Group #20244550 — subgroups

Subgroups — #20244550

Order 6016.

246 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 246 conjugacy classes of subgroups — too many to draw. Only its 67 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	#15689754	C₂	1	1	yes	maximal, characteristic
2	#15689133	C₂	1	2	yes	maximal
3	#15689028	C₂	1	1	yes	maximal, characteristic
4	#15688884	C₂	1	2	yes	maximal
5	C₄₇×#343	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₉₄	C₂²	1	1	yes	characteristic
7	#13094023	—	2	2	no	—
8	#13094015	—	2	2	no	—
9	#13094003	—	2	2	no	—
10	#13093983	C₂²	1	2	yes	—
11	C₂×C₄×Dic₄₇	C₂²	1	1	yes	characteristic
12	#13093949	—	2	2	no	—
13	#13093949	—	2	2	no	—
14	C₄₇⋊#95	—	2	2	no	—
15	C₂×M₁₆×C₄₇	C₂²	1	2	yes	—
16	C₂×C₄⋊C₄×C₄₇	C₂²	1	1	yes	characteristic
17	C₂²×C₄×C₄₇	D₄	1	2	yes	—
18	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
19	C₂²×Dic₄₇	—	2	1	no	—
20	C₂²×Dic₄₇	—	2	1	no	—
21	C₂×Dic₉₄	—	2	2	no	—
22	C₂×Dic₉₄	—	2	2	no	—
23	C₂×Dic₉₄	—	2	1	no	—
24	C₂×Dic₉₄	—	2	2	no	—
25	C₂×Dic₉₄	Q₈	1	2	yes	complemented
26	M₁₆×C₄₇	—	2	2	no	—
27	M₁₆×C₄₇	—	2	2	no	—
28	C₂×C₈×C₄₇	—	2	2	no	—
29	C₄⋊C₄×C₄₇	—	2	2	no	—
30	C₂²⋊Dic₄₇	—	2	2	no	—
31	C₄⋊Dic₄₇	—	2	2	no	—
32	C₄⋊Dic₄₇	D₄	1	2	yes	—
33	C₄⋊Dic₄₇	D₄	1	2	yes	—
34	Dic₄₇⋊C₄	—	2	4	no	—
35	C₄×Dic₄₇	—	2	2	no	—
36	C₄×Dic₄₇	—	2	2	no	—
37	C₂³×C₄₇	C₂×D₄	1	1	yes	characteristic
38	C₂×C₄×C₄₇	—	2	2	no	—
39	C₂×C₄×C₄₇	—	4	2	no	—
40	C₂×C₄×C₄₇	C₂×D₄	1	2	yes	—
41	C₂×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
42	C₂×C₄×C₄₇	#41	1	2	yes	—
43	C₂×C₄×C₄₇	#41	1	1	yes	characteristic
44	C₂×Dic₄₇	—	2	2	no	—
45	C₂×Dic₄₇	—	2	2	no	—
46	C₂×Dic₄₇	—	2	2	no	—
47	C₂×Dic₄₇	—	2	2	no	—
48	C₂×Dic₄₇	—	4	1	no	—
49	C₂×Dic₄₇	—	4	1	no	—
50	C₂×Dic₄₇	—	4	1	no	—
51	C₂×Dic₄₇	—	4	1	no	—
52	Dic₉₄	—	2	4	no	—
53	Dic₉₄	—	4	2	no	—
54	Dic₉₄	—	8	2	no	—
55	C₈×C₄₇	—	4	2	no	—
56	C₂²×C₄₇	—	2	1	no	—
57	C₂²×C₄₇	—	2	1	no	—
58	C₂²×C₄₇	#120	1	1	yes	characteristic
59	C₂²×C₄₇	C₂²⋊Q₈	1	1	yes	characteristic
60	C₂²×C₄₇	#124	1	1	yes	characteristic
61	C₄×C₄₇	—	4	2	no	—
62	C₄×C₄₇	C₂²⋊Q₈	1	2	yes	—
63	C₄×C₄₇	#124	1	2	yes	—
64	Dic₄₇	—	4	2	no	—
65	Dic₄₇	—	4	2	no	—
66	Dic₄₇	—	4	2	no	—
67	Dic₄₇	—	4	2	no	—
68	#2028	—	47	1	no	maximal, Sylow
69	C₂×C₄₇	—	2	1	no	—
70	C₂×C₄₇	—	2	1	no	—
71	C₂×C₄₇	#393	1	1	yes	characteristic
72	C₂×C₄₇	#455	1	1	yes	characteristic
73	C₂×C₄₇	#456	1	1	yes	characteristic
74	#544	—	47	1	no	—
75	#419	—	47	2	no	—
76	#391	—	47	1	no	—
77	#343	D₄₇	1	1	yes	characteristic
78	#339	—	47	2	no	—
79	C₄₇	#2028	1	1	yes	characteristic, complemented, Sylow
80	C₂²×Q₈	—	47	1	no	—
81	C₂×M₁₆	C₂×D₄₇	1	2	yes	—
82	C₄⋊Q₈	—	94	2	no	—
83	#125	—	94	2	no	—
84	C₂²⋊Q₈	—	94	2	no	—
85	#117	—	47	2	no	—
86	C₂×C₄⋊C₄	C₂×D₄₇	1	1	yes	characteristic
87	C₂×C₄²	—	47	1	no	—
88	#103	—	94	2	no	—
89	#103	—	94	2	no	—
90	#95	—	94	2	no	—
91	C₂×Q₈	—	47	2	no	—
92	C₂×Q₈	—	94	2	no	—
93	C₂×Q₈	—	94	1	no	—
94	C₂×Q₈	—	94	2	no	—
95	C₂×Q₈	—	94	2	no	—
96	C₂²×C₄	—	94	1	no	—
97	C₂²×C₄	—	94	1	no	—
98	C₂²×C₄	D₁₈₈	1	2	yes	—
99	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
100	M₁₆	—	2	2	no	—
101	M₁₆	—	2	2	no	—
102	C₂×C₈	—	2	2	no	—
103	C₄⋊C₄	—	2	2	no	—
104	C₄⋊C₄	—	47	2	no	—
105	C₄⋊C₄	—	47	2	no	—
106	C₄⋊C₄	—	94	4	no	—
107	C₄⋊C₄	—	94	2	no	—
108	C₂²⋊C₄	—	94	2	no	—
109	C₄²	—	94	2	no	—
110	C₄²	—	94	2	no	—
111	C₂³	C₂×D₁₈₈	1	1	yes	characteristic
112	Q₈	—	94	4	no	—
113	Q₈	—	376	2	no	—
114	Q₈	—	188	2	no	—
115	C₂×C₄	—	2	2	no	—
116	C₂×C₄	—	4	2	no	—
117	C₂×C₄	—	94	2	no	—
118	C₂×C₄	—	94	2	no	—
119	C₂×C₄	—	94	2	no	—
120	C₂×C₄	—	94	2	no	—
121	C₂×C₄	—	188	1	no	—
122	C₂×C₄	—	188	1	no	—
123	C₂×C₄	—	188	1	no	—
124	C₂×C₄	—	188	1	no	—
125	C₂×C₄	#10625350	1	2	yes	—
126	C₂×C₄	D₄×D₄₇	1	2	yes	—
127	C₂×C₄	#10625352	1	1	yes	characteristic
128	C₂×C₄	Q₈×D₄₇	1	1	yes	characteristic
129	C₈	—	4	2	no	—
130	C₂²	—	2	1	no	—
131	C₂²	—	2	1	no	—
132	C₂²	#13093994	1	1	yes	characteristic
133	C₂²	#13094004	1	1	yes	characteristic
134	C₂²	#13094012	1	1	yes	characteristic
135	C₄	—	4	2	no	—
136	C₄	—	188	2	no	—
137	C₄	—	188	2	no	—
138	C₄	—	188	2	no	—
139	C₄	—	188	2	no	—
140	C₄	#13094002	1	2	yes	—
141	C₄	#13094010	1	2	yes	—
142	C₂	—	2	1	no	—
143	C₂	—	2	1	no	—
144	C₂	#15689036	1	1	yes	characteristic
145	C₂	#15689303	1	1	yes	characteristic
146	C₂	#15689308	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#15689754

C₂

yes

maximal, characteristic

#15689133

C₂

yes

maximal

#15689028

C₂

yes

maximal, characteristic

#15688884

C₂

yes

maximal

C₄₇×#343

C₂

yes

maximal, characteristic

C₂²×Dic₉₄

C₂²

yes

—

—

—

C₂²

yes

—

C₂×C₄×Dic₄₇

C₂²

yes

characteristic

#13093949

—

#13093949

—

C₄₇⋊#95

—

C₂×M₁₆×C₄₇

C₂²

yes

—

C₂×C₄⋊C₄×C₄₇

C₂²

yes

characteristic

C₂²×C₄×C₄₇

D₄

yes

—

C₂²×C₄×C₄₇

C₂³

yes

characteristic

C₂²×Dic₄₇

—

C₂²×Dic₄₇

—

C₂×Dic₉₄

—

C₂×Dic₉₄

—

C₂×Dic₉₄

—

C₂×Dic₉₄

—

C₂×Dic₉₄

Q₈

yes

complemented

M₁₆×C₄₇

—

M₁₆×C₄₇

—

C₂×C₈×C₄₇

—

C₄⋊C₄×C₄₇

—

C₂²⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

D₄

yes

—

C₄⋊Dic₄₇

D₄

yes

—

Dic₄₇⋊C₄

—

C₄×Dic₄₇

—

C₄×Dic₄₇

—

C₂³×C₄₇

C₂×D₄

yes

characteristic

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂×D₄

yes

—

C₂×C₄×C₄₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₄₇

#41

yes

—

C₂×C₄×C₄₇

#41

yes

characteristic

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

Dic₉₄

—

Dic₉₄

—

Dic₉₄

—

C₈×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

#120

yes

characteristic

C₂²×C₄₇

C₂²⋊Q₈

yes

characteristic

C₂²×C₄₇

#124

yes

characteristic

C₄×C₄₇

—

C₄×C₄₇

C₂²⋊Q₈

yes

—

C₄×C₄₇

#124

yes

—

Dic₄₇

—

Dic₄₇

—

Dic₄₇

—

Dic₄₇

—

#2028

—

maximal, Sylow

C₂×C₄₇

—

C₂×C₄₇

—

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

—

—

—

D₄₇

yes

characteristic

#339

—

C₄₇

#2028

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₂×M₁₆

C₂×D₄₇

yes

—

C₄⋊Q₈

—

#125

—

C₂²⋊Q₈

—

#117

—

C₂×C₄⋊C₄

C₂×D₄₇

yes

characteristic

C₂×C₄²

—

#103

—

#103

—

#95

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

D₁₈₈

yes

—

C₂²×C₄

C₂²×D₄₇

yes

characteristic

100

M₁₆

—

101

M₁₆

—

102

C₂×C₈

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

106

C₄⋊C₄

—

107

C₄⋊C₄

—

108

C₂²⋊C₄

—

109

C₄²

—

110

C₄²

—

111

C₂³

C₂×D₁₈₈

yes

characteristic

112

Q₈

—

113

Q₈

—

376

—

114

Q₈

—

188

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

188

—

122

C₂×C₄

—

188

—

123

C₂×C₄

—

188

—

124

C₂×C₄

—

188

—

125

C₂×C₄

#10625350

yes

—

126

C₂×C₄

D₄×D₄₇

yes

—

127

C₂×C₄

#10625352

yes

characteristic

128

C₂×C₄

Q₈×D₄₇

yes

characteristic

129

C₈

—

130

C₂²

—

131

C₂²

—

132

C₂²

#13093994

yes

characteristic

133

C₂²

#13094004

yes

characteristic

134

C₂²

#13094012

yes

characteristic

135

C₄

—

136

C₄

—

188

—

137

C₄

—

188

—

138

C₄

—

188

—

139

C₄

—

188

—

140

C₄

#13094002

yes

—

141

C₄

#13094010

yes

—

142

C₂

—

143

C₂

—

144

C₂

#15689036

yes

characteristic

145

C₂

#15689303

yes

characteristic

146

C₂

#15689308

yes

characteristic

	subgroup	G/H
center	C₂²	#13093994
commutator	C₂²×C₄×C₄₇	C₂³
Frattini	C₂²×C₄	C₂²×D₄₇
Fitting	C₄₇×#343	C₂
socle	C₂²×C₄₇	#120
hypercenter	#343	D₄₇
hypocenter	C₄₇	#2028
LCS	G ▷ C₂²×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #343

subgroup

G/H

center

C₂²

#13093994

commutator

C₂²×C₄×C₄₇

C₂³

Frattini

C₂²×C₄

C₂²×D₄₇

Fitting

C₄₇×#343

C₂

C₂²×C₄₇

D₄₇

C₄₇

G ▷ C₂²×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #343

N	H ≅ G/N	type
25. C₂×Dic₉₄	Q₈	semidirect
79. C₄₇	#2028	semidirect

H ≅ G/N

type

25. C₂×Dic₉₄

Q₈

semidirect

79. C₄₇

#2028

semidirect

Subgroups — #20244550

Conjugacy classes of subgroups

Complements