Group #38770153 — subgroups

Subgroups — #38770153

Order 25472.

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

normal subgroups only

This group has 264 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	#27164046	C₂	1	1	yes	maximal, characteristic, complemented
2	#27164014	C₂	1	2	yes	maximal, complemented
3	#27163992	C₂	1	2	yes	maximal, complemented
4	#27163775	C₂	1	1	yes	maximal, characteristic, complemented
5	#575×C₁₉₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₁₉₉⋊C₈	C₂²	1	1	yes	characteristic
7	#20263914	C₂²	1	2	yes	—
8	#20263914	C₂²	1	1	yes	characteristic
9	Dic₁₉₉⋊C₈	—	2	2	no	—
10	#20263879	—	2	2	no	—
11	#20263876	—	2	2	no	—
12	#20263870	—	2	4	no	—
13	C₂×#41×C₁₉₉	C₂²	1	2	yes	—
14	#137×C₁₉₉	—	2	2	no	—
15	AGL(1,ℤ/8)×C₁₉₉	—	2	2	no	—
16	#135×C₁₉₉	—	2	2	no	—
17	C₂×M₁₆×C₁₉₉	C₂²	1	1	yes	characteristic
18	C₂×C₁₉₉⋊C₈	—	2	2	no	—
19	C₂×C₁₉₉⋊C₈	—	2	1	no	—
20	C₄⋊Dic₁₉₉	—	2	2	no	—
21	C₄⋊Dic₁₉₉	—	2	1	no	—
22	C₂²⋊Dic₁₉₉	—	2	2	no	—
23	C₂²⋊Dic₁₉₉	D₄	1	2	yes	—
24	C₄×Dic₁₉₉	—	2	1	no	—
25	C₄×Dic₁₉₉	—	2	2	no	—
26	#41×C₁₉₉	—	2	2	no	—
27	#41×C₁₉₉	—	2	2	no	—
28	#41×C₁₉₉	—	2	2	no	—
29	#41×C₁₉₉	—	2	2	no	—
30	C₂×Q₈×C₁₉₉	D₄	1	2	yes	—
31	C₂×D₄×C₁₉₉	—	2	2	no	—
32	C₂×D₄×C₁₉₉	D₄	1	2	yes	—
33	C₂²×C₄×C₁₉₉	—	2	2	no	—
34	C₂²×C₄×C₁₉₉	C₂³	1	1	yes	characteristic
35	Q₁₆×C₁₉₉	—	2	2	no	—
36	SD₁₆×C₁₉₉	—	2	4	no	—
37	D₈×C₁₉₉	—	2	2	no	—
38	M₁₆×C₁₉₉	—	2	2	no	—
39	C₂×C₈×C₁₉₉	D₄	1	2	yes	—
40	C₂×Dic₁₉₉	—	2	2	no	—
41	C₂×Dic₁₉₉	—	4	2	no	—
42	C₁₉₉⋊C₈	—	4	1	no	—
43	C₂³×C₁₉₉	—	2	2	no	—
44	C₂³×C₁₉₉	C₂×D₄	1	1	yes	characteristic
45	Q₈×C₁₉₉	—	2	2	no	—
46	Q₈×C₁₉₉	—	2	2	no	—
47	D₄×C₁₉₉	—	2	2	no	—
48	D₄×C₁₉₉	—	2	2	no	—
49	D₄×C₁₉₉	—	4	2	no	—
50	D₄×C₁₉₉	—	4	2	no	—
51	C₂×C₄×C₁₉₉	—	2	2	no	—
52	C₂×C₄×C₁₉₉	—	4	2	no	—
53	C₂×C₄×C₁₉₉	—	4	2	no	—
54	C₂×C₄×C₁₉₉	C₂×D₄	1	2	yes	—
55	C₂×C₄×C₁₉₉	C₂×D₄	1	1	yes	characteristic
56	C₂×C₄×C₁₉₉	#41	1	1	yes	characteristic
57	C₂×C₄×C₁₉₉	#41	1	2	yes	—
58	C₈×C₁₉₉	—	2	2	no	—
59	C₂²×C₁₉₉	—	4	1	no	—
60	C₂²×C₁₉₉	—	4	2	no	—
61	C₂²×C₁₉₉	—	8	2	no	—
62	C₂²×C₁₉₉	#121	1	1	yes	characteristic
63	C₂²×C₁₉₉	#121	1	2	yes	—
64	C₄×C₁₉₉	—	4	2	no	—
65	C₄×C₁₉₉	#120	1	1	yes	characteristic
66	C₄×C₁₉₉	#123	1	2	yes	—
67	C₄×C₁₉₉	#124	1	1	yes	characteristic
68	Dic₁₉₉	—	4	2	no	—
69	Dic₁₉₉	—	8	2	no	—
70	C₂×C₁₉₉	—	2	1	no	—
71	C₂×C₁₉₉	—	2	2	no	—
72	C₂×C₁₉₉	—	8	2	no	—
73	C₂×C₁₉₉	#394	1	1	yes	characteristic
74	C₁₉₉	#2053	1	1	yes	characteristic, complemented, Sylow
75	#2053	—	199	1	no	maximal, Sylow
76	#575	D₁₉₉	1	1	yes	characteristic
77	#424	—	199	1	no	—
78	#416	—	199	2	no	—
79	#410	—	199	2	no	—
80	H(3,ℤ/4)	—	199	1	no	—
81	C₂×#41	C₂×D₁₉₉	1	2	yes	—
82	#137	—	2	2	no	—
83	AGL(1,ℤ/8)	—	2	2	no	—
84	#135	—	2	2	no	—
85	C₂×M₁₆	C₂×D₁₉₉	1	1	yes	characteristic
86	C₂²×C₈	—	199	1	no	—
87	#117	—	199	2	no	—
88	#117	—	199	1	no	—
89	C₄⋊C₈	—	398	2	no	—
90	#103	—	398	2	no	—
91	#102	—	398	2	no	—
92	#99	—	398	4	no	—
93	#41	—	2	2	no	—
94	#41	—	2	2	no	—
95	#41	—	2	2	no	—
96	#41	—	2	2	no	—
97	C₂×Q₈	C₂²⋊D₁₉₉	1	2	yes	—
98	C₂×D₄	—	2	2	no	—
99	C₂×D₄	C₂²⋊D₁₉₉	1	2	yes	—
100	C₂²×C₄	—	2	2	no	—
101	C₂²×C₄	C₂²×D₁₉₉	1	1	yes	characteristic
102	Q₁₆	—	2	2	no	—
103	SD₁₆	—	2	4	no	—
104	D₈	—	2	2	no	—
105	M₁₆	—	2	2	no	—
106	C₂×C₈	—	398	2	no	—
107	C₂×C₈	—	398	1	no	—
108	C₂×C₈	C₂²⋊D₁₉₉	1	2	yes	—
109	C₄⋊C₄	—	398	2	no	—
110	C₄⋊C₄	—	398	1	no	—
111	C₂²⋊C₄	—	398	2	no	—
112	C₂²⋊C₄	—	199	2	no	—
113	C₄²	—	398	2	no	—
114	C₄²	—	398	1	no	—
115	C₂³	—	2	2	no	—
116	C₂³	D₄×D₁₉₉	1	1	yes	characteristic
117	Q₈	—	2	2	no	—
118	Q₈	—	2	2	no	—
119	D₄	—	2	2	no	—
120	D₄	—	2	2	no	—
121	D₄	—	4	2	no	—
122	D₄	—	4	2	no	—
123	C₂×C₄	—	2	2	no	—
124	C₂×C₄	—	4	2	no	—
125	C₂×C₄	—	4	2	no	—
126	C₂×C₄	—	796	2	no	—
127	C₂×C₄	—	398	2	no	—
128	C₂×C₄	C₂×C₂²⋊D₁₉₉	1	2	yes	—
129	C₂×C₄	C₂×C₂²⋊D₁₉₉	1	1	yes	characteristic
130	C₂×C₄	#15691885	1	1	yes	characteristic
131	C₂×C₄	#15691885	1	2	yes	—
132	C₈	—	2	2	no	—
133	C₈	—	796	1	no	—
134	C₂²	—	4	1	no	—
135	C₂²	—	4	2	no	—
136	C₂²	—	8	2	no	—
137	C₂²	#20263926	1	1	yes	characteristic
138	C₂²	#20263927	1	2	yes	—
139	C₄	—	4	2	no	—
140	C₄	—	796	2	no	—
141	C₄	—	1592	2	no	—
142	C₄	#20263924	1	1	yes	characteristic
143	C₄	#20263936	1	2	yes	—
144	C₄	#20263942	1	1	yes	characteristic
145	C₂	—	2	1	no	—
146	C₂	—	2	2	no	—
147	C₂	—	8	2	no	—
148	C₂	#27163932	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#27164046

C₂

yes

maximal, characteristic, complemented

#27164014

C₂

yes

maximal, complemented

#27163992

C₂

yes

maximal, complemented

#27163775

C₂

yes

maximal, characteristic, complemented

#575×C₁₉₉

C₂

yes

maximal, characteristic

C₂²×C₁₉₉⋊C₈

C₂²

yes

characteristic

#20263914

C₂²

yes

—

#20263914

C₂²

yes

characteristic

Dic₁₉₉⋊C₈

—

#20263879

—

#20263876

—

#20263870

—

C₂×#41×C₁₉₉

C₂²

yes

—

#137×C₁₉₉

—

AGL(1,ℤ/8)×C₁₉₉

—

#135×C₁₉₉

—

C₂×M₁₆×C₁₉₉

C₂²

yes

characteristic

C₂×C₁₉₉⋊C₈

—

C₂×C₁₉₉⋊C₈

—

C₄⋊Dic₁₉₉

—

C₄⋊Dic₁₉₉

—

C₂²⋊Dic₁₉₉

—

C₂²⋊Dic₁₉₉

D₄

yes

—

C₄×Dic₁₉₉

—

C₄×Dic₁₉₉

—

#41×C₁₉₉

—

#41×C₁₉₉

—

#41×C₁₉₉

—

#41×C₁₉₉

—

C₂×Q₈×C₁₉₉

D₄

yes

—

C₂×D₄×C₁₉₉

—

C₂×D₄×C₁₉₉

D₄

yes

—

C₂²×C₄×C₁₉₉

—

C₂²×C₄×C₁₉₉

C₂³

yes

characteristic

Q₁₆×C₁₉₉

—

SD₁₆×C₁₉₉

—

D₈×C₁₉₉

—

M₁₆×C₁₉₉

—

C₂×C₈×C₁₉₉

D₄

yes

—

C₂×Dic₁₉₉

—

C₂×Dic₁₉₉

—

C₁₉₉⋊C₈

—

C₂³×C₁₉₉

—

C₂³×C₁₉₉

C₂×D₄

yes

characteristic

Q₈×C₁₉₉

—

Q₈×C₁₉₉

—

D₄×C₁₉₉

—

D₄×C₁₉₉

—

D₄×C₁₉₉

—

D₄×C₁₉₉

—

C₂×C₄×C₁₉₉

—

C₂×C₄×C₁₉₉

—

C₂×C₄×C₁₉₉

—

C₂×C₄×C₁₉₉

C₂×D₄

yes

—

C₂×C₄×C₁₉₉

C₂×D₄

yes

C₂×C₄×C₁₉₉

yes

C₂×C₄×C₁₉₉

yes

—

C₈×C₁₉₉

—

C₂²×C₁₉₉

—

C₂²×C₁₉₉

—

C₂²×C₁₉₉

—

C₂²×C₁₉₉

#121

yes

characteristic

C₂²×C₁₉₉

#121

yes

—

C₄×C₁₉₉

—

C₄×C₁₉₉

#120

yes

characteristic

C₄×C₁₉₉

#123

yes

—

C₄×C₁₉₉

#124

yes

characteristic

Dic₁₉₉

—

Dic₁₉₉

—

C₂×C₁₉₉

—

C₂×C₁₉₉

—

C₂×C₁₉₉

—

C₂×C₁₉₉

#394

yes

characteristic

C₁₉₉

#2053

yes

characteristic, complemented, Sylow

#2053

—

199

maximal, Sylow

#575

D₁₉₉

yes

characteristic

#424

—

199

—

#416

—

199

—

#410

—

199

—

H(3,ℤ/4)

—

199

—

C₂×#41

C₂×D₁₉₉

yes

—

#137

—

AGL(1,ℤ/8)

—

#135

—

C₂×M₁₆

C₂×D₁₉₉

yes

characteristic

C₂²×C₈

—

199

—

#117

—

199

—

#117

—

199

—

C₄⋊C₈

—

398

—

#103

—

398

—

#102

—

398

—

#99

—

398

—

—

—

—

—

C₂×Q₈

C₂²⋊D₁₉₉

yes

—

C₂×D₄

—

C₂×D₄

C₂²⋊D₁₉₉

yes

—

100

C₂²×C₄

—

101

C₂²×C₄

C₂²×D₁₉₉

yes

characteristic

102

Q₁₆

—

103

SD₁₆

—

104

D₈

—

105

M₁₆

—

106

C₂×C₈

—

398

—

107

C₂×C₈

—

398

—

108

C₂×C₈

C₂²⋊D₁₉₉

yes

—

109

C₄⋊C₄

—

398

—

110

C₄⋊C₄

—

398

—

111

C₂²⋊C₄

—

398

—

112

C₂²⋊C₄

—

199

—

113

C₄²

—

398

—

114

C₄²

—

398

—

115

C₂³

—

116

C₂³

D₄×D₁₉₉

yes

characteristic

117

Q₈

—

118

Q₈

—

119

D₄

—

120

D₄

—

121

D₄

—

122

D₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

796

—

127

C₂×C₄

—

398

—

128

C₂×C₄

C₂×C₂²⋊D₁₉₉

yes

—

129

C₂×C₄

C₂×C₂²⋊D₁₉₉

yes

characteristic

130

C₂×C₄

#15691885

yes

characteristic

131

C₂×C₄

#15691885

yes

—

132

C₈

—

133

C₈

—

796

—

134

C₂²

—

135

C₂²

—

136

C₂²

—

137

C₂²

#20263926

yes

characteristic

138

C₂²

#20263927

yes

—

139

C₄

—

140

C₄

—

796

—

141

C₄

—

1592

—

142

C₄

#20263924

yes

characteristic

143

C₄

#20263936

yes

—

144

C₄

#20263942

yes

characteristic

145

C₂

—

146

C₂

—

147

C₂

—

148

C₂

#27163932

yes

characteristic

	subgroup	G/H
center	C₄	#20263942
commutator, cosocle	C₂²×C₄×C₁₉₉	C₂³
Frattini	C₂²×C₄	C₂²×D₁₉₉
Fitting, prime radical	#575×C₁₉₉	C₂
socle	C₂×C₁₉₉	#394
hypercenter	#575	D₁₉₉
hypocenter	C₁₉₉	#2053
LCS	G ▷ C₂²×C₄×C₁₉₉ ▷ C₂×C₁₉₉ ▷ C₁₉₉
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #575

subgroup

G/H

center

C₄

#20263942

commutator, cosocle

C₂²×C₄×C₁₉₉

C₂³

Frattini

C₂²×C₄

C₂²×D₁₉₉

Fitting, prime radical

#575×C₁₉₉

C₂

C₂×C₁₉₉

D₁₉₉

C₁₉₉

G ▷ C₂²×C₄×C₁₉₉ ▷ C₂×C₁₉₉ ▷ C₁₉₉

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #575

N	H ≅ G/N	type
1. #27164046	C₂	semidirect
2. #27164014	C₂	semidirect
3. #27163992	C₂	semidirect
4. #27163775	C₂	semidirect
74. C₁₉₉	#2053	semidirect

H ≅ G/N

type

1. #27164046

C₂

semidirect

2. #27164014

C₂

semidirect

3. #27163992

C₂

semidirect

4. #27163775

C₂

semidirect

74. C₁₉₉

#2053

semidirect

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

subgroup	index
#27164046	2
#27164014	2
#27163992	2
#27163775	2
#575×C₁₉₉	2
#2053	199

subgroup

index

#575×C₁₉₉

199

Subgroups — #38770153

Conjugacy classes of subgroups

Complements

Maximal subgroups