Group #20251966 — subgroups

Subgroups — C₂×#15689159 #20251966

Order 6016.

512 conjugacy classes of subgroups; only the 269 normal subgroups are shown.

normal subgroups only

This group has 512 conjugacy classes of subgroups — too many to draw. Only its 269 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	C₂³×C₄₇⋊C₈	C₂	1	1	yes	maximal, characteristic
2	C₂×#13093983	C₂	1	1	yes	maximal, characteristic
3	#15689159	C₂	1	8	yes	maximal, direct product factor
4	C₂×#13093957	C₂	1	2	yes	maximal, complemented
5	C₂×#13093955	C₂	1	2	yes	maximal, complemented
6	C₂×#117×C₄₇	C₂	1	1	yes	maximal, characteristic
7	C₂²×C₄₇⋊C₈	C₄	1	8	yes	complemented
8	C₂²×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
9	C₂²×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
10	C₂²×C₄₇⋊C₈	C₂²	1	4	yes	—
11	#13093983	—	2	2	no	—
12	#13093983	C₂²	1	4	yes	—
13	C₂×C₄⋊Dic₄₇	C₂²	1	2	yes	—
14	C₂×C₂²⋊Dic₄₇	—	2	1	no	—
15	C₂×C₄×Dic₄₇	—	2	1	no	—
16	#13093957	C₂²	1	8	yes	complemented
17	#13093955	C₂²	1	8	yes	complemented
18	C₂³×C₄×C₄₇	C₂²	1	1	yes	characteristic
19	#117×C₄₇	—	2	2	no	—
20	#117×C₄₇	C₂²	1	4	yes	—
21	C₂×C₄⋊C₄×C₄₇	C₂²	1	2	yes	—
22	C₂×C₂²⋊C₄×C₄₇	—	2	1	no	—
23	C₂×C₄²×C₄₇	—	2	1	no	—
24	C₂⁴×C₄₇	D₄	1	1	yes	characteristic
25	C₂²×C₄×C₄₇	—	2	2	no	—
26	C₂²×C₄×C₄₇	C₂×C₄	1	4	yes	—
27	C₂²×C₄×C₄₇	D₄	1	2	yes	—
28	C₂²×C₄×C₄₇	D₄	1	1	yes	characteristic
29	C₂²×C₄×C₄₇	Q₈	1	4	yes	—
30	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
31	C₂²×C₄×C₄₇	C₂³	1	2	yes	—
32	C₂²×Dic₄₇	—	2	2	no	—
33	C₄⋊C₄×C₄₇	—	2	2	no	—
34	C₄⋊C₄×C₄₇	C₂³	1	4	yes	—
35	C₂²⋊C₄×C₄₇	—	2	4	no	—
36	C₄²×C₄₇	—	2	4	no	—
37	C₂²⋊Dic₄₇	—	2	4	no	—
38	C₄⋊Dic₄₇	—	2	2	no	—
39	C₄⋊Dic₄₇	C₂³	1	4	yes	—
40	C₄×Dic₄₇	—	2	4	no	—
41	C₂×C₄₇⋊C₈	C₂×C₄	1	4	yes	—
42	C₂×C₄₇⋊C₈	C₂×C₄	1	4	yes	—
43	C₂×C₄₇⋊C₈	C₂×C₄	1	16	yes	complemented
44	C₂×C₄₇⋊C₈	C₂³	1	2	yes	—
45	C₂×C₄₇⋊C₈	C₂³	1	2	yes	—
46	C₂³×C₄₇	—	2	4	no	—
47	C₂³×C₄₇	C₄⋊C₄	1	4	yes	—
48	C₂³×C₄₇	C₂×D₄	1	2	yes	—
49	C₂³×C₄₇	C₂×D₄	1	1	yes	characteristic
50	C₂×C₄×C₄₇	—	2	4	no	—
51	C₂×C₄×C₄₇	—	4	4	no	—
52	C₂×C₄×C₄₇	C₄⋊C₄	1	8	yes	—
53	C₂×C₄×C₄₇	C₄⋊C₄	1	4	yes	—
54	C₂×C₄×C₄₇	C₂²×C₄	1	2	yes	—
55	C₂×C₄×C₄₇	C₂²×C₄	1	4	yes	—
56	C₂×C₄×C₄₇	C₂×D₄	1	2	yes	—
57	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
58	C₂×C₄×C₄₇	C₂×Q₈	1	4	yes	—
59	C₂×C₄×C₄₇	C₂×Q₈	1	2	yes	—
60	C₂×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
61	C₂×Dic₄₇	—	2	4	no	—
62	C₂×Dic₄₇	—	4	4	no	—
63	C₄₇⋊C₈	C₂²×C₄	1	4	yes	—
64	C₄₇⋊C₈	C₂²×C₄	1	4	yes	—
65	C₂²×C₄₇	—	2	4	no	—
66	C₂²×C₄₇	—	2	8	no	—
67	C₂²×C₄₇	C₂×C₄⋊C₄	1	4	yes	—
68	C₂²×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
69	C₂²×C₄₇	#135	1	4	yes	—
70	C₂²×C₄₇	C₂²×D₄	1	1	yes	characteristic
71	C₄×C₄₇	—	4	4	no	—
72	C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
73	C₄×C₄₇	C₂×C₄⋊C₄	1	4	yes	—
74	C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
75	C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
76	Dic₄₇	—	4	4	no	—
77	C₂×#427	—	47	1	no	maximal, Sylow
78	C₂×C₄₇	—	2	4	no	—
79	C₂×C₄₇	#427	1	4	yes	complemented
80	C₂×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
81	C₂×C₄₇	C₂×#135	1	2	yes	—
82	C₂³×C₈	—	47	1	no	—
83	C₂×#117	—	47	1	no	—
84	C₂×#117	D₄₇	1	1	yes	characteristic
85	#427	—	47	8	no	—
86	C₂×#107	—	47	2	no	—
87	C₂×#106	—	47	2	no	—
88	C₄₇	C₂×#427	1	1	yes	characteristic, complemented, Sylow
89	C₂³×C₄	C₂×D₄₇	1	1	yes	characteristic
90	C₂²×C₈	—	47	8	no	—
91	C₂²×C₈	—	47	1	no	—
92	C₂²×C₈	—	47	1	no	—
93	C₂²×C₈	—	47	4	no	—
94	#117	—	2	2	no	—
95	#117	—	47	4	no	—
96	#117	—	94	2	no	—
97	#117	C₂×D₄₇	1	4	yes	—
98	C₂×C₄⋊C₄	—	47	2	no	—
99	C₂×C₄⋊C₄	C₂×D₄₇	1	2	yes	—
100	C₂×C₂²⋊C₄	—	2	1	no	—
101	C₂×C₂²⋊C₄	—	94	1	no	—
102	C₂×C₄²	—	2	1	no	—
103	C₂×C₄²	—	94	1	no	—
104	#107	—	47	8	no	—
105	#106	—	47	8	no	—
106	C₂⁴	C₂²⋊D₄₇	1	1	yes	characteristic
107	C₂²×C₄	—	2	2	no	—
108	C₂²×C₄	—	94	2	no	—
109	C₂²×C₄	Dic₉₄	1	4	yes	—
110	C₂²×C₄	C₄×D₄₇	1	4	yes	—
111	C₂²×C₄	C₂²⋊D₄₇	1	2	yes	—
112	C₂²×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
113	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
114	C₂²×C₄	C₂²×D₄₇	1	2	yes	—
115	C₂×C₈	—	47	4	no	—
116	C₂×C₈	—	47	4	no	—
117	C₂×C₈	—	47	16	no	—
118	C₂×C₈	—	47	2	no	—
119	C₂×C₈	—	47	2	no	—
120	C₄⋊C₄	—	2	2	no	—
121	C₄⋊C₄	—	47	4	no	—
122	C₄⋊C₄	—	94	2	no	—
123	C₄⋊C₄	C₂²×D₄₇	1	4	yes	—
124	C₂²⋊C₄	—	2	4	no	—
125	C₂²⋊C₄	—	94	4	no	—
126	C₄²	—	2	4	no	—
127	C₄²	—	94	4	no	—
128	C₂³	—	2	4	no	—
129	C₂³	Dic₄₇⋊C₄	1	4	yes	—
130	C₂³	C₂×C₂²⋊D₄₇	1	2	yes	—
131	C₂³	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
132	C₂×C₄	—	2	4	no	—
133	C₂×C₄	—	4	4	no	—
134	C₂×C₄	—	94	4	no	—
135	C₂×C₄	—	188	4	no	—
136	C₂×C₄	Dic₄₇⋊C₄	1	8	yes	—
137	C₂×C₄	Dic₄₇⋊C₄	1	4	yes	—
138	C₂×C₄	C₂×Dic₉₄	1	4	yes	—
139	C₂×C₄	C₂×Dic₉₄	1	2	yes	—
140	C₂×C₄	C₂×C₄×D₄₇	1	4	yes	—
141	C₂×C₄	C₂×C₄×D₄₇	1	2	yes	—
142	C₂×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
143	C₂×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
144	C₂×C₄	C₂³×D₄₇	1	1	yes	characteristic
145	C₈	—	47	4	no	—
146	C₈	—	47	4	no	—
147	C₂²	—	2	4	no	—
148	C₂²	—	2	8	no	—
149	C₂²	C₂×Dic₄₇⋊C₄	1	4	yes	—
150	C₂²	C₂×Dic₄₇⋊C₄	1	2	yes	—
151	C₂²	#13094044	1	4	yes	—
152	C₂²	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
153	C₄	—	4	4	no	—
154	C₄	—	188	4	no	—
155	C₄	C₂×Dic₄₇⋊C₄	1	2	yes	—
156	C₄	C₂×Dic₄₇⋊C₄	1	4	yes	—
157	C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
158	C₄	C₂²×Dic₉₄	1	1	yes	characteristic
159	C₂	—	2	4	no	—
160	C₂	#15689159	1	4	yes	direct product factor
161	C₂	C₂²×Dic₄₇⋊C₄	1	1	yes	characteristic
162	C₂	C₂×#13094044	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂³×C₄₇⋊C₈

C₂

yes

maximal, characteristic

C₂×#13093983

C₂

yes

maximal, characteristic

#15689159

C₂

yes

maximal, direct product factor

C₂×#13093957

C₂

yes

maximal, complemented

C₂×#13093955

C₂

yes

maximal, complemented

C₂×#117×C₄₇

C₂

yes

maximal, characteristic

C₂²×C₄₇⋊C₈

C₄

yes

complemented

C₂²×C₄₇⋊C₈

C₂²

yes

characteristic

C₂²×C₄₇⋊C₈

C₂²

yes

characteristic

C₂²×C₄₇⋊C₈

C₂²

yes

—

#13093983

—

#13093983

C₂²

yes

—

C₂×C₄⋊Dic₄₇

C₂²

yes

—

C₂×C₂²⋊Dic₄₇

—

C₂×C₄×Dic₄₇

—

#13093957

C₂²

yes

complemented

#13093955

C₂²

yes

complemented

C₂³×C₄×C₄₇

C₂²

yes

characteristic

#117×C₄₇

—

#117×C₄₇

C₂²

yes

—

C₂×C₄⋊C₄×C₄₇

C₂²

yes

—

C₂×C₂²⋊C₄×C₄₇

—

C₂×C₄²×C₄₇

—

C₂⁴×C₄₇

D₄

yes

characteristic

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

C₂×C₄

yes

—

C₂²×C₄×C₄₇

D₄

yes

—

C₂²×C₄×C₄₇

D₄

yes

characteristic

C₂²×C₄×C₄₇

Q₈

yes

—

C₂²×C₄×C₄₇

C₂³

yes

characteristic

C₂²×C₄×C₄₇

C₂³

yes

—

C₂²×Dic₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

C₂³

yes

—

C₂²⋊C₄×C₄₇

—

C₄²×C₄₇

—

C₂²⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

C₂³

yes

—

C₄×Dic₄₇

—

C₂×C₄₇⋊C₈

C₂×C₄

yes

—

C₂×C₄₇⋊C₈

C₂×C₄

yes

—

C₂×C₄₇⋊C₈

C₂×C₄

yes

complemented

C₂×C₄₇⋊C₈

C₂³

yes

—

C₂×C₄₇⋊C₈

C₂³

yes

—

C₂³×C₄₇

—

C₂³×C₄₇

C₄⋊C₄

yes

—

C₂³×C₄₇

C₂×D₄

yes

—

C₂³×C₄₇

C₂×D₄

yes

characteristic

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₄⋊C₄

yes

—

C₂×C₄×C₄₇

C₄⋊C₄

yes

—

C₂×C₄×C₄₇

C₂²×C₄

yes

—

C₂×C₄×C₄₇

C₂²×C₄

yes

—

C₂×C₄×C₄₇

C₂×D₄

yes

—

C₂×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂×C₄×C₄₇

C₂×Q₈

yes

—

C₂×C₄×C₄₇

C₂×Q₈

yes

—

C₂×C₄×C₄₇

C₂⁴

yes

characteristic

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₄₇⋊C₈

C₂²×C₄

yes

—

C₄₇⋊C₈

C₂²×C₄

yes

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

C₂×C₄⋊C₄

yes

—

C₂²×C₄₇

C₂×C₄⋊C₄

yes

—

C₂²×C₄₇

#135

yes

—

C₂²×C₄₇

C₂²×D₄

yes

characteristic

C₄×C₄₇

—

C₄×C₄₇

C₂×C₄⋊C₄

yes

—

C₄×C₄₇

C₂×C₄⋊C₄

yes

—

C₄×C₄₇

C₂³×C₄

yes

characteristic

C₄×C₄₇

C₂²×Q₈

yes

characteristic

Dic₄₇

—

C₂×#427

—

maximal, Sylow

C₂×C₄₇

—

C₂×C₄₇

#427

yes

complemented

C₂×C₄₇

C₂²×C₄⋊C₄

yes

characteristic

C₂×C₄₇

C₂×#135

yes

—

C₂³×C₈

—

C₂×#117

—

C₂×#117

D₄₇

yes

—

—

—

C₄₇

C₂×#427

yes

characteristic, complemented, Sylow

C₂³×C₄

C₂×D₄₇

yes

characteristic

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

—

—

—

C₂×D₄₇

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₄₇

yes

—

100

C₂×C₂²⋊C₄

—

101

C₂×C₂²⋊C₄

—

102

C₂×C₄²

—

103

C₂×C₄²

—

104

#107

—

105

#106

—

106

C₂⁴

C₂²⋊D₄₇

yes

characteristic

107

C₂²×C₄

—

108

C₂²×C₄

—

109

C₂²×C₄

Dic₉₄

yes

—

110

C₂²×C₄

C₄×D₄₇

yes

—

111

C₂²×C₄

C₂²⋊D₄₇

yes

—

112

C₂²×C₄

C₂²⋊D₄₇

yes

characteristic

113

C₂²×C₄

C₂²×D₄₇

yes

characteristic

114

C₂²×C₄

C₂²×D₄₇

yes

—

115

C₂×C₈

—

116

C₂×C₈

—

117

C₂×C₈

—

118

C₂×C₈

—

119

C₂×C₈

—

120

C₄⋊C₄

—

121

C₄⋊C₄

—

122

C₄⋊C₄

—

123

C₄⋊C₄

C₂²×D₄₇

yes

—

124

C₂²⋊C₄

—

125

C₂²⋊C₄

—

126

C₄²

—

127

C₄²

—

128

C₂³

—

129

C₂³

Dic₄₇⋊C₄

yes

—

130

C₂³

C₂×C₂²⋊D₄₇

yes

—

131

C₂³

C₂×C₂²⋊D₄₇

yes

characteristic

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

188

—

136

C₂×C₄

Dic₄₇⋊C₄

yes

—

137

C₂×C₄

Dic₄₇⋊C₄

yes

—

138

C₂×C₄

C₂×Dic₉₄

yes

—

139

C₂×C₄

C₂×Dic₉₄

yes

—

140

C₂×C₄

C₂×C₄×D₄₇

yes

—

141

C₂×C₄

C₂×C₄×D₄₇

yes

—

142

C₂×C₄

C₂×C₂²⋊D₄₇

yes

—

143

C₂×C₄

C₂×C₂²⋊D₄₇

yes

characteristic

144

C₂×C₄

C₂³×D₄₇

yes

characteristic

145

C₈

—

146

C₈

—

147

C₂²

—

148

C₂²

—

149

C₂²

C₂×Dic₄₇⋊C₄

yes

—

150

C₂²

C₂×Dic₄₇⋊C₄

yes

—

151

C₂²

#13094044

yes

—

152

C₂²

C₂²×C₂²⋊D₄₇

yes

characteristic

153

C₄

—

154

C₄

—

188

—

155

C₄

C₂×Dic₄₇⋊C₄

yes

—

156

C₄

C₂×Dic₄₇⋊C₄

yes

—

157

C₄

C₂²×C₄×D₄₇

yes

characteristic

158

C₄

C₂²×Dic₉₄

yes

characteristic

159

C₂

—

160

C₂

#15689159

yes

direct product factor

161

C₂

C₂²×Dic₄₇⋊C₄

yes

characteristic

162

C₂

C₂×#13094044

yes

—

	subgroup	G/H
center	C₂²×C₄	C₂²⋊D₄₇
commutator	C₄×C₄₇	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₄₇
Fitting	C₂×#117×C₄₇	C₂
socle	C₂³×C₄₇	C₂×D₄
hypercenter	C₂×#117	D₄₇
hypocenter	C₄₇	C₂×#427
LCS	G ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ C₂×#117

subgroup

G/H

center

C₂²×C₄

C₂²⋊D₄₇

commutator

C₄×C₄₇

C₂³×C₄

Frattini

C₂×C₄

C₂³×D₄₇

Fitting

C₂×#117×C₄₇

C₂

socle

C₂³×C₄₇

C₂×D₄

D₄₇

C₄₇

G ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ C₂×#117

N	H ≅ G/N	type
3. #15689159	C₂	direct (N × H)
4. C₂×#13093957	C₂	semidirect
5. C₂×#13093955	C₂	semidirect
7. C₂²×C₄₇⋊C₈	C₄	semidirect
16. #13093957	C₂²	semidirect
17. #13093955	C₂²	semidirect
43. C₂×C₄₇⋊C₈	C₂×C₄	semidirect
79. C₂×C₄₇	#427	semidirect
88. C₄₇	C₂×#427	semidirect
160. C₂	#15689159	direct (N × H)

H ≅ G/N

type

3. #15689159

C₂

direct (N × H)

4. C₂×#13093957

C₂

semidirect

5. C₂×#13093955

C₂

semidirect

7. C₂²×C₄₇⋊C₈

C₄

semidirect

16. #13093957

C₂²

semidirect

17. #13093955

C₂²

semidirect

43. C₂×C₄₇⋊C₈

C₂×C₄

semidirect

79. C₂×C₄₇

#427

semidirect

88. C₄₇

C₂×#427

semidirect

160. C₂

#15689159

direct (N × H)

Subgroups — C₂×#15689159 #20251966

Conjugacy classes of subgroups

Complements

Subgroups — C2×#15689159 #20251966

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#15689159 #20251966