Group #17997315 — subgroups

Subgroups — #17997315

Order 4736.

516 conjugacy classes of subgroups; only the 205 normal subgroups are shown.

normal subgroups only

This group has 516 conjugacy classes of subgroups — too many to draw. Only its 205 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×#11924536	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×Dic₇₄	C₂	1	2	yes	maximal
3	#14557439	C₂	1	2	yes	maximal
4	#14557418	C₂	1	8	yes	maximal
5	#14557399	C₂	1	1	yes	maximal, characteristic
6	C₃₇×#376	C₂	1	1	yes	maximal, characteristic
7	C₂²×Dic₇₄	C₂²	1	2	yes	—
8	#11924536	C₄	1	8	yes	complemented
9	C₄×Dic₇₄	—	2	8	no	—
10	C₂×C₄⋊Dic₃₇	C₂²	1	2	yes	—
11	C₂×C₄⋊Dic₃₇	C₂²	1	4	yes	—
12	C₂×#10596936	C₂²	1	4	yes	—
13	C₂×C₄×Dic₃₇	C₂²	1	4	yes	—
14	#11924443	C₂²	1	4	yes	—
15	#11924443	C₂²	1	8	yes	—
16	C₂×C₄²×C₃₇	C₂²	1	1	yes	characteristic
17	C₂×C₄²×C₃₇	C₂²	1	1	yes	characteristic
18	C₂×C₄²×C₃₇	C₂²	1	1	yes	characteristic
19	#95×C₃₇	C₂²	1	4	yes	—
20	C₂²×C₄×C₃₇	C₂³	1	2	yes	—
21	C₂²×C₄×C₃₇	C₂³	1	2	yes	—
22	C₂²×C₄×C₃₇	C₂³	1	2	yes	—
23	C₂²×C₄×C₃₇	C₂³	1	1	yes	characteristic
24	C₂²×Dic₃₇	C₂³	1	4	yes	—
25	C₂²×Dic₃₇	C₂³	1	4	yes	—
26	C₂×Dic₇₄	—	2	4	no	—
27	C₂×Dic₇₄	—	2	4	no	—
28	C₂×Dic₇₄	C₂×C₄	1	8	yes	—
29	C₄²×C₃₇	—	2	2	no	—
30	C₄²×C₃₇	C₂×C₄	1	4	yes	—
31	C₄²×C₃₇	D₄	1	4	yes	—
32	C₄⋊Dic₃₇	—	2	4	no	—
33	C₄⋊Dic₃₇	C₂×C₄	1	16	yes	—
34	#10596936	—	2	8	no	—
35	C₄×Dic₃₇	—	2	8	no	—
36	C₂³×C₃₇	C₂⁴	1	1	yes	characteristic
37	C₂×C₄×C₃₇	—	2	4	no	—
38	C₂×C₄×C₃₇	—	2	4	no	—
39	C₂×C₄×C₃₇	—	2	2	no	—
40	C₂×C₄×C₃₇	—	2	4	no	—
41	C₂×C₄×C₃₇	C₂²×C₄	1	4	yes	—
42	C₂×C₄×C₃₇	C₂²×C₄	1	2	yes	—
43	C₂×C₄×C₃₇	C₂×D₄	1	4	yes	—
44	C₂×C₄×C₃₇	C₂×D₄	1	2	yes	—
45	C₂×C₄×C₃₇	#41	1	2	yes	—
46	C₂×Dic₃₇	—	2	4	no	—
47	C₂×Dic₃₇	—	2	4	no	—
48	C₂×Dic₃₇	—	2	4	no	—
49	C₂×Dic₃₇	—	2	8	no	—
50	C₂×Dic₃₇	C₂²×C₄	1	8	yes	—
51	Dic₇₄	—	2	16	no	—
52	C₂²×C₃₇	C₂³×C₄	1	1	yes	characteristic
53	C₂²×C₃₇	C₂²×D₄	1	1	yes	characteristic
54	C₂²×C₃₇	C₂×#41	1	1	yes	characteristic
55	C₂²×C₃₇	2₋¹⁺⁴	1	4	yes	—
56	C₄×C₃₇	—	2	4	no	—
57	C₄×C₃₇	—	2	4	no	—
58	C₄×C₃₇	—	4	2	no	—
59	C₄×C₃₇	C₄×D₄	1	4	yes	—
60	Dic₃₇	—	2	8	no	—
61	Dic₃₇	—	4	4	no	—
62	#2327	—	37	1	no	maximal, Sylow
63	C₂×C₃₇	C₂×C₄×D₄	1	1	yes	characteristic
64	C₂×C₃₇	#519	1	2	yes	—
65	C₂×C₃₇	#536	1	2	yes	—
66	C₂×C₃₇	#541	1	2	yes	—
67	C₂×C₄⋊Q₈	—	37	1	no	—
68	C₂×C₄×Q₈	—	37	2	no	—
69	#391	—	37	2	no	—
70	#387	—	37	8	no	—
71	#382	—	37	1	no	—
72	#376	D₃₇	1	1	yes	characteristic
73	C₃₇	#2327	1	1	yes	characteristic, complemented, Sylow
74	C₂²×Q₈	—	37	2	no	—
75	C₄⋊Q₈	—	37	8	no	—
76	C₄×Q₈	—	74	8	no	—
77	C₂×C₄⋊C₄	—	37	4	no	—
78	C₂×C₄⋊C₄	—	37	4	no	—
79	C₂×C₄⋊C₄	—	37	2	no	—
80	C₂×C₄²	—	37	4	no	—
81	C₂×C₄²	C₂×D₃₇	1	1	yes	characteristic
82	C₂×C₄²	C₂×D₃₇	1	1	yes	characteristic
83	C₂×C₄²	C₂×D₃₇	1	1	yes	characteristic
84	#95	—	37	4	no	—
85	#95	—	37	8	no	—
86	#95	C₂×D₃₇	1	4	yes	—
87	C₂×Q₈	—	37	8	no	—
88	C₂×Q₈	—	74	4	no	—
89	C₂×Q₈	—	74	4	no	—
90	C₂²×C₄	—	37	4	no	—
91	C₂²×C₄	—	37	4	no	—
92	C₂²×C₄	C₂²×D₃₇	1	2	yes	—
93	C₂²×C₄	C₂²×D₃₇	1	2	yes	—
94	C₂²×C₄	C₂²×D₃₇	1	2	yes	—
95	C₂²×C₄	C₂²×D₃₇	1	1	yes	characteristic
96	C₄⋊C₄	—	37	16	no	—
97	C₄⋊C₄	—	74	8	no	—
98	C₄⋊C₄	—	74	4	no	—
99	C₄²	—	2	2	no	—
100	C₄²	—	74	8	no	—
101	C₄²	C₄×D₃₇	1	4	yes	—
102	C₄²	D₁₄₈	1	4	yes	—
103	C₂³	C₂³×D₃₇	1	1	yes	characteristic
104	Q₈	—	74	16	no	—
105	C₂×C₄	—	2	4	no	—
106	C₂×C₄	—	2	4	no	—
107	C₂×C₄	—	2	2	no	—
108	C₂×C₄	—	2	4	no	—
109	C₂×C₄	—	37	8	no	—
110	C₂×C₄	—	74	4	no	—
111	C₂×C₄	—	74	4	no	—
112	C₂×C₄	—	74	4	no	—
113	C₂×C₄	—	74	8	no	—
114	C₂×C₄	C₂×C₄×D₃₇	1	4	yes	—
115	C₂×C₄	C₂×C₄×D₃₇	1	2	yes	—
116	C₂×C₄	C₂×D₁₄₈	1	4	yes	—
117	C₂×C₄	C₂×D₁₄₈	1	2	yes	—
118	C₂×C₄	#10596962	1	2	yes	—
119	C₂²	C₂²×C₄×D₃₇	1	1	yes	characteristic
120	C₂²	C₂²×D₁₄₈	1	1	yes	characteristic
121	C₂²	C₂×#10596962	1	1	yes	characteristic
122	C₂²	#11924583	1	4	yes	—
123	C₄	—	2	4	no	—
124	C₄	—	2	4	no	—
125	C₄	—	4	2	no	—
126	C₄	—	74	8	no	—
127	C₄	—	148	4	no	—
128	C₄	C₄×D₁₄₈	1	4	yes	—
129	C₂	C₂×C₄×D₁₄₈	1	1	yes	characteristic
130	C₂	#14558010	1	2	yes	—
131	C₂	#14558109	1	2	yes	—
132	C₂	#14558145	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#11924536

C₂

yes

maximal, characteristic

C₂×C₄×Dic₇₄

C₂

yes

maximal

#14557439

C₂

yes

maximal

#14557418

C₂

yes

maximal

#14557399

C₂

yes

maximal, characteristic

C₃₇×#376

C₂

yes

maximal, characteristic

C₂²×Dic₇₄

C₂²

yes

—

#11924536

C₄

yes

complemented

C₄×Dic₇₄

—

C₂×C₄⋊Dic₃₇

C₂²

yes

—

C₂×C₄⋊Dic₃₇

C₂²

yes

—

C₂×#10596936

C₂²

yes

—

C₂×C₄×Dic₃₇

C₂²

yes

—

#11924443

C₂²

yes

—

#11924443

C₂²

yes

—

C₂×C₄²×C₃₇

C₂²

yes

characteristic

C₂×C₄²×C₃₇

C₂²

yes

characteristic

C₂×C₄²×C₃₇

C₂²

yes

characteristic

#95×C₃₇

C₂²

yes

—

C₂²×C₄×C₃₇

C₂³

yes

—

C₂²×C₄×C₃₇

C₂³

yes

—

C₂²×C₄×C₃₇

C₂³

yes

—

C₂²×C₄×C₃₇

C₂³

yes

characteristic

C₂²×Dic₃₇

C₂³

yes

—

C₂²×Dic₃₇

C₂³

yes

—

C₂×Dic₇₄

—

C₂×Dic₇₄

—

C₂×Dic₇₄

C₂×C₄

yes

—

C₄²×C₃₇

—

C₄²×C₃₇

C₂×C₄

yes

—

C₄²×C₃₇

D₄

yes

—

C₄⋊Dic₃₇

—

C₄⋊Dic₃₇

C₂×C₄

yes

—

#10596936

—

C₄×Dic₃₇

—

C₂³×C₃₇

C₂⁴

yes

characteristic

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

C₂²×C₄

yes

—

C₂×C₄×C₃₇

C₂²×C₄

yes

—

C₂×C₄×C₃₇

C₂×D₄

yes

—

C₂×C₄×C₃₇

C₂×D₄

yes

—

C₂×C₄×C₃₇

#41

yes

—

C₂×Dic₃₇

—

C₂×Dic₃₇

—

C₂×Dic₃₇

—

C₂×Dic₃₇

—

C₂×Dic₃₇

C₂²×C₄

yes

—

Dic₇₄

—

C₂²×C₃₇

C₂³×C₄

yes

characteristic

C₂²×C₃₇

C₂²×D₄

yes

characteristic

C₂²×C₃₇

C₂×#41

yes

characteristic

C₂²×C₃₇

2₋¹⁺⁴

yes

—

C₄×C₃₇

—

C₄×C₃₇

—

C₄×C₃₇

—

C₄×C₃₇

C₄×D₄

yes

—

Dic₃₇

—

Dic₃₇

—

#2327

—

maximal, Sylow

C₂×C₃₇

C₂×C₄×D₄

yes

characteristic

C₂×C₃₇

#519

yes

—

C₂×C₃₇

#536

yes

—

C₂×C₃₇

#541

yes

—

C₂×C₄⋊Q₈

—

C₂×C₄×Q₈

—

—

—

—

D₃₇

yes

characteristic

C₃₇

#2327

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₄⋊Q₈

—

C₄×Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

—

C₂×C₄²

C₂×D₃₇

yes

characteristic

C₂×C₄²

C₂×D₃₇

yes

characteristic

C₂×C₄²

C₂×D₃₇

yes

—

—

C₂×D₃₇

yes

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₃₇

yes

—

C₂²×C₄

C₂²×D₃₇

yes

—

C₂²×C₄

C₂²×D₃₇

yes

—

C₂²×C₄

C₂²×D₃₇

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

100

C₄²

—

101

C₄²

C₄×D₃₇

yes

—

102

C₄²

D₁₄₈

yes

—

103

C₂³

C₂³×D₃₇

yes

characteristic

104

Q₈

—

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

C₂×C₄×D₃₇

yes

—

115

C₂×C₄

C₂×C₄×D₃₇

yes

—

116

C₂×C₄

C₂×D₁₄₈

yes

—

117

C₂×C₄

C₂×D₁₄₈

yes

—

118

C₂×C₄

#10596962

yes

—

119

C₂²

C₂²×C₄×D₃₇

yes

characteristic

120

C₂²

C₂²×D₁₄₈

yes

characteristic

121

C₂²

C₂×#10596962

yes

characteristic

122

C₂²

#11924583

yes

—

123

C₄

—

124

C₄

—

125

C₄

—

126

C₄

—

127

C₄

—

148

—

128

C₄

C₄×D₁₄₈

yes

—

129

C₂

C₂×C₄×D₁₄₈

yes

characteristic

130

C₂

#14558010

yes

—

131

C₂

#14558109

yes

—

132

C₂

#14558145

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₃₇
commutator	C₂²×C₃₇	C₂³×C₄
Fitting	C₃₇×#376	C₂
socle, prime radical, cosocle	C₂³×C₃₇	C₂⁴
hypercenter	#376	D₃₇
hypocenter	C₃₇	#2327
LCS	G ▷ C₂²×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂³ ◁ #376

subgroup

G/H

center, Frattini

C₂³

C₂³×D₃₇

commutator

C₂²×C₃₇

C₂³×C₄

Fitting

C₃₇×#376

C₂

socle, prime radical, cosocle

C₂³×C₃₇

C₂⁴

D₃₇

C₃₇

G ▷ C₂²×C₃₇ ▷ C₃₇

C₁ ◁ C₂³ ◁ #376

N	H ≅ G/N	type
8. #11924536	C₄	semidirect
73. C₃₇	#2327	semidirect

H ≅ G/N

type

8. #11924536

C₄

semidirect

73. C₃₇

#2327

semidirect

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

subgroup	index
C₂×#11924536	2
C₂×C₄×Dic₇₄	2
#14557439	2
#14557418	2
#14557399	2
C₃₇×#376	2
#2327	37

subgroup

index

C₂×C₄×Dic₇₄

C₃₇×#376

Subgroups — #17997315

Conjugacy classes of subgroups

Complements

Maximal subgroups