Group #21433868 — subgroups

Subgroups — C₂×#16837698 #21433868

Order 7552.

416 conjugacy classes of subgroups; only the 173 normal subgroups are shown.

normal subgroups only

This group has 416 conjugacy classes of subgroups — too many to draw. Only its 173 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	C₂×#14273727	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×Dic₁₁₈	C₂	1	1	yes	maximal, characteristic
3	#16837698	C₂	1	8	yes	maximal, direct product factor
4	C₂×#14273661	C₂	1	2	yes	maximal
5	C₂×#14273652	C₂	1	2	yes	maximal
6	C₂×C₄⋊C₈×C₅₉	C₂	1	1	yes	maximal, characteristic
7	C₂²×Dic₁₁₈	—	2	1	no	—
8	C₂²×Dic₁₁₈	C₂²	1	1	yes	characteristic
9	#14273727	—	2	2	no	—
10	#14273727	C₂²	1	4	yes	—
11	C₄×Dic₁₁₈	—	2	2	no	—
12	C₄×Dic₁₁₈	C₂²	1	4	yes	—
13	C₂×C₄⋊Dic₅₉	—	2	1	no	—
14	C₂×C₄⋊Dic₅₉	C₂²	1	2	yes	—
15	C₂×C₄⋊Dic₅₉	C₂²	1	1	yes	characteristic
16	C₂×Dic₅₉⋊C₄	—	2	1	no	—
17	C₂×C₄×Dic₅₉	—	2	1	no	—
18	#14273661	C₂²	1	8	yes	—
19	#14273652	C₂²	1	8	yes	—
20	C₂²×C₈×C₅₉	C₂²	1	2	yes	—
21	C₂×C₄²×C₅₉	C₂²	1	1	yes	characteristic
22	C₄⋊C₈×C₅₉	C₂²	1	4	yes	—
23	C₂²×C₄×C₅₉	D₄	1	1	yes	characteristic
24	C₂²×C₄×C₅₉	D₄	1	1	yes	characteristic
25	C₂²×C₄×C₅₉	C₂³	1	1	yes	characteristic
26	C₂²×Dic₅₉	—	2	1	no	—
27	C₂²×Dic₅₉	—	2	1	no	—
28	C₂²×Dic₅₉	—	2	2	no	—
29	C₂×Dic₁₁₈	—	2	1	no	—
30	C₂×Dic₁₁₈	—	2	2	no	—
31	C₂×Dic₁₁₈	—	2	4	no	—
32	C₂×Dic₁₁₈	—	4	4	no	—
33	C₂×Dic₁₁₈	Q₈	1	4	yes	complemented
34	C₂×Dic₁₁₈	C₂³	1	2	yes	—
35	C₂×C₈×C₅₉	—	2	4	no	—
36	C₂×C₈×C₅₉	C₂³	1	4	yes	—
37	C₄²×C₅₉	D₄	1	2	yes	—
38	C₄²×C₅₉	C₂³	1	2	yes	—
39	C₄⋊Dic₅₉	—	2	4	no	—
40	C₄⋊Dic₅₉	—	2	1	no	—
41	C₄⋊Dic₅₉	—	2	2	no	—
42	C₄⋊Dic₅₉	C₂³	1	4	yes	—
43	C₄⋊Dic₅₉	C₂³	1	2	yes	—
44	Dic₅₉⋊C₄	—	2	4	no	—
45	C₄×Dic₅₉	—	2	4	no	—
46	C₂³×C₅₉	C₂×D₄	1	1	yes	characteristic
47	C₂×C₄×C₅₉	—	2	2	no	—
48	C₂×C₄×C₅₉	Q₁₆	1	4	yes	—
49	C₂×C₄×C₅₉	C₂×D₄	1	2	yes	—
50	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₅₉	C₂×D₄	1	2	yes	—
52	C₂×C₄×C₅₉	C₂×Q₈	1	2	yes	—
53	C₂×C₄×C₅₉	#41	1	2	yes	—
54	C₂×C₄×C₅₉	C₂⁴	1	1	yes	characteristic
55	C₂×Dic₅₉	—	2	4	no	—
56	C₂×Dic₅₉	—	2	2	no	—
57	C₂×Dic₅₉	—	2	2	no	—
58	C₂×Dic₅₉	—	2	4	no	—
59	C₂×Dic₅₉	—	4	4	no	—
60	C₂×Dic₅₉	—	4	2	no	—
61	Dic₁₁₈	—	2	4	no	—
62	Dic₁₁₈	—	2	2	no	—
63	Dic₁₁₈	—	4	8	no	—
64	Dic₁₁₈	C₂×Q₈	1	4	yes	complemented
65	C₈×C₅₉	—	2	4	no	—
66	C₂²×C₅₉	C₂²⋊Q₈	1	2	yes	—
67	C₂²×C₅₉	C₂×Q₁₆	1	2	yes	—
68	C₂²×C₅₉	AGL(1,ℤ/8)	1	2	yes	—
69	C₂²×C₅₉	C₂²×D₄	1	1	yes	characteristic
70	C₄×C₅₉	—	2	2	no	—
71	C₄×C₅₉	C₂²⋊Q₈	1	2	yes	—
72	C₄×C₅₉	C₂×Q₁₆	1	4	yes	—
73	C₄×C₅₉	C₂²×Q₈	1	1	yes	characteristic
74	C₄×C₅₉	C₂×#41	1	1	yes	characteristic
75	Dic₅₉	—	2	4	no	—
76	Dic₅₉	—	4	4	no	—
77	Dic₅₉	—	4	2	no	—
78	C₂×#477	—	59	1	no	maximal, Sylow
79	C₂×C₅₉	#477	1	4	yes	complemented
80	C₂×C₅₉	C₂×C₂²⋊Q₈	1	1	yes	characteristic
81	C₂×C₅₉	C₂²×Q₁₆	1	1	yes	characteristic
82	C₂×C₅₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
83	C₂×C₄⋊Q₈	—	59	1	no	—
84	C₂×C₄×Q₈	—	59	1	no	—
85	#477	—	59	8	no	—
86	C₂×#107	—	59	2	no	—
87	C₂×C₄⋊C₈	D₅₉	1	1	yes	characteristic
88	C₂×#103	—	59	2	no	—
89	C₅₉	C₂×#477	1	1	yes	characteristic, complemented, Sylow
90	C₂²×Q₈	—	59	1	no	—
91	C₂²×Q₈	—	118	1	no	—
92	C₂²×C₈	C₂×D₅₉	1	2	yes	—
93	C₄⋊Q₈	—	59	4	no	—
94	C₄⋊Q₈	—	118	2	no	—
95	C₄×Q₈	—	59	4	no	—
96	C₄×Q₈	—	118	2	no	—
97	C₂×C₄⋊C₄	—	59	1	no	—
98	C₂×C₄⋊C₄	—	59	2	no	—
99	C₂×C₄⋊C₄	—	118	1	no	—
100	C₂×C₄⋊C₄	—	118	1	no	—
101	C₂×C₄²	—	118	1	no	—
102	C₂×C₄²	C₂×D₅₉	1	1	yes	characteristic
103	#107	—	59	8	no	—
104	C₄⋊C₈	C₂×D₅₉	1	4	yes	—
105	#103	—	59	8	no	—
106	C₂×Q₈	—	59	4	no	—
107	C₂×Q₈	—	59	2	no	—
108	C₂×Q₈	—	118	1	no	—
109	C₂×Q₈	—	118	4	no	—
110	C₂×Q₈	—	118	2	no	—
111	C₂×Q₈	—	236	4	no	—
112	C₂²×C₄	—	118	1	no	—
113	C₂²×C₄	—	118	1	no	—
114	C₂²×C₄	—	118	2	no	—
115	C₂²×C₄	D₂₃₆	1	1	yes	characteristic
116	C₂²×C₄	D₂₃₆	1	1	yes	characteristic
117	C₂²×C₄	C₂²×D₅₉	1	1	yes	characteristic
118	C₂×C₈	—	2	4	no	—
119	C₂×C₈	C₂²×D₅₉	1	4	yes	—
120	C₄⋊C₄	—	59	4	no	—
121	C₄⋊C₄	—	59	2	no	—
122	C₄⋊C₄	—	118	1	no	—
123	C₄⋊C₄	—	118	2	no	—
124	C₄⋊C₄	—	118	4	no	—
125	C₄⋊C₄	—	118	4	no	—
126	C₄²	—	118	4	no	—
127	C₄²	D₂₃₆	1	2	yes	—
128	C₄²	C₂²×D₅₉	1	2	yes	—
129	C₂³	C₂×D₂₃₆	1	1	yes	characteristic
130	Q₈	—	59	4	no	—
131	Q₈	—	118	4	no	—
132	Q₈	—	118	2	no	—
133	Q₈	—	236	8	no	—
134	C₂×C₄	—	2	2	no	—
135	C₂×C₄	—	118	4	no	—
136	C₂×C₄	—	118	2	no	—
137	C₂×C₄	—	118	2	no	—
138	C₂×C₄	—	118	4	no	—
139	C₂×C₄	—	236	2	no	—
140	C₂×C₄	—	236	4	no	—
141	C₂×C₄	Dic₂₃₆	1	4	yes	—
142	C₂×C₄	C₂×D₂₃₆	1	2	yes	—
143	C₂×C₄	C₂×D₂₃₆	1	1	yes	characteristic
144	C₂×C₄	C₂×D₂₃₆	1	2	yes	—
145	C₂×C₄	#11745687	1	2	yes	—
146	C₂×C₄	Q₈×D₅₉	1	2	yes	—
147	C₂×C₄	C₂³×D₅₉	1	1	yes	characteristic
148	C₈	—	2	4	no	—
149	C₂²	#14273707	1	2	yes	—
150	C₂²	C₂×Dic₂₃₆	1	2	yes	—
151	C₂²	#14273755	1	2	yes	—
152	C₂²	C₂²×D₂₃₆	1	1	yes	characteristic
153	C₄	—	2	2	no	—
154	C₄	—	118	4	no	—
155	C₄	—	236	4	no	—
156	C₄	—	236	2	no	—
157	C₄	#14273707	1	2	yes	—
158	C₄	C₂×Dic₂₃₆	1	4	yes	—
159	C₄	C₂×Q₈×D₅₉	1	1	yes	characteristic
160	C₄	C₂×#11745687	1	1	yes	characteristic
161	C₂	#16837698	1	4	yes	direct product factor
162	C₂	C₂×#14273707	1	1	yes	characteristic
163	C₂	C₂²×Dic₂₃₆	1	1	yes	characteristic
164	C₂	C₂×#14273755	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14273727

C₂

yes

maximal, characteristic

C₂×C₄×Dic₁₁₈

C₂

yes

maximal, characteristic

#16837698

C₂

yes

maximal, direct product factor

C₂×#14273661

C₂

yes

maximal

C₂×#14273652

C₂

yes

maximal

C₂×C₄⋊C₈×C₅₉

C₂

yes

maximal, characteristic

C₂²×Dic₁₁₈

—

C₂²×Dic₁₁₈

C₂²

yes

characteristic

#14273727

—

#14273727

C₂²

yes

—

C₄×Dic₁₁₈

—

C₄×Dic₁₁₈

C₂²

yes

—

C₂×C₄⋊Dic₅₉

—

C₂×C₄⋊Dic₅₉

C₂²

yes

—

C₂×C₄⋊Dic₅₉

C₂²

yes

characteristic

C₂×Dic₅₉⋊C₄

—

C₂×C₄×Dic₅₉

—

#14273661

C₂²

yes

—

#14273652

C₂²

yes

—

C₂²×C₈×C₅₉

C₂²

yes

—

C₂×C₄²×C₅₉

C₂²

yes

characteristic

C₄⋊C₈×C₅₉

C₂²

yes

—

C₂²×C₄×C₅₉

D₄

yes

characteristic

C₂²×C₄×C₅₉

D₄

yes

characteristic

C₂²×C₄×C₅₉

C₂³

yes

characteristic

C₂²×Dic₅₉

—

C₂²×Dic₅₉

—

C₂²×Dic₅₉

—

C₂×Dic₁₁₈

—

C₂×Dic₁₁₈

—

C₂×Dic₁₁₈

—

C₂×Dic₁₁₈

—

C₂×Dic₁₁₈

Q₈

yes

complemented

C₂×Dic₁₁₈

C₂³

yes

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

C₂³

yes

—

C₄²×C₅₉

D₄

yes

—

C₄²×C₅₉

C₂³

yes

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

C₂³

yes

—

C₄⋊Dic₅₉

C₂³

yes

—

Dic₅₉⋊C₄

—

C₄×Dic₅₉

—

C₂³×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

Q₁₆

yes

—

C₂×C₄×C₅₉

C₂×D₄

yes

—

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

C₂×D₄

yes

—

C₂×C₄×C₅₉

C₂×Q₈

yes

—

C₂×C₄×C₅₉

#41

yes

—

C₂×C₄×C₅₉

C₂⁴

yes

characteristic

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

Dic₁₁₈

—

Dic₁₁₈

—

Dic₁₁₈

—

Dic₁₁₈

C₂×Q₈

yes

complemented

C₈×C₅₉

—

C₂²×C₅₉

C₂²⋊Q₈

yes

—

C₂²×C₅₉

C₂×Q₁₆

yes

—

C₂²×C₅₉

AGL(1,ℤ/8)

yes

—

C₂²×C₅₉

C₂²×D₄

yes

characteristic

C₄×C₅₉

—

C₄×C₅₉

C₂²⋊Q₈

yes

—

C₄×C₅₉

C₂×Q₁₆

yes

—

C₄×C₅₉

C₂²×Q₈

yes

characteristic

C₄×C₅₉

C₂×#41

yes

characteristic

Dic₅₉

—

Dic₅₉

—

Dic₅₉

—

—

C₂×C₅₉

yes

C₂×C₅₉

C₂×C₂²⋊Q₈

yes

characteristic

C₂×C₅₉

C₂²×Q₁₆

yes

characteristic

C₂×C₅₉

C₂×AGL(1,ℤ/8)

yes

characteristic

C₂×C₄⋊Q₈

—

C₂×C₄×Q₈

—

#477

—

C₂×#107

—

C₂×C₄⋊C₈

D₅₉

yes

characteristic

C₂×#103

—

C₅₉

C₂×#477

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₂²×Q₈

—

118

—

C₂²×C₈

C₂×D₅₉

yes

—

C₄⋊Q₈

—

C₄⋊Q₈

—

118

—

C₄×Q₈

—

C₄×Q₈

—

118

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

118

—

100

C₂×C₄⋊C₄

—

118

—

101

C₂×C₄²

—

118

—

102

C₂×C₄²

C₂×D₅₉

yes

characteristic

103

#107

—

104

C₄⋊C₈

C₂×D₅₉

yes

—

105

#103

—

106

C₂×Q₈

—

107

C₂×Q₈

—

108

C₂×Q₈

—

118

—

109

C₂×Q₈

—

118

—

110

C₂×Q₈

—

118

—

111

C₂×Q₈

—

236

—

112

C₂²×C₄

—

118

—

113

C₂²×C₄

—

118

—

114

C₂²×C₄

—

118

—

115

C₂²×C₄

D₂₃₆

yes

characteristic

116

C₂²×C₄

D₂₃₆

yes

characteristic

117

C₂²×C₄

C₂²×D₅₉

yes

characteristic

118

C₂×C₈

—

119

C₂×C₈

C₂²×D₅₉

yes

—

120

C₄⋊C₄

—

121

C₄⋊C₄

—

122

C₄⋊C₄

—

118

—

123

C₄⋊C₄

—

118

—

124

C₄⋊C₄

—

118

—

125

C₄⋊C₄

—

118

—

126

C₄²

—

118

—

127

C₄²

D₂₃₆

yes

—

128

C₄²

C₂²×D₅₉

yes

—

129

C₂³

C₂×D₂₃₆

yes

characteristic

130

Q₈

—

131

Q₈

—

118

—

132

Q₈

—

118

—

133

Q₈

—

236

—

134

C₂×C₄

—

135

C₂×C₄

—

118

—

136

C₂×C₄

—

118

—

137

C₂×C₄

—

118

—

138

C₂×C₄

—

118

—

139

C₂×C₄

—

236

—

140

C₂×C₄

—

236

—

141

C₂×C₄

Dic₂₃₆

yes

—

142

C₂×C₄

C₂×D₂₃₆

yes

—

143

C₂×C₄

C₂×D₂₃₆

yes

characteristic

144

C₂×C₄

C₂×D₂₃₆

yes

—

145

C₂×C₄

#11745687

yes

—

146

C₂×C₄

Q₈×D₅₉

yes

—

147

C₂×C₄

C₂³×D₅₉

yes

characteristic

148

C₈

—

149

C₂²

#14273707

yes

—

150

C₂²

C₂×Dic₂₃₆

yes

—

151

C₂²

#14273755

yes

—

152

C₂²

C₂²×D₂₃₆

yes

characteristic

153

C₄

—

154

C₄

—

118

—

155

C₄

—

236

—

156

C₄

—

236

—

157

C₄

#14273707

yes

—

158

C₄

C₂×Dic₂₃₆

yes

—

159

C₄

C₂×Q₈×D₅₉

yes

characteristic

160

C₄

C₂×#11745687

yes

characteristic

161

C₂

#16837698

yes

direct product factor

162

C₂

C₂×#14273707

yes

characteristic

163

C₂

C₂²×Dic₂₃₆

yes

characteristic

164

C₂

C₂×#14273755

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×D₂₃₆
commutator	C₂×C₄×C₅₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₅₉
Fitting	C₂×C₄⋊C₈×C₅₉	C₂
socle	C₂³×C₅₉	C₂×D₄
hypercenter	C₂×C₄⋊C₈	D₅₉
hypocenter	C₅₉	C₂×#477
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

subgroup

G/H

center

C₂³

C₂×D₂₃₆

commutator

C₂×C₄×C₅₉

C₂⁴

Frattini

C₂×C₄

C₂³×D₅₉

Fitting

C₂×C₄⋊C₈×C₅₉

C₂

socle

C₂³×C₅₉

C₂×D₄

hypercenter

C₂×C₄⋊C₈

D₅₉

hypocenter

C₅₉

C₂×#477

LCS

G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

N	H ≅ G/N	type
3. #16837698	C₂	direct (N × H)
33. C₂×Dic₁₁₈	Q₈	semidirect
64. Dic₁₁₈	C₂×Q₈	semidirect
79. C₂×C₅₉	#477	semidirect
89. C₅₉	C₂×#477	semidirect
161. C₂	#16837698	direct (N × H)

H ≅ G/N

type

3. #16837698

C₂

direct (N × H)

33. C₂×Dic₁₁₈

Q₈

semidirect

64. Dic₁₁₈

C₂×Q₈

semidirect

79. C₂×C₅₉

#477

semidirect

89. C₅₉

C₂×#477

semidirect

161. C₂

#16837698

direct (N × H)

Subgroups — C₂×#16837698 #21433868

Conjugacy classes of subgroups

Complements

Subgroups — C2×#16837698 #21433868

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#16837698 #21433868