Group #18011141 — subgroups

Subgroups — #18011141

Order 4736.

424 conjugacy classes of subgroups; only the 152 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 152 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	(C₂×#3866)⋊Q₈	C₂	1	4	yes	maximal
2	#14558482	C₂	1	8	yes	maximal
3	#14558478	C₂	1	2	yes	maximal
4	C₄⋊Q₈×D₃₇	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊#3866	C₂²	1	4	yes	—
6	C₂×C₄×#3866	C₂²	1	4	yes	—
7	(C₂×#3866)⋊C₄	C₂²	1	8	yes	—
8	(C₂×#3866)⋊C₄	C₂²	1	4	yes	—
9	(C₂×#3866)⋊C₄	C₂²	1	8	yes	—
10	C₂×Q₈×D₃₇	C₂²	1	2	yes	—
11	#11924536	C₄	1	1	yes	characteristic, complemented
12	#11924535	C₄	1	4	yes	complemented
13	#11924534	C₄	1	2	yes	complemented
14	C₄⋊C₄×D₃₇	C₂²	1	4	yes	—
15	C₄²×D₃₇	C₂²	1	1	yes	characteristic
16	C₄⋊Q₈×C₃₇	C₄	1	1	yes	characteristic, complemented
17	C₂²×#3866	C₂³	1	8	yes	—
18	C₂×Q₈×C₃₇	C₂×C₄	1	2	yes	—
19	Q₈×D₃₇	—	4	2	no	—
20	Q₈×D₃₇	—	4	2	no	—
21	C₂×C₄×D₃₇	C₂³	1	2	yes	—
22	C₂×C₄×D₃₇	C₂³	1	1	yes	characteristic
23	C₂×C₄×D₃₇	C₂³	1	4	yes	—
24	C₂×Dic₇₄	C₂×C₄	1	2	yes	—
25	C₂×Dic₇₄	C₂×C₄	1	4	yes	—
26	C₄⋊#3866	—	4	4	no	—
27	C₄×#3866	—	4	4	no	—
28	C₄⋊C₄×C₃₇	C₂×C₄	1	4	yes	—
29	C₄²×C₃₇	C₂×C₄	1	1	yes	characteristic
30	C₄⋊Dic₃₇	C₂×C₄	1	4	yes	—
31	#10596936	C₂×C₄	1	8	yes	—
32	C₄×Dic₃₇	C₂×C₄	1	1	yes	characteristic
33	C₄×Dic₃₇	C₂×C₄	1	2	yes	—
34	C₂²×D₃₇	C₂⁴	1	1	yes	characteristic
35	C₂×#3866	—	2	8	no	—
36	C₂×#3866	—	2	16	no	—
37	Q₈×C₃₇	—	4	2	no	—
38	C₂×C₄×C₃₇	C₂²×C₄	1	2	yes	—
39	C₂×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
40	C₂×C₄×C₃₇	C₂²×C₄	1	4	yes	—
41	C₂×Dic₃₇	C₂²×C₄	1	2	yes	—
42	C₂×Dic₃₇	C₂²×C₄	1	1	yes	characteristic
43	C₂×Dic₃₇	C₂²×C₄	1	4	yes	—
44	C₄×D₃₇	—	2	4	no	—
45	C₄×D₃₇	—	2	4	no	—
46	C₄×D₃₇	—	2	2	no	—
47	C₄×D₃₇	—	2	4	no	—
48	Dic₇₄	—	4	4	no	—
49	Dic₇₄	—	4	2	no	—
50	C₂²×C₃₇	C₂³×C₄	1	1	yes	characteristic
51	C₂×D₃₇	H(5,𝔽₂)	1	2	yes	—
52	C₂×D₃₇	2₋¹⁺⁴	1	4	yes	—
53	#3866	—	4	8	no	—
54	C₄×C₃₇	—	2	1	no	—
55	C₄×C₃₇	—	2	4	no	—
56	C₄×C₃₇	—	2	2	no	—
57	Dic₃₇	—	2	1	no	—
58	Dic₃₇	—	2	4	no	—
59	Dic₃₇	—	2	2	no	—
60	#2381	—	37	1	no	maximal, Sylow
61	C₂×C₃₇	#518	1	1	yes	characteristic
62	C₂×C₃₇	#519	1	2	yes	—
63	D₃₇	#563	1	4	yes	—
64	C₂×C₄⋊Q₈	—	37	1	no	—
65	#391	—	37	4	no	—
66	#387	—	37	8	no	—
67	#383	—	37	2	no	—
68	C₃₇	#2381	1	1	yes	characteristic, complemented, Sylow
69	C₂²×Q₈	—	37	2	no	—
70	C₄⋊Q₈	—	37	2	no	—
71	C₄⋊Q₈	—	37	4	no	—
72	C₄⋊Q₈	—	37	1	no	—
73	C₄⋊Q₈	#3866	1	1	yes	characteristic, complemented
74	C₂×C₄⋊C₄	—	37	4	no	—
75	C₂×C₄⋊C₄	—	37	4	no	—
76	C₂×C₄²	—	37	1	no	—
77	C₂×C₄²	—	37	4	no	—
78	#95	—	37	8	no	—
79	#95	—	37	4	no	—
80	#95	—	37	8	no	—
81	C₂×Q₈	—	37	4	no	—
82	C₂×Q₈	—	37	2	no	—
83	C₂×Q₈	—	148	2	no	—
84	C₂×Q₈	—	148	2	no	—
85	C₂×Q₈	C₂×#3866	1	2	yes	—
86	C₂²×C₄	—	37	2	no	—
87	C₂²×C₄	—	37	1	no	—
88	C₂²×C₄	—	37	4	no	—
89	C₂²×C₄	—	37	8	no	—
90	C₄⋊C₄	—	37	8	no	—
91	C₄⋊C₄	—	37	4	no	—
92	C₄⋊C₄	—	148	4	no	—
93	C₄⋊C₄	C₂×#3866	1	4	yes	—
94	C₄²	—	37	1	no	—
95	C₄²	—	37	2	no	—
96	C₄²	—	148	4	no	—
97	C₄²	C₂×#3866	1	1	yes	characteristic
98	C₂³	—	37	1	no	—
99	Q₈	—	4	2	no	—
100	Q₈	—	148	2	no	—
101	Q₈	—	148	4	no	—
102	C₂×C₄	—	37	2	no	—
103	C₂×C₄	—	37	1	no	—
104	C₂×C₄	—	37	4	no	—
105	C₂×C₄	—	74	4	no	—
106	C₂×C₄	—	74	4	no	—
107	C₂×C₄	—	74	2	no	—
108	C₂×C₄	—	74	4	no	—
109	C₂×C₄	—	74	8	no	—
110	C₂×C₄	—	74	16	no	—
111	C₂×C₄	C₂²×#3866	1	2	yes	—
112	C₂×C₄	C₂²×#3866	1	1	yes	characteristic
113	C₂×C₄	C₂²×#3866	1	4	yes	—
114	C₂²	—	37	2	no	—
115	C₂²	—	37	4	no	—
116	C₂²	C₂³×#3866	1	1	yes	characteristic
117	C₄	—	2	1	no	—
118	C₄	—	2	4	no	—
119	C₄	—	2	2	no	—
120	C₄	—	74	1	no	—
121	C₄	—	74	4	no	—
122	C₄	—	74	2	no	—
123	C₄	—	148	8	no	—
124	C₂	—	37	4	no	—
125	C₂	#14558600	1	1	yes	characteristic
126	C₂	#14558601	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

(C₂×#3866)⋊Q₈

C₂

yes

maximal

#14558482

C₂

yes

maximal

#14558478

C₂

yes

maximal

C₄⋊Q₈×D₃₇

C₂

yes

maximal, characteristic

C₂×C₄⋊#3866

C₂²

yes

—

C₂×C₄×#3866

C₂²

yes

—

(C₂×#3866)⋊C₄

C₂²

yes

—

(C₂×#3866)⋊C₄

C₂²

yes

—

(C₂×#3866)⋊C₄

C₂²

yes

—

C₂×Q₈×D₃₇

C₂²

yes

—

#11924536

C₄

yes

characteristic, complemented

#11924535

C₄

yes

complemented

#11924534

C₄

yes

complemented

C₄⋊C₄×D₃₇

C₂²

yes

—

C₄²×D₃₇

C₂²

yes

characteristic

C₄⋊Q₈×C₃₇

C₄

yes

characteristic, complemented

C₂²×#3866

C₂³

yes

—

C₂×Q₈×C₃₇

C₂×C₄

yes

—

Q₈×D₃₇

—

Q₈×D₃₇

—

C₂×C₄×D₃₇

C₂³

yes

—

C₂×C₄×D₃₇

C₂³

yes

characteristic

C₂×C₄×D₃₇

C₂³

yes

—

C₂×Dic₇₄

C₂×C₄

yes

—

C₂×Dic₇₄

C₂×C₄

yes

—

C₄⋊#3866

—

C₄×#3866

—

C₄⋊C₄×C₃₇

C₂×C₄

yes

—

C₄²×C₃₇

C₂×C₄

yes

characteristic

C₄⋊Dic₃₇

C₂×C₄

yes

—

#10596936

C₂×C₄

yes

—

C₄×Dic₃₇

C₂×C₄

yes

characteristic

C₄×Dic₃₇

C₂×C₄

yes

—

C₂²×D₃₇

C₂⁴

yes

characteristic

C₂×#3866

—

C₂×#3866

—

Q₈×C₃₇

—

C₂×C₄×C₃₇

C₂²×C₄

yes

—

C₂×C₄×C₃₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₃₇

C₂²×C₄

yes

—

C₂×Dic₃₇

C₂²×C₄

yes

—

C₂×Dic₃₇

C₂²×C₄

yes

characteristic

C₂×Dic₃₇

C₂²×C₄

yes

—

C₄×D₃₇

—

C₄×D₃₇

—

C₄×D₃₇

—

C₄×D₃₇

—

Dic₇₄

—

Dic₇₄

—

C₂²×C₃₇

C₂³×C₄

yes

characteristic

C₂×D₃₇

H(5,𝔽₂)

yes

—

C₂×D₃₇

2₋¹⁺⁴

yes

—

#3866

—

C₄×C₃₇

—

C₄×C₃₇

—

C₄×C₃₇

—

Dic₃₇

—

Dic₃₇

—

Dic₃₇

—

—

C₂×C₃₇

yes

C₂×C₃₇

yes

—

D₃₇

#563

yes

—

C₂×C₄⋊Q₈

—

#391

—

#387

—

#383

—

C₃₇

#2381

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

#3866

yes

characteristic, complemented

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

—

C₂×C₄²

—

#95

—

#95

—

#95

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

148

—

C₂×Q₈

—

148

—

C₂×Q₈

C₂×#3866

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

148

—

C₄⋊C₄

C₂×#3866

yes

—

C₄²

—

C₄²

—

C₄²

—

148

—

C₄²

C₂×#3866

yes

characteristic

C₂³

—

Q₈

—

100

Q₈

—

148

—

101

Q₈

—

148

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

C₂²×#3866

yes

—

112

C₂×C₄

C₂²×#3866

yes

characteristic

113

C₂×C₄

C₂²×#3866

yes

—

114

C₂²

—

115

C₂²

—

116

C₂²

C₂³×#3866

yes

characteristic

117

C₄

—

118

C₄

—

119

C₄

—

120

C₄

—

121

C₄

—

122

C₄

—

123

C₄

—

148

—

124

C₂

—

125

C₂

#14558600

yes

characteristic

126

C₂

#14558601

yes

—

	subgroup	G/H
center, Frattini	C₂²	C₂³×#3866
commutator, socle	C₂²×C₃₇	C₂³×C₄
Fitting	C₄⋊Q₈×C₃₇	C₄
prime radical, cosocle	C₂²×D₃₇	C₂⁴
hypercenter	C₄⋊Q₈	#3866
hypocenter	C₃₇	#2381
LCS	G ▷ C₂²×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂² ◁ C₄⋊Q₈

subgroup

G/H

center, Frattini

C₂²

C₂³×#3866

commutator, socle

C₂²×C₃₇

C₂³×C₄

Fitting

C₄⋊Q₈×C₃₇

C₄

prime radical, cosocle

C₂²×D₃₇

C₂⁴

C₄⋊Q₈

C₃₇

G ▷ C₂²×C₃₇ ▷ C₃₇

C₁ ◁ C₂² ◁ C₄⋊Q₈

N	H ≅ G/N	type
11. #11924536	C₄	semidirect
12. #11924535	C₄	semidirect
13. #11924534	C₄	semidirect
16. C₄⋊Q₈×C₃₇	C₄	semidirect
68. C₃₇	#2381	semidirect
73. C₄⋊Q₈	#3866	semidirect

H ≅ G/N

type

11. #11924536

C₄

semidirect

12. #11924535

C₄

semidirect

13. #11924534

C₄

semidirect

16. C₄⋊Q₈×C₃₇

C₄

semidirect

68. C₃₇

#2381

semidirect

73. C₄⋊Q₈

#3866

semidirect

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

subgroup	index
(C₂×#3866)⋊Q₈	2
#14558482	2
#14558478	2
C₄⋊Q₈×D₃₇	2
#2381	37

subgroup

index

C₄⋊Q₈×D₃₇

Subgroups — #18011141

Conjugacy classes of subgroups

Complements

Maximal subgroups