Group #18010317 — subgroups

Subgroups — #18010317

Order 4736.

200 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 78 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	#14558496	C₂	1	2	yes	maximal
2	#14558402	C₂	1	1	yes	maximal, characteristic
3	#14558401	C₂	1	1	yes	maximal, characteristic
4	#14558400	C₂	1	2	yes	maximal
5	#14557487	C₂	1	1	yes	maximal, characteristic
6	D₃₇⋊M₁₆	C₂²	1	2	yes	—
7	#11924615	C₂²	1	2	yes	—
8	#11924590	C₄	1	4	yes	complemented
9	C₈×#3866	C₄	1	4	yes	complemented
10	C₂×#10596929	C₂²	1	1	yes	characteristic
11	C₂×C₈×D₃₇	C₂²	1	1	yes	characteristic
12	#11924493	C₂²	1	1	yes	characteristic
13	#11924447	C₄	1	1	yes	characteristic, complemented
14	#11924446	C₄	1	1	yes	characteristic, complemented
15	C₈×Dic₃₇	C₄	1	1	yes	characteristic, complemented
16	C₄×C₈×C₃₇	C₄	1	1	yes	characteristic, complemented
17	C₂×C₄×D₃₇	C₂³	1	1	yes	characteristic
18	C₂²⋊#3866	—	2	2	no	—
19	#10596957	—	2	2	no	—
20	C₄⋊#3866	—	2	2	no	—
21	C₄×#3866	C₂×C₄	1	4	yes	—
22	#10596953	—	2	2	no	—
23	D₃₇⋊C₈	C₂×C₄	1	4	yes	—
24	C₂×C₈×C₃₇	C₂×C₄	1	1	yes	characteristic
25	C₂×C₈×C₃₇	C₂×C₄	1	1	yes	characteristic
26	C₄²×C₃₇	C₂×C₄	1	1	yes	characteristic
27	#10596938	—	2	1	no	—
28	#10596936	—	2	1	no	—
29	C₄×Dic₃₇	C₂×C₄	1	1	yes	characteristic
30	C₂×#64436	C₂×C₄	1	1	yes	characteristic
31	C₂×#64436	C₂×C₄	1	1	yes	characteristic
32	#10596929	—	2	2	no	—
33	C₈×D₃₇	C₂×C₄	1	2	yes	—
34	C₈×D₃₇	C₂×C₄	1	2	yes	—
35	C₂²×D₃₇	#41	1	1	yes	characteristic
36	C₂×#3866	—	2	4	no	—
37	C₂×C₄×C₃₇	—	2	1	no	—
38	C₂×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
39	C₂×Dic₃₇	—	2	1	no	—
40	C₂×Dic₃₇	#41	1	1	yes	characteristic
41	C₄×D₃₇	C₂²×C₄	1	2	yes	—
42	C₄×D₃₇	#41	1	2	yes	—
43	#64438	—	2	4	no	—
44	C₈×C₃₇	—	2	1	no	—
45	C₈×C₃₇	C₄²	1	1	yes	characteristic
46	C₈×C₃₇	C₄²	1	1	yes	characteristic
47	#64436	—	2	1	no	—
48	#64436	C₄²	1	2	yes	—
49	C₂²×C₃₇	#117	1	1	yes	characteristic
50	C₂×D₃₇	—	4	1	no	—
51	C₂×D₃₇	#117	1	2	yes	—
52	#3866	—	4	4	no	—
53	C₄×C₃₇	—	4	1	no	—
54	C₄×C₃₇	C₂×C₄²	1	1	yes	characteristic
55	C₄×C₃₇	#117	1	1	yes	characteristic
56	Dic₃₇	—	4	1	no	—
57	Dic₃₇	#117	1	2	yes	—
58	#1747	—	37	1	no	maximal, Sylow
59	C₂×C₃₇	—	2	1	no	—
60	C₂×C₃₇	#376	1	1	yes	characteristic
61	D₃₇	—	2	2	no	—
62	#405	—	37	2	no	—
63	#405	—	37	1	no	—
64	#338	—	37	1	no	—
65	H(3,ℤ/4)	—	37	2	no	—
66	H(3,ℤ/4)	—	37	1	no	—
67	C₃₇	#1747	1	1	yes	characteristic, complemented, Sylow
68	C₂×M₁₆	—	37	2	no	—
69	C₂×M₁₆	—	37	1	no	—
70	C₂²×C₈	—	37	1	no	—
71	#117	—	37	2	no	—
72	#117	—	37	1	no	—
73	#97	—	37	4	no	—
74	#97	—	37	1	no	—
75	#97	—	37	1	no	—
76	C₄×C₈	—	37	4	no	—
77	C₄×C₈	—	37	1	no	—
78	C₄×C₈	#3866	1	1	yes	characteristic, complemented
79	C₂²×C₄	—	37	1	no	—
80	M₁₆	—	74	2	no	—
81	M₁₆	—	74	2	no	—
82	M₁₆	—	74	2	no	—
83	C₂×C₈	—	37	1	no	—
84	C₂×C₈	—	37	2	no	—
85	C₂×C₈	—	37	2	no	—
86	C₂×C₈	—	37	4	no	—
87	C₂×C₈	—	37	1	no	—
88	C₂×C₈	C₂×#3866	1	1	yes	characteristic
89	C₂×C₈	C₂×#3866	1	1	yes	characteristic
90	C₄⋊C₄	—	74	1	no	—
91	C₄⋊C₄	—	74	2	no	—
92	C₂²⋊C₄	—	74	1	no	—
93	C₂²⋊C₄	—	74	2	no	—
94	C₄²	—	37	4	no	—
95	C₄²	—	37	1	no	—
96	C₄²	C₂×#3866	1	1	yes	characteristic
97	C₂³	—	37	1	no	—
98	C₂×C₄	—	2	1	no	—
99	C₂×C₄	—	37	1	no	—
100	C₂×C₄	—	37	2	no	—
101	C₂×C₄	—	37	2	no	—
102	C₂×C₄	—	74	4	no	—
103	C₂×C₄	—	74	1	no	—
104	C₂×C₄	C₂²×#3866	1	1	yes	characteristic
105	C₈	—	2	1	no	—
106	C₈	—	37	2	no	—
107	C₈	—	74	4	no	—
108	C₈	—	74	1	no	—
109	C₈	C₄×#3866	1	1	yes	characteristic
110	C₈	C₄×#3866	1	1	yes	characteristic
111	C₂²	—	37	2	no	—
112	C₂²	—	148	1	no	—
113	C₂²	#11924615	1	1	yes	characteristic
114	C₄	—	4	1	no	—
115	C₄	—	37	2	no	—
116	C₄	—	148	1	no	—
117	C₄	—	148	4	no	—
118	C₄	C₂×C₄×#3866	1	1	yes	characteristic
119	C₄	#11924615	1	1	yes	characteristic
120	C₂	—	2	1	no	—
121	C₂	—	74	2	no	—
122	C₂	#14558469	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14558496

C₂

yes

maximal

#14558402

C₂

yes

maximal, characteristic

#14558401

C₂

yes

maximal, characteristic

#14558400

C₂

yes

maximal

#14557487

C₂

yes

maximal, characteristic

D₃₇⋊M₁₆

C₂²

yes

—

#11924615

C₂²

yes

—

#11924590

C₄

yes

complemented

C₈×#3866

C₄

yes

complemented

C₂×#10596929

C₂²

yes

characteristic

C₂×C₈×D₃₇

C₂²

yes

characteristic

#11924493

C₂²

yes

characteristic

#11924447

C₄

yes

characteristic, complemented

#11924446

C₄

yes

characteristic, complemented

C₈×Dic₃₇

C₄

yes

characteristic, complemented

C₄×C₈×C₃₇

C₄

yes

characteristic, complemented

C₂×C₄×D₃₇

C₂³

yes

characteristic

C₂²⋊#3866

—

#10596957

—

C₄⋊#3866

—

C₄×#3866

C₂×C₄

yes

—

#10596953

—

D₃₇⋊C₈

C₂×C₄

yes

—

C₂×C₈×C₃₇

C₂×C₄

yes

characteristic

C₂×C₈×C₃₇

C₂×C₄

yes

characteristic

C₄²×C₃₇

C₂×C₄

yes

characteristic

#10596938

—

#10596936

—

C₄×Dic₃₇

C₂×C₄

yes

characteristic

C₂×#64436

C₂×C₄

yes

characteristic

C₂×#64436

C₂×C₄

yes

characteristic

#10596929

—

C₈×D₃₇

C₂×C₄

yes

—

C₈×D₃₇

C₂×C₄

yes

—

C₂²×D₃₇

#41

yes

characteristic

C₂×#3866

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

C₂²×C₄

yes

characteristic

C₂×Dic₃₇

—

C₂×Dic₃₇

#41

yes

characteristic

C₄×D₃₇

C₂²×C₄

yes

—

C₄×D₃₇

#41

yes

—

#64438

—

C₈×C₃₇

—

C₈×C₃₇

C₄²

yes

characteristic

C₈×C₃₇

C₄²

yes

characteristic

#64436

—

#64436

C₄²

yes

—

C₂²×C₃₇

#117

yes

characteristic

C₂×D₃₇

—

C₂×D₃₇

#117

yes

—

#3866

—

C₄×C₃₇

—

C₄×C₃₇

C₂×C₄²

yes

characteristic

C₄×C₃₇

#117

yes

characteristic

Dic₃₇

—

Dic₃₇

#117

yes

—

#1747

—

maximal, Sylow

C₂×C₃₇

—

C₂×C₃₇

#376

yes

characteristic

D₃₇

—

—

—

—

—

—

C₃₇

#1747

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

—

—

—

—

—

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

#3866

yes

characteristic, complemented

C₂²×C₄

—

M₁₆

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

yes

C₂×C₈

yes

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

—

C₄²

C₂×#3866

yes

characteristic

C₂³

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

C₂²×#3866

yes

characteristic

105

C₈

—

106

C₈

—

107

C₈

—

108

C₈

—

109

C₈

C₄×#3866

yes

characteristic

110

C₈

C₄×#3866

yes

characteristic

111

C₂²

—

112

C₂²

—

148

—

113

C₂²

#11924615

yes

characteristic

114

C₄

—

115

C₄

—

116

C₄

—

148

—

117

C₄

—

148

—

118

C₄

C₂×C₄×#3866

yes

characteristic

119

C₄

#11924615

yes

characteristic

120

C₂

—

121

C₂

—

122

C₂

#14558469

yes

characteristic

	subgroup	G/H
center	C₈	C₄×#3866
commutator	C₄×C₃₇	C₂×C₄²
Frattini	C₂×C₄	C₂²×#3866
Fitting	C₄×C₈×C₃₇	C₄
socle	C₂×C₃₇	#376
prime radical	C₂²×D₃₇	#41
cosocle	C₂×C₄×D₃₇	C₂³
hypercenter	C₄×C₈	#3866
hypocenter	C₃₇	#1747
LCS	G ▷ C₄×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₈ ◁ C₄×C₈

subgroup

G/H

center

C₈

C₄×#3866

commutator

C₄×C₃₇

C₂×C₄²

Frattini

C₂×C₄

C₂²×#3866

Fitting

C₄×C₈×C₃₇

C₄

C₂×C₃₇

C₂²×D₃₇

C₂×C₄×D₃₇

C₂³

C₄×C₈

C₃₇

G ▷ C₄×C₃₇ ▷ C₃₇

C₁ ◁ C₈ ◁ C₄×C₈

N	H ≅ G/N	type
8. #11924590	C₄	semidirect
9. C₈×#3866	C₄	semidirect
13. #11924447	C₄	semidirect
14. #11924446	C₄	semidirect
15. C₈×Dic₃₇	C₄	semidirect
16. C₄×C₈×C₃₇	C₄	semidirect
67. C₃₇	#1747	semidirect
78. C₄×C₈	#3866	semidirect

H ≅ G/N

type

8. #11924590

C₄

semidirect

9. C₈×#3866

C₄

semidirect

13. #11924447

C₄

semidirect

14. #11924446

C₄

semidirect

15. C₈×Dic₃₇

C₄

semidirect

16. C₄×C₈×C₃₇

C₄

semidirect

67. C₃₇

#1747

semidirect

78. C₄×C₈

#3866

semidirect

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

subgroup	index
#14558496	2
#14558402	2
#14558401	2
#14558400	2
#14557487	2
#1747	37

subgroup

index

Subgroups — #18010317

Conjugacy classes of subgroups

Complements

Maximal subgroups