Group #18005094 — subgroups

Subgroups — C₂×#14557930 #18005094

Order 4736.

424 conjugacy classes of subgroups; only the 181 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 181 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×#11924535	C₂	1	2	yes	maximal
2	#14557930	C₂	1	8	yes	maximal, direct product factor
3	C₂×#11924468	C₂	1	1	yes	maximal, characteristic
4	C₂×#11924467	C₂	1	2	yes	maximal
5	C₂×C₈×Dic₃₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#106×C₃₇	C₂	1	1	yes	maximal, characteristic
7	C₂²×Dic₇₄	—	2	2	no	—
8	C₂²×#64436	C₂²	1	1	yes	characteristic
9	#11924535	—	2	4	no	—
10	#11924535	C₂²	1	8	yes	—
11	C₂×C₄⋊Dic₃₇	C₂²	1	2	yes	—
12	C₂×#10596936	—	2	2	no	—
13	C₂×C₄×Dic₃₇	C₂²	1	1	yes	characteristic
14	#11924468	C₂²	1	4	yes	—
15	#11924467	C₂²	1	8	yes	—
16	C₈×Dic₃₇	C₂²	1	4	yes	—
17	C₂²×C₈×C₃₇	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊C₄×C₃₇	C₂²	1	2	yes	—
19	#106×C₃₇	C₂²	1	4	yes	—
20	C₂²×C₄×C₃₇	—	2	2	no	—
21	C₂²×C₄×C₃₇	C₂³	1	1	yes	characteristic
22	C₂²×Dic₃₇	—	2	2	no	—
23	C₂²×Dic₃₇	D₄	1	2	yes	—
24	C₂×Dic₇₄	—	2	8	no	—
25	C₂×Dic₇₄	—	4	8	no	—
26	C₂×C₈×C₃₇	Q₈	1	4	yes	complemented
27	C₂×C₈×C₃₇	C₂³	1	2	yes	—
28	C₄⋊C₄×C₃₇	—	2	2	no	—
29	C₄⋊C₄×C₃₇	C₂³	1	4	yes	—
30	C₄⋊Dic₃₇	—	2	2	no	—
31	C₄⋊Dic₃₇	C₂³	1	4	yes	—
32	#10596936	—	2	8	no	—
33	C₄×Dic₃₇	D₄	1	2	yes	—
34	C₄×Dic₃₇	C₂³	1	2	yes	—
35	C₂×#64436	Q₈	1	4	yes	complemented
36	C₂×#64436	C₂³	1	2	yes	—
37	C₂³×C₃₇	C₂×D₄	1	1	yes	characteristic
38	C₂×C₄×C₃₇	—	2	4	no	—
39	C₂×C₄×C₃₇	—	4	4	no	—
40	C₂×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
41	C₂×C₄×C₃₇	C₂×Q₈	1	2	yes	—
42	C₂×C₄×C₃₇	C₂×Q₈	1	2	yes	—
43	C₂×C₄×C₃₇	C₂⁴	1	1	yes	characteristic
44	C₂×Dic₃₇	—	2	4	no	—
45	C₂×Dic₃₇	—	4	4	no	—
46	C₂×Dic₃₇	SD₁₆	1	8	yes	—
47	C₂×Dic₃₇	C₂×D₄	1	4	yes	—
48	Dic₇₄	—	4	16	no	—
49	C₈×C₃₇	C₂×Q₈	1	4	yes	complemented
50	#64436	C₂×Q₈	1	4	yes	complemented
51	C₂²×C₃₇	C₄⋊Q₈	1	2	yes	—
52	C₂²×C₃₇	C₂×SD₁₆	1	4	yes	—
53	C₂²×C₃₇	C₂²×D₄	1	1	yes	characteristic
54	C₄×C₃₇	—	4	4	no	—
55	C₄×C₃₇	C₄⋊Q₈	1	2	yes	—
56	C₄×C₃₇	C₂²×Q₈	1	1	yes	characteristic
57	C₄×C₃₇	C₂²×Q₈	1	1	yes	characteristic
58	Dic₃₇	—	4	4	no	—
59	Dic₃₇	C₂×SD₁₆	1	8	yes	—
60	C₂×#498	—	37	1	no	maximal, Sylow
61	C₂×C₃₇	#498	1	4	yes	complemented
62	C₂×C₃₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
63	C₂×C₃₇	C₂²×SD₁₆	1	2	yes	—
64	C₂×C₄⋊Q₈	—	37	2	no	—
65	#498	—	37	8	no	—
66	C₂×#106	—	37	1	no	—
67	C₂×#106	—	37	2	no	—
68	C₂×#106	D₃₇	1	1	yes	characteristic
69	C₂×C₄×C₈	—	37	1	no	—
70	C₃₇	C₂×#498	1	1	yes	characteristic, complemented, Sylow
71	C₂²×Q₈	—	74	2	no	—
72	C₂²×C₈	—	37	1	no	—
73	C₂²×C₈	C₂×D₃₇	1	1	yes	characteristic
74	C₄⋊Q₈	—	37	8	no	—
75	C₄⋊Q₈	—	74	4	no	—
76	C₂×C₄⋊C₄	—	37	2	no	—
77	C₂×C₄⋊C₄	—	74	2	no	—
78	C₂×C₄⋊C₄	C₂×D₃₇	1	2	yes	—
79	C₂×C₄²	—	37	1	no	—
80	#106	—	37	4	no	—
81	#106	—	37	8	no	—
82	#106	C₂×D₃₇	1	4	yes	—
83	C₄×C₈	—	37	4	no	—
84	C₂×Q₈	—	74	8	no	—
85	C₂×Q₈	—	148	8	no	—
86	C₂²×C₄	—	2	2	no	—
87	C₂²×C₄	—	37	2	no	—
88	C₂²×C₄	—	74	2	no	—
89	C₂²×C₄	C₂²×D₃₇	1	1	yes	characteristic
90	C₂×C₈	—	37	4	no	—
91	C₂×C₈	—	37	2	no	—
92	C₂×C₈	Dic₇₄	1	4	yes	complemented
93	C₂×C₈	C₂²×D₃₇	1	2	yes	—
94	C₄⋊C₄	—	2	2	no	—
95	C₄⋊C₄	—	37	4	no	—
96	C₄⋊C₄	—	74	8	no	—
97	C₄⋊C₄	—	74	2	no	—
98	C₄⋊C₄	C₂²×D₃₇	1	4	yes	—
99	C₄²	—	37	2	no	—
100	C₄²	—	37	2	no	—
101	C₂³	D₄×D₃₇	1	1	yes	characteristic
102	Q₈	—	148	16	no	—
103	C₂×C₄	—	2	4	no	—
104	C₂×C₄	—	4	4	no	—
105	C₂×C₄	—	37	8	no	—
106	C₂×C₄	—	37	4	no	—
107	C₂×C₄	—	74	4	no	—
108	C₂×C₄	—	148	4	no	—
109	C₂×C₄	C₂×Dic₇₄	1	2	yes	—
110	C₂×C₄	D₄×D₃₇	1	1	yes	characteristic
111	C₂×C₄	Q₈×D₃₇	1	2	yes	—
112	C₂×C₄	C₂³×D₃₇	1	1	yes	characteristic
113	C₈	—	37	4	no	—
114	C₈	C₂×Dic₇₄	1	4	yes	complemented
115	C₂²	#11924535	1	2	yes	—
116	C₂²	SD₁₆×D₃₇	1	4	yes	—
117	C₂²	C₂×D₄×D₃₇	1	1	yes	characteristic
118	C₄	—	4	4	no	—
119	C₄	—	37	8	no	—
120	C₄	—	148	4	no	—
121	C₄	#11924535	1	2	yes	—
122	C₄	C₂×Q₈×D₃₇	1	1	yes	characteristic
123	C₄	C₂²×Dic₇₄	1	1	yes	characteristic
124	C₂	#14557930	1	4	yes	direct product factor
125	C₂	C₂×#11924535	1	1	yes	characteristic
126	C₂	C₂×SD₁₆×D₃₇	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#11924535

C₂

yes

maximal

#14557930

C₂

yes

maximal, direct product factor

C₂×#11924468

C₂

yes

maximal, characteristic

C₂×#11924467

C₂

yes

maximal

C₂×C₈×Dic₃₇

C₂

yes

maximal, characteristic

C₂×#106×C₃₇

C₂

yes

maximal, characteristic

C₂²×Dic₇₄

—

C₂²×#64436

C₂²

yes

characteristic

#11924535

—

#11924535

C₂²

yes

—

C₂×C₄⋊Dic₃₇

C₂²

yes

—

C₂×#10596936

—

C₂×C₄×Dic₃₇

C₂²

yes

characteristic

#11924468

C₂²

yes

—

#11924467

C₂²

yes

—

C₈×Dic₃₇

C₂²

yes

—

C₂²×C₈×C₃₇

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₃₇

C₂²

yes

—

#106×C₃₇

C₂²

yes

—

C₂²×C₄×C₃₇

—

C₂²×C₄×C₃₇

C₂³

yes

characteristic

C₂²×Dic₃₇

—

C₂²×Dic₃₇

D₄

yes

—

C₂×Dic₇₄

—

C₂×Dic₇₄

—

C₂×C₈×C₃₇

Q₈

yes

complemented

C₂×C₈×C₃₇

C₂³

yes

—

C₄⋊C₄×C₃₇

—

C₄⋊C₄×C₃₇

C₂³

yes

—

C₄⋊Dic₃₇

—

C₄⋊Dic₃₇

C₂³

yes

—

#10596936

—

C₄×Dic₃₇

D₄

yes

—

C₄×Dic₃₇

C₂³

yes

—

C₂×#64436

Q₈

yes

complemented

C₂×#64436

C₂³

yes

—

C₂³×C₃₇

C₂×D₄

yes

characteristic

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

C₂×D₄

yes

characteristic

C₂×C₄×C₃₇

C₂×Q₈

yes

—

C₂×C₄×C₃₇

C₂×Q₈

yes

—

C₂×C₄×C₃₇

C₂⁴

yes

characteristic

C₂×Dic₃₇

—

C₂×Dic₃₇

—

C₂×Dic₃₇

SD₁₆

yes

—

C₂×Dic₃₇

C₂×D₄

yes

—

Dic₇₄

—

C₈×C₃₇

C₂×Q₈

yes

complemented

#64436

C₂×Q₈

yes

complemented

C₂²×C₃₇

C₄⋊Q₈

yes

—

C₂²×C₃₇

C₂×SD₁₆

yes

—

C₂²×C₃₇

C₂²×D₄

yes

characteristic

C₄×C₃₇

—

C₄×C₃₇

C₄⋊Q₈

yes

—

C₄×C₃₇

C₂²×Q₈

yes

characteristic

C₄×C₃₇

C₂²×Q₈

yes

characteristic

Dic₃₇

—

Dic₃₇

C₂×SD₁₆

yes

—

—

C₂×C₃₇

yes

C₂×C₃₇

C₂×C₄⋊Q₈

yes

characteristic

C₂×C₃₇

C₂²×SD₁₆

yes

—

C₂×C₄⋊Q₈

—

—

—

—

D₃₇

yes

characteristic

C₂×C₄×C₈

—

C₃₇

C₂×#498

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₂²×C₈

—

C₂²×C₈

C₂×D₃₇

yes

characteristic

C₄⋊Q₈

—

C₄⋊Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₃₇

yes

—

C₂×C₄²

—

#106

—

#106

—

#106

C₂×D₃₇

yes

—

C₄×C₈

—

C₂×Q₈

—

C₂×Q₈

—

148

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₃₇

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

Dic₇₄

yes

complemented

C₂×C₈

C₂²×D₃₇

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂²×D₃₇

yes

—

C₄²

—

100

C₄²

—

101

C₂³

D₄×D₃₇

yes

characteristic

102

Q₈

—

148

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

148

—

109

C₂×C₄

C₂×Dic₇₄

yes

—

110

C₂×C₄

D₄×D₃₇

yes

characteristic

111

C₂×C₄

Q₈×D₃₇

yes

—

112

C₂×C₄

C₂³×D₃₇

yes

characteristic

113

C₈

—

114

C₈

C₂×Dic₇₄

yes

complemented

115

C₂²

#11924535

yes

—

116

C₂²

SD₁₆×D₃₇

yes

—

117

C₂²

C₂×D₄×D₃₇

yes

characteristic

118

C₄

—

119

C₄

—

120

C₄

—

148

—

121

C₄

#11924535

yes

—

122

C₄

C₂×Q₈×D₃₇

yes

characteristic

123

C₄

C₂²×Dic₇₄

yes

characteristic

124

C₂

#14557930

yes

direct product factor

125

C₂

C₂×#11924535

yes

characteristic

126

C₂

C₂×SD₁₆×D₃₇

yes

—

	subgroup	G/H
center	C₂³	D₄×D₃₇
commutator, cosocle	C₂×C₄×C₃₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₃₇
Fitting	C₂×#106×C₃₇	C₂
socle, prime radical	C₂³×C₃₇	C₂×D₄
hypercenter	C₂×#106	D₃₇
hypocenter	C₃₇	C₂×#498
LCS	G ▷ C₂×C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#106

subgroup

G/H

center

C₂³

D₄×D₃₇

commutator, cosocle

C₂×C₄×C₃₇

C₂⁴

Frattini

C₂×C₄

C₂³×D₃₇

Fitting

C₂×#106×C₃₇

C₂

socle, prime radical

C₂³×C₃₇

C₂×D₄

D₃₇

C₃₇

G ▷ C₂×C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#106

N	H ≅ G/N	type
2. #14557930	C₂	direct (N × H)
26. C₂×C₈×C₃₇	Q₈	semidirect
35. C₂×#64436	Q₈	semidirect
49. C₈×C₃₇	C₂×Q₈	semidirect
50. #64436	C₂×Q₈	semidirect
61. C₂×C₃₇	#498	semidirect
70. C₃₇	C₂×#498	semidirect
92. C₂×C₈	Dic₇₄	semidirect
114. C₈	C₂×Dic₇₄	semidirect
124. C₂	#14557930	direct (N × H)

H ≅ G/N

type

2. #14557930

C₂

direct (N × H)

26. C₂×C₈×C₃₇

Q₈

semidirect

35. C₂×#64436

Q₈

semidirect

49. C₈×C₃₇

C₂×Q₈

semidirect

50. #64436

C₂×Q₈

semidirect

61. C₂×C₃₇

#498

semidirect

70. C₃₇

C₂×#498

semidirect

92. C₂×C₈

Dic₇₄

semidirect

114. C₈

C₂×Dic₇₄

semidirect

124. C₂

#14557930

direct (N × H)

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

subgroup	index
C₂×#11924535	2
#14557930	2
C₂×#11924468	2
C₂×#11924467	2
C₂×C₈×Dic₃₇	2
C₂×#106×C₃₇	2
C₂×#498	37

subgroup

index

C₂×C₈×Dic₃₇

C₂×#106×C₃₇

C₂×#498

Subgroups — C₂×#14557930 #18005094

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×#14557930 #18005094

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#14557930 #18005094