Group #37510307 — subgroups

Subgroups — #37510307

Order 22912.

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	#26039239	C₂	1	1	yes	maximal, characteristic
2	#26038467	C₂	1	1	yes	maximal, characteristic, complemented
3	#26038433	C₂	1	2	yes	maximal, complemented
4	#26038412	C₂	1	2	yes	maximal, complemented
5	H(3,ℤ/4)×C₁₇₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#15668259	C₂²	1	2	yes	—
7	#19153146	—	2	2	no	—
8	#19153141	—	2	2	no	—
9	#19153135	—	2	2	no	—
10	C₂²×C₁₇₉⋊C₈	C₂²	1	1	yes	characteristic
11	#19153072	C₂²	1	1	yes	characteristic
12	Dic₁₇₉⋊C₈	—	2	2	no	—
13	#19153036	—	2	2	no	—
14	#19153033	—	2	2	no	—
15	#19153027	—	2	4	no	—
16	C₂×M₁₆×C₁₇₉	C₂²	1	1	yes	characteristic
17	#117×C₁₇₉	C₂²	1	2	yes	—
18	#15668259	—	2	2	no	—
19	#15668259	—	2	2	no	—
20	#15668259	—	2	2	no	—
21	#15668259	—	2	2	no	—
22	C₂×Dic₃₅₈	D₄	1	2	yes	—
23	C₂×D₇₁₆	D₄	1	2	yes	—
24	C₂×C₂²⋊D₁₇₉	—	2	2	no	—
25	C₂×C₄×D₁₇₉	—	2	2	no	—
26	Dic₇₁₆	—	2	2	no	—
27	#15668248	—	2	4	no	—
28	D₁₄₃₂	—	2	2	no	—
29	C₂×C₁₇₉⋊C₈	—	2	2	no	—
30	C₂×C₁₇₉⋊C₈	—	2	1	no	—
31	C₄⋊Dic₁₇₉	—	2	1	no	—
32	C₂²⋊Dic₁₇₉	D₄	1	2	yes	—
33	C₄×Dic₁₇₉	—	2	1	no	—
34	C₂²×C₄×C₁₇₉	C₂³	1	1	yes	characteristic
35	M₁₆×C₁₇₉	—	2	2	no	—
36	C₂×C₈×C₁₇₉	D₄	1	2	yes	—
37	C₄⋊C₄×C₁₇₉	—	2	2	no	—
38	C₂²⋊C₄×C₁₇₉	—	2	2	no	—
39	C₄²×C₁₇₉	—	2	2	no	—
40	C₂²×D₁₇₉	—	2	2	no	—
41	Dic₃₅₈	—	2	2	no	—
42	Dic₃₅₈	—	2	2	no	—
43	D₇₁₆	—	2	2	no	—
44	D₇₁₆	—	2	2	no	—
45	C₂²⋊D₁₇₉	—	4	2	no	—
46	C₂²⋊D₁₇₉	—	4	2	no	—
47	C₂×Dic₁₇₉	—	2	2	no	—
48	C₂×Dic₁₇₉	—	2	2	no	—
49	C₄×D₁₇₉	—	4	2	no	—
50	C₄×D₁₇₉	—	4	2	no	—
51	C₁₇₉⋊C₈	—	4	1	no	—
52	C₂³×C₁₇₉	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄×C₁₇₉	—	4	2	no	—
54	C₂×C₄×C₁₇₉	C₂×D₄	1	2	yes	—
55	C₂×C₄×C₁₇₉	C₂×D₄	1	1	yes	characteristic
56	C₂×C₄×C₁₇₉	#41	1	2	yes	—
57	C₂×C₄×C₁₇₉	#41	1	1	yes	characteristic
58	C₈×C₁₇₉	—	2	2	no	—
59	C₂²×C₁₇₉	—	4	1	no	—
60	C₂²×C₁₇₉	#121	1	2	yes	—
61	C₂²×C₁₇₉	#121	1	1	yes	characteristic
62	C₂×D₁₇₉	—	4	2	no	—
63	C₂×D₁₇₉	—	8	2	no	—
64	C₄×C₁₇₉	—	8	2	no	—
65	C₄×C₁₇₉	#120	1	1	yes	characteristic
66	C₄×C₁₇₉	#123	1	2	yes	—
67	C₄×C₁₇₉	#124	1	1	yes	characteristic
68	Dic₁₇₉	—	4	2	no	—
69	Dic₁₇₉	—	4	2	no	—
70	C₂×C₁₇₉	—	2	1	no	—
71	C₂×C₁₇₉	—	2	2	no	—
72	C₂×C₁₇₉	#394	1	1	yes	characteristic
73	D₁₇₉	—	8	2	no	—
74	C₁₇₉	#2053	1	1	yes	characteristic, complemented, Sylow
75	#2053	—	179	1	no	maximal, Sylow
76	#575	—	179	1	no	—
77	#424	—	179	1	no	—
78	#416	—	179	2	no	—
79	#410	—	179	2	no	—
80	H(3,ℤ/4)	D₁₇₉	1	1	yes	characteristic, complemented
81	C₂×#41	—	179	2	no	—
82	#137	—	358	2	no	—
83	AGL(1,ℤ/8)	—	358	2	no	—
84	#135	—	358	2	no	—
85	C₂×M₁₆	C₂×D₁₇₉	1	1	yes	characteristic
86	C₂²×C₈	—	179	1	no	—
87	#117	—	179	1	no	—
88	#117	C₂×D₁₇₉	1	2	yes	—
89	C₄⋊C₈	—	358	2	no	—
90	#103	—	358	2	no	—
91	#102	—	358	2	no	—
92	#99	—	358	4	no	—
93	#41	—	358	2	no	—
94	#41	—	358	2	no	—
95	#41	—	358	2	no	—
96	#41	—	358	2	no	—
97	C₂×Q₈	—	179	2	no	—
98	C₂×D₄	—	358	2	no	—
99	C₂×D₄	—	179	2	no	—
100	C₂²×C₄	—	358	2	no	—
101	C₂²×C₄	C₂²×D₁₇₉	1	1	yes	characteristic
102	Q₁₆	—	358	2	no	—
103	SD₁₆	—	358	4	no	—
104	D₈	—	358	2	no	—
105	M₁₆	—	2	2	no	—
106	C₂×C₈	—	358	2	no	—
107	C₂×C₈	—	358	1	no	—
108	C₂×C₈	D₇₁₆	1	2	yes	—
109	C₄⋊C₄	—	2	2	no	—
110	C₄⋊C₄	—	358	1	no	—
111	C₂²⋊C₄	—	2	2	no	—
112	C₂²⋊C₄	—	179	2	no	—
113	C₄²	—	2	2	no	—
114	C₄²	—	358	1	no	—
115	C₂³	—	358	2	no	—
116	C₂³	D₄×D₁₇₉	1	1	yes	characteristic
117	Q₈	—	358	2	no	—
118	Q₈	—	358	2	no	—
119	D₄	—	358	2	no	—
120	D₄	—	358	2	no	—
121	D₄	—	716	2	no	—
122	D₄	—	716	2	no	—
123	C₂×C₄	—	4	2	no	—
124	C₂×C₄	—	358	2	no	—
125	C₂×C₄	—	358	2	no	—
126	C₂×C₄	—	716	2	no	—
127	C₂×C₄	—	716	2	no	—
128	C₂×C₄	D₄×D₁₇₉	1	2	yes	—
129	C₂×C₄	C₂×D₇₁₆	1	1	yes	characteristic
130	C₂×C₄	#15668259	1	2	yes	—
131	C₂×C₄	#15668260	1	1	yes	characteristic
132	C₈	—	2	2	no	—
133	C₈	—	716	1	no	—
134	C₂²	—	4	1	no	—
135	C₂²	—	1432	2	no	—
136	C₂²	—	716	2	no	—
137	C₂²	#19153086	1	2	yes	—
138	C₂²	#19153088	1	1	yes	characteristic
139	C₄	—	8	2	no	—
140	C₄	—	716	2	no	—
141	C₄	—	716	2	no	—
142	C₄	#19153083	1	1	yes	characteristic
143	C₄	#19153098	1	2	yes	—
144	C₄	#19153101	1	1	yes	characteristic
145	C₂	—	2	2	no	—
146	C₂	—	2	1	no	—
147	C₂	—	1432	2	no	—
148	C₂	#26038357	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#26039239

C₂

yes

maximal, characteristic

#26038467

C₂

yes

maximal, characteristic, complemented

#26038433

C₂

yes

maximal, complemented

#26038412

C₂

yes

maximal, complemented

H(3,ℤ/4)×C₁₇₉

C₂

yes

maximal, characteristic, complemented

C₂×#15668259

C₂²

yes

—

#19153146

—

#19153141

—

#19153135

—

C₂²×C₁₇₉⋊C₈

C₂²

yes

characteristic

#19153072

C₂²

yes

characteristic

Dic₁₇₉⋊C₈

—

#19153036

—

#19153033

—

#19153027

—

C₂×M₁₆×C₁₇₉

C₂²

yes

characteristic

#117×C₁₇₉

C₂²

yes

—

—

—

—

—

C₂×Dic₃₅₈

D₄

yes

—

C₂×D₇₁₆

D₄

yes

—

C₂×C₂²⋊D₁₇₉

—

C₂×C₄×D₁₇₉

—

Dic₇₁₆

—

#15668248

—

D₁₄₃₂

—

C₂×C₁₇₉⋊C₈

—

C₂×C₁₇₉⋊C₈

—

C₄⋊Dic₁₇₉

—

C₂²⋊Dic₁₇₉

D₄

yes

—

C₄×Dic₁₇₉

—

C₂²×C₄×C₁₇₉

C₂³

yes

characteristic

M₁₆×C₁₇₉

—

C₂×C₈×C₁₇₉

D₄

yes

—

C₄⋊C₄×C₁₇₉

—

C₂²⋊C₄×C₁₇₉

—

C₄²×C₁₇₉

—

C₂²×D₁₇₉

—

Dic₃₅₈

—

Dic₃₅₈

—

D₇₁₆

—

D₇₁₆

—

C₂²⋊D₁₇₉

—

C₂²⋊D₁₇₉

—

C₂×Dic₁₇₉

—

C₂×Dic₁₇₉

—

C₄×D₁₇₉

—

C₄×D₁₇₉

—

C₁₇₉⋊C₈

—

C₂³×C₁₇₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₇₉

—

C₂×C₄×C₁₇₉

C₂×D₄

yes

—

C₂×C₄×C₁₇₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₇₉

#41

yes

—

C₂×C₄×C₁₇₉

#41

yes

characteristic

C₈×C₁₇₉

—

C₂²×C₁₇₉

—

C₂²×C₁₇₉

#121

yes

—

C₂²×C₁₇₉

#121

yes

characteristic

C₂×D₁₇₉

—

C₂×D₁₇₉

—

C₄×C₁₇₉

—

C₄×C₁₇₉

#120

yes

characteristic

C₄×C₁₇₉

#123

yes

—

C₄×C₁₇₉

#124

yes

characteristic

Dic₁₇₉

—

Dic₁₇₉

—

C₂×C₁₇₉

—

C₂×C₁₇₉

—

C₂×C₁₇₉

#394

yes

characteristic

D₁₇₉

—

C₁₇₉

#2053

yes

characteristic, complemented, Sylow

#2053

—

179

maximal, Sylow

#575

—

179

—

#424

—

179

—

#416

—

179

—

#410

—

179

—

H(3,ℤ/4)

D₁₇₉

yes

characteristic, complemented

C₂×#41

—

179

—

#137

—

358

—

AGL(1,ℤ/8)

—

358

—

#135

—

358

—

C₂×M₁₆

C₂×D₁₇₉

yes

characteristic

C₂²×C₈

—

179

—

#117

—

179

—

#117

C₂×D₁₇₉

yes

—

C₄⋊C₈

—

358

—

#103

—

358

—

#102

—

358

—

#99

—

358

—

#41

—

358

—

#41

—

358

—

#41

—

358

—

#41

—

358

—

C₂×Q₈

—

179

—

C₂×D₄

—

358

—

C₂×D₄

—

179

—

100

C₂²×C₄

—

358

—

101

C₂²×C₄

C₂²×D₁₇₉

yes

characteristic

102

Q₁₆

—

358

—

103

SD₁₆

—

358

—

104

D₈

—

358

—

105

M₁₆

—

106

C₂×C₈

—

358

—

107

C₂×C₈

—

358

—

108

C₂×C₈

D₇₁₆

yes

—

109

C₄⋊C₄

—

110

C₄⋊C₄

—

358

—

111

C₂²⋊C₄

—

112

C₂²⋊C₄

—

179

—

113

C₄²

—

114

C₄²

—

358

—

115

C₂³

—

358

—

116

C₂³

D₄×D₁₇₉

yes

characteristic

117

Q₈

—

358

—

118

Q₈

—

358

—

119

D₄

—

358

—

120

D₄

—

358

—

121

D₄

—

716

—

122

D₄

—

716

—

123

C₂×C₄

—

124

C₂×C₄

—

358

—

125

C₂×C₄

—

358

—

126

C₂×C₄

—

716

—

127

C₂×C₄

—

716

—

128

C₂×C₄

D₄×D₁₇₉

yes

—

129

C₂×C₄

C₂×D₇₁₆

yes

characteristic

130

C₂×C₄

#15668259

yes

—

131

C₂×C₄

#15668260

yes

characteristic

132

C₈

—

133

C₈

—

716

—

134

C₂²

—

135

C₂²

—

1432

—

136

C₂²

—

716

—

137

C₂²

#19153086

yes

—

138

C₂²

#19153088

yes

characteristic

139

C₄

—

140

C₄

—

716

—

141

C₄

—

716

—

142

C₄

#19153083

yes

characteristic

143

C₄

#19153098

yes

—

144

C₄

#19153101

yes

characteristic

145

C₂

—

146

C₂

—

147

C₂

—

1432

—

148

C₂

#26038357

yes

characteristic

	subgroup	G/H
center	C₄	#19153101
commutator, cosocle	C₂²×C₄×C₁₇₉	C₂³
Frattini	C₂²×C₄	C₂²×D₁₇₉
Fitting	H(3,ℤ/4)×C₁₇₉	C₂
socle	C₂×C₁₇₉	#394
prime radical	#26039239	C₂
hypercenter	H(3,ℤ/4)	D₁₇₉
hypocenter	C₁₇₉	#2053
LCS	G ▷ C₂²×C₄×C₁₇₉ ▷ C₂×C₁₇₉ ▷ C₁₇₉
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ H(3,ℤ/4)

subgroup

G/H

center

C₄

#19153101

commutator, cosocle

C₂²×C₄×C₁₇₉

C₂³

Frattini

C₂²×C₄

C₂²×D₁₇₉

Fitting

H(3,ℤ/4)×C₁₇₉

C₂

C₂×C₁₇₉

C₂

D₁₇₉

C₁₇₉

G ▷ C₂²×C₄×C₁₇₉ ▷ C₂×C₁₇₉ ▷ C₁₇₉

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ H(3,ℤ/4)

N	H ≅ G/N	type
2. #26038467	C₂	semidirect
3. #26038433	C₂	semidirect
4. #26038412	C₂	semidirect
5. H(3,ℤ/4)×C₁₇₉	C₂	semidirect
74. C₁₇₉	#2053	semidirect
80. H(3,ℤ/4)	D₁₇₉	semidirect

H ≅ G/N

type

2. #26038467

C₂

semidirect

3. #26038433

C₂

semidirect

4. #26038412

C₂

semidirect

5. H(3,ℤ/4)×C₁₇₉

C₂

semidirect

74. C₁₇₉

#2053

semidirect

80. H(3,ℤ/4)

D₁₇₉

semidirect

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

subgroup	index
#26039239	2
#26038467	2
#26038433	2
#26038412	2
H(3,ℤ/4)×C₁₇₉	2
#2053	179

subgroup

index

H(3,ℤ/4)×C₁₇₉

179

Subgroups — #37510307

Conjugacy classes of subgroups

Complements

Maximal subgroups