Group #20244227 — subgroups

Subgroups — #20244227

Order 6016.

396 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 396 conjugacy classes of subgroups — too many to draw. Only its 97 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×#13094021	C₂	1	1	yes	maximal, characteristic
2	C₂×#13093940	C₂	1	4	yes	maximal, complemented
3	C₄×#10625329	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄₇×#423	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×D₁₈₈	C₄	1	4	yes	—
6	C₂²×D₁₈₈	C₂²	1	2	yes	—
7	C₂×#10625329	C₂²	1	2	yes	—
8	#13094021	—	2	1	no	—
9	#13094021	—	2	1	no	—
10	#13094021	—	2	2	no	—
11	#13093940	—	2	8	no	—
12	#13093935	—	2	1	no	—
13	C₄×C₄₇⋊C₈	—	2	1	no	—
14	C₂×M₁₆×C₄₇	C₂²	1	2	yes	—
15	C₂×C₄²×C₄₇	C₂²	1	1	yes	characteristic
16	C₄⋊C₈×C₄₇	—	2	2	no	—
17	C₂³×D₄₇	C₂×C₄	1	4	yes	—
18	C₂²×C₄×C₄₇	C₂×C₄	1	2	yes	—
19	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
20	C₂×D₁₈₈	—	2	4	no	—
21	C₂×D₁₈₈	—	2	4	no	—
22	C₂×D₁₈₈	—	2	4	no	—
23	C₂×D₁₈₈	—	2	2	no	—
24	C₂×D₁₈₈	—	2	2	no	—
25	C₂×D₁₈₈	—	2	4	no	—
26	C₂×D₁₈₈	—	4	8	no	—
27	M₁₆×C₄₇	—	2	4	no	—
28	C₂×C₈×C₄₇	—	2	2	no	—
29	C₄²×C₄₇	D₄	1	1	yes	characteristic
30	C₄²×C₄₇	D₄	1	1	yes	characteristic
31	C₄²×C₄₇	D₄	1	2	yes	—
32	#10625329	—	2	2	no	—
33	#10625329	D₄	1	4	yes	complemented
34	C₂×C₄₇⋊C₈	—	2	2	no	—
35	C₂³×C₄₇	C₂²×C₄	1	1	yes	characteristic
36	C₂²×D₄₇	—	2	4	no	—
37	C₂²×D₄₇	—	2	8	no	—
38	C₂²×D₄₇	—	4	8	no	—
39	C₂×C₄×C₄₇	—	2	4	no	—
40	C₂×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
41	C₂×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
42	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
43	C₂×C₄×C₄₇	C₂×D₄	1	2	yes	—
44	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₄₇	#41	1	2	yes	—
46	D₁₈₈	—	4	8	no	—
47	D₁₈₈	—	4	8	no	—
48	D₁₈₈	—	4	4	no	—
49	D₁₈₈	—	4	2	no	—
50	D₁₈₈	—	4	2	no	—
51	C₈×C₄₇	—	4	2	no	—
52	C₄₇⋊C₈	—	2	4	no	—
53	C₂²×C₄₇	—	2	2	no	—
54	C₂²×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
55	C₂²×C₄₇	C₄×D₄	1	2	yes	—
56	C₂×D₄₇	—	4	4	no	—
57	C₂×D₄₇	—	4	8	no	—
58	C₂×D₄₇	—	8	8	no	—
59	C₄×C₄₇	—	2	2	no	—
60	C₄×C₄₇	#100	1	4	yes	complemented
61	C₄×C₄₇	#121	1	2	yes	—
62	C₄×C₄₇	#124	1	1	yes	characteristic
63	C₄×C₄₇	#127	1	1	yes	characteristic
64	#1965	—	47	1	no	maximal, Sylow
65	C₂×C₄₇	—	2	2	no	—
66	C₂×C₄₇	#390	1	1	yes	characteristic
67	C₂×C₄₇	C₂×#100	1	2	yes	—
68	D₄₇	—	8	4	no	—
69	C₂×#127	—	47	1	no	—
70	#423	D₄₇	1	1	yes	characteristic, complemented
71	C₂×#100	—	47	4	no	—
72	C₄×M₁₆	—	47	1	no	—
73	C₄₇	#1965	1	1	yes	characteristic, complemented, Sylow
74	C₂²×D₄	—	47	4	no	—
75	C₂²×D₄	—	47	2	no	—
76	C₂×M₁₆	—	47	2	no	—
77	C₂×M₁₆	C₂×D₄₇	1	2	yes	—
78	#127	—	94	1	no	—
79	#127	—	94	1	no	—
80	#127	—	94	2	no	—
81	C₂×C₄²	C₂×D₄₇	1	1	yes	characteristic
82	C₄⋊C₈	—	2	2	no	—
83	#100	—	94	8	no	—
84	#97	—	94	1	no	—
85	C₄×C₈	—	94	1	no	—
86	C₂⁴	—	47	4	no	—
87	C₂×D₄	—	94	4	no	—
88	C₂×D₄	—	94	4	no	—
89	C₂×D₄	—	94	2	no	—
90	C₂×D₄	—	94	4	no	—
91	C₂×D₄	—	94	4	no	—
92	C₂×D₄	—	94	2	no	—
93	C₂×D₄	—	188	8	no	—
94	C₂²×C₄	C₄×D₄₇	1	2	yes	—
95	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
96	M₁₆	—	2	4	no	—
97	M₁₆	—	47	4	no	—
98	M₁₆	—	94	2	no	—
99	C₂×C₈	—	2	2	no	—
100	C₂×C₈	—	94	2	no	—
101	C₄²	D₁₈₈	1	1	yes	characteristic
102	C₄²	D₁₈₈	1	1	yes	characteristic
103	C₄²	C₂²⋊D₄₇	1	2	yes	—
104	C₂³	—	94	8	no	—
105	C₂³	—	94	4	no	—
106	C₂³	—	188	8	no	—
107	C₂³	C₂×C₄×D₄₇	1	1	yes	characteristic
108	D₄	—	188	8	no	—
109	D₄	—	188	8	no	—
110	D₄	—	188	4	no	—
111	D₄	—	188	2	no	—
112	D₄	—	188	2	no	—
113	C₂×C₄	—	2	4	no	—
114	C₂×C₄	(C₂×D₄₇)⋊C₄	1	2	yes	—
115	C₂×C₄	(C₂×D₄₇)⋊C₄	1	2	yes	—
116	C₂×C₄	C₂×D₁₈₈	1	1	yes	characteristic
117	C₂×C₄	D₄×D₄₇	1	2	yes	—
118	C₂×C₄	#10625354	1	2	yes	—
119	C₂×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
120	C₈	—	4	2	no	—
121	C₈	—	94	4	no	—
122	C₂²	—	2	2	no	—
123	C₂²	—	376	8	no	—
124	C₂²	—	188	4	no	—
125	C₂²	—	188	8	no	—
126	C₂²	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
127	C₂²	#13093986	1	2	yes	—
128	C₄	—	2	2	no	—
129	C₄	#13093940	1	4	yes	complemented
130	C₄	#13093999	1	2	yes	—
131	C₄	#13094013	1	1	yes	characteristic
132	C₄	#13094020	1	1	yes	characteristic
133	C₂	—	2	2	no	—
134	C₂	—	376	4	no	—
135	C₂	#15689024	1	1	yes	characteristic
136	C₂	C₂×#13093940	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13094021

C₂

yes

maximal, characteristic

C₂×#13093940

C₂

yes

maximal, complemented

C₄×#10625329

C₂

yes

maximal, characteristic, complemented

C₄₇×#423

C₂

yes

maximal, characteristic, complemented

C₂²×D₁₈₈

C₄

yes

—

C₂²×D₁₈₈

C₂²

yes

—

C₂×#10625329

C₂²

yes

—

—

—

—

—

—

C₄×C₄₇⋊C₈

—

C₂×M₁₆×C₄₇

C₂²

yes

—

C₂×C₄²×C₄₇

C₂²

yes

characteristic

C₄⋊C₈×C₄₇

—

C₂³×D₄₇

C₂×C₄

yes

—

C₂²×C₄×C₄₇

C₂×C₄

yes

—

C₂²×C₄×C₄₇

C₂³

yes

characteristic

C₂×D₁₈₈

—

C₂×D₁₈₈

—

C₂×D₁₈₈

—

C₂×D₁₈₈

—

C₂×D₁₈₈

—

C₂×D₁₈₈

—

C₂×D₁₈₈

—

M₁₆×C₄₇

—

C₂×C₈×C₄₇

—

C₄²×C₄₇

D₄

yes

characteristic

C₄²×C₄₇

D₄

yes

characteristic

C₄²×C₄₇

D₄

yes

—

#10625329

—

#10625329

D₄

yes

complemented

C₂×C₄₇⋊C₈

—

C₂³×C₄₇

C₂²×C₄

yes

characteristic

C₂²×D₄₇

—

C₂²×D₄₇

—

C₂²×D₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂²⋊C₄

yes

—

C₂×C₄×C₄₇

C₂²⋊C₄

yes

—

C₂×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂×C₄×C₄₇

C₂×D₄

yes

—

C₂×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂×C₄×C₄₇

#41

yes

—

D₁₈₈

—

D₁₈₈

—

D₁₈₈

—

D₁₈₈

—

D₁₈₈

—

C₈×C₄₇

—

C₄₇⋊C₈

—

C₂²×C₄₇

—

C₂²×C₄₇

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₄₇

C₄×D₄

yes

—

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₄×C₄₇

—

C₄×C₄₇

#100

yes

complemented

C₄×C₄₇

#121

yes

—

C₄×C₄₇

yes

C₄×C₄₇

yes

—

C₂×C₄₇

—

C₂×C₄₇

#390

yes

characteristic

C₂×C₄₇

C₂×#100

yes

—

D₄₇

—

C₂×#127

—

#423

D₄₇

yes

characteristic, complemented

C₂×#100

—

C₄×M₁₆

—

C₄₇

#1965

yes

characteristic, complemented, Sylow

C₂²×D₄

—

C₂²×D₄

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₄₇

yes

—

#127

—

#127

—

#127

—

C₂×C₄²

C₂×D₄₇

yes

characteristic

C₄⋊C₈

—

#100

—

#97

—

C₄×C₈

—

C₂⁴

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

188

—

C₂²×C₄

C₄×D₄₇

yes

—

C₂²×C₄

C₂²×D₄₇

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

C₂×C₈

—

100

C₂×C₈

—

101

C₄²

D₁₈₈

yes

characteristic

102

C₄²

D₁₈₈

yes

characteristic

103

C₄²

C₂²⋊D₄₇

yes

—

104

C₂³

—

105

C₂³

—

106

C₂³

—

188

—

107

C₂³

C₂×C₄×D₄₇

yes

characteristic

108

D₄

—

188

—

109

D₄

—

188

—

110

D₄

—

188

—

111

D₄

—

188

—

112

D₄

—

188

—

113

C₂×C₄

—

114

C₂×C₄

(C₂×D₄₇)⋊C₄

yes

—

115

C₂×C₄

(C₂×D₄₇)⋊C₄

yes

—

116

C₂×C₄

C₂×D₁₈₈

yes

characteristic

117

C₂×C₄

D₄×D₄₇

yes

—

118

C₂×C₄

#10625354

yes

—

119

C₂×C₄

C₂×C₂²⋊D₄₇

yes

characteristic

120

C₈

—

121

C₈

—

122

C₂²

—

123

C₂²

—

376

—

124

C₂²

—

188

—

125

C₂²

—

188

—

126

C₂²

C₂×(C₂×D₄₇)⋊C₄

yes

characteristic

127

C₂²

#13093986

yes

—

128

C₄

—

129

C₄

#13093940

yes

complemented

130

C₄

#13093999

yes

—

131

C₄

#13094013

yes

characteristic

132

C₄

#13094020

yes

characteristic

133

C₂

—

134

C₂

—

376

—

135

C₂

#15689024

yes

characteristic

136

C₂

C₂×#13093940

yes

—

	subgroup	G/H
center	C₂²	C₂×(C₂×D₄₇)⋊C₄
commutator	C₂³×C₄₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₄₇
Fitting	C₄₇×#423	C₂
socle	C₂²×C₄₇	C₂×C₂²⋊C₄
hypercenter	#423	D₄₇
hypocenter	C₄₇	#1965
LCS	G ▷ C₂³×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #423

subgroup

G/H

center

C₂²

C₂×(C₂×D₄₇)⋊C₄

commutator

C₂³×C₄₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₄₇

Fitting

C₄₇×#423

C₂

socle

C₂²×C₄₇

C₂×C₂²⋊C₄

D₄₇

C₄₇

G ▷ C₂³×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #423

N	H ≅ G/N	type
2. C₂×#13093940	C₂	semidirect
3. C₄×#10625329	C₂	semidirect
4. C₄₇×#423	C₂	semidirect
33. #10625329	D₄	semidirect
60. C₄×C₄₇	#100	semidirect
70. #423	D₄₇	semidirect
73. C₄₇	#1965	semidirect
129. C₄	#13093940	semidirect

H ≅ G/N

type

2. C₂×#13093940

C₂

semidirect

3. C₄×#10625329

C₂

semidirect

4. C₄₇×#423

C₂

semidirect

33. #10625329

D₄

semidirect

60. C₄×C₄₇

#100

semidirect

70. #423

D₄₇

semidirect

73. C₄₇

#1965

semidirect

129. C₄

#13093940

semidirect

Subgroups — #20244227

Conjugacy classes of subgroups

Complements