Group #38770157 — subgroups

Subgroups — #38770157

Order 25472.

264 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 264 conjugacy classes of subgroups — too many to draw. Only its 63 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	#27164818	C₂	1	1	yes	maximal, characteristic
2	#27164046	C₂	1	1	yes	maximal, characteristic, complemented
3	#27164012	C₂	1	2	yes	maximal, complemented
4	#27163991	C₂	1	2	yes	maximal, complemented
5	H(3,ℤ/4)×C₁₉₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#15691886	C₂²	1	2	yes	—
7	#20263988	—	2	2	no	—
8	#20263983	—	2	2	no	—
9	#20263977	—	2	2	no	—
10	C₂²×C₁₉₉⋊C₈	C₂²	1	1	yes	characteristic
11	#20263914	C₂²	1	1	yes	characteristic
12	Dic₁₉₉⋊C₈	—	2	2	no	—
13	#20263878	—	2	2	no	—
14	#20263875	—	2	2	no	—
15	#20263869	—	2	4	no	—
16	C₂×M₁₆×C₁₉₉	C₂²	1	1	yes	characteristic
17	#117×C₁₉₉	C₂²	1	2	yes	—
18	#15691886	—	2	2	no	—
19	#15691886	—	2	2	no	—
20	#15691886	—	2	2	no	—
21	#15691886	—	2	2	no	—
22	C₂×Dic₃₉₈	D₄	1	2	yes	—
23	C₂×D₇₉₆	D₄	1	2	yes	—
24	C₂×C₂²⋊D₁₉₉	—	2	2	no	—
25	C₂×C₄×D₁₉₉	—	2	2	no	—
26	Dic₇₉₆	—	2	2	no	—
27	#15691875	—	2	4	no	—
28	D₁₅₉₂	—	2	2	no	—
29	C₂×C₁₉₉⋊C₈	—	2	2	no	—
30	C₂×C₁₉₉⋊C₈	—	2	1	no	—
31	C₄⋊Dic₁₉₉	—	2	1	no	—
32	C₂²⋊Dic₁₉₉	D₄	1	2	yes	—
33	C₄×Dic₁₉₉	—	2	1	no	—
34	C₂²×C₄×C₁₉₉	C₂³	1	1	yes	characteristic
35	M₁₆×C₁₉₉	—	2	2	no	—
36	C₂×C₈×C₁₉₉	D₄	1	2	yes	—
37	C₄⋊C₄×C₁₉₉	—	2	2	no	—
38	C₂²⋊C₄×C₁₉₉	—	2	2	no	—
39	C₄²×C₁₉₉	—	2	2	no	—
40	C₂²×D₁₉₉	—	2	2	no	—
41	Dic₃₉₈	—	2	2	no	—
42	Dic₃₉₈	—	2	2	no	—
43	D₇₉₆	—	2	2	no	—
44	D₇₉₆	—	2	2	no	—
45	C₂²⋊D₁₉₉	—	4	2	no	—
46	C₂²⋊D₁₉₉	—	4	2	no	—
47	C₂×Dic₁₉₉	—	2	2	no	—
48	C₂×Dic₁₉₉	—	2	2	no	—
49	C₄×D₁₉₉	—	4	2	no	—
50	C₄×D₁₉₉	—	4	2	no	—
51	C₁₉₉⋊C₈	—	4	1	no	—
52	C₂³×C₁₉₉	C₂×D₄	1	1	yes	characteristic
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	2	yes	—
57	C₂×C₄×C₁₉₉	#41	1	1	yes	characteristic
58	C₈×C₁₉₉	—	2	2	no	—
59	C₂²×C₁₉₉	—	4	1	no	—
60	C₂²×C₁₉₉	#121	1	2	yes	—
61	C₂²×C₁₉₉	#121	1	1	yes	characteristic
62	C₂×D₁₉₉	—	4	2	no	—
63	C₂×D₁₉₉	—	8	2	no	—
64	C₄×C₁₉₉	—	8	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	—	4	2	no	—
70	C₂×C₁₉₉	—	2	1	no	—
71	C₂×C₁₉₉	—	2	2	no	—
72	C₂×C₁₉₉	#394	1	1	yes	characteristic
73	D₁₉₉	—	8	2	no	—
74	C₁₉₉	#2053	1	1	yes	characteristic, complemented, Sylow
75	#2053	—	199	1	no	maximal, Sylow
76	#575	—	199	1	no	—
77	#424	—	199	1	no	—
78	#416	—	199	2	no	—
79	#410	—	199	2	no	—
80	H(3,ℤ/4)	D₁₉₉	1	1	yes	characteristic, complemented
81	C₂×#41	—	199	2	no	—
82	#137	—	398	2	no	—
83	AGL(1,ℤ/8)	—	398	2	no	—
84	#135	—	398	2	no	—
85	C₂×M₁₆	C₂×D₁₉₉	1	1	yes	characteristic
86	C₂²×C₈	—	199	1	no	—
87	#117	—	199	1	no	—
88	#117	C₂×D₁₉₉	1	2	yes	—
89	C₄⋊C₈	—	398	2	no	—
90	#103	—	398	2	no	—
91	#102	—	398	2	no	—
92	#99	—	398	4	no	—
93	#41	—	398	2	no	—
94	#41	—	398	2	no	—
95	#41	—	398	2	no	—
96	#41	—	398	2	no	—
97	C₂×Q₈	—	199	2	no	—
98	C₂×D₄	—	398	2	no	—
99	C₂×D₄	—	199	2	no	—
100	C₂²×C₄	—	398	2	no	—
101	C₂²×C₄	C₂²×D₁₉₉	1	1	yes	characteristic
102	Q₁₆	—	398	2	no	—
103	SD₁₆	—	398	4	no	—
104	D₈	—	398	2	no	—
105	M₁₆	—	2	2	no	—
106	C₂×C₈	—	398	2	no	—
107	C₂×C₈	—	398	1	no	—
108	C₂×C₈	D₇₉₆	1	2	yes	—
109	C₄⋊C₄	—	2	2	no	—
110	C₄⋊C₄	—	398	1	no	—
111	C₂²⋊C₄	—	2	2	no	—
112	C₂²⋊C₄	—	199	2	no	—
113	C₄²	—	2	2	no	—
114	C₄²	—	398	1	no	—
115	C₂³	—	398	2	no	—
116	C₂³	D₄×D₁₉₉	1	1	yes	characteristic
117	Q₈	—	398	2	no	—
118	Q₈	—	398	2	no	—
119	D₄	—	796	2	no	—
120	D₄	—	796	2	no	—
121	D₄	—	398	2	no	—
122	D₄	—	398	2	no	—
123	C₂×C₄	—	4	2	no	—
124	C₂×C₄	—	796	2	no	—
125	C₂×C₄	—	796	2	no	—
126	C₂×C₄	—	398	2	no	—
127	C₂×C₄	—	398	2	no	—
128	C₂×C₄	D₄×D₁₉₉	1	2	yes	—
129	C₂×C₄	C₂×D₇₉₆	1	1	yes	characteristic
130	C₂×C₄	#15691886	1	2	yes	—
131	C₂×C₄	#15691887	1	1	yes	characteristic
132	C₈	—	2	2	no	—
133	C₈	—	796	1	no	—
134	C₂²	—	4	1	no	—
135	C₂²	—	796	2	no	—
136	C₂²	—	1592	2	no	—
137	C₂²	#20263928	1	2	yes	—
138	C₂²	#20263930	1	1	yes	characteristic
139	C₄	—	8	2	no	—
140	C₄	—	796	2	no	—
141	C₄	—	796	2	no	—
142	C₄	#20263925	1	1	yes	characteristic
143	C₄	#20263940	1	2	yes	—
144	C₄	#20263943	1	1	yes	characteristic
145	C₂	—	2	2	no	—
146	C₂	—	2	1	no	—
147	C₂	—	1592	2	no	—
148	C₂	#27163936	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#27164818

C₂

yes

maximal, characteristic

#27164046

C₂

yes

maximal, characteristic, complemented

#27164012

C₂

yes

maximal, complemented

#27163991

C₂

yes

maximal, complemented

H(3,ℤ/4)×C₁₉₉

C₂

yes

maximal, characteristic, complemented

C₂×#15691886

C₂²

yes

—

#20263988

—

#20263983

—

#20263977

—

C₂²×C₁₉₉⋊C₈

C₂²

yes

characteristic

#20263914

C₂²

yes

characteristic

Dic₁₉₉⋊C₈

—

#20263878

—

#20263875

—

#20263869

—

C₂×M₁₆×C₁₉₉

C₂²

yes

characteristic

#117×C₁₉₉

C₂²

yes

—

—

—

—

—

C₂×Dic₃₉₈

D₄

yes

—

C₂×D₇₉₆

D₄

yes

—

C₂×C₂²⋊D₁₉₉

—

C₂×C₄×D₁₉₉

—

Dic₇₉₆

—

#15691875

—

D₁₅₉₂

—

C₂×C₁₉₉⋊C₈

—

C₂×C₁₉₉⋊C₈

—

C₄⋊Dic₁₉₉

—

C₂²⋊Dic₁₉₉

D₄

yes

—

C₄×Dic₁₉₉

—

C₂²×C₄×C₁₉₉

C₂³

yes

characteristic

M₁₆×C₁₉₉

—

C₂×C₈×C₁₉₉

D₄

yes

—

C₄⋊C₄×C₁₉₉

—

C₂²⋊C₄×C₁₉₉

—

C₄²×C₁₉₉

—

C₂²×D₁₉₉

—

Dic₃₉₈

—

Dic₃₉₈

—

D₇₉₆

—

D₇₉₆

—

C₂²⋊D₁₉₉

—

C₂²⋊D₁₉₉

—

C₂×Dic₁₉₉

—

C₂×Dic₁₉₉

—

C₄×D₁₉₉

—

C₄×D₁₉₉

—

C₁₉₉⋊C₈

—

C₂³×C₁₉₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₉₉

—

C₂×C₄×C₁₉₉

C₂×D₄

yes

—

C₂×C₄×C₁₉₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₉₉

#41

yes

—

C₂×C₄×C₁₉₉

#41

yes

characteristic

C₈×C₁₉₉

—

C₂²×C₁₉₉

—

C₂²×C₁₉₉

#121

yes

—

C₂²×C₁₉₉

#121

yes

characteristic

C₂×D₁₉₉

—

C₂×D₁₉₉

—

C₄×C₁₉₉

—

C₄×C₁₉₉

#120

yes

characteristic

C₄×C₁₉₉

#123

yes

—

C₄×C₁₉₉

#124

yes

characteristic

Dic₁₉₉

—

Dic₁₉₉

—

C₂×C₁₉₉

—

C₂×C₁₉₉

—

C₂×C₁₉₉

#394

yes

characteristic

D₁₉₉

—

C₁₉₉

#2053

yes

characteristic, complemented, Sylow

#2053

—

199

maximal, Sylow

#575

—

199

—

#424

—

199

—

#416

—

199

—

#410

—

199

—

H(3,ℤ/4)

D₁₉₉

yes

characteristic, complemented

C₂×#41

—

199

—

#137

—

398

—

AGL(1,ℤ/8)

—

398

—

#135

—

398

—

C₂×M₁₆

C₂×D₁₉₉

yes

characteristic

C₂²×C₈

—

199

—

#117

—

199

—

#117

C₂×D₁₉₉

yes

—

C₄⋊C₈

—

398

—

#103

—

398

—

#102

—

398

—

#99

—

398

—

#41

—

398

—

#41

—

398

—

#41

—

398

—

#41

—

398

—

C₂×Q₈

—

199

—

C₂×D₄

—

398

—

C₂×D₄

—

199

—

100

C₂²×C₄

—

398

—

101

C₂²×C₄

C₂²×D₁₉₉

yes

characteristic

102

Q₁₆

—

398

—

103

SD₁₆

—

398

—

104

D₈

—

398

—

105

M₁₆

—

106

C₂×C₈

—

398

—

107

C₂×C₈

—

398

—

108

C₂×C₈

D₇₉₆

yes

—

109

C₄⋊C₄

—

110

C₄⋊C₄

—

398

—

111

C₂²⋊C₄

—

112

C₂²⋊C₄

—

199

—

113

C₄²

—

114

C₄²

—

398

—

115

C₂³

—

398

—

116

C₂³

D₄×D₁₉₉

yes

characteristic

117

Q₈

—

398

—

118

Q₈

—

398

—

119

D₄

—

796

—

120

D₄

—

796

—

121

D₄

—

398

—

122

D₄

—

398

—

123

C₂×C₄

—

124

C₂×C₄

—

796

—

125

C₂×C₄

—

796

—

126

C₂×C₄

—

398

—

127

C₂×C₄

—

398

—

128

C₂×C₄

D₄×D₁₉₉

yes

—

129

C₂×C₄

C₂×D₇₉₆

yes

characteristic

130

C₂×C₄

#15691886

yes

—

131

C₂×C₄

#15691887

yes

characteristic

132

C₈

—

133

C₈

—

796

—

134

C₂²

—

135

C₂²

—

796

—

136

C₂²

—

1592

—

137

C₂²

#20263928

yes

—

138

C₂²

#20263930

yes

characteristic

139

C₄

—

140

C₄

—

796

—

141

C₄

—

796

—

142

C₄

#20263925

yes

characteristic

143

C₄

#20263940

yes

—

144

C₄

#20263943

yes

characteristic

145

C₂

—

146

C₂

—

147

C₂

—

1592

—

148

C₂

#27163936

yes

characteristic

	subgroup	G/H
center	C₄	#20263943
commutator, cosocle	C₂²×C₄×C₁₉₉	C₂³
Frattini	C₂²×C₄	C₂²×D₁₉₉
Fitting	H(3,ℤ/4)×C₁₉₉	C₂
socle	C₂×C₁₉₉	#394
prime radical	#27164818	C₂
hypercenter	H(3,ℤ/4)	D₁₉₉
hypocenter	C₁₉₉	#2053
LCS	G ▷ C₂²×C₄×C₁₉₉ ▷ C₂×C₁₉₉ ▷ C₁₉₉
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ H(3,ℤ/4)

subgroup

G/H

center

C₄

#20263943

commutator, cosocle

C₂²×C₄×C₁₉₉

C₂³

Frattini

C₂²×C₄

C₂²×D₁₉₉

Fitting

H(3,ℤ/4)×C₁₉₉

C₂

C₂×C₁₉₉

C₂

D₁₉₉

C₁₉₉

G ▷ C₂²×C₄×C₁₉₉ ▷ C₂×C₁₉₉ ▷ C₁₉₉

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ H(3,ℤ/4)

N	H ≅ G/N	type
2. #27164046	C₂	semidirect
3. #27164012	C₂	semidirect
4. #27163991	C₂	semidirect
5. H(3,ℤ/4)×C₁₉₉	C₂	semidirect
74. C₁₉₉	#2053	semidirect
80. H(3,ℤ/4)	D₁₉₉	semidirect

H ≅ G/N

type

2. #27164046

C₂

semidirect

3. #27164012

C₂

semidirect

4. #27163991

C₂

semidirect

5. H(3,ℤ/4)×C₁₉₉

C₂

semidirect

74. C₁₉₉

#2053

semidirect

80. H(3,ℤ/4)

D₁₉₉

semidirect

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

subgroup	index
#27164818	2
#27164046	2
#27164012	2
#27163991	2
H(3,ℤ/4)×C₁₉₉	2
#2053	199

subgroup

index

H(3,ℤ/4)×C₁₉₉

199

Subgroups — #38770157

Conjugacy classes of subgroups

Complements

Maximal subgroups