Group #37510554 — subgroups

Subgroups — #37510554

Order 22912.

210 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 210 conjugacy classes of subgroups — too many to draw. Only its 67 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	#26038488	C₂	1	1	yes	maximal, characteristic, complemented
2	#26038466	C₂	1	2	yes	maximal, complemented
3	#26038457	C₂	1	2	yes	maximal
4	#26038407	C₂	1	1	yes	maximal, characteristic
5	H(3,ℤ/4)×C₁₇₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#15668259	C₂²	1	1	yes	characteristic
7	C₂×#15668243	C₂²	1	2	yes	—
8	C₂²×C₁₇₉⋊C₈	C₂²	1	1	yes	characteristic
9	#19153049	—	2	2	no	—
10	C₄⋊(C₁₇₉⋊C₈)	—	2	4	no	—
11	#19153041	—	2	4	no	—
12	(C₂×D₁₇₉)⋊C₈	—	2	2	no	—
13	C₂×M₁₆×C₁₇₉	C₂²	1	1	yes	characteristic
14	#117×C₁₇₉	C₂²	1	2	yes	—
15	#15668259	—	2	2	no	—
16	#15668259	—	2	2	no	—
17	C₂×Dic₃₅₈	Q₈	1	1	yes	characteristic
18	C₂×D₇₁₆	Q₈	1	1	yes	characteristic
19	C₂×C₂²⋊D₁₇₉	—	2	1	no	—
20	C₂×C₄×D₁₇₉	—	2	1	no	—
21	#15668243	—	2	2	no	—
22	#15668243	—	2	2	no	—
23	C₂×C₁₇₉⋊C₈	—	2	1	no	—
24	C₂×C₁₇₉⋊C₈	—	2	2	no	—
25	C₂×C₁₇₉⋊C₈	—	2	2	no	—
26	C₂²×C₄×C₁₇₉	C₂³	1	1	yes	characteristic
27	M₁₆×C₁₇₉	—	2	2	no	—
28	C₂×C₈×C₁₇₉	D₄	1	2	yes	—
29	C₄⋊C₄×C₁₇₉	—	2	2	no	—
30	C₂²⋊C₄×C₁₇₉	—	2	2	no	—
31	C₄²×C₁₇₉	D₄	1	4	yes	—
32	C₂²×D₁₇₉	—	2	1	no	—
33	Dic₃₅₈	—	2	2	no	—
34	D₇₁₆	—	2	2	no	—
35	C₂²⋊D₁₇₉	—	4	2	no	—
36	C₂×Dic₁₇₉	—	2	1	no	—
37	C₄×D₁₇₉	—	4	1	no	—
38	C₄×D₁₇₉	—	4	1	no	—
39	C₁₇₉⋊C₈	—	4	1	no	—
40	C₁₇₉⋊C₈	—	4	2	no	—
41	C₂³×C₁₇₉	#41	1	1	yes	characteristic
42	C₂×C₄×C₁₇₉	—	2	4	no	—
43	C₂×C₄×C₁₇₉	C₂×D₄	1	2	yes	—
44	C₂×C₄×C₁₇₉	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₁₇₉	C₂×Q₈	1	1	yes	characteristic
46	C₂×C₄×C₁₇₉	#41	1	2	yes	—
47	C₈×C₁₇₉	—	2	2	no	—
48	C₂²×C₁₇₉	—	4	1	no	—
49	C₂²×C₁₇₉	C₂²⋊Q₈	1	1	yes	characteristic
50	C₂²×C₁₇₉	#123	1	2	yes	—
51	C₂×D₁₇₉	—	4	2	no	—
52	C₂×D₁₇₉	—	4	1	no	—
53	C₄×C₁₇₉	—	4	4	no	—
54	C₄×C₁₇₉	C₂²⋊Q₈	1	2	yes	—
55	C₄×C₁₇₉	#123	1	1	yes	characteristic
56	C₄×C₁₇₉	#127	1	1	yes	characteristic
57	Dic₁₇₉	—	4	1	no	—
58	C₂×C₁₇₉	—	2	2	no	—
59	C₂×C₁₇₉	—	2	1	no	—
60	C₂×C₁₇₉	#399	1	1	yes	characteristic
61	D₁₇₉	—	8	1	no	—
62	C₁₇₉	#2098	1	1	yes	characteristic, complemented, Sylow
63	#2098	—	179	1	no	maximal, Sylow
64	#430	—	179	1	no	—
65	#424	—	179	2	no	—
66	#421	—	179	2	no	—
67	#408	—	179	1	no	—
68	H(3,ℤ/4)	D₁₇₉	1	1	yes	characteristic, complemented
69	C₂×#41	—	179	1	no	—
70	C₂×M₁₆	—	179	2	no	—
71	C₂×M₁₆	C₂×D₁₇₉	1	1	yes	characteristic
72	C₂²×C₈	—	179	1	no	—
73	#117	C₂×D₁₇₉	1	2	yes	—
74	#108	—	358	2	no	—
75	C₄⋊C₈	—	358	4	no	—
76	#104	—	358	4	no	—
77	C₂²⋊C₈	—	358	2	no	—
78	#41	—	358	2	no	—
79	#41	—	358	2	no	—
80	C₂×Q₈	—	179	1	no	—
81	C₂×D₄	—	358	1	no	—
82	C₂×D₄	—	179	1	no	—
83	C₂²×C₄	—	358	1	no	—
84	C₂²×C₄	C₂²×D₁₇₉	1	1	yes	characteristic
85	M₁₆	—	2	2	no	—
86	M₁₆	—	358	2	no	—
87	M₁₆	—	358	2	no	—
88	C₂×C₈	—	358	1	no	—
89	C₂×C₈	—	358	2	no	—
90	C₂×C₈	—	358	2	no	—
91	C₂×C₈	D₇₁₆	1	2	yes	—
92	C₄⋊C₄	—	2	2	no	—
93	C₂²⋊C₄	—	2	2	no	—
94	C₄²	D₇₁₆	1	4	yes	—
95	C₂³	—	358	1	no	—
96	C₂³	#15668258	1	1	yes	characteristic
97	Q₈	—	358	2	no	—
98	D₄	—	358	2	no	—
99	D₄	—	716	2	no	—
100	C₂×C₄	—	2	4	no	—
101	C₂×C₄	—	358	1	no	—
102	C₂×C₄	—	716	1	no	—
103	C₂×C₄	—	716	1	no	—
104	C₂×C₄	C₂×D₇₁₆	1	2	yes	—
105	C₂×C₄	C₂×D₇₁₆	1	1	yes	characteristic
106	C₂×C₄	Q₈×D₁₇₉	1	1	yes	characteristic
107	C₂×C₄	#15668258	1	2	yes	—
108	C₈	—	2	2	no	—
109	C₈	—	716	1	no	—
110	C₈	—	716	2	no	—
111	C₂²	—	4	1	no	—
112	C₂²	—	716	2	no	—
113	C₂²	—	716	1	no	—
114	C₂²	#19153093	1	1	yes	characteristic
115	C₂²	#19153097	1	2	yes	—
116	C₄	—	4	4	no	—
117	C₄	—	716	1	no	—
118	C₄	#19153093	1	2	yes	—
119	C₄	#19153097	1	1	yes	characteristic
120	C₄	#19153110	1	1	yes	characteristic
121	C₂	—	2	2	no	—
122	C₂	—	2	1	no	—
123	C₂	—	1432	1	no	—
124	C₂	#26038376	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#26038488

C₂

yes

maximal, characteristic, complemented

#26038466

C₂

yes

maximal, complemented

#26038457

C₂

yes

maximal

#26038407

C₂

yes

maximal, characteristic

H(3,ℤ/4)×C₁₇₉

C₂

yes

maximal, characteristic, complemented

C₂×#15668259

C₂²

yes

characteristic

C₂×#15668243

C₂²

yes

—

C₂²×C₁₇₉⋊C₈

C₂²

yes

characteristic

#19153049

—

C₄⋊(C₁₇₉⋊C₈)

—

#19153041

—

(C₂×D₁₇₉)⋊C₈

—

C₂×M₁₆×C₁₇₉

C₂²

yes

characteristic

#117×C₁₇₉

C₂²

yes

—

#15668259

—

#15668259

—

C₂×Dic₃₅₈

Q₈

yes

characteristic

C₂×D₇₁₆

Q₈

yes

characteristic

C₂×C₂²⋊D₁₇₉

—

C₂×C₄×D₁₇₉

—

#15668243

—

#15668243

—

C₂×C₁₇₉⋊C₈

—

C₂×C₁₇₉⋊C₈

—

C₂×C₁₇₉⋊C₈

—

C₂²×C₄×C₁₇₉

C₂³

yes

characteristic

M₁₆×C₁₇₉

—

C₂×C₈×C₁₇₉

D₄

yes

—

C₄⋊C₄×C₁₇₉

—

C₂²⋊C₄×C₁₇₉

—

C₄²×C₁₇₉

D₄

yes

—

C₂²×D₁₇₉

—

Dic₃₅₈

—

D₇₁₆

—

C₂²⋊D₁₇₉

—

C₂×Dic₁₇₉

—

C₄×D₁₇₉

—

C₄×D₁₇₉

—

C₁₇₉⋊C₈

—

C₁₇₉⋊C₈

—

C₂³×C₁₇₉

#41

yes

characteristic

C₂×C₄×C₁₇₉

—

C₂×C₄×C₁₇₉

C₂×D₄

yes

—

C₂×C₄×C₁₇₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₇₉

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₇₉

#41

yes

—

C₈×C₁₇₉

—

C₂²×C₁₇₉

—

C₂²×C₁₇₉

C₂²⋊Q₈

yes

characteristic

C₂²×C₁₇₉

#123

yes

—

C₂×D₁₇₉

—

C₂×D₁₇₉

—

C₄×C₁₇₉

—

C₄×C₁₇₉

C₂²⋊Q₈

yes

—

C₄×C₁₇₉

yes

C₄×C₁₇₉

yes

Dic₁₇₉

—

C₂×C₁₇₉

—

C₂×C₁₇₉

—

C₂×C₁₇₉

#399

yes

characteristic

D₁₇₉

—

C₁₇₉

#2098

yes

characteristic, complemented, Sylow

#2098

—

179

maximal, Sylow

#430

—

179

—

#424

—

179

—

#421

—

179

—

#408

—

179

—

H(3,ℤ/4)

D₁₇₉

yes

characteristic, complemented

C₂×#41

—

179

—

C₂×M₁₆

—

179

—

C₂×M₁₆

C₂×D₁₇₉

yes

characteristic

C₂²×C₈

—

179

—

#117

C₂×D₁₇₉

yes

—

#108

—

358

—

C₄⋊C₈

—

358

—

#104

—

358

—

C₂²⋊C₈

—

358

—

#41

—

358

—

#41

—

358

—

C₂×Q₈

—

179

—

C₂×D₄

—

358

—

C₂×D₄

—

179

—

C₂²×C₄

—

358

—

C₂²×C₄

C₂²×D₁₇₉

yes

characteristic

M₁₆

—

M₁₆

—

358

—

M₁₆

—

358

—

C₂×C₈

—

358

—

C₂×C₈

—

358

—

C₂×C₈

—

358

—

C₂×C₈

D₇₁₆

yes

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

D₇₁₆

yes

—

C₂³

—

358

—

C₂³

#15668258

yes

characteristic

Q₈

—

358

—

D₄

—

358

—

D₄

—

716

—

100

C₂×C₄

—

101

C₂×C₄

—

358

—

102

C₂×C₄

—

716

—

103

C₂×C₄

—

716

—

104

C₂×C₄

C₂×D₇₁₆

yes

—

105

C₂×C₄

C₂×D₇₁₆

yes

characteristic

106

C₂×C₄

Q₈×D₁₇₉

yes

characteristic

107

C₂×C₄

#15668258

yes

—

108

C₈

—

109

C₈

—

716

—

110

C₈

—

716

—

111

C₂²

—

112

C₂²

—

716

—

113

C₂²

—

716

—

114

C₂²

#19153093

yes

characteristic

115

C₂²

#19153097

yes

—

116

C₄

—

117

C₄

—

716

—

118

C₄

#19153093

yes

—

119

C₄

#19153097

yes

characteristic

120

C₄

#19153110

yes

characteristic

121

C₂

—

122

C₂

—

123

C₂

—

1432

—

124

C₂

#26038376

yes

characteristic

	subgroup	G/H
center	C₄	#19153110
commutator, cosocle	C₂²×C₄×C₁₇₉	C₂³
Frattini	C₂²×C₄	C₂²×D₁₇₉
Fitting	H(3,ℤ/4)×C₁₇₉	C₂
socle	C₂×C₁₇₉	#399
prime radical	C₂×#15668259	C₂²
hypercenter	H(3,ℤ/4)	D₁₇₉
hypocenter	C₁₇₉	#2098
LCS	G ▷ C₂²×C₄×C₁₇₉ ▷ C₂×C₁₇₉ ▷ C₁₇₉
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ H(3,ℤ/4)

subgroup

G/H

center

C₄

#19153110

commutator, cosocle

C₂²×C₄×C₁₇₉

C₂³

Frattini

C₂²×C₄

C₂²×D₁₇₉

Fitting

H(3,ℤ/4)×C₁₇₉

C₂

C₂×C₁₇₉

C₂²

D₁₇₉

C₁₇₉

G ▷ C₂²×C₄×C₁₇₉ ▷ C₂×C₁₇₉ ▷ C₁₇₉

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ H(3,ℤ/4)

N	H ≅ G/N	type
1. #26038488	C₂	semidirect
2. #26038466	C₂	semidirect
5. H(3,ℤ/4)×C₁₇₉	C₂	semidirect
62. C₁₇₉	#2098	semidirect
68. H(3,ℤ/4)	D₁₇₉	semidirect

H ≅ G/N

type

1. #26038488

C₂

semidirect

2. #26038466

C₂

semidirect

5. H(3,ℤ/4)×C₁₇₉

C₂

semidirect

62. C₁₇₉

#2098

semidirect

68. 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
#26038488	2
#26038466	2
#26038457	2
#26038407	2
H(3,ℤ/4)×C₁₇₉	2
#2098	179

subgroup

index

H(3,ℤ/4)×C₁₇₉

179

Subgroups — #37510554

Conjugacy classes of subgroups

Complements

Maximal subgroups