Group #37509831 — subgroups

Subgroups — #37509831

Order 22912.

320 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 320 conjugacy classes of subgroups — too many to draw. Only its 89 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	#26038989	C₂	1	1	yes	maximal, characteristic
2	C₂×#19153039	C₂	1	4	yes	maximal, complemented
3	#26038285	C₂	1	1	yes	maximal, characteristic, complemented
4	#423×C₁₇₉	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#15668259	C₂²	1	2	yes	—
6	#19153113	C₄	1	1	yes	characteristic, complemented
7	#19153110	C₄	1	1	yes	characteristic, complemented
8	#19153101	C₄	1	2	yes	complemented
9	#19153087	—	2	2	no	—
10	C₄×D₇₁₆	—	2	2	no	—
11	C₂×C₄×Dic₁₇₉	C₂²	1	2	yes	—
12	#19153039	—	2	8	no	—
13	C₁₇₉⋊#95	—	2	2	no	—
14	C₂×M₁₆×C₁₇₉	C₂²	1	2	yes	—
15	C₂×C₄²×C₁₇₉	C₂²	1	1	yes	characteristic
16	C₄⋊C₈×C₁₇₉	—	2	2	no	—
17	#15668259	—	2	4	no	—
18	#15668259	—	2	4	no	—
19	C₂×Dic₃₅₈	C₂×C₄	1	2	yes	—
20	C₂×D₇₁₆	—	2	2	no	—
21	C₂×D₇₁₆	C₂×C₄	1	2	yes	—
22	C₂×C₂²⋊D₁₇₉	—	2	2	no	—
23	C₂²×Dic₁₇₉	—	2	2	no	—
24	C₂×C₄×D₁₇₉	—	2	2	no	—
25	C₄⋊Dic₁₇₉	—	2	2	no	—
26	(C₂×D₁₇₉)⋊C₄	—	2	4	no	—
27	C₄×Dic₁₇₉	—	2	2	no	—
28	C₄×Dic₁₇₉	D₄	1	4	yes	complemented
29	C₂²×C₄×C₁₇₉	D₄	1	2	yes	—
30	C₂²×C₄×C₁₇₉	C₂³	1	1	yes	characteristic
31	M₁₆×C₁₇₉	—	2	4	no	—
32	C₂×C₈×C₁₇₉	—	2	2	no	—
33	C₄²×C₁₇₉	C₂×C₄	1	1	yes	characteristic
34	C₄²×C₁₇₉	C₂×C₄	1	1	yes	characteristic
35	C₄²×C₁₇₉	D₄	1	2	yes	—
36	C₂²×D₁₇₉	—	2	2	no	—
37	Dic₃₅₈	—	2	4	no	—
38	D₇₁₆	—	2	4	no	—
39	D₇₁₆	—	4	4	no	—
40	C₂²⋊D₁₇₉	—	4	4	no	—
41	C₂×Dic₁₇₉	—	2	2	no	—
42	C₂×Dic₁₇₉	—	2	4	no	—
43	C₂×Dic₁₇₉	—	4	4	no	—
44	C₄×D₁₇₉	—	4	2	no	—
45	C₄×D₁₇₉	—	4	2	no	—
46	C₂³×C₁₇₉	C₂×D₄	1	1	yes	characteristic
47	C₂×C₄×C₁₇₉	—	4	2	no	—
48	C₂×C₄×C₁₇₉	C₂²⋊C₄	1	2	yes	—
49	C₂×C₄×C₁₇₉	C₂²⋊C₄	1	2	yes	—
50	C₂×C₄×C₁₇₉	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₄×C₁₇₉	C₂×D₄	1	2	yes	—
52	C₂×C₄×C₁₇₉	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄×C₁₇₉	#41	1	2	yes	—
54	C₈×C₁₇₉	—	4	2	no	—
55	C₂²×C₁₇₉	—	2	2	no	—
56	C₂²×C₁₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
57	C₂²×C₁₇₉	#121	1	2	yes	—
58	C₂×D₁₇₉	—	4	2	no	—
59	C₂×D₁₇₉	—	4	4	no	—
60	C₄×C₁₇₉	—	2	2	no	—
61	C₄×C₁₇₉	—	2	2	no	—
62	C₄×C₁₇₉	C₄×D₄	1	2	yes	—
63	C₄×C₁₇₉	#124	1	1	yes	characteristic
64	C₄×C₁₇₉	#127	1	1	yes	characteristic
65	Dic₁₇₉	—	4	4	no	—
66	Dic₁₇₉	—	4	2	no	—
67	C₂×C₁₇₉	—	2	2	no	—
68	C₂×C₁₇₉	#390	1	1	yes	characteristic
69	C₂×C₁₇₉	#421	1	2	yes	—
70	D₁₇₉	—	8	2	no	—
71	C₁₇₉	#1963	1	1	yes	characteristic, complemented, Sylow
72	#1963	—	179	1	no	maximal, Sylow
73	#532	—	179	1	no	—
74	#423	D₁₇₉	1	1	yes	characteristic, complemented
75	C₂×#104	—	179	4	no	—
76	#376	—	179	1	no	—
77	C₂×#41	—	179	2	no	—
78	C₂×M₁₆	C₂×D₁₇₉	1	2	yes	—
79	C₄⋊Q₈	—	179	1	no	—
80	#127	—	179	1	no	—
81	#124	—	179	2	no	—
82	#121	—	358	2	no	—
83	C₄×D₄	—	358	2	no	—
84	C₂×C₄²	—	179	2	no	—
85	C₂×C₄²	C₂×D₁₇₉	1	1	yes	characteristic
86	C₄⋊C₈	—	2	2	no	—
87	#104	—	358	8	no	—
88	#95	—	358	2	no	—
89	#41	—	358	4	no	—
90	#41	—	358	4	no	—
91	C₂×Q₈	—	179	2	no	—
92	C₂×D₄	—	358	2	no	—
93	C₂×D₄	—	358	2	no	—
94	C₂×D₄	—	179	2	no	—
95	C₂²×C₄	—	358	2	no	—
96	C₂²×C₄	—	358	2	no	—
97	C₂²×C₄	D₇₁₆	1	2	yes	—
98	C₂²×C₄	C₂²×D₁₇₉	1	1	yes	characteristic
99	M₁₆	—	2	4	no	—
100	C₂×C₈	—	2	2	no	—
101	C₄⋊C₄	—	358	2	no	—
102	C₂²⋊C₄	—	358	4	no	—
103	C₄²	—	358	2	no	—
104	C₄²	—	179	4	no	—
105	C₄²	C₄×D₁₇₉	1	1	yes	characteristic
106	C₄²	C₄×D₁₇₉	1	1	yes	characteristic
107	C₄²	C₂²⋊D₁₇₉	1	2	yes	—
108	C₂³	—	358	2	no	—
109	C₂³	C₂×D₇₁₆	1	1	yes	characteristic
110	Q₈	—	358	4	no	—
111	D₄	—	358	4	no	—
112	D₄	—	716	4	no	—
113	D₄	—	716	4	no	—
114	C₂×C₄	—	4	2	no	—
115	C₂×C₄	—	358	2	no	—
116	C₂×C₄	—	358	4	no	—
117	C₂×C₄	—	716	4	no	—
118	C₂×C₄	—	716	2	no	—
119	C₂×C₄	—	716	2	no	—
120	C₂×C₄	(C₂×D₁₇₉)⋊C₄	1	2	yes	—
121	C₂×C₄	(C₂×D₁₇₉)⋊C₄	1	2	yes	—
122	C₂×C₄	C₂×C₄×D₁₇₉	1	1	yes	characteristic
123	C₂×C₄	D₄×D₁₇₉	1	2	yes	—
124	C₂×C₄	C₂×C₂²⋊D₁₇₉	1	1	yes	characteristic
125	C₂×C₄	#15668260	1	2	yes	—
126	C₈	—	4	2	no	—
127	C₂²	—	2	2	no	—
128	C₂²	—	716	2	no	—
129	C₂²	—	716	4	no	—
130	C₂²	C₂×(C₂×D₁₇₉)⋊C₄	1	1	yes	characteristic
131	C₂²	#19153088	1	2	yes	—
132	C₄	—	2	2	no	—
133	C₄	—	2	2	no	—
134	C₄	—	716	4	no	—
135	C₄	—	716	2	no	—
136	C₄	#19153075	1	2	yes	—
137	C₄	#19153102	1	1	yes	characteristic
138	C₄	#19153109	1	1	yes	characteristic
139	C₂	—	2	2	no	—
140	C₂	—	1432	2	no	—
141	C₂	#26038340	1	1	yes	characteristic
142	C₂	#26038455	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#26038989

C₂

yes

maximal, characteristic

C₂×#19153039

C₂

yes

maximal, complemented

#26038285

C₂

yes

maximal, characteristic, complemented

#423×C₁₇₉

C₂

yes

maximal, characteristic, complemented

C₂×#15668259

C₂²

yes

—

#19153113

C₄

yes

characteristic, complemented

#19153110

C₄

yes

characteristic, complemented

#19153101

C₄

yes

complemented

#19153087

—

C₄×D₇₁₆

—

C₂×C₄×Dic₁₇₉

C₂²

yes

—

#19153039

—

C₁₇₉⋊#95

—

C₂×M₁₆×C₁₇₉

C₂²

yes

—

C₂×C₄²×C₁₇₉

C₂²

yes

characteristic

C₄⋊C₈×C₁₇₉

—

#15668259

—

#15668259

—

C₂×Dic₃₅₈

C₂×C₄

yes

—

C₂×D₇₁₆

—

C₂×D₇₁₆

C₂×C₄

yes

—

C₂×C₂²⋊D₁₇₉

—

C₂²×Dic₁₇₉

—

C₂×C₄×D₁₇₉

—

C₄⋊Dic₁₇₉

—

(C₂×D₁₇₉)⋊C₄

—

C₄×Dic₁₇₉

—

C₄×Dic₁₇₉

D₄

yes

complemented

C₂²×C₄×C₁₇₉

D₄

yes

—

C₂²×C₄×C₁₇₉

C₂³

yes

characteristic

M₁₆×C₁₇₉

—

C₂×C₈×C₁₇₉

—

C₄²×C₁₇₉

C₂×C₄

yes

characteristic

C₄²×C₁₇₉

C₂×C₄

yes

characteristic

C₄²×C₁₇₉

D₄

yes

—

C₂²×D₁₇₉

—

Dic₃₅₈

—

D₇₁₆

—

D₇₁₆

—

C₂²⋊D₁₇₉

—

C₂×Dic₁₇₉

—

C₂×Dic₁₇₉

—

C₂×Dic₁₇₉

—

C₄×D₁₇₉

—

C₄×D₁₇₉

—

C₂³×C₁₇₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₇₉

—

C₂×C₄×C₁₇₉

C₂²⋊C₄

yes

—

C₂×C₄×C₁₇₉

C₂²⋊C₄

yes

—

C₂×C₄×C₁₇₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₇₉

C₂×D₄

yes

—

C₂×C₄×C₁₇₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₇₉

#41

yes

—

C₈×C₁₇₉

—

C₂²×C₁₇₉

—

C₂²×C₁₇₉

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₁₇₉

#121

yes

—

C₂×D₁₇₉

—

C₂×D₁₇₉

—

C₄×C₁₇₉

—

C₄×C₁₇₉

—

C₄×C₁₇₉

C₄×D₄

yes

—

C₄×C₁₇₉

yes

C₄×C₁₇₉

yes

Dic₁₇₉

—

Dic₁₇₉

—

C₂×C₁₇₉

—

C₂×C₁₇₉

#390

yes

characteristic

C₂×C₁₇₉

#421

yes

—

D₁₇₉

—

C₁₇₉

#1963

yes

characteristic, complemented, Sylow

#1963

—

179

maximal, Sylow

#532

—

179

—

#423

D₁₇₉

yes

characteristic, complemented

C₂×#104

—

179

—

#376

—

179

—

C₂×#41

—

179

—

C₂×M₁₆

C₂×D₁₇₉

yes

—

C₄⋊Q₈

—

179

—

#127

—

179

—

#124

—

179

—

#121

—

358

—

C₄×D₄

—

358

—

C₂×C₄²

—

179

—

C₂×C₄²

C₂×D₁₇₉

yes

characteristic

C₄⋊C₈

—

#104

—

358

—

#95

—

358

—

#41

—

358

—

#41

—

358

—

C₂×Q₈

—

179

—

C₂×D₄

—

358

—

C₂×D₄

—

358

—

C₂×D₄

—

179

—

C₂²×C₄

—

358

—

C₂²×C₄

—

358

—

C₂²×C₄

D₇₁₆

yes

—

C₂²×C₄

C₂²×D₁₇₉

yes

characteristic

M₁₆

—

100

C₂×C₈

—

101

C₄⋊C₄

—

358

—

102

C₂²⋊C₄

—

358

—

103

C₄²

—

358

—

104

C₄²

—

179

—

105

C₄²

C₄×D₁₇₉

yes

characteristic

106

C₄²

C₄×D₁₇₉

yes

characteristic

107

C₄²

C₂²⋊D₁₇₉

yes

—

108

C₂³

—

358

—

109

C₂³

C₂×D₇₁₆

yes

characteristic

110

Q₈

—

358

—

111

D₄

—

358

—

112

D₄

—

716

—

113

D₄

—

716

—

114

C₂×C₄

—

115

C₂×C₄

—

358

—

116

C₂×C₄

—

358

—

117

C₂×C₄

—

716

—

118

C₂×C₄

—

716

—

119

C₂×C₄

—

716

—

120

C₂×C₄

(C₂×D₁₇₉)⋊C₄

yes

—

121

C₂×C₄

(C₂×D₁₇₉)⋊C₄

yes

—

122

C₂×C₄

C₂×C₄×D₁₇₉

yes

characteristic

123

C₂×C₄

D₄×D₁₇₉

yes

—

124

C₂×C₄

C₂×C₂²⋊D₁₇₉

yes

characteristic

125

C₂×C₄

#15668260

yes

—

126

C₈

—

127

C₂²

—

128

C₂²

—

716

—

129

C₂²

—

716

—

130

C₂²

C₂×(C₂×D₁₇₉)⋊C₄

yes

characteristic

131

C₂²

#19153088

yes

—

132

C₄

—

133

C₄

—

134

C₄

—

716

—

135

C₄

—

716

—

136

C₄

#19153075

yes

—

137

C₄

#19153102

yes

characteristic

138

C₄

#19153109

yes

characteristic

139

C₂

—

140

C₂

—

1432

—

141

C₂

#26038340

yes

characteristic

142

C₂

#26038455

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₁₇₉
commutator	C₂×C₄×C₁₇₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₇₉
Fitting	#423×C₁₇₉	C₂
socle	C₂²×C₁₇₉	C₂×C₂²⋊C₄
prime radical	#26038989	C₂
cosocle	C₂²×C₄×C₁₇₉	C₂³
hypercenter	#423	D₁₇₉
hypocenter	C₁₇₉	#1963
LCS	G ▷ C₂×C₄×C₁₇₉ ▷ C₂×C₁₇₉ ▷ C₁₇₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #423

subgroup

G/H

center

C₂×C₄

C₂×C₂²⋊D₁₇₉

commutator

C₂×C₄×C₁₇₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₇₉

Fitting

#423×C₁₇₉

C₂

socle

C₂²×C₁₇₉

C₂×C₂²⋊C₄

prime radical

#26038989

C₂

cosocle

C₂²×C₄×C₁₇₉

C₂³

D₁₇₉

C₁₇₉

G ▷ C₂×C₄×C₁₇₉ ▷ C₂×C₁₇₉ ▷ C₁₇₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #423

N	H ≅ G/N	type
2. C₂×#19153039	C₂	semidirect
3. #26038285	C₂	semidirect
4. #423×C₁₇₉	C₂	semidirect
6. #19153113	C₄	semidirect
7. #19153110	C₄	semidirect
8. #19153101	C₄	semidirect
28. C₄×Dic₁₇₉	D₄	semidirect
71. C₁₇₉	#1963	semidirect
74. #423	D₁₇₉	semidirect

H ≅ G/N

type

2. C₂×#19153039

C₂

semidirect

3. #26038285

C₂

semidirect

4. #423×C₁₇₉

C₂

semidirect

6. #19153113

C₄

semidirect

7. #19153110

C₄

semidirect

8. #19153101

C₄

semidirect

28. C₄×Dic₁₇₉

D₄

semidirect

71. C₁₇₉

#1963

semidirect

74. #423

D₁₇₉

semidirect

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

subgroup	index
#26038989	2
C₂×#19153039	2
#26038285	2
#423×C₁₇₉	2
#1963	179

subgroup

index

#423×C₁₇₉

179

Subgroups — #37509831

Conjugacy classes of subgroups

Complements

Maximal subgroups