Group #37509173 — subgroups

Subgroups — #37509173

Order 22912.

212 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 212 conjugacy classes of subgroups — too many to draw. Only its 101 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×#19153049	C₂	1	1	yes	maximal, characteristic
2	C₂×#19153048	C₂	1	2	yes	maximal
3	#26038462	C₂	1	2	yes	maximal
4	C₂×#19153023	C₂	1	1	yes	maximal, characteristic
5	C₂×#108×C₁₇₉	C₂	1	1	yes	maximal, characteristic
6	C₂×#15668243	C₂²	1	2	yes	—
7	C₂²×C₁₇₉⋊C₈	C₂²	1	1	yes	characteristic
8	C₂×C₄×Dic₁₇₉	C₂²	1	1	yes	characteristic
9	#19153049	—	2	2	no	—
10	#19153048	—	2	4	no	—
11	Dic₁₇₉⋊C₈	—	2	4	no	—
12	#19153023	C₄	1	4	yes	—
13	C₂×M₁₆×C₁₇₉	C₂²	1	2	yes	—
14	C₂²×C₈×C₁₇₉	C₂²	1	1	yes	characteristic
15	#108×C₁₇₉	—	2	2	no	—
16	C₂²×Dic₁₇₉	Q₈	1	2	yes	—
17	#15668243	—	2	2	no	—
18	#15668243	—	2	2	no	—
19	C₂×C₁₇₉⋊C₈	—	2	2	no	—
20	C₂×C₁₇₉⋊C₈	C₂×C₄	1	2	yes	—
21	C₂×C₁₇₉⋊C₈	D₄	1	2	yes	—
22	C₂×C₁₇₉⋊C₈	Q₈	1	2	yes	—
23	C₄×Dic₁₇₉	C₂×C₄	1	2	yes	—
24	C₄×Dic₁₇₉	D₄	1	2	yes	—
25	C₂²×C₄×C₁₇₉	C₂³	1	1	yes	characteristic
26	M₁₆×C₁₇₉	—	2	2	no	—
27	M₁₆×C₁₇₉	—	2	2	no	—
28	C₂×C₈×C₁₇₉	—	2	2	no	—
29	C₂×C₈×C₁₇₉	C₂×C₄	1	2	yes	—
30	C₂×C₈×C₁₇₉	D₄	1	2	yes	—
31	C₂×C₈×C₁₇₉	Q₈	1	2	yes	—
32	C₂×Dic₁₇₉	—	4	2	no	—
33	C₂×Dic₁₇₉	C₄⋊C₄	1	4	yes	—
34	C₁₇₉⋊C₈	—	4	2	no	—
35	C₁₇₉⋊C₈	C₄⋊C₄	1	4	yes	—
36	C₂³×C₁₇₉	C₂×Q₈	1	1	yes	characteristic
37	C₂×C₄×C₁₇₉	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₄×C₁₇₉	C₂×D₄	1	1	yes	characteristic
39	C₂×C₄×C₁₇₉	C₂×D₄	1	1	yes	characteristic
40	C₂×C₄×C₁₇₉	C₂×D₄	1	1	yes	characteristic
41	C₂×C₄×C₁₇₉	C₂×Q₈	1	1	yes	characteristic
42	C₂×C₄×C₁₇₉	C₂×Q₈	1	1	yes	characteristic
43	C₈×C₁₇₉	—	4	2	no	—
44	C₈×C₁₇₉	C₄⋊C₄	1	4	yes	—
45	C₂²×C₁₇₉	—	2	2	no	—
46	C₂²×C₁₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
47	C₂²×C₁₇₉	C₄⋊Q₈	1	1	yes	characteristic
48	C₂²×C₁₇₉	C₄⋊Q₈	1	1	yes	characteristic
49	C₄×C₁₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
50	C₄×C₁₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
51	C₄×C₁₇₉	#127	1	1	yes	characteristic
52	C₄×C₁₇₉	C₄⋊Q₈	1	1	yes	characteristic
53	Dic₁₇₉	—	2	4	no	—
54	C₂×C₁₇₉	—	2	1	no	—
55	C₂×C₁₇₉	—	2	1	no	—
56	C₂×C₁₇₉	#384	1	1	yes	characteristic
57	C₂×C₁₇₉	#430	1	2	yes	—
58	C₁₇₉	#1849	1	1	yes	characteristic, complemented, Sylow
59	#1849	—	179	1	no	maximal, Sylow
60	C₂×#108	—	179	2	no	—
61	C₂×#108	—	179	1	no	—
62	C₂×#108	D₁₇₉	1	1	yes	characteristic
63	#423	—	179	2	no	—
64	C₂×#97	—	179	1	no	—
65	C₂×M₁₆	—	179	2	no	—
66	C₂×M₁₆	C₂×D₁₇₉	1	2	yes	—
67	C₂²×C₈	—	179	1	no	—
68	C₂²×C₈	C₂×D₁₇₉	1	1	yes	characteristic
69	C₂×C₄²	—	179	1	no	—
70	#108	—	2	2	no	—
71	#108	—	358	4	no	—
72	#108	—	358	2	no	—
73	C₄⋊C₈	—	358	4	no	—
74	#97	—	179	4	no	—
75	C₂²×C₄	—	179	2	no	—
76	C₂²×C₄	C₂²×D₁₇₉	1	1	yes	characteristic
77	M₁₆	—	2	2	no	—
78	M₁₆	—	2	2	no	—
79	M₁₆	—	358	2	no	—
80	M₁₆	—	358	2	no	—
81	C₂×C₈	—	2	2	no	—
82	C₂×C₈	—	358	2	no	—
83	C₂×C₈	—	179	2	no	—
84	C₂×C₈	—	179	2	no	—
85	C₂×C₈	—	179	2	no	—
86	C₂×C₈	C₄×D₁₇₉	1	2	yes	—
87	C₂×C₈	C₂²⋊D₁₇₉	1	2	yes	—
88	C₂×C₈	Dic₃₅₈	1	2	yes	—
89	C₄²	—	179	2	no	—
90	C₄²	—	179	2	no	—
91	C₂³	Q₈×D₁₇₉	1	1	yes	characteristic
92	C₂×C₄	—	179	4	no	—
93	C₂×C₄	—	716	2	no	—
94	C₂×C₄	C₂×C₄×D₁₇₉	1	1	yes	characteristic
95	C₂×C₄	D₄×D₁₇₉	1	1	yes	characteristic
96	C₂×C₄	D₄×D₁₇₉	1	1	yes	characteristic
97	C₂×C₄	C₂×C₂²⋊D₁₇₉	1	1	yes	characteristic
98	C₂×C₄	Q₈×D₁₇₉	1	1	yes	characteristic
99	C₂×C₄	C₂×Dic₃₅₈	1	1	yes	characteristic
100	C₈	—	4	2	no	—
101	C₈	—	179	4	no	—
102	C₈	—	716	2	no	—
103	C₈	Dic₁₇₉⋊C₄	1	4	yes	—
104	C₂²	—	2	2	no	—
105	C₂²	C₄⋊C₄×D₁₇₉	1	1	yes	characteristic
106	C₂²	#19153111	1	1	yes	characteristic
107	C₂²	#19153112	1	1	yes	characteristic
108	C₄	—	358	4	no	—
109	C₄	C₄⋊C₄×D₁₇₉	1	1	yes	characteristic
110	C₄	C₂×Dic₁₇₉⋊C₄	1	1	yes	characteristic
111	C₄	#19153109	1	1	yes	characteristic
112	C₄	#19153112	1	1	yes	characteristic
113	C₂	—	2	1	no	—
114	C₂	—	2	1	no	—
115	C₂	#26038308	1	1	yes	characteristic
116	C₂	#26038486	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#19153049

C₂

yes

maximal, characteristic

C₂×#19153048

C₂

yes

maximal

#26038462

C₂

yes

maximal

C₂×#19153023

C₂

yes

maximal, characteristic

C₂×#108×C₁₇₉

C₂

yes

maximal, characteristic

C₂×#15668243

C₂²

yes

—

C₂²×C₁₇₉⋊C₈

C₂²

yes

characteristic

C₂×C₄×Dic₁₇₉

C₂²

yes

characteristic

#19153049

—

#19153048

—

Dic₁₇₉⋊C₈

—

#19153023

C₄

yes

—

C₂×M₁₆×C₁₇₉

C₂²

yes

—

C₂²×C₈×C₁₇₉

C₂²

yes

characteristic

#108×C₁₇₉

—

C₂²×Dic₁₇₉

Q₈

yes

—

#15668243

—

#15668243

—

C₂×C₁₇₉⋊C₈

—

C₂×C₁₇₉⋊C₈

C₂×C₄

yes

—

C₂×C₁₇₉⋊C₈

D₄

yes

—

C₂×C₁₇₉⋊C₈

Q₈

yes

—

C₄×Dic₁₇₉

C₂×C₄

yes

—

C₄×Dic₁₇₉

D₄

yes

—

C₂²×C₄×C₁₇₉

C₂³

yes

characteristic

M₁₆×C₁₇₉

—

M₁₆×C₁₇₉

—

C₂×C₈×C₁₇₉

—

C₂×C₈×C₁₇₉

C₂×C₄

yes

—

C₂×C₈×C₁₇₉

D₄

yes

—

C₂×C₈×C₁₇₉

Q₈

yes

—

C₂×Dic₁₇₉

—

C₂×Dic₁₇₉

C₄⋊C₄

yes

—

C₁₇₉⋊C₈

—

C₁₇₉⋊C₈

C₄⋊C₄

yes

—

C₂³×C₁₇₉

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₇₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₇₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₇₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₇₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₇₉

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₇₉

C₂×Q₈

yes

characteristic

C₈×C₁₇₉

—

C₈×C₁₇₉

C₄⋊C₄

yes

—

C₂²×C₁₇₉

—

C₂²×C₁₇₉

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₁₇₉

C₄⋊Q₈

yes

characteristic

C₂²×C₁₇₉

C₄⋊Q₈

yes

characteristic

C₄×C₁₇₉

C₂×C₄⋊C₄

yes

characteristic

C₄×C₁₇₉

C₂×C₄⋊C₄

yes

characteristic

C₄×C₁₇₉

#127

yes

characteristic

C₄×C₁₇₉

C₄⋊Q₈

yes

characteristic

Dic₁₇₉

—

C₂×C₁₇₉

—

C₂×C₁₇₉

—

C₂×C₁₇₉

#384

yes

characteristic

C₂×C₁₇₉

#430

yes

—

C₁₇₉

#1849

yes

characteristic, complemented, Sylow

#1849

—

179

maximal, Sylow

C₂×#108

—

179

—

C₂×#108

—

179

—

C₂×#108

D₁₇₉

yes

characteristic

#423

—

179

—

C₂×#97

—

179

—

C₂×M₁₆

—

179

—

C₂×M₁₆

C₂×D₁₇₉

yes

—

C₂²×C₈

—

179

—

C₂²×C₈

C₂×D₁₇₉

yes

characteristic

C₂×C₄²

—

179

—

#108

—

#108

—

358

—

#108

—

358

—

C₄⋊C₈

—

358

—

#97

—

179

—

C₂²×C₄

—

179

—

C₂²×C₄

C₂²×D₁₇₉

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

358

—

M₁₆

—

358

—

C₂×C₈

—

C₂×C₈

—

358

—

C₂×C₈

—

179

—

C₂×C₈

—

179

—

C₂×C₈

—

179

—

C₂×C₈

C₄×D₁₇₉

yes

—

C₂×C₈

C₂²⋊D₁₇₉

yes

—

C₂×C₈

Dic₃₅₈

yes

—

C₄²

—

179

—

C₄²

—

179

—

C₂³

Q₈×D₁₇₉

yes

characteristic

C₂×C₄

—

179

—

C₂×C₄

—

716

—

C₂×C₄

C₂×C₄×D₁₇₉

yes

characteristic

C₂×C₄

D₄×D₁₇₉

yes

characteristic

C₂×C₄

D₄×D₁₇₉

yes

characteristic

C₂×C₄

C₂×C₂²⋊D₁₇₉

yes

characteristic

C₂×C₄

Q₈×D₁₇₉

yes

characteristic

C₂×C₄

C₂×Dic₃₅₈

yes

characteristic

100

C₈

—

101

C₈

—

179

—

102

C₈

—

716

—

103

C₈

Dic₁₇₉⋊C₄

yes

—

104

C₂²

—

105

C₂²

C₄⋊C₄×D₁₇₉

yes

characteristic

106

C₂²

#19153111

yes

characteristic

107

C₂²

#19153112

yes

characteristic

108

C₄

—

358

—

109

C₄

C₄⋊C₄×D₁₇₉

yes

characteristic

110

C₄

C₂×Dic₁₇₉⋊C₄

yes

characteristic

111

C₄

#19153109

yes

characteristic

112

C₄

#19153112

yes

characteristic

113

C₂

—

114

C₂

—

115

C₂

#26038308

yes

characteristic

116

C₂

#26038486

yes

—

	subgroup	G/H
center	C₂×C₄	D₄×D₁₇₉
commutator	C₂×C₄×C₁₇₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₇₉
Fitting	C₂×#108×C₁₇₉	C₂
socle	C₂²×C₁₇₉	C₂×C₄⋊C₄
prime radical	C₂³×C₁₇₉	C₂×Q₈
cosocle	C₂²×C₄×C₁₇₉	C₂³
hypercenter	C₂×#108	D₁₇₉
hypocenter	C₁₇₉	#1849
LCS	G ▷ C₂×C₄×C₁₇₉ ▷ C₂×C₁₇₉ ▷ C₁₇₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×#108

subgroup

G/H

center

C₂×C₄

D₄×D₁₇₉

commutator

C₂×C₄×C₁₇₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₇₉

Fitting

C₂×#108×C₁₇₉

C₂

socle

C₂²×C₁₇₉

C₂×C₄⋊C₄

prime radical

C₂³×C₁₇₉

C₂×Q₈

cosocle

C₂²×C₄×C₁₇₉

C₂³

D₁₇₉

C₁₇₉

G ▷ C₂×C₄×C₁₇₉ ▷ C₂×C₁₇₉ ▷ C₁₇₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×#108

N	H ≅ G/N	type
58. C₁₇₉	#1849	semidirect

H ≅ G/N

type

58. C₁₇₉

#1849

semidirect

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

subgroup	index
C₂×#19153049	2
C₂×#19153048	2
#26038462	2
C₂×#19153023	2
C₂×#108×C₁₇₉	2
#1849	179

subgroup

index

C₂×#108×C₁₇₉

179

Subgroups — #37509173

Conjugacy classes of subgroups

Complements

Maximal subgroups