Group #14561236 — subgroups

Subgroups — Q₈×SD₁₆×C₁₉ #14561236

Order 2432.

370 conjugacy classes of subgroups; only the 208 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 208 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	Q₈²×C₁₉	C₂	1	1	yes	maximal, characteristic, complemented
2	D₄×Q₈×C₁₉	C₂	1	1	yes	maximal, characteristic
3	C₁₉×#498	C₂	1	3	yes	maximal, complemented
4	C₁₉×#476	C₂	1	3	yes	maximal
5	C₁₉×#475	C₂	1	3	yes	maximal, complemented
6	C₈×Q₈×C₁₉	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄×SD₁₆×C₁₉	C₂	1	3	yes	maximal
8	C₂²×Q₈×C₁₉	—	2	1	no	—
9	C₂×SD₁₆×C₁₉	C₂²	1	1	yes	characteristic
10	C₁₉×C₄⋊Q₈	—	2	3	no	—
11	C₁₉×C₄⋊Q₈	C₂²	1	3	yes	—
12	C₁₉×C₄⋊Q₈	C₂²	1	3	yes	—
13	C₁₉×C₂²⋊Q₈	—	2	3	no	—
14	C₄×Q₈×C₁₉	—	2	1	no	—
15	C₄×Q₈×C₁₉	C₂²	1	3	yes	—
16	C₄×Q₈×C₁₉	C₂²	1	1	yes	characteristic
17	C₄×D₄×C₁₉	C₂²	1	3	yes	—
18	C₁₉×#106	C₂²	1	3	yes	—
19	C₁₉×#106	C₂²	1	6	yes	—
20	C₁₉×C₄⋊C₈	C₂²	1	3	yes	—
21	C₁₉×#103	C₂²	1	3	yes	—
22	C₁₉×#102	C₂²	1	3	yes	—
23	C₄×C₈×C₁₉	C₂²	1	3	yes	—
24	C₂×Q₈×C₁₉	—	2	3	no	—
25	C₂×Q₈×C₁₉	—	2	3	no	—
26	C₂×Q₈×C₁₉	—	4	4	no	—
27	C₂×Q₈×C₁₉	D₄	1	1	yes	characteristic
28	C₂×Q₈×C₁₉	C₂³	1	1	yes	characteristic
29	C₂×D₄×C₁₉	C₂³	1	1	yes	characteristic
30	C₂²×C₄×C₁₉	—	2	3	no	—
31	SD₁₆×C₁₉	Q₈	1	4	yes	direct product factor
32	C₂×C₈×C₁₉	C₂³	1	3	yes	—
33	C₂×C₈×C₁₉	C₂³	1	1	yes	characteristic
34	C₄⋊C₄×C₁₉	—	2	3	no	—
35	C₄⋊C₄×C₁₉	—	2	3	no	—
36	C₄⋊C₄×C₁₉	—	2	3	no	—
37	C₄⋊C₄×C₁₉	D₄	1	3	yes	—
38	C₄⋊C₄×C₁₉	C₂³	1	3	yes	—
39	C₄⋊C₄×C₁₉	C₂³	1	3	yes	—
40	C₂²⋊C₄×C₁₉	—	2	3	no	—
41	C₄²×C₁₉	—	2	3	no	—
42	C₄²×C₁₉	C₂³	1	3	yes	—
43	C₂³×C₁₉	—	2	1	no	—
44	Q₈×C₁₉	—	2	1	no	—
45	Q₈×C₁₉	—	4	6	no	—
46	Q₈×C₁₉	—	8	3	no	—
47	Q₈×C₁₉	SD₁₆	1	4	yes	direct product factor
48	Q₈×C₁₉	C₂×Q₈	1	2	yes	complemented
49	D₄×C₁₉	—	2	1	no	—
50	D₄×C₁₉	C₂×Q₈	1	2	yes	—
51	C₂×C₄×C₁₉	—	2	1	no	—
52	C₂×C₄×C₁₉	—	2	3	no	—
53	C₂×C₄×C₁₉	—	2	3	no	—
54	C₂×C₄×C₁₉	—	4	6	no	—
55	C₂×C₄×C₁₉	C₂×D₄	1	3	yes	—
56	C₂×C₄×C₁₉	C₂×D₄	1	3	yes	—
57	C₂×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
58	C₈×C₁₉	—	2	3	no	—
59	C₈×C₁₉	C₂×Q₈	1	2	yes	complemented
60	Q₈×SD₁₆	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
61	C₂²×C₁₉	—	2	2	no	—
62	C₂²×C₁₉	—	4	1	no	—
63	C₂²×C₁₉	—	4	1	no	—
64	C₂²×C₁₉	C₂²×D₄	1	1	yes	characteristic
65	C₄×C₁₉	—	2	2	no	—
66	C₄×C₁₉	—	2	3	no	—
67	C₄×C₁₉	—	4	3	no	—
68	C₄×C₁₉	—	4	3	no	—
69	C₄×C₁₉	C₂×SD₁₆	1	6	yes	—
70	C₄×C₁₉	C₂²×Q₈	1	1	yes	characteristic
71	C₄×C₁₉	2₋¹⁺⁴	1	1	yes	characteristic
72	Q₈²	C₂×C₁₉	1	1	yes	characteristic, complemented
73	D₄×Q₈	C₂×C₁₉	1	1	yes	characteristic
74	#498	C₂×C₁₉	1	3	yes	complemented
75	#476	C₂×C₁₉	1	3	yes	—
76	#475	C₂×C₁₉	1	3	yes	complemented
77	C₈×Q₈	C₂×C₁₉	1	1	yes	characteristic, complemented
78	C₄×SD₁₆	C₂×C₁₉	1	3	yes	—
79	C₂×C₁₉	—	4	2	no	—
80	C₂×C₁₉	D₄×Q₈	1	1	yes	characteristic
81	C₂×C₁₉	C₂²×SD₁₆	1	1	yes	characteristic
82	C₂×C₁₉	#577	1	1	yes	characteristic
83	C₂²×Q₈	—	2	1	no	—
84	C₂×SD₁₆	C₂²×C₁₉	1	1	yes	characteristic
85	C₄⋊Q₈	—	2	3	no	—
86	C₄⋊Q₈	C₂²×C₁₉	1	3	yes	—
87	C₄⋊Q₈	C₂²×C₁₉	1	3	yes	—
88	C₂²⋊Q₈	—	2	3	no	—
89	C₄×Q₈	—	2	1	no	—
90	C₄×Q₈	C₂²×C₁₉	1	3	yes	—
91	C₄×Q₈	C₂²×C₁₉	1	1	yes	characteristic
92	C₄×D₄	C₂²×C₁₉	1	3	yes	—
93	#106	C₂²×C₁₉	1	3	yes	—
94	#106	C₂²×C₁₉	1	6	yes	—
95	C₄⋊C₈	C₂²×C₁₉	1	3	yes	—
96	#103	C₂²×C₁₉	1	3	yes	—
97	#102	C₂²×C₁₉	1	3	yes	—
98	C₄×C₈	C₂²×C₁₉	1	3	yes	—
99	C₁₉	Q₈×SD₁₆	1	1	yes	characteristic, direct product factor, Sylow
100	C₂×Q₈	—	2	3	no	—
101	C₂×Q₈	—	2	3	no	—
102	C₂×Q₈	—	4	4	no	—
103	C₂×Q₈	D₄×C₁₉	1	1	yes	characteristic
104	C₂×Q₈	C₂³×C₁₉	1	1	yes	characteristic
105	C₂×D₄	C₂³×C₁₉	1	1	yes	characteristic
106	C₂²×C₄	—	2	3	no	—
107	SD₁₆	Q₈×C₁₉	1	4	yes	direct product factor
108	C₂×C₈	C₂³×C₁₉	1	3	yes	—
109	C₂×C₈	C₂³×C₁₉	1	1	yes	characteristic
110	C₄⋊C₄	—	2	3	no	—
111	C₄⋊C₄	—	2	3	no	—
112	C₄⋊C₄	—	2	3	no	—
113	C₄⋊C₄	D₄×C₁₉	1	3	yes	—
114	C₄⋊C₄	C₂³×C₁₉	1	3	yes	—
115	C₄⋊C₄	C₂³×C₁₉	1	3	yes	—
116	C₂²⋊C₄	—	2	3	no	—
117	C₄²	—	2	3	no	—
118	C₄²	C₂³×C₁₉	1	3	yes	—
119	C₂³	—	2	1	no	—
120	Q₈	—	2	1	no	—
121	Q₈	—	4	6	no	—
122	Q₈	—	8	3	no	—
123	Q₈	SD₁₆×C₁₉	1	4	yes	direct product factor
124	Q₈	C₂×Q₈×C₁₉	1	2	yes	complemented
125	D₄	—	2	1	no	—
126	D₄	C₂×Q₈×C₁₉	1	2	yes	—
127	C₂×C₄	—	2	1	no	—
128	C₂×C₄	—	2	3	no	—
129	C₂×C₄	—	2	3	no	—
130	C₂×C₄	—	4	6	no	—
131	C₂×C₄	C₂×D₄×C₁₉	1	3	yes	—
132	C₂×C₄	C₂×D₄×C₁₉	1	3	yes	—
133	C₂×C₄	C₂⁴×C₁₉	1	1	yes	characteristic
134	C₈	—	2	3	no	—
135	C₈	C₂×Q₈×C₁₉	1	2	yes	complemented
136	C₂²	—	2	2	no	—
137	C₂²	—	4	1	no	—
138	C₂²	—	4	1	no	—
139	C₂²	C₂²×D₄×C₁₉	1	1	yes	characteristic
140	C₄	—	2	2	no	—
141	C₄	—	2	3	no	—
142	C₄	—	4	3	no	—
143	C₄	—	4	3	no	—
144	C₄	C₂×SD₁₆×C₁₉	1	6	yes	—
145	C₄	C₂²×Q₈×C₁₉	1	1	yes	characteristic
146	C₄	C₁₉×2₋¹⁺⁴	1	1	yes	characteristic
147	C₂	—	4	2	no	—
148	C₂	D₄×Q₈×C₁₉	1	1	yes	characteristic
149	C₂	C₂²×SD₁₆×C₁₉	1	1	yes	characteristic
150	C₂	C₁₉×#577	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

Q₈²×C₁₉

C₂

yes

maximal, characteristic, complemented

D₄×Q₈×C₁₉

C₂

yes

maximal, characteristic

C₁₉×#498

C₂

yes

maximal, complemented

C₁₉×#476

C₂

yes

maximal

C₁₉×#475

C₂

yes

maximal, complemented

C₈×Q₈×C₁₉

C₂

yes

maximal, characteristic, complemented

C₄×SD₁₆×C₁₉

C₂

yes

maximal

C₂²×Q₈×C₁₉

—

C₂×SD₁₆×C₁₉

C₂²

yes

characteristic

C₁₉×C₄⋊Q₈

—

C₁₉×C₄⋊Q₈

C₂²

yes

—

C₁₉×C₄⋊Q₈

C₂²

yes

—

C₁₉×C₂²⋊Q₈

—

C₄×Q₈×C₁₉

—

C₄×Q₈×C₁₉

C₂²

yes

—

C₄×Q₈×C₁₉

C₂²

yes

characteristic

C₄×D₄×C₁₉

C₂²

yes

—

C₁₉×#106

C₂²

yes

—

C₁₉×#106

C₂²

yes

—

C₁₉×C₄⋊C₈

C₂²

yes

—

C₁₉×#103

C₂²

yes

—

C₁₉×#102

C₂²

yes

—

C₄×C₈×C₁₉

C₂²

yes

—

C₂×Q₈×C₁₉

—

C₂×Q₈×C₁₉

—

C₂×Q₈×C₁₉

—

C₂×Q₈×C₁₉

D₄

yes

characteristic

C₂×Q₈×C₁₉

C₂³

yes

characteristic

C₂×D₄×C₁₉

C₂³

yes

characteristic

C₂²×C₄×C₁₉

—

SD₁₆×C₁₉

Q₈

yes

direct product factor

C₂×C₈×C₁₉

C₂³

yes

—

C₂×C₈×C₁₉

C₂³

yes

characteristic

C₄⋊C₄×C₁₉

—

C₄⋊C₄×C₁₉

—

C₄⋊C₄×C₁₉

—

C₄⋊C₄×C₁₉

D₄

yes

—

C₄⋊C₄×C₁₉

C₂³

yes

—

C₄⋊C₄×C₁₉

C₂³

yes

—

C₂²⋊C₄×C₁₉

—

C₄²×C₁₉

—

C₄²×C₁₉

C₂³

yes

—

C₂³×C₁₉

—

Q₈×C₁₉

—

Q₈×C₁₉

—

Q₈×C₁₉

—

Q₈×C₁₉

SD₁₆

yes

direct product factor

Q₈×C₁₉

C₂×Q₈

yes

complemented

D₄×C₁₉

—

D₄×C₁₉

C₂×Q₈

yes

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

C₂×D₄

yes

—

C₂×C₄×C₁₉

C₂×D₄

yes

—

C₂×C₄×C₁₉

C₂⁴

yes

characteristic

C₈×C₁₉

—

C₈×C₁₉

C₂×Q₈

yes

complemented

Q₈×SD₁₆

C₁₉

yes

maximal, characteristic, direct product factor, Sylow

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

C₂²×D₄

yes

characteristic

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

C₂×SD₁₆

yes

—

C₄×C₁₉

C₂²×Q₈

yes

characteristic

C₄×C₁₉

2₋¹⁺⁴

yes

characteristic

Q₈²

C₂×C₁₉

yes

characteristic, complemented

D₄×Q₈

C₂×C₁₉

yes

characteristic

#498

C₂×C₁₉

yes

complemented

#476

C₂×C₁₉

yes

—

#475

C₂×C₁₉

yes

complemented

C₈×Q₈

C₂×C₁₉

yes

characteristic, complemented

C₄×SD₁₆

C₂×C₁₉

yes

—

C₂×C₁₉

—

C₂×C₁₉

D₄×Q₈

yes

characteristic

C₂×C₁₉

C₂²×SD₁₆

yes

characteristic

C₂×C₁₉

#577

yes

characteristic

C₂²×Q₈

—

C₂×SD₁₆

C₂²×C₁₉

yes

characteristic

C₄⋊Q₈

—

C₄⋊Q₈

C₂²×C₁₉

yes

—

C₄⋊Q₈

C₂²×C₁₉

yes

—

C₂²⋊Q₈

—

C₄×Q₈

—

C₄×Q₈

C₂²×C₁₉

yes

—

C₄×Q₈

C₂²×C₁₉

yes

characteristic

C₄×D₄

C₂²×C₁₉

yes

—

#106

C₂²×C₁₉

yes

—

#106

C₂²×C₁₉

yes

—

C₄⋊C₈

C₂²×C₁₉

yes

—

#103

C₂²×C₁₉

yes

—

#102

C₂²×C₁₉

yes

—

C₄×C₈

C₂²×C₁₉

yes

—

C₁₉

Q₈×SD₁₆

yes

characteristic, direct product factor, Sylow

100

C₂×Q₈

—

101

C₂×Q₈

—

102

C₂×Q₈

—

103

C₂×Q₈

D₄×C₁₉

yes

characteristic

104

C₂×Q₈

C₂³×C₁₉

yes

characteristic

105

C₂×D₄

C₂³×C₁₉

yes

characteristic

106

C₂²×C₄

—

107

SD₁₆

Q₈×C₁₉

yes

direct product factor

108

C₂×C₈

C₂³×C₁₉

yes

—

109

C₂×C₈

C₂³×C₁₉

yes

characteristic

110

C₄⋊C₄

—

111

C₄⋊C₄

—

112

C₄⋊C₄

—

113

C₄⋊C₄

D₄×C₁₉

yes

—

114

C₄⋊C₄

C₂³×C₁₉

yes

—

115

C₄⋊C₄

C₂³×C₁₉

yes

—

116

C₂²⋊C₄

—

117

C₄²

—

118

C₄²

C₂³×C₁₉

yes

—

119

C₂³

—

120

Q₈

—

121

Q₈

—

122

Q₈

—

123

Q₈

SD₁₆×C₁₉

yes

direct product factor

124

Q₈

C₂×Q₈×C₁₉

yes

complemented

125

D₄

—

126

D₄

C₂×Q₈×C₁₉

yes

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

C₂×D₄×C₁₉

yes

—

132

C₂×C₄

C₂×D₄×C₁₉

yes

—

133

C₂×C₄

C₂⁴×C₁₉

yes

characteristic

134

C₈

—

135

C₈

C₂×Q₈×C₁₉

yes

complemented

136

C₂²

—

137

C₂²

—

138

C₂²

—

139

C₂²

C₂²×D₄×C₁₉

yes

characteristic

140

C₄

—

141

C₄

—

142

C₄

—

143

C₄

—

144

C₄

C₂×SD₁₆×C₁₉

yes

—

145

C₄

C₂²×Q₈×C₁₉

yes

characteristic

146

C₄

C₁₉×2₋¹⁺⁴

yes

characteristic

147

C₂

—

148

C₂

D₄×Q₈×C₁₉

yes

characteristic

149

C₂

C₂²×SD₁₆×C₁₉

yes

characteristic

150

C₂

C₁₉×#577

yes

characteristic

	subgroup	G/H
center, socle	C₂²×C₁₉	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₁₉
prime radical	C₂×D₄×C₁₉	C₂³
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₉ ◁ C₄×Q₈×C₁₉ ◁ G

subgroup

G/H

center, socle

C₂²×C₁₉

C₂²×D₄

commutator, Frattini, cosocle

C₂×C₄

C₂⁴×C₁₉

prime radical

C₂×D₄×C₁₉

C₂³

LCS

G ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₁₉ ◁ C₄×Q₈×C₁₉ ◁ G

N	H ≅ G/N	type
1. Q₈²×C₁₉	C₂	semidirect
3. C₁₉×#498	C₂	semidirect
5. C₁₉×#475	C₂	semidirect
6. C₈×Q₈×C₁₉	C₂	semidirect
31. SD₁₆×C₁₉	Q₈	direct (N × H)
47. Q₈×C₁₉	SD₁₆	direct (N × H)
48. Q₈×C₁₉	C₂×Q₈	semidirect
59. C₈×C₁₉	C₂×Q₈	semidirect
60. Q₈×SD₁₆	C₁₉	direct (N × H)
72. Q₈²	C₂×C₁₉	semidirect
74. #498	C₂×C₁₉	semidirect
76. #475	C₂×C₁₉	semidirect
77. C₈×Q₈	C₂×C₁₉	semidirect
99. C₁₉	Q₈×SD₁₆	direct (N × H)
107. SD₁₆	Q₈×C₁₉	direct (N × H)
123. Q₈	SD₁₆×C₁₉	direct (N × H)
124. Q₈	C₂×Q₈×C₁₉	semidirect
135. C₈	C₂×Q₈×C₁₉	semidirect

H ≅ G/N

type

1. Q₈²×C₁₉

C₂

semidirect

3. C₁₉×#498

C₂

semidirect

5. C₁₉×#475

C₂

semidirect

6. C₈×Q₈×C₁₉

C₂

semidirect

31. SD₁₆×C₁₉

Q₈

direct (N × H)

47. Q₈×C₁₉

SD₁₆

direct (N × H)

48. Q₈×C₁₉

C₂×Q₈

semidirect

59. C₈×C₁₉

C₂×Q₈

semidirect

60. Q₈×SD₁₆

C₁₉

direct (N × H)

72. Q₈²

C₂×C₁₉

semidirect

74. #498

C₂×C₁₉

semidirect

76. #475

C₂×C₁₉

semidirect

77. C₈×Q₈

C₂×C₁₉

semidirect

99. C₁₉

Q₈×SD₁₆

direct (N × H)

107. SD₁₆

Q₈×C₁₉

direct (N × H)

123. Q₈

SD₁₆×C₁₉

direct (N × H)

124. Q₈

C₂×Q₈×C₁₉

semidirect

135. C₈

C₂×Q₈×C₁₉

semidirect

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

subgroup	index
Q₈²×C₁₉	2
D₄×Q₈×C₁₉	2
C₁₉×#498	2
C₁₉×#476	2
C₁₉×#475	2
C₈×Q₈×C₁₉	2
C₄×SD₁₆×C₁₉	2
Q₈×SD₁₆	19

subgroup

index

Q₈²×C₁₉

D₄×Q₈×C₁₉

C₁₉×#498

C₁₉×#476

C₁₉×#475

C₈×Q₈×C₁₉

C₄×SD₁₆×C₁₉

Q₈×SD₁₆

Subgroups — Q₈×SD₁₆×C₁₉ #14561236

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q8×SD16×C19 #14561236

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×SD₁₆×C₁₉ #14561236