Group #18039904 — subgroups

Subgroups — Q₈×C₁₉×D₁₆ #18039904

Order 4864.

528 conjugacy classes of subgroups; only the 234 normal subgroups are shown.

normal subgroups only

This group has 528 conjugacy classes of subgroups — too many to draw. Only its 234 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	Q₈×D₈×C₁₉	C₂	1	2	yes	maximal, complemented
2	C₁₉×#2252	C₂	1	3	yes	maximal, complemented
3	C₁₉×#2224	C₂	1	6	yes	maximal, complemented
4	Q₈×C₁₆×C₁₉	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄×C₁₉×D₁₆	C₂	1	3	yes	maximal
6	D₄×Q₈×C₁₉	—	2	2	no	—
7	C₂×C₁₉×D₁₆	C₂²	1	1	yes	characteristic
8	C₁₉×#500	C₂²	1	6	yes	—
9	C₁₉×#474	—	2	6	no	—
10	C₈×Q₈×C₁₉	C₂²	1	1	yes	characteristic
11	C₄×D₈×C₁₉	C₂²	1	6	yes	—
12	C₁₉×#366	C₂²	1	3	yes	—
13	C₁₉×#366	C₂²	1	6	yes	—
14	C₁₉×C₄⋊C₁₆	C₂²	1	3	yes	—
15	C₁₉×#357	C₂²	1	6	yes	—
16	C₄×C₁₆×C₁₉	C₂²	1	3	yes	—
17	C₂²×Q₈×C₁₉	—	4	2	no	—
18	C₂×D₈×C₁₉	C₂³	1	2	yes	—
19	C₁₉×C₄⋊Q₈	—	2	6	no	—
20	C₁₉×C₂²⋊Q₈	—	4	6	no	—
21	C₄×Q₈×C₁₉	D₄	1	1	yes	characteristic
22	C₄×D₄×C₁₉	—	2	6	no	—
23	C₁₉×D₁₆	Q₈	1	4	yes	direct product factor
24	C₂×C₁₆×C₁₉	C₂³	1	3	yes	—
25	C₂×C₁₆×C₁₉	C₂³	1	1	yes	characteristic
26	C₁₉×#107	—	2	6	no	—
27	C₁₉×#107	C₂³	1	6	yes	—
28	C₁₉×C₄⋊C₈	D₄	1	3	yes	—
29	C₁₉×#102	—	2	6	no	—
30	C₄×C₈×C₁₉	C₂³	1	3	yes	—
31	C₂×Q₈×C₁₉	—	4	6	no	—
32	C₂×Q₈×C₁₉	—	8	8	no	—
33	C₂×Q₈×C₁₉	D₈	1	1	yes	characteristic
34	C₂×D₄×C₁₉	—	2	2	no	—
35	C₂²×C₄×C₁₉	—	4	6	no	—
36	D₈×C₁₉	—	2	2	no	—
37	D₈×C₁₉	C₂×Q₈	1	4	yes	complemented
38	C₂×C₈×C₁₉	C₂×D₄	1	3	yes	—
39	C₂×C₈×C₁₉	C₂⁴	1	1	yes	characteristic
40	C₄⋊C₄×C₁₉	—	2	6	no	—
41	C₄⋊C₄×C₁₉	—	4	6	no	—
42	C₄⋊C₄×C₁₉	D₈	1	3	yes	—
43	C₂²⋊C₄×C₁₉	—	4	6	no	—
44	C₄²×C₁₉	C₂×D₄	1	3	yes	—
45	C₁₆×C₁₉	—	2	3	no	—
46	C₁₆×C₁₉	C₂×Q₈	1	2	yes	complemented
47	Q₈×D₁₆	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
48	C₂³×C₁₉	—	4	2	no	—
49	Q₈×C₁₉	—	8	12	no	—
50	Q₈×C₁₉	D₁₆	1	4	yes	direct product factor
51	D₄×C₁₉	—	2	4	no	—
52	D₄×C₁₉	—	4	2	no	—
53	C₂×C₄×C₁₉	—	4	6	no	—
54	C₂×C₄×C₁₉	—	8	12	no	—
55	C₂×C₄×C₁₉	C₂×D₈	1	3	yes	—
56	C₂×C₄×C₁₉	C₂×D₈	1	3	yes	—
57	C₂×C₄×C₁₉	C₂²×D₄	1	1	yes	characteristic
58	C₈×C₁₉	—	2	3	no	—
59	C₈×C₁₉	C₂²×Q₈	1	1	yes	characteristic
60	C₈×C₁₉	2₋¹⁺⁴	1	1	yes	characteristic
61	Q₈×D₈	C₂×C₁₉	1	2	yes	complemented
62	#2252	C₂×C₁₉	1	3	yes	complemented
63	#2224	C₂×C₁₉	1	6	yes	complemented
64	Q₈×C₁₆	C₂×C₁₉	1	1	yes	characteristic, complemented
65	C₄×D₁₆	C₂×C₁₉	1	3	yes	—
66	C₂²×C₁₉	—	4	4	no	—
67	C₂²×C₁₉	—	8	2	no	—
68	C₂²×C₁₉	—	8	2	no	—
69	C₂²×C₁₉	C₂²×D₈	1	1	yes	characteristic
70	C₄×C₁₉	—	2	3	no	—
71	C₄×C₁₉	—	8	6	no	—
72	C₄×C₁₉	C₂×D₁₆	1	6	yes	—
73	C₄×C₁₉	D₄×Q₈	1	1	yes	characteristic
74	C₄×C₁₉	#578	1	1	yes	characteristic
75	D₄×Q₈	—	2	2	no	—
76	C₂×D₁₆	C₂²×C₁₉	1	1	yes	characteristic
77	#500	C₂²×C₁₉	1	6	yes	—
78	#474	—	2	6	no	—
79	C₈×Q₈	C₂²×C₁₉	1	1	yes	characteristic
80	C₄×D₈	C₂²×C₁₉	1	6	yes	—
81	#366	C₂²×C₁₉	1	3	yes	—
82	#366	C₂²×C₁₉	1	6	yes	—
83	C₄⋊C₁₆	C₂²×C₁₉	1	3	yes	—
84	#357	C₂²×C₁₉	1	6	yes	—
85	C₄×C₁₆	C₂²×C₁₉	1	3	yes	—
86	C₂×C₁₉	—	8	4	no	—
87	C₂×C₁₉	Q₈×D₈	1	1	yes	characteristic
88	C₂×C₁₉	C₂²×D₁₆	1	1	yes	characteristic
89	C₂×C₁₉	#3417	1	1	yes	characteristic
90	C₂²×Q₈	—	4	2	no	—
91	C₂×D₈	C₂³×C₁₉	1	2	yes	—
92	C₄⋊Q₈	—	2	6	no	—
93	C₂²⋊Q₈	—	4	6	no	—
94	C₄×Q₈	D₄×C₁₉	1	1	yes	characteristic
95	C₄×D₄	—	2	6	no	—
96	D₁₆	Q₈×C₁₉	1	4	yes	direct product factor
97	C₂×C₁₆	C₂³×C₁₉	1	3	yes	—
98	C₂×C₁₆	C₂³×C₁₉	1	1	yes	characteristic
99	#107	—	2	6	no	—
100	#107	C₂³×C₁₉	1	6	yes	—
101	C₄⋊C₈	D₄×C₁₉	1	3	yes	—
102	#102	—	2	6	no	—
103	C₄×C₈	C₂³×C₁₉	1	3	yes	—
104	C₁₉	Q₈×D₁₆	1	1	yes	characteristic, direct product factor, Sylow
105	C₂×Q₈	—	4	6	no	—
106	C₂×Q₈	—	8	8	no	—
107	C₂×Q₈	D₈×C₁₉	1	1	yes	characteristic
108	C₂×D₄	—	2	2	no	—
109	C₂²×C₄	—	4	6	no	—
110	D₈	—	2	2	no	—
111	D₈	C₂×Q₈×C₁₉	1	4	yes	complemented
112	C₂×C₈	C₂×D₄×C₁₉	1	3	yes	—
113	C₂×C₈	C₂⁴×C₁₉	1	1	yes	characteristic
114	C₄⋊C₄	—	2	6	no	—
115	C₄⋊C₄	—	4	6	no	—
116	C₄⋊C₄	D₈×C₁₉	1	3	yes	—
117	C₂²⋊C₄	—	4	6	no	—
118	C₄²	C₂×D₄×C₁₉	1	3	yes	—
119	C₁₆	—	2	3	no	—
120	C₁₆	C₂×Q₈×C₁₉	1	2	yes	complemented
121	C₂³	—	4	2	no	—
122	Q₈	—	8	12	no	—
123	Q₈	C₁₉×D₁₆	1	4	yes	direct product factor
124	D₄	—	2	4	no	—
125	D₄	—	4	2	no	—
126	C₂×C₄	—	4	6	no	—
127	C₂×C₄	—	8	12	no	—
128	C₂×C₄	C₂×D₈×C₁₉	1	3	yes	—
129	C₂×C₄	C₂×D₈×C₁₉	1	3	yes	—
130	C₂×C₄	C₂²×D₄×C₁₉	1	1	yes	characteristic
131	C₈	—	2	3	no	—
132	C₈	C₂²×Q₈×C₁₉	1	1	yes	characteristic
133	C₈	C₁₉×2₋¹⁺⁴	1	1	yes	characteristic
134	C₂²	—	4	4	no	—
135	C₂²	—	8	2	no	—
136	C₂²	—	8	2	no	—
137	C₂²	C₂²×D₈×C₁₉	1	1	yes	characteristic
138	C₄	—	2	3	no	—
139	C₄	—	8	6	no	—
140	C₄	C₂×C₁₉×D₁₆	1	6	yes	—
141	C₄	D₄×Q₈×C₁₉	1	1	yes	characteristic
142	C₄	C₁₉×#578	1	1	yes	characteristic
143	C₂	—	8	4	no	—
144	C₂	Q₈×D₈×C₁₉	1	1	yes	characteristic
145	C₂	C₂²×C₁₉×D₁₆	1	1	yes	characteristic
146	C₂	C₁₉×#3417	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

Q₈×D₈×C₁₉

C₂

yes

maximal, complemented

C₁₉×#2252

C₂

yes

maximal, complemented

C₁₉×#2224

C₂

yes

maximal, complemented

Q₈×C₁₆×C₁₉

C₂

yes

maximal, characteristic, complemented

C₄×C₁₉×D₁₆

C₂

yes

maximal

D₄×Q₈×C₁₉

—

C₂×C₁₉×D₁₆

C₂²

yes

characteristic

C₁₉×#500

C₂²

yes

—

C₁₉×#474

—

C₈×Q₈×C₁₉

C₂²

yes

characteristic

C₄×D₈×C₁₉

C₂²

yes

—

C₁₉×#366

C₂²

yes

—

C₁₉×#366

C₂²

yes

—

C₁₉×C₄⋊C₁₆

C₂²

yes

—

C₁₉×#357

C₂²

yes

—

C₄×C₁₆×C₁₉

C₂²

yes

—

C₂²×Q₈×C₁₉

—

C₂×D₈×C₁₉

C₂³

yes

—

C₁₉×C₄⋊Q₈

—

C₁₉×C₂²⋊Q₈

—

C₄×Q₈×C₁₉

D₄

yes

characteristic

C₄×D₄×C₁₉

—

C₁₉×D₁₆

Q₈

yes

direct product factor

C₂×C₁₆×C₁₉

C₂³

yes

—

C₂×C₁₆×C₁₉

C₂³

yes

characteristic

C₁₉×#107

—

C₁₉×#107

C₂³

yes

—

C₁₉×C₄⋊C₈

D₄

yes

—

C₁₉×#102

—

C₄×C₈×C₁₉

C₂³

yes

—

C₂×Q₈×C₁₉

—

C₂×Q₈×C₁₉

—

C₂×Q₈×C₁₉

D₈

yes

characteristic

C₂×D₄×C₁₉

—

C₂²×C₄×C₁₉

—

D₈×C₁₉

—

D₈×C₁₉

C₂×Q₈

yes

complemented

C₂×C₈×C₁₉

C₂×D₄

yes

—

C₂×C₈×C₁₉

C₂⁴

yes

characteristic

C₄⋊C₄×C₁₉

—

C₄⋊C₄×C₁₉

—

C₄⋊C₄×C₁₉

D₈

yes

—

C₂²⋊C₄×C₁₉

—

C₄²×C₁₉

C₂×D₄

yes

—

C₁₆×C₁₉

—

C₁₆×C₁₉

C₂×Q₈

yes

complemented

Q₈×D₁₆

C₁₉

yes

maximal, characteristic, direct product factor, Sylow

C₂³×C₁₉

—

Q₈×C₁₉

—

Q₈×C₁₉

D₁₆

yes

direct product factor

D₄×C₁₉

—

D₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

C₂×D₈

yes

—

C₂×C₄×C₁₉

C₂×D₈

yes

—

C₂×C₄×C₁₉

C₂²×D₄

yes

characteristic

C₈×C₁₉

—

C₈×C₁₉

C₂²×Q₈

yes

characteristic

C₈×C₁₉

2₋¹⁺⁴

yes

characteristic

Q₈×D₈

C₂×C₁₉

yes

complemented

#2252

C₂×C₁₉

yes

complemented

#2224

C₂×C₁₉

yes

complemented

Q₈×C₁₆

C₂×C₁₉

yes

characteristic, complemented

C₄×D₁₆

C₂×C₁₉

yes

—

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

C₂²×D₈

yes

characteristic

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

C₂×D₁₆

yes

—

C₄×C₁₉

D₄×Q₈

yes

characteristic

C₄×C₁₉

#578

yes

characteristic

D₄×Q₈

—

C₂×D₁₆

C₂²×C₁₉

yes

characteristic

#500

C₂²×C₁₉

yes

—

#474

—

C₈×Q₈

C₂²×C₁₉

yes

characteristic

C₄×D₈

C₂²×C₁₉

yes

—

#366

C₂²×C₁₉

yes

—

#366

C₂²×C₁₉

yes

—

C₄⋊C₁₆

C₂²×C₁₉

yes

—

#357

C₂²×C₁₉

yes

—

C₄×C₁₆

C₂²×C₁₉

yes

—

C₂×C₁₉

—

C₂×C₁₉

Q₈×D₈

yes

characteristic

C₂×C₁₉

C₂²×D₁₆

yes

characteristic

C₂×C₁₉

#3417

yes

characteristic

C₂²×Q₈

—

C₂×D₈

C₂³×C₁₉

yes

—

C₄⋊Q₈

—

C₂²⋊Q₈

—

C₄×Q₈

D₄×C₁₉

yes

characteristic

C₄×D₄

—

D₁₆

Q₈×C₁₉

yes

direct product factor

C₂×C₁₆

C₂³×C₁₉

yes

—

C₂×C₁₆

C₂³×C₁₉

yes

characteristic

#107

—

100

#107

C₂³×C₁₉

yes

—

101

C₄⋊C₈

D₄×C₁₉

yes

—

102

#102

—

103

C₄×C₈

C₂³×C₁₉

yes

—

104

C₁₉

Q₈×D₁₆

yes

characteristic, direct product factor, Sylow

105

C₂×Q₈

—

106

C₂×Q₈

—

107

C₂×Q₈

D₈×C₁₉

yes

characteristic

108

C₂×D₄

—

109

C₂²×C₄

—

110

D₈

—

111

D₈

C₂×Q₈×C₁₉

yes

complemented

112

C₂×C₈

C₂×D₄×C₁₉

yes

—

113

C₂×C₈

C₂⁴×C₁₉

yes

characteristic

114

C₄⋊C₄

—

115

C₄⋊C₄

—

116

C₄⋊C₄

D₈×C₁₉

yes

—

117

C₂²⋊C₄

—

118

C₄²

C₂×D₄×C₁₉

yes

—

119

C₁₆

—

120

C₁₆

C₂×Q₈×C₁₉

yes

complemented

121

C₂³

—

122

Q₈

—

123

Q₈

C₁₉×D₁₆

yes

direct product factor

124

D₄

—

125

D₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

C₂×D₈×C₁₉

yes

—

129

C₂×C₄

C₂×D₈×C₁₉

yes

—

130

C₂×C₄

C₂²×D₄×C₁₉

yes

characteristic

131

C₈

—

132

C₈

C₂²×Q₈×C₁₉

yes

characteristic

133

C₈

C₁₉×2₋¹⁺⁴

yes

characteristic

134

C₂²

—

135

C₂²

—

136

C₂²

—

137

C₂²

C₂²×D₈×C₁₉

yes

characteristic

138

C₄

—

139

C₄

—

140

C₄

C₂×C₁₉×D₁₆

yes

—

141

C₄

D₄×Q₈×C₁₉

yes

characteristic

142

C₄

C₁₉×#578

yes

characteristic

143

C₂

—

144

C₂

Q₈×D₈×C₁₉

yes

characteristic

145

C₂

C₂²×C₁₉×D₁₆

yes

characteristic

146

C₂

C₁₉×#3417

yes

characteristic

	subgroup	G/H
center, socle	C₂²×C₁₉	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₁₉
prime radical	C₂×C₁₉×D₁₆	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₉ ◁ C₄×Q₈×C₁₉ ◁ C₈×Q₈×C₁₉ ◁ G

subgroup

G/H

center, socle

C₂²×C₁₉

C₂²×D₈

commutator, Frattini, cosocle

C₂×C₈

C₂⁴×C₁₉

prime radical

C₂×C₁₉×D₁₆

C₂²

LCS

G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₁₉ ◁ C₄×Q₈×C₁₉ ◁ C₈×Q₈×C₁₉ ◁ G

N	H ≅ G/N	type
1. Q₈×D₈×C₁₉	C₂	semidirect
2. C₁₉×#2252	C₂	semidirect
3. C₁₉×#2224	C₂	semidirect
4. Q₈×C₁₆×C₁₉	C₂	semidirect
23. C₁₉×D₁₆	Q₈	direct (N × H)
37. D₈×C₁₉	C₂×Q₈	semidirect
46. C₁₆×C₁₉	C₂×Q₈	semidirect
47. Q₈×D₁₆	C₁₉	direct (N × H)
50. Q₈×C₁₉	D₁₆	direct (N × H)
61. Q₈×D₈	C₂×C₁₉	semidirect
62. #2252	C₂×C₁₉	semidirect
63. #2224	C₂×C₁₉	semidirect
64. Q₈×C₁₆	C₂×C₁₉	semidirect
96. D₁₆	Q₈×C₁₉	direct (N × H)
104. C₁₉	Q₈×D₁₆	direct (N × H)
111. D₈	C₂×Q₈×C₁₉	semidirect
120. C₁₆	C₂×Q₈×C₁₉	semidirect
123. Q₈	C₁₉×D₁₆	direct (N × H)

H ≅ G/N

type

1. Q₈×D₈×C₁₉

C₂

semidirect

2. C₁₉×#2252

C₂

semidirect

3. C₁₉×#2224

C₂

semidirect

4. Q₈×C₁₆×C₁₉

C₂

semidirect

23. C₁₉×D₁₆

Q₈

direct (N × H)

37. D₈×C₁₉

C₂×Q₈

semidirect

46. C₁₆×C₁₉

C₂×Q₈

semidirect

47. Q₈×D₁₆

C₁₉

direct (N × H)

50. Q₈×C₁₉

D₁₆

direct (N × H)

61. Q₈×D₈

C₂×C₁₉

semidirect

62. #2252

C₂×C₁₉

semidirect

63. #2224

C₂×C₁₉

semidirect

64. Q₈×C₁₆

C₂×C₁₉

semidirect

96. D₁₆

Q₈×C₁₉

direct (N × H)

104. C₁₉

Q₈×D₁₆

direct (N × H)

111. D₈

C₂×Q₈×C₁₉

semidirect

120. C₁₆

C₂×Q₈×C₁₉

semidirect

123. Q₈

C₁₉×D₁₆

direct (N × H)

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

subgroup	index
Q₈×D₈×C₁₉	2
C₁₉×#2252	2
C₁₉×#2224	2
Q₈×C₁₆×C₁₉	2
C₄×C₁₉×D₁₆	2
Q₈×D₁₆	19

subgroup

index

Q₈×D₈×C₁₉

C₁₉×#2252

C₁₉×#2224

Q₈×C₁₆×C₁₉

C₄×C₁₉×D₁₆

Q₈×D₁₆

Subgroups — Q₈×C₁₉×D₁₆ #18039904

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q8×C19×D16 #18039904

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×C₁₉×D₁₆ #18039904