Group #14577002 — subgroups

Subgroups — C₂×#11927155 #14577002

Order 2432.

1004 conjugacy classes of subgroups; only the 485 normal subgroups are shown.

normal subgroups only

This group has 1004 conjugacy classes of subgroups — too many to draw. Only its 485 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	#11927155	C₂	1	16	yes	maximal, direct product factor
2	C₂×#10597344	C₂	1	4	yes	maximal
3	C₂×#10597316	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#10597335	C₂	1	4	yes	maximal, complemented
5	C₂×#10597392	C₂	1	2	yes	maximal
6	C₂×Q₈×Dic₁₉	C₂	1	2	yes	maximal, complemented
7	C₂×#10597318	C₂	1	1	yes	maximal, characteristic
8	C₂×C₁₉×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×Q₈×C₁₉	C₂²	1	2	yes	—
10	C₂²×C₄×D₁₉	C₂²	1	1	yes	characteristic
11	C₁₉×C₄⋊Q₈	C₂²	1	8	yes	complemented
12	C₂×C₄⋊C₄×C₁₉	C₂²	1	4	yes	—
13	C₂×C₄²×C₁₉	C₂²	1	1	yes	characteristic
14	#10597392	C₂²	1	16	yes	—
15	Q₈×Dic₁₉	C₂²	1	16	yes	complemented
16	C₂×(C₂×D₁₉)⋊C₄	C₂²	1	2	yes	—
17	C₂×(C₂×D₁₉)⋊C₄	C₂²	1	4	yes	—
18	C₂×C₄⋊Dic₁₉	C₂²	1	4	yes	—
19	C₂×C₄⋊Dic₁₉	C₂²	1	2	yes	—
20	C₂×Dic₁₉⋊C₄	C₂²	1	2	yes	—
21	C₂×Dic₁₉⋊C₄	C₂²	1	4	yes	—
22	C₂×Dic₁₉⋊C₄	C₂²	1	4	yes	—
23	C₂×C₄×Dic₁₉	C₂²	1	4	yes	—
24	C₂×C₄×Dic₁₉	C₂²	1	1	yes	characteristic
25	#10597344	C₂²	1	32	yes	—
26	#10597335	C₂²	1	32	yes	complemented
27	#10597318	C₂²	1	8	yes	—
28	#10597316	C₂²	1	8	yes	complemented
29	C₂³×D₁₉	C₂³	1	1	yes	characteristic
30	C₂×Q₈×C₁₉	—	2	8	no	—
31	C₂×Q₈×C₁₉	C₂³	1	8	yes	—
32	C₂²×C₄×C₁₉	C₂³	1	2	yes	—
33	C₂²×C₄×C₁₉	C₂³	1	4	yes	—
34	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
35	C₂²×Dic₁₉	C₂³	1	1	yes	characteristic
36	C₂²×Dic₁₉	C₂³	1	2	yes	—
37	C₂²×Dic₁₉	C₂³	1	4	yes	—
38	C₂×C₄×D₁₉	—	2	4	no	—
39	C₂×C₄×D₁₉	C₂³	1	4	yes	—
40	C₄⋊C₄×C₁₉	C₂³	1	16	yes	—
41	C₄²×C₁₉	C₂³	1	4	yes	—
42	(C₂×D₁₉)⋊C₄	C₂³	1	16	yes	—
43	(C₂×D₁₉)⋊C₄	C₂³	1	8	yes	—
44	C₄⋊Dic₁₉	C₂³	1	8	yes	—
45	C₄⋊Dic₁₉	C₂³	1	16	yes	—
46	Dic₁₉⋊C₄	C₂³	1	16	yes	—
47	Dic₁₉⋊C₄	C₂³	1	16	yes	—
48	Dic₁₉⋊C₄	C₂³	1	8	yes	—
49	C₄×Dic₁₉	C₂³	1	4	yes	—
50	C₄×Dic₁₉	C₂³	1	16	yes	—
51	C₂³×C₁₉	C₂⁴	1	1	yes	characteristic
52	C₂²×D₁₉	—	2	2	no	—
53	C₂²×D₁₉	—	2	4	no	—
54	C₂²×D₁₉	C₂⁴	1	2	yes	—
55	Q₈×C₁₉	—	2	16	no	—
56	C₂×C₄×C₁₉	—	2	8	no	—
57	C₂×C₄×C₁₉	—	2	4	no	—
58	C₂×C₄×C₁₉	#41	1	4	yes	—
59	C₂×C₄×C₁₉	C₂⁴	1	8	yes	—
60	C₂×C₄×C₁₉	C₂⁴	1	4	yes	—
61	C₂×C₄×C₁₉	C₂⁴	1	2	yes	—
62	C₂×Dic₁₉	—	2	2	no	—
63	C₂×Dic₁₉	—	2	4	no	—
64	C₂×Dic₁₉	—	2	8	no	—
65	C₂×Dic₁₉	C₂⁴	1	4	yes	—
66	C₂×Dic₁₉	C₂⁴	1	8	yes	—
67	C₂×Dic₁₉	C₂⁴	1	2	yes	—
68	C₄×D₁₉	—	2	8	no	—
69	C₂×#541	—	19	1	no	maximal, Sylow
70	C₂²×C₁₉	C₂×#41	1	2	yes	—
71	C₂²×C₁₉	2₋¹⁺⁴	1	4	yes	—
72	C₂²×C₁₉	C₂⁵	1	1	yes	characteristic
73	C₂×D₁₉	—	2	2	no	—
74	C₂×D₁₉	—	2	4	no	—
75	C₂×D₁₉	—	4	4	no	—
76	C₄×C₁₉	—	2	8	no	—
77	C₄×C₁₉	—	2	4	no	—
78	C₄×C₁₉	C₂×#41	1	4	yes	—
79	Dic₁₉	—	2	2	no	—
80	Dic₁₉	—	2	4	no	—
81	Dic₁₉	—	2	8	no	—
82	#541	—	19	16	no	—
83	C₂×C₄⋊Q₈	D₁₉	1	1	yes	characteristic, complemented
84	C₂×#126	—	19	4	no	—
85	C₂×#125	—	19	4	no	—
86	C₂×#125	—	19	1	no	—
87	C₂×C₂²⋊Q₈	—	19	2	no	—
88	C₂×C₄×Q₈	—	19	2	no	—
89	C₂×#117	—	19	1	no	—
90	C₂×C₁₉	#541	1	4	yes	complemented
91	C₂×C₁₉	C₂²×#41	1	1	yes	characteristic
92	C₂×C₁₉	C₂×2₋¹⁺⁴	1	2	yes	—
93	D₁₉	—	4	2	no	—
94	C₂²×Q₈	C₂×D₁₉	1	2	yes	—
95	C₂³×C₄	—	19	1	no	—
96	C₄⋊Q₈	C₂×D₁₉	1	8	yes	complemented
97	#126	—	19	32	no	—
98	#125	—	19	32	no	—
99	#125	—	19	8	no	—
100	C₂²⋊Q₈	—	19	16	no	—
101	C₄×Q₈	—	19	16	no	—
102	#117	—	19	8	no	—
103	C₂×C₄⋊C₄	—	19	2	no	—
104	C₂×C₄⋊C₄	—	19	2	no	—
105	C₂×C₄⋊C₄	—	19	4	no	—
106	C₂×C₄⋊C₄	—	19	4	no	—
107	C₂×C₄⋊C₄	—	19	4	no	—
108	C₂×C₄⋊C₄	C₂×D₁₉	1	4	yes	—
109	C₂×C₂²⋊C₄	—	19	2	no	—
110	C₂×C₂²⋊C₄	—	19	4	no	—
111	C₂×C₄²	—	19	1	no	—
112	C₂×C₄²	—	19	4	no	—
113	C₂×C₄²	C₂×D₁₉	1	1	yes	characteristic
114	C₁₉	C₂×#541	1	1	yes	characteristic, complemented, Sylow
115	C₂⁴	—	19	1	no	—
116	C₂×Q₈	—	2	8	no	—
117	C₂×Q₈	C₂²×D₁₉	1	8	yes	—
118	C₂²×C₄	—	19	2	no	—
119	C₂²×C₄	—	19	4	no	—
120	C₂²×C₄	—	19	1	no	—
121	C₂²×C₄	—	19	4	no	—
122	C₂²×C₄	—	38	4	no	—
123	C₂²×C₄	C₂²×D₁₉	1	4	yes	—
124	C₂²×C₄	C₂²×D₁₉	1	2	yes	—
125	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
126	C₄⋊C₄	—	19	8	no	—
127	C₄⋊C₄	—	19	16	no	—
128	C₄⋊C₄	—	19	16	no	—
129	C₄⋊C₄	—	19	8	no	—
130	C₄⋊C₄	—	19	16	no	—
131	C₄⋊C₄	C₂²×D₁₉	1	16	yes	—
132	C₂²⋊C₄	—	19	8	no	—
133	C₂²⋊C₄	—	19	16	no	—
134	C₄²	—	19	4	no	—
135	C₄²	—	19	16	no	—
136	C₄²	C₂²×D₁₉	1	4	yes	—
137	C₂³	—	19	2	no	—
138	C₂³	—	38	2	no	—
139	C₂³	—	38	4	no	—
140	C₂³	C₂³×D₁₉	1	1	yes	characteristic
141	Q₈	—	2	16	no	—
142	C₂×C₄	—	2	8	no	—
143	C₂×C₄	—	2	4	no	—
144	C₂×C₄	—	19	4	no	—
145	C₂×C₄	—	19	8	no	—
146	C₂×C₄	—	19	2	no	—
147	C₂×C₄	—	38	2	no	—
148	C₂×C₄	—	38	4	no	—
149	C₂×C₄	—	38	8	no	—
150	C₂×C₄	—	38	8	no	—
151	C₂×C₄	#64543	1	4	yes	—
152	C₂×C₄	C₂³×D₁₉	1	8	yes	—
153	C₂×C₄	C₂³×D₁₉	1	4	yes	—
154	C₂×C₄	C₂³×D₁₉	1	2	yes	—
155	C₂²	—	38	2	no	—
156	C₂²	—	38	4	no	—
157	C₂²	—	76	4	no	—
158	C₂²	C₂×#64543	1	2	yes	—
159	C₂²	#10597433	1	4	yes	—
160	C₂²	C₂⁴×D₁₉	1	1	yes	characteristic
161	C₄	—	2	8	no	—
162	C₄	—	2	4	no	—
163	C₄	—	38	2	no	—
164	C₄	—	38	4	no	—
165	C₄	—	38	8	no	—
166	C₄	C₂×#64543	1	4	yes	—
167	C₂	—	76	2	no	—
168	C₂	#11927155	1	4	yes	direct product factor
169	C₂	C₂²×#64543	1	1	yes	characteristic
170	C₂	C₂×#10597433	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#11927155

C₂

yes

maximal, direct product factor

C₂×#10597344

C₂

yes

maximal

C₂×#10597316

C₂

yes

maximal, characteristic, complemented

C₂×#10597335

C₂

yes

maximal, complemented

C₂×#10597392

C₂

yes

maximal

C₂×Q₈×Dic₁₉

C₂

yes

maximal, complemented

C₂×#10597318

C₂

yes

maximal, characteristic

C₂×C₁₉×C₄⋊Q₈

C₂

yes

maximal, characteristic, complemented

C₂²×Q₈×C₁₉

C₂²

yes

—

C₂²×C₄×D₁₉

C₂²

yes

characteristic

C₁₉×C₄⋊Q₈

C₂²

yes

complemented

C₂×C₄⋊C₄×C₁₉

C₂²

yes

—

C₂×C₄²×C₁₉

C₂²

yes

characteristic

#10597392

C₂²

yes

—

Q₈×Dic₁₉

C₂²

yes

complemented

C₂×(C₂×D₁₉)⋊C₄

C₂²

yes

—

C₂×(C₂×D₁₉)⋊C₄

C₂²

yes

—

C₂×C₄⋊Dic₁₉

C₂²

yes

—

C₂×C₄⋊Dic₁₉

C₂²

yes

—

C₂×Dic₁₉⋊C₄

C₂²

yes

—

C₂×Dic₁₉⋊C₄

C₂²

yes

—

C₂×Dic₁₉⋊C₄

C₂²

yes

—

C₂×C₄×Dic₁₉

C₂²

yes

—

C₂×C₄×Dic₁₉

C₂²

yes

characteristic

#10597344

C₂²

yes

—

#10597335

C₂²

yes

complemented

#10597318

C₂²

yes

—

#10597316

C₂²

yes

complemented

C₂³×D₁₉

C₂³

yes

characteristic

C₂×Q₈×C₁₉

—

C₂×Q₈×C₁₉

C₂³

yes

—

C₂²×C₄×C₁₉

C₂³

yes

—

C₂²×C₄×C₁₉

C₂³

yes

—

C₂²×C₄×C₁₉

C₂³

yes

characteristic

C₂²×Dic₁₉

C₂³

yes

characteristic

C₂²×Dic₁₉

C₂³

yes

—

C₂²×Dic₁₉

C₂³

yes

—

C₂×C₄×D₁₉

—

C₂×C₄×D₁₉

C₂³

yes

—

C₄⋊C₄×C₁₉

C₂³

yes

—

C₄²×C₁₉

C₂³

yes

—

(C₂×D₁₉)⋊C₄

C₂³

yes

—

(C₂×D₁₉)⋊C₄

C₂³

yes

—

C₄⋊Dic₁₉

C₂³

yes

—

C₄⋊Dic₁₉

C₂³

yes

—

Dic₁₉⋊C₄

C₂³

yes

—

Dic₁₉⋊C₄

C₂³

yes

—

Dic₁₉⋊C₄

C₂³

yes

—

C₄×Dic₁₉

C₂³

yes

—

C₄×Dic₁₉

C₂³

yes

—

C₂³×C₁₉

C₂⁴

yes

characteristic

C₂²×D₁₉

—

C₂²×D₁₉

—

C₂²×D₁₉

C₂⁴

yes

—

Q₈×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

#41

yes

—

C₂×C₄×C₁₉

C₂⁴

yes

—

C₂×C₄×C₁₉

C₂⁴

yes

—

C₂×C₄×C₁₉

C₂⁴

yes

—

C₂×Dic₁₉

—

C₂×Dic₁₉

—

C₂×Dic₁₉

—

C₂×Dic₁₉

C₂⁴

yes

—

C₂×Dic₁₉

C₂⁴

yes

—

C₂×Dic₁₉

C₂⁴

yes

—

C₄×D₁₉

—

C₂×#541

—

maximal, Sylow

C₂²×C₁₉

C₂×#41

yes

—

C₂²×C₁₉

2₋¹⁺⁴

yes

—

C₂²×C₁₉

C₂⁵

yes

characteristic

C₂×D₁₉

—

C₂×D₁₉

—

C₂×D₁₉

—

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

C₂×#41

yes

—

Dic₁₉

—

Dic₁₉

—

Dic₁₉

—

#541

—

C₂×C₄⋊Q₈

D₁₉

yes

characteristic, complemented

C₂×#126

—

C₂×#125

—

C₂×#125

—

C₂×C₂²⋊Q₈

—

C₂×C₄×Q₈

—

C₂×#117

—

C₂×C₁₉

#541

yes

complemented

C₂×C₁₉

C₂²×#41

yes

characteristic

C₂×C₁₉

C₂×2₋¹⁺⁴

yes

—

D₁₉

—

C₂²×Q₈

C₂×D₁₉

yes

—

C₂³×C₄

—

C₄⋊Q₈

C₂×D₁₉

yes

—

—

—

100

C₂²⋊Q₈

—

101

C₄×Q₈

—

102

#117

—

103

C₂×C₄⋊C₄

—

104

C₂×C₄⋊C₄

—

105

C₂×C₄⋊C₄

—

106

C₂×C₄⋊C₄

—

107

C₂×C₄⋊C₄

—

108

C₂×C₄⋊C₄

C₂×D₁₉

yes

—

109

C₂×C₂²⋊C₄

—

110

C₂×C₂²⋊C₄

—

111

C₂×C₄²

—

112

C₂×C₄²

—

113

C₂×C₄²

C₂×D₁₉

yes

characteristic

114

C₁₉

C₂×#541

yes

characteristic, complemented, Sylow

115

C₂⁴

—

116

C₂×Q₈

—

117

C₂×Q₈

C₂²×D₁₉

yes

—

118

C₂²×C₄

—

119

C₂²×C₄

—

120

C₂²×C₄

—

121

C₂²×C₄

—

122

C₂²×C₄

—

123

C₂²×C₄

C₂²×D₁₉

yes

—

124

C₂²×C₄

C₂²×D₁₉

yes

—

125

C₂²×C₄

C₂²×D₁₉

yes

characteristic

126

C₄⋊C₄

—

127

C₄⋊C₄

—

128

C₄⋊C₄

—

129

C₄⋊C₄

—

130

C₄⋊C₄

—

131

C₄⋊C₄

C₂²×D₁₉

yes

—

132

C₂²⋊C₄

—

133

C₂²⋊C₄

—

134

C₄²

—

135

C₄²

—

136

C₄²

C₂²×D₁₉

yes

—

137

C₂³

—

138

C₂³

—

139

C₂³

—

140

C₂³

C₂³×D₁₉

yes

characteristic

141

Q₈

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

—

151

C₂×C₄

#64543

yes

—

152

C₂×C₄

C₂³×D₁₉

yes

—

153

C₂×C₄

C₂³×D₁₉

yes

—

154

C₂×C₄

C₂³×D₁₉

yes

—

155

C₂²

—

156

C₂²

—

157

C₂²

—

158

C₂²

C₂×#64543

yes

—

159

C₂²

#10597433

yes

—

160

C₂²

C₂⁴×D₁₉

yes

characteristic

161

C₄

—

162

C₄

—

163

C₄

—

164

C₄

—

165

C₄

—

166

C₄

C₂×#64543

yes

—

167

C₂

—

168

C₂

#11927155

yes

direct product factor

169

C₂

C₂²×#64543

yes

characteristic

170

C₂

C₂×#10597433

yes

—

	subgroup	G/H
center	C₂³	C₂³×D₁₉
commutator, cosocle	C₂²×C₁₉	C₂⁵
Frattini	C₂²	C₂⁴×D₁₉
Fitting	C₂×C₁₉×C₄⋊Q₈	C₂
socle	C₂³×C₁₉	C₂⁴
prime radical	C₂³×D₁₉	C₂³
hypercenter	C₂×C₄⋊Q₈	D₁₉
hypocenter	C₁₉	C₂×#541
LCS	G ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄⋊Q₈

subgroup

G/H

center

C₂³

C₂³×D₁₉

commutator, cosocle

C₂²×C₁₉

C₂⁵

Frattini

C₂²

C₂⁴×D₁₉

Fitting

C₂×C₁₉×C₄⋊Q₈

C₂

socle

C₂³×C₁₉

C₂⁴

prime radical

C₂³×D₁₉

C₂³

hypercenter

C₂×C₄⋊Q₈

D₁₉

C₁₉

G ▷ C₂²×C₁₉ ▷ C₁₉

C₁ ◁ C₂³ ◁ C₂×C₄⋊Q₈

N	H ≅ G/N	type
1. #11927155	C₂	direct (N × H)
3. C₂×#10597316	C₂	semidirect
4. C₂×#10597335	C₂	semidirect
6. C₂×Q₈×Dic₁₉	C₂	semidirect
8. C₂×C₁₉×C₄⋊Q₈	C₂	semidirect
11. C₁₉×C₄⋊Q₈	C₂²	semidirect
15. Q₈×Dic₁₉	C₂²	semidirect
26. #10597335	C₂²	semidirect
28. #10597316	C₂²	semidirect
83. C₂×C₄⋊Q₈	D₁₉	semidirect
90. C₂×C₁₉	#541	semidirect
96. C₄⋊Q₈	C₂×D₁₉	semidirect
114. C₁₉	C₂×#541	semidirect
168. C₂	#11927155	direct (N × H)

H ≅ G/N

type

1. #11927155

C₂

direct (N × H)

3. C₂×#10597316

C₂

semidirect

4. C₂×#10597335

C₂

semidirect

6. C₂×Q₈×Dic₁₉

C₂

semidirect

8. C₂×C₁₉×C₄⋊Q₈

C₂

semidirect

11. C₁₉×C₄⋊Q₈

C₂²

semidirect

15. Q₈×Dic₁₉

C₂²

semidirect

26. #10597335

C₂²

semidirect

28. #10597316

C₂²

semidirect

83. C₂×C₄⋊Q₈

D₁₉

semidirect

90. C₂×C₁₉

#541

semidirect

96. C₄⋊Q₈

C₂×D₁₉

semidirect

114. C₁₉

C₂×#541

semidirect

168. C₂

#11927155

direct (N × H)

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

subgroup	index
#11927155	2
C₂×#10597344	2
C₂×#10597316	2
C₂×#10597335	2
C₂×#10597392	2
C₂×Q₈×Dic₁₉	2
C₂×#10597318	2
C₂×C₁₉×C₄⋊Q₈	2
C₂×#541	19

subgroup

index

C₂×Q₈×Dic₁₉

C₂×C₁₉×C₄⋊Q₈

Subgroups — C₂×#11927155 #14577002

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×#11927155 #14577002

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#11927155 #14577002