Group #18010416 — subgroups

Subgroups — #18010416

Order 4736.

220 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 92 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×#11924607	C₂	1	1	yes	maximal, characteristic
2	C₂×#11924606	C₂	1	1	yes	maximal, characteristic
3	C₄×#10596953	C₂	1	2	yes	maximal
4	#14558409	C₂	1	2	yes	maximal
5	#97×D₃₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#10596953	C₂²	1	2	yes	—
7	C₂×#10596953	C₂²	1	2	yes	—
8	#11924607	C₄	1	4	yes	complemented
9	#11924606	C₄	1	4	yes	complemented
10	#11924591	—	2	2	no	—
11	C₄×#64438	—	2	2	no	—
12	C₂×C₈×D₃₇	C₂²	1	1	yes	characteristic
13	C₂×C₈×D₃₇	C₂²	1	1	yes	characteristic
14	C₄²×D₃₇	C₂²	1	1	yes	characteristic
15	#11924447	C₄	1	1	yes	characteristic
16	#11924446	C₄	1	1	yes	characteristic
17	#11924446	C₄	1	1	yes	characteristic
18	#97×C₃₇	C₄	1	1	yes	characteristic
19	C₂×C₄×D₃₇	Q₈	1	2	yes	—
20	C₂×C₄×D₃₇	C₂³	1	1	yes	characteristic
21	C₂×#64438	—	2	2	no	—
22	C₂×#64438	—	2	2	no	—
23	#10596953	—	2	2	no	—
24	#10596953	—	2	2	no	—
25	#10596953	C₂×C₄	1	4	yes	—
26	#10596953	C₂×C₄	1	4	yes	—
27	C₂×C₈×C₃₇	C₂×C₄	1	1	yes	characteristic
28	C₂×C₈×C₃₇	C₂×C₄	1	1	yes	characteristic
29	C₄²×C₃₇	C₂×C₄	1	1	yes	characteristic
30	C₄×Dic₃₇	C₂×C₄	1	1	yes	characteristic
31	C₄×Dic₃₇	D₄	1	2	yes	—
32	C₂×#64436	C₂×C₄	1	1	yes	characteristic
33	C₂×#64436	C₂×C₄	1	1	yes	characteristic
34	C₈×D₃₇	—	2	1	no	—
35	C₈×D₃₇	—	2	1	no	—
36	C₈×D₃₇	C₂×C₄	1	2	yes	—
37	C₈×D₃₇	C₂×C₄	1	2	yes	—
38	C₂²×D₃₇	C₂×Q₈	1	1	yes	characteristic
39	C₂×C₄×C₃₇	C₄⋊C₄	1	2	yes	—
40	C₂×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
41	C₂×Dic₃₇	C₄⋊C₄	1	2	yes	—
42	C₂×Dic₃₇	C₂×D₄	1	1	yes	characteristic
43	C₄×D₃₇	—	4	2	no	—
44	C₄×D₃₇	C₂²×C₄	1	1	yes	characteristic
45	C₄×D₃₇	C₂²×C₄	1	1	yes	characteristic
46	C₄×D₃₇	#41	1	1	yes	characteristic
47	C₄×D₃₇	#41	1	1	yes	characteristic
48	#64438	—	2	4	no	—
49	#64438	—	2	4	no	—
50	C₈×C₃₇	—	2	1	no	—
51	C₈×C₃₇	C₄²	1	2	yes	—
52	#64436	—	2	1	no	—
53	#64436	C₄²	1	2	yes	—
54	C₂²×C₃₇	C₂×C₄⋊C₄	1	1	yes	characteristic
55	C₂×D₃₇	—	2	2	no	—
56	C₂×D₃₇	C₄×Q₈	1	1	yes	characteristic
57	C₂×D₃₇	C₄×Q₈	1	1	yes	characteristic
58	C₄×C₃₇	—	2	2	no	—
59	C₄×C₃₇	C₂×C₄²	1	1	yes	characteristic
60	C₄×C₃₇	#117	1	1	yes	characteristic
61	Dic₃₇	—	2	2	no	—
62	Dic₃₇	C₄×D₄	1	1	yes	characteristic
63	Dic₃₇	C₄×D₄	1	1	yes	characteristic
64	#1778	—	37	1	no	maximal, Sylow
65	C₂×C₃₇	C₄×C₄⋊C₄	1	1	yes	characteristic
66	C₂×C₃₇	#430	1	2	yes	—
67	D₃₇	—	2	1	no	—
68	D₃₇	—	2	1	no	—
69	C₂×#108	—	37	1	no	—
70	C₂×#108	—	37	1	no	—
71	C₄×M₁₆	—	37	2	no	—
72	C₂×#97	—	37	1	no	—
73	#341	—	37	2	no	—
74	C₃₇	#1778	1	1	yes	characteristic, complemented, Sylow
75	C₂×M₁₆	—	37	2	no	—
76	C₂×M₁₆	—	37	2	no	—
77	C₂²×C₈	—	37	1	no	—
78	C₂²×C₈	—	37	1	no	—
79	C₂×C₄²	—	37	1	no	—
80	#108	—	37	4	no	—
81	#108	—	37	4	no	—
82	#97	—	37	1	no	—
83	#97	—	37	1	no	—
84	#97	—	37	1	no	—
85	#97	—	74	2	no	—
86	#97	#3866	1	1	yes	characteristic
87	C₄×C₈	—	74	2	no	—
88	C₂²×C₄	—	37	1	no	—
89	C₂²×C₄	—	37	2	no	—
90	M₁₆	—	37	4	no	—
91	M₁₆	—	37	4	no	—
92	M₁₆	—	74	2	no	—
93	M₁₆	—	74	2	no	—
94	C₂×C₈	—	37	1	no	—
95	C₂×C₈	—	37	2	no	—
96	C₂×C₈	—	37	2	no	—
97	C₂×C₈	—	37	1	no	—
98	C₂×C₈	—	74	1	no	—
99	C₂×C₈	—	74	1	no	—
100	C₂×C₈	—	74	2	no	—
101	C₂×C₈	—	74	2	no	—
102	C₂×C₈	C₂×#3866	1	1	yes	characteristic
103	C₂×C₈	C₂×#3866	1	1	yes	characteristic
104	C₄²	—	37	2	no	—
105	C₄²	—	37	1	no	—
106	C₄²	C₂×#3866	1	1	yes	characteristic
107	C₂³	—	37	1	no	—
108	C₂×C₄	—	37	1	no	—
109	C₂×C₄	—	37	1	no	—
110	C₂×C₄	—	37	1	no	—
111	C₂×C₄	—	37	1	no	—
112	C₂×C₄	—	37	1	no	—
113	C₂×C₄	—	37	2	no	—
114	C₂×C₄	—	148	2	no	—
115	C₂×C₄	C₄⋊#3866	1	2	yes	—
116	C₂×C₄	C₂²×#3866	1	1	yes	characteristic
117	C₈	—	2	1	no	—
118	C₈	—	37	2	no	—
119	C₈	—	74	4	no	—
120	C₈	—	74	4	no	—
121	C₈	—	74	1	no	—
122	C₈	C₄×#3866	1	2	yes	—
123	C₂²	—	37	1	no	—
124	C₂²	—	37	1	no	—
125	C₂²	—	74	2	no	—
126	C₂²	C₂×C₄⋊#3866	1	1	yes	characteristic
127	C₄	—	2	2	no	—
128	C₄	—	37	1	no	—
129	C₄	—	37	1	no	—
130	C₄	—	74	2	no	—
131	C₄	C₂×C₄×#3866	1	1	yes	characteristic
132	C₄	#11924615	1	1	yes	characteristic
133	C₂	—	74	1	no	—
134	C₂	—	74	1	no	—
135	C₂	C₄×C₄⋊#3866	1	1	yes	characteristic
136	C₂	#14558548	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#11924607

C₂

yes

maximal, characteristic

C₂×#11924606

C₂

yes

maximal, characteristic

C₄×#10596953

C₂

yes

maximal

#14558409

C₂

yes

maximal

#97×D₃₇

C₂

yes

maximal, characteristic

C₂×#10596953

C₂²

yes

—

C₂×#10596953

C₂²

yes

—

#11924607

C₄

yes

complemented

#11924606

C₄

yes

complemented

#11924591

—

C₄×#64438

—

C₂×C₈×D₃₇

C₂²

yes

characteristic

C₂×C₈×D₃₇

C₂²

yes

characteristic

C₄²×D₃₇

C₂²

yes

characteristic

#11924447

C₄

yes

characteristic

#11924446

C₄

yes

characteristic

#11924446

C₄

yes

characteristic

#97×C₃₇

C₄

yes

characteristic

C₂×C₄×D₃₇

Q₈

yes

—

C₂×C₄×D₃₇

C₂³

yes

—

—

—

—

C₂×C₄

yes

—

#10596953

C₂×C₄

yes

—

C₂×C₈×C₃₇

C₂×C₄

yes

characteristic

C₂×C₈×C₃₇

C₂×C₄

yes

characteristic

C₄²×C₃₇

C₂×C₄

yes

characteristic

C₄×Dic₃₇

C₂×C₄

yes

characteristic

C₄×Dic₃₇

D₄

yes

—

C₂×#64436

C₂×C₄

yes

characteristic

C₂×#64436

C₂×C₄

yes

characteristic

C₈×D₃₇

—

C₈×D₃₇

—

C₈×D₃₇

C₂×C₄

yes

—

C₈×D₃₇

C₂×C₄

yes

—

C₂²×D₃₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₃₇

C₄⋊C₄

yes

—

C₂×C₄×C₃₇

C₂²×C₄

yes

characteristic

C₂×Dic₃₇

C₄⋊C₄

yes

—

C₂×Dic₃₇

C₂×D₄

yes

characteristic

C₄×D₃₇

—

C₄×D₃₇

C₂²×C₄

yes

characteristic

C₄×D₃₇

C₂²×C₄

yes

C₄×D₃₇

yes

C₄×D₃₇

yes

—

—

C₈×C₃₇

—

C₈×C₃₇

C₄²

yes

—

#64436

—

#64436

C₄²

yes

—

C₂²×C₃₇

C₂×C₄⋊C₄

yes

characteristic

C₂×D₃₇

—

C₂×D₃₇

C₄×Q₈

yes

characteristic

C₂×D₃₇

C₄×Q₈

yes

characteristic

C₄×C₃₇

—

C₄×C₃₇

C₂×C₄²

yes

characteristic

C₄×C₃₇

#117

yes

characteristic

Dic₃₇

—

Dic₃₇

C₄×D₄

yes

characteristic

Dic₃₇

C₄×D₄

yes

characteristic

#1778

—

maximal, Sylow

C₂×C₃₇

C₄×C₄⋊C₄

yes

characteristic

C₂×C₃₇

#430

yes

—

D₃₇

—

D₃₇

—

C₂×#108

—

C₂×#108

—

C₄×M₁₆

—

C₂×#97

—

#341

—

C₃₇

#1778

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄²

—

—

—

—

—

—

—

yes

C₄×C₈

—

C₂²×C₄

—

C₂²×C₄

—

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

100

C₂×C₈

—

101

C₂×C₈

—

102

C₂×C₈

C₂×#3866

yes

characteristic

103

C₂×C₈

C₂×#3866

yes

characteristic

104

C₄²

—

105

C₄²

—

106

C₄²

C₂×#3866

yes

characteristic

107

C₂³

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

148

—

115

C₂×C₄

C₄⋊#3866

yes

—

116

C₂×C₄

C₂²×#3866

yes

characteristic

117

C₈

—

118

C₈

—

119

C₈

—

120

C₈

—

121

C₈

—

122

C₈

C₄×#3866

yes

—

123

C₂²

—

124

C₂²

—

125

C₂²

—

126

C₂²

C₂×C₄⋊#3866

yes

characteristic

127

C₄

—

128

C₄

—

129

C₄

—

130

C₄

—

131

C₄

C₂×C₄×#3866

yes

characteristic

132

C₄

#11924615

yes

characteristic

133

C₂

—

134

C₂

—

135

C₂

C₄×C₄⋊#3866

yes

characteristic

136

C₂

#14558548

yes

—

	subgroup	G/H
center	C₂²	C₂×C₄⋊#3866
commutator	C₄×C₃₇	C₂×C₄²
Frattini	C₂×C₄	C₂²×#3866
Fitting	#97×C₃₇	C₄
socle	C₂²×C₃₇	C₂×C₄⋊C₄
prime radical	C₂²×D₃₇	C₂×Q₈
cosocle	C₂×C₄×D₃₇	C₂³
hypercenter	#97	#3866
hypocenter	C₃₇	#1778
LCS	G ▷ C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #97

subgroup

G/H

center

C₂²

C₂×C₄⋊#3866

commutator

C₄×C₃₇

C₂×C₄²

Frattini

C₂×C₄

C₂²×#3866

Fitting

#97×C₃₇

C₄

socle

C₂²×C₃₇

C₂×C₄⋊C₄

prime radical

C₂²×D₃₇

C₂×Q₈

cosocle

C₂×C₄×D₃₇

C₂³

C₃₇

G ▷ C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #97

N	H ≅ G/N	type
8. #11924607	C₄	semidirect
9. #11924606	C₄	semidirect
74. C₃₇	#1778	semidirect

H ≅ G/N

type

8. #11924607

C₄

semidirect

9. #11924606

C₄

semidirect

74. C₃₇

#1778

semidirect

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

subgroup	index
C₂×#11924607	2
C₂×#11924606	2
C₄×#10596953	2
#14558409	2
#97×D₃₇	2
#1778	37

subgroup

index

#97×D₃₇

Subgroups — #18010416

Conjugacy classes of subgroups

Complements

Maximal subgroups