Group #17995744 — subgroups

Subgroups — #17995744

Order 4736.

258 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 258 conjugacy classes of subgroups — too many to draw. Only its 87 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×#11924459	C₂	1	2	yes	maximal
2	#14557439	C₂	1	1	yes	maximal, characteristic
3	#14557295	C₂	1	2	yes	maximal
4	#14557288	C₂	1	1	yes	maximal, characteristic
5	C₂×#125×C₃₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₇₄	C₂²	1	1	yes	characteristic
7	C₂²×#64436	C₂²	1	2	yes	—
8	C₂×C₄⋊Dic₃₇	C₂²	1	1	yes	characteristic
9	#11924459	—	2	4	no	—
10	#11924443	—	2	2	no	—
11	#125×C₃₇	—	2	4	no	—
12	C₂×C₄⋊C₄×C₃₇	—	2	2	no	—
13	C₂×C₄⋊C₄×C₃₇	C₂²	1	2	yes	—
14	C₂×C₄²×C₃₇	C₂²	1	1	yes	characteristic
15	C₂²×C₄×C₃₇	—	2	2	no	—
16	C₂²×C₄×C₃₇	D₄	1	2	yes	—
17	C₂²×C₄×C₃₇	C₂³	1	1	yes	characteristic
18	C₂²×Dic₃₇	—	2	1	no	—
19	C₂²×Dic₃₇	—	2	1	no	—
20	C₂×Dic₇₄	—	2	2	no	—
21	C₂×Dic₇₄	—	2	2	no	—
22	C₂×Dic₇₄	—	2	1	no	—
23	C₂×Dic₇₄	Q₈	1	2	yes	—
24	C₄⋊C₄×C₃₇	—	2	8	no	—
25	C₄⋊C₄×C₃₇	—	2	2	no	—
26	C₄⋊C₄×C₃₇	D₄	1	4	yes	—
27	C₄²×C₃₇	—	2	2	no	—
28	C₄⋊Dic₃₇	—	2	2	no	—
29	C₂×#64436	—	2	2	no	—
30	C₂×#64436	—	2	2	no	—
31	C₂×#64436	—	2	2	no	—
32	C₂³×C₃₇	C₂×D₄	1	1	yes	characteristic
33	C₂×C₄×C₃₇	—	2	2	no	—
34	C₂×C₄×C₃₇	—	2	4	no	—
35	C₂×C₄×C₃₇	—	2	2	no	—
36	C₂×C₄×C₃₇	—	2	2	no	—
37	C₂×C₄×C₃₇	—	4	2	no	—
38	C₂×C₄×C₃₇	—	4	2	no	—
39	C₂×C₄×C₃₇	C₂×D₄	1	2	yes	—
40	C₂×C₄×C₃₇	C₂×Q₈	1	1	yes	characteristic
41	C₂×C₄×C₃₇	#41	1	2	yes	—
42	C₂×C₄×C₃₇	#41	1	1	yes	characteristic
43	C₂×Dic₃₇	—	2	2	no	—
44	C₂×Dic₃₇	—	4	1	no	—
45	C₂×Dic₃₇	—	4	1	no	—
46	C₂×Dic₃₇	—	4	1	no	—
47	C₂×Dic₃₇	—	4	2	no	—
48	Dic₇₄	—	2	4	no	—
49	Dic₇₄	—	4	2	no	—
50	#64436	—	4	2	no	—
51	C₂²×C₃₇	#120	1	1	yes	characteristic
52	C₂²×C₃₇	C₂²⋊Q₈	1	1	yes	characteristic
53	C₂²×C₃₇	#124	1	1	yes	characteristic
54	C₂²×C₃₇	#135	1	2	yes	—
55	C₂²×C₃₇	#137	1	2	yes	—
56	C₄×C₃₇	—	4	2	no	—
57	C₄×C₃₇	—	4	4	no	—
58	C₄×C₃₇	C₂²⋊Q₈	1	2	yes	—
59	C₄×C₃₇	#124	1	2	yes	—
60	Dic₃₇	—	4	2	no	—
61	Dic₃₇	—	8	1	no	—
62	#2026	—	37	1	no	maximal, Sylow
63	C₂×C₃₇	#393	1	1	yes	characteristic
64	C₂×C₃₇	#452	1	2	yes	—
65	C₂×C₃₇	#479	1	2	yes	—
66	C₂×C₃₇	#488	1	1	yes	characteristic
67	C₂×C₃₇	#491	1	1	yes	characteristic
68	C₂×#125	D₃₇	1	1	yes	characteristic
69	C₂×#103	—	37	2	no	—
70	#391	—	37	1	no	—
71	#340	—	37	2	no	—
72	#336	—	37	1	no	—
73	C₃₇	#2026	1	1	yes	characteristic, complemented, Sylow
74	C₂²×Q₈	—	37	1	no	—
75	C₂²×C₈	—	37	2	no	—
76	#125	—	2	4	no	—
77	C₂×C₄⋊C₄	—	2	2	no	—
78	C₂×C₄⋊C₄	—	37	1	no	—
79	C₂×C₄⋊C₄	C₂×D₃₇	1	2	yes	—
80	C₂×C₄²	C₂×D₃₇	1	1	yes	characteristic
81	#103	—	74	4	no	—
82	#95	—	74	2	no	—
83	C₂×Q₈	—	37	2	no	—
84	C₂×Q₈	—	74	1	no	—
85	C₂×Q₈	—	74	2	no	—
86	C₂×Q₈	—	74	2	no	—
87	C₂²×C₄	—	2	2	no	—
88	C₂²×C₄	—	74	1	no	—
89	C₂²×C₄	—	74	1	no	—
90	C₂²×C₄	C₂²⋊D₃₇	1	2	yes	—
91	C₂²×C₄	C₂²×D₃₇	1	1	yes	characteristic
92	C₂×C₈	—	74	2	no	—
93	C₂×C₈	—	74	2	no	—
94	C₂×C₈	—	74	2	no	—
95	C₄⋊C₄	—	2	8	no	—
96	C₄⋊C₄	—	2	2	no	—
97	C₄⋊C₄	—	74	2	no	—
98	C₄⋊C₄	C₂²⋊D₃₇	1	4	yes	—
99	C₄²	—	2	2	no	—
100	C₂³	C₂×C₂²⋊D₃₇	1	1	yes	characteristic
101	Q₈	—	74	4	no	—
102	Q₈	—	148	2	no	—
103	C₂×C₄	—	2	2	no	—
104	C₂×C₄	—	2	4	no	—
105	C₂×C₄	—	2	2	no	—
106	C₂×C₄	—	2	2	no	—
107	C₂×C₄	—	4	2	no	—
108	C₂×C₄	—	4	2	no	—
109	C₂×C₄	—	74	2	no	—
110	C₂×C₄	—	148	1	no	—
111	C₂×C₄	—	148	2	no	—
112	C₂×C₄	—	148	1	no	—
113	C₂×C₄	—	148	1	no	—
114	C₂×C₄	Q₈×D₃₇	1	1	yes	characteristic
115	C₂×C₄	#10596966	1	2	yes	—
116	C₂×C₄	#10596966	1	1	yes	characteristic
117	C₂×C₄	C₂×C₂²⋊D₃₇	1	2	yes	—
118	C₈	—	148	2	no	—
119	C₂²	#11924505	1	1	yes	characteristic
120	C₂²	#11924512	1	1	yes	characteristic
121	C₂²	#11924525	1	1	yes	characteristic
122	C₂²	#11924556	1	2	yes	—
123	C₂²	#11924566	1	2	yes	—
124	C₄	—	4	2	no	—
125	C₄	—	4	4	no	—
126	C₄	—	296	1	no	—
127	C₄	—	148	2	no	—
128	C₄	#11924512	1	2	yes	—
129	C₄	#11924525	1	2	yes	—
130	C₂	#14557444	1	1	yes	characteristic
131	C₂	#14557695	1	2	yes	—
132	C₂	#14557837	1	2	yes	—
133	C₂	#14557890	1	1	yes	characteristic
134	C₂	#14557903	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#11924459

C₂

yes

maximal

#14557439

C₂

yes

maximal, characteristic

#14557295

C₂

yes

maximal

#14557288

C₂

yes

maximal, characteristic

C₂×#125×C₃₇

C₂

yes

maximal, characteristic

C₂²×Dic₇₄

C₂²

yes

characteristic

C₂²×#64436

C₂²

yes

—

C₂×C₄⋊Dic₃₇

C₂²

yes

characteristic

#11924459

—

#11924443

—

#125×C₃₇

—

C₂×C₄⋊C₄×C₃₇

—

C₂×C₄⋊C₄×C₃₇

C₂²

yes

—

C₂×C₄²×C₃₇

C₂²

yes

characteristic

C₂²×C₄×C₃₇

—

C₂²×C₄×C₃₇

D₄

yes

—

C₂²×C₄×C₃₇

C₂³

yes

characteristic

C₂²×Dic₃₇

—

C₂²×Dic₃₇

—

C₂×Dic₇₄

—

C₂×Dic₇₄

—

C₂×Dic₇₄

—

C₂×Dic₇₄

Q₈

yes

—

C₄⋊C₄×C₃₇

—

C₄⋊C₄×C₃₇

—

C₄⋊C₄×C₃₇

D₄

yes

—

C₄²×C₃₇

—

C₄⋊Dic₃₇

—

C₂×#64436

—

C₂×#64436

—

C₂×#64436

—

C₂³×C₃₇

C₂×D₄

yes

characteristic

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

C₂×D₄

yes

—

C₂×C₄×C₃₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₃₇

#41

yes

—

C₂×C₄×C₃₇

#41

yes

characteristic

C₂×Dic₃₇

—

C₂×Dic₃₇

—

C₂×Dic₃₇

—

C₂×Dic₃₇

—

C₂×Dic₃₇

—

Dic₇₄

—

Dic₇₄

—

#64436

—

C₂²×C₃₇

#120

yes

characteristic

C₂²×C₃₇

C₂²⋊Q₈

yes

C₂²×C₃₇

yes

C₂²×C₃₇

yes

—

C₂²×C₃₇

#137

yes

—

C₄×C₃₇

—

C₄×C₃₇

—

C₄×C₃₇

C₂²⋊Q₈

yes

—

C₄×C₃₇

#124

yes

—

Dic₃₇

—

Dic₃₇

—

—

C₂×C₃₇

yes

C₂×C₃₇

yes

—

C₂×C₃₇

#479

yes

—

C₂×C₃₇

yes

C₂×C₃₇

yes

D₃₇

yes

—

—

—

—

C₃₇

#2026

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₂²×C₈

—

#125

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₃₇

yes

—

C₂×C₄²

C₂×D₃₇

yes

characteristic

#103

—

#95

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊D₃₇

yes

—

C₂²×C₄

C₂²×D₃₇

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂²⋊D₃₇

yes

—

C₄²

—

100

C₂³

C₂×C₂²⋊D₃₇

yes

characteristic

101

Q₈

—

102

Q₈

—

148

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

148

—

111

C₂×C₄

—

148

—

112

C₂×C₄

—

148

—

113

C₂×C₄

—

148

—

114

C₂×C₄

Q₈×D₃₇

yes

characteristic

115

C₂×C₄

#10596966

yes

—

116

C₂×C₄

#10596966

yes

characteristic

117

C₂×C₄

C₂×C₂²⋊D₃₇

yes

—

118

C₈

—

148

—

119

C₂²

#11924505

yes

characteristic

120

C₂²

#11924512

yes

characteristic

121

C₂²

#11924525

yes

characteristic

122

C₂²

#11924556

yes

—

123

C₂²

#11924566

yes

—

124

C₄

—

125

C₄

—

126

C₄

—

296

—

127

C₄

—

148

—

128

C₄

#11924512

yes

—

129

C₄

#11924525

yes

—

130

C₂

#14557444

yes

characteristic

131

C₂

#14557695

yes

—

132

C₂

#14557837

yes

—

133

C₂

#14557890

yes

characteristic

134

C₂

#14557903

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₃₇
commutator, cosocle	C₂²×C₄×C₃₇	C₂³
Frattini	C₂²×C₄	C₂²×D₃₇
Fitting	C₂×#125×C₃₇	C₂
socle, prime radical	C₂³×C₃₇	C₂×D₄
hypercenter	C₂×#125	D₃₇
hypocenter	C₃₇	#2026
LCS	G ▷ C₂²×C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#125

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₃₇

commutator, cosocle

C₂²×C₄×C₃₇

C₂³

Frattini

C₂²×C₄

C₂²×D₃₇

Fitting

C₂×#125×C₃₇

C₂

socle, prime radical

C₂³×C₃₇

C₂×D₄

D₃₇

C₃₇

G ▷ C₂²×C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#125

N	H ≅ G/N	type
73. C₃₇	#2026	semidirect

H ≅ G/N

type

73. C₃₇

#2026

semidirect

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

subgroup	index
C₂×#11924459	2
#14557439	2
#14557295	2
#14557288	2
C₂×#125×C₃₇	2
#2026	37

subgroup

index

C₂×#125×C₃₇

Subgroups — #17995744

Conjugacy classes of subgroups

Complements

Maximal subgroups