Group #37507093 — subgroups

Subgroups — #37507093

Order 22912.

128 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 128 conjugacy classes of subgroups — too many to draw. Only its 73 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	C₂²×C₁₇₉⋊C₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂×#19153049	C₂	1	1	yes	maximal, characteristic
3	C₂×#108×C₁₇₉	C₂	1	1	yes	maximal, characteristic
4	C₂×#15668243	—	2	1	no	—
5	C₂×C₁₇₉⋊C₁₆	—	2	1	no	—
6	C₂×C₁₇₉⋊C₁₆	—	2	1	no	—
7	C₂×C₁₇₉⋊C₁₆	C₄	1	2	yes	—
8	#19153049	—	2	1	no	—
9	#19153049	C₄	1	2	yes	—
10	C₂×M₁₆×C₁₇₉	—	2	1	no	—
11	C₂²×C₈×C₁₇₉	C₂²	1	1	yes	characteristic
12	#108×C₁₇₉	—	2	1	no	—
13	#108×C₁₇₉	C₄	1	2	yes	—
14	#15668243	—	2	2	no	—
15	#15668243	—	4	1	no	—
16	C₂×C₁₇₉⋊C₈	—	4	1	no	—
17	C₁₇₉⋊C₁₆	—	2	2	no	—
18	C₂²×C₄×C₁₇₉	D₄	1	1	yes	characteristic
19	M₁₆×C₁₇₉	—	2	2	no	—
20	M₁₆×C₁₇₉	—	4	1	no	—
21	C₂×C₈×C₁₇₉	—	4	1	no	—
22	C₂×C₈×C₁₇₉	C₂×C₄	1	1	yes	characteristic
23	C₂×C₈×C₁₇₉	C₂×C₄	1	1	yes	characteristic
24	C₂×C₈×C₁₇₉	C₂×C₄	1	1	yes	characteristic
25	C₂×C₈×C₁₇₉	D₄	1	1	yes	characteristic
26	C₂×C₈×C₁₇₉	D₄	1	1	yes	characteristic
27	C₂×C₈×C₁₇₉	Q₈	1	1	yes	characteristic
28	C₁₇₉⋊C₈	—	4	2	no	—
29	C₂³×C₁₇₉	Q₁₆	1	1	yes	characteristic
30	C₂×C₄×C₁₇₉	C₂²⋊C₄	1	1	yes	characteristic
31	C₂×C₄×C₁₇₉	C₂²⋊C₄	1	1	yes	characteristic
32	C₂×C₄×C₁₇₉	C₄⋊C₄	1	1	yes	characteristic
33	C₂×C₄×C₁₇₉	D₈	1	1	yes	characteristic
34	C₂×C₄×C₁₇₉	SD₁₆	1	1	yes	characteristic
35	C₂×C₄×C₁₇₉	SD₁₆	1	1	yes	characteristic
36	C₈×C₁₇₉	—	4	2	no	—
37	C₈×C₁₇₉	C₄²	1	1	yes	characteristic
38	C₈×C₁₇₉	C₂²⋊C₄	1	1	yes	characteristic
39	C₈×C₁₇₉	C₄⋊C₄	1	1	yes	characteristic
40	C₈×C₁₇₉	C₄⋊C₄	1	1	yes	characteristic
41	C₂²×C₁₇₉	—	2	2	no	—
42	C₂²×C₁₇₉	#103	1	1	yes	characteristic
43	C₂²×C₁₇₉	#103	1	1	yes	characteristic
44	C₂²×C₁₇₉	#107	1	1	yes	characteristic
45	C₄×C₁₇₉	#95	1	1	yes	characteristic
46	C₄×C₁₇₉	#102	1	1	yes	characteristic
47	C₄×C₁₇₉	#102	1	1	yes	characteristic
48	C₄×C₁₇₉	#106	1	1	yes	characteristic
49	C₂×C₁₇₉	—	2	1	no	—
50	C₂×C₁₇₉	—	2	1	no	—
51	C₂×C₁₇₉	#340	1	1	yes	characteristic
52	C₂×C₁₇₉	#368	1	2	yes	—
53	C₁₇₉	#1381	1	1	yes	characteristic, complemented, Sylow
54	#1381	—	179	1	no	maximal, Sylow
55	C₂²×C₁₆	—	179	1	no	—
56	C₂×#108	—	179	1	no	—
57	C₂×#108	D₁₇₉	1	1	yes	characteristic
58	C₂×M₁₆	—	2	1	no	—
59	C₂×M₁₆	—	358	1	no	—
60	C₂²×C₈	C₂×D₁₇₉	1	1	yes	characteristic
61	C₂×C₁₆	—	358	1	no	—
62	C₂×C₁₆	—	358	1	no	—
63	C₂×C₁₆	—	179	2	no	—
64	#108	—	2	1	no	—
65	#108	—	358	1	no	—
66	#108	—	179	2	no	—
67	#108	Dic₁₇₉	1	2	yes	—
68	C₂²×C₄	C₂²⋊D₁₇₉	1	1	yes	characteristic
69	M₁₆	—	2	2	no	—
70	M₁₆	—	4	1	no	—
71	M₁₆	—	358	2	no	—
72	M₁₆	—	716	1	no	—
73	C₂×C₈	—	4	1	no	—
74	C₂×C₈	—	716	1	no	—
75	C₂×C₈	C₄×D₁₇₉	1	1	yes	characteristic
76	C₂×C₈	C₄×D₁₇₉	1	1	yes	characteristic
77	C₂×C₈	C₂×Dic₁₇₉	1	1	yes	characteristic
78	C₂×C₈	C₂²⋊D₁₇₉	1	1	yes	characteristic
79	C₂×C₈	D₇₁₆	1	1	yes	characteristic
80	C₂×C₈	Dic₃₅₈	1	1	yes	characteristic
81	C₁₆	—	358	2	no	—
82	C₂³	#15668249	1	1	yes	characteristic
83	C₂×C₄	(C₂×D₁₇₉)⋊C₄	1	1	yes	characteristic
84	C₂×C₄	C₂²⋊Dic₁₇₉	1	1	yes	characteristic
85	C₂×C₄	Dic₁₇₉⋊C₄	1	1	yes	characteristic
86	C₂×C₄	#15668244	1	1	yes	characteristic
87	C₂×C₄	#15668246	1	1	yes	characteristic
88	C₂×C₄	#15668247	1	1	yes	characteristic
89	C₈	—	4	2	no	—
90	C₈	—	716	2	no	—
91	C₈	C₄×Dic₁₇₉	1	1	yes	characteristic
92	C₈	(C₂×D₁₇₉)⋊C₄	1	1	yes	characteristic
93	C₈	Dic₁₇₉⋊C₄	1	1	yes	characteristic
94	C₈	C₄⋊Dic₁₇₉	1	1	yes	characteristic
95	C₂²	—	2	2	no	—
96	C₂²	#19153036	1	1	yes	characteristic
97	C₂²	#19153037	1	1	yes	characteristic
98	C₂²	#19153046	1	1	yes	characteristic
99	C₄	C₁₇₉⋊#95	1	1	yes	characteristic
100	C₄	#19153033	1	1	yes	characteristic
101	C₄	#19153034	1	1	yes	characteristic
102	C₄	#19153044	1	1	yes	characteristic
103	C₂	—	2	1	no	—
104	C₂	—	2	1	no	—
105	C₂	#26038201	1	1	yes	characteristic
106	C₂	#26038268	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×C₁₇₉⋊C₁₆

C₂

yes

maximal, characteristic

C₂×#19153049

C₂

yes

maximal, characteristic

C₂×#108×C₁₇₉

C₂

yes

maximal, characteristic

C₂×#15668243

—

C₂×C₁₇₉⋊C₁₆

—

C₂×C₁₇₉⋊C₁₆

—

C₂×C₁₇₉⋊C₁₆

C₄

yes

—

#19153049

—

#19153049

C₄

yes

—

C₂×M₁₆×C₁₇₉

—

C₂²×C₈×C₁₇₉

C₂²

yes

characteristic

#108×C₁₇₉

—

#108×C₁₇₉

C₄

yes

—

#15668243

—

#15668243

—

C₂×C₁₇₉⋊C₈

—

C₁₇₉⋊C₁₆

—

C₂²×C₄×C₁₇₉

D₄

yes

characteristic

M₁₆×C₁₇₉

—

M₁₆×C₁₇₉

—

C₂×C₈×C₁₇₉

—

C₂×C₈×C₁₇₉

C₂×C₄

yes

characteristic

C₂×C₈×C₁₇₉

C₂×C₄

yes

characteristic

C₂×C₈×C₁₇₉

C₂×C₄

yes

characteristic

C₂×C₈×C₁₇₉

D₄

yes

characteristic

C₂×C₈×C₁₇₉

D₄

yes

characteristic

C₂×C₈×C₁₇₉

Q₈

yes

characteristic

C₁₇₉⋊C₈

—

C₂³×C₁₇₉

Q₁₆

yes

characteristic

C₂×C₄×C₁₇₉

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₇₉

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₇₉

C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₇₉

D₈

yes

characteristic

C₂×C₄×C₁₇₉

SD₁₆

yes

characteristic

C₂×C₄×C₁₇₉

SD₁₆

yes

characteristic

C₈×C₁₇₉

—

C₈×C₁₇₉

C₄²

yes

characteristic

C₈×C₁₇₉

C₂²⋊C₄

yes

characteristic

C₈×C₁₇₉

C₄⋊C₄

yes

characteristic

C₈×C₁₇₉

C₄⋊C₄

yes

characteristic

C₂²×C₁₇₉

—

C₂²×C₁₇₉

yes

C₂²×C₁₇₉

yes

C₂²×C₁₇₉

yes

C₄×C₁₇₉

yes

C₄×C₁₇₉

yes

C₄×C₁₇₉

yes

C₄×C₁₇₉

yes

C₂×C₁₇₉

—

C₂×C₁₇₉

—

C₂×C₁₇₉

#340

yes

characteristic

C₂×C₁₇₉

#368

yes

—

C₁₇₉

#1381

yes

characteristic, complemented, Sylow

#1381

—

179

maximal, Sylow

C₂²×C₁₆

—

179

—

C₂×#108

—

179

—

C₂×#108

D₁₇₉

yes

characteristic

C₂×M₁₆

—

C₂×M₁₆

—

358

—

C₂²×C₈

C₂×D₁₇₉

yes

characteristic

C₂×C₁₆

—

358

—

C₂×C₁₆

—

358

—

C₂×C₁₆

—

179

—

#108

—

#108

—

358

—

#108

—

179

—

#108

Dic₁₇₉

yes

—

C₂²×C₄

C₂²⋊D₁₇₉

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

358

—

M₁₆

—

716

—

C₂×C₈

—

C₂×C₈

—

716

—

C₂×C₈

C₄×D₁₇₉

yes

characteristic

C₂×C₈

C₄×D₁₇₉

yes

characteristic

C₂×C₈

C₂×Dic₁₇₉

yes

characteristic

C₂×C₈

C₂²⋊D₁₇₉

yes

characteristic

C₂×C₈

D₇₁₆

yes

characteristic

C₂×C₈

Dic₃₅₈

yes

characteristic

C₁₆

—

358

—

C₂³

#15668249

yes

characteristic

C₂×C₄

(C₂×D₁₇₉)⋊C₄

yes

characteristic

C₂×C₄

C₂²⋊Dic₁₇₉

yes

characteristic

C₂×C₄

Dic₁₇₉⋊C₄

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

C₈

—

C₈

—

716

—

C₈

C₄×Dic₁₇₉

yes

characteristic

C₈

(C₂×D₁₇₉)⋊C₄

yes

characteristic

C₈

Dic₁₇₉⋊C₄

yes

characteristic

C₈

C₄⋊Dic₁₇₉

yes

characteristic

C₂²

—

C₂²

yes

C₂²

yes

C₂²

yes

C₄

C₁₇₉⋊#95

yes

characteristic

100

C₄

#19153033

yes

characteristic

101

C₄

#19153034

yes

characteristic

102

C₄

#19153044

yes

characteristic

103

C₂

—

104

C₂

—

105

C₂

#26038201

yes

characteristic

106

C₂

#26038268

yes

—

	subgroup	G/H
center	C₂×C₄	#15668244
commutator	C₈×C₁₇₉	C₄²
Frattini	C₂²×C₈	C₂×D₁₇₉
Fitting	C₂×#108×C₁₇₉	C₂
socle	C₂²×C₁₇₉	#107
prime radical	C₂³×C₁₇₉	Q₁₆
cosocle	C₂²×C₈×C₁₇₉	C₂²
hypercenter	C₂×#108	D₁₇₉
hypocenter	C₁₇₉	#1381
LCS	G ▷ C₈×C₁₇₉ ▷ C₄×C₁₇₉ ▷ C₂×C₁₇₉ ▷ C₁₇₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#108

subgroup

G/H

center

C₂×C₄

#15668244

commutator

C₈×C₁₇₉

C₄²

Frattini

C₂²×C₈

C₂×D₁₇₉

Fitting

C₂×#108×C₁₇₉

C₂

socle

C₂²×C₁₇₉

#107

prime radical

C₂³×C₁₇₉

Q₁₆

cosocle

C₂²×C₈×C₁₇₉

C₂²

D₁₇₉

C₁₇₉

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

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#108

N	H ≅ G/N	type
53. C₁₇₉	#1381	semidirect

H ≅ G/N

type

53. C₁₇₉

#1381

semidirect

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

subgroup	index
C₂²×C₁₇₉⋊C₁₆	2
C₂×#19153049	2
C₂×#108×C₁₇₉	2
#1381	179

subgroup

index

C₂²×C₁₇₉⋊C₁₆

C₂×#19153049

C₂×#108×C₁₇₉

#1381

179

Subgroups — #37507093

Conjugacy classes of subgroups

Complements

Maximal subgroups