Group #18010770 — subgroups

Subgroups — #18010770

Order 4736.

280 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 280 conjugacy classes of subgroups — too many to draw. Only its 94 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	1	yes	maximal, characteristic
2	C₂×#11924591	C₂	1	1	yes	maximal, characteristic
3	#14558399	C₂	1	4	yes	maximal
4	C₂×Q₈×Dic₃₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×#64438	C₂²	1	2	yes	—
6	C₂²×#64438	C₂²	1	2	yes	—
7	#11924603	—	2	2	no	—
8	#11924591	—	2	2	no	—
9	Q₈×Dic₃₇	—	2	2	no	—
10	Q₈×Dic₃₇	—	2	2	no	—
11	C₂×C₄⋊Dic₃₇	C₄	1	1	yes	characteristic
12	C₂×C₄⋊Dic₃₇	C₄	1	2	yes	—
13	C₂×C₄×Dic₃₇	C₂²	1	2	yes	—
14	C₂×C₄×Dic₃₇	C₂²	1	1	yes	characteristic
15	C₂²×Q₈×C₃₇	C₄	1	1	yes	characteristic
16	C₂×Q₈×C₃₇	—	2	2	no	—
17	C₂×Q₈×C₃₇	—	2	4	no	—
18	C₂²×C₄×C₃₇	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₄×C₃₇	C₂×C₄	1	2	yes	—
20	C₂²×Dic₃₇	C₂×C₄	1	2	yes	—
21	C₂²×Dic₃₇	C₂×C₄	1	1	yes	characteristic
22	C₂²×Dic₃₇	C₂³	1	1	yes	characteristic
23	C₂×#64438	—	2	2	no	—
24	C₂×#64438	—	2	2	no	—
25	C₂×#64438	—	2	2	no	—
26	C₂×#64438	—	2	4	no	—
27	C₂×#64438	Q₈	1	4	yes	complemented
28	C₄⋊Dic₃₇	—	2	2	no	—
29	C₄⋊Dic₃₇	—	2	4	no	—
30	C₄×Dic₃₇	—	2	2	no	—
31	C₄×Dic₃₇	—	2	2	no	—
32	C₄×Dic₃₇	D₄	1	2	yes	—
33	C₄×Dic₃₇	D₄	1	2	yes	—
34	C₂³×C₃₇	C₂²×C₄	1	1	yes	characteristic
35	Q₈×C₃₇	—	2	8	no	—
36	C₂×C₄×C₃₇	—	2	2	no	—
37	C₂×C₄×C₃₇	—	2	2	no	—
38	C₂×C₄×C₃₇	—	2	2	no	—
39	C₂×C₄×C₃₇	C₂²⋊C₄	1	2	yes	—
40	C₂×C₄×C₃₇	M₁₆	1	4	yes	—
41	C₂×Dic₃₇	—	2	2	no	—
42	C₂×Dic₃₇	—	2	2	no	—
43	C₂×Dic₃₇	—	2	4	no	—
44	C₂×Dic₃₇	C₂²⋊C₄	1	2	yes	—
45	C₂×Dic₃₇	C₂×D₄	1	1	yes	characteristic
46	C₂×Dic₃₇	C₂×D₄	1	1	yes	characteristic
47	C₂×Dic₃₇	C₂×Q₈	1	2	yes	—
48	C₂×Dic₃₇	#41	1	2	yes	—
49	#64438	—	2	4	no	—
50	#64438	—	4	2	no	—
51	C₂²×C₃₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
52	C₂²×C₃₇	C₄×Q₈	1	2	yes	—
53	C₂²×C₃₇	C₂×M₁₆	1	2	yes	—
54	C₂²×C₃₇	#131	1	2	yes	—
55	C₄×C₃₇	—	2	2	no	—
56	C₄×C₃₇	—	2	4	no	—
57	Dic₃₇	—	2	2	no	—
58	Dic₃₇	—	4	2	no	—
59	Dic₃₇	C₂²⋊Q₈	1	2	yes	—
60	Dic₃₇	#124	1	1	yes	characteristic
61	Dic₃₇	C₄⋊Q₈	1	1	yes	characteristic
62	#1985	—	37	1	no	maximal, Sylow
63	C₂×C₃₇	#391	1	1	yes	characteristic
64	C₂×C₃₇	#407	1	1	yes	characteristic
65	C₂×C₃₇	#408	1	1	yes	characteristic
66	C₂×C₃₇	#446	1	4	yes	—
67	C₂×C₄×Q₈	—	37	1	no	—
68	C₂×C₄⋊C₈	—	37	1	no	—
69	C₂×#97	—	37	1	no	—
70	#336	—	37	4	no	—
71	C₃₇	#1985	1	1	yes	characteristic, complemented, Sylow
72	C₂²×Q₈	#3866	1	1	yes	characteristic
73	C₂²×C₈	—	37	2	no	—
74	C₂²×C₈	—	37	2	no	—
75	C₄×Q₈	—	74	2	no	—
76	C₄×Q₈	—	74	2	no	—
77	C₂×C₄⋊C₄	—	37	2	no	—
78	C₂×C₄⋊C₄	—	37	1	no	—
79	C₂×C₄²	—	37	1	no	—
80	C₂×C₄²	—	37	2	no	—
81	C₄⋊C₈	—	74	2	no	—
82	#97	—	74	2	no	—
83	C₂×Q₈	—	2	4	no	—
84	C₂×Q₈	—	2	2	no	—
85	C₂²×C₄	—	37	1	no	—
86	C₂²×C₄	—	37	1	no	—
87	C₂²×C₄	—	37	2	no	—
88	C₂²×C₄	C₂×#3866	1	2	yes	—
89	C₂²×C₄	C₂×#3866	1	1	yes	characteristic
90	C₂×C₈	—	37	4	no	—
91	C₂×C₈	—	74	2	no	—
92	C₂×C₈	—	74	2	no	—
93	C₂×C₈	—	74	2	no	—
94	C₂×C₈	—	74	4	no	—
95	C₄⋊C₄	—	74	4	no	—
96	C₄⋊C₄	—	74	2	no	—
97	C₄²	—	37	2	no	—
98	C₄²	—	37	2	no	—
99	C₄²	—	74	2	no	—
100	C₄²	—	74	2	no	—
101	C₂³	C₂²×#3866	1	1	yes	characteristic
102	Q₈	—	2	8	no	—
103	C₂×C₄	—	2	2	no	—
104	C₂×C₄	—	2	2	no	—
105	C₂×C₄	—	2	2	no	—
106	C₂×C₄	—	37	1	no	—
107	C₂×C₄	—	37	1	no	—
108	C₂×C₄	—	37	2	no	—
109	C₂×C₄	—	37	2	no	—
110	C₂×C₄	—	37	2	no	—
111	C₂×C₄	—	74	2	no	—
112	C₂×C₄	—	74	2	no	—
113	C₂×C₄	—	74	4	no	—
114	C₂×C₄	#10596957	1	4	yes	—
115	C₂×C₄	C₂²⋊#3866	1	2	yes	—
116	C₈	—	74	4	no	—
117	C₈	—	148	2	no	—
118	C₂²	C₂×C₂²⋊#3866	1	1	yes	characteristic
119	C₂²	Q₈×#3866	1	2	yes	—
120	C₂²	C₂×#10596957	1	2	yes	—
121	C₂²	#11924624	1	2	yes	—
122	C₄	—	2	4	no	—
123	C₄	—	2	2	no	—
124	C₄	—	37	2	no	—
125	C₄	—	37	1	no	—
126	C₄	—	37	1	no	—
127	C₄	—	74	2	no	—
128	C₄	—	148	2	no	—
129	C₂	(C₂×#3866)⋊Q₈	1	1	yes	characteristic
130	C₂	#14558503	1	1	yes	characteristic
131	C₂	#14558508	1	1	yes	characteristic
132	C₂	#14558584	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#11924603

C₂

yes

maximal, characteristic

C₂×#11924591

C₂

yes

maximal, characteristic

#14558399

C₂

yes

maximal

C₂×Q₈×Dic₃₇

C₂

yes

maximal, characteristic

C₂²×#64438

C₂²

yes

—

C₂²×#64438

C₂²

yes

—

#11924603

—

#11924591

—

Q₈×Dic₃₇

—

Q₈×Dic₃₇

—

C₂×C₄⋊Dic₃₇

C₄

yes

characteristic

C₂×C₄⋊Dic₃₇

C₄

yes

—

C₂×C₄×Dic₃₇

C₂²

yes

—

C₂×C₄×Dic₃₇

C₂²

yes

characteristic

C₂²×Q₈×C₃₇

C₄

yes

characteristic

C₂×Q₈×C₃₇

—

C₂×Q₈×C₃₇

—

C₂²×C₄×C₃₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₃₇

C₂×C₄

yes

—

C₂²×Dic₃₇

C₂×C₄

yes

—

C₂²×Dic₃₇

C₂×C₄

yes

characteristic

C₂²×Dic₃₇

C₂³

yes

—

—

—

—

Q₈

yes

complemented

C₄⋊Dic₃₇

—

C₄⋊Dic₃₇

—

C₄×Dic₃₇

—

C₄×Dic₃₇

—

C₄×Dic₃₇

D₄

yes

—

C₄×Dic₃₇

D₄

yes

—

C₂³×C₃₇

C₂²×C₄

yes

characteristic

Q₈×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

C₂²⋊C₄

yes

—

C₂×C₄×C₃₇

M₁₆

yes

—

C₂×Dic₃₇

—

C₂×Dic₃₇

—

C₂×Dic₃₇

—

C₂×Dic₃₇

C₂²⋊C₄

yes

—

C₂×Dic₃₇

C₂×D₄

yes

characteristic

C₂×Dic₃₇

C₂×D₄

yes

characteristic

C₂×Dic₃₇

C₂×Q₈

yes

—

C₂×Dic₃₇

#41

yes

—

#64438

—

#64438

—

C₂²×C₃₇

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₃₇

C₄×Q₈

yes

—

C₂²×C₃₇

C₂×M₁₆

yes

—

C₂²×C₃₇

#131

yes

—

C₄×C₃₇

—

C₄×C₃₇

—

Dic₃₇

—

Dic₃₇

—

Dic₃₇

C₂²⋊Q₈

yes

—

Dic₃₇

#124

yes

characteristic

Dic₃₇

C₄⋊Q₈

yes

—

C₂×C₃₇

yes

C₂×C₃₇

yes

C₂×C₃₇

yes

C₂×C₃₇

yes

—

C₂×C₄×Q₈

—

C₂×C₄⋊C₈

—

C₂×#97

—

#336

—

C₃₇

#1985

yes

characteristic, complemented, Sylow

C₂²×Q₈

#3866

yes

characteristic

C₂²×C₈

—

C₂²×C₈

—

C₄×Q₈

—

C₄×Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

—

C₂×C₄²

—

C₄⋊C₈

—

#97

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×#3866

yes

—

C₂²×C₄

C₂×#3866

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₄²

—

C₄²

—

100

C₄²

—

101

C₂³

C₂²×#3866

yes

characteristic

102

Q₈

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

#10596957

yes

—

115

C₂×C₄

C₂²⋊#3866

yes

—

116

C₈

—

117

C₈

—

148

—

118

C₂²

C₂×C₂²⋊#3866

yes

characteristic

119

C₂²

Q₈×#3866

yes

—

120

C₂²

C₂×#10596957

yes

—

121

C₂²

#11924624

yes

—

122

C₄

—

123

C₄

—

124

C₄

—

125

C₄

—

126

C₄

—

127

C₄

—

128

C₄

—

148

—

129

C₂

(C₂×#3866)⋊Q₈

yes

characteristic

130

C₂

#14558503

yes

characteristic

131

C₂

#14558508

yes

characteristic

132

C₂

#14558584

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂²×#3866
commutator, socle, prime radical	C₂³×C₃₇	C₂²×C₄
Fitting	C₂²×Q₈×C₃₇	C₄
cosocle	C₂²×Dic₃₇	C₂³
hypercenter	C₂²×Q₈	#3866
hypocenter	C₃₇	#1985
LCS	G ▷ C₂³×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂³ ◁ C₂²×Q₈

subgroup

G/H

center, Frattini

C₂³

C₂²×#3866

commutator, socle, prime radical

C₂³×C₃₇

C₂²×C₄

Fitting

C₂²×Q₈×C₃₇

C₄

cosocle

C₂²×Dic₃₇

C₂³

C₂²×Q₈

C₃₇

G ▷ C₂³×C₃₇ ▷ C₃₇

C₁ ◁ C₂³ ◁ C₂²×Q₈

N	H ≅ G/N	type
27. C₂×#64438	Q₈	semidirect
71. C₃₇	#1985	semidirect

H ≅ G/N

type

27. C₂×#64438

Q₈

semidirect

71. C₃₇

#1985

semidirect

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

subgroup	index
C₂×#11924603	2
C₂×#11924591	2
#14558399	2
C₂×Q₈×Dic₃₇	2
#1985	37

subgroup

index

C₂×Q₈×Dic₃₇

Subgroups — #18010770

Conjugacy classes of subgroups

Complements

Maximal subgroups