Group #18010529 — subgroups

Subgroups — #18010529

Order 4736.

300 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 300 conjugacy classes of subgroups — too many to draw. Only its 138 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×#11924603	C₂	1	4	yes	maximal
2	C₂×C₄⋊#64438	C₂	1	2	yes	maximal
3	C₄²×Dic₃₇	C₂	1	1	yes	maximal, characteristic
4	C₂²×#64438	C₂²	1	4	yes	—
5	#11924603	—	2	8	no	—
6	C₄⋊#64438	—	2	4	no	—
7	C₂×C₄×Dic₃₇	C₄	1	2	yes	—
8	C₂×C₄×Dic₃₇	C₄	1	1	yes	characteristic
9	C₂×C₄×Dic₃₇	C₂²	1	1	yes	characteristic
10	C₂×C₄×Dic₃₇	C₂²	1	2	yes	—
11	C₂×C₄²×C₃₇	C₄	1	1	yes	characteristic
12	C₂²×C₄×C₃₇	C₂×C₄	1	2	yes	—
13	C₂²×C₄×C₃₇	C₂×C₄	1	1	yes	characteristic
14	C₂²×Dic₃₇	C₂×C₄	1	1	yes	characteristic
15	C₂²×Dic₃₇	C₂×C₄	1	2	yes	—
16	C₂²×Dic₃₇	C₂³	1	1	yes	characteristic
17	C₂×#64438	—	2	4	no	—
18	C₂×#64438	—	2	8	no	—
19	C₄²×C₃₇	—	2	2	no	—
20	C₄×Dic₃₇	—	2	4	no	—
21	C₄×Dic₃₇	C₈	1	4	yes	complemented
22	C₄×Dic₃₇	D₄	1	2	yes	—
23	C₄×Dic₃₇	D₄	1	4	yes	—
24	C₄×Dic₃₇	Q₈	1	2	yes	—
25	C₄×Dic₃₇	Q₈	1	4	yes	—
26	C₂³×C₃₇	C₂²×C₄	1	1	yes	characteristic
27	C₂×C₄×C₃₇	—	2	2	no	—
28	C₂×C₄×C₃₇	—	2	2	no	—
29	C₂×C₄×C₃₇	C₄⋊C₄	1	4	yes	—
30	C₂×C₄×C₃₇	C₄⋊C₄	1	2	yes	—
31	C₂×C₄×C₃₇	C₂×C₈	1	2	yes	—
32	C₂×C₄×C₃₇	M₁₆	1	2	yes	—
33	C₂×Dic₃₇	—	2	2	no	—
34	C₂×Dic₃₇	C₄⋊C₄	1	2	yes	—
35	C₂×Dic₃₇	C₄⋊C₄	1	4	yes	—
36	C₂×Dic₃₇	C₂×C₈	1	4	yes	—
37	C₂×Dic₃₇	M₁₆	1	4	yes	—
38	C₂×Dic₃₇	C₂×D₄	1	1	yes	characteristic
39	C₂×Dic₃₇	C₂×D₄	1	2	yes	—
40	C₂×Dic₃₇	C₂×Q₈	1	2	yes	—
41	C₂×Dic₃₇	C₂×Q₈	1	1	yes	characteristic
42	#64438	—	4	4	no	—
43	C₂²×C₃₇	C₂×C₄⋊C₄	1	2	yes	—
44	C₂²×C₃₇	C₂×C₄⋊C₄	1	1	yes	characteristic
45	C₂²×C₃₇	C₂²×C₈	1	1	yes	characteristic
46	C₂²×C₃₇	C₂×M₁₆	1	1	yes	characteristic
47	C₂²×C₃₇	C₂×M₁₆	1	2	yes	—
48	C₄×C₃₇	—	2	4	no	—
49	C₄×C₃₇	C₄⋊C₈	1	4	yes	complemented
50	Dic₃₇	—	2	2	no	—
51	Dic₃₇	C₄⋊C₈	1	8	yes	complemented
52	Dic₃₇	#127	1	1	yes	characteristic
53	Dic₃₇	C₄⋊Q₈	1	1	yes	characteristic
54	Dic₃₇	C₄⋊Q₈	1	2	yes	—
55	#1842	—	37	1	no	maximal, Sylow
56	C₂×C₃₇	#384	1	1	yes	characteristic
57	C₂×C₃₇	C₂×C₄⋊C₈	1	2	yes	—
58	C₂×C₃₇	C₂×C₄⋊C₈	1	1	yes	characteristic
59	C₂×C₃₇	#423	1	2	yes	—
60	C₂×C₃₇	#423	1	1	yes	characteristic
61	C₂×C₄⋊C₈	—	37	2	no	—
62	C₂×C₄⋊C₈	—	37	4	no	—
63	C₄³	—	37	1	no	—
64	C₃₇	#1842	1	1	yes	characteristic, complemented, Sylow
65	C₂²×C₈	—	37	4	no	—
66	C₂×C₄²	—	37	1	no	—
67	C₂×C₄²	—	37	2	no	—
68	C₂×C₄²	—	37	1	no	—
69	C₂×C₄²	—	37	2	no	—
70	C₂×C₄²	#3866	1	1	yes	characteristic
71	C₄⋊C₈	—	74	4	no	—
72	C₄⋊C₈	—	74	8	no	—
73	C₂²×C₄	—	37	2	no	—
74	C₂²×C₄	—	37	1	no	—
75	C₂²×C₄	—	37	1	no	—
76	C₂²×C₄	C₂×#3866	1	2	yes	—
77	C₂²×C₄	C₂×#3866	1	1	yes	characteristic
78	C₂×C₈	—	74	4	no	—
79	C₂×C₈	—	74	8	no	—
80	C₄²	—	2	2	no	—
81	C₄²	—	37	4	no	—
82	C₄²	—	37	4	no	—
83	C₄²	—	37	2	no	—
84	C₄²	—	37	4	no	—
85	C₄²	—	37	2	no	—
86	C₄²	—	74	4	no	—
87	C₂³	C₂²×#3866	1	1	yes	characteristic
88	C₂×C₄	—	2	2	no	—
89	C₂×C₄	—	2	2	no	—
90	C₂×C₄	—	37	2	no	—
91	C₂×C₄	—	37	1	no	—
92	C₂×C₄	—	37	4	no	—
93	C₂×C₄	—	37	2	no	—
94	C₂×C₄	—	37	1	no	—
95	C₂×C₄	—	37	4	no	—
96	C₂×C₄	—	37	4	no	—
97	C₂×C₄	—	37	2	no	—
98	C₂×C₄	—	74	2	no	—
99	C₂×C₄	D₃₇⋊C₈	1	2	yes	—
100	C₂×C₄	C₄⋊#3866	1	4	yes	—
101	C₂×C₄	C₄⋊#3866	1	2	yes	—
102	C₂×C₄	#10596957	1	2	yes	—
103	C₈	—	148	4	no	—
104	C₂²	C₂×C₄⋊#3866	1	2	yes	—
105	C₂²	C₂×C₄⋊#3866	1	1	yes	characteristic
106	C₂²	C₂×D₃₇⋊C₈	1	1	yes	characteristic
107	C₂²	C₂×#10596957	1	1	yes	characteristic
108	C₂²	D₃₇⋊M₁₆	1	2	yes	—
109	C₄	—	2	4	no	—
110	C₄	—	37	1	no	—
111	C₄	—	37	2	no	—
112	C₄	—	37	1	no	—
113	C₄	—	37	8	no	—
114	C₄	—	74	2	no	—
115	C₄	#11924603	1	4	yes	complemented
116	C₂	#14558479	1	1	yes	characteristic
117	C₂	D₃₇⋊(C₄⋊C₈)	1	2	yes	—
118	C₂	C₂×#11924603	1	1	yes	characteristic
119	C₂	#14558536	1	2	yes	—
120	C₂	#14558538	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#11924603

C₂

yes

maximal

C₂×C₄⋊#64438

C₂

yes

maximal

C₄²×Dic₃₇

C₂

yes

maximal, characteristic

C₂²×#64438

C₂²

yes

—

#11924603

—

C₄⋊#64438

—

C₂×C₄×Dic₃₇

C₄

yes

—

C₂×C₄×Dic₃₇

C₄

yes

characteristic

C₂×C₄×Dic₃₇

C₂²

yes

characteristic

C₂×C₄×Dic₃₇

C₂²

yes

—

C₂×C₄²×C₃₇

C₄

yes

characteristic

C₂²×C₄×C₃₇

C₂×C₄

yes

—

C₂²×C₄×C₃₇

C₂×C₄

yes

characteristic

C₂²×Dic₃₇

C₂×C₄

yes

characteristic

C₂²×Dic₃₇

C₂×C₄

yes

—

C₂²×Dic₃₇

C₂³

yes

characteristic

C₂×#64438

—

C₂×#64438

—

C₄²×C₃₇

—

C₄×Dic₃₇

—

C₄×Dic₃₇

C₈

yes

complemented

C₄×Dic₃₇

D₄

yes

—

C₄×Dic₃₇

D₄

yes

—

C₄×Dic₃₇

Q₈

yes

—

C₄×Dic₃₇

Q₈

yes

—

C₂³×C₃₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

C₄⋊C₄

yes

—

C₂×C₄×C₃₇

C₄⋊C₄

yes

—

C₂×C₄×C₃₇

C₂×C₈

yes

—

C₂×C₄×C₃₇

M₁₆

yes

—

C₂×Dic₃₇

—

C₂×Dic₃₇

C₄⋊C₄

yes

—

C₂×Dic₃₇

C₄⋊C₄

yes

—

C₂×Dic₃₇

C₂×C₈

yes

—

C₂×Dic₃₇

M₁₆

yes

—

C₂×Dic₃₇

C₂×D₄

yes

characteristic

C₂×Dic₃₇

C₂×D₄

yes

—

C₂×Dic₃₇

C₂×Q₈

yes

—

C₂×Dic₃₇

C₂×Q₈

yes

characteristic

#64438

—

C₂²×C₃₇

C₂×C₄⋊C₄

yes

—

C₂²×C₃₇

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₃₇

C₂²×C₈

yes

characteristic

C₂²×C₃₇

C₂×M₁₆

yes

characteristic

C₂²×C₃₇

C₂×M₁₆

yes

—

C₄×C₃₇

—

C₄×C₃₇

C₄⋊C₈

yes

complemented

Dic₃₇

—

Dic₃₇

C₄⋊C₈

yes

complemented

Dic₃₇

#127

yes

characteristic

Dic₃₇

C₄⋊Q₈

yes

characteristic

Dic₃₇

C₄⋊Q₈

yes

—

—

C₂×C₃₇

yes

C₂×C₃₇

C₂×C₄⋊C₈

yes

—

C₂×C₃₇

C₂×C₄⋊C₈

yes

characteristic

C₂×C₃₇

#423

yes

—

C₂×C₃₇

#423

yes

characteristic

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

C₄³

—

C₃₇

#1842

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

#3866

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×#3866

yes

—

C₂²×C₄

C₂×#3866

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₂³

C₂²×#3866

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

D₃₇⋊C₈

yes

—

100

C₂×C₄

C₄⋊#3866

yes

—

101

C₂×C₄

C₄⋊#3866

yes

—

102

C₂×C₄

#10596957

yes

—

103

C₈

—

148

—

104

C₂²

C₂×C₄⋊#3866

yes

—

105

C₂²

C₂×C₄⋊#3866

yes

characteristic

106

C₂²

C₂×D₃₇⋊C₈

yes

characteristic

107

C₂²

C₂×#10596957

yes

characteristic

108

C₂²

D₃₇⋊M₁₆

yes

—

109

C₄

—

110

C₄

—

111

C₄

—

112

C₄

—

113

C₄

—

114

C₄

—

115

C₄

#11924603

yes

complemented

116

C₂

#14558479

yes

characteristic

117

C₂

D₃₇⋊(C₄⋊C₈)

yes

—

118

C₂

C₂×#11924603

yes

characteristic

119

C₂

#14558536

yes

—

120

C₂

#14558538

yes

characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂²×#3866
commutator	C₂²×C₃₇	C₂²×C₈
Fitting	C₂×C₄²×C₃₇	C₄
socle, prime radical	C₂³×C₃₇	C₂²×C₄
cosocle	C₂²×Dic₃₇	C₂³
hypercenter	C₂×C₄²	#3866
hypocenter	C₃₇	#1842
LCS	G ▷ C₂²×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄²

subgroup

G/H

center, Frattini

C₂³

C₂²×#3866

commutator

C₂²×C₃₇

C₂²×C₈

Fitting

C₂×C₄²×C₃₇

C₄

socle, prime radical

C₂³×C₃₇

C₂²×C₄

cosocle

C₂²×Dic₃₇

C₂³

C₂×C₄²

C₃₇

G ▷ C₂²×C₃₇ ▷ C₃₇

C₁ ◁ C₂³ ◁ C₂×C₄²

N	H ≅ G/N	type
21. C₄×Dic₃₇	C₈	semidirect
49. C₄×C₃₇	C₄⋊C₈	semidirect
51. Dic₃₇	C₄⋊C₈	semidirect
64. C₃₇	#1842	semidirect
115. C₄	#11924603	semidirect

H ≅ G/N

type

21. C₄×Dic₃₇

C₈

semidirect

49. C₄×C₃₇

C₄⋊C₈

semidirect

51. Dic₃₇

C₄⋊C₈

semidirect

64. C₃₇

#1842

semidirect

115. C₄

#11924603

semidirect

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

subgroup	index
C₂×#11924603	2
C₂×C₄⋊#64438	2
C₄²×Dic₃₇	2
#1842	37

subgroup

index

C₂×#11924603

C₂×C₄⋊#64438

C₄²×Dic₃₇

#1842

Subgroups — #18010529

Conjugacy classes of subgroups

Complements

Maximal subgroups