Group #37508664 — subgroups

Subgroups — #37508664

Order 22912.

242 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 242 conjugacy classes of subgroups — too many to draw. Only its 65 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	#26038816	C₂	1	1	yes	maximal, characteristic, complemented
2	#26038815	C₂	1	2	yes	maximal, complemented
3	#26038787	C₂	1	2	yes	maximal, complemented
4	#26038160	C₂	1	1	yes	maximal, characteristic, complemented
5	#492×C₁₇₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₁₄₃₂	—	2	2	no	—
7	C₂×#15668244	—	2	4	no	—
8	#19153110	C₂²	1	2	yes	—
9	#19153033	—	2	4	no	—
10	C₄×C₁₇₉⋊C₈	C₂²	1	2	yes	—
11	C₂×SD₁₆×C₁₇₉	—	2	2	no	—
12	C₄⋊Q₈×C₁₇₉	C₂²	1	1	yes	characteristic
13	#127×C₁₇₉	C₂²	1	1	yes	characteristic
14	C₄×C₈×C₁₇₉	C₂²	1	1	yes	characteristic
15	C₂×D₇₁₆	—	2	2	no	—
16	C₂×D₇₁₆	—	2	2	no	—
17	C₂×D₇₁₆	—	2	2	no	—
18	D₁₄₃₂	—	4	4	no	—
19	#15668244	—	4	8	no	—
20	C₂×C₁₇₉⋊C₈	D₄	1	4	yes	—
21	C₂×Q₈×C₁₇₉	—	2	1	no	—
22	C₂×D₄×C₁₇₉	—	2	1	no	—
23	C₂×D₄×C₁₇₉	—	2	2	no	—
24	SD₁₆×C₁₇₉	—	4	4	no	—
25	C₂×C₈×C₁₇₉	D₄	1	2	yes	—
26	C₄⋊C₄×C₁₇₉	—	2	2	no	—
27	C₄²×C₁₇₉	C₂³	1	1	yes	characteristic
28	C₂²×D₁₇₉	—	4	2	no	—
29	D₇₁₆	—	4	4	no	—
30	D₇₁₆	—	4	4	no	—
31	D₇₁₆	—	4	4	no	—
32	C₁₇₉⋊C₈	—	2	4	no	—
33	C₂³×C₁₇₉	—	4	1	no	—
34	Q₈×C₁₇₉	—	4	2	no	—
35	D₄×C₁₇₉	—	4	2	no	—
36	D₄×C₁₇₉	—	4	4	no	—
37	C₂×C₄×C₁₇₉	—	4	1	no	—
38	C₂×C₄×C₁₇₉	C₂×D₄	1	2	yes	—
39	C₂×C₄×C₁₇₉	C₂×D₄	1	1	yes	characteristic
40	C₈×C₁₇₉	D₈	1	4	yes	complemented
41	C₂²×C₁₇₉	—	8	1	no	—
42	C₂²×C₁₇₉	—	8	2	no	—
43	C₂²×C₁₇₉	#127	1	1	yes	characteristic
44	C₂×D₁₇₉	—	8	2	no	—
45	C₂×D₁₇₉	—	8	2	no	—
46	C₂×D₁₇₉	—	8	2	no	—
47	C₄×C₁₇₉	—	8	1	no	—
48	C₄×C₁₇₉	C₂×D₈	1	2	yes	—
49	C₄×C₁₇₉	AGL(1,ℤ/8)	1	4	yes	—
50	C₂×C₁₇₉	—	16	1	no	—
51	C₂×C₁₇₉	#493	1	1	yes	characteristic
52	C₂×C₁₇₉	#496	1	2	yes	—
53	D₁₇₉	—	16	2	no	—
54	C₁₇₉	#1721	1	1	yes	characteristic, complemented, Sylow
55	#1721	—	179	1	no	maximal, Sylow
56	#493	—	179	2	no	—
57	#493	—	179	1	no	—
58	#492	D₁₇₉	1	1	yes	characteristic, complemented
59	#486	—	179	2	no	—
60	#322	—	179	1	no	—
61	C₂×SD₁₆	—	2	2	no	—
62	C₂×D₈	—	358	4	no	—
63	C₂×D₈	—	358	2	no	—
64	C₄⋊Q₈	C₂×D₁₇₉	1	1	yes	characteristic
65	#127	—	179	2	no	—
66	#127	C₂×D₁₇₉	1	1	yes	characteristic
67	#102	—	358	4	no	—
68	C₄×C₈	—	179	2	no	—
69	C₄×C₈	C₂×D₁₇₉	1	1	yes	characteristic
70	C₂×Q₈	—	2	1	no	—
71	C₂×D₄	—	2	2	no	—
72	C₂×D₄	—	2	1	no	—
73	C₂×D₄	—	358	2	no	—
74	C₂×D₄	—	358	2	no	—
75	C₂×D₄	—	358	2	no	—
76	SD₁₆	—	4	4	no	—
77	D₈	—	716	8	no	—
78	D₈	—	716	4	no	—
79	C₂×C₈	—	179	4	no	—
80	C₂×C₈	C₂²⋊D₁₇₉	1	2	yes	—
81	C₄⋊C₄	—	2	2	no	—
82	C₄²	C₂²×D₁₇₉	1	1	yes	characteristic
83	C₂³	—	4	1	no	—
84	C₂³	—	716	2	no	—
85	Q₈	—	4	2	no	—
86	D₄	—	4	2	no	—
87	D₄	—	4	4	no	—
88	D₄	—	716	4	no	—
89	D₄	—	716	4	no	—
90	D₄	—	716	4	no	—
91	C₂×C₄	—	4	1	no	—
92	C₂×C₄	D₄×D₁₇₉	1	2	yes	—
93	C₂×C₄	C₂×C₂²⋊D₁₇₉	1	1	yes	characteristic
94	C₈	—	358	4	no	—
95	C₈	#15668244	1	4	yes	complemented
96	C₂²	—	8	1	no	—
97	C₂²	—	8	2	no	—
98	C₂²	—	1432	2	no	—
99	C₂²	—	1432	2	no	—
100	C₂²	—	1432	2	no	—
101	C₂²	#19153109	1	1	yes	characteristic
102	C₄	—	8	1	no	—
103	C₄	C₂×#15668244	1	2	yes	—
104	C₄	#19153140	1	4	yes	—
105	C₂	—	16	1	no	—
106	C₂	—	2864	2	no	—
107	C₂	#26038815	1	1	yes	characteristic
108	C₂	#26038825	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#26038816

C₂

yes

maximal, characteristic, complemented

#26038815

C₂

yes

maximal, complemented

#26038787

C₂

yes

maximal, complemented

#26038160

C₂

yes

maximal, characteristic, complemented

#492×C₁₇₉

C₂

yes

maximal, characteristic, complemented

C₂×D₁₄₃₂

—

C₂×#15668244

—

#19153110

C₂²

yes

—

#19153033

—

C₄×C₁₇₉⋊C₈

C₂²

yes

—

C₂×SD₁₆×C₁₇₉

—

C₄⋊Q₈×C₁₇₉

C₂²

yes

characteristic

#127×C₁₇₉

C₂²

yes

characteristic

C₄×C₈×C₁₇₉

C₂²

yes

characteristic

C₂×D₇₁₆

—

C₂×D₇₁₆

—

C₂×D₇₁₆

—

D₁₄₃₂

—

#15668244

—

C₂×C₁₇₉⋊C₈

D₄

yes

—

C₂×Q₈×C₁₇₉

—

C₂×D₄×C₁₇₉

—

C₂×D₄×C₁₇₉

—

SD₁₆×C₁₇₉

—

C₂×C₈×C₁₇₉

D₄

yes

—

C₄⋊C₄×C₁₇₉

—

C₄²×C₁₇₉

C₂³

yes

characteristic

C₂²×D₁₇₉

—

D₇₁₆

—

D₇₁₆

—

D₇₁₆

—

C₁₇₉⋊C₈

—

C₂³×C₁₇₉

—

Q₈×C₁₇₉

—

D₄×C₁₇₉

—

D₄×C₁₇₉

—

C₂×C₄×C₁₇₉

—

C₂×C₄×C₁₇₉

C₂×D₄

yes

—

C₂×C₄×C₁₇₉

C₂×D₄

yes

characteristic

C₈×C₁₇₉

D₈

yes

complemented

C₂²×C₁₇₉

—

C₂²×C₁₇₉

—

C₂²×C₁₇₉

#127

yes

characteristic

C₂×D₁₇₉

—

C₂×D₁₇₉

—

C₂×D₁₇₉

—

C₄×C₁₇₉

—

C₄×C₁₇₉

C₂×D₈

yes

—

C₄×C₁₇₉

AGL(1,ℤ/8)

yes

—

C₂×C₁₇₉

—

C₂×C₁₇₉

#493

yes

characteristic

C₂×C₁₇₉

#496

yes

—

D₁₇₉

—

C₁₇₉

#1721

yes

characteristic, complemented, Sylow

#1721

—

179

maximal, Sylow

#493

—

179

—

#493

—

179

—

#492

D₁₇₉

yes

characteristic, complemented

#486

—

179

—

#322

—

179

—

C₂×SD₁₆

—

C₂×D₈

—

358

—

C₂×D₈

—

358

—

C₄⋊Q₈

C₂×D₁₇₉

yes

characteristic

#127

—

179

—

#127

C₂×D₁₇₉

yes

characteristic

#102

—

358

—

C₄×C₈

—

179

—

C₄×C₈

C₂×D₁₇₉

yes

characteristic

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

358

—

C₂×D₄

—

358

—

C₂×D₄

—

358

—

SD₁₆

—

D₈

—

716

—

D₈

—

716

—

C₂×C₈

—

179

—

C₂×C₈

C₂²⋊D₁₇₉

yes

—

C₄⋊C₄

—

C₄²

C₂²×D₁₇₉

yes

characteristic

C₂³

—

C₂³

—

716

—

Q₈

—

D₄

—

D₄

—

D₄

—

716

—

D₄

—

716

—

D₄

—

716

—

C₂×C₄

—

C₂×C₄

D₄×D₁₇₉

yes

—

C₂×C₄

C₂×C₂²⋊D₁₇₉

yes

characteristic

C₈

—

358

—

C₈

#15668244

yes

complemented

C₂²

—

C₂²

—

C₂²

—

1432

—

C₂²

—

1432

—

100

C₂²

—

1432

—

101

C₂²

#19153109

yes

characteristic

102

C₄

—

103

C₄

C₂×#15668244

yes

—

104

C₄

#19153140

yes

—

105

C₂

—

106

C₂

—

2864

—

107

C₂

#26038815

yes

characteristic

108

C₂

#26038825

yes

—

	subgroup	G/H
center	C₂²	#19153109
commutator, cosocle	C₄²×C₁₇₉	C₂³
Frattini	C₄²	C₂²×D₁₇₉
Fitting	#492×C₁₇₉	C₂
socle	C₂²×C₁₇₉	#127
hypercenter	#492	D₁₇₉
hypocenter	C₁₇₉	#1721
LCS	G ▷ C₄²×C₁₇₉ ▷ C₂²×C₁₇₉ ▷ C₁₇₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #492

subgroup

G/H

center

C₂²

#19153109

commutator, cosocle

C₄²×C₁₇₉

C₂³

Frattini

C₄²

C₂²×D₁₇₉

Fitting

#492×C₁₇₉

C₂

C₂²×C₁₇₉

D₁₇₉

C₁₇₉

G ▷ C₄²×C₁₇₉ ▷ C₂²×C₁₇₉ ▷ C₁₇₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #492

N	H ≅ G/N	type
1. #26038816	C₂	semidirect
2. #26038815	C₂	semidirect
3. #26038787	C₂	semidirect
4. #26038160	C₂	semidirect
5. #492×C₁₇₉	C₂	semidirect
40. C₈×C₁₇₉	D₈	semidirect
54. C₁₇₉	#1721	semidirect
58. #492	D₁₇₉	semidirect
95. C₈	#15668244	semidirect

H ≅ G/N

type

1. #26038816

C₂

semidirect

2. #26038815

C₂

semidirect

3. #26038787

C₂

semidirect

4. #26038160

C₂

semidirect

5. #492×C₁₇₉

C₂

semidirect

40. C₈×C₁₇₉

D₈

semidirect

54. C₁₇₉

#1721

semidirect

58. #492

D₁₇₉

semidirect

95. C₈

#15668244

semidirect

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

subgroup	index
#26038816	2
#26038815	2
#26038787	2
#26038160	2
#492×C₁₇₉	2
#1721	179

subgroup

index

#492×C₁₇₉

179

Subgroups — #37508664

Conjugacy classes of subgroups

Complements

Maximal subgroups