Group #37588906 — subgroups

Subgroups — #37588906

Order 24512.

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

normal subgroups only

This group has 222 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	#27133841	C₂	1	1	yes	maximal, characteristic, complemented
2	#27133832	C₂	1	2	yes	maximal, complemented
3	#27133821	C₂	1	2	yes	maximal
4	#27133817	C₂	1	2	yes	maximal
5	Q₈×Dic₃₈₃	C₂	1	2	yes	maximal, complemented
6	#27133806	C₂	1	2	yes	maximal, complemented
7	#27133799	C₂	1	2	yes	maximal
8	C₄²×D₃₈₃	C₂	1	1	yes	maximal, characteristic
9	C₄⋊Q₈×C₃₈₃	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×Dic₇₆₆	C₂²	1	2	yes	—
11	C₂×C₄×D₃₈₃	C₂²	1	1	yes	characteristic
12	C₂×C₄×D₃₈₃	C₂²	1	1	yes	characteristic
13	C₂×C₄×D₃₈₃	C₂²	1	1	yes	characteristic
14	C₄⋊Dic₃₈₃	C₂²	1	2	yes	—
15	C₄⋊Dic₃₈₃	C₂²	1	2	yes	—
16	C₄⋊Dic₃₈₃	C₂²	1	4	yes	—
17	Dic₃₈₃⋊C₄	C₂²	1	4	yes	—
18	(C₂×D₃₈₃)⋊C₄	C₂²	1	4	yes	—
19	C₄×Dic₃₈₃	C₂²	1	1	yes	characteristic
20	C₄×Dic₃₈₃	C₂²	1	1	yes	characteristic
21	C₄×Dic₃₈₃	C₂²	1	1	yes	characteristic
22	C₄×Dic₃₈₃	C₂²	1	4	yes	—
23	C₂×Q₈×C₃₈₃	C₂²	1	2	yes	—
24	C₄⋊C₄×C₃₈₃	C₂²	1	2	yes	—
25	C₄⋊C₄×C₃₈₃	C₂²	1	2	yes	—
26	C₄²×C₃₈₃	C₂²	1	1	yes	characteristic
27	C₂²×D₃₈₃	C₂³	1	1	yes	characteristic
28	Dic₇₆₆	—	2	4	no	—
29	C₂×Dic₃₈₃	C₂³	1	1	yes	characteristic
30	C₂×Dic₃₈₃	C₂³	1	1	yes	characteristic
31	C₂×Dic₃₈₃	C₂³	1	4	yes	—
32	C₂×Dic₃₈₃	C₂³	1	1	yes	characteristic
33	C₄×D₃₈₃	—	2	2	no	—
34	C₄×D₃₈₃	—	2	2	no	—
35	C₄×D₃₈₃	Q₈	1	4	yes	complemented
36	Q₈×C₃₈₃	—	2	4	no	—
37	C₂×C₄×C₃₈₃	C₂³	1	4	yes	—
38	C₂×C₄×C₃₈₃	C₂³	1	1	yes	characteristic
39	C₂×C₄×C₃₈₃	C₂³	1	1	yes	characteristic
40	C₂×C₄×C₃₈₃	C₂³	1	1	yes	characteristic
41	C₂²×C₃₈₃	C₂⁴	1	1	yes	characteristic
42	C₂×D₃₈₃	—	2	1	no	—
43	C₂×D₃₈₃	—	2	1	no	—
44	C₂×D₃₈₃	C₂×Q₈	1	2	yes	—
45	C₄×C₃₈₃	—	2	4	no	—
46	C₄×C₃₈₃	C₂×Q₈	1	2	yes	—
47	C₄×C₃₈₃	#41	1	2	yes	—
48	C₄×C₃₈₃	#41	1	2	yes	—
49	Dic₃₈₃	—	2	1	no	—
50	Dic₃₈₃	—	2	1	no	—
51	Dic₃₈₃	—	2	4	no	—
52	Dic₃₈₃	C₂×Q₈	1	2	yes	—
53	C₂×C₃₈₃	C₂²×Q₈	1	1	yes	characteristic
54	C₂×C₃₈₃	C₂×#41	1	1	yes	characteristic
55	C₂×C₃₈₃	C₂×#41	1	1	yes	characteristic
56	D₃₈₃	—	2	2	no	—
57	C₃₈₃	#533	1	1	yes	characteristic, complemented, Sylow
58	#533	—	383	1	no	maximal, Sylow
59	C₄⋊Q₈	—	383	1	no	—
60	C₄⋊Q₈	D₃₈₃	1	1	yes	characteristic, complemented
61	#125	—	383	2	no	—
62	C₂²⋊Q₈	—	383	2	no	—
63	C₂²⋊Q₈	—	383	2	no	—
64	C₄×Q₈	—	383	2	no	—
65	C₄×Q₈	—	383	2	no	—
66	#117	—	383	2	no	—
67	C₂×C₄²	—	383	1	no	—
68	C₂×Q₈	—	383	2	no	—
69	C₂×Q₈	C₂×D₃₈₃	1	2	yes	—
70	C₂²×C₄	—	383	1	no	—
71	C₂²×C₄	—	383	1	no	—
72	C₂²×C₄	—	383	1	no	—
73	C₄⋊C₄	—	383	4	no	—
74	C₄⋊C₄	—	383	2	no	—
75	C₄⋊C₄	—	383	4	no	—
76	C₄⋊C₄	—	383	2	no	—
77	C₄⋊C₄	C₂×D₃₈₃	1	2	yes	—
78	C₄⋊C₄	C₂×D₃₈₃	1	2	yes	—
79	C₂²⋊C₄	—	383	4	no	—
80	C₄²	—	383	1	no	—
81	C₄²	—	383	1	no	—
82	C₄²	—	383	1	no	—
83	C₄²	—	383	4	no	—
84	C₄²	C₂×D₃₈₃	1	1	yes	characteristic
85	C₂³	—	383	1	no	—
86	Q₈	—	2	4	no	—
87	Q₈	—	766	4	no	—
88	C₂×C₄	—	383	1	no	—
89	C₂×C₄	—	383	4	no	—
90	C₂×C₄	—	383	1	no	—
91	C₂×C₄	—	383	1	no	—
92	C₂×C₄	—	383	4	no	—
93	C₂×C₄	—	766	2	no	—
94	C₂×C₄	—	766	2	no	—
95	C₂×C₄	C₂²×D₃₈₃	1	4	yes	—
96	C₂×C₄	C₂²×D₃₈₃	1	1	yes	characteristic
97	C₂×C₄	C₂²×D₃₈₃	1	1	yes	characteristic
98	C₂×C₄	C₂²×D₃₈₃	1	1	yes	characteristic
99	C₂²	—	383	2	no	—
100	C₂²	—	766	1	no	—
101	C₂²	—	766	1	no	—
102	C₂²	C₂³×D₃₈₃	1	1	yes	characteristic
103	C₄	—	2	4	no	—
104	C₄	—	383	2	no	—
105	C₄	—	766	1	no	—
106	C₄	—	766	1	no	—
107	C₄	—	766	4	no	—
108	C₄	Q₈×D₃₈₃	1	2	yes	—
109	C₄	#20259456	1	2	yes	—
110	C₄	#20259458	1	2	yes	—
111	C₂	—	766	2	no	—
112	C₂	C₂×Q₈×D₃₈₃	1	1	yes	characteristic
113	C₂	C₂×#20259456	1	1	yes	characteristic
114	C₂	C₂×#20259458	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#27133841

C₂

yes

maximal, characteristic, complemented

#27133832

C₂

yes

maximal, complemented

#27133821

C₂

yes

maximal

#27133817

C₂

yes

maximal

Q₈×Dic₃₈₃

C₂

yes

maximal, complemented

#27133806

C₂

yes

maximal, complemented

#27133799

C₂

yes

maximal

C₄²×D₃₈₃

C₂

yes

maximal, characteristic

C₄⋊Q₈×C₃₈₃

C₂

yes

maximal, characteristic, complemented

C₂×Dic₇₆₆

C₂²

yes

—

C₂×C₄×D₃₈₃

C₂²

yes

characteristic

C₂×C₄×D₃₈₃

C₂²

yes

characteristic

C₂×C₄×D₃₈₃

C₂²

yes

characteristic

C₄⋊Dic₃₈₃

C₂²

yes

—

C₄⋊Dic₃₈₃

C₂²

yes

—

C₄⋊Dic₃₈₃

C₂²

yes

—

Dic₃₈₃⋊C₄

C₂²

yes

—

(C₂×D₃₈₃)⋊C₄

C₂²

yes

—

C₄×Dic₃₈₃

C₂²

yes

characteristic

C₄×Dic₃₈₃

C₂²

yes

characteristic

C₄×Dic₃₈₃

C₂²

yes

characteristic

C₄×Dic₃₈₃

C₂²

yes

—

C₂×Q₈×C₃₈₃

C₂²

yes

—

C₄⋊C₄×C₃₈₃

C₂²

yes

—

C₄⋊C₄×C₃₈₃

C₂²

yes

—

C₄²×C₃₈₃

C₂²

yes

characteristic

C₂²×D₃₈₃

C₂³

yes

characteristic

Dic₇₆₆

—

C₂×Dic₃₈₃

C₂³

yes

characteristic

C₂×Dic₃₈₃

C₂³

yes

characteristic

C₂×Dic₃₈₃

C₂³

yes

—

C₂×Dic₃₈₃

C₂³

yes

characteristic

C₄×D₃₈₃

—

C₄×D₃₈₃

—

C₄×D₃₈₃

Q₈

yes

complemented

Q₈×C₃₈₃

—

C₂×C₄×C₃₈₃

C₂³

yes

—

C₂×C₄×C₃₈₃

C₂³

yes

characteristic

C₂×C₄×C₃₈₃

C₂³

yes

characteristic

C₂×C₄×C₃₈₃

C₂³

yes

characteristic

C₂²×C₃₈₃

C₂⁴

yes

characteristic

C₂×D₃₈₃

—

C₂×D₃₈₃

—

C₂×D₃₈₃

C₂×Q₈

yes

—

C₄×C₃₈₃

—

C₄×C₃₈₃

C₂×Q₈

yes

—

C₄×C₃₈₃

#41

yes

—

C₄×C₃₈₃

#41

yes

—

Dic₃₈₃

—

Dic₃₈₃

—

Dic₃₈₃

—

Dic₃₈₃

C₂×Q₈

yes

—

C₂×C₃₈₃

C₂²×Q₈

yes

characteristic

C₂×C₃₈₃

C₂×#41

yes

characteristic

C₂×C₃₈₃

C₂×#41

yes

characteristic

D₃₈₃

—

C₃₈₃

#533

yes

characteristic, complemented, Sylow

#533

—

383

maximal, Sylow

C₄⋊Q₈

—

383

—

C₄⋊Q₈

D₃₈₃

yes

characteristic, complemented

#125

—

383

—

C₂²⋊Q₈

—

383

—

C₂²⋊Q₈

—

383

—

C₄×Q₈

—

383

—

C₄×Q₈

—

383

—

#117

—

383

—

C₂×C₄²

—

383

—

C₂×Q₈

—

383

—

C₂×Q₈

C₂×D₃₈₃

yes

—

C₂²×C₄

—

383

—

C₂²×C₄

—

383

—

C₂²×C₄

—

383

—

C₄⋊C₄

—

383

—

C₄⋊C₄

—

383

—

C₄⋊C₄

—

383

—

C₄⋊C₄

—

383

—

C₄⋊C₄

C₂×D₃₈₃

yes

—

C₄⋊C₄

C₂×D₃₈₃

yes

—

C₂²⋊C₄

—

383

—

C₄²

—

383

—

C₄²

—

383

—

C₄²

—

383

—

C₄²

—

383

—

C₄²

C₂×D₃₈₃

yes

characteristic

C₂³

—

383

—

Q₈

—

Q₈

—

766

—

C₂×C₄

—

383

—

C₂×C₄

—

383

—

C₂×C₄

—

383

—

C₂×C₄

—

383

—

C₂×C₄

—

383

—

C₂×C₄

—

766

—

C₂×C₄

—

766

—

C₂×C₄

C₂²×D₃₈₃

yes

—

C₂×C₄

C₂²×D₃₈₃

yes

characteristic

C₂×C₄

C₂²×D₃₈₃

yes

characteristic

C₂×C₄

C₂²×D₃₈₃

yes

characteristic

C₂²

—

383

—

100

C₂²

—

766

—

101

C₂²

—

766

—

102

C₂²

C₂³×D₃₈₃

yes

characteristic

103

C₄

—

104

C₄

—

383

—

105

C₄

—

766

—

106

C₄

—

766

—

107

C₄

—

766

—

108

C₄

Q₈×D₃₈₃

yes

—

109

C₄

#20259456

yes

—

110

C₄

#20259458

yes

—

111

C₂

—

766

—

112

C₂

C₂×Q₈×D₃₈₃

yes

characteristic

113

C₂

C₂×#20259456

yes

characteristic

114

C₂

C₂×#20259458

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₃₈₃
commutator, socle, cosocle	C₂²×C₃₈₃	C₂⁴
Fitting	C₄⋊Q₈×C₃₈₃	C₂
prime radical	C₂²×D₃₈₃	C₂³
hypercenter	C₄⋊Q₈	D₃₈₃
hypocenter	C₃₈₃	#533
LCS	G ▷ C₂²×C₃₈₃ ▷ C₃₈₃
UCS	C₁ ◁ C₂² ◁ C₄⋊Q₈

subgroup

G/H

center, Frattini

C₂²

C₂³×D₃₈₃

commutator, socle, cosocle

C₂²×C₃₈₃

C₂⁴

Fitting

C₄⋊Q₈×C₃₈₃

C₂

prime radical

C₂²×D₃₈₃

C₂³

hypercenter

C₄⋊Q₈

D₃₈₃

C₃₈₃

G ▷ C₂²×C₃₈₃ ▷ C₃₈₃

C₁ ◁ C₂² ◁ C₄⋊Q₈

N	H ≅ G/N	type
1. #27133841	C₂	semidirect
2. #27133832	C₂	semidirect
5. Q₈×Dic₃₈₃	C₂	semidirect
6. #27133806	C₂	semidirect
9. C₄⋊Q₈×C₃₈₃	C₂	semidirect
35. C₄×D₃₈₃	Q₈	semidirect
57. C₃₈₃	#533	semidirect
60. C₄⋊Q₈	D₃₈₃	semidirect

H ≅ G/N

type

1. #27133841

C₂

semidirect

2. #27133832

C₂

semidirect

5. Q₈×Dic₃₈₃

C₂

semidirect

6. #27133806

C₂

semidirect

9. C₄⋊Q₈×C₃₈₃

C₂

semidirect

35. C₄×D₃₈₃

Q₈

semidirect

57. C₃₈₃

#533

semidirect

60. C₄⋊Q₈

D₃₈₃

semidirect

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

subgroup	index
#27133841	2
#27133832	2
#27133821	2
#27133817	2
Q₈×Dic₃₈₃	2
#27133806	2
#27133799	2
C₄²×D₃₈₃	2
C₄⋊Q₈×C₃₈₃	2
#533	383

subgroup

index

Q₈×Dic₃₈₃

C₄²×D₃₈₃

C₄⋊Q₈×C₃₈₃

#533

383

Subgroups — #37588906

Conjugacy classes of subgroups

Complements

Maximal subgroups