Group #16820045 — subgroups

Subgroups — #16820045

Order 3712.

308 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 308 conjugacy classes of subgroups — too many to draw. Only its 93 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	#14271930	C₂	1	1	yes	maximal, characteristic
2	#14271147	C₂	1	1	yes	maximal, characteristic
3	C₂×#11745184	C₂	1	2	yes	maximal, complemented
4	#14270933	C₂	1	2	yes	maximal, complemented
5	C₂₉×#536	C₂	1	1	yes	maximal, characteristic
6	C₂×#41×C₂₉	C₂²	1	1	yes	characteristic
7	C₂²×Q₈×C₂₉	C₂²	1	1	yes	characteristic
8	C₂₉×C₄⋊Q₈	—	2	1	no	—
9	C₂₉×#124	—	2	1	no	—
10	C₂₉×#123	C₄	1	4	yes	—
11	C₂₉×C₂²⋊Q₈	—	2	2	no	—
12	C₂₉×#117	C₂²	1	1	yes	characteristic
13	#11745310	—	2	4	no	—
14	C₂×#90777	—	2	2	no	—
15	C₂×#90777	C₂²	1	2	yes	—
16	C₂×#90777	C₂²	1	2	yes	—
17	#11745184	—	2	1	no	—
18	#11745184	—	2	4	no	—
19	#11745181	—	2	1	no	—
20	#41×C₂₉	—	2	2	no	—
21	#41×C₂₉	—	2	2	no	—
22	C₂×Q₈×C₂₉	—	2	1	no	—
23	C₂×Q₈×C₂₉	—	2	1	no	—
24	C₂×Q₈×C₂₉	—	2	2	no	—
25	C₂×Q₈×C₂₉	—	2	2	no	—
26	C₂×Q₈×C₂₉	D₄	1	1	yes	characteristic
27	C₂×D₄×C₂₉	C₂×C₄	1	2	yes	—
28	C₂×D₄×C₂₉	D₄	1	1	yes	characteristic
29	C₂²×C₄×C₂₉	C₂×C₄	1	2	yes	—
30	C₂²×C₄×C₂₉	D₄	1	2	yes	—
31	C₂²×C₄×C₂₉	C₂³	1	1	yes	characteristic
32	C₄⋊C₄×C₂₉	—	2	4	no	—
33	C₄⋊C₄×C₂₉	—	2	1	no	—
34	C₂²⋊C₄×C₂₉	—	2	4	no	—
35	C₂²⋊C₄×C₂₉	C₂×C₄	1	2	yes	—
36	C₄²×C₂₉	—	2	1	no	—
37	#90777	—	2	4	no	—
38	#90777	—	2	2	no	—
39	#90777	—	2	2	no	—
40	#90777	—	2	2	no	—
41	#90777	D₄	1	4	yes	—
42	C₂×#6597	—	2	4	no	—
43	C₂×#6597	—	2	2	no	—
44	C₂×#6597	—	2	2	no	—
45	C₂³×C₂₉	C₂²⋊C₄	1	2	yes	—
46	C₂³×C₂₉	C₂²×C₄	1	1	yes	characteristic
47	Q₈×C₂₉	—	2	2	no	—
48	Q₈×C₂₉	—	4	1	no	—
49	Q₈×C₂₉	—	4	2	no	—
50	Q₈×C₂₉	—	4	1	no	—
51	D₄×C₂₉	—	2	4	no	—
52	D₄×C₂₉	—	2	2	no	—
53	C₂×C₄×C₂₉	—	2	2	no	—
54	C₂×C₄×C₂₉	—	2	2	no	—
55	C₂×C₄×C₂₉	—	2	2	no	—
56	C₂×C₄×C₂₉	—	2	2	no	—
57	C₂×C₄×C₂₉	—	2	4	no	—
58	C₂×C₄×C₂₉	C₂²⋊C₄	1	2	yes	—
59	C₂×C₄×C₂₉	C₂×D₄	1	2	yes	—
60	C₂×C₄×C₂₉	C₂×D₄	1	1	yes	characteristic
61	C₂×C₄×C₂₉	C₂×D₄	1	1	yes	characteristic
62	C₂×C₄×C₂₉	#41	1	2	yes	—
63	#6597	—	2	4	no	—
64	#6597	—	4	2	no	—
65	#1915	—	29	1	no	maximal, Sylow
66	C₂²×C₂₉	—	2	2	no	—
67	C₂²×C₂₉	—	4	1	no	—
68	C₂²×C₂₉	—	4	2	no	—
69	C₂²×C₂₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
70	C₂²×C₂₉	C₄×D₄	1	2	yes	—
71	C₄×C₂₉	—	2	2	no	—
72	C₄×C₂₉	—	4	2	no	—
73	C₄×C₂₉	—	4	2	no	—
74	C₄×C₂₉	#120	1	1	yes	characteristic
75	C₄×C₂₉	#121	1	2	yes	—
76	C₄×C₂₉	#123	1	1	yes	characteristic
77	#568	—	29	1	no	—
78	#536	D₂₉	1	1	yes	characteristic
79	#413	—	29	1	no	—
80	C₂×#101	—	29	2	no	—
81	#344	—	29	2	no	—
82	C₂×C₂₉	—	2	1	no	—
83	C₂×C₂₉	—	2	2	no	—
84	C₂×C₂₉	—	4	2	no	—
85	C₂×C₂₉	#386	1	1	yes	characteristic
86	C₂×#41	C₂×D₂₉	1	1	yes	characteristic
87	C₂²×Q₈	C₂×D₂₉	1	1	yes	characteristic
88	#131	—	58	4	no	—
89	C₂×M₁₆	—	29	2	no	—
90	C₂×M₁₆	—	29	2	no	—
91	C₂×M₁₆	—	58	2	no	—
92	C₄⋊Q₈	—	2	1	no	—
93	#124	—	2	1	no	—
94	#123	Dic₂₉	1	4	yes	—
95	C₂²⋊Q₈	—	2	2	no	—
96	#117	C₂×D₂₉	1	1	yes	characteristic
97	#101	—	58	4	no	—
98	#101	—	58	1	no	—
99	#100	—	58	1	no	—
100	C₂₉	#1915	1	1	yes	characteristic, complemented, Sylow
101	#41	—	2	2	no	—
102	#41	—	2	2	no	—
103	C₂×Q₈	—	2	2	no	—
104	C₂×Q₈	—	2	2	no	—
105	C₂×Q₈	—	2	1	no	—
106	C₂×Q₈	—	2	1	no	—
107	C₂×Q₈	C₂²⋊D₂₉	1	1	yes	characteristic
108	C₂×D₄	C₂×Dic₂₉	1	2	yes	—
109	C₂×D₄	C₂²⋊D₂₉	1	1	yes	characteristic
110	C₂²×C₄	C₂×Dic₂₉	1	2	yes	—
111	C₂²×C₄	C₂²⋊D₂₉	1	2	yes	—
112	C₂²×C₄	C₂²×D₂₉	1	1	yes	characteristic
113	M₁₆	—	29	4	no	—
114	M₁₆	—	58	4	no	—
115	M₁₆	—	58	2	no	—
116	M₁₆	—	58	2	no	—
117	M₁₆	—	58	2	no	—
118	C₂×C₈	—	58	4	no	—
119	C₂×C₈	—	58	2	no	—
120	C₂×C₈	—	58	2	no	—
121	C₄⋊C₄	—	2	4	no	—
122	C₄⋊C₄	—	2	1	no	—
123	C₂²⋊C₄	—	2	4	no	—
124	C₂²⋊C₄	C₂×Dic₂₉	1	2	yes	—
125	C₄²	—	2	1	no	—
126	C₂³	C₂²⋊Dic₂₉	1	2	yes	—
127	C₂³	C₂²×Dic₂₉	1	1	yes	characteristic
128	Q₈	—	2	2	no	—
129	Q₈	—	4	1	no	—
130	Q₈	—	4	1	no	—
131	Q₈	—	4	2	no	—
132	D₄	—	2	4	no	—
133	D₄	—	2	2	no	—
134	C₂×C₄	—	2	2	no	—
135	C₂×C₄	—	2	2	no	—
136	C₂×C₄	—	2	4	no	—
137	C₂×C₄	—	2	2	no	—
138	C₂×C₄	—	2	2	no	—
139	C₂×C₄	C₂²⋊Dic₂₉	1	2	yes	—
140	C₂×C₄	D₄×D₂₉	1	2	yes	—
141	C₂×C₄	#90807	1	2	yes	—
142	C₂×C₄	C₂×C₂²⋊D₂₉	1	1	yes	characteristic
143	C₂×C₄	C₂×C₂²⋊D₂₉	1	1	yes	characteristic
144	C₈	—	58	4	no	—
145	C₈	—	116	2	no	—
146	C₂²	—	2	2	no	—
147	C₂²	—	4	1	no	—
148	C₂²	—	4	2	no	—
149	C₂²	D₄×Dic₂₉	1	2	yes	—
150	C₂²	C₂×C₂²⋊Dic₂₉	1	1	yes	characteristic
151	C₄	—	2	2	no	—
152	C₄	—	4	2	no	—
153	C₄	—	4	2	no	—
154	C₄	#11745297	1	1	yes	characteristic
155	C₄	#11745299	1	1	yes	characteristic
156	C₄	#11745301	1	2	yes	—
157	C₂	—	2	1	no	—
158	C₂	—	2	2	no	—
159	C₂	—	4	2	no	—
160	C₂	#14271043	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14271930

C₂

yes

maximal, characteristic

#14271147

C₂

yes

maximal, characteristic

C₂×#11745184

C₂

yes

maximal, complemented

#14270933

C₂

yes

maximal, complemented

C₂₉×#536

C₂

yes

maximal, characteristic

C₂×#41×C₂₉

C₂²

yes

characteristic

C₂²×Q₈×C₂₉

C₂²

yes

characteristic

C₂₉×C₄⋊Q₈

—

C₂₉×#124

—

C₂₉×#123

C₄

yes

—

C₂₉×C₂²⋊Q₈

—

C₂₉×#117

C₂²

yes

—

—

C₂²

yes

—

C₂×#90777

C₂²

yes

—

#11745184

—

#11745184

—

#11745181

—

#41×C₂₉

—

#41×C₂₉

—

C₂×Q₈×C₂₉

—

C₂×Q₈×C₂₉

—

C₂×Q₈×C₂₉

—

C₂×Q₈×C₂₉

—

C₂×Q₈×C₂₉

D₄

yes

characteristic

C₂×D₄×C₂₉

C₂×C₄

yes

—

C₂×D₄×C₂₉

D₄

yes

characteristic

C₂²×C₄×C₂₉

C₂×C₄

yes

—

C₂²×C₄×C₂₉

D₄

yes

—

C₂²×C₄×C₂₉

C₂³

yes

characteristic

C₄⋊C₄×C₂₉

—

C₄⋊C₄×C₂₉

—

C₂²⋊C₄×C₂₉

—

C₂²⋊C₄×C₂₉

C₂×C₄

yes

—

C₄²×C₂₉

—

—

—

—

—

D₄

yes

—

C₂×#6597

—

C₂×#6597

—

C₂×#6597

—

C₂³×C₂₉

C₂²⋊C₄

yes

—

C₂³×C₂₉

C₂²×C₄

yes

characteristic

Q₈×C₂₉

—

Q₈×C₂₉

—

Q₈×C₂₉

—

Q₈×C₂₉

—

D₄×C₂₉

—

D₄×C₂₉

—

C₂×C₄×C₂₉

—

C₂×C₄×C₂₉

—

C₂×C₄×C₂₉

—

C₂×C₄×C₂₉

—

C₂×C₄×C₂₉

—

C₂×C₄×C₂₉

C₂²⋊C₄

yes

—

C₂×C₄×C₂₉

C₂×D₄

yes

—

C₂×C₄×C₂₉

C₂×D₄

yes

characteristic

C₂×C₄×C₂₉

C₂×D₄

yes

characteristic

C₂×C₄×C₂₉

#41

yes

—

—

—

—

C₂²×C₂₉

—

C₂²×C₂₉

—

C₂²×C₂₉

—

C₂²×C₂₉

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₂₉

C₄×D₄

yes

—

C₄×C₂₉

—

C₄×C₂₉

—

C₄×C₂₉

—

C₄×C₂₉

#120

yes

characteristic

C₄×C₂₉

#121

yes

—

C₄×C₂₉

yes

—

D₂₉

yes

—

—

—

C₂×C₂₉

—

C₂×C₂₉

—

C₂×C₂₉

—

C₂×C₂₉

#386

yes

characteristic

C₂×#41

C₂×D₂₉

yes

characteristic

C₂²×Q₈

C₂×D₂₉

yes

characteristic

#131

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

—

C₄⋊Q₈

—

#124

—

#123

Dic₂₉

yes

—

C₂²⋊Q₈

—

#117

C₂×D₂₉

yes

—

—

—

100

C₂₉

#1915

yes

characteristic, complemented, Sylow

101

#41

—

102

#41

—

103

C₂×Q₈

—

104

C₂×Q₈

—

105

C₂×Q₈

—

106

C₂×Q₈

—

107

C₂×Q₈

C₂²⋊D₂₉

yes

characteristic

108

C₂×D₄

C₂×Dic₂₉

yes

—

109

C₂×D₄

C₂²⋊D₂₉

yes

characteristic

110

C₂²×C₄

C₂×Dic₂₉

yes

—

111

C₂²×C₄

C₂²⋊D₂₉

yes

—

112

C₂²×C₄

C₂²×D₂₉

yes

characteristic

113

M₁₆

—

114

M₁₆

—

115

M₁₆

—

116

M₁₆

—

117

M₁₆

—

118

C₂×C₈

—

119

C₂×C₈

—

120

C₂×C₈

—

121

C₄⋊C₄

—

122

C₄⋊C₄

—

123

C₂²⋊C₄

—

124

C₂²⋊C₄

C₂×Dic₂₉

yes

—

125

C₄²

—

126

C₂³

C₂²⋊Dic₂₉

yes

—

127

C₂³

C₂²×Dic₂₉

yes

characteristic

128

Q₈

—

129

Q₈

—

130

Q₈

—

131

Q₈

—

132

D₄

—

133

D₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

C₂²⋊Dic₂₉

yes

—

140

C₂×C₄

D₄×D₂₉

yes

—

141

C₂×C₄

#90807

yes

—

142

C₂×C₄

C₂×C₂²⋊D₂₉

yes

characteristic

143

C₂×C₄

C₂×C₂²⋊D₂₉

yes

characteristic

144

C₈

—

145

C₈

—

116

—

146

C₂²

—

147

C₂²

—

148

C₂²

—

149

C₂²

D₄×Dic₂₉

yes

—

150

C₂²

C₂×C₂²⋊Dic₂₉

yes

characteristic

151

C₄

—

152

C₄

—

153

C₄

—

154

C₄

#11745297

yes

characteristic

155

C₄

#11745299

yes

characteristic

156

C₄

#11745301

yes

—

157

C₂

—

158

C₂

—

159

C₂

—

160

C₂

#14271043

yes

characteristic

	subgroup	G/H
center	C₂	#14271043
commutator	C₂³×C₂₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₉
Fitting	C₂₉×#536	C₂
socle	C₂×C₂₉	#386
prime radical	C₂×#41×C₂₉	C₂²
cosocle	C₂²×C₄×C₂₉	C₂³
hypercenter	#536	D₂₉
hypocenter	C₂₉	#1915
LCS	G ▷ C₂³×C₂₉ ▷ C₂×C₂₉ ▷ C₂₉
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #536

subgroup

G/H

center

C₂

#14271043

commutator

C₂³×C₂₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₂₉

Fitting

C₂₉×#536

C₂

socle

C₂×C₂₉

#386

prime radical

C₂×#41×C₂₉

C₂²

cosocle

C₂²×C₄×C₂₉

C₂³

D₂₉

C₂₉

G ▷ C₂³×C₂₉ ▷ C₂×C₂₉ ▷ C₂₉

UCS

C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #536

N	H ≅ G/N	type
3. C₂×#11745184	C₂	semidirect
4. #14270933	C₂	semidirect
100. C₂₉	#1915	semidirect

H ≅ G/N

type

3. C₂×#11745184

C₂

semidirect

4. #14270933

C₂

semidirect

100. C₂₉

#1915

semidirect

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

subgroup	index
#14271930	2
#14271147	2
C₂×#11745184	2
#14270933	2
C₂₉×#536	2
#1915	29

subgroup

index

C₂₉×#536

Subgroups — #16820045

Conjugacy classes of subgroups

Complements

Maximal subgroups