Group #37511077 — subgroups

Subgroups — #37511077

Order 22912.

170 conjugacy classes of subgroups; only the 47 normal subgroups are shown.

normal subgroups only

This group has 170 conjugacy classes of subgroups — too many to draw. Only its 47 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	#26038877	C₂	1	2	yes	maximal, complemented
2	#26038816	C₂	1	1	yes	maximal, characteristic, complemented
3	#26038557	C₂	1	1	yes	maximal, characteristic, complemented
4	#26038253	C₂	1	2	yes	maximal, complemented
5	#364×C₁₇₉	C₂	1	1	yes	maximal, characteristic, complemented
6	#19153135	C₂²	1	1	yes	characteristic
7	C₂×D₁₄₃₂	—	2	1	no	—
8	C₂×D₁₄₃₂	C₂²	1	2	yes	—
9	#19153119	—	2	1	no	—
10	#19153110	—	2	1	no	—
11	#19153058	—	2	2	no	—
12	D₂₈₆₄	—	2	2	no	—
13	#19153049	C₂²	1	1	yes	characteristic
14	#19153041	—	2	1	no	—
15	M₃₂×C₁₇₉	C₂²	1	2	yes	—
16	C₄×C₈×C₁₇₉	C₂²	1	1	yes	characteristic
17	#15668259	—	2	1	no	—
18	C₂×D₇₁₆	—	2	2	no	—
19	C₂×D₇₁₆	—	4	2	no	—
20	Dic₇₁₆	D₄	1	1	yes	characteristic, complemented
21	#15668248	—	2	1	no	—
22	D₁₄₃₂	—	2	2	no	—
23	D₁₄₃₂	—	2	2	no	—
24	D₁₄₃₂	—	4	1	no	—
25	D₁₄₃₂	—	4	1	no	—
26	D₁₄₃₂	D₄	1	1	yes	characteristic, complemented
27	#15668243	—	2	1	no	—
28	#15668242	—	4	1	no	—
29	C₈×D₁₇₉	—	4	1	no	—
30	C₂×C₈×C₁₇₉	D₄	1	1	yes	characteristic
31	C₂×C₈×C₁₇₉	C₂³	1	1	yes	characteristic
32	C₄²×C₁₇₉	D₄	1	1	yes	characteristic
33	C₁₆×C₁₇₉	—	2	2	no	—
34	C₂²×D₁₇₉	—	4	2	no	—
35	Dic₃₅₈	—	2	1	no	—
36	D₇₁₆	—	2	1	no	—
37	D₇₁₆	—	4	2	no	—
38	D₇₁₆	—	4	2	no	—
39	D₇₁₆	—	8	4	no	—
40	C₂²⋊D₁₇₉	—	4	1	no	—
41	C₄×D₁₇₉	—	4	1	no	—
42	C₁₇₉⋊C₈	—	4	1	no	—
43	C₂×C₄×C₁₇₉	—	2	1	no	—
44	C₂×C₄×C₁₇₉	C₂×D₄	1	1	yes	characteristic
45	C₈×C₁₇₉	D₈	1	1	yes	characteristic
46	C₈×C₁₇₉	D₈	1	1	yes	characteristic
47	C₈×C₁₇₉	C₂×D₄	1	1	yes	characteristic
48	C₈×C₁₇₉	#41	1	1	yes	characteristic
49	C₂²×C₁₇₉	AGL(1,ℤ/8)	1	1	yes	characteristic
50	C₂×D₁₇₉	—	4	1	no	—
51	C₂×D₁₇₉	—	8	4	no	—
52	C₂×D₁₇₉	—	8	2	no	—
53	C₄×C₁₇₉	—	2	2	no	—
54	C₄×C₁₇₉	#121	1	1	yes	characteristic
55	C₄×C₁₇₉	C₂×D₈	1	1	yes	characteristic
56	Dic₁₇₉	—	4	1	no	—
57	C₂×C₁₇₉	—	2	1	no	—
58	C₂×C₁₇₉	#459	1	1	yes	characteristic
59	D₁₇₉	—	8	1	no	—
60	D₁₇₉	—	16	2	no	—
61	C₁₇₉	#2213	1	1	yes	characteristic, complemented, Sylow
62	#2213	—	179	1	no	maximal, Sylow
63	#509	—	179	2	no	—
64	#493	—	179	1	no	—
65	#443	—	179	1	no	—
66	#364	D₁₇₉	1	1	yes	characteristic, complemented
67	#361	—	179	2	no	—
68	#135	—	179	1	no	—
69	C₂×D₈	—	358	1	no	—
70	C₂×D₈	—	179	2	no	—
71	#131	—	358	1	no	—
72	#127	—	358	1	no	—
73	SD₃₂	—	358	2	no	—
74	D₁₆	—	358	2	no	—
75	M₃₂	C₂×D₁₇₉	1	2	yes	—
76	#108	—	179	1	no	—
77	#104	—	358	1	no	—
78	C₄×C₈	C₂×D₁₇₉	1	1	yes	characteristic
79	#41	—	358	1	no	—
80	C₂×D₄	—	358	2	no	—
81	C₂×D₄	—	716	2	no	—
82	Q₁₆	—	179	1	no	—
83	SD₁₆	—	358	1	no	—
84	D₈	—	358	2	no	—
85	D₈	—	358	2	no	—
86	D₈	—	179	1	no	—
87	D₈	—	716	1	no	—
88	D₈	—	716	1	no	—
89	M₁₆	—	358	1	no	—
90	M₁₆	—	716	1	no	—
91	C₂×C₈	—	716	1	no	—
92	C₂×C₈	D₇₁₆	1	1	yes	characteristic
93	C₂×C₈	C₂²×D₁₇₉	1	1	yes	characteristic
94	C₄²	D₇₁₆	1	1	yes	characteristic
95	C₁₆	—	2	2	no	—
96	C₂³	—	716	2	no	—
97	Q₈	—	358	1	no	—
98	D₄	—	358	1	no	—
99	D₄	—	1432	4	no	—
100	D₄	—	716	1	no	—
101	D₄	—	716	2	no	—
102	D₄	—	716	2	no	—
103	C₂×C₄	—	2	1	no	—
104	C₂×C₄	—	716	1	no	—
105	C₂×C₄	C₂×D₇₁₆	1	1	yes	characteristic
106	C₈	—	716	1	no	—
107	C₈	D₁₄₃₂	1	1	yes	characteristic
108	C₈	D₁₄₃₂	1	1	yes	characteristic
109	C₈	D₄×D₁₇₉	1	1	yes	characteristic
110	C₈	#15668260	1	1	yes	characteristic
111	C₂²	—	1432	4	no	—
112	C₂²	—	1432	2	no	—
113	C₂²	—	716	1	no	—
114	C₂²	#19153141	1	1	yes	characteristic
115	C₄	—	2	2	no	—
116	C₄	—	716	1	no	—
117	C₄	#19153088	1	1	yes	characteristic
118	C₄	C₂×D₁₄₃₂	1	1	yes	characteristic
119	C₂	—	2	1	no	—
120	C₂	—	2864	2	no	—
121	C₂	—	1432	1	no	—
122	C₂	#26038635	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#26038877

C₂

yes

maximal, complemented

#26038816

C₂

yes

maximal, characteristic, complemented

#26038557

C₂

yes

maximal, characteristic, complemented

#26038253

C₂

yes

maximal, complemented

#364×C₁₇₉

C₂

yes

maximal, characteristic, complemented

#19153135

C₂²

yes

characteristic

C₂×D₁₄₃₂

—

C₂×D₁₄₃₂

C₂²

yes

—

#19153119

—

#19153110

—

#19153058

—

D₂₈₆₄

—

#19153049

C₂²

yes

characteristic

#19153041

—

M₃₂×C₁₇₉

C₂²

yes

—

C₄×C₈×C₁₇₉

C₂²

yes

characteristic

#15668259

—

C₂×D₇₁₆

—

C₂×D₇₁₆

—

Dic₇₁₆

D₄

yes

characteristic, complemented

#15668248

—

D₁₄₃₂

—

D₁₄₃₂

—

D₁₄₃₂

—

D₁₄₃₂

—

D₁₄₃₂

D₄

yes

characteristic, complemented

#15668243

—

#15668242

—

C₈×D₁₇₉

—

C₂×C₈×C₁₇₉

D₄

yes

characteristic

C₂×C₈×C₁₇₉

C₂³

yes

characteristic

C₄²×C₁₇₉

D₄

yes

characteristic

C₁₆×C₁₇₉

—

C₂²×D₁₇₉

—

Dic₃₅₈

—

D₇₁₆

—

D₇₁₆

—

D₇₁₆

—

D₇₁₆

—

C₂²⋊D₁₇₉

—

C₄×D₁₇₉

—

C₁₇₉⋊C₈

—

C₂×C₄×C₁₇₉

—

C₂×C₄×C₁₇₉

C₂×D₄

yes

characteristic

C₈×C₁₇₉

D₈

yes

characteristic

C₈×C₁₇₉

D₈

yes

characteristic

C₈×C₁₇₉

C₂×D₄

yes

C₈×C₁₇₉

yes

C₂²×C₁₇₉

yes

C₂×D₁₇₉

—

C₂×D₁₇₉

—

C₂×D₁₇₉

—

C₄×C₁₇₉

—

C₄×C₁₇₉

#121

yes

characteristic

C₄×C₁₇₉

C₂×D₈

yes

characteristic

Dic₁₇₉

—

C₂×C₁₇₉

—

C₂×C₁₇₉

#459

yes

characteristic

D₁₇₉

—

D₁₇₉

—

C₁₇₉

#2213

yes

characteristic, complemented, Sylow

#2213

—

179

maximal, Sylow

#509

—

179

—

#493

—

179

—

#443

—

179

—

#364

D₁₇₉

yes

characteristic, complemented

#361

—

179

—

#135

—

179

—

C₂×D₈

—

358

—

C₂×D₈

—

179

—

#131

—

358

—

#127

—

358

—

SD₃₂

—

358

—

D₁₆

—

358

—

M₃₂

C₂×D₁₇₉

yes

—

#108

—

179

—

#104

—

358

—

C₄×C₈

C₂×D₁₇₉

yes

characteristic

#41

—

358

—

C₂×D₄

—

358

—

C₂×D₄

—

716

—

Q₁₆

—

179

—

SD₁₆

—

358

—

D₈

—

358

—

D₈

—

358

—

D₈

—

179

—

D₈

—

716

—

D₈

—

716

—

M₁₆

—

358

—

M₁₆

—

716

—

C₂×C₈

—

716

—

C₂×C₈

D₇₁₆

yes

characteristic

C₂×C₈

C₂²×D₁₇₉

yes

characteristic

C₄²

D₇₁₆

yes

characteristic

C₁₆

—

C₂³

—

716

—

Q₈

—

358

—

D₄

—

358

—

D₄

—

1432

—

100

D₄

—

716

—

101

D₄

—

716

—

102

D₄

—

716

—

103

C₂×C₄

—

104

C₂×C₄

—

716

—

105

C₂×C₄

C₂×D₇₁₆

yes

characteristic

106

C₈

—

716

—

107

C₈

D₁₄₃₂

yes

characteristic

108

C₈

D₁₄₃₂

yes

characteristic

109

C₈

D₄×D₁₇₉

yes

characteristic

110

C₈

#15668260

yes

characteristic

111

C₂²

—

1432

—

112

C₂²

—

1432

—

113

C₂²

—

716

—

114

C₂²

#19153141

yes

characteristic

115

C₄

—

116

C₄

—

716

—

117

C₄

#19153088

yes

characteristic

118

C₄

C₂×D₁₄₃₂

yes

characteristic

119

C₂

—

120

C₂

—

2864

—

121

C₂

—

1432

—

122

C₂

#26038635

yes

characteristic

	subgroup	G/H
center	C₂	#26038635
commutator, cosocle	C₂×C₈×C₁₇₉	C₂³
Frattini	C₂×C₈	C₂²×D₁₇₉
Fitting	#364×C₁₇₉	C₂
socle	C₂×C₁₇₉	#459
hypercenter	#364	D₁₇₉
hypocenter	C₁₇₉	#2213
LCS	G ▷ C₂×C₈×C₁₇₉ ▷ C₄×C₁₇₉ ▷ C₂×C₁₇₉ ▷ C₁₇₉
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #364

subgroup

G/H

center

C₂

#26038635

commutator, cosocle

C₂×C₈×C₁₇₉

C₂³

Frattini

C₂×C₈

C₂²×D₁₇₉

Fitting

#364×C₁₇₉

C₂

C₂×C₁₇₉

D₁₇₉

C₁₇₉

G ▷ C₂×C₈×C₁₇₉ ▷ C₄×C₁₇₉ ▷ C₂×C₁₇₉ ▷ C₁₇₉

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #364

N	H ≅ G/N	type
1. #26038877	C₂	semidirect
2. #26038816	C₂	semidirect
3. #26038557	C₂	semidirect
4. #26038253	C₂	semidirect
5. #364×C₁₇₉	C₂	semidirect
20. Dic₇₁₆	D₄	semidirect
26. D₁₄₃₂	D₄	semidirect
61. C₁₇₉	#2213	semidirect
66. #364	D₁₇₉	semidirect

H ≅ G/N

type

1. #26038877

C₂

semidirect

2. #26038816

C₂

semidirect

3. #26038557

C₂

semidirect

4. #26038253

C₂

semidirect

5. #364×C₁₇₉

C₂

semidirect

20. Dic₇₁₆

D₄

semidirect

26. D₁₄₃₂

D₄

semidirect

61. C₁₇₉

#2213

semidirect

66. #364

D₁₇₉

semidirect

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

subgroup	index
#26038877	2
#26038816	2
#26038557	2
#26038253	2
#364×C₁₇₉	2
#2213	179

subgroup

index

#364×C₁₇₉

179

Subgroups — #37511077

Conjugacy classes of subgroups

Complements

Maximal subgroups