Group #17994056 — subgroups

Subgroups — #17994056

Order 4736.

144 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 144 conjugacy classes of subgroups — too many to draw. Only its 55 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	#14557885	C₂	1	1	yes	maximal, characteristic
2	#14557818	C₂	1	1	yes	maximal, characteristic
3	Q₈×#64436	C₂	1	1	yes	maximal, characteristic
4	#14557285	C₂	1	1	yes	maximal, characteristic
5	#14557282	C₂	1	1	yes	maximal, characteristic
6	#14557278	C₂	1	1	yes	maximal, characteristic
7	C₃₇×#477	C₂	1	1	yes	maximal, characteristic
8	#11924536	C₂²	1	1	yes	characteristic
9	#11924467	—	2	1	no	—
10	C₄⋊#64436	—	2	1	no	—
11	C₄⋊#64436	C₂²	1	1	yes	characteristic
12	C₄⋊#64436	C₂²	1	1	yes	characteristic
13	#11924460	—	2	1	no	—
14	#11924459	—	2	1	no	—
15	#11924459	—	2	1	no	—
16	C₄×#64436	—	2	1	no	—
17	C₄×#64436	C₂²	1	1	yes	characteristic
18	C₄⋊Q₈×C₃₇	C₂²	1	1	yes	characteristic
19	C₄×Q₈×C₃₇	C₂²	1	1	yes	characteristic
20	#107×C₃₇	—	2	1	no	—
21	C₄⋊C₈×C₃₇	C₂²	1	1	yes	characteristic
22	#103×C₃₇	—	2	1	no	—
23	C₂×Q₈×C₃₇	—	2	1	no	—
24	C₂×Q₈×C₃₇	D₄	1	1	yes	characteristic
25	C₂×Dic₇₄	—	2	1	no	—
26	C₂×C₈×C₃₇	—	2	1	no	—
27	C₄⋊C₄×C₃₇	—	2	1	no	—
28	C₄⋊C₄×C₃₇	—	2	1	no	—
29	C₄⋊C₄×C₃₇	—	2	1	no	—
30	C₄⋊C₄×C₃₇	D₄	1	1	yes	characteristic
31	C₄²×C₃₇	—	2	1	no	—
32	C₄²×C₃₇	C₂³	1	1	yes	characteristic
33	C₄⋊Dic₃₇	—	2	1	no	—
34	C₄⋊Dic₃₇	—	2	1	no	—
35	C₂×#64436	—	2	1	no	—
36	C₂×#64436	—	2	1	no	—
37	C₂×#64436	D₄	1	1	yes	characteristic
38	C₂×#64436	D₄	1	1	yes	characteristic
39	Q₈×C₃₇	—	2	1	no	—
40	Q₈×C₃₇	—	8	1	no	—
41	Q₈×C₃₇	Q₁₆	1	2	yes	—
42	C₂×C₄×C₃₇	—	2	1	no	—
43	C₂×C₄×C₃₇	—	2	1	no	—
44	C₂×C₄×C₃₇	—	4	1	no	—
45	C₂×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
47	C₂×C₄×C₃₇	#41	1	1	yes	characteristic
48	C₂×Dic₃₇	—	4	1	no	—
49	Dic₇₄	—	4	1	no	—
50	Dic₇₄	—	4	1	no	—
51	C₈×C₃₇	—	4	1	no	—
52	#64436	—	2	1	no	—
53	#64436	—	2	1	no	—
54	#64436	—	4	1	no	—
55	#64436	—	4	1	no	—
56	C₂²×C₃₇	#121	1	1	yes	characteristic
57	C₄×C₃₇	—	2	2	no	—
58	C₄×C₃₇	—	2	1	no	—
59	C₄×C₃₇	—	4	1	no	—
60	C₄×C₃₇	—	8	1	no	—
61	C₄×C₃₇	C₂×Q₁₆	1	1	yes	characteristic
62	C₄×C₃₇	#135	1	1	yes	characteristic
63	C₄×C₃₇	#135	1	1	yes	characteristic
64	C₄×C₃₇	#137	1	1	yes	characteristic
65	Dic₃₇	—	8	1	no	—
66	#1684	—	37	1	no	maximal, Sylow
67	C₂×C₃₇	#464	1	1	yes	characteristic
68	C₂×C₃₇	#467	1	1	yes	characteristic
69	C₂×C₃₇	#471	1	1	yes	characteristic
70	#487	—	37	1	no	—
71	#477	D₃₇	1	1	yes	characteristic
72	#475	—	37	1	no	—
73	C₈×Q₈	—	37	1	no	—
74	#335	—	37	1	no	—
75	#333	—	37	1	no	—
76	#332	—	37	1	no	—
77	C₃₇	#1684	1	1	yes	characteristic, complemented, Sylow
78	C₄⋊Q₈	—	37	1	no	—
79	C₄⋊Q₈	C₂×D₃₇	1	1	yes	characteristic
80	C₄×Q₈	C₂×D₃₇	1	1	yes	characteristic
81	#107	—	2	1	no	—
82	#106	—	74	1	no	—
83	C₄⋊C₈	—	37	1	no	—
84	C₄⋊C₈	—	37	1	no	—
85	C₄⋊C₈	—	74	1	no	—
86	C₄⋊C₈	C₂×D₃₇	1	1	yes	characteristic
87	#103	—	2	1	no	—
88	#103	—	74	1	no	—
89	#103	—	74	1	no	—
90	#103	—	74	1	no	—
91	C₄×C₈	—	37	1	no	—
92	C₄×C₈	—	74	1	no	—
93	C₂×Q₈	—	2	1	no	—
94	C₂×Q₈	—	74	1	no	—
95	C₂×Q₈	C₂²⋊D₃₇	1	1	yes	characteristic
96	C₂×C₈	—	2	1	no	—
97	C₂×C₈	—	37	1	no	—
98	C₂×C₈	—	37	1	no	—
99	C₂×C₈	—	74	1	no	—
100	C₂×C₈	—	74	1	no	—
101	C₄⋊C₄	—	2	1	no	—
102	C₄⋊C₄	—	2	1	no	—
103	C₄⋊C₄	—	2	1	no	—
104	C₄⋊C₄	—	74	1	no	—
105	C₄⋊C₄	—	74	1	no	—
106	C₄⋊C₄	C₂²⋊D₃₇	1	1	yes	characteristic
107	C₄²	—	2	1	no	—
108	C₄²	C₂²×D₃₇	1	1	yes	characteristic
109	Q₈	—	2	1	no	—
110	Q₈	—	8	1	no	—
111	Q₈	—	148	1	no	—
112	Q₈	—	148	1	no	—
113	Q₈	#10596942	1	2	yes	—
114	C₂×C₄	—	2	1	no	—
115	C₂×C₄	—	2	1	no	—
116	C₂×C₄	—	4	1	no	—
117	C₂×C₄	—	148	1	no	—
118	C₂×C₄	D₄×D₃₇	1	1	yes	characteristic
119	C₂×C₄	#10596964	1	1	yes	characteristic
120	C₂×C₄	C₂×C₂²⋊D₃₇	1	1	yes	characteristic
121	C₈	—	4	1	no	—
122	C₈	—	74	1	no	—
123	C₈	—	74	1	no	—
124	C₈	—	148	1	no	—
125	C₈	—	148	1	no	—
126	C₂²	#11924508	1	1	yes	characteristic
127	C₄	—	2	2	no	—
128	C₄	—	2	1	no	—
129	C₄	—	4	1	no	—
130	C₄	—	8	1	no	—
131	C₄	—	296	1	no	—
132	C₄	C₂×#10596942	1	1	yes	characteristic
133	C₄	#11924556	1	1	yes	characteristic
134	C₄	#11924559	1	1	yes	characteristic
135	C₄	#11924568	1	1	yes	characteristic
136	C₂	#14557757	1	1	yes	characteristic
137	C₂	#14557772	1	1	yes	characteristic
138	C₂	#14557794	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14557885

C₂

yes

maximal, characteristic

#14557818

C₂

yes

maximal, characteristic

Q₈×#64436

C₂

yes

maximal, characteristic

#14557285

C₂

yes

maximal, characteristic

#14557282

C₂

yes

maximal, characteristic

#14557278

C₂

yes

maximal, characteristic

C₃₇×#477

C₂

yes

maximal, characteristic

#11924536

C₂²

yes

—

—

C₂²

yes

characteristic

C₄⋊#64436

C₂²

yes

—

—

—

—

C₂²

yes

characteristic

C₄⋊Q₈×C₃₇

C₂²

yes

characteristic

C₄×Q₈×C₃₇

C₂²

yes

characteristic

#107×C₃₇

—

C₄⋊C₈×C₃₇

C₂²

yes

characteristic

#103×C₃₇

—

C₂×Q₈×C₃₇

—

C₂×Q₈×C₃₇

D₄

yes

characteristic

C₂×Dic₇₄

—

C₂×C₈×C₃₇

—

C₄⋊C₄×C₃₇

—

C₄⋊C₄×C₃₇

—

C₄⋊C₄×C₃₇

—

C₄⋊C₄×C₃₇

D₄

yes

characteristic

C₄²×C₃₇

—

C₄²×C₃₇

C₂³

yes

characteristic

C₄⋊Dic₃₇

—

C₄⋊Dic₃₇

—

C₂×#64436

—

C₂×#64436

—

C₂×#64436

D₄

yes

characteristic

C₂×#64436

D₄

yes

characteristic

Q₈×C₃₇

—

Q₈×C₃₇

—

Q₈×C₃₇

Q₁₆

yes

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

C₂×D₄

yes

characteristic

C₂×C₄×C₃₇

C₂×D₄

yes

characteristic

C₂×C₄×C₃₇

#41

yes

characteristic

C₂×Dic₃₇

—

Dic₇₄

—

Dic₇₄

—

C₈×C₃₇

—

—

—

—

—

C₂²×C₃₇

#121

yes

characteristic

C₄×C₃₇

—

C₄×C₃₇

—

C₄×C₃₇

—

C₄×C₃₇

—

C₄×C₃₇

C₂×Q₁₆

yes

C₄×C₃₇

yes

C₄×C₃₇

yes

C₄×C₃₇

yes

Dic₃₇

—

—

C₂×C₃₇

yes

C₂×C₃₇

yes

C₂×C₃₇

yes

—

D₃₇

yes

characteristic

#475

—

C₈×Q₈

—

#335

—

#333

—

#332

—

C₃₇

#1684

yes

characteristic, complemented, Sylow

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₃₇

yes

characteristic

C₄×Q₈

C₂×D₃₇

yes

characteristic

#107

—

#106

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

C₂×D₃₇

yes

—

—

—

—

C₄×C₈

—

C₄×C₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂²⋊D₃₇

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

100

C₂×C₈

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

106

C₄⋊C₄

C₂²⋊D₃₇

yes

characteristic

107

C₄²

—

108

C₄²

C₂²×D₃₇

yes

characteristic

109

Q₈

—

110

Q₈

—

111

Q₈

—

148

—

112

Q₈

—

148

—

113

Q₈

#10596942

yes

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

148

—

118

C₂×C₄

D₄×D₃₇

yes

characteristic

119

C₂×C₄

#10596964

yes

characteristic

120

C₂×C₄

C₂×C₂²⋊D₃₇

yes

characteristic

121

C₈

—

122

C₈

—

123

C₈

—

124

C₈

—

148

—

125

C₈

—

148

—

126

C₂²

#11924508

yes

characteristic

127

C₄

—

128

C₄

—

129

C₄

—

130

C₄

—

131

C₄

—

296

—

132

C₄

C₂×#10596942

yes

characteristic

133

C₄

#11924556

yes

characteristic

134

C₄

#11924559

yes

characteristic

135

C₄

#11924568

yes

characteristic

136

C₂

#14557757

yes

characteristic

137

C₂

#14557772

yes

characteristic

138

C₂

#14557794

yes

characteristic

	subgroup	G/H
center	C₂²	#11924508
commutator, cosocle	C₄²×C₃₇	C₂³
Frattini	C₄²	C₂²×D₃₇
Fitting	C₃₇×#477	C₂
socle, prime radical	C₂²×C₃₇	#121
hypercenter	#477	D₃₇
hypocenter	C₃₇	#1684
LCS	G ▷ C₄²×C₃₇ ▷ C₂²×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #477

subgroup

G/H

center

C₂²

#11924508

commutator, cosocle

C₄²×C₃₇

C₂³

Frattini

C₄²

C₂²×D₃₇

Fitting

C₃₇×#477

C₂

C₂²×C₃₇

D₃₇

C₃₇

G ▷ C₄²×C₃₇ ▷ C₂²×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #477

N	H ≅ G/N	type
77. C₃₇	#1684	semidirect

H ≅ G/N

type

77. C₃₇

#1684

semidirect

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

subgroup	index
#14557885	2
#14557818	2
Q₈×#64436	2
#14557285	2
#14557282	2
#14557278	2
C₃₇×#477	2
#1684	37

subgroup

index

C₃₇×#477

Subgroups — #17994056

Conjugacy classes of subgroups

Complements

Maximal subgroups