Group #18011590 — subgroups

Subgroups — C₂×#14558613 #18011590

Order 4736.

1460 conjugacy classes of subgroups; only the 754 normal subgroups are shown.

normal subgroups only

This group has 1460 conjugacy classes of subgroups — too many to draw. Only its 754 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	#14558613	C₂	1	16	yes	maximal, direct product factor
2	C₂×#11924624	C₂	1	8	yes	maximal, complemented
3	C₂×D₃₇⋊M₁₆	C₂	1	6	yes	maximal, complemented
4	C₂²×#10596966	C₂	1	1	yes	maximal, characteristic
5	#11924624	C₂²	1	64	yes	complemented
6	C₂×#10596953	C₂²	1	12	yes	complemented
7	D₃₇⋊M₁₆	C₂²	1	48	yes	complemented
8	C₂×#10596957	C₂²	1	4	yes	—
9	C₂×D₃₇⋊C₈	C₂²	1	12	yes	complemented
10	C₂×#10596966	C₄	1	4	yes	—
11	C₂×#10596966	C₄	1	8	yes	—
12	C₂×#10596966	C₂²	1	4	yes	—
13	C₂×#10596966	C₂²	1	8	yes	—
14	C₂²×D₁₄₈	C₄	1	3	yes	—
15	C₂²×C₄×D₃₇	C₂²	1	3	yes	—
16	C₂²×Q₈×C₃₇	C₄	1	1	yes	characteristic
17	C₂³×D₃₇	C₂×C₄	1	3	yes	—
18	C₂×Q₈×C₃₇	C₂×C₄	1	4	yes	—
19	C₂×Q₈×C₃₇	C₂×C₄	1	8	yes	—
20	C₂²×C₄×C₃₇	C₂×C₄	1	3	yes	—
21	C₂²×Dic₃₇	C₂³	1	1	yes	characteristic
22	#10596966	C₂×C₄	1	16	yes	—
23	#10596966	C₂×C₄	1	32	yes	—
24	#10596966	C₂³	1	16	yes	—
25	C₂×D₁₄₈	C₂×C₄	1	12	yes	—
26	C₂×D₁₄₈	C₂×C₄	1	24	yes	—
27	C₂×C₄×D₃₇	C₂×C₄	1	6	yes	—
28	C₂×C₄×D₃₇	C₂×C₄	1	12	yes	—
29	C₂×C₄×D₃₇	C₂³	1	6	yes	—
30	C₂×C₄×D₃₇	C₂³	1	12	yes	—
31	#10596957	C₂³	1	16	yes	—
32	C₂×#64438	C₂³	1	8	yes	—
33	#10596953	C₂³	1	48	yes	complemented
34	D₃₇⋊C₈	C₂³	1	48	yes	complemented
35	C₂³×C₃₇	C₂²×C₄	1	1	yes	characteristic
36	C₂²×D₃₇	—	4	6	no	—
37	C₂²×D₃₇	C₂²×C₄	1	6	yes	—
38	C₂²×D₃₇	C₂²×C₄	1	12	yes	—
39	Q₈×C₃₇	C₂²×C₄	1	16	yes	—
40	C₂×C₄×C₃₇	C₂²×C₄	1	12	yes	—
41	C₂×C₄×C₃₇	C₂²×C₄	1	6	yes	—
42	C₂×Dic₃₇	C₂²×C₄	1	2	yes	—
43	C₂×Dic₃₇	C₂²×C₄	1	1	yes	characteristic
44	C₂×Dic₃₇	C₂⁴	1	1	yes	characteristic
45	C₂×Dic₃₇	C₂⁴	1	2	yes	—
46	D₁₄₈	C₂²×C₄	1	48	yes	—
47	C₄×D₃₇	C₂²×C₄	1	12	yes	—
48	C₄×D₃₇	C₂²×C₄	1	24	yes	—
49	C₄×D₃₇	C₂⁴	1	12	yes	—
50	#64438	C₂⁴	1	16	yes	—
51	C₂²×C₃₇	—	2	2	no	—
52	C₂²×C₃₇	C₂³×C₄	1	2	yes	—
53	C₂²×C₃₇	C₂³×C₄	1	1	yes	characteristic
54	C₂×D₃₇	—	2	12	no	—
55	C₂×D₃₇	—	4	12	no	—
56	C₂×D₃₇	C₂³×C₄	1	12	yes	—
57	C₄×C₃₇	C₂³×C₄	1	12	yes	—
58	Dic₃₇	C₂³×C₄	1	1	yes	characteristic
59	Dic₃₇	C₂³×C₄	1	2	yes	—
60	Dic₃₇	C₂⁵	1	1	yes	characteristic
61	C₂×#568	—	37	1	no	maximal, Sylow
62	C₂×C₃₇	—	2	2	no	—
63	C₂×C₃₇	#568	1	2	yes	complemented
64	C₂×C₃₇	C₂⁴×C₄	1	1	yes	characteristic
65	D₃₇	—	2	12	no	—
66	C₂²×#41	—	37	1	no	—
67	#568	—	37	16	no	—
68	C₂×#131	—	37	8	no	—
69	C₂²×M₁₆	—	37	6	no	—
70	C₃₇	C₂×#568	1	1	yes	characteristic, complemented, Sylow
71	C₂×#41	—	37	4	no	—
72	C₂×#41	—	37	8	no	—
73	C₂×#41	—	37	8	no	—
74	C₂×#41	—	37	4	no	—
75	C₂²×Q₈	#3866	1	1	yes	characteristic
76	C₂²×D₄	—	37	3	no	—
77	C₂³×C₄	—	37	3	no	—
78	#131	—	37	64	no	—
79	C₂×M₁₆	—	37	4	no	—
80	C₂×M₁₆	—	37	12	no	—
81	C₂×M₁₆	—	37	48	no	—
82	C₂²×C₈	—	37	12	no	—
83	C₂⁴	—	37	3	no	—
84	#41	—	37	16	no	—
85	#41	—	37	32	no	—
86	#41	—	37	16	no	—
87	C₂×Q₈	C₂×#3866	1	8	yes	—
88	C₂×Q₈	C₂×#3866	1	4	yes	—
89	C₂×D₄	—	37	24	no	—
90	C₂×D₄	—	37	12	no	—
91	C₂²×C₄	—	37	1	no	—
92	C₂²×C₄	—	37	12	no	—
93	C₂²×C₄	—	37	6	no	—
94	C₂²×C₄	—	37	6	no	—
95	C₂²×C₄	—	37	12	no	—
96	C₂²×C₄	C₂×#3866	1	3	yes	—
97	M₁₆	—	37	48	no	—
98	M₁₆	—	37	16	no	—
99	C₂×C₈	—	37	48	no	—
100	C₂×C₈	—	37	8	no	—
101	C₂³	—	37	6	no	—
102	C₂³	—	37	12	no	—
103	C₂³	—	148	6	no	—
104	C₂³	C₂²×#3866	1	1	yes	characteristic
105	Q₈	C₂²×#3866	1	16	yes	—
106	D₄	—	37	48	no	—
107	C₂×C₄	—	37	2	no	—
108	C₂×C₄	—	37	2	no	—
109	C₂×C₄	—	37	1	no	—
110	C₂×C₄	—	37	1	no	—
111	C₂×C₄	—	37	12	no	—
112	C₂×C₄	—	37	12	no	—
113	C₂×C₄	—	37	24	no	—
114	C₂×C₄	C₂²×#3866	1	12	yes	—
115	C₂×C₄	C₂²×#3866	1	6	yes	—
116	C₈	—	37	16	no	—
117	C₂²	—	2	2	no	—
118	C₂²	—	37	12	no	—
119	C₂²	—	74	12	no	—
120	C₂²	—	148	12	no	—
121	C₂²	C₂³×#3866	1	2	yes	—
122	C₂²	C₂³×#3866	1	1	yes	characteristic
123	C₄	—	37	1	no	—
124	C₄	—	37	2	no	—
125	C₄	—	37	1	no	—
126	C₄	C₂³×#3866	1	12	yes	—
127	C₂	—	2	2	no	—
128	C₂	—	74	12	no	—
129	C₂	#14558613	1	2	yes	direct product factor
130	C₂	C₂⁴×#3866	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14558613

C₂

yes

maximal, direct product factor

C₂×#11924624

C₂

yes

maximal, complemented

C₂×D₃₇⋊M₁₆

C₂

yes

maximal, complemented

C₂²×#10596966

C₂

yes

maximal, characteristic

#11924624

C₂²

yes

complemented

C₂×#10596953

C₂²

yes

complemented

D₃₇⋊M₁₆

C₂²

yes

complemented

C₂×#10596957

C₂²

yes

—

C₂×D₃₇⋊C₈

C₂²

yes

complemented

C₂×#10596966

C₄

yes

—

C₂×#10596966

C₄

yes

—

C₂×#10596966

C₂²

yes

—

C₂×#10596966

C₂²

yes

—

C₂²×D₁₄₈

C₄

yes

—

C₂²×C₄×D₃₇

C₂²

yes

—

C₂²×Q₈×C₃₇

C₄

yes

characteristic

C₂³×D₃₇

C₂×C₄

yes

—

C₂×Q₈×C₃₇

C₂×C₄

yes

—

C₂×Q₈×C₃₇

C₂×C₄

yes

—

C₂²×C₄×C₃₇

C₂×C₄

yes

—

C₂²×Dic₃₇

C₂³

yes

characteristic

#10596966

C₂×C₄

yes

—

#10596966

C₂×C₄

yes

—

#10596966

C₂³

yes

—

C₂×D₁₄₈

C₂×C₄

yes

—

C₂×D₁₄₈

C₂×C₄

yes

—

C₂×C₄×D₃₇

C₂×C₄

yes

—

C₂×C₄×D₃₇

C₂×C₄

yes

—

C₂×C₄×D₃₇

C₂³

yes

—

C₂×C₄×D₃₇

C₂³

yes

—

#10596957

C₂³

yes

—

C₂×#64438

C₂³

yes

—

#10596953

C₂³

yes

complemented

D₃₇⋊C₈

C₂³

yes

complemented

C₂³×C₃₇

C₂²×C₄

yes

characteristic

C₂²×D₃₇

—

C₂²×D₃₇

C₂²×C₄

yes

—

C₂²×D₃₇

C₂²×C₄

yes

—

Q₈×C₃₇

C₂²×C₄

yes

—

C₂×C₄×C₃₇

C₂²×C₄

yes

—

C₂×C₄×C₃₇

C₂²×C₄

yes

—

C₂×Dic₃₇

C₂²×C₄

yes

—

C₂×Dic₃₇

C₂²×C₄

yes

characteristic

C₂×Dic₃₇

C₂⁴

yes

characteristic

C₂×Dic₃₇

C₂⁴

yes

—

D₁₄₈

C₂²×C₄

yes

—

C₄×D₃₇

C₂²×C₄

yes

—

C₄×D₃₇

C₂²×C₄

yes

—

C₄×D₃₇

C₂⁴

yes

—

#64438

C₂⁴

yes

—

C₂²×C₃₇

—

C₂²×C₃₇

C₂³×C₄

yes

—

C₂²×C₃₇

C₂³×C₄

yes

characteristic

C₂×D₃₇

—

C₂×D₃₇

—

C₂×D₃₇

C₂³×C₄

yes

—

C₄×C₃₇

C₂³×C₄

yes

—

Dic₃₇

C₂³×C₄

yes

characteristic

Dic₃₇

C₂³×C₄

yes

—

Dic₃₇

C₂⁵

yes

characteristic

C₂×#568

—

maximal, Sylow

C₂×C₃₇

—

C₂×C₃₇

#568

yes

complemented

C₂×C₃₇

C₂⁴×C₄

yes

characteristic

D₃₇

—

C₂²×#41

—

#568

—

C₂×#131

—

C₂²×M₁₆

—

C₃₇

C₂×#568

yes

characteristic, complemented, Sylow

C₂×#41

—

C₂×#41

—

C₂×#41

—

C₂×#41

—

C₂²×Q₈

#3866

yes

characteristic

C₂²×D₄

—

C₂³×C₄

—

#131

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

—

C₂⁴

—

#41

—

#41

—

#41

—

C₂×Q₈

C₂×#3866

yes

—

C₂×Q₈

C₂×#3866

yes

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×#3866

yes

—

M₁₆

—

M₁₆

—

C₂×C₈

—

100

C₂×C₈

—

101

C₂³

—

102

C₂³

—

103

C₂³

—

148

—

104

C₂³

C₂²×#3866

yes

characteristic

105

Q₈

C₂²×#3866

yes

—

106

D₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

C₂²×#3866

yes

—

115

C₂×C₄

C₂²×#3866

yes

—

116

C₈

—

117

C₂²

—

118

C₂²

—

119

C₂²

—

120

C₂²

—

148

—

121

C₂²

C₂³×#3866

yes

—

122

C₂²

C₂³×#3866

yes

characteristic

123

C₄

—

124

C₄

—

125

C₄

—

126

C₄

C₂³×#3866

yes

—

127

C₂

—

128

C₂

—

129

C₂

#14558613

yes

direct product factor

130

C₂

C₂⁴×#3866

yes

characteristic

	subgroup	G/H
center	C₂²	C₂³×#3866
commutator	C₂×C₃₇	C₂⁴×C₄
Frattini	C₂	C₂⁴×#3866
Fitting	C₂²×Q₈×C₃₇	C₄
socle	C₂²×C₃₇	C₂³×C₄
prime radical	C₂²×#10596966	C₂
cosocle	Dic₃₇	C₂⁵
hypercenter	C₂²×Q₈	#3866
hypocenter	C₃₇	C₂×#568
LCS	G ▷ C₂×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂² ◁ C₂²×Q₈

subgroup

G/H

center

C₂²

C₂³×#3866

commutator

C₂×C₃₇

C₂⁴×C₄

Frattini

C₂

C₂⁴×#3866

Fitting

C₂²×Q₈×C₃₇

C₄

socle

C₂²×C₃₇

C₂³×C₄

prime radical

C₂²×#10596966

C₂

cosocle

Dic₃₇

C₂⁵

C₂²×Q₈

C₃₇

G ▷ C₂×C₃₇ ▷ C₃₇

C₁ ◁ C₂² ◁ C₂²×Q₈

N	H ≅ G/N	type
1. #14558613	C₂	direct (N × H)
2. C₂×#11924624	C₂	semidirect
3. C₂×D₃₇⋊M₁₆	C₂	semidirect
5. #11924624	C₂²	semidirect
6. C₂×#10596953	C₂²	semidirect
7. D₃₇⋊M₁₆	C₂²	semidirect
9. C₂×D₃₇⋊C₈	C₂²	semidirect
33. #10596953	C₂³	semidirect
34. D₃₇⋊C₈	C₂³	semidirect
63. C₂×C₃₇	#568	semidirect
70. C₃₇	C₂×#568	semidirect
129. C₂	#14558613	direct (N × H)

H ≅ G/N

type

1. #14558613

C₂

direct (N × H)

2. C₂×#11924624

C₂

semidirect

3. C₂×D₃₇⋊M₁₆

C₂

semidirect

5. #11924624

C₂²

semidirect

6. C₂×#10596953

C₂²

semidirect

7. D₃₇⋊M₁₆

C₂²

semidirect

9. C₂×D₃₇⋊C₈

C₂²

semidirect

33. #10596953

C₂³

semidirect

34. D₃₇⋊C₈

C₂³

semidirect

63. C₂×C₃₇

#568

semidirect

70. C₃₇

C₂×#568

semidirect

129. C₂

#14558613

direct (N × H)

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

subgroup	index
#14558613	2
C₂×#11924624	2
C₂×D₃₇⋊M₁₆	2
C₂²×#10596966	2
C₂×#568	37

subgroup

index

C₂×D₃₇⋊M₁₆

Subgroups — C₂×#14558613 #18011590

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×#14558613 #18011590

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#14558613 #18011590