Group #37508678 — subgroups

Subgroups — #37508678

Order 22912.

176 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 176 conjugacy classes of subgroups — too many to draw. Only its 65 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	#26038842	C₂	1	1	yes	maximal, characteristic
2	#26038837	C₂	1	1	yes	maximal, characteristic
3	#26038818	C₂	1	1	yes	maximal, characteristic
4	#26038791	C₂	1	1	yes	maximal, characteristic
5	#26038789	C₂	1	1	yes	maximal, characteristic
6	#26038160	C₂	1	1	yes	maximal, characteristic
7	#500×C₁₇₉	C₂	1	1	yes	maximal, characteristic
8	C₂×#15668249	—	2	2	no	—
9	#19153113	C₂²	1	1	yes	characteristic
10	#19153113	C₂²	1	1	yes	characteristic
11	#19153046	—	2	2	no	—
12	#19153045	—	2	2	no	—
13	#19153037	—	2	2	no	—
14	#19153036	—	2	2	no	—
15	C₄×C₁₇₉⋊C₈	C₂²	1	1	yes	characteristic
16	C₄×C₁₇₉⋊C₈	C₂²	1	1	yes	characteristic
17	C₄⋊Q₈×C₁₇₉	C₂²	1	1	yes	characteristic
18	C₄⋊Q₈×C₁₇₉	C₂²	1	1	yes	characteristic
19	#107×C₁₇₉	—	2	2	no	—
20	C₄×C₈×C₁₇₉	C₂²	1	1	yes	characteristic
21	C₂×Dic₃₅₈	—	2	1	no	—
22	C₂×Dic₃₅₈	—	2	1	no	—
23	#15668249	—	4	4	no	—
24	C₂×C₁₇₉⋊C₈	D₄	1	2	yes	—
25	C₂×C₁₇₉⋊C₈	D₄	1	2	yes	—
26	C₄⋊Dic₁₇₉	—	2	1	no	—
27	C₄⋊Dic₁₇₉	—	2	1	no	—
28	C₄⋊Dic₁₇₉	—	2	1	no	—
29	C₄⋊Dic₁₇₉	—	2	1	no	—
30	C₂×Q₈×C₁₇₉	—	2	1	no	—
31	C₂×Q₈×C₁₇₉	—	2	1	no	—
32	C₂×C₈×C₁₇₉	D₄	1	2	yes	—
33	C₄⋊C₄×C₁₇₉	—	2	1	no	—
34	C₄⋊C₄×C₁₇₉	—	2	1	no	—
35	C₄⋊C₄×C₁₇₉	—	2	1	no	—
36	C₄⋊C₄×C₁₇₉	—	2	1	no	—
37	C₄²×C₁₇₉	C₂³	1	1	yes	characteristic
38	Dic₃₅₈	—	4	2	no	—
39	Dic₃₅₈	—	4	2	no	—
40	C₂×Dic₁₇₉	—	4	1	no	—
41	C₂×Dic₁₇₉	—	4	1	no	—
42	C₁₇₉⋊C₈	—	2	2	no	—
43	C₁₇₉⋊C₈	—	2	2	no	—
44	Q₈×C₁₇₉	—	4	2	no	—
45	Q₈×C₁₇₉	—	4	2	no	—
46	C₂×C₄×C₁₇₉	—	4	1	no	—
47	C₂×C₄×C₁₇₉	—	4	1	no	—
48	C₂×C₄×C₁₇₉	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₁₇₉	C₂×D₄	1	1	yes	characteristic
50	C₂×C₄×C₁₇₉	C₂×D₄	1	1	yes	characteristic
51	C₈×C₁₇₉	Q₁₆	1	4	yes	complemented
52	C₂²×C₁₇₉	#127	1	1	yes	characteristic
53	C₄×C₁₇₉	—	8	1	no	—
54	C₄×C₁₇₉	—	8	1	no	—
55	C₄×C₁₇₉	C₂×Q₁₆	1	2	yes	—
56	C₄×C₁₇₉	AGL(1,ℤ/8)	1	2	yes	—
57	C₄×C₁₇₉	#137	1	2	yes	—
58	Dic₁₇₉	—	8	1	no	—
59	Dic₁₇₉	—	8	1	no	—
60	C₂×C₁₇₉	#494	1	1	yes	characteristic
61	C₂×C₁₇₉	#496	1	1	yes	characteristic
62	C₂×C₁₇₉	#497	1	1	yes	characteristic
63	C₁₇₉	#1723	1	1	yes	characteristic, complemented, Sylow
64	#1723	—	179	1	no	maximal, Sylow
65	#500	—	179	1	no	—
66	#500	D₁₇₉	1	1	yes	characteristic
67	#498	—	179	1	no	—
68	#494	—	179	1	no	—
69	#487	—	179	1	no	—
70	#487	—	179	1	no	—
71	#322	—	179	1	no	—
72	C₂×Q₁₆	—	358	2	no	—
73	C₄⋊Q₈	—	179	1	no	—
74	C₄⋊Q₈	—	179	1	no	—
75	C₄⋊Q₈	C₂×D₁₇₉	1	1	yes	characteristic
76	C₄⋊Q₈	C₂×D₁₇₉	1	1	yes	characteristic
77	#107	—	2	2	no	—
78	#107	—	358	2	no	—
79	#106	—	358	2	no	—
80	#103	—	358	2	no	—
81	#103	—	358	2	no	—
82	C₄×C₈	—	179	1	no	—
83	C₄×C₈	—	179	1	no	—
84	C₄×C₈	C₂×D₁₇₉	1	1	yes	characteristic
85	C₂×Q₈	—	2	1	no	—
86	C₂×Q₈	—	2	1	no	—
87	C₂×Q₈	—	358	1	no	—
88	C₂×Q₈	—	358	1	no	—
89	Q₁₆	—	716	4	no	—
90	C₂×C₈	—	179	2	no	—
91	C₂×C₈	—	179	2	no	—
92	C₂×C₈	C₂²⋊D₁₇₉	1	2	yes	—
93	C₄⋊C₄	—	2	1	no	—
94	C₄⋊C₄	—	2	1	no	—
95	C₄⋊C₄	—	2	1	no	—
96	C₄⋊C₄	—	2	1	no	—
97	C₄⋊C₄	—	358	1	no	—
98	C₄⋊C₄	—	358	1	no	—
99	C₄⋊C₄	—	358	1	no	—
100	C₄⋊C₄	—	358	1	no	—
101	C₄²	C₂²×D₁₇₉	1	1	yes	characteristic
102	Q₈	—	4	2	no	—
103	Q₈	—	4	2	no	—
104	Q₈	—	716	2	no	—
105	Q₈	—	716	2	no	—
106	C₂×C₄	—	4	1	no	—
107	C₂×C₄	—	4	1	no	—
108	C₂×C₄	—	716	1	no	—
109	C₂×C₄	—	716	1	no	—
110	C₂×C₄	D₄×D₁₇₉	1	1	yes	characteristic
111	C₂×C₄	D₄×D₁₇₉	1	1	yes	characteristic
112	C₂×C₄	C₂×C₂²⋊D₁₇₉	1	1	yes	characteristic
113	C₈	—	358	2	no	—
114	C₈	—	358	2	no	—
115	C₈	#15668249	1	4	yes	complemented
116	C₂²	#19153109	1	1	yes	characteristic
117	C₄	—	8	1	no	—
118	C₄	—	8	1	no	—
119	C₄	—	1432	1	no	—
120	C₄	—	1432	1	no	—
121	C₄	C₂×#15668249	1	2	yes	—
122	C₄	#19153137	1	2	yes	—
123	C₄	#19153145	1	2	yes	—
124	C₂	#26038818	1	1	yes	characteristic
125	C₂	#26038827	1	1	yes	characteristic
126	C₂	#26038832	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#26038842

C₂

yes

maximal, characteristic

#26038837

C₂

yes

maximal, characteristic

#26038818

C₂

yes

maximal, characteristic

#26038791

C₂

yes

maximal, characteristic

#26038789

C₂

yes

maximal, characteristic

#26038160

C₂

yes

maximal, characteristic

#500×C₁₇₉

C₂

yes

maximal, characteristic

C₂×#15668249

—

#19153113

C₂²

yes

characteristic

#19153113

C₂²

yes

—

—

—

—

C₄×C₁₇₉⋊C₈

C₂²

yes

characteristic

C₄×C₁₇₉⋊C₈

C₂²

yes

characteristic

C₄⋊Q₈×C₁₇₉

C₂²

yes

characteristic

C₄⋊Q₈×C₁₇₉

C₂²

yes

characteristic

#107×C₁₇₉

—

C₄×C₈×C₁₇₉

C₂²

yes

characteristic

C₂×Dic₃₅₈

—

C₂×Dic₃₅₈

—

#15668249

—

C₂×C₁₇₉⋊C₈

D₄

yes

—

C₂×C₁₇₉⋊C₈

D₄

yes

—

C₄⋊Dic₁₇₉

—

C₄⋊Dic₁₇₉

—

C₄⋊Dic₁₇₉

—

C₄⋊Dic₁₇₉

—

C₂×Q₈×C₁₇₉

—

C₂×Q₈×C₁₇₉

—

C₂×C₈×C₁₇₉

D₄

yes

—

C₄⋊C₄×C₁₇₉

—

C₄⋊C₄×C₁₇₉

—

C₄⋊C₄×C₁₇₉

—

C₄⋊C₄×C₁₇₉

—

C₄²×C₁₇₉

C₂³

yes

characteristic

Dic₃₅₈

—

Dic₃₅₈

—

C₂×Dic₁₇₉

—

C₂×Dic₁₇₉

—

C₁₇₉⋊C₈

—

C₁₇₉⋊C₈

—

Q₈×C₁₇₉

—

Q₈×C₁₇₉

—

C₂×C₄×C₁₇₉

—

C₂×C₄×C₁₇₉

—

C₂×C₄×C₁₇₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₇₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₇₉

C₂×D₄

yes

characteristic

C₈×C₁₇₉

Q₁₆

yes

complemented

C₂²×C₁₇₉

#127

yes

characteristic

C₄×C₁₇₉

—

C₄×C₁₇₉

—

C₄×C₁₇₉

C₂×Q₁₆

yes

—

C₄×C₁₇₉

AGL(1,ℤ/8)

yes

—

C₄×C₁₇₉

#137

yes

—

Dic₁₇₉

—

Dic₁₇₉

—

C₂×C₁₇₉

yes

C₂×C₁₇₉

yes

C₂×C₁₇₉

yes

C₁₇₉

yes

characteristic, complemented, Sylow

#1723

—

179

maximal, Sylow

#500

—

179

—

#500

D₁₇₉

yes

characteristic

#498

—

179

—

#494

—

179

—

#487

—

179

—

#487

—

179

—

#322

—

179

—

C₂×Q₁₆

—

358

—

C₄⋊Q₈

—

179

—

C₄⋊Q₈

—

179

—

C₄⋊Q₈

C₂×D₁₇₉

yes

characteristic

C₄⋊Q₈

C₂×D₁₇₉

yes

characteristic

#107

—

#107

—

358

—

#106

—

358

—

#103

—

358

—

#103

—

358

—

C₄×C₈

—

179

—

C₄×C₈

—

179

—

C₄×C₈

C₂×D₁₇₉

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

358

—

C₂×Q₈

—

358

—

Q₁₆

—

716

—

C₂×C₈

—

179

—

C₂×C₈

—

179

—

C₂×C₈

C₂²⋊D₁₇₉

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

358

—

C₄⋊C₄

—

358

—

C₄⋊C₄

—

358

—

100

C₄⋊C₄

—

358

—

101

C₄²

C₂²×D₁₇₉

yes

characteristic

102

Q₈

—

103

Q₈

—

104

Q₈

—

716

—

105

Q₈

—

716

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

716

—

109

C₂×C₄

—

716

—

110

C₂×C₄

D₄×D₁₇₉

yes

characteristic

111

C₂×C₄

D₄×D₁₇₉

yes

characteristic

112

C₂×C₄

C₂×C₂²⋊D₁₇₉

yes

characteristic

113

C₈

—

358

—

114

C₈

—

358

—

115

C₈

#15668249

yes

complemented

116

C₂²

#19153109

yes

characteristic

117

C₄

—

118

C₄

—

119

C₄

—

1432

—

120

C₄

—

1432

—

121

C₄

C₂×#15668249

yes

—

122

C₄

#19153137

yes

—

123

C₄

#19153145

yes

—

124

C₂

#26038818

yes

characteristic

125

C₂

#26038827

yes

characteristic

126

C₂

#26038832

yes

characteristic

	subgroup	G/H
center	C₂²	#19153109
commutator, cosocle	C₄²×C₁₇₉	C₂³
Frattini	C₄²	C₂²×D₁₇₉
Fitting	#500×C₁₇₉	C₂
socle, prime radical	C₂²×C₁₇₉	#127
hypercenter	#500	D₁₇₉
hypocenter	C₁₇₉	#1723
LCS	G ▷ C₄²×C₁₇₉ ▷ C₂²×C₁₇₉ ▷ C₁₇₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #500

subgroup

G/H

center

C₂²

#19153109

commutator, cosocle

C₄²×C₁₇₉

C₂³

Frattini

C₄²

C₂²×D₁₇₉

Fitting

#500×C₁₇₉

C₂

C₂²×C₁₇₉

D₁₇₉

C₁₇₉

G ▷ C₄²×C₁₇₉ ▷ C₂²×C₁₇₉ ▷ C₁₇₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #500

N	H ≅ G/N	type
51. C₈×C₁₇₉	Q₁₆	semidirect
63. C₁₇₉	#1723	semidirect
115. C₈	#15668249	semidirect

H ≅ G/N

type

51. C₈×C₁₇₉

Q₁₆

semidirect

63. C₁₇₉

#1723

semidirect

115. C₈

#15668249

semidirect

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

subgroup	index
#26038842	2
#26038837	2
#26038818	2
#26038791	2
#26038789	2
#26038160	2
#500×C₁₇₉	2
#1723	179

subgroup

index

#500×C₁₇₉

179

Subgroups — #37508678

Conjugacy classes of subgroups

Complements

Maximal subgroups