Group #20252257 — subgroups

Subgroups — #20252257

Order 6016.

456 conjugacy classes of subgroups; only the 189 normal subgroups are shown.

normal subgroups only

This group has 456 conjugacy classes of subgroups — too many to draw. Only its 189 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	C₂²×Dic₁₈₈	C₂	1	1	yes	maximal, characteristic
2	#15689600	C₂	1	2	yes	maximal
3	#15689231	C₂	1	4	yes	maximal, complemented
4	C₄×Dic₁₈₈	C₂	1	4	yes	maximal, complemented
5	#15689161	C₂	1	1	yes	maximal, characteristic
6	#15689133	C₂	1	2	yes	maximal
7	C₄₇×#405	C₂	1	1	yes	maximal, characteristic
8	C₂²×Dic₉₄	C₂²	1	2	yes	—
9	C₂×Dic₁₈₈	C₄	1	8	yes	complemented
10	C₂×Dic₁₈₈	C₂²	1	2	yes	—
11	C₂×Dic₁₈₈	C₂²	1	2	yes	—
12	C₄×Dic₉₄	C₂²	1	8	yes	—
13	#13093989	—	2	4	no	—
14	#13093983	C₂²	1	2	yes	—
15	#13093980	—	2	4	no	—
16	#13093958	C₂²	1	2	yes	—
17	#13093956	C₂²	1	2	yes	—
18	#13093949	C₂²	1	8	yes	—
19	C₂×M₁₆×C₄₇	C₂²	1	1	yes	characteristic
20	C₂²×C₈×C₄₇	C₂²	1	1	yes	characteristic
21	#117×C₄₇	C₂²	1	1	yes	characteristic
22	#97×C₄₇	C₂²	1	2	yes	—
23	C₄×C₈×C₄₇	C₂²	1	2	yes	—
24	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
25	C₂²×Dic₄₇	—	2	2	no	—
26	C₂×Dic₉₄	—	2	2	no	—
27	C₂×Dic₉₄	—	2	4	no	—
28	C₂×Dic₉₄	C₂×C₄	1	8	yes	—
29	C₂×Dic₉₄	C₂³	1	4	yes	—
30	M₁₆×C₄₇	—	2	2	no	—
31	C₂×C₈×C₄₇	C₂×C₄	1	4	yes	—
32	C₂×C₈×C₄₇	C₂³	1	2	yes	—
33	C₂×C₈×C₄₇	C₂³	1	1	yes	characteristic
34	C₂×C₈×C₄₇	C₂³	1	1	yes	characteristic
35	C₄⋊C₄×C₄₇	D₄	1	2	yes	—
36	C₂²⋊C₄×C₄₇	D₄	1	2	yes	—
37	C₄²×C₄₇	C₂³	1	2	yes	—
38	C₂²⋊Dic₄₇	—	2	2	no	—
39	C₄⋊Dic₄₇	C₂³	1	4	yes	—
40	Dic₄₇⋊C₄	—	2	8	no	—
41	C₄×Dic₄₇	—	2	8	no	—
42	C₄×Dic₄₇	—	2	2	no	—
43	Dic₁₈₈	C₂×C₄	1	8	yes	—
44	Dic₁₈₈	C₂×C₄	1	8	yes	—
45	C₂³×C₄₇	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×C₄₇	C₂²×C₄	1	2	yes	—
47	C₂×C₄×C₄₇	C₂×D₄	1	4	yes	—
48	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₄₇	#41	1	2	yes	—
50	C₂×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
51	C₂×Dic₄₇	—	2	8	no	—
52	C₂×Dic₄₇	—	2	4	no	—
53	C₂×Dic₄₇	—	2	4	no	—
54	Dic₉₄	—	2	8	no	—
55	Dic₉₄	—	2	4	no	—
56	Dic₉₄	C₂²×C₄	1	8	yes	—
57	C₈×C₄₇	—	2	2	no	—
58	C₈×C₄₇	C₂²×C₄	1	2	yes	—
59	C₈×C₄₇	C₂²×C₄	1	2	yes	—
60	C₂²×C₄₇	—	2	2	no	—
61	C₂²×C₄₇	C₄×D₄	1	2	yes	—
62	C₂²×C₄₇	C₂²×D₄	1	1	yes	characteristic
63	C₄×C₄₇	—	2	4	no	—
64	C₄×C₄₇	C₄×D₄	1	2	yes	—
65	C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
66	C₄×C₄₇	C₂×#41	1	1	yes	characteristic
67	Dic₄₇	—	2	8	no	—
68	Dic₄₇	—	4	4	no	—
69	#2950	—	47	1	no	maximal, Sylow
70	C₂×C₄₇	—	2	2	no	—
71	C₂×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
72	C₂×C₄₇	#578	1	2	yes	—
73	C₂²×Q₁₆	—	47	1	no	—
74	#519	—	47	2	no	—
75	#441	—	47	4	no	—
76	C₄×Q₁₆	—	47	4	no	—
77	#427	—	47	1	no	—
78	#419	—	47	2	no	—
79	#405	D₄₇	1	1	yes	characteristic
80	C₄₇	#2950	1	1	yes	characteristic, complemented, Sylow
81	C₂²×Q₈	—	47	2	no	—
82	C₂×Q₁₆	—	47	8	no	—
83	C₂×Q₁₆	—	47	2	no	—
84	C₂×Q₁₆	—	47	2	no	—
85	C₂×M₁₆	C₂×D₄₇	1	1	yes	characteristic
86	C₂²×C₈	C₂×D₄₇	1	1	yes	characteristic
87	C₄×Q₈	—	47	8	no	—
88	C₄×Q₈	—	94	4	no	—
89	#117	—	47	2	no	—
90	#117	—	94	4	no	—
91	#117	C₂×D₄₇	1	1	yes	characteristic
92	#107	—	47	2	no	—
93	#106	—	47	2	no	—
94	#103	—	47	8	no	—
95	#97	C₂×D₄₇	1	2	yes	—
96	C₄×C₈	C₂×D₄₇	1	2	yes	—
97	C₂×Q₈	—	47	8	no	—
98	C₂×Q₈	—	47	4	no	—
99	C₂×Q₈	—	94	2	no	—
100	C₂×Q₈	—	94	4	no	—
101	C₂²×C₄	—	94	2	no	—
102	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
103	Q₁₆	—	47	8	no	—
104	Q₁₆	—	47	8	no	—
105	M₁₆	—	2	2	no	—
106	C₂×C₈	C₄×D₄₇	1	4	yes	—
107	C₂×C₈	C₂²×D₄₇	1	2	yes	—
108	C₂×C₈	C₂²×D₄₇	1	1	yes	characteristic
109	C₂×C₈	C₂²×D₄₇	1	1	yes	characteristic
110	C₄⋊C₄	—	47	4	no	—
111	C₄⋊C₄	—	94	8	no	—
112	C₄⋊C₄	D₁₈₈	1	2	yes	—
113	C₂²⋊C₄	—	94	2	no	—
114	C₂²⋊C₄	D₁₈₈	1	2	yes	—
115	C₄²	—	94	2	no	—
116	C₄²	—	94	8	no	—
117	C₄²	C₂²×D₄₇	1	2	yes	—
118	C₂³	C₂×D₁₈₈	1	1	yes	characteristic
119	Q₈	—	47	8	no	—
120	Q₈	—	94	8	no	—
121	Q₈	—	94	4	no	—
122	C₂×C₄	—	94	8	no	—
123	C₂×C₄	—	94	4	no	—
124	C₂×C₄	—	94	4	no	—
125	C₂×C₄	C₂×C₄×D₄₇	1	2	yes	—
126	C₂×C₄	C₂×D₁₈₈	1	4	yes	—
127	C₂×C₄	C₂×D₁₈₈	1	1	yes	characteristic
128	C₂×C₄	#10625350	1	2	yes	—
129	C₂×C₄	C₂³×D₄₇	1	1	yes	characteristic
130	C₈	—	2	2	no	—
131	C₈	C₂×C₄×D₄₇	1	2	yes	—
132	C₈	C₂×C₄×D₄₇	1	2	yes	—
133	C₂²	—	2	2	no	—
134	C₂²	C₄×D₁₈₈	1	2	yes	—
135	C₂²	C₂²×D₁₈₈	1	1	yes	characteristic
136	C₄	—	2	4	no	—
137	C₄	—	94	8	no	—
138	C₄	—	188	4	no	—
139	C₄	C₄×D₁₈₈	1	2	yes	—
140	C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
141	C₄	C₂×#10625350	1	1	yes	characteristic
142	C₂	—	2	2	no	—
143	C₂	C₂×C₄×D₁₈₈	1	1	yes	characteristic
144	C₂	#15689940	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×Dic₁₈₈

C₂

yes

maximal, characteristic

#15689600

C₂

yes

maximal

#15689231

C₂

yes

maximal, complemented

C₄×Dic₁₈₈

C₂

yes

maximal, complemented

#15689161

C₂

yes

maximal, characteristic

#15689133

C₂

yes

maximal

C₄₇×#405

C₂

yes

maximal, characteristic

C₂²×Dic₉₄

C₂²

yes

—

C₂×Dic₁₈₈

C₄

yes

complemented

C₂×Dic₁₈₈

C₂²

yes

—

C₂×Dic₁₈₈

C₂²

yes

—

C₄×Dic₉₄

C₂²

yes

—

#13093989

—

#13093983

C₂²

yes

—

#13093980

—

#13093958

C₂²

yes

—

#13093956

C₂²

yes

—

#13093949

C₂²

yes

—

C₂×M₁₆×C₄₇

C₂²

yes

characteristic

C₂²×C₈×C₄₇

C₂²

yes

characteristic

#117×C₄₇

C₂²

yes

characteristic

#97×C₄₇

C₂²

yes

—

C₄×C₈×C₄₇

C₂²

yes

—

C₂²×C₄×C₄₇

C₂³

yes

characteristic

C₂²×Dic₄₇

—

C₂×Dic₉₄

—

C₂×Dic₉₄

—

C₂×Dic₉₄

C₂×C₄

yes

—

C₂×Dic₉₄

C₂³

yes

—

M₁₆×C₄₇

—

C₂×C₈×C₄₇

C₂×C₄

yes

—

C₂×C₈×C₄₇

C₂³

yes

—

C₂×C₈×C₄₇

C₂³

yes

characteristic

C₂×C₈×C₄₇

C₂³

yes

characteristic

C₄⋊C₄×C₄₇

D₄

yes

—

C₂²⋊C₄×C₄₇

D₄

yes

—

C₄²×C₄₇

C₂³

yes

—

C₂²⋊Dic₄₇

—

C₄⋊Dic₄₇

C₂³

yes

—

Dic₄₇⋊C₄

—

C₄×Dic₄₇

—

C₄×Dic₄₇

—

Dic₁₈₈

C₂×C₄

yes

—

Dic₁₈₈

C₂×C₄

yes

—

C₂³×C₄₇

C₂×D₄

yes

characteristic

C₂×C₄×C₄₇

C₂²×C₄

yes

—

C₂×C₄×C₄₇

C₂×D₄

yes

—

C₂×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂×C₄×C₄₇

#41

yes

—

C₂×C₄×C₄₇

C₂⁴

yes

characteristic

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

Dic₉₄

—

Dic₉₄

—

Dic₉₄

C₂²×C₄

yes

—

C₈×C₄₇

—

C₈×C₄₇

C₂²×C₄

yes

—

C₈×C₄₇

C₂²×C₄

yes

—

C₂²×C₄₇

—

C₂²×C₄₇

C₄×D₄

yes

—

C₂²×C₄₇

C₂²×D₄

yes

characteristic

C₄×C₄₇

—

C₄×C₄₇

C₄×D₄

yes

—

C₄×C₄₇

C₂³×C₄

yes

characteristic

C₄×C₄₇

C₂×#41

yes

characteristic

Dic₄₇

—

Dic₄₇

—

#2950

—

maximal, Sylow

C₂×C₄₇

—

C₂×C₄₇

C₂×C₄×D₄

yes

characteristic

C₂×C₄₇

#578

yes

—

C₂²×Q₁₆

—

#519

—

#441

—

C₄×Q₁₆

—

#427

—

#419

—

#405

D₄₇

yes

characteristic

C₄₇

#2950

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×M₁₆

C₂×D₄₇

yes

characteristic

C₂²×C₈

C₂×D₄₇

yes

characteristic

C₄×Q₈

—

C₄×Q₈

—

#117

—

#117

—

#117

C₂×D₄₇

yes

—

—

—

C₂×D₄₇

yes

—

C₄×C₈

C₂×D₄₇

yes

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

100

C₂×Q₈

—

101

C₂²×C₄

—

102

C₂²×C₄

C₂²×D₄₇

yes

characteristic

103

Q₁₆

—

104

Q₁₆

—

105

M₁₆

—

106

C₂×C₈

C₄×D₄₇

yes

—

107

C₂×C₈

C₂²×D₄₇

yes

—

108

C₂×C₈

C₂²×D₄₇

yes

characteristic

109

C₂×C₈

C₂²×D₄₇

yes

characteristic

110

C₄⋊C₄

—

111

C₄⋊C₄

—

112

C₄⋊C₄

D₁₈₈

yes

—

113

C₂²⋊C₄

—

114

C₂²⋊C₄

D₁₈₈

yes

—

115

C₄²

—

116

C₄²

—

117

C₄²

C₂²×D₄₇

yes

—

118

C₂³

C₂×D₁₈₈

yes

characteristic

119

Q₈

—

120

Q₈

—

121

Q₈

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

C₂×C₄×D₄₇

yes

—

126

C₂×C₄

C₂×D₁₈₈

yes

—

127

C₂×C₄

C₂×D₁₈₈

yes

characteristic

128

C₂×C₄

#10625350

yes

—

129

C₂×C₄

C₂³×D₄₇

yes

characteristic

130

C₈

—

131

C₈

C₂×C₄×D₄₇

yes

—

132

C₈

C₂×C₄×D₄₇

yes

—

133

C₂²

—

134

C₂²

C₄×D₁₈₈

yes

—

135

C₂²

C₂²×D₁₈₈

yes

characteristic

136

C₄

—

137

C₄

—

138

C₄

—

188

—

139

C₄

C₄×D₁₈₈

yes

—

140

C₄

C₂²×C₄×D₄₇

yes

characteristic

141

C₄

C₂×#10625350

yes

characteristic

142

C₂

—

143

C₂

C₂×C₄×D₁₈₈

yes

characteristic

144

C₂

#15689940

yes

—

	subgroup	G/H
center	C₂²	C₂²×D₁₈₈
commutator	C₄×C₄₇	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₄₇
Fitting	C₄₇×#405	C₂
socle	C₂²×C₄₇	C₂²×D₄
hypercenter	#405	D₄₇
hypocenter	C₄₇	#2950
LCS	G ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ #117 ◁ #405

subgroup

G/H

center

C₂²

C₂²×D₁₈₈

commutator

C₄×C₄₇

C₂³×C₄

Frattini

C₂×C₄

C₂³×D₄₇

Fitting

C₄₇×#405

C₂

socle

C₂²×C₄₇

C₂²×D₄

D₄₇

C₄₇

G ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂² ◁ #117 ◁ #405

N	H ≅ G/N	type
3. #15689231	C₂	semidirect
4. C₄×Dic₁₈₈	C₂	semidirect
9. C₂×Dic₁₈₈	C₄	semidirect
80. C₄₇	#2950	semidirect

H ≅ G/N

type

3. #15689231

C₂

semidirect

4. C₄×Dic₁₈₈

C₂

semidirect

9. C₂×Dic₁₈₈

C₄

semidirect

80. C₄₇

#2950

semidirect

Subgroups — #20252257

Conjugacy classes of subgroups

Complements