Group #37511170 — subgroups

Subgroups — #37511170

Order 22912.

126 conjugacy classes of subgroups; only the 43 normal subgroups are shown.

normal subgroups only

This group has 126 conjugacy classes of subgroups — too many to draw. Only its 43 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	#26038836	C₂	1	1	yes	maximal, characteristic, complemented
2	#26038555	C₂	1	1	yes	maximal, characteristic
3	#26038263	C₂	1	1	yes	maximal, characteristic, complemented
4	#26038259	C₂	1	1	yes	maximal, characteristic, complemented
5	#26038250	C₂	1	1	yes	maximal, characteristic
6	#26038248	C₂	1	1	yes	maximal, characteristic
7	#365×C₁₇₉	C₂	1	1	yes	maximal, characteristic, complemented
8	#19153135	C₂²	1	1	yes	characteristic
9	#19153121	—	2	1	no	—
10	#19153112	—	2	1	no	—
11	#19153053	C₂²	1	1	yes	characteristic
12	#19153045	C₂²	1	1	yes	characteristic
13	#19153044	—	2	1	no	—
14	#19153039	—	2	1	no	—
15	C₈×Dic₁₇₉	C₂²	1	1	yes	characteristic
16	M₃₂×C₁₇₉	C₂²	1	1	yes	characteristic
17	#108×C₁₇₉	C₂²	1	1	yes	characteristic
18	#106×C₁₇₉	C₂²	1	1	yes	characteristic
19	#15668259	—	2	1	no	—
20	C₂×Dic₃₅₈	—	4	1	no	—
21	Dic₇₁₆	Q₈	1	1	yes	characteristic, complemented
22	#15668248	—	2	1	no	—
23	D₁₄₃₂	Q₈	1	1	yes	characteristic, complemented
24	#15668242	—	4	1	no	—
25	C₂×C₁₇₉⋊C₈	D₄	1	1	yes	characteristic
26	C₈×D₁₇₉	—	4	1	no	—
27	C₄⋊Dic₁₇₉	—	2	1	no	—
28	Dic₁₇₉⋊C₄	—	4	1	no	—
29	C₄×Dic₁₇₉	D₄	1	1	yes	characteristic
30	C₁₇₉⋊C₁₆	—	2	1	no	—
31	M₁₆×C₁₇₉	—	2	1	no	—
32	C₂×C₈×C₁₇₉	C₂³	1	1	yes	characteristic
33	C₄⋊C₄×C₁₇₉	—	2	1	no	—
34	C₁₆×C₁₇₉	—	2	1	no	—
35	Dic₃₅₈	—	2	1	no	—
36	Dic₃₅₈	—	8	2	no	—
37	D₇₁₆	—	2	1	no	—
38	C₂²⋊D₁₇₉	—	4	1	no	—
39	C₂×Dic₁₇₉	—	2	1	no	—
40	C₂×Dic₁₇₉	—	4	1	no	—
41	C₄×D₁₇₉	—	4	1	no	—
42	C₁₇₉⋊C₈	SD₁₆	1	1	yes	characteristic
43	C₁₇₉⋊C₈	SD₁₆	1	1	yes	characteristic
44	C₂×C₄×C₁₇₉	—	4	1	no	—
45	C₂×C₄×C₁₇₉	C₂×D₄	1	1	yes	characteristic
46	C₈×C₁₇₉	—	4	1	no	—
47	C₈×C₁₇₉	C₂×Q₈	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	C₄×C₁₇₉	—	8	1	no	—
52	C₄×C₁₇₉	C₂²⋊Q₈	1	1	yes	characteristic
53	C₄×C₁₇₉	C₂×SD₁₆	1	1	yes	characteristic
54	Dic₁₇₉	—	2	2	no	—
55	Dic₁₇₉	—	4	1	no	—
56	Dic₁₇₉	—	8	1	no	—
57	C₂×C₁₇₉	—	2	1	no	—
58	C₂×C₁₇₉	#476	1	1	yes	characteristic
59	D₁₇₉	—	8	1	no	—
60	C₁₇₉	#2230	1	1	yes	characteristic, complemented, Sylow
61	#2230	—	179	1	no	maximal, Sylow
62	#498	—	179	1	no	—
63	#443	—	179	1	no	—
64	#365	—	179	1	no	—
65	#365	D₁₇₉	1	1	yes	characteristic, complemented
66	#364	—	179	1	no	—
67	#360	—	179	1	no	—
68	#360	—	179	1	no	—
69	#135	—	179	1	no	—
70	#131	—	358	1	no	—
71	C₄⋊Q₈	—	358	1	no	—
72	M₃₂	—	179	1	no	—
73	M₃₂	C₂×D₁₇₉	1	1	yes	characteristic
74	#108	C₂×D₁₇₉	1	1	yes	characteristic
75	#106	—	358	1	no	—
76	#106	—	179	1	no	—
77	#106	C₂×D₁₇₉	1	1	yes	characteristic
78	#104	—	358	1	no	—
79	C₄×C₈	—	179	1	no	—
80	#41	—	358	1	no	—
81	C₂×Q₈	—	716	1	no	—
82	Q₁₆	—	179	1	no	—
83	SD₁₆	—	358	1	no	—
84	D₈	—	179	1	no	—
85	M₁₆	—	2	1	no	—
86	M₁₆	—	716	1	no	—
87	C₂×C₈	—	179	1	no	—
88	C₂×C₈	—	716	1	no	—
89	C₂×C₈	C₂²×D₁₇₉	1	1	yes	characteristic
90	C₄⋊C₄	—	2	1	no	—
91	C₄⋊C₄	—	358	1	no	—
92	C₄⋊C₄	—	716	1	no	—
93	C₄²	—	179	1	no	—
94	C₁₆	—	2	1	no	—
95	C₁₆	—	358	1	no	—
96	Q₈	—	358	1	no	—
97	Q₈	—	1432	2	no	—
98	D₄	—	358	1	no	—
99	D₄	—	716	1	no	—
100	C₂×C₄	—	4	1	no	—
101	C₂×C₄	—	358	1	no	—
102	C₂×C₄	—	716	1	no	—
103	C₂×C₄	—	716	1	no	—
104	C₂×C₄	D₄×D₁₇₉	1	1	yes	characteristic
105	C₈	—	4	1	no	—
106	C₈	—	179	1	no	—
107	C₈	—	179	1	no	—
108	C₈	Q₈×D₁₇₉	1	1	yes	characteristic
109	C₈	#15668259	1	1	yes	characteristic
110	C₂²	—	716	1	no	—
111	C₂²	#19153140	1	1	yes	characteristic
112	C₄	—	8	1	no	—
113	C₄	—	358	2	no	—
114	C₄	—	1432	1	no	—
115	C₄	—	716	1	no	—
116	C₄	#19153091	1	1	yes	characteristic
117	C₄	SD₁₆×D₁₇₉	1	1	yes	characteristic
118	C₂	—	2	1	no	—
119	C₂	—	1432	1	no	—
120	C₂	#26038728	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#26038836

C₂

yes

maximal, characteristic, complemented

#26038555

C₂

yes

maximal, characteristic

#26038263

C₂

yes

maximal, characteristic, complemented

#26038259

C₂

yes

maximal, characteristic, complemented

#26038250

C₂

yes

maximal, characteristic

#26038248

C₂

yes

maximal, characteristic

#365×C₁₇₉

C₂

yes

maximal, characteristic, complemented

#19153135

C₂²

yes

—

—

C₂²

yes

characteristic

#19153045

C₂²

yes

characteristic

#19153044

—

#19153039

—

C₈×Dic₁₇₉

C₂²

yes

characteristic

M₃₂×C₁₇₉

C₂²

yes

characteristic

#108×C₁₇₉

C₂²

yes

characteristic

#106×C₁₇₉

C₂²

yes

characteristic

#15668259

—

C₂×Dic₃₅₈

—

Dic₇₁₆

Q₈

yes

characteristic, complemented

#15668248

—

D₁₄₃₂

Q₈

yes

characteristic, complemented

#15668242

—

C₂×C₁₇₉⋊C₈

D₄

yes

characteristic

C₈×D₁₇₉

—

C₄⋊Dic₁₇₉

—

Dic₁₇₉⋊C₄

—

C₄×Dic₁₇₉

D₄

yes

characteristic

C₁₇₉⋊C₁₆

—

M₁₆×C₁₇₉

—

C₂×C₈×C₁₇₉

C₂³

yes

characteristic

C₄⋊C₄×C₁₇₉

—

C₁₆×C₁₇₉

—

Dic₃₅₈

—

Dic₃₅₈

—

D₇₁₆

—

C₂²⋊D₁₇₉

—

C₂×Dic₁₇₉

—

C₂×Dic₁₇₉

—

C₄×D₁₇₉

—

C₁₇₉⋊C₈

SD₁₆

yes

characteristic

C₁₇₉⋊C₈

SD₁₆

yes

characteristic

C₂×C₄×C₁₇₉

—

C₂×C₄×C₁₇₉

C₂×D₄

yes

characteristic

C₈×C₁₇₉

—

C₈×C₁₇₉

C₂×Q₈

yes

C₈×C₁₇₉

yes

C₂²×C₁₇₉

yes

C₂×D₁₇₉

—

C₄×C₁₇₉

—

C₄×C₁₇₉

C₂²⋊Q₈

yes

characteristic

C₄×C₁₇₉

C₂×SD₁₆

yes

characteristic

Dic₁₇₉

—

Dic₁₇₉

—

Dic₁₇₉

—

C₂×C₁₇₉

—

C₂×C₁₇₉

#476

yes

characteristic

D₁₇₉

—

C₁₇₉

#2230

yes

characteristic, complemented, Sylow

#2230

—

179

maximal, Sylow

#498

—

179

—

#443

—

179

—

#365

—

179

—

#365

D₁₇₉

yes

characteristic, complemented

#364

—

179

—

#360

—

179

—

#360

—

179

—

#135

—

179

—

#131

—

358

—

C₄⋊Q₈

—

358

—

M₃₂

—

179

—

M₃₂

C₂×D₁₇₉

yes

characteristic

#108

C₂×D₁₇₉

yes

characteristic

#106

—

358

—

#106

—

179

—

#106

C₂×D₁₇₉

yes

characteristic

#104

—

358

—

C₄×C₈

—

179

—

#41

—

358

—

C₂×Q₈

—

716

—

Q₁₆

—

179

—

SD₁₆

—

358

—

D₈

—

179

—

M₁₆

—

M₁₆

—

716

—

C₂×C₈

—

179

—

C₂×C₈

—

716

—

C₂×C₈

C₂²×D₁₇₉

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

358

—

C₄⋊C₄

—

716

—

C₄²

—

179

—

C₁₆

—

C₁₆

—

358

—

Q₈

—

358

—

Q₈

—

1432

—

D₄

—

358

—

D₄

—

716

—

100

C₂×C₄

—

101

C₂×C₄

—

358

—

102

C₂×C₄

—

716

—

103

C₂×C₄

—

716

—

104

C₂×C₄

D₄×D₁₇₉

yes

characteristic

105

C₈

—

106

C₈

—

179

—

107

C₈

—

179

—

108

C₈

Q₈×D₁₇₉

yes

characteristic

109

C₈

#15668259

yes

characteristic

110

C₂²

—

716

—

111

C₂²

#19153140

yes

characteristic

112

C₄

—

113

C₄

—

358

—

114

C₄

—

1432

—

115

C₄

—

716

—

116

C₄

#19153091

yes

characteristic

117

C₄

SD₁₆×D₁₇₉

yes

characteristic

118

C₂

—

119

C₂

—

1432

—

120

C₂

#26038728

yes

characteristic

	subgroup	G/H
center	C₂	#26038728
commutator, cosocle	C₂×C₈×C₁₇₉	C₂³
Frattini	C₂×C₈	C₂²×D₁₇₉
Fitting	#365×C₁₇₉	C₂
socle	C₂×C₁₇₉	#476
prime radical	#19153135	C₂²
hypercenter	#365	D₁₇₉
hypocenter	C₁₇₉	#2230
LCS	G ▷ C₂×C₈×C₁₇₉ ▷ C₄×C₁₇₉ ▷ C₂×C₁₇₉ ▷ C₁₇₉
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #365

subgroup

G/H

center

C₂

#26038728

commutator, cosocle

C₂×C₈×C₁₇₉

C₂³

Frattini

C₂×C₈

C₂²×D₁₇₉

Fitting

#365×C₁₇₉

C₂

C₂×C₁₇₉

C₂²

D₁₇₉

C₁₇₉

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

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #365

N	H ≅ G/N	type
1. #26038836	C₂	semidirect
3. #26038263	C₂	semidirect
4. #26038259	C₂	semidirect
7. #365×C₁₇₉	C₂	semidirect
21. Dic₇₁₆	Q₈	semidirect
23. D₁₄₃₂	Q₈	semidirect
60. C₁₇₉	#2230	semidirect
65. #365	D₁₇₉	semidirect

H ≅ G/N

type

1. #26038836

C₂

semidirect

3. #26038263

C₂

semidirect

4. #26038259

C₂

semidirect

7. #365×C₁₇₉

C₂

semidirect

21. Dic₇₁₆

Q₈

semidirect

23. D₁₄₃₂

Q₈

semidirect

60. C₁₇₉

#2230

semidirect

65. #365

D₁₇₉

semidirect

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

subgroup	index
#26038836	2
#26038555	2
#26038263	2
#26038259	2
#26038250	2
#26038248	2
#365×C₁₇₉	2
#2230	179

subgroup

index

#365×C₁₇₉

179

Subgroups — #37511170

Conjugacy classes of subgroups

Complements

Maximal subgroups