Group #37527014 — subgroups

Subgroups — #37527014

Order 22976.

270 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 270 conjugacy classes of subgroups — too many to draw. Only its 111 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×Dic₇₁₈	C₂	1	1	yes	maximal, characteristic, complemented
2	#26040013	C₂	1	2	yes	maximal, complemented
3	#26040002	C₂	1	2	yes	maximal, complemented
4	#26039996	C₂	1	4	yes	maximal, complemented
5	#26039994	C₂	1	4	yes	maximal, complemented
6	#26039974	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#41×C₃₅₉	C₂	1	1	yes	maximal, characteristic
8	C₂×Dic₇₁₈	—	2	4	no	—
9	C₂×Dic₇₁₈	C₂²	1	2	yes	—
10	C₂×Dic₇₁₈	C₂²	1	2	yes	—
11	C₂²×Dic₃₅₉	C₂²	1	2	yes	—
12	C₄⋊Dic₃₅₉	C₂²	1	2	yes	—
13	Dic₃₅₉⋊C₄	C₂²	1	8	yes	—
14	C₂²⋊Dic₃₅₉	C₂²	1	2	yes	—
15	C₂²⋊Dic₃₅₉	C₂²	1	8	yes	—
16	C₄×Dic₃₅₉	C₂²	1	2	yes	—
17	#41×C₃₅₉	—	2	4	no	—
18	C₂×Q₈×C₃₅₉	C₂²	1	1	yes	characteristic
19	C₂×D₄×C₃₅₉	C₂²	1	2	yes	—
20	C₂×D₄×C₃₅₉	C₂²	1	1	yes	characteristic
21	C₂²×C₄×C₃₅₉	C₂²	1	2	yes	—
22	C₂²×C₄×C₃₅₉	C₂²	1	1	yes	characteristic
23	Dic₇₁₈	—	2	4	no	—
24	Dic₇₁₈	—	2	4	no	—
25	C₂×Dic₃₅₉	—	2	4	no	—
26	C₂×Dic₃₅₉	C₂³	1	4	yes	—
27	C₂×Dic₃₅₉	C₂³	1	4	yes	—
28	C₂³×C₃₅₉	C₂³	1	2	yes	—
29	C₂³×C₃₅₉	C₂³	1	1	yes	characteristic
30	Q₈×C₃₅₉	—	2	2	no	—
31	D₄×C₃₅₉	—	2	4	no	—
32	D₄×C₃₅₉	—	2	2	no	—
33	C₂×C₄×C₃₅₉	—	2	4	no	—
34	C₂×C₄×C₃₅₉	D₄	1	4	yes	—
35	C₂×C₄×C₃₅₉	C₂³	1	2	yes	—
36	C₂×C₄×C₃₅₉	C₂³	1	1	yes	characteristic
37	C₂×C₄×C₃₅₉	C₂³	1	1	yes	characteristic
38	C₂²×C₃₅₉	—	2	2	no	—
39	C₂²×C₃₅₉	—	2	2	no	—
40	C₂²×C₃₅₉	—	2	4	no	—
41	C₂²×C₃₅₉	C₂×D₄	1	2	yes	—
42	C₂²×C₃₅₉	C₂⁴	1	1	yes	characteristic
43	C₄×C₃₅₉	—	2	2	no	—
44	C₄×C₃₅₉	C₂×D₄	1	2	yes	—
45	C₄×C₃₅₉	C₂×D₄	1	2	yes	—
46	Dic₃₅₉	—	2	4	no	—
47	Dic₃₅₉	—	2	4	no	—
48	C₂×C₃₅₉	—	2	2	no	—
49	C₂×C₃₅₉	—	4	2	no	—
50	C₂×C₃₅₉	C₂²×D₄	1	1	yes	characteristic
51	C₂×C₃₅₉	2₋¹⁺⁴	1	2	yes	—
52	C₃₅₉	#536	1	1	yes	characteristic, complemented, Sylow
53	#536	—	359	1	no	maximal, Sylow
54	C₂×#41	D₃₅₉	1	1	yes	characteristic
55	C₂²×Q₈	—	359	1	no	—
56	C₄⋊Q₈	—	359	2	no	—
57	#124	—	359	2	no	—
58	#123	—	359	4	no	—
59	C₂²⋊Q₈	—	359	4	no	—
60	#117	—	359	1	no	—
61	#41	—	2	4	no	—
62	C₂×Q₈	—	359	2	no	—
63	C₂×Q₈	—	359	2	no	—
64	C₂×Q₈	—	718	4	no	—
65	C₂×Q₈	C₂×D₃₅₉	1	1	yes	characteristic
66	C₂×D₄	C₂×D₃₅₉	1	2	yes	—
67	C₂×D₄	C₂×D₃₅₉	1	1	yes	characteristic
68	C₂²×C₄	—	359	2	no	—
69	C₂²×C₄	C₂×D₃₅₉	1	2	yes	—
70	C₂²×C₄	C₂×D₃₅₉	1	1	yes	characteristic
71	C₄⋊C₄	—	359	2	no	—
72	C₄⋊C₄	—	359	8	no	—
73	C₂²⋊C₄	—	359	2	no	—
74	C₂²⋊C₄	—	359	8	no	—
75	C₄²	—	359	2	no	—
76	C₂³	C₂²×D₃₅₉	1	2	yes	—
77	C₂³	C₂²×D₃₅₉	1	1	yes	characteristic
78	Q₈	—	2	2	no	—
79	Q₈	—	718	4	no	—
80	Q₈	—	718	4	no	—
81	D₄	—	2	4	no	—
82	D₄	—	2	2	no	—
83	C₂×C₄	—	2	4	no	—
84	C₂×C₄	—	359	4	no	—
85	C₂×C₄	—	359	4	no	—
86	C₂×C₄	—	718	4	no	—
87	C₂×C₄	C₂²⋊D₃₅₉	1	4	yes	—
88	C₂×C₄	C₂²×D₃₅₉	1	2	yes	—
89	C₂×C₄	C₂²×D₃₅₉	1	1	yes	characteristic
90	C₂×C₄	C₂²×D₃₅₉	1	1	yes	characteristic
91	C₂²	—	2	2	no	—
92	C₂²	—	2	2	no	—
93	C₂²	—	2	4	no	—
94	C₂²	C₂×C₂²⋊D₃₅₉	1	2	yes	—
95	C₂²	C₂³×D₃₅₉	1	1	yes	characteristic
96	C₄	—	2	2	no	—
97	C₄	—	718	4	no	—
98	C₄	—	718	4	no	—
99	C₄	C₂×C₂²⋊D₃₅₉	1	2	yes	—
100	C₄	C₂×C₂²⋊D₃₅₉	1	2	yes	—
101	C₂	—	2	2	no	—
102	C₂	—	4	2	no	—
103	C₂	C₂²×C₂²⋊D₃₅₉	1	1	yes	characteristic
104	C₂	#26040062	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×Dic₇₁₈

C₂

yes

maximal, characteristic, complemented

#26040013

C₂

yes

maximal, complemented

#26040002

C₂

yes

maximal, complemented

#26039996

C₂

yes

maximal, complemented

#26039994

C₂

yes

maximal, complemented

#26039974

C₂

yes

maximal, characteristic, complemented

C₂×#41×C₃₅₉

C₂

yes

maximal, characteristic

C₂×Dic₇₁₈

—

C₂×Dic₇₁₈

C₂²

yes

—

C₂×Dic₇₁₈

C₂²

yes

—

C₂²×Dic₃₅₉

C₂²

yes

—

C₄⋊Dic₃₅₉

C₂²

yes

—

Dic₃₅₉⋊C₄

C₂²

yes

—

C₂²⋊Dic₃₅₉

C₂²

yes

—

C₂²⋊Dic₃₅₉

C₂²

yes

—

C₄×Dic₃₅₉

C₂²

yes

—

#41×C₃₅₉

—

C₂×Q₈×C₃₅₉

C₂²

yes

characteristic

C₂×D₄×C₃₅₉

C₂²

yes

—

C₂×D₄×C₃₅₉

C₂²

yes

characteristic

C₂²×C₄×C₃₅₉

C₂²

yes

—

C₂²×C₄×C₃₅₉

C₂²

yes

characteristic

Dic₇₁₈

—

Dic₇₁₈

—

C₂×Dic₃₅₉

—

C₂×Dic₃₅₉

C₂³

yes

—

C₂×Dic₃₅₉

C₂³

yes

—

C₂³×C₃₅₉

C₂³

yes

—

C₂³×C₃₅₉

C₂³

yes

characteristic

Q₈×C₃₅₉

—

D₄×C₃₅₉

—

D₄×C₃₅₉

—

C₂×C₄×C₃₅₉

—

C₂×C₄×C₃₅₉

D₄

yes

—

C₂×C₄×C₃₅₉

C₂³

yes

—

C₂×C₄×C₃₅₉

C₂³

yes

characteristic

C₂×C₄×C₃₅₉

C₂³

yes

characteristic

C₂²×C₃₅₉

—

C₂²×C₃₅₉

—

C₂²×C₃₅₉

—

C₂²×C₃₅₉

C₂×D₄

yes

—

C₂²×C₃₅₉

C₂⁴

yes

characteristic

C₄×C₃₅₉

—

C₄×C₃₅₉

C₂×D₄

yes

—

C₄×C₃₅₉

C₂×D₄

yes

—

Dic₃₅₉

—

Dic₃₅₉

—

C₂×C₃₅₉

—

C₂×C₃₅₉

—

C₂×C₃₅₉

C₂²×D₄

yes

characteristic

C₂×C₃₅₉

2₋¹⁺⁴

yes

—

C₃₅₉

#536

yes

characteristic, complemented, Sylow

#536

—

359

maximal, Sylow

C₂×#41

D₃₅₉

yes

characteristic

C₂²×Q₈

—

359

—

C₄⋊Q₈

—

359

—

#124

—

359

—

#123

—

359

—

C₂²⋊Q₈

—

359

—

#117

—

359

—

#41

—

C₂×Q₈

—

359

—

C₂×Q₈

—

359

—

C₂×Q₈

—

718

—

C₂×Q₈

C₂×D₃₅₉

yes

characteristic

C₂×D₄

C₂×D₃₅₉

yes

—

C₂×D₄

C₂×D₃₅₉

yes

characteristic

C₂²×C₄

—

359

—

C₂²×C₄

C₂×D₃₅₉

yes

—

C₂²×C₄

C₂×D₃₅₉

yes

characteristic

C₄⋊C₄

—

359

—

C₄⋊C₄

—

359

—

C₂²⋊C₄

—

359

—

C₂²⋊C₄

—

359

—

C₄²

—

359

—

C₂³

C₂²×D₃₅₉

yes

—

C₂³

C₂²×D₃₅₉

yes

characteristic

Q₈

—

Q₈

—

718

—

Q₈

—

718

—

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

—

359

—

C₂×C₄

—

359

—

C₂×C₄

—

718

—

C₂×C₄

C₂²⋊D₃₅₉

yes

—

C₂×C₄

C₂²×D₃₅₉

yes

—

C₂×C₄

C₂²×D₃₅₉

yes

characteristic

C₂×C₄

C₂²×D₃₅₉

yes

characteristic

C₂²

—

C₂²

—

C₂²

—

C₂²

C₂×C₂²⋊D₃₅₉

yes

—

C₂²

C₂³×D₃₅₉

yes

characteristic

C₄

—

C₄

—

718

—

C₄

—

718

—

C₄

C₂×C₂²⋊D₃₅₉

yes

—

100

C₄

C₂×C₂²⋊D₃₅₉

yes

—

101

C₂

—

102

C₂

—

103

C₂

C₂²×C₂²⋊D₃₅₉

yes

characteristic

104

C₂

#26040062

yes

—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₃₅₉
commutator, socle, cosocle	C₂²×C₃₅₉	C₂⁴
Fitting, prime radical	C₂×#41×C₃₅₉	C₂
hypercenter	C₂×#41	D₃₅₉
hypocenter	C₃₅₉	#536
LCS	G ▷ C₂²×C₃₅₉ ▷ C₃₅₉
UCS	C₁ ◁ C₂² ◁ C₂×#41

subgroup

G/H

center, Frattini

C₂²

C₂³×D₃₅₉

commutator, socle, cosocle

C₂²×C₃₅₉

C₂⁴

Fitting, prime radical

C₂×#41×C₃₅₉

C₂

hypercenter

C₂×#41

D₃₅₉

C₃₅₉

G ▷ C₂²×C₃₅₉ ▷ C₃₅₉

C₁ ◁ C₂² ◁ C₂×#41

N	H ≅ G/N	type
1. C₂²×Dic₇₁₈	C₂	semidirect
2. #26040013	C₂	semidirect
3. #26040002	C₂	semidirect
4. #26039996	C₂	semidirect
5. #26039994	C₂	semidirect
6. #26039974	C₂	semidirect
52. C₃₅₉	#536	semidirect

H ≅ G/N

type

1. C₂²×Dic₇₁₈

C₂

semidirect

2. #26040013

C₂

semidirect

3. #26040002

C₂

semidirect

4. #26039996

C₂

semidirect

5. #26039994

C₂

semidirect

6. #26039974

C₂

semidirect

52. C₃₅₉

#536

semidirect

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

subgroup	index
C₂²×Dic₇₁₈	2
#26040013	2
#26040002	2
#26039996	2
#26039994	2
#26039974	2
C₂×#41×C₃₅₉	2
#536	359

subgroup

index

C₂²×Dic₇₁₈

C₂×#41×C₃₅₉

359

Subgroups — #37527014

Conjugacy classes of subgroups

Complements

Maximal subgroups