	398	2	no	—
100	C₂×D₄	—	199	2	no	—
101	C₂×D₄	C₄×D₁₉₉	1	2	yes	—
102	C₂²×C₄	—	199	1	no	—
103	C₂²×C₄	—	199	2	no	—
104	C₂×C₈	—	2	2	no	—
105	C₂×C₈	—	398	2	no	—
106	C₄²	—	199	1	no	—
107	C₄²	—	199	2	no	—
108	C₄²	C₂²×D₁₉₉	1	1	yes	characteristic
109	C₂³	—	2	2	no	—
110	C₂³	—	796	8	no	—
111	C₂³	—	796	4	no	—
112	C₂³	—	398	2	no	—
113	C₂³	—	199	1	no	—
114	D₄	—	4	2	no	—
115	D₄	—	4	2	no	—
116	D₄	—	4	2	no	—
117	D₄	—	796	2	no	—
118	D₄	—	796	4	no	—
119	D₄	—	796	8	no	—
120	D₄	—	796	2	no	—
121	D₄	—	796	2	no	—
122	C₂×C₄	—	398	2	no	—
123	C₂×C₄	—	398	2	no	—
124	C₂×C₄	—	398	2	no	—
125	C₂×C₄	—	199	1	no	—
126	C₂×C₄	—	199	2	no	—
127	C₂×C₄	C₂×C₄×D₁₉₉	1	1	yes	characteristic
128	C₂×C₄	D₄×D₁₉₉	1	2	yes	—
129	C₈	—	4	2	no	—
130	C₈	—	796	2	no	—
131	C₂²	—	4	2	no	—
132	C₂²	—	4	4	no	—
133	C₂²	—	796	2	no	—
134	C₂²	—	796	4	no	—
135	C₂²	—	1592	8	no	—
136	C₂²	—	398	2	no	—
137	C₂²	—	199	2	no	—
138	C₂²	C₂²⋊C₄×D₁₉₉	1	1	yes	characteristic
139	C₄	—	2	1	no	—
140	C₄	—	398	1	no	—
141	C₄	—	398	2	no	—
142	C₄	#20263979	1	2	yes	—
143	C₄	#20263982	1	2	yes	—
144	C₂	—	8	2	no	—
145	C₂	—	1592	2	no	—
146	C₂	—	398	2	no	—
147	C₂	#100×D₁₉₉	1	1	yes	characteristic
148	C₂	#27164021	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#127×D₁₉₉

C₂

yes

maximal, characteristic

#27164074

C₂

yes

maximal, complemented

#27163762

C₂

yes

maximal, characteristic, complemented

#27163761

C₂

yes

maximal, complemented

#331×C₁₉₉

C₂

yes

maximal, characteristic, complemented

C₂×D₄×D₁₉₉

—

C₂×D₄×D₁₉₉

C₄

yes

—

#20263952

C₂²

yes

characteristic

#20263951

—

#20263951

C₄

yes

—

C₄²×D₁₉₉

C₂²

yes

characteristic

C₄⋊(C₁₉₉⋊C₈)

C₂²

yes

complemented

(C₂×D₁₉₉)⋊C₈

—

#20263865

—

#127×C₁₉₉

C₂²

yes

characteristic

C₄⋊C₈×C₁₉₉

C₂²

yes

complemented

C₂³×D₁₉₉

—

C₂×D₇₉₆

—

C₂×D₇₉₆

C₂×C₄

yes

—

C₂×C₂²⋊D₁₉₉

—

C₂×C₂²⋊D₁₉₉

—

D₄×D₁₉₉

—

D₄×D₁₉₉

—

D₄×D₁₉₉

—

C₂×C₄×D₁₉₉

C₂×C₄

yes

characteristic

C₂×C₄×D₁₉₉

D₄

yes

—

C₂×C₁₉₉⋊C₈

—

C₄×Dic₁₉₉

C₂×C₄

yes

characteristic

C₄×Dic₁₉₉

D₄

yes

—

C₂×D₄×C₁₉₉

—

C₂×D₄×C₁₉₉

C₂×C₄

yes

—

C₂×C₈×C₁₉₉

—

C₄²×C₁₉₉

C₂³

yes

characteristic

C₂²×D₁₉₉

—

C₂²×D₁₉₉

—

C₂²×D₁₉₉

—

C₂²×D₁₉₉

C₂²⋊C₄

yes

characteristic

D₇₉₆

—

D₇₉₆

—

D₇₉₆

—

C₂²⋊D₁₉₉

—

C₂²⋊D₁₉₉

—

C₂×Dic₁₉₉

C₂²⋊C₄

yes

—

C₂×Dic₁₉₉

C₂²⋊C₄

yes

characteristic

C₄×D₁₉₉

—

C₄×D₁₉₉

—

C₄×D₁₉₉

—

C₁₉₉⋊C₈

—

C₂³×C₁₉₉

—

D₄×C₁₉₉

—

D₄×C₁₉₉

—

D₄×C₁₉₉

—

C₂×C₄×C₁₉₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₉₉

C₂×D₄

yes

—

C₈×C₁₉₉

—

C₂²×C₁₉₉

—

C₂²×C₁₉₉

—

C₂²×C₁₉₉

C₂×C₂²⋊C₄

yes

characteristic

C₂×D₁₉₉

—

C₂×D₁₉₉

—

C₂×D₁₉₉

—

C₂×D₁₉₉

—

C₂×D₁₉₉

#100

yes

—

C₄×C₁₉₉

—

C₄×C₁₉₉

AGL(1,ℤ/8)

yes

—

C₄×C₁₉₉

AGL(1,ℤ/8)

yes

—

Dic₁₉₉

—

Dic₁₉₉

—

C₂×C₁₉₉

—

C₂×C₁₉₉

C₂×#100

yes

characteristic

C₂×C₁₉₉

#418

yes

—

D₁₉₉

—

D₁₉₉

—

C₁₉₉

#1555

yes

characteristic, complemented, Sylow

#1555

—

199

maximal, Sylow

C₂×#127

—

199

—

#432

—

199

—

#331

—

199

—

#331

—

199

—

#331

D₁₉₉

yes

characteristic, complemented

C₂²×D₄

—

398

—

C₂²×D₄

—

199

—

#127

—

398

—

#127

—

199

—

#127

—

199

—

#127

C₂×D₁₉₉

yes

characteristic

C₂×C₄²

—

199

—

C₄⋊C₈

—

199

—

C₄⋊C₈

C₂×D₁₉₉

yes

complemented

C₂²⋊C₈

—

398

—

#97

—

398

—

C₂⁴

—

398

—

C₂×D₄

—

C₂×D₄

—

796

—

C₂×D₄

—

796

—

C₂×D₄

—

796

—

C₂×D₄

—

398

—

C₂×D₄

—

398

—

C₂×D₄

—

398

—

100

C₂×D₄

—

199

—

101

C₂×D₄

C₄×D₁₉₉

yes

—

102

C₂²×C₄

—

199

—

103

C₂²×C₄

—

199

—

104

C₂×C₈

—

105

C₂×C₈

—

398

—

106

C₄²

—

199

—

107

C₄²

—

199

—

108

C₄²

C₂²×D₁₉₉

yes

characteristic

109

C₂³

—

110

C₂³

—

796

—

111

C₂³

—

796

—

112

C₂³

—

398

—

113

C₂³

—

199

—

114

D₄

—

115

D₄

—

116

D₄

—

117

D₄

—

796

—

118

D₄

—

796

—

119

D₄

—

796

—

120

D₄

—

796

—

121

D₄

—

796

—

122

C₂×C₄

—

398

—

123

C₂×C₄

—

398

—

124

C₂×C₄

—

398

—

125

C₂×C₄

—

199

—

126

C₂×C₄

—

199

—

127

C₂×C₄

C₂×C₄×D₁₉₉

yes

characteristic

128

C₂×C₄

D₄×D₁₉₉

yes

—

129

C₈

—

130

C₈

—

796

—

131

C₂²

—

132

C₂²

—

133

C₂²

—

796

—

134

C₂²

—

796

—

135

C₂²

—

1592

—

136

C₂²

—

398

—

137

C₂²

—

199

—

138

C₂²

C₂²⋊C₄×D₁₉₉

yes

characteristic

139

C₄

—

140

C₄

—

398

—

141

C₄

—

398

—

142

C₄

#20263979

yes

—

143

C₄

#20263982

yes

—

144

C₂

—

145

C₂

—

1592

—

146

C₂

—

398

—

147

C₂

#100×D₁₉₉

yes

characteristic

148

C₂

#27164021

yes

—

	subgroup	G/H
center	C₂²	C₂²⋊C₄×D₁₉₉
commutator	C₂×C₄×C₁₉₉	C₂²×C₄
Frattini	C₄²	C₂²×D₁₉₉
Fitting	#331×C₁₉₉	C₂
socle	C₂²×C₁₉₉	C₂×C₂²⋊C₄
prime radical	#127×D₁₉₉	C₂
cosocle	C₄²×C₁₉₉	C₂³
hypercenter	#331	D₁₉₉
hypocenter	C₁₉₉	#1555
LCS	G ▷ C₂×C₄×C₁₉₉ ▷ C₂²×C₁₉₉ ▷ C₁₉₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #331

subgroup

G/H

center

C₂²

C₂²⋊C₄×D₁₉₉

commutator

C₂×C₄×C₁₉₉

C₂²×C₄

Frattini

C₄²

C₂²×D₁₉₉

Fitting

#331×C₁₉₉

C₂

socle

C₂²×C₁₉₉

C₂×C₂²⋊C₄

prime radical

#127×D₁₉₉

C₂

cosocle

C₄²×C₁₉₉

C₂³

D₁₉₉

C₁₉₉

G ▷ C₂×C₄×C₁₉₉ ▷ C₂²×C₁₉₉ ▷ C₁₉₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #331

N	H ≅ G/N	type
2. #27164074	C₂	semidirect
3. #27163762	C₂	semidirect
4. #27163761	C₂	semidirect
5. #331×C₁₉₉	C₂	semidirect
12. C₄⋊(C₁₉₉⋊C₈)	C₂²	semidirect
16. C₄⋊C₈×C₁₉₉	C₂²	semidirect
74. C₁₉₉	#1555	semidirect
80. #331	D₁₉₉	semidirect
89. C₄⋊C₈	C₂×D₁₉₉	semidirect

H ≅ G/N

type

2. #27164074

C₂

semidirect

3. #27163762

C₂

semidirect

4. #27163761

C₂

semidirect

5. #331×C₁₉₉

C₂

semidirect

12. C₄⋊(C₁₉₉⋊C₈)

C₂²

semidirect

16. C₄⋊C₈×C₁₉₉

C₂²

semidirect

74. C₁₉₉

#1555

semidirect

80. #331

D₁₉₉

semidirect

89. C₄⋊C₈

C₂×D₁₉₉

semidirect

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

subgroup	index
#127×D₁₉₉	2
#27164074	2
#27163762	2
#27163761	2
#331×C₁₉₉	2
#1555	199

subgroup

index

#127×D₁₉₉

#331×C₁₉₉

199

Subgroups — #38767562

Conjugacy classes of subgroups

Complements

Maximal subgroups