Group #16835643 — subgroups

Subgroups — #16835643

Order 3712.

300 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 300 conjugacy classes of subgroups — too many to draw. Only its 70 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×#11745212	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#11745226	C₂	1	2	yes	maximal, complemented
3	#14272138	C₂	1	2	yes	maximal, complemented
4	C₂×C₈×#1181	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#11745281	C₂	1	1	yes	maximal, characteristic
6	C₂×#90809	C₂²	1	2	yes	—
7	C₂×C₄×#1181	C₂²	1	1	yes	characteristic
8	C₂×#90796	C₂²	1	2	yes	—
9	C₂×D₂₉⋊C₈	C₂²	1	1	yes	characteristic
10	C₂×Q₁₆×C₂₉	C₄	1	1	yes	characteristic, complemented
11	C₂×#90784	C₄	1	2	yes	complemented
12	#11745281	—	2	2	no	—
13	#11745281	—	2	2	no	—
14	C₂×D₂₃₂	C₄	1	1	yes	characteristic, complemented
15	C₂×C₈×D₂₉	C₂²	1	1	yes	characteristic
16	#11745226	—	2	4	no	—
17	#11745219	—	2	4	no	—
18	#11745212	—	2	2	no	—
19	C₈×#1181	—	2	2	no	—
20	C₂²×#1181	—	2	1	no	—
21	C₂×Q₈×C₂₉	C₂×C₄	1	2	yes	—
22	#90809	—	2	2	no	—
23	#90809	—	2	2	no	—
24	#90809	—	2	2	no	—
25	#90809	—	2	2	no	—
26	C₂×D₁₁₆	—	2	2	no	—
27	C₂×D₁₁₆	C₂×C₄	1	2	yes	—
28	C₂×C₄×D₂₉	—	2	2	no	—
29	C₂×C₄×D₂₉	C₂³	1	1	yes	characteristic
30	C₂×#6599	—	2	2	no	—
31	C₂×#6599	D₄	1	2	yes	—
32	C₄×#1181	—	2	1	no	—
33	C₄×#1181	D₄	1	2	yes	—
34	#90796	—	2	2	no	—
35	#90796	—	2	2	no	—
36	D₂₉⋊C₈	—	2	1	no	—
37	D₂₉⋊C₈	—	2	1	no	—
38	Q₁₆×C₂₉	—	2	2	no	—
39	C₂×C₈×C₂₉	C₂×C₄	1	1	yes	characteristic
40	#90784	—	2	4	no	—
41	C₂×#6597	C₂×C₄	1	1	yes	characteristic
42	D₂₃₂	—	2	2	no	—
43	C₈×D₂₉	D₄	1	2	yes	—
44	C₈×D₂₉	D₄	1	2	yes	—
45	C₂²×D₂₉	—	2	2	no	—
46	C₂²×D₂₉	#41	1	1	yes	characteristic
47	C₂×#1181	—	2	4	no	—
48	C₂×#1181	—	2	2	no	—
49	Q₈×C₂₉	—	2	2	no	—
50	Q₈×C₂₉	—	2	2	no	—
51	C₂×C₄×C₂₉	—	2	2	no	—
52	C₂×C₄×C₂₉	C₂²×C₄	1	1	yes	characteristic
53	C₂×Dic₂₉	C₂×D₄	1	1	yes	characteristic
54	D₁₁₆	—	2	2	no	—
55	D₁₁₆	—	2	2	no	—
56	D₁₁₆	—	4	2	no	—
57	D₁₁₆	—	4	2	no	—
58	C₄×D₂₉	—	4	2	no	—
59	C₄×D₂₉	—	4	2	no	—
60	C₄×D₂₉	C₂×D₄	1	1	yes	characteristic
61	C₄×D₂₉	C₂×D₄	1	1	yes	characteristic
62	C₄×D₂₉	C₂×D₄	1	1	yes	characteristic
63	C₄×D₂₉	#41	1	1	yes	characteristic
64	#6599	—	2	2	no	—
65	#6599	—	4	2	no	—
66	C₈×C₂₉	C₂²⋊C₄	1	2	yes	—
67	#6597	C₂²⋊C₄	1	2	yes	—
68	#1974	—	29	1	no	maximal, Sylow
69	C₂²×C₂₉	C₄×D₄	1	1	yes	characteristic
70	C₂×D₂₉	—	2	2	no	—
71	C₂×D₂₉	—	4	2	no	—
72	C₂×D₂₉	—	4	4	no	—
73	C₂×D₂₉	#121	1	1	yes	characteristic
74	C₂×D₂₉	#124	1	1	yes	characteristic
75	#1181	—	2	4	no	—
76	C₄×C₂₉	—	4	2	no	—
77	C₄×C₂₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
78	C₄×C₂₉	C₄×D₄	1	1	yes	characteristic
79	Dic₂₉	#121	1	1	yes	characteristic
80	Dic₂₉	#127	1	1	yes	characteristic
81	C₂×#135	—	29	1	no	—
82	C₂×#108	—	29	1	no	—
83	C₂×#104	—	29	2	no	—
84	#408	—	29	2	no	—
85	C₂×C₄×C₈	—	29	1	no	—
86	C₂×C₂₉	#390	1	1	yes	characteristic
87	C₂×C₂₉	#443	1	2	yes	—
88	D₂₉	—	2	1	no	—
89	D₂₉	—	2	1	no	—
90	D₂₉	—	8	2	no	—
91	C₂×#41	—	29	2	no	—
92	#135	—	58	2	no	—
93	#135	—	58	2	no	—
94	C₂×Q₁₆	#1181	1	1	yes	characteristic, complemented
95	C₂×SD₁₆	—	29	2	no	—
96	C₂×D₈	—	29	1	no	—
97	C₂×M₁₆	—	29	2	no	—
98	C₂²×C₈	—	29	1	no	—
99	C₂²×C₈	—	29	1	no	—
100	C₂×C₄²	—	29	1	no	—
101	#108	—	58	2	no	—
102	#104	—	58	4	no	—
103	C₂²⋊C₈	—	58	4	no	—
104	C₄×C₈	—	58	2	no	—
105	C₂₉	#1974	1	1	yes	characteristic, complemented, Sylow
106	#41	—	58	2	no	—
107	#41	—	58	2	no	—
108	#41	—	58	2	no	—
109	#41	—	58	2	no	—
110	C₂×Q₈	C₂×#1181	1	2	yes	—
111	C₂×D₄	—	29	2	no	—
112	C₂×D₄	—	58	2	no	—
113	C₂²×C₄	—	29	1	no	—
114	C₂²×C₄	—	58	2	no	—
115	C₂²×C₄	—	58	1	no	—
116	Q₁₆	—	2	2	no	—
117	SD₁₆	—	58	4	no	—
118	D₈	—	58	2	no	—
119	M₁₆	—	58	2	no	—
120	M₁₆	—	58	2	no	—
121	C₂×C₈	—	29	1	no	—
122	C₂×C₈	—	29	2	no	—
123	C₂×C₈	—	29	2	no	—
124	C₂×C₈	—	29	2	no	—
125	C₂×C₈	—	58	1	no	—
126	C₂×C₈	—	58	1	no	—
127	C₂×C₈	—	58	2	no	—
128	C₂×C₈	C₂×#1181	1	1	yes	characteristic
129	C₄²	—	29	2	no	—
130	C₄²	—	58	1	no	—
131	C₂³	—	29	1	no	—
132	C₂³	—	58	2	no	—
133	Q₈	—	2	2	no	—
134	Q₈	—	2	2	no	—
135	D₄	—	58	2	no	—
136	D₄	—	58	2	no	—
137	D₄	—	116	2	no	—
138	D₄	—	116	2	no	—
139	C₂×C₄	—	2	2	no	—
140	C₂×C₄	—	29	1	no	—
141	C₂×C₄	—	29	1	no	—
142	C₂×C₄	—	29	1	no	—
143	C₂×C₄	—	29	1	no	—
144	C₂×C₄	—	29	1	no	—
145	C₂×C₄	—	58	2	no	—
146	C₂×C₄	—	58	4	no	—
147	C₂×C₄	—	116	2	no	—
148	C₂×C₄	—	116	2	no	—
149	C₂×C₄	C₂²×#1181	1	1	yes	characteristic
150	C₈	—	29	2	no	—
151	C₈	—	58	2	no	—
152	C₈	—	116	2	no	—
153	C₈	C₂²⋊#1181	1	2	yes	—
154	C₂²	—	29	1	no	—
155	C₂²	—	29	1	no	—
156	C₂²	—	58	2	no	—
157	C₂²	—	116	2	no	—
158	C₂²	—	116	4	no	—
159	C₂²	D₄×#1181	1	1	yes	characteristic
160	C₄	—	4	2	no	—
161	C₄	—	29	1	no	—
162	C₄	—	29	1	no	—
163	C₄	—	58	4	no	—
164	C₄	D₄×#1181	1	1	yes	characteristic
165	C₄	C₂×C₂²⋊#1181	1	1	yes	characteristic
166	C₂	—	58	1	no	—
167	C₂	—	58	1	no	—
168	C₂	—	232	2	no	—
169	C₂	#14272115	1	1	yes	characteristic
170	C₂	#14272207	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#11745212

C₂

yes

maximal, characteristic, complemented

C₂×#11745226

C₂

yes

maximal, complemented

#14272138

C₂

yes

maximal, complemented

C₂×C₈×#1181

C₂

yes

maximal, characteristic, complemented

C₂×#11745281

C₂

yes

maximal, characteristic

C₂×#90809

C₂²

yes

—

C₂×C₄×#1181

C₂²

yes

characteristic

C₂×#90796

C₂²

yes

—

C₂×D₂₉⋊C₈

C₂²

yes

characteristic

C₂×Q₁₆×C₂₉

C₄

yes

characteristic, complemented

C₂×#90784

C₄

yes

complemented

#11745281

—

#11745281

—

C₂×D₂₃₂

C₄

yes

characteristic, complemented

C₂×C₈×D₂₉

C₂²

yes

—

—

—

—

C₂²×#1181

—

C₂×Q₈×C₂₉

C₂×C₄

yes

—

—

—

—

—

C₂×D₁₁₆

—

C₂×D₁₁₆

C₂×C₄

yes

—

C₂×C₄×D₂₉

—

C₂×C₄×D₂₉

C₂³

yes

characteristic

C₂×#6599

—

C₂×#6599

D₄

yes

—

C₄×#1181

—

C₄×#1181

D₄

yes

—

#90796

—

#90796

—

D₂₉⋊C₈

—

D₂₉⋊C₈

—

Q₁₆×C₂₉

—

C₂×C₈×C₂₉

C₂×C₄

yes

characteristic

#90784

—

C₂×#6597

C₂×C₄

yes

characteristic

D₂₃₂

—

C₈×D₂₉

D₄

yes

—

C₈×D₂₉

D₄

yes

—

C₂²×D₂₉

—

C₂²×D₂₉

yes

—

—

Q₈×C₂₉

—

Q₈×C₂₉

—

C₂×C₄×C₂₉

—

C₂×C₄×C₂₉

C₂²×C₄

yes

characteristic

C₂×Dic₂₉

C₂×D₄

yes

characteristic

D₁₁₆

—

D₁₁₆

—

D₁₁₆

—

D₁₁₆

—

C₄×D₂₉

—

C₄×D₂₉

—

C₄×D₂₉

C₂×D₄

yes

characteristic

C₄×D₂₉

C₂×D₄

yes

characteristic

C₄×D₂₉

C₂×D₄

yes

C₄×D₂₉

yes

—

—

C₈×C₂₉

C₂²⋊C₄

yes

—

#6597

C₂²⋊C₄

yes

—

#1974

—

maximal, Sylow

C₂²×C₂₉

C₄×D₄

yes

characteristic

C₂×D₂₉

—

C₂×D₂₉

—

C₂×D₂₉

—

C₂×D₂₉

yes

C₂×D₂₉

yes

—

C₄×C₂₉

—

C₄×C₂₉

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₂₉

C₄×D₄

yes

Dic₂₉

yes

Dic₂₉

yes

—

—

—

—

C₂×C₄×C₈

—

C₂×C₂₉

#390

yes

characteristic

C₂×C₂₉

#443

yes

—

D₂₉

—

D₂₉

—

D₂₉

—

C₂×#41

—

#135

—

#135

—

C₂×Q₁₆

#1181

yes

characteristic, complemented

C₂×SD₁₆

—

C₂×D₈

—

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

—

100

C₂×C₄²

—

101

#108

—

102

#104

—

103

C₂²⋊C₈

—

104

C₄×C₈

—

105

C₂₉

#1974

yes

characteristic, complemented, Sylow

106

#41

—

107

#41

—

108

#41

—

109

#41

—

110

C₂×Q₈

C₂×#1181

yes

—

111

C₂×D₄

—

112

C₂×D₄

—

113

C₂²×C₄

—

114

C₂²×C₄

—

115

C₂²×C₄

—

116

Q₁₆

—

117

SD₁₆

—

118

D₈

—

119

M₁₆

—

120

M₁₆

—

121

C₂×C₈

—

122

C₂×C₈

—

123

C₂×C₈

—

124

C₂×C₈

—

125

C₂×C₈

—

126

C₂×C₈

—

127

C₂×C₈

—

128

C₂×C₈

C₂×#1181

yes

characteristic

129

C₄²

—

130

C₄²

—

131

C₂³

—

132

C₂³

—

133

Q₈

—

134

Q₈

—

135

D₄

—

136

D₄

—

137

D₄

—

116

—

138

D₄

—

116

—

139

C₂×C₄

—

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

—

116

—

148

C₂×C₄

—

116

—

149

C₂×C₄

C₂²×#1181

yes

characteristic

150

C₈

—

151

C₈

—

152

C₈

—

116

—

153

C₈

C₂²⋊#1181

yes

—

154

C₂²

—

155

C₂²

—

156

C₂²

—

157

C₂²

—

116

—

158

C₂²

—

116

—

159

C₂²

D₄×#1181

yes

characteristic

160

C₄

—

161

C₄

—

162

C₄

—

163

C₄

—

164

C₄

D₄×#1181

yes

characteristic

165

C₄

C₂×C₂²⋊#1181

yes

characteristic

166

C₂

—

167

C₂

—

168

C₂

—

232

—

169

C₂

#14272115

yes

characteristic

170

C₂

#14272207

yes

—

	subgroup	G/H
center	C₂²	D₄×#1181
commutator	C₂×C₄×C₂₉	C₂²×C₄
Frattini	C₂×C₄	C₂²×#1181
Fitting	C₂×Q₁₆×C₂₉	C₄
socle	C₂²×C₂₉	C₄×D₄
prime radical	C₂×#11745281	C₂
cosocle	C₂×C₄×D₂₉	C₂³
hypercenter	C₂×Q₁₆	#1181
hypocenter	C₂₉	#1974
LCS	G ▷ C₂×C₄×C₂₉ ▷ C₂×C₂₉ ▷ C₂₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×Q₁₆

subgroup

G/H

center

C₂²

D₄×#1181

commutator

C₂×C₄×C₂₉

C₂²×C₄

Frattini

C₂×C₄

C₂²×#1181

Fitting

C₂×Q₁₆×C₂₉

C₄

socle

C₂²×C₂₉

C₄×D₄

prime radical

C₂×#11745281

C₂

cosocle

C₂×C₄×D₂₉

C₂³

C₂×Q₁₆

C₂₉

G ▷ C₂×C₄×C₂₉ ▷ C₂×C₂₉ ▷ C₂₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×Q₁₆

N	H ≅ G/N	type
1. C₂×#11745212	C₂	semidirect
2. C₂×#11745226	C₂	semidirect
3. #14272138	C₂	semidirect
4. C₂×C₈×#1181	C₂	semidirect
10. C₂×Q₁₆×C₂₉	C₄	semidirect
11. C₂×#90784	C₄	semidirect
14. C₂×D₂₃₂	C₄	semidirect
94. C₂×Q₁₆	#1181	semidirect
105. C₂₉	#1974	semidirect

H ≅ G/N

type

1. C₂×#11745212

C₂

semidirect

2. C₂×#11745226

C₂

semidirect

3. #14272138

C₂

semidirect

4. C₂×C₈×#1181

C₂

semidirect

10. C₂×Q₁₆×C₂₉

C₄

semidirect

11. C₂×#90784

C₄

semidirect

14. C₂×D₂₃₂

C₄

semidirect

94. C₂×Q₁₆

#1181

semidirect

105. C₂₉

#1974

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×#11745212	2
C₂×#11745226	2
#14272138	2
C₂×C₈×#1181	2
C₂×#11745281	2
#1974	29

subgroup

index

C₂×C₈×#1181

Subgroups — #16835643

Conjugacy classes of subgroups

Complements

Maximal subgroups