Group #18011591 — subgroups

Subgroups — #18011591

Order 4736.

1424 conjugacy classes of subgroups; only the 750 normal subgroups are shown.

normal subgroups only

This group has 1424 conjugacy classes of subgroups — too many to draw. Only its 750 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	#14558612	C₂	1	6	yes	maximal, complemented
2	#14558611	C₂	1	9	yes	maximal, complemented
3	D₃₇⋊#131	C₂	1	6	yes	maximal, complemented
4	C₂×#11924623	C₂	1	9	yes	maximal, complemented
5	#14558368	C₂	1	1	yes	maximal, characteristic
6	#11924624	C₂²	1	2	yes	complemented
7	#11924624	C₂²	1	6	yes	complemented
8	#11924623	C₂²	1	18	yes	complemented
9	#11924623	C₂²	1	36	yes	complemented
10	#11924623	C₂²	1	18	yes	complemented
11	C₂×#10596953	C₂²	1	9	yes	complemented
12	D₃₇⋊M₁₆	C₂²	1	18	yes	complemented
13	C₂×#10596957	C₂²	1	18	yes	complemented
14	C₂×D₃₇⋊C₈	C₂²	1	9	yes	complemented
15	C₂²×#64438	C₂²	1	6	yes	complemented
16	#11924583	C₄	1	6	yes	—
17	#11924581	C₄	1	9	yes	—
18	C₂×#10596964	C₂²	1	9	yes	—
19	#41×D₃₇	C₂²	1	6	yes	—
20	H(5,𝔽₂)×C₃₇	C₄	1	1	yes	characteristic
21	#41×C₃₇	C₂×C₄	1	6	yes	—
22	C₂×D₄×C₃₇	C₂×C₄	1	9	yes	—
23	C₂×C₂²⋊D₃₇	C₂×C₄	1	18	yes	—
24	C₂²×Dic₃₇	C₂³	1	6	yes	—
25	#10596966	C₂³	1	2	yes	—
26	Q₈×D₃₇	C₂×C₄	1	6	yes	—
27	#10596964	C₂×C₄	1	36	yes	—
28	#10596964	C₂³	1	18	yes	—
29	D₄×D₃₇	C₂×C₄	1	18	yes	—
30	#10596962	C₂×C₄	1	18	yes	—
31	C₂×C₄×D₃₇	C₂³	1	9	yes	—
32	C₂×Dic₇₄	C₂×C₄	1	9	yes	—
33	#10596957	C₂³	1	36	yes	—
34	C₂×#64438	C₂³	1	9	yes	—
35	C₂×#64438	C₂³	1	18	yes	—
36	C₂×#64438	C₂³	1	9	yes	—
37	#10596953	C₂³	1	6	yes	—
38	#10596953	C₂³	1	18	yes	—
39	D₃₇⋊C₈	C₂³	1	6	yes	—
40	D₃₇⋊C₈	C₂³	1	18	yes	—
41	C₂³×C₃₇	C₂²×C₄	1	6	yes	—
42	C₂²×D₃₇	C₂²×C₄	1	9	yes	—
43	Q₈×C₃₇	C₂²×C₄	1	2	yes	—
44	D₄×C₃₇	C₂²×C₄	1	18	yes	—
45	C₂×C₄×C₃₇	C₂²×C₄	1	9	yes	—
46	C₂²⋊D₃₇	C₂²×C₄	1	36	yes	—
47	C₂×Dic₃₇	C₂²×C₄	1	18	yes	—
48	C₂×Dic₃₇	C₂⁴	1	9	yes	—
49	D₁₄₈	C₂²×C₄	1	6	yes	—
50	C₄×D₃₇	C₂²×C₄	1	18	yes	—
51	C₄×D₃₇	C₂⁴	1	6	yes	—
52	Dic₇₄	C₂²×C₄	1	18	yes	—
53	#64438	C₂⁴	1	1	yes	characteristic
54	#64438	C₂⁴	1	6	yes	—
55	#64438	C₂⁴	1	9	yes	—
56	C₂²×C₃₇	—	4	6	no	—
57	C₂²×C₃₇	C₂³×C₄	1	9	yes	—
58	C₂×D₃₇	—	4	9	no	—
59	C₂×D₃₇	C₂³×C₄	1	6	yes	—
60	C₄×C₃₇	C₂³×C₄	1	6	yes	—
61	Dic₃₇	C₂³×C₄	1	9	yes	—
62	Dic₃₇	C₂⁵	1	1	yes	characteristic
63	#3573	—	37	1	no	maximal, Sylow
64	C₂×C₃₇	—	2	9	no	—
65	C₂×C₃₇	C₂⁴×C₄	1	1	yes	characteristic
66	D₃₇	—	2	6	no	—
67	#585	—	37	1	no	—
68	#568	—	37	9	no	—
69	#568	—	37	6	no	—
70	C₂×#131	—	37	9	no	—
71	C₂×#131	—	37	6	no	—
72	C₃₇	#3573	1	1	yes	characteristic, complemented, Sylow
73	2₋¹⁺⁴	—	37	6	no	—
74	H(5,𝔽₂)	—	37	9	no	—
75	H(5,𝔽₂)	#3866	1	1	yes	characteristic
76	C₂×#41	—	37	6	no	—
77	C₂×#41	—	37	9	no	—
78	#131	—	37	18	no	—
79	#131	—	37	2	no	—
80	#131	—	37	36	no	—
81	#131	—	37	18	no	—
82	#131	—	37	6	no	—
83	C₂×M₁₆	—	37	18	no	—
84	C₂×M₁₆	—	37	18	no	—
85	C₂×M₁₆	—	37	9	no	—
86	C₂²×C₈	—	37	6	no	—
87	C₂²×C₈	—	37	9	no	—
88	#41	—	37	2	no	—
89	#41	—	37	36	no	—
90	#41	—	37	18	no	—
91	#41	—	37	18	no	—
92	#41	C₂×#3866	1	6	yes	—
93	C₂×Q₈	—	37	6	no	—
94	C₂×Q₈	—	37	9	no	—
95	C₂×D₄	—	37	18	no	—
96	C₂×D₄	—	37	18	no	—
97	C₂×D₄	C₂×#3866	1	9	yes	—
98	C₂²×C₄	—	37	6	no	—
99	C₂²×C₄	—	37	9	no	—
100	M₁₆	—	37	6	no	—
101	M₁₆	—	37	18	no	—
102	M₁₆	—	37	36	no	—
103	C₂×C₈	—	37	6	no	—
104	C₂×C₈	—	37	18	no	—
105	C₂×C₈	—	37	9	no	—
106	C₂×C₈	—	37	18	no	—
107	C₂×C₈	—	37	9	no	—
108	C₂³	—	37	9	no	—
109	C₂³	C₂²×#3866	1	6	yes	—
110	Q₈	—	37	18	no	—
111	Q₈	C₂²×#3866	1	2	yes	—
112	D₄	—	37	36	no	—
113	D₄	—	37	6	no	—
114	D₄	C₂²×#3866	1	18	yes	—
115	C₂×C₄	—	37	9	no	—
116	C₂×C₄	—	37	18	no	—
117	C₂×C₄	—	37	6	no	—
118	C₂×C₄	—	37	18	no	—
119	C₂×C₄	C₂²×#3866	1	9	yes	—
120	C₈	—	37	1	no	—
121	C₈	—	37	6	no	—
122	C₈	—	37	9	no	—
123	C₂²	—	4	6	no	—
124	C₂²	—	37	6	no	—
125	C₂²	—	148	9	no	—
126	C₂²	C₂³×#3866	1	9	yes	—
127	C₄	—	37	1	no	—
128	C₄	—	37	9	no	—
129	C₄	C₂³×#3866	1	6	yes	—
130	C₂	—	2	9	no	—
131	C₂	—	74	6	no	—
132	C₂	C₂⁴×#3866	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14558612

C₂

yes

maximal, complemented

#14558611

C₂

yes

maximal, complemented

D₃₇⋊#131

C₂

yes

maximal, complemented

C₂×#11924623

C₂

yes

maximal, complemented

#14558368

C₂

yes

maximal, characteristic

#11924624

C₂²

yes

complemented

#11924624

C₂²

yes

complemented

#11924623

C₂²

yes

complemented

#11924623

C₂²

yes

complemented

#11924623

C₂²

yes

complemented

C₂×#10596953

C₂²

yes

complemented

D₃₇⋊M₁₆

C₂²

yes

complemented

C₂×#10596957

C₂²

yes

complemented

C₂×D₃₇⋊C₈

C₂²

yes

complemented

C₂²×#64438

C₂²

yes

complemented

#11924583

C₄

yes

—

#11924581

C₄

yes

—

C₂×#10596964

C₂²

yes

—

#41×D₃₇

C₂²

yes

—

H(5,𝔽₂)×C₃₇

C₄

yes

characteristic

#41×C₃₇

C₂×C₄

yes

—

C₂×D₄×C₃₇

C₂×C₄

yes

—

C₂×C₂²⋊D₃₇

C₂×C₄

yes

—

C₂²×Dic₃₇

C₂³

yes

—

#10596966

C₂³

yes

—

Q₈×D₃₇

C₂×C₄

yes

—

#10596964

C₂×C₄

yes

—

#10596964

C₂³

yes

—

D₄×D₃₇

C₂×C₄

yes

—

#10596962

C₂×C₄

yes

—

C₂×C₄×D₃₇

C₂³

yes

—

C₂×Dic₇₄

C₂×C₄

yes

—

#10596957

C₂³

yes

—

C₂×#64438

C₂³

yes

—

C₂×#64438

C₂³

yes

—

C₂×#64438

C₂³

yes

—

#10596953

C₂³

yes

—

#10596953

C₂³

yes

—

D₃₇⋊C₈

C₂³

yes

—

D₃₇⋊C₈

C₂³

yes

—

C₂³×C₃₇

C₂²×C₄

yes

—

C₂²×D₃₇

C₂²×C₄

yes

—

Q₈×C₃₇

C₂²×C₄

yes

—

D₄×C₃₇

C₂²×C₄

yes

—

C₂×C₄×C₃₇

C₂²×C₄

yes

—

C₂²⋊D₃₇

C₂²×C₄

yes

—

C₂×Dic₃₇

C₂²×C₄

yes

—

C₂×Dic₃₇

C₂⁴

yes

—

D₁₄₈

C₂²×C₄

yes

—

C₄×D₃₇

C₂²×C₄

yes

—

C₄×D₃₇

C₂⁴

yes

—

Dic₇₄

C₂²×C₄

yes

—

#64438

C₂⁴

yes

characteristic

#64438

C₂⁴

yes

—

#64438

C₂⁴

yes

—

C₂²×C₃₇

—

C₂²×C₃₇

C₂³×C₄

yes

—

C₂×D₃₇

—

C₂×D₃₇

C₂³×C₄

yes

—

C₄×C₃₇

C₂³×C₄

yes

—

Dic₃₇

C₂³×C₄

yes

—

Dic₃₇

C₂⁵

yes

characteristic

#3573

—

maximal, Sylow

C₂×C₃₇

—

C₂×C₃₇

C₂⁴×C₄

yes

characteristic

D₃₇

—

—

—

—

—

—

C₃₇

#3573

yes

characteristic, complemented, Sylow

2₋¹⁺⁴

—

—

yes

C₂×#41

—

C₂×#41

—

—

—

—

—

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

—

—

—

—

—

yes

—

C₂×Q₈

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

C₂×#3866

yes

—

C₂²×C₄

—

C₂²×C₄

—

100

M₁₆

—

101

M₁₆

—

102

M₁₆

—

103

C₂×C₈

—

104

C₂×C₈

—

105

C₂×C₈

—

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂³

—

109

C₂³

C₂²×#3866

yes

—

110

Q₈

—

111

Q₈

C₂²×#3866

yes

—

112

D₄

—

113

D₄

—

114

D₄

C₂²×#3866

yes

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

C₂²×#3866

yes

—

120

C₈

—

121

C₈

—

122

C₈

—

123

C₂²

—

124

C₂²

—

125

C₂²

—

148

—

126

C₂²

C₂³×#3866

yes

—

127

C₄

—

128

C₄

—

129

C₄

C₂³×#3866

yes

—

130

C₂

—

131

C₂

—

132

C₂

C₂⁴×#3866

yes

characteristic

	subgroup	G/H
center, Frattini	C₂	C₂⁴×#3866
commutator, socle	C₂×C₃₇	C₂⁴×C₄
Fitting	H(5,𝔽₂)×C₃₇	C₄
prime radical	#14558368	C₂
cosocle	Dic₃₇	C₂⁵
hypercenter	H(5,𝔽₂)	#3866
hypocenter	C₃₇	#3573
LCS	G ▷ C₂×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂ ◁ H(5,𝔽₂)

subgroup

G/H

center, Frattini

C₂

C₂⁴×#3866

commutator, socle

C₂×C₃₇

C₂⁴×C₄

Fitting

H(5,𝔽₂)×C₃₇

C₄

prime radical

#14558368

C₂

cosocle

Dic₃₇

C₂⁵

C₃₇

G ▷ C₂×C₃₇ ▷ C₃₇

C₁ ◁ C₂ ◁ H(5,𝔽₂)

N	H ≅ G/N	type
1. #14558612	C₂	semidirect
2. #14558611	C₂	semidirect
3. D₃₇⋊#131	C₂	semidirect
4. C₂×#11924623	C₂	semidirect
6. #11924624	C₂²	semidirect
7. #11924624	C₂²	semidirect
8. #11924623	C₂²	semidirect
9. #11924623	C₂²	semidirect
10. #11924623	C₂²	semidirect
11. C₂×#10596953	C₂²	semidirect
12. D₃₇⋊M₁₆	C₂²	semidirect
13. C₂×#10596957	C₂²	semidirect
14. C₂×D₃₇⋊C₈	C₂²	semidirect
15. C₂²×#64438	C₂²	semidirect
72. C₃₇	#3573	semidirect

H ≅ G/N

type

1. #14558612

C₂

semidirect

2. #14558611

C₂

semidirect

3. D₃₇⋊#131

C₂

semidirect

4. C₂×#11924623

C₂

semidirect

6. #11924624

C₂²

semidirect

7. #11924624

C₂²

semidirect

8. #11924623

C₂²

semidirect

9. #11924623

C₂²

semidirect

10. #11924623

C₂²

semidirect

11. C₂×#10596953

C₂²

semidirect

12. D₃₇⋊M₁₆

C₂²

semidirect

13. C₂×#10596957

C₂²

semidirect

14. C₂×D₃₇⋊C₈

C₂²

semidirect

15. C₂²×#64438

C₂²

semidirect

72. C₃₇

#3573

semidirect

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

subgroup	index
#14558612	2
#14558611	2
D₃₇⋊#131	2
C₂×#11924623	2
#14558368	2
#3573	37

subgroup

index

D₃₇⋊#131

Subgroups — #18011591

Conjugacy classes of subgroups

Complements

Maximal subgroups