Group #37508589 — subgroups

Subgroups — #37508589

Order 22912.

224 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 224 conjugacy classes of subgroups — too many to draw. Only its 69 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	#26038844	C₂	1	1	yes	maximal, characteristic, complemented
2	#26038811	C₂	1	2	yes	maximal, complemented
3	#26038785	C₂	1	2	yes	maximal, complemented
4	C₈×C₁₇₉⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
5	#493×C₁₇₉	C₂	1	1	yes	maximal, characteristic
6	C₂×#15668247	—	2	4	no	—
7	#19153113	C₂²	1	2	yes	—
8	#19153047	—	2	2	no	—
9	#19153034	—	2	4	no	—
10	C₄×C₁₇₉⋊C₈	C₂²	1	1	yes	characteristic
11	C₄×C₁₇₉⋊C₈	C₂²	1	1	yes	characteristic
12	C₂×D₈×C₁₇₉	—	2	2	no	—
13	#127×C₁₇₉	C₂²	1	2	yes	—
14	C₄×C₈×C₁₇₉	C₂²	1	1	yes	characteristic
15	C₂×Dic₃₅₈	—	2	2	no	—
16	#15668247	—	4	8	no	—
17	C₂×C₁₇₉⋊C₈	D₄	1	2	yes	—
18	C₂×C₁₇₉⋊C₈	D₄	1	2	yes	—
19	C₄⋊Dic₁₇₉	—	2	2	no	—
20	C₄⋊Dic₁₇₉	—	2	2	no	—
21	C₂×D₄×C₁₇₉	—	2	2	no	—
22	C₂×D₄×C₁₇₉	—	2	2	no	—
23	C₂×D₄×C₁₇₉	—	2	2	no	—
24	D₈×C₁₇₉	—	4	4	no	—
25	C₂×C₈×C₁₇₉	D₄	1	2	yes	—
26	C₄²×C₁₇₉	C₂³	1	1	yes	characteristic
27	Dic₃₅₈	—	4	4	no	—
28	C₂×Dic₁₇₉	—	4	2	no	—
29	C₁₇₉⋊C₈	—	2	2	no	—
30	C₁₇₉⋊C₈	D₈	1	4	yes	complemented
31	C₂³×C₁₇₉	—	4	2	no	—
32	D₄×C₁₇₉	—	4	4	no	—
33	D₄×C₁₇₉	—	4	4	no	—
34	D₄×C₁₇₉	—	4	4	no	—
35	C₂×C₄×C₁₇₉	C₂×D₄	1	1	yes	characteristic
36	C₂×C₄×C₁₇₉	C₂×D₄	1	1	yes	characteristic
37	C₂×C₄×C₁₇₉	C₂×D₄	1	1	yes	characteristic
38	C₈×C₁₇₉	SD₁₆	1	4	yes	complemented
39	C₂²×C₁₇₉	—	8	2	no	—
40	C₂²×C₁₇₉	—	8	2	no	—
41	C₂²×C₁₇₉	—	8	2	no	—
42	C₂²×C₁₇₉	#127	1	1	yes	characteristic
43	C₄×C₁₇₉	C₂×D₈	1	2	yes	—
44	C₄×C₁₇₉	C₂×SD₁₆	1	2	yes	—
45	C₄×C₁₇₉	#135	1	2	yes	—
46	Dic₁₇₉	—	8	2	no	—
47	C₂×C₁₇₉	—	16	2	no	—
48	C₂×C₁₇₉	#492	1	1	yes	characteristic
49	C₂×C₁₇₉	#493	1	1	yes	characteristic
50	C₂×C₁₇₉	#495	1	1	yes	characteristic
51	C₁₇₉	#1706	1	1	yes	characteristic, complemented, Sylow
52	#1706	—	179	1	no	maximal, Sylow
53	#500	—	179	1	no	—
54	#493	D₁₇₉	1	1	yes	characteristic
55	#492	—	179	2	no	—
56	#486	—	179	2	no	—
57	C₈²	—	179	1	no	—
58	C₂×SD₁₆	—	358	4	no	—
59	C₂×D₈	—	2	2	no	—
60	C₄⋊Q₈	—	179	2	no	—
61	#127	C₂×D₁₇₉	1	2	yes	—
62	#107	—	358	2	no	—
63	#102	—	358	4	no	—
64	C₄×C₈	—	179	1	no	—
65	C₄×C₈	—	179	1	no	—
66	C₄×C₈	C₂×D₁₇₉	1	1	yes	characteristic
67	C₂×Q₈	—	358	2	no	—
68	C₂×D₄	—	2	2	no	—
69	C₂×D₄	—	2	2	no	—
70	C₂×D₄	—	2	2	no	—
71	SD₁₆	—	716	8	no	—
72	D₈	—	4	4	no	—
73	C₂×C₈	—	179	2	no	—
74	C₂×C₈	—	179	2	no	—
75	C₂×C₈	C₂²⋊D₁₇₉	1	2	yes	—
76	C₄⋊C₄	—	358	2	no	—
77	C₄⋊C₄	—	358	2	no	—
78	C₄²	C₂²×D₁₇₉	1	1	yes	characteristic
79	C₂³	—	4	2	no	—
80	Q₈	—	716	4	no	—
81	D₄	—	4	4	no	—
82	D₄	—	4	4	no	—
83	D₄	—	4	4	no	—
84	C₂×C₄	—	716	2	no	—
85	C₂×C₄	D₄×D₁₇₉	1	1	yes	characteristic
86	C₂×C₄	D₄×D₁₇₉	1	1	yes	characteristic
87	C₂×C₄	C₂×C₂²⋊D₁₇₉	1	1	yes	characteristic
88	C₈	—	358	2	no	—
89	C₈	—	179	4	no	—
90	C₈	#15668247	1	4	yes	complemented
91	C₂²	—	8	2	no	—
92	C₂²	—	8	2	no	—
93	C₂²	—	8	2	no	—
94	C₂²	#19153109	1	1	yes	characteristic
95	C₄	—	1432	2	no	—
96	C₄	D₈×D₁₇₉	1	2	yes	—
97	C₄	C₂×#15668247	1	2	yes	—
98	C₄	#19153132	1	2	yes	—
99	C₂	—	16	2	no	—
100	C₂	#26038811	1	1	yes	characteristic
101	C₂	#26038814	1	1	yes	characteristic
102	C₂	#26038822	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#26038844

C₂

yes

maximal, characteristic, complemented

#26038811

C₂

yes

maximal, complemented

#26038785

C₂

yes

maximal, complemented

C₈×C₁₇₉⋊C₈

C₂

yes

maximal, characteristic, complemented

#493×C₁₇₉

C₂

yes

maximal, characteristic

C₂×#15668247

—

#19153113

C₂²

yes

—

#19153047

—

#19153034

—

C₄×C₁₇₉⋊C₈

C₂²

yes

characteristic

C₄×C₁₇₉⋊C₈

C₂²

yes

characteristic

C₂×D₈×C₁₇₉

—

#127×C₁₇₉

C₂²

yes

—

C₄×C₈×C₁₇₉

C₂²

yes

characteristic

C₂×Dic₃₅₈

—

#15668247

—

C₂×C₁₇₉⋊C₈

D₄

yes

—

C₂×C₁₇₉⋊C₈

D₄

yes

—

C₄⋊Dic₁₇₉

—

C₄⋊Dic₁₇₉

—

C₂×D₄×C₁₇₉

—

C₂×D₄×C₁₇₉

—

C₂×D₄×C₁₇₉

—

D₈×C₁₇₉

—

C₂×C₈×C₁₇₉

D₄

yes

—

C₄²×C₁₇₉

C₂³

yes

characteristic

Dic₃₅₈

—

C₂×Dic₁₇₉

—

C₁₇₉⋊C₈

—

C₁₇₉⋊C₈

D₈

yes

complemented

C₂³×C₁₇₉

—

D₄×C₁₇₉

—

D₄×C₁₇₉

—

D₄×C₁₇₉

—

C₂×C₄×C₁₇₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₇₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₇₉

C₂×D₄

yes

characteristic

C₈×C₁₇₉

SD₁₆

yes

complemented

C₂²×C₁₇₉

—

C₂²×C₁₇₉

—

C₂²×C₁₇₉

—

C₂²×C₁₇₉

#127

yes

characteristic

C₄×C₁₇₉

C₂×D₈

yes

—

C₄×C₁₇₉

C₂×SD₁₆

yes

—

C₄×C₁₇₉

#135

yes

—

Dic₁₇₉

—

C₂×C₁₇₉

—

C₂×C₁₇₉

yes

C₂×C₁₇₉

yes

C₂×C₁₇₉

yes

C₁₇₉

yes

characteristic, complemented, Sylow

#1706

—

179

maximal, Sylow

#500

—

179

—

#493

D₁₇₉

yes

characteristic

#492

—

179

—

#486

—

179

—

C₈²

—

179

—

C₂×SD₁₆

—

358

—

C₂×D₈

—

C₄⋊Q₈

—

179

—

#127

C₂×D₁₇₉

yes

—

#107

—

358

—

#102

—

358

—

C₄×C₈

—

179

—

C₄×C₈

—

179

—

C₄×C₈

C₂×D₁₇₉

yes

characteristic

C₂×Q₈

—

358

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

SD₁₆

—

716

—

D₈

—

C₂×C₈

—

179

—

C₂×C₈

—

179

—

C₂×C₈

C₂²⋊D₁₇₉

yes

—

C₄⋊C₄

—

358

—

C₄⋊C₄

—

358

—

C₄²

C₂²×D₁₇₉

yes

characteristic

C₂³

—

Q₈

—

716

—

D₄

—

D₄

—

D₄

—

C₂×C₄

—

716

—

C₂×C₄

D₄×D₁₇₉

yes

characteristic

C₂×C₄

D₄×D₁₇₉

yes

characteristic

C₂×C₄

C₂×C₂²⋊D₁₇₉

yes

characteristic

C₈

—

358

—

C₈

—

179

—

C₈

#15668247

yes

complemented

C₂²

—

C₂²

—

C₂²

—

C₂²

#19153109

yes

characteristic

C₄

—

1432

—

C₄

D₈×D₁₇₉

yes

—

C₄

C₂×#15668247

yes

—

C₄

#19153132

yes

—

C₂

—

100

C₂

#26038811

yes

characteristic

101

C₂

#26038814

yes

characteristic

102

C₂

#26038822

yes

characteristic

	subgroup	G/H
center	C₂²	#19153109
commutator, cosocle	C₄²×C₁₇₉	C₂³
Frattini	C₄²	C₂²×D₁₇₉
Fitting, prime radical	#493×C₁₇₉	C₂
socle	C₂²×C₁₇₉	#127
hypercenter	#493	D₁₇₉
hypocenter	C₁₇₉	#1706
LCS	G ▷ C₄²×C₁₇₉ ▷ C₂²×C₁₇₉ ▷ C₁₇₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #493

subgroup

G/H

center

C₂²

#19153109

commutator, cosocle

C₄²×C₁₇₉

C₂³

Frattini

C₄²

C₂²×D₁₇₉

Fitting, prime radical

#493×C₁₇₉

C₂

C₂²×C₁₇₉

D₁₇₉

C₁₇₉

G ▷ C₄²×C₁₇₉ ▷ C₂²×C₁₇₉ ▷ C₁₇₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #493

N	H ≅ G/N	type
1. #26038844	C₂	semidirect
2. #26038811	C₂	semidirect
3. #26038785	C₂	semidirect
4. C₈×C₁₇₉⋊C₈	C₂	semidirect
30. C₁₇₉⋊C₈	D₈	semidirect
38. C₈×C₁₇₉	SD₁₆	semidirect
51. C₁₇₉	#1706	semidirect
90. C₈	#15668247	semidirect

H ≅ G/N

type

1. #26038844

C₂

semidirect

2. #26038811

C₂

semidirect

3. #26038785

C₂

semidirect

4. C₈×C₁₇₉⋊C₈

C₂

semidirect

30. C₁₇₉⋊C₈

D₈

semidirect

38. C₈×C₁₇₉

SD₁₆

semidirect

51. C₁₇₉

#1706

semidirect

90. C₈

#15668247

semidirect

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

subgroup	index
#26038844	2
#26038811	2
#26038785	2
C₈×C₁₇₉⋊C₈	2
#493×C₁₇₉	2
#1706	179

subgroup

index

#26038844

#26038811

#26038785

C₈×C₁₇₉⋊C₈

#493×C₁₇₉

#1706

179

Subgroups — #37508589

Conjugacy classes of subgroups

Complements

Maximal subgroups