Group #20253898 — subgroups

Subgroups — #20253898

Order 6016.

356 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 356 conjugacy classes of subgroups — too many to draw. Only its 157 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	#15689680	C₂	1	1	yes	maximal, characteristic
2	#15689526	C₂	1	2	yes	maximal, complemented
3	#15689522	C₂	1	4	yes	maximal, complemented
4	#15689519	C₂	1	2	yes	maximal, complemented
5	#15689159	C₂	1	4	yes	maximal
6	C₂×C₄×C₄₇⋊C₈	C₂	1	1	yes	maximal, characteristic
7	C₄₇×#533	C₂	1	1	yes	maximal, characteristic
8	C₂²×C₄₇⋊C₈	C₂²	1	2	yes	—
9	#13094024	C₂²	1	2	yes	—
10	#13094016	C₂²	1	2	yes	—
11	#13094003	—	2	2	no	—
12	C₄×Dic₉₄	—	2	2	no	—
13	#13093983	C₂²	1	2	yes	—
14	#13093957	C₂²	1	8	yes	—
15	#13093955	C₂²	1	8	yes	—
16	C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
17	C₄×C₄₇⋊C₈	C₂²	1	2	yes	—
18	C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
19	C₄⋊Q₈×C₄₇	C₂²	1	2	yes	—
20	#125×C₄₇	C₂²	1	2	yes	—
21	C₂²⋊Q₈×C₄₇	—	2	2	no	—
22	C₄×Q₈×C₄₇	—	2	2	no	—
23	#117×C₄₇	C₂²	1	2	yes	—
24	C₂×C₄²×C₄₇	C₂²	1	1	yes	characteristic
25	C₂×Q₈×C₄₇	—	2	2	no	—
26	C₂²×C₄×C₄₇	D₄	1	2	yes	—
27	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
28	C₂×Dic₉₄	—	2	2	no	—
29	C₄⋊C₄×C₄₇	—	2	4	no	—
30	C₄⋊C₄×C₄₇	—	2	2	no	—
31	C₄⋊C₄×C₄₇	C₂³	1	4	yes	—
32	C₂²⋊C₄×C₄₇	—	2	2	no	—
33	C₄²×C₄₇	—	2	2	no	—
34	C₄²×C₄₇	D₄	1	2	yes	—
35	C₄²×C₄₇	C₂³	1	1	yes	characteristic
36	C₄²×C₄₇	C₂³	1	1	yes	characteristic
37	C₂²⋊Dic₄₇	—	2	2	no	—
38	C₄⋊Dic₄₇	—	2	2	no	—
39	C₄⋊Dic₄₇	C₂³	1	4	yes	—
40	Dic₄₇⋊C₄	—	2	4	no	—
41	C₄×Dic₄₇	—	2	2	no	—
42	C₂×C₄₇⋊C₈	Q₈	1	8	yes	complemented
43	C₂×C₄₇⋊C₈	C₂³	1	2	yes	—
44	C₂×C₄₇⋊C₈	C₂³	1	2	yes	—
45	C₂³×C₄₇	C₂×D₄	1	1	yes	characteristic
46	Q₈×C₄₇	—	4	4	no	—
47	C₂×C₄×C₄₇	—	2	4	no	—
48	C₂×C₄×C₄₇	—	2	4	no	—
49	C₂×C₄×C₄₇	C₂×D₄	1	2	yes	—
50	C₂×C₄×C₄₇	C₂×D₄	1	2	yes	—
51	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
52	C₂×C₄×C₄₇	C₂×Q₈	1	4	yes	—
53	C₂×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
54	C₂×Dic₄₇	—	2	4	no	—
55	Dic₉₄	—	4	4	no	—
56	C₄₇⋊C₈	C₂×Q₈	1	4	yes	—
57	C₄₇⋊C₈	C₂×Q₈	1	4	yes	—
58	C₂²×C₄₇	—	2	2	no	—
59	C₂²×C₄₇	C₄⋊Q₈	1	2	yes	—
60	C₂²×C₄₇	C₂²×D₄	1	1	yes	characteristic
61	C₄×C₄₇	—	2	2	no	—
62	C₄×C₄₇	—	4	4	no	—
63	C₄×C₄₇	C₄⋊Q₈	1	2	yes	—
64	C₄×C₄₇	#135	1	4	yes	—
65	C₄×C₄₇	C₂²×Q₈	1	2	yes	—
66	Dic₄₇	—	4	4	no	—
67	#3160	—	47	1	no	maximal, Sylow
68	C₂×C₄₇	—	2	2	no	—
69	C₂×C₄₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
70	C₂×C₄₇	C₂×#135	1	2	yes	—
71	#533	—	47	1	no	—
72	#533	D₄₇	1	1	yes	characteristic
73	#500	—	47	2	no	—
74	#499	—	47	4	no	—
75	#498	—	47	2	no	—
76	#427	—	47	4	no	—
77	C₂×C₄×C₈	—	47	1	no	—
78	C₄₇	#3160	1	1	yes	characteristic, complemented, Sylow
79	C₂²×C₈	—	47	2	no	—
80	C₄⋊Q₈	—	47	2	no	—
81	C₄⋊Q₈	C₂×D₄₇	1	2	yes	—
82	#125	—	47	2	no	—
83	#125	C₂×D₄₇	1	2	yes	—
84	C₂²⋊Q₈	—	2	2	no	—
85	C₂²⋊Q₈	—	94	2	no	—
86	C₄×Q₈	—	2	2	no	—
87	C₄×Q₈	—	94	2	no	—
88	#117	—	47	2	no	—
89	#117	C₂×D₄₇	1	2	yes	—
90	C₂×C₄²	C₂×D₄₇	1	1	yes	characteristic
91	#107	—	47	8	no	—
92	#106	—	47	8	no	—
93	C₄×C₈	—	47	1	no	—
94	C₄×C₈	—	47	2	no	—
95	C₄×C₈	—	47	1	no	—
96	C₂×Q₈	—	2	2	no	—
97	C₂×Q₈	—	94	2	no	—
98	C₂²×C₄	C₂²⋊D₄₇	1	2	yes	—
99	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
100	C₂×C₈	—	47	8	no	—
101	C₂×C₈	—	47	2	no	—
102	C₂×C₈	—	47	2	no	—
103	C₄⋊C₄	—	2	4	no	—
104	C₄⋊C₄	—	2	2	no	—
105	C₄⋊C₄	—	47	4	no	—
106	C₄⋊C₄	—	94	4	no	—
107	C₄⋊C₄	—	94	2	no	—
108	C₄⋊C₄	C₂²×D₄₇	1	4	yes	—
109	C₂²⋊C₄	—	2	2	no	—
110	C₂²⋊C₄	—	94	2	no	—
111	C₄²	—	2	2	no	—
112	C₄²	—	94	2	no	—
113	C₄²	C₂²⋊D₄₇	1	2	yes	—
114	C₄²	C₂²×D₄₇	1	1	yes	characteristic
115	C₄²	C₂²×D₄₇	1	1	yes	characteristic
116	C₂³	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
117	Q₈	—	4	4	no	—
118	Q₈	—	188	4	no	—
119	C₂×C₄	—	2	4	no	—
120	C₂×C₄	—	2	4	no	—
121	C₂×C₄	—	94	4	no	—
122	C₂×C₄	Q₈×D₄₇	1	4	yes	—
123	C₂×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
124	C₂×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
125	C₂×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
126	C₂×C₄	C₂³×D₄₇	1	1	yes	characteristic
127	C₈	—	47	4	no	—
128	C₈	—	47	4	no	—
129	C₂²	—	2	2	no	—
130	C₂²	#13094022	1	2	yes	—
131	C₂²	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
132	C₄	—	2	2	no	—
133	C₄	—	4	4	no	—
134	C₄	—	188	4	no	—
135	C₄	#13094022	1	2	yes	—
136	C₄	#13094044	1	4	yes	—
137	C₄	C₂×Q₈×D₄₇	1	2	yes	—
138	C₂	—	2	2	no	—
139	C₂	C₂×#13094022	1	1	yes	characteristic
140	C₂	C₂×#13094044	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#15689680

C₂

yes

maximal, characteristic

#15689526

C₂

yes

maximal, complemented

#15689522

C₂

yes

maximal, complemented

#15689519

C₂

yes

maximal, complemented

#15689159

C₂

yes

maximal

C₂×C₄×C₄₇⋊C₈

C₂

yes

maximal, characteristic

C₄₇×#533

C₂

yes

maximal, characteristic

C₂²×C₄₇⋊C₈

C₂²

yes

—

#13094024

C₂²

yes

—

#13094016

C₂²

yes

—

#13094003

—

C₄×Dic₉₄

—

#13093983

C₂²

yes

—

#13093957

C₂²

yes

—

#13093955

C₂²

yes

—

C₄×C₄₇⋊C₈

C₂²

yes

characteristic

C₄×C₄₇⋊C₈

C₂²

yes

—

C₄×C₄₇⋊C₈

C₂²

yes

characteristic

C₄⋊Q₈×C₄₇

C₂²

yes

—

#125×C₄₇

C₂²

yes

—

C₂²⋊Q₈×C₄₇

—

C₄×Q₈×C₄₇

—

#117×C₄₇

C₂²

yes

—

C₂×C₄²×C₄₇

C₂²

yes

characteristic

C₂×Q₈×C₄₇

—

C₂²×C₄×C₄₇

D₄

yes

—

C₂²×C₄×C₄₇

C₂³

yes

characteristic

C₂×Dic₉₄

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

C₂³

yes

—

C₂²⋊C₄×C₄₇

—

C₄²×C₄₇

—

C₄²×C₄₇

D₄

yes

—

C₄²×C₄₇

C₂³

yes

characteristic

C₄²×C₄₇

C₂³

yes

characteristic

C₂²⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

C₂³

yes

—

Dic₄₇⋊C₄

—

C₄×Dic₄₇

—

C₂×C₄₇⋊C₈

Q₈

yes

complemented

C₂×C₄₇⋊C₈

C₂³

yes

—

C₂×C₄₇⋊C₈

C₂³

yes

—

C₂³×C₄₇

C₂×D₄

yes

characteristic

Q₈×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂×D₄

yes

—

C₂×C₄×C₄₇

C₂×D₄

yes

—

C₂×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂×C₄×C₄₇

C₂×Q₈

yes

—

C₂×C₄×C₄₇

C₂⁴

yes

characteristic

C₂×Dic₄₇

—

Dic₉₄

—

C₄₇⋊C₈

C₂×Q₈

yes

—

C₄₇⋊C₈

C₂×Q₈

yes

—

C₂²×C₄₇

—

C₂²×C₄₇

C₄⋊Q₈

yes

—

C₂²×C₄₇

C₂²×D₄

yes

characteristic

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

C₄⋊Q₈

yes

—

C₄×C₄₇

#135

yes

—

C₄×C₄₇

C₂²×Q₈

yes

—

Dic₄₇

—

#3160

—

maximal, Sylow

C₂×C₄₇

—

C₂×C₄₇

C₂×C₄⋊Q₈

yes

characteristic

C₂×C₄₇

C₂×#135

yes

—

#533

—

#533

D₄₇

yes

—

—

—

—

C₂×C₄×C₈

—

C₄₇

#3160

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₄₇

yes

—

#125

—

#125

C₂×D₄₇

yes

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

C₄×Q₈

—

C₄×Q₈

—

#117

—

#117

C₂×D₄₇

yes

—

C₂×C₄²

C₂×D₄₇

yes

characteristic

#107

—

#106

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

C₂²⋊D₄₇

yes

—

C₂²×C₄

C₂²×D₄₇

yes

characteristic

100

C₂×C₈

—

101

C₂×C₈

—

102

C₂×C₈

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

106

C₄⋊C₄

—

107

C₄⋊C₄

—

108

C₄⋊C₄

C₂²×D₄₇

yes

—

109

C₂²⋊C₄

—

110

C₂²⋊C₄

—

111

C₄²

—

112

C₄²

—

113

C₄²

C₂²⋊D₄₇

yes

—

114

C₄²

C₂²×D₄₇

yes

characteristic

115

C₄²

C₂²×D₄₇

yes

characteristic

116

C₂³

C₂×C₂²⋊D₄₇

yes

characteristic

117

Q₈

—

118

Q₈

—

188

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

Q₈×D₄₇

yes

—

123

C₂×C₄

C₂×C₂²⋊D₄₇

yes

—

124

C₂×C₄

C₂×C₂²⋊D₄₇

yes

—

125

C₂×C₄

C₂×C₂²⋊D₄₇

yes

characteristic

126

C₂×C₄

C₂³×D₄₇

yes

characteristic

127

C₈

—

128

C₈

—

129

C₂²

—

130

C₂²

#13094022

yes

—

131

C₂²

C₂²×C₂²⋊D₄₇

yes

characteristic

132

C₄

—

133

C₄

—

134

C₄

—

188

—

135

C₄

#13094022

yes

—

136

C₄

#13094044

yes

—

137

C₄

C₂×Q₈×D₄₇

yes

—

138

C₂

—

139

C₂

C₂×#13094022

yes

characteristic

140

C₂

C₂×#13094044

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₄₇
commutator	C₂×C₄×C₄₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₄₇
Fitting	C₄₇×#533	C₂
socle	C₂²×C₄₇	C₂²×D₄
hypercenter	#533	D₄₇
hypocenter	C₄₇	#3160
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #533

subgroup

G/H

center

C₂×C₄

C₂×C₂²⋊D₄₇

commutator

C₂×C₄×C₄₇

C₂⁴

Frattini

C₂×C₄

C₂³×D₄₇

Fitting

C₄₇×#533

C₂

socle

C₂²×C₄₇

C₂²×D₄

D₄₇

C₄₇

G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #533

N	H ≅ G/N	type
2. #15689526	C₂	semidirect
3. #15689522	C₂	semidirect
4. #15689519	C₂	semidirect
42. C₂×C₄₇⋊C₈	Q₈	semidirect
78. C₄₇	#3160	semidirect

H ≅ G/N

type

2. #15689526

C₂

semidirect

3. #15689522

C₂

semidirect

4. #15689519

C₂

semidirect

42. C₂×C₄₇⋊C₈

Q₈

semidirect

78. C₄₇

#3160

semidirect

Subgroups — #20253898

Conjugacy classes of subgroups

Complements