Group #20251998 — subgroups

Subgroups — #20251998

Order 6016.

440 conjugacy classes of subgroups; only the 209 normal subgroups are shown.

normal subgroups only

This group has 440 conjugacy classes of subgroups — too many to draw. Only its 209 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	#15689889	C₂	1	2	yes	maximal
2	C₂×#13094032	C₂	1	1	yes	maximal, characteristic
3	#15689172	C₂	1	1	yes	maximal, characteristic, complemented
4	#15689170	C₂	1	4	yes	maximal, complemented
5	C₂×#13093959	C₂	1	2	yes	maximal, complemented
6	#108×D₄₇	C₂	1	4	yes	maximal, complemented
7	C₄₇×#430	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#10625350	C₂²	1	1	yes	characteristic
9	#13094032	—	2	4	no	—
10	#13094032	C₄	1	8	yes	—
11	#13094030	—	2	4	no	—
12	C₂×#10625329	C₂²	1	2	yes	—
13	C₂×#10625324	C₂²	1	2	yes	—
14	M₁₆×D₄₇	C₂²	1	8	yes	—
15	C₂²×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
16	C₂×C₈×D₄₇	C₂²	1	2	yes	—
17	#13093960	C₂²	1	4	yes	—
18	#13093959	C₂²	1	4	yes	—
19	#13093959	C₂²	1	4	yes	—
20	C₂×M₁₆×C₄₇	C₂²	1	2	yes	—
21	C₂×M₁₆×C₄₇	C₂²	1	1	yes	characteristic
22	#108×C₄₇	C₂²	1	4	yes	—
23	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
24	C₂×C₂²⋊D₄₇	Q₈	1	2	yes	—
25	#10625350	C₂×C₄	1	4	yes	—
26	#10625350	D₄	1	4	yes	—
27	C₂×D₁₈₈	Q₈	1	1	yes	characteristic
28	C₂×C₄×D₄₇	C₂³	1	2	yes	—
29	C₂×Dic₉₄	Q₈	1	1	yes	characteristic
30	M₁₆×C₄₇	—	2	2	no	—
31	M₁₆×C₄₇	C₂×C₄	1	4	yes	—
32	M₁₆×C₄₇	C₂³	1	4	yes	—
33	C₂×C₈×C₄₇	—	2	2	no	—
34	C₂×C₈×C₄₇	C₂³	1	2	yes	—
35	#10625329	—	2	2	no	—
36	#10625329	C₂³	1	4	yes	—
37	C₂×C₄₇⋊C₈	—	2	2	no	—
38	C₂×C₄₇⋊C₈	C₂×C₄	1	4	yes	—
39	C₂×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
40	C₂×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
41	#10625324	—	2	8	no	—
42	#10625324	C₂×C₄	1	8	yes	—
43	C₈×D₄₇	—	2	8	no	—
44	C₈×D₄₇	C₂×C₄	1	8	yes	—
45	C₂³×C₄₇	C₂×Q₈	1	1	yes	characteristic
46	C₂²×D₄₇	C₂×Q₈	1	2	yes	—
47	C₂×C₄×C₄₇	C₂²×C₄	1	2	yes	—
48	C₂×C₄×C₄₇	C₂×D₄	1	2	yes	—
49	C₂×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
50	C₂×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
51	C₂²⋊D₄₇	C₄⋊C₄	1	8	yes	—
52	C₂×Dic₄₇	C₂×Q₈	1	2	yes	—
53	D₁₈₈	C₄⋊C₄	1	4	yes	—
54	C₄×D₄₇	C₂²×C₄	1	4	yes	—
55	C₄×D₄₇	C₂×D₄	1	4	yes	—
56	Dic₉₄	C₄⋊C₄	1	4	yes	—
57	C₈×C₄₇	—	2	4	no	—
58	C₈×C₄₇	C₂²×C₄	1	4	yes	—
59	C₄₇⋊C₈	—	2	4	no	—
60	C₄₇⋊C₈	C₂²×C₄	1	2	yes	—
61	C₄₇⋊C₈	C₂²×C₄	1	2	yes	—
62	C₂²×C₄₇	—	4	1	no	—
63	C₂²×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
64	C₂²×C₄₇	C₂²×Q₈	1	1	yes	characteristic
65	C₂×D₄₇	—	4	2	no	—
66	C₂×D₄₇	C₂×C₄⋊C₄	1	4	yes	—
67	C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
68	C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
69	C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
70	Dic₄₇	C₂×C₄⋊C₄	1	4	yes	—
71	#2916	—	47	1	no	maximal, Sylow
72	C₂×C₄₇	—	2	1	no	—
73	C₂×C₄₇	—	2	2	no	—
74	C₂×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
75	D₄₇	—	2	4	no	—
76	#568	—	47	2	no	—
77	C₂×#131	—	47	1	no	—
78	#430	—	47	1	no	—
79	#430	—	47	4	no	—
80	#430	D₄₇	1	1	yes	characteristic, complemented
81	C₂×#108	—	47	2	no	—
82	C₂×#108	—	47	4	no	—
83	C₄₇	#2916	1	1	yes	characteristic, complemented, Sylow
84	C₂×#41	—	47	1	no	—
85	#131	—	47	8	no	—
86	#131	—	94	4	no	—
87	#131	—	94	4	no	—
88	C₂×M₁₆	—	47	2	no	—
89	C₂×M₁₆	—	47	2	no	—
90	C₂×M₁₆	—	47	8	no	—
91	C₂×M₁₆	C₂×D₄₇	1	2	yes	—
92	C₂×M₁₆	C₂×D₄₇	1	1	yes	characteristic
93	C₂²×C₈	—	47	1	no	—
94	C₂²×C₈	—	47	2	no	—
95	#108	—	47	4	no	—
96	#108	—	47	4	no	—
97	#108	—	47	4	no	—
98	#108	C₂×D₄₇	1	4	yes	—
99	#41	—	47	4	no	—
100	#41	—	47	4	no	—
101	C₂×Q₈	—	47	1	no	—
102	C₂×D₄	—	47	2	no	—
103	C₂×D₄	—	47	1	no	—
104	C₂²×C₄	—	47	2	no	—
105	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
106	M₁₆	—	2	2	no	—
107	M₁₆	—	47	8	no	—
108	M₁₆	—	47	4	no	—
109	M₁₆	—	94	8	no	—
110	M₁₆	—	94	2	no	—
111	M₁₆	C₄×D₄₇	1	4	yes	—
112	M₁₆	C₂²×D₄₇	1	4	yes	—
113	C₂×C₈	—	2	2	no	—
114	C₂×C₈	—	47	4	no	—
115	C₂×C₈	—	47	1	no	—
116	C₂×C₈	—	47	1	no	—
117	C₂×C₈	—	47	8	no	—
118	C₂×C₈	—	94	8	no	—
119	C₂×C₈	—	94	2	no	—
120	C₂×C₈	C₂²×D₄₇	1	2	yes	—
121	C₂³	—	47	2	no	—
122	C₂³	Q₈×D₄₇	1	1	yes	characteristic
123	Q₈	—	47	4	no	—
124	D₄	—	47	8	no	—
125	D₄	—	47	4	no	—
126	C₂×C₄	—	47	2	no	—
127	C₂×C₄	—	47	4	no	—
128	C₂×C₄	—	47	4	no	—
129	C₂×C₄	C₂×C₄×D₄₇	1	2	yes	—
130	C₂×C₄	D₄×D₄₇	1	2	yes	—
131	C₂×C₄	Q₈×D₄₇	1	1	yes	characteristic
132	C₂×C₄	C₂³×D₄₇	1	1	yes	characteristic
133	C₈	—	2	4	no	—
134	C₈	—	47	2	no	—
135	C₈	—	47	2	no	—
136	C₈	—	94	4	no	—
137	C₈	C₂×C₄×D₄₇	1	4	yes	—
138	C₂²	—	4	1	no	—
139	C₂²	—	47	4	no	—
140	C₂²	—	188	2	no	—
141	C₂²	C₄⋊C₄×D₄₇	1	2	yes	—
142	C₂²	C₂×Q₈×D₄₇	1	1	yes	characteristic
143	C₄	—	47	4	no	—
144	C₄	C₄⋊C₄×D₄₇	1	2	yes	—
145	C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
146	C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
147	C₂	—	2	1	no	—
148	C₂	—	2	2	no	—
149	C₂	—	94	4	no	—
150	C₂	C₂×C₄⋊C₄×D₄₇	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#15689889

C₂

yes

maximal

C₂×#13094032

C₂

yes

maximal, characteristic

#15689172

C₂

yes

maximal, characteristic, complemented

#15689170

C₂

yes

maximal, complemented

C₂×#13093959

C₂

yes

maximal, complemented

#108×D₄₇

C₂

yes

maximal, complemented

C₄₇×#430

C₂

yes

maximal, characteristic, complemented

C₂×#10625350

C₂²

yes

characteristic

#13094032

—

#13094032

C₄

yes

—

#13094030

—

C₂×#10625329

C₂²

yes

—

C₂×#10625324

C₂²

yes

—

M₁₆×D₄₇

C₂²

yes

—

C₂²×C₄₇⋊C₈

C₂²

yes

characteristic

C₂×C₈×D₄₇

C₂²

yes

—

#13093960

C₂²

yes

—

#13093959

C₂²

yes

—

#13093959

C₂²

yes

—

C₂×M₁₆×C₄₇

C₂²

yes

—

C₂×M₁₆×C₄₇

C₂²

yes

characteristic

#108×C₄₇

C₂²

yes

—

C₂²×C₄×C₄₇

C₂³

yes

characteristic

C₂×C₂²⋊D₄₇

Q₈

yes

—

#10625350

C₂×C₄

yes

—

#10625350

D₄

yes

—

C₂×D₁₈₈

Q₈

yes

characteristic

C₂×C₄×D₄₇

C₂³

yes

—

C₂×Dic₉₄

Q₈

yes

characteristic

M₁₆×C₄₇

—

M₁₆×C₄₇

C₂×C₄

yes

—

M₁₆×C₄₇

C₂³

yes

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

C₂³

yes

—

#10625329

—

#10625329

C₂³

yes

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

C₂×C₄

yes

—

C₂×C₄₇⋊C₈

C₂³

yes

characteristic

C₂×C₄₇⋊C₈

C₂³

yes

characteristic

#10625324

—

#10625324

C₂×C₄

yes

—

C₈×D₄₇

—

C₈×D₄₇

C₂×C₄

yes

—

C₂³×C₄₇

C₂×Q₈

yes

characteristic

C₂²×D₄₇

C₂×Q₈

yes

—

C₂×C₄×C₄₇

C₂²×C₄

yes

—

C₂×C₄×C₄₇

C₂×D₄

yes

—

C₂×C₄×C₄₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₄₇

C₂⁴

yes

characteristic

C₂²⋊D₄₇

C₄⋊C₄

yes

—

C₂×Dic₄₇

C₂×Q₈

yes

—

D₁₈₈

C₄⋊C₄

yes

—

C₄×D₄₇

C₂²×C₄

yes

—

C₄×D₄₇

C₂×D₄

yes

—

Dic₉₄

C₄⋊C₄

yes

—

C₈×C₄₇

—

C₈×C₄₇

C₂²×C₄

yes

—

C₄₇⋊C₈

—

C₄₇⋊C₈

C₂²×C₄

yes

—

C₄₇⋊C₈

C₂²×C₄

yes

—

C₂²×C₄₇

—

C₂²×C₄₇

C₂×C₄⋊C₄

yes

—

C₂²×C₄₇

C₂²×Q₈

yes

characteristic

C₂×D₄₇

—

C₂×D₄₇

C₂×C₄⋊C₄

yes

—

C₄×C₄₇

C₂×C₄⋊C₄

yes

—

C₄×C₄₇

C₂³×C₄

yes

characteristic

C₄×C₄₇

C₂²×D₄

yes

characteristic

Dic₄₇

C₂×C₄⋊C₄

yes

—

#2916

—

maximal, Sylow

C₂×C₄₇

—

C₂×C₄₇

—

C₂×C₄₇

C₂²×C₄⋊C₄

yes

characteristic

D₄₇

—

—

—

—

—

D₄₇

yes

characteristic, complemented

C₂×#108

—

C₂×#108

—

C₄₇

#2916

yes

characteristic, complemented, Sylow

C₂×#41

—

#131

—

#131

—

#131

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₄₇

yes

—

C₂×M₁₆

C₂×D₄₇

yes

characteristic

C₂²×C₈

—

C₂²×C₈

—

—

—

—

C₂×D₄₇

yes

—

#41

—

100

#41

—

101

C₂×Q₈

—

102

C₂×D₄

—

103

C₂×D₄

—

104

C₂²×C₄

—

105

C₂²×C₄

C₂²×D₄₇

yes

characteristic

106

M₁₆

—

107

M₁₆

—

108

M₁₆

—

109

M₁₆

—

110

M₁₆

—

111

M₁₆

C₄×D₄₇

yes

—

112

M₁₆

C₂²×D₄₇

yes

—

113

C₂×C₈

—

114

C₂×C₈

—

115

C₂×C₈

—

116

C₂×C₈

—

117

C₂×C₈

—

118

C₂×C₈

—

119

C₂×C₈

—

120

C₂×C₈

C₂²×D₄₇

yes

—

121

C₂³

—

122

C₂³

Q₈×D₄₇

yes

characteristic

123

Q₈

—

124

D₄

—

125

D₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

C₂×C₄×D₄₇

yes

—

130

C₂×C₄

D₄×D₄₇

yes

—

131

C₂×C₄

Q₈×D₄₇

yes

characteristic

132

C₂×C₄

C₂³×D₄₇

yes

characteristic

133

C₈

—

134

C₈

—

135

C₈

—

136

C₈

—

137

C₈

C₂×C₄×D₄₇

yes

—

138

C₂²

—

139

C₂²

—

140

C₂²

—

188

—

141

C₂²

C₄⋊C₄×D₄₇

yes

—

142

C₂²

C₂×Q₈×D₄₇

yes

characteristic

143

C₄

—

144

C₄

C₄⋊C₄×D₄₇

yes

—

145

C₄

C₂²×C₄×D₄₇

yes

characteristic

146

C₄

C₂×D₄×D₄₇

yes

characteristic

147

C₂

—

148

C₂

—

149

C₂

—

150

C₂

C₂×C₄⋊C₄×D₄₇

yes

characteristic

	subgroup	G/H
center	C₄	C₂×D₄×D₄₇
commutator	C₄×C₄₇	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₄₇
Fitting	C₄₇×#430	C₂
socle	C₂×C₄₇	C₂²×C₄⋊C₄
hypercenter	#430	D₄₇
hypocenter	C₄₇	#2916
LCS	G ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #430

subgroup

G/H

center

C₄

C₂×D₄×D₄₇

commutator

C₄×C₄₇

C₂³×C₄

Frattini

C₂×C₄

C₂³×D₄₇

Fitting

C₄₇×#430

C₂

socle

C₂×C₄₇

C₂²×C₄⋊C₄

D₄₇

C₄₇

G ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #430

N	H ≅ G/N	type
3. #15689172	C₂	semidirect
4. #15689170	C₂	semidirect
5. C₂×#13093959	C₂	semidirect
6. #108×D₄₇	C₂	semidirect
7. C₄₇×#430	C₂	semidirect
80. #430	D₄₇	semidirect
83. C₄₇	#2916	semidirect

H ≅ G/N

type

3. #15689172

C₂

semidirect

4. #15689170

C₂

semidirect

5. C₂×#13093959

C₂

semidirect

6. #108×D₄₇

C₂

semidirect

7. C₄₇×#430

C₂

semidirect

80. #430

D₄₇

semidirect

83. C₄₇

#2916

semidirect

Subgroups — #20251998

Conjugacy classes of subgroups

Complements