Group #37509126 — subgroups

Subgroups — #37509126

Order 22912.

248 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 109 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	C₂×C₈×Dic₁₇₉	C₂	1	1	yes	maximal, characteristic
2	#26038308	C₂	1	2	yes	maximal
3	#26038202	C₂	1	1	yes	maximal, characteristic
4	#26038201	C₂	1	2	yes	maximal
5	#340×C₁₇₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₁₇₉⋊C₈	C₂²	1	1	yes	characteristic
7	C₂×C₄⋊Dic₁₇₉	C₂²	1	2	yes	—
8	C₂×Dic₁₇₉⋊C₄	—	2	2	no	—
9	C₂×Dic₁₇₉⋊C₄	—	2	2	no	—
10	C₂×C₄×Dic₁₇₉	C₂²	1	1	yes	characteristic
11	C₈×Dic₁₇₉	C₄	1	4	yes	complemented
12	C₂²×C₈×C₁₇₉	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊C₄×C₁₇₉	C₂²	1	2	yes	—
14	C₂²×Dic₁₇₉	—	2	2	no	—
15	C₂²×Dic₁₇₉	D₄	1	1	yes	characteristic
16	C₂²×Dic₁₇₉	D₄	1	1	yes	characteristic
17	C₂×C₁₇₉⋊C₈	—	2	2	no	—
18	C₂×C₁₇₉⋊C₈	C₂×C₄	1	2	yes	—
19	C₄⋊Dic₁₇₉	—	2	2	no	—
20	C₄⋊Dic₁₇₉	—	2	2	no	—
21	Dic₁₇₉⋊C₄	—	4	4	no	—
22	Dic₁₇₉⋊C₄	—	4	4	no	—
23	C₄×Dic₁₇₉	C₂×C₄	1	2	yes	—
24	C₄×Dic₁₇₉	Q₈	1	2	yes	—
25	C₂²×C₄×C₁₇₉	—	2	2	no	—
26	C₂²×C₄×C₁₇₉	C₂³	1	1	yes	characteristic
27	C₂×C₈×C₁₇₉	—	2	2	no	—
28	C₂×C₈×C₁₇₉	C₂×C₄	1	2	yes	—
29	C₄⋊C₄×C₁₇₉	—	2	2	no	—
30	C₄⋊C₄×C₁₇₉	—	2	2	no	—
31	C₂×Dic₁₇₉	—	4	2	no	—
32	C₂×Dic₁₇₉	—	4	2	no	—
33	C₂×Dic₁₇₉	—	4	2	no	—
34	C₂×Dic₁₇₉	C₄⋊C₄	1	2	yes	—
35	C₂×Dic₁₇₉	C₄⋊C₄	1	2	yes	—
36	C₂×Dic₁₇₉	D₈	1	2	yes	—
37	C₂×Dic₁₇₉	SD₁₆	1	4	yes	—
38	C₂×Dic₁₇₉	Q₁₆	1	2	yes	—
39	C₁₇₉⋊C₈	—	2	2	no	—
40	C₂³×C₁₇₉	C₂×D₄	1	1	yes	characteristic
41	C₂×C₄×C₁₇₉	—	4	2	no	—
42	C₂×C₄×C₁₇₉	—	4	2	no	—
43	C₂×C₄×C₁₇₉	—	4	2	no	—
44	C₂×C₄×C₁₇₉	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₄×C₁₇₉	C₂×Q₈	1	1	yes	characteristic
46	C₂×C₄×C₁₇₉	#41	1	2	yes	—
47	C₂×C₄×C₁₇₉	#41	1	2	yes	—
48	C₈×C₁₇₉	—	2	2	no	—
49	C₂²×C₁₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
50	C₂²×C₁₇₉	#124	1	2	yes	—
51	C₂²×C₁₇₉	C₂×D₈	1	1	yes	characteristic
52	C₂²×C₁₇₉	C₂×SD₁₆	1	2	yes	—
53	C₂²×C₁₇₉	C₂×Q₁₆	1	1	yes	characteristic
54	C₄×C₁₇₉	—	8	2	no	—
55	C₄×C₁₇₉	#117	1	1	yes	characteristic
56	C₄×C₁₇₉	#117	1	1	yes	characteristic
57	C₄×C₁₇₉	#125	1	2	yes	—
58	Dic₁₇₉	—	8	2	no	—
59	Dic₁₇₉	#106	1	4	yes	—
60	Dic₁₇₉	#107	1	4	yes	—
61	C₂×C₁₇₉	#382	1	1	yes	characteristic
62	C₂×C₁₇₉	C₂×#106	1	1	yes	characteristic
63	C₂×C₁₇₉	C₂×#107	1	1	yes	characteristic
64	C₂×C₁₇₉	#486	1	2	yes	—
65	C₂×C₁₇₉	#487	1	2	yes	—
66	C₁₇₉	#1834	1	1	yes	characteristic, complemented, Sylow
67	#1834	—	179	1	no	maximal, Sylow
68	C₂×C₄×C₈	—	179	1	no	—
69	#384	—	179	2	no	—
70	#340	—	179	1	no	—
71	#340	—	179	2	no	—
72	#340	D₁₇₉	1	1	yes	characteristic
73	C₂²×C₈	—	179	1	no	—
74	C₂²×C₈	C₂×D₁₇₉	1	1	yes	characteristic
75	C₂×C₄⋊C₄	—	358	2	no	—
76	C₂×C₄⋊C₄	—	358	2	no	—
77	C₂×C₄⋊C₄	—	179	2	no	—
78	C₂×C₄⋊C₄	C₂×D₁₇₉	1	2	yes	—
79	C₂×C₄²	—	179	1	no	—
80	C₄×C₈	—	179	4	no	—
81	C₂²×C₄	—	2	2	no	—
82	C₂²×C₄	—	358	2	no	—
83	C₂²×C₄	—	179	1	no	—
84	C₂²×C₄	—	179	1	no	—
85	C₂²×C₄	C₂²×D₁₇₉	1	1	yes	characteristic
86	C₂×C₈	—	2	2	no	—
87	C₂×C₈	—	358	2	no	—
88	C₂×C₈	—	179	2	no	—
89	C₂×C₈	C₄×D₁₇₉	1	2	yes	—
90	C₄⋊C₄	—	2	2	no	—
91	C₄⋊C₄	—	2	2	no	—
92	C₄⋊C₄	—	358	2	no	—
93	C₄⋊C₄	—	358	2	no	—
94	C₄⋊C₄	—	716	4	no	—
95	C₄⋊C₄	—	716	4	no	—
96	C₄²	—	179	2	no	—
97	C₄²	—	179	2	no	—
98	C₂³	D₄×D₁₇₉	1	1	yes	characteristic
99	C₂×C₄	—	4	2	no	—
100	C₂×C₄	—	4	2	no	—
101	C₂×C₄	—	4	2	no	—
102	C₂×C₄	—	179	2	no	—
103	C₂×C₄	—	179	2	no	—
104	C₂×C₄	—	179	2	no	—
105	C₂×C₄	—	179	4	no	—
106	C₂×C₄	—	179	2	no	—
107	C₂×C₄	—	716	2	no	—
108	C₂×C₄	—	716	2	no	—
109	C₂×C₄	—	716	2	no	—
110	C₂×C₄	C₂×C₄×D₁₇₉	1	1	yes	characteristic
111	C₂×C₄	Q₈×D₁₇₉	1	1	yes	characteristic
112	C₂×C₄	#15668258	1	2	yes	—
113	C₂×C₄	#15668259	1	2	yes	—
114	C₈	—	2	2	no	—
115	C₈	—	358	2	no	—
116	C₂²	C₄⋊C₄×D₁₇₉	1	1	yes	characteristic
117	C₂²	#19153099	1	2	yes	—
118	C₂²	D₈×D₁₇₉	1	1	yes	characteristic
119	C₂²	SD₁₆×D₁₇₉	1	2	yes	—
120	C₂²	Q₁₆×D₁₇₉	1	1	yes	characteristic
121	C₄	—	8	2	no	—
122	C₄	—	1432	2	no	—
123	C₄	—	179	4	no	—
124	C₄	—	179	4	no	—
125	C₄	#19153069	1	1	yes	characteristic
126	C₄	#19153070	1	1	yes	characteristic
127	C₄	#19153103	1	2	yes	—
128	C₂	#26038303	1	1	yes	characteristic
129	C₂	#106×D₁₇₉	1	1	yes	characteristic
130	C₂	#107×D₁₇₉	1	1	yes	characteristic
131	C₂	#26038784	1	2	yes	—
132	C₂	#26038788	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₈×Dic₁₇₉

C₂

yes

maximal, characteristic

#26038308

C₂

yes

maximal

#26038202

C₂

yes

maximal, characteristic

#26038201

C₂

yes

maximal

#340×C₁₇₉

C₂

yes

maximal, characteristic

C₂²×C₁₇₉⋊C₈

C₂²

yes

characteristic

C₂×C₄⋊Dic₁₇₉

C₂²

yes

—

C₂×Dic₁₇₉⋊C₄

—

C₂×Dic₁₇₉⋊C₄

—

C₂×C₄×Dic₁₇₉

C₂²

yes

characteristic

C₈×Dic₁₇₉

C₄

yes

complemented

C₂²×C₈×C₁₇₉

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₁₇₉

C₂²

yes

—

C₂²×Dic₁₇₉

—

C₂²×Dic₁₇₉

D₄

yes

characteristic

C₂²×Dic₁₇₉

D₄

yes

characteristic

C₂×C₁₇₉⋊C₈

—

C₂×C₁₇₉⋊C₈

C₂×C₄

yes

—

C₄⋊Dic₁₇₉

—

C₄⋊Dic₁₇₉

—

Dic₁₇₉⋊C₄

—

Dic₁₇₉⋊C₄

—

C₄×Dic₁₇₉

C₂×C₄

yes

—

C₄×Dic₁₇₉

Q₈

yes

—

C₂²×C₄×C₁₇₉

—

C₂²×C₄×C₁₇₉

C₂³

yes

characteristic

C₂×C₈×C₁₇₉

—

C₂×C₈×C₁₇₉

C₂×C₄

yes

—

C₄⋊C₄×C₁₇₉

—

C₄⋊C₄×C₁₇₉

—

C₂×Dic₁₇₉

—

C₂×Dic₁₇₉

—

C₂×Dic₁₇₉

—

C₂×Dic₁₇₉

C₄⋊C₄

yes

—

C₂×Dic₁₇₉

C₄⋊C₄

yes

—

C₂×Dic₁₇₉

D₈

yes

—

C₂×Dic₁₇₉

SD₁₆

yes

—

C₂×Dic₁₇₉

Q₁₆

yes

—

C₁₇₉⋊C₈

—

C₂³×C₁₇₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₇₉

—

C₂×C₄×C₁₇₉

—

C₂×C₄×C₁₇₉

—

C₂×C₄×C₁₇₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₇₉

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₇₉

#41

yes

—

C₂×C₄×C₁₇₉

#41

yes

—

C₈×C₁₇₉

—

C₂²×C₁₇₉

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₁₇₉

#124

yes

—

C₂²×C₁₇₉

C₂×D₈

yes

characteristic

C₂²×C₁₇₉

C₂×SD₁₆

yes

—

C₂²×C₁₇₉

C₂×Q₁₆

yes

characteristic

C₄×C₁₇₉

—

C₄×C₁₇₉

yes

C₄×C₁₇₉

yes

C₄×C₁₇₉

yes

—

Dic₁₇₉

—

Dic₁₇₉

#106

yes

—

Dic₁₇₉

#107

yes

—

C₂×C₁₇₉

yes

C₂×C₁₇₉

yes

C₂×C₁₇₉

yes

C₂×C₁₇₉

yes

—

C₂×C₁₇₉

#487

yes

—

C₁₇₉

#1834

yes

characteristic, complemented, Sylow

#1834

—

179

maximal, Sylow

C₂×C₄×C₈

—

179

—

#384

—

179

—

#340

—

179

—

#340

—

179

—

#340

D₁₇₉

yes

characteristic

C₂²×C₈

—

179

—

C₂²×C₈

C₂×D₁₇₉

yes

characteristic

C₂×C₄⋊C₄

—

358

—

C₂×C₄⋊C₄

—

358

—

C₂×C₄⋊C₄

—

179

—

C₂×C₄⋊C₄

C₂×D₁₇₉

yes

—

C₂×C₄²

—

179

—

C₄×C₈

—

179

—

C₂²×C₄

—

C₂²×C₄

—

358

—

C₂²×C₄

—

179

—

C₂²×C₄

—

179

—

C₂²×C₄

C₂²×D₁₇₉

yes

characteristic

C₂×C₈

—

C₂×C₈

—

358

—

C₂×C₈

—

179

—

C₂×C₈

C₄×D₁₇₉

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

358

—

C₄⋊C₄

—

358

—

C₄⋊C₄

—

716

—

C₄⋊C₄

—

716

—

C₄²

—

179

—

C₄²

—

179

—

C₂³

D₄×D₁₇₉

yes

characteristic

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

179

—

103

C₂×C₄

—

179

—

104

C₂×C₄

—

179

—

105

C₂×C₄

—

179

—

106

C₂×C₄

—

179

—

107

C₂×C₄

—

716

—

108

C₂×C₄

—

716

—

109

C₂×C₄

—

716

—

110

C₂×C₄

C₂×C₄×D₁₇₉

yes

characteristic

111

C₂×C₄

Q₈×D₁₇₉

yes

characteristic

112

C₂×C₄

#15668258

yes

—

113

C₂×C₄

#15668259

yes

—

114

C₈

—

115

C₈

—

358

—

116

C₂²

C₄⋊C₄×D₁₇₉

yes

characteristic

117

C₂²

#19153099

yes

—

118

C₂²

D₈×D₁₇₉

yes

characteristic

119

C₂²

SD₁₆×D₁₇₉

yes

—

120

C₂²

Q₁₆×D₁₇₉

yes

characteristic

121

C₄

—

122

C₄

—

1432

—

123

C₄

—

179

—

124

C₄

—

179

—

125

C₄

#19153069

yes

characteristic

126

C₄

#19153070

yes

characteristic

127

C₄

#19153103

yes

—

128

C₂

#26038303

yes

characteristic

129

C₂

#106×D₁₇₉

yes

characteristic

130

C₂

#107×D₁₇₉

yes

characteristic

131

C₂

#26038784

yes

—

132

C₂

#26038788

yes

—

	subgroup	G/H
center	C₂³	D₄×D₁₇₉
commutator	C₂×C₄×C₁₇₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₇₉
Fitting	#340×C₁₇₉	C₂
socle, prime radical	C₂³×C₁₇₉	C₂×D₄
cosocle	C₂²×C₄×C₁₇₉	C₂³
hypercenter	#340	D₁₇₉
hypocenter	C₁₇₉	#1834
LCS	G ▷ C₂×C₄×C₁₇₉ ▷ C₂×C₁₇₉ ▷ C₁₇₉
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ #340

subgroup

G/H

center

C₂³

D₄×D₁₇₉

commutator

C₂×C₄×C₁₇₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₇₉

Fitting

#340×C₁₇₉

C₂

socle, prime radical

C₂³×C₁₇₉

C₂×D₄

cosocle

C₂²×C₄×C₁₇₉

C₂³

D₁₇₉

C₁₇₉

G ▷ C₂×C₄×C₁₇₉ ▷ C₂×C₁₇₉ ▷ C₁₇₉

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ #340

N	H ≅ G/N	type
11. C₈×Dic₁₇₉	C₄	semidirect
66. C₁₇₉	#1834	semidirect

H ≅ G/N

type

11. C₈×Dic₁₇₉

C₄

semidirect

66. C₁₇₉

#1834

semidirect

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

subgroup	index
C₂×C₈×Dic₁₇₉	2
#26038308	2
#26038202	2
#26038201	2
#340×C₁₇₉	2
#1834	179

subgroup

index

C₂×C₈×Dic₁₇₉

#340×C₁₇₉

179

Subgroups — #37509126

Conjugacy classes of subgroups

Complements

Maximal subgroups