Group #18010043 — subgroups

Subgroups — C₂×#14558389 #18010043

Order 4736.

284 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 284 conjugacy classes of subgroups — too many to draw. Only its 74 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×#11924591	C₂	1	2	yes	maximal, complemented
2	#14558389	C₂	1	4	yes	maximal, direct product factor
3	C₂×#11924525	C₂	1	1	yes	maximal, characteristic
4	C₂²×#64438	—	2	2	no	—
5	#11924591	—	2	2	no	—
6	#11924591	C₂²	1	4	yes	complemented
7	C₂²×D₁₄₈	C₄	1	1	yes	characteristic
8	#11924525	—	2	2	no	—
9	#11924525	C₄	1	2	yes	—
10	#11924525	C₂²	1	2	yes	—
11	C₂×#10596938	—	2	2	no	—
12	C₂×C₄×Dic₃₇	C₂²	1	1	yes	characteristic
13	C₂²×Q₈×C₃₇	C₄	1	1	yes	characteristic
14	C₂³×D₃₇	—	2	1	no	—
15	C₂×Q₈×C₃₇	—	2	1	no	—
16	C₂×Q₈×C₃₇	—	4	2	no	—
17	C₂×Q₈×C₃₇	C₂×C₄	1	2	yes	—
18	C₂²×C₄×C₃₇	—	2	1	no	—
19	C₂²×C₄×C₃₇	C₂×C₄	1	1	yes	characteristic
20	C₂²×Dic₃₇	D₄	1	2	yes	—
21	C₂×D₁₄₈	—	2	1	no	—
22	C₂×D₁₄₈	—	4	2	no	—
23	C₂×D₁₄₈	C₂×C₄	1	2	yes	—
24	C₂×#64438	—	2	8	no	—
25	C₂×#64438	—	2	4	no	—
26	#10596938	—	2	4	no	—
27	#10596938	—	2	4	no	—
28	C₄×Dic₃₇	C₂×C₄	1	1	yes	characteristic
29	C₄×Dic₃₇	D₄	1	2	yes	—
30	C₄×Dic₃₇	C₂³	1	1	yes	characteristic
31	C₂³×C₃₇	C₂²⋊C₄	1	1	yes	characteristic
32	C₂²×D₃₇	—	2	2	no	—
33	C₂²×D₃₇	—	4	2	no	—
34	C₂²×D₃₇	—	4	4	no	—
35	Q₈×C₃₇	—	4	2	no	—
36	Q₈×C₃₇	—	4	2	no	—
37	C₂×C₄×C₃₇	—	2	2	no	—
38	C₂×C₄×C₃₇	—	2	2	no	—
39	C₂×C₄×C₃₇	—	4	2	no	—
40	C₂×C₄×C₃₇	C₂²⋊C₄	1	1	yes	characteristic
41	C₂×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
42	C₂×Dic₃₇	—	2	4	no	—
43	C₂×Dic₃₇	C₂²⋊C₄	1	2	yes	—
44	C₂×Dic₃₇	C₂×D₄	1	2	yes	—
45	D₁₄₈	—	4	2	no	—
46	D₁₄₈	—	4	2	no	—
47	#64438	—	2	8	no	—
48	C₂²×C₃₇	#100	1	2	yes	—
49	C₂²×C₃₇	#104	1	4	yes	—
50	C₂²×C₃₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
51	C₂×D₃₇	—	4	2	no	—
52	C₂×D₃₇	—	4	4	no	—
53	C₂×D₃₇	—	8	4	no	—
54	C₄×C₃₇	—	2	1	no	—
55	C₄×C₃₇	—	2	1	no	—
56	C₄×C₃₇	—	4	2	no	—
57	Dic₃₇	—	2	2	no	—
58	Dic₃₇	—	2	2	no	—
59	C₂×#329	—	37	1	no	maximal, Sylow
60	C₂×C₃₇	#329	1	4	yes	complemented
61	C₂×C₃₇	C₂×#100	1	1	yes	characteristic
62	C₂×C₃₇	C₂×#104	1	2	yes	—
63	D₃₇	—	8	2	no	—
64	C₂×#124	—	37	1	no	—
65	C₂×#97	—	37	2	no	—
66	#329	—	37	4	no	—
67	C₃₇	C₂×#329	1	1	yes	characteristic, complemented, Sylow
68	C₂²×Q₈	#3866	1	1	yes	characteristic
69	C₂²×D₄	—	37	1	no	—
70	C₂²×C₈	—	74	2	no	—
71	#124	—	37	2	no	—
72	#124	—	37	2	no	—
73	#124	—	74	2	no	—
74	C₂×C₂²⋊C₄	—	74	2	no	—
75	C₂×C₄²	—	37	1	no	—
76	#97	—	37	4	no	—
77	#97	—	74	2	no	—
78	C₂⁴	—	74	1	no	—
79	C₂×Q₈	—	2	1	no	—
80	C₂×Q₈	—	4	2	no	—
81	C₂×Q₈	C₂×#3866	1	2	yes	—
82	C₂×D₄	—	37	2	no	—
83	C₂×D₄	—	74	1	no	—
84	C₂×D₄	—	148	2	no	—
85	C₂²×C₄	—	2	1	no	—
86	C₂²×C₄	—	37	2	no	—
87	C₂²×C₄	C₂×#3866	1	1	yes	characteristic
88	C₂×C₈	—	74	4	no	—
89	C₂×C₈	—	74	8	no	—
90	C₂²⋊C₄	—	74	4	no	—
91	C₂²⋊C₄	—	74	4	no	—
92	C₄²	—	37	2	no	—
93	C₄²	—	37	1	no	—
94	C₄²	—	37	1	no	—
95	C₂³	—	74	2	no	—
96	C₂³	—	148	2	no	—
97	C₂³	—	148	4	no	—
98	C₂³	C₂²⋊#3866	1	1	yes	characteristic
99	Q₈	—	4	2	no	—
100	Q₈	—	4	2	no	—
101	D₄	—	148	2	no	—
102	D₄	—	148	2	no	—
103	C₂×C₄	—	2	2	no	—
104	C₂×C₄	—	2	2	no	—
105	C₂×C₄	—	4	2	no	—
106	C₂×C₄	—	37	2	no	—
107	C₂×C₄	—	37	2	no	—
108	C₂×C₄	—	74	4	no	—
109	C₂×C₄	C₂²⋊#3866	1	1	yes	characteristic
110	C₂×C₄	C₂²×#3866	1	1	yes	characteristic
111	C₈	—	74	8	no	—
112	C₂²	—	296	4	no	—
113	C₂²	—	148	2	no	—
114	C₂²	—	148	4	no	—
115	C₂²	#11924596	1	2	yes	—
116	C₂²	#11924601	1	4	yes	—
117	C₂²	C₂×C₂²⋊#3866	1	1	yes	characteristic
118	C₄	—	2	1	no	—
119	C₄	—	2	1	no	—
120	C₄	—	4	2	no	—
121	C₄	—	74	2	no	—
122	C₄	—	74	2	no	—
123	C₂	—	296	2	no	—
124	C₂	#14558389	1	4	yes	direct product factor
125	C₂	C₂×#11924596	1	1	yes	characteristic
126	C₂	C₂×#11924601	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#11924591

C₂

yes

maximal, complemented

#14558389

C₂

yes

maximal, direct product factor

C₂×#11924525

C₂

yes

maximal, characteristic

C₂²×#64438

—

#11924591

—

#11924591

C₂²

yes

complemented

C₂²×D₁₄₈

C₄

yes

characteristic

#11924525

—

#11924525

C₄

yes

—

#11924525

C₂²

yes

—

C₂×#10596938

—

C₂×C₄×Dic₃₇

C₂²

yes

characteristic

C₂²×Q₈×C₃₇

C₄

yes

characteristic

C₂³×D₃₇

—

C₂×Q₈×C₃₇

—

C₂×Q₈×C₃₇

—

C₂×Q₈×C₃₇

C₂×C₄

yes

—

C₂²×C₄×C₃₇

—

C₂²×C₄×C₃₇

C₂×C₄

yes

characteristic

C₂²×Dic₃₇

D₄

yes

—

C₂×D₁₄₈

—

C₂×D₁₄₈

—

C₂×D₁₄₈

C₂×C₄

yes

—

—

—

—

—

C₄×Dic₃₇

C₂×C₄

yes

characteristic

C₄×Dic₃₇

D₄

yes

—

C₄×Dic₃₇

C₂³

yes

characteristic

C₂³×C₃₇

C₂²⋊C₄

yes

characteristic

C₂²×D₃₇

—

C₂²×D₃₇

—

C₂²×D₃₇

—

Q₈×C₃₇

—

Q₈×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₃₇

C₂²×C₄

yes

characteristic

C₂×Dic₃₇

—

C₂×Dic₃₇

C₂²⋊C₄

yes

—

C₂×Dic₃₇

C₂×D₄

yes

—

D₁₄₈

—

D₁₄₈

—

#64438

—

C₂²×C₃₇

#100

yes

—

C₂²×C₃₇

#104

yes

—

C₂²×C₃₇

C₂×C₂²⋊C₄

yes

characteristic

C₂×D₃₇

—

C₂×D₃₇

—

C₂×D₃₇

—

C₄×C₃₇

—

C₄×C₃₇

—

C₄×C₃₇

—

Dic₃₇

—

Dic₃₇

—

—

C₂×C₃₇

yes

C₂×C₃₇

yes

C₂×C₃₇

yes

—

D₃₇

—

C₂×#124

—

C₂×#97

—

#329

—

C₃₇

C₂×#329

yes

characteristic, complemented, Sylow

C₂²×Q₈

#3866

yes

characteristic

C₂²×D₄

—

C₂²×C₈

—

#124

—

#124

—

#124

—

C₂×C₂²⋊C₄

—

C₂×C₄²

—

#97

—

#97

—

C₂⁴

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂×#3866

yes

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

148

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×#3866

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

—

C₄²

—

C₂³

—

C₂³

—

148

—

C₂³

—

148

—

C₂³

C₂²⋊#3866

yes

characteristic

Q₈

—

100

Q₈

—

101

D₄

—

148

—

102

D₄

—

148

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

C₂²⋊#3866

yes

characteristic

110

C₂×C₄

C₂²×#3866

yes

characteristic

111

C₈

—

112

C₂²

—

296

—

113

C₂²

—

148

—

114

C₂²

—

148

—

115

C₂²

#11924596

yes

—

116

C₂²

#11924601

yes

—

117

C₂²

C₂×C₂²⋊#3866

yes

characteristic

118

C₄

—

119

C₄

—

120

C₄

—

121

C₄

—

122

C₄

—

123

C₂

—

296

—

124

C₂

#14558389

yes

direct product factor

125

C₂

C₂×#11924596

yes

characteristic

126

C₂

C₂×#11924601

yes

—

	subgroup	G/H
center	C₂³	C₂²⋊#3866
commutator	C₂×C₄×C₃₇	C₂²×C₄
Frattini	C₂×C₄	C₂²×#3866
Fitting	C₂²×Q₈×C₃₇	C₄
socle	C₂³×C₃₇	C₂²⋊C₄
prime radical	C₂²×D₁₄₈	C₄
cosocle	C₄×Dic₃₇	C₂³
hypercenter	C₂²×Q₈	#3866
hypocenter	C₃₇	C₂×#329
LCS	G ▷ C₂×C₄×C₃₇ ▷ C₂²×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×Q₈

subgroup

G/H

center

C₂³

C₂²⋊#3866

commutator

C₂×C₄×C₃₇

C₂²×C₄

Frattini

C₂×C₄

C₂²×#3866

Fitting

C₂²×Q₈×C₃₇

C₄

socle

C₂³×C₃₇

C₂²⋊C₄

prime radical

C₂²×D₁₄₈

C₄

cosocle

C₄×Dic₃₇

C₂³

C₂²×Q₈

C₃₇

G ▷ C₂×C₄×C₃₇ ▷ C₂²×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×Q₈

N	H ≅ G/N	type
1. C₂×#11924591	C₂	semidirect
2. #14558389	C₂	direct (N × H)
6. #11924591	C₂²	semidirect
60. C₂×C₃₇	#329	semidirect
67. C₃₇	C₂×#329	semidirect
124. C₂	#14558389	direct (N × H)

H ≅ G/N

type

1. C₂×#11924591

C₂

semidirect

2. #14558389

C₂

direct (N × H)

6. #11924591

C₂²

semidirect

60. C₂×C₃₇

#329

semidirect

67. C₃₇

C₂×#329

semidirect

124. C₂

#14558389

direct (N × H)

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

subgroup	index
C₂×#11924591	2
#14558389	2
C₂×#11924525	2
C₂×#329	37

subgroup

index

Subgroups — C₂×#14558389 #18010043

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×#14558389 #18010043

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#14558389 #18010043