Group #20241939 — subgroups

Subgroups — #20241939

Order 6016.

208 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 208 conjugacy classes of subgroups — too many to draw. Only its 77 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	#15689180	C₂	1	1	yes	maximal, characteristic
2	#15688991	C₂	1	1	yes	maximal, characteristic
3	#15688860	C₂	1	4	yes	maximal
4	C₄₇×#533	C₂	1	1	yes	maximal, characteristic
5	C₄⋊(C₄₇⋊C₈)	—	2	1	no	—
6	C₂²⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
7	#13093935	—	2	1	no	—
8	C₄₇⋊#95	—	2	1	no	—
9	C₄₇⋊#95	—	2	1	no	—
10	C₄₇⋊#95	C₂²	1	2	yes	—
11	C₄⋊Q₈×C₄₇	C₄	1	1	yes	characteristic, complemented
12	C₄⋊Q₈×C₄₇	C₄	1	1	yes	characteristic, complemented
13	#125×C₄₇	—	2	1	no	—
14	C₂²⋊Q₈×C₄₇	—	2	1	no	—
15	C₂²⋊Q₈×C₄₇	C₂²	1	2	yes	—
16	C₄×Q₈×C₄₇	—	2	1	no	—
17	C₄×Q₈×C₄₇	C₄	1	2	yes	complemented
18	#117×C₄₇	—	2	1	no	—
19	C₂×C₄²×C₄₇	C₂²	1	1	yes	characteristic
20	C₂×Q₈×C₄₇	—	2	1	no	—
21	C₂×Q₈×C₄₇	C₂×C₄	1	2	yes	—
22	C₂²×C₄×C₄₇	D₄	1	1	yes	characteristic
23	C₂²×C₄×C₄₇	D₄	1	1	yes	characteristic
24	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
25	C₂²×Dic₄₇	—	2	2	no	—
26	C₄⋊C₄×C₄₇	—	2	1	no	—
27	C₄⋊C₄×C₄₇	—	2	1	no	—
28	C₄⋊C₄×C₄₇	—	2	1	no	—
29	C₄⋊C₄×C₄₇	—	2	2	no	—
30	C₄⋊C₄×C₄₇	—	2	2	no	—
31	C₄⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
32	C₂²⋊C₄×C₄₇	—	2	2	no	—
33	C₄²×C₄₇	—	2	2	no	—
34	C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
35	C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
36	C₄²×C₄₇	D₄	1	1	yes	characteristic
37	C₄²×C₄₇	D₄	1	1	yes	characteristic
38	C₂×C₄₇⋊C₈	—	2	2	no	—
39	C₂³×C₄₇	C₂×D₄	1	1	yes	characteristic
40	Q₈×C₄₇	—	4	2	no	—
41	Q₈×C₄₇	—	4	2	no	—
42	C₂×C₄×C₄₇	—	2	2	no	—
43	C₂×C₄×C₄₇	—	2	2	no	—
44	C₂×C₄×C₄₇	—	2	2	no	—
45	C₂×C₄×C₄₇	—	4	1	no	—
46	C₂×C₄×C₄₇	—	4	1	no	—
47	C₂×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
48	C₂×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
49	C₂×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
50	C₂×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
51	C₂×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
53	C₂×Dic₄₇	—	4	2	no	—
54	C₂×Dic₄₇	—	4	2	no	—
55	C₂×Dic₄₇	—	4	2	no	—
56	C₄₇⋊C₈	—	4	2	no	—
57	C₂²×C₄₇	—	2	1	no	—
58	C₂²×C₄₇	—	2	1	no	—
59	C₂²×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
60	C₂²×C₄₇	#137	1	2	yes	—
61	C₄×C₄₇	—	2	1	no	—
62	C₄×C₄₇	—	2	1	no	—
63	C₄×C₄₇	—	2	1	no	—
64	C₄×C₄₇	—	2	1	no	—
65	C₄×C₄₇	—	2	1	no	—
66	C₄×C₄₇	—	2	1	no	—
67	C₄×C₄₇	—	4	2	no	—
68	C₄×C₄₇	—	4	2	no	—
69	Dic₄₇	—	8	2	no	—
70	#1515	—	47	1	no	maximal, Sylow
71	C₂×C₄₇	—	2	2	no	—
72	C₂×C₄₇	#410	1	1	yes	characteristic
73	C₂×C₄₇	#419	1	1	yes	characteristic
74	C₂×C₄₇	#421	1	1	yes	characteristic
75	#533	D₄₇	1	1	yes	characteristic
76	#432	—	47	1	no	—
77	#383	—	47	1	no	—
78	#328	—	47	4	no	—
79	C₄₇	#1515	1	1	yes	characteristic, complemented, Sylow
80	C₄⋊Q₈	Dic₄₇	1	1	yes	characteristic, complemented
81	C₄⋊Q₈	Dic₄₇	1	1	yes	characteristic, complemented
82	#125	—	2	1	no	—
83	C₂²⋊Q₈	—	2	1	no	—
84	C₂²⋊Q₈	C₂×D₄₇	1	2	yes	—
85	C₄×Q₈	—	2	1	no	—
86	C₄×Q₈	Dic₄₇	1	2	yes	complemented
87	#117	—	2	1	no	—
88	C₂×C₄²	C₂×D₄₇	1	1	yes	characteristic
89	C₄⋊C₈	—	94	1	no	—
90	C₂²⋊C₈	—	47	2	no	—
91	#97	—	94	1	no	—
92	#95	—	47	2	no	—
93	#95	—	94	1	no	—
94	#95	—	94	1	no	—
95	C₂×Q₈	—	2	1	no	—
96	C₂×Q₈	C₂×Dic₄₇	1	2	yes	—
97	C₂²×C₄	—	94	2	no	—
98	C₂²×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
99	C₂²×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
100	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
101	C₂×C₈	—	94	2	no	—
102	C₄⋊C₄	—	2	1	no	—
103	C₄⋊C₄	—	2	1	no	—
104	C₄⋊C₄	—	2	1	no	—
105	C₄⋊C₄	—	2	2	no	—
106	C₄⋊C₄	—	2	2	no	—
107	C₄⋊C₄	C₂×Dic₄₇	1	2	yes	—
108	C₂²⋊C₄	—	2	2	no	—
109	C₄²	—	2	2	no	—
110	C₄²	C₂×Dic₄₇	1	1	yes	characteristic
111	C₄²	C₂×Dic₄₇	1	1	yes	characteristic
112	C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
113	C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
114	C₂³	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
115	Q₈	—	4	2	no	—
116	Q₈	—	4	2	no	—
117	C₂×C₄	—	2	2	no	—
118	C₂×C₄	—	2	2	no	—
119	C₂×C₄	—	2	2	no	—
120	C₂×C₄	—	4	1	no	—
121	C₂×C₄	—	4	1	no	—
122	C₂×C₄	—	188	2	no	—
123	C₂×C₄	—	188	2	no	—
124	C₂×C₄	—	188	2	no	—
125	C₂×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
126	C₂×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
127	C₂×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
128	C₂×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
129	C₂×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
130	C₂×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
131	C₈	—	188	2	no	—
132	C₂²	—	2	1	no	—
133	C₂²	—	2	1	no	—
134	C₂²	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
135	C₂²	#13094055	1	2	yes	—
136	C₄	—	2	1	no	—
137	C₄	—	2	1	no	—
138	C₄	—	2	1	no	—
139	C₄	—	2	1	no	—
140	C₄	—	2	1	no	—
141	C₄	—	2	1	no	—
142	C₄	—	4	2	no	—
143	C₄	—	4	2	no	—
144	C₄	—	376	2	no	—
145	C₂	—	2	2	no	—
146	C₂	#15689097	1	1	yes	characteristic
147	C₂	#15689132	1	1	yes	characteristic
148	C₂	#15689140	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#15689180

C₂

yes

maximal, characteristic

#15688991

C₂

yes

maximal, characteristic

#15688860

C₂

yes

maximal

C₄₇×#533

C₂

yes

maximal, characteristic

C₄⋊(C₄₇⋊C₈)

—

C₂²⋊(C₄₇⋊C₈)

C₂²

yes

—

#13093935

—

C₄₇⋊#95

—

C₄₇⋊#95

—

C₄₇⋊#95

C₂²

yes

—

C₄⋊Q₈×C₄₇

C₄

yes

characteristic, complemented

C₄⋊Q₈×C₄₇

C₄

yes

characteristic, complemented

#125×C₄₇

—

C₂²⋊Q₈×C₄₇

—

C₂²⋊Q₈×C₄₇

C₂²

yes

—

C₄×Q₈×C₄₇

—

C₄×Q₈×C₄₇

C₄

yes

complemented

#117×C₄₇

—

C₂×C₄²×C₄₇

C₂²

yes

characteristic

C₂×Q₈×C₄₇

—

C₂×Q₈×C₄₇

C₂×C₄

yes

—

C₂²×C₄×C₄₇

D₄

yes

characteristic

C₂²×C₄×C₄₇

D₄

yes

characteristic

C₂²×C₄×C₄₇

C₂³

yes

characteristic

C₂²×Dic₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

C₂×C₄

yes

—

C₂²⋊C₄×C₄₇

—

C₄²×C₄₇

—

C₄²×C₄₇

C₂×C₄

yes

characteristic

C₄²×C₄₇

C₂×C₄

yes

characteristic

C₄²×C₄₇

D₄

yes

characteristic

C₄²×C₄₇

D₄

yes

characteristic

C₂×C₄₇⋊C₈

—

C₂³×C₄₇

C₂×D₄

yes

characteristic

Q₈×C₄₇

—

Q₈×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₄₇

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₄₇

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₄₇

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₄₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₄₇⋊C₈

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₄₇

#137

yes

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

Dic₄₇

—

#1515

—

maximal, Sylow

C₂×C₄₇

—

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

D₄₇

yes

—

—

—

C₄₇

#1515

yes

characteristic, complemented, Sylow

C₄⋊Q₈

Dic₄₇

yes

characteristic, complemented

C₄⋊Q₈

Dic₄₇

yes

characteristic, complemented

#125

—

C₂²⋊Q₈

—

C₂²⋊Q₈

C₂×D₄₇

yes

—

C₄×Q₈

—

C₄×Q₈

Dic₄₇

yes

complemented

#117

—

C₂×C₄²

C₂×D₄₇

yes

characteristic

C₄⋊C₈

—

C₂²⋊C₈

—

—

—

—

—

C₂×Q₈

—

C₂×Q₈

C₂×Dic₄₇

yes

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊D₄₇

yes

characteristic

C₂²×C₄

C₂²⋊D₄₇

yes

characteristic

100

C₂²×C₄

C₂²×D₄₇

yes

characteristic

101

C₂×C₈

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

106

C₄⋊C₄

—

107

C₄⋊C₄

C₂×Dic₄₇

yes

—

108

C₂²⋊C₄

—

109

C₄²

—

110

C₄²

C₂×Dic₄₇

yes

characteristic

111

C₄²

C₂×Dic₄₇

yes

characteristic

112

C₄²

C₂²⋊D₄₇

yes

characteristic

113

C₄²

C₂²⋊D₄₇

yes

characteristic

114

C₂³

C₂×C₂²⋊D₄₇

yes

characteristic

115

Q₈

—

116

Q₈

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

188

—

123

C₂×C₄

—

188

—

124

C₂×C₄

—

188

—

125

C₂×C₄

C₂²⋊Dic₄₇

yes

characteristic

126

C₂×C₄

C₂²⋊Dic₄₇

yes

characteristic

127

C₂×C₄

C₂²⋊Dic₄₇

yes

characteristic

128

C₂×C₄

C₂²⋊Dic₄₇

yes

characteristic

129

C₂×C₄

C₂²×Dic₄₇

yes

characteristic

130

C₂×C₄

C₂×C₂²⋊D₄₇

yes

characteristic

131

C₈

—

188

—

132

C₂²

—

133

C₂²

—

134

C₂²

C₂×C₂²⋊Dic₄₇

yes

characteristic

135

C₂²

#13094055

yes

—

136

C₄

—

137

C₄

—

138

C₄

—

139

C₄

—

140

C₄

—

141

C₄

—

142

C₄

—

143

C₄

—

144

C₄

—

376

—

145

C₂

—

146

C₂

#15689097

yes

characteristic

147

C₂

#15689132

yes

characteristic

148

C₂

#15689140

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊Dic₄₇
commutator	C₂×C₄×C₄₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₄₇
Fitting	C₄₇×#533	C₂
socle	C₂²×C₄₇	C₂×C₂²⋊C₄
hypercenter	#533	D₄₇
hypocenter	C₄₇	#1515
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #533

subgroup

G/H

center

C₂²

C₂×C₂²⋊Dic₄₇

commutator

C₂×C₄×C₄₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₄₇

Fitting

C₄₇×#533

C₂

socle

C₂²×C₄₇

C₂×C₂²⋊C₄

D₄₇

C₄₇

G ▷ C₂×C₄×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #533

N	H ≅ G/N	type
11. C₄⋊Q₈×C₄₇	C₄	semidirect
12. C₄⋊Q₈×C₄₇	C₄	semidirect
17. C₄×Q₈×C₄₇	C₄	semidirect
79. C₄₇	#1515	semidirect
80. C₄⋊Q₈	Dic₄₇	semidirect
81. C₄⋊Q₈	Dic₄₇	semidirect
86. C₄×Q₈	Dic₄₇	semidirect

H ≅ G/N

type

11. C₄⋊Q₈×C₄₇

C₄

semidirect

12. C₄⋊Q₈×C₄₇

C₄

semidirect

17. C₄×Q₈×C₄₇

C₄

semidirect

79. C₄₇

#1515

semidirect

80. C₄⋊Q₈

Dic₄₇

semidirect

81. C₄⋊Q₈

Dic₄₇

semidirect

86. C₄×Q₈

Dic₄₇

semidirect

Subgroups — #20241939

Conjugacy classes of subgroups

Complements