Group #18008223 — subgroups

Subgroups — #18008223

Order 4736.

1202 conjugacy classes of subgroups; only the 491 normal subgroups are shown.

normal subgroups only

This group has 1202 conjugacy classes of subgroups — too many to draw. Only its 491 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	#14558249	C₂	1	9	yes	maximal, complemented
2	#14558205	C₂	1	6	yes	maximal, complemented
3	#14558084	C₂	1	9	yes	maximal, complemented
4	C₄×#10596966	C₂	1	6	yes	maximal, complemented
5	Q₈²×C₃₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#10596966	C₂²	1	6	yes	—
7	C₂×#10596966	C₂²	1	6	yes	—
8	#11924533	C₂²	1	9	yes	—
9	#11924525	C₂²	1	18	yes	—
10	#11924511	C₂²	1	36	yes	—
11	Q₈×Dic₃₇	C₂²	1	2	yes	—
12	C₄×D₁₄₈	C₂²	1	18	yes	—
13	#11924498	C₂²	1	18	yes	—
14	#11924494	C₂²	1	18	yes	—
15	C₄²×D₃₇	C₂²	1	9	yes	—
16	C₄⋊Q₈×C₃₇	C₂²	1	9	yes	—
17	C₄×Q₈×C₃₇	C₂²	1	6	yes	—
18	C₂×Q₈×C₃₇	C₂³	1	6	yes	—
19	C₂×Q₈×C₃₇	C₂³	1	2	yes	—
20	#10596966	—	2	24	no	—
21	#10596966	—	4	12	no	—
22	C₂×D₁₄₈	C₂³	1	18	yes	—
23	C₂×D₁₄₈	C₂³	1	18	yes	—
24	C₂×C₄×D₃₇	C₂³	1	18	yes	—
25	C₂×C₄×D₃₇	C₂³	1	9	yes	—
26	C₄⋊C₄×C₃₇	C₂³	1	18	yes	—
27	C₄²×C₃₇	C₂³	1	9	yes	—
28	#10596938	C₂³	1	36	yes	—
29	C₄⋊Dic₃₇	C₂³	1	6	yes	—
30	C₄×Dic₃₇	C₂³	1	6	yes	—
31	C₄×Dic₃₇	C₂³	1	9	yes	—
32	C₂²×D₃₇	C₂⁴	1	9	yes	—
33	Q₈×C₃₇	—	4	6	no	—
34	Q₈×C₃₇	#41	1	8	yes	complemented
35	C₂×C₄×C₃₇	C₂⁴	1	9	yes	—
36	C₂×C₄×C₃₇	C₂⁴	1	6	yes	—
37	C₂×Dic₃₇	C₂⁴	1	6	yes	—
38	C₂×Dic₃₇	C₂⁴	1	1	yes	characteristic
39	D₁₄₈	—	2	36	no	—
40	D₁₄₈	—	4	18	no	—
41	C₄×D₃₇	—	2	36	no	—
42	C₄×D₃₇	—	2	18	no	—
43	C₂²×C₃₇	C₂⁵	1	1	yes	characteristic
44	C₂×D₃₇	—	2	18	no	—
45	C₂×D₃₇	—	2	9	no	—
46	C₄×C₃₇	—	2	9	no	—
47	C₄×C₃₇	C₂×#41	1	12	yes	—
48	Dic₃₇	—	2	6	no	—
49	Dic₃₇	H(5,𝔽₂)	1	2	yes	—
50	#3483	—	37	1	no	maximal, Sylow
51	C₂×C₃₇	C₂²×#41	1	2	yes	—
52	C₂×C₃₇	C₂×H(5,𝔽₂)	1	1	yes	characteristic
53	D₃₇	—	4	9	no	—
54	Q₈²	D₃₇	1	1	yes	characteristic, complemented
55	#555	—	37	9	no	—
56	#550	—	37	6	no	—
57	#532	—	37	9	no	—
58	C₄×#41	—	37	6	no	—
59	C₃₇	#3483	1	1	yes	characteristic, complemented, Sylow
60	C₂×#41	—	37	6	no	—
61	C₂×#41	—	37	6	no	—
62	C₄⋊Q₈	C₂×D₃₇	1	9	yes	—
63	#127	—	37	9	no	—
64	#124	—	37	18	no	—
65	#121	—	37	36	no	—
66	C₄×Q₈	—	37	2	no	—
67	C₄×Q₈	C₂×D₃₇	1	6	yes	—
68	C₄×D₄	—	37	18	no	—
69	C₄×D₄	—	37	18	no	—
70	#117	—	37	18	no	—
71	C₂×C₄²	—	37	9	no	—
72	#41	—	74	24	no	—
73	#41	—	148	12	no	—
74	C₂×Q₈	C₂²×D₃₇	1	6	yes	—
75	C₂×Q₈	C₂²×D₃₇	1	2	yes	—
76	C₂×D₄	—	37	18	no	—
77	C₂×D₄	—	37	18	no	—
78	C₂²×C₄	—	37	18	no	—
79	C₂²×C₄	—	37	9	no	—
80	C₄⋊C₄	—	37	6	no	—
81	C₄⋊C₄	C₂²×D₃₇	1	18	yes	—
82	C₂²⋊C₄	—	37	36	no	—
83	C₄²	—	37	6	no	—
84	C₄²	—	37	9	no	—
85	C₄²	C₂²×D₃₇	1	9	yes	—
86	C₂³	—	37	9	no	—
87	Q₈	—	4	6	no	—
88	Q₈	#10596966	1	8	yes	complemented
89	D₄	—	74	36	no	—
90	D₄	—	148	18	no	—
91	C₂×C₄	—	37	1	no	—
92	C₂×C₄	—	37	6	no	—
93	C₂×C₄	—	74	18	no	—
94	C₂×C₄	—	74	36	no	—
95	C₂×C₄	C₂³×D₃₇	1	9	yes	—
96	C₂×C₄	C₂³×D₃₇	1	6	yes	—
97	C₂²	—	74	18	no	—
98	C₂²	—	74	9	no	—
99	C₂²	C₂⁴×D₃₇	1	1	yes	characteristic
100	C₄	—	2	9	no	—
101	C₄	—	37	2	no	—
102	C₄	—	74	6	no	—
103	C₄	C₂×#10596966	1	12	yes	—
104	C₂	—	148	9	no	—
105	C₂	C₂²×#10596966	1	2	yes	—
106	C₂	H(5,𝔽₂)×D₃₇	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14558249

C₂

yes

maximal, complemented

#14558205

C₂

yes

maximal, complemented

#14558084

C₂

yes

maximal, complemented

C₄×#10596966

C₂

yes

maximal, complemented

Q₈²×C₃₇

C₂

yes

maximal, characteristic, complemented

C₂×#10596966

C₂²

yes

—

C₂×#10596966

C₂²

yes

—

#11924533

C₂²

yes

—

#11924525

C₂²

yes

—

#11924511

C₂²

yes

—

Q₈×Dic₃₇

C₂²

yes

—

C₄×D₁₄₈

C₂²

yes

—

#11924498

C₂²

yes

—

#11924494

C₂²

yes

—

C₄²×D₃₇

C₂²

yes

—

C₄⋊Q₈×C₃₇

C₂²

yes

—

C₄×Q₈×C₃₇

C₂²

yes

—

C₂×Q₈×C₃₇

C₂³

yes

—

C₂×Q₈×C₃₇

C₂³

yes

—

#10596966

—

#10596966

—

C₂×D₁₄₈

C₂³

yes

—

C₂×D₁₄₈

C₂³

yes

—

C₂×C₄×D₃₇

C₂³

yes

—

C₂×C₄×D₃₇

C₂³

yes

—

C₄⋊C₄×C₃₇

C₂³

yes

—

C₄²×C₃₇

C₂³

yes

—

#10596938

C₂³

yes

—

C₄⋊Dic₃₇

C₂³

yes

—

C₄×Dic₃₇

C₂³

yes

—

C₄×Dic₃₇

C₂³

yes

—

C₂²×D₃₇

C₂⁴

yes

—

Q₈×C₃₇

—

Q₈×C₃₇

#41

yes

complemented

C₂×C₄×C₃₇

C₂⁴

yes

—

C₂×C₄×C₃₇

C₂⁴

yes

—

C₂×Dic₃₇

C₂⁴

yes

—

C₂×Dic₃₇

C₂⁴

yes

characteristic

D₁₄₈

—

D₁₄₈

—

C₄×D₃₇

—

C₄×D₃₇

—

C₂²×C₃₇

C₂⁵

yes

characteristic

C₂×D₃₇

—

C₂×D₃₇

—

C₄×C₃₇

—

C₄×C₃₇

C₂×#41

yes

—

Dic₃₇

—

Dic₃₇

H(5,𝔽₂)

yes

—

#3483

—

maximal, Sylow

C₂×C₃₇

C₂²×#41

yes

—

C₂×C₃₇

C₂×H(5,𝔽₂)

yes

characteristic

D₃₇

—

Q₈²

D₃₇

yes

characteristic, complemented

#555

—

#550

—

#532

—

C₄×#41

—

C₃₇

#3483

yes

characteristic, complemented, Sylow

C₂×#41

—

C₂×#41

—

C₄⋊Q₈

C₂×D₃₇

yes

—

#127

—

#124

—

#121

—

C₄×Q₈

—

C₄×Q₈

C₂×D₃₇

yes

—

C₄×D₄

—

C₄×D₄

—

#117

—

C₂×C₄²

—

#41

—

#41

—

148

—

C₂×Q₈

C₂²×D₃₇

yes

—

C₂×Q₈

C₂²×D₃₇

yes

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂²×D₃₇

yes

—

C₂²⋊C₄

—

C₄²

—

C₄²

—

C₄²

C₂²×D₃₇

yes

—

C₂³

—

Q₈

—

Q₈

#10596966

yes

complemented

D₄

—

D₄

—

148

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂³×D₃₇

yes

—

C₂×C₄

C₂³×D₃₇

yes

—

C₂²

—

C₂²

—

C₂²

C₂⁴×D₃₇

yes

characteristic

100

C₄

—

101

C₄

—

102

C₄

—

103

C₄

C₂×#10596966

yes

—

104

C₂

—

148

—

105

C₂

C₂²×#10596966

yes

—

106

C₂

H(5,𝔽₂)×D₃₇

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₃₇
commutator, socle, cosocle	C₂²×C₃₇	C₂⁵
Fitting	Q₈²×C₃₇	C₂
hypercenter	Q₈²	D₃₇
hypocenter	C₃₇	#3483
LCS	G ▷ C₂²×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂² ◁ Q₈²

subgroup

G/H

center, Frattini

C₂²

C₂⁴×D₃₇

commutator, socle, cosocle

C₂²×C₃₇

C₂⁵

Fitting

Q₈²×C₃₇

C₂

hypercenter

Q₈²

D₃₇

C₃₇

G ▷ C₂²×C₃₇ ▷ C₃₇

C₁ ◁ C₂² ◁ Q₈²

N	H ≅ G/N	type
1. #14558249	C₂	semidirect
2. #14558205	C₂	semidirect
3. #14558084	C₂	semidirect
4. C₄×#10596966	C₂	semidirect
5. Q₈²×C₃₇	C₂	semidirect
34. Q₈×C₃₇	#41	semidirect
54. Q₈²	D₃₇	semidirect
59. C₃₇	#3483	semidirect
88. Q₈	#10596966	semidirect

H ≅ G/N

type

1. #14558249

C₂

semidirect

2. #14558205

C₂

semidirect

3. #14558084

C₂

semidirect

4. C₄×#10596966

C₂

semidirect

5. Q₈²×C₃₇

C₂

semidirect

34. Q₈×C₃₇

#41

semidirect

54. Q₈²

D₃₇

semidirect

59. C₃₇

#3483

semidirect

88. Q₈

#10596966

semidirect

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

subgroup	index
#14558249	2
#14558205	2
#14558084	2
C₄×#10596966	2
Q₈²×C₃₇	2
#3483	37

subgroup

index

Q₈²×C₃₇

Subgroups — #18008223

Conjugacy classes of subgroups

Complements

Maximal subgroups