Group #19147456 — subgroups

Subgroups — Q₈×#10621873 #19147456

Order 5504.

370 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 149 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×Dic₈₆	C₂	1	1	yes	maximal, characteristic, complemented
2	Q₈×D₁₇₂	C₂	1	1	yes	maximal, characteristic
3	#14583297	C₂	1	3	yes	maximal, complemented
4	#14583190	C₂	1	3	yes	maximal
5	#14583185	C₂	1	3	yes	maximal, complemented
6	C₄×#10621873	C₂	1	3	yes	maximal
7	C₈×Q₈×C₄₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×Q₈×D₄₃	—	2	1	no	—
9	C₂×#10621873	C₂²	1	1	yes	characteristic
10	#13063325	C₂²	1	3	yes	—
11	#13063325	C₂²	1	3	yes	—
12	#13063324	—	2	3	no	—
13	#13063305	—	2	3	no	—
14	Q₈×Dic₄₃	—	2	1	no	—
15	C₄×Dic₈₆	C₂²	1	3	yes	—
16	C₄×D₁₇₂	C₂²	1	3	yes	—
17	#13063257	C₂²	1	3	yes	—
18	#13063257	C₂²	1	6	yes	—
19	#13063250	C₂²	1	3	yes	—
20	#13063247	C₂²	1	3	yes	—
21	C₄×Q₈×C₄₃	C₂²	1	1	yes	characteristic
22	C₄⋊C₈×C₄₃	C₂²	1	3	yes	—
23	C₄×C₈×C₄₃	C₂²	1	3	yes	—
24	C₂×Q₈×C₄₃	D₄	1	1	yes	characteristic
25	Q₈×D₄₃	—	4	4	no	—
26	C₂×D₁₇₂	C₂³	1	1	yes	characteristic
27	C₂×C₄×D₄₃	—	2	3	no	—
28	C₂×Dic₈₆	—	2	3	no	—
29	C₂×Dic₈₆	—	2	3	no	—
30	C₂×Dic₈₆	C₂³	1	1	yes	characteristic
31	C₂×C₈×C₄₃	C₂³	1	3	yes	—
32	C₂×C₈×C₄₃	C₂³	1	1	yes	characteristic
33	C₄⋊C₄×C₄₃	D₄	1	3	yes	—
34	C₄²×C₄₃	C₂³	1	3	yes	—
35	(C₂×D₄₃)⋊C₄	—	2	3	no	—
36	C₄⋊Dic₄₃	—	2	3	no	—
37	C₄⋊Dic₄₃	—	2	3	no	—
38	C₄⋊Dic₄₃	C₂³	1	3	yes	—
39	C₄⋊Dic₄₃	C₂³	1	3	yes	—
40	Dic₄₃⋊C₄	—	2	3	no	—
41	C₄×Dic₄₃	—	2	3	no	—
42	#10621873	Q₈	1	4	yes	direct product factor
43	C₂²×D₄₃	—	2	1	no	—
44	Q₈×C₄₃	SD₁₆	1	4	yes	complemented
45	C₂×C₄×C₄₃	C₂×D₄	1	3	yes	—
46	C₂×C₄×C₄₃	C₂×D₄	1	3	yes	—
47	C₂×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
48	C₂×Dic₄₃	—	2	1	no	—
49	C₂×Dic₄₃	—	2	3	no	—
50	C₂×Dic₄₃	—	2	3	no	—
51	D₁₇₂	—	2	1	no	—
52	D₁₇₂	C₂×Q₈	1	2	yes	—
53	C₄×D₄₃	—	4	6	no	—
54	Dic₈₆	—	2	1	no	—
55	Dic₈₆	—	4	6	no	—
56	Dic₈₆	—	8	3	no	—
57	Dic₈₆	C₂×Q₈	1	2	yes	complemented
58	C₈×C₄₃	—	2	3	no	—
59	C₈×C₄₃	C₂×Q₈	1	2	yes	complemented
60	C₂²×C₄₃	C₂²×D₄	1	1	yes	characteristic
61	C₂×D₄₃	—	2	2	no	—
62	C₂×D₄₃	—	4	1	no	—
63	C₂×D₄₃	—	4	1	no	—
64	C₄×C₄₃	—	2	3	no	—
65	C₄×C₄₃	C₂×SD₁₆	1	6	yes	—
66	C₄×C₄₃	C₂²×Q₈	1	1	yes	characteristic
67	C₄×C₄₃	2₋¹⁺⁴	1	1	yes	characteristic
68	Dic₄₃	—	2	2	no	—
69	Dic₄₃	—	4	3	no	—
70	Dic₄₃	—	4	3	no	—
71	Q₈×SD₁₆	—	43	1	no	maximal, Sylow
72	C₂×C₄₃	D₄×Q₈	1	1	yes	characteristic
73	C₂×C₄₃	C₂²×SD₁₆	1	1	yes	characteristic
74	C₂×C₄₃	#577	1	1	yes	characteristic
75	D₄₃	—	4	2	no	—
76	Q₈²	—	43	1	no	—
77	D₄×Q₈	—	43	1	no	—
78	#498	—	43	3	no	—
79	#476	—	43	3	no	—
80	#475	—	43	3	no	—
81	C₈×Q₈	D₄₃	1	1	yes	characteristic, complemented
82	C₄×SD₁₆	—	43	3	no	—
83	C₄₃	Q₈×SD₁₆	1	1	yes	characteristic, complemented, Sylow
84	C₂²×Q₈	—	86	1	no	—
85	C₂×SD₁₆	—	43	1	no	—
86	C₄⋊Q₈	—	43	3	no	—
87	C₄⋊Q₈	—	43	3	no	—
88	C₄⋊Q₈	—	86	3	no	—
89	C₂²⋊Q₈	—	86	3	no	—
90	C₄×Q₈	—	43	3	no	—
91	C₄×Q₈	—	86	1	no	—
92	C₄×Q₈	C₂×D₄₃	1	1	yes	characteristic
93	C₄×D₄	—	43	3	no	—
94	#106	—	43	3	no	—
95	#106	—	43	6	no	—
96	C₄⋊C₈	C₂×D₄₃	1	3	yes	—
97	#103	—	43	3	no	—
98	#102	—	43	3	no	—
99	C₄×C₈	C₂×D₄₃	1	3	yes	—
100	C₂×Q₈	—	43	1	no	—
101	C₂×Q₈	—	86	3	no	—
102	C₂×Q₈	—	86	3	no	—
103	C₂×Q₈	—	172	4	no	—
104	C₂×Q₈	D₁₇₂	1	1	yes	characteristic
105	C₂×D₄	—	43	1	no	—
106	C₂²×C₄	—	86	3	no	—
107	SD₁₆	—	43	4	no	—
108	C₂×C₈	C₂²×D₄₃	1	3	yes	—
109	C₂×C₈	C₂²×D₄₃	1	1	yes	characteristic
110	C₄⋊C₄	—	43	3	no	—
111	C₄⋊C₄	—	43	3	no	—
112	C₄⋊C₄	—	86	3	no	—
113	C₄⋊C₄	—	86	3	no	—
114	C₄⋊C₄	—	86	3	no	—
115	C₄⋊C₄	D₁₇₂	1	3	yes	—
116	C₂²⋊C₄	—	86	3	no	—
117	C₄²	—	86	3	no	—
118	C₄²	C₂²×D₄₃	1	3	yes	—
119	C₂³	—	86	1	no	—
120	Q₈	—	43	2	no	—
121	Q₈	—	86	1	no	—
122	Q₈	—	344	3	no	—
123	Q₈	—	172	6	no	—
124	Q₈	#10621873	1	4	yes	direct product factor
125	D₄	—	43	2	no	—
126	D₄	—	86	1	no	—
127	C₂×C₄	—	86	1	no	—
128	C₂×C₄	—	86	3	no	—
129	C₂×C₄	—	86	3	no	—
130	C₂×C₄	—	172	6	no	—
131	C₂×C₄	C₂×D₁₇₂	1	3	yes	—
132	C₂×C₄	C₂×D₁₇₂	1	3	yes	—
133	C₂×C₄	C₂³×D₄₃	1	1	yes	characteristic
134	C₈	—	2	3	no	—
135	C₈	Q₈×D₄₃	1	2	yes	complemented
136	C₂²	—	86	2	no	—
137	C₂²	—	172	1	no	—
138	C₂²	—	172	1	no	—
139	C₂²	C₂²×D₁₇₂	1	1	yes	characteristic
140	C₄	—	2	3	no	—
141	C₄	—	86	2	no	—
142	C₄	—	172	3	no	—
143	C₄	—	172	3	no	—
144	C₄	C₂×#10621873	1	6	yes	—
145	C₄	C₂×Q₈×D₄₃	1	1	yes	characteristic
146	C₄	#13063372	1	1	yes	characteristic
147	C₂	—	172	2	no	—
148	C₂	Q₈×D₁₇₂	1	1	yes	characteristic
149	C₂	C₂²×#10621873	1	1	yes	characteristic
150	C₂	#14583711	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

Q₈×Dic₈₆

C₂

yes

maximal, characteristic, complemented

Q₈×D₁₇₂

C₂

yes

maximal, characteristic

#14583297

C₂

yes

maximal, complemented

#14583190

C₂

yes

maximal

#14583185

C₂

yes

maximal, complemented

C₄×#10621873

C₂

yes

maximal

C₈×Q₈×C₄₃

C₂

yes

maximal, characteristic, complemented

C₂×Q₈×D₄₃

—

C₂×#10621873

C₂²

yes

characteristic

#13063325

C₂²

yes

—

#13063325

C₂²

yes

—

#13063324

—

#13063305

—

Q₈×Dic₄₃

—

C₄×Dic₈₆

C₂²

yes

—

C₄×D₁₇₂

C₂²

yes

—

#13063257

C₂²

yes

—

#13063257

C₂²

yes

—

#13063250

C₂²

yes

—

#13063247

C₂²

yes

—

C₄×Q₈×C₄₃

C₂²

yes

characteristic

C₄⋊C₈×C₄₃

C₂²

yes

—

C₄×C₈×C₄₃

C₂²

yes

—

C₂×Q₈×C₄₃

D₄

yes

characteristic

Q₈×D₄₃

—

C₂×D₁₇₂

C₂³

yes

characteristic

C₂×C₄×D₄₃

—

C₂×Dic₈₆

—

C₂×Dic₈₆

—

C₂×Dic₈₆

C₂³

yes

characteristic

C₂×C₈×C₄₃

C₂³

yes

—

C₂×C₈×C₄₃

C₂³

yes

characteristic

C₄⋊C₄×C₄₃

D₄

yes

—

C₄²×C₄₃

C₂³

yes

—

(C₂×D₄₃)⋊C₄

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

C₂³

yes

—

C₄⋊Dic₄₃

C₂³

yes

—

Dic₄₃⋊C₄

—

C₄×Dic₄₃

—

#10621873

Q₈

yes

direct product factor

C₂²×D₄₃

—

Q₈×C₄₃

SD₁₆

yes

complemented

C₂×C₄×C₄₃

C₂×D₄

yes

—

C₂×C₄×C₄₃

C₂×D₄

yes

—

C₂×C₄×C₄₃

C₂⁴

yes

characteristic

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

D₁₇₂

—

D₁₇₂

C₂×Q₈

yes

—

C₄×D₄₃

—

Dic₈₆

—

Dic₈₆

—

Dic₈₆

—

Dic₈₆

C₂×Q₈

yes

complemented

C₈×C₄₃

—

C₈×C₄₃

C₂×Q₈

yes

complemented

C₂²×C₄₃

C₂²×D₄

yes

characteristic

C₂×D₄₃

—

C₂×D₄₃

—

C₂×D₄₃

—

C₄×C₄₃

—

C₄×C₄₃

C₂×SD₁₆

yes

—

C₄×C₄₃

C₂²×Q₈

yes

characteristic

C₄×C₄₃

2₋¹⁺⁴

yes

characteristic

Dic₄₃

—

Dic₄₃

—

Dic₄₃

—

Q₈×SD₁₆

—

maximal, Sylow

C₂×C₄₃

D₄×Q₈

yes

characteristic

C₂×C₄₃

C₂²×SD₁₆

yes

characteristic

C₂×C₄₃

#577

yes

characteristic

D₄₃

—

Q₈²

—

D₄×Q₈

—

#498

—

#476

—

#475

—

C₈×Q₈

D₄₃

yes

characteristic, complemented

C₄×SD₁₆

—

C₄₃

Q₈×SD₁₆

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₂×SD₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₂²⋊Q₈

—

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

C₂×D₄₃

yes

characteristic

C₄×D₄

—

#106

—

#106

—

C₄⋊C₈

C₂×D₄₃

yes

—

#103

—

#102

—

C₄×C₈

C₂×D₄₃

yes

—

100

C₂×Q₈

—

101

C₂×Q₈

—

102

C₂×Q₈

—

103

C₂×Q₈

—

172

—

104

C₂×Q₈

D₁₇₂

yes

characteristic

105

C₂×D₄

—

106

C₂²×C₄

—

107

SD₁₆

—

108

C₂×C₈

C₂²×D₄₃

yes

—

109

C₂×C₈

C₂²×D₄₃

yes

characteristic

110

C₄⋊C₄

—

111

C₄⋊C₄

—

112

C₄⋊C₄

—

113

C₄⋊C₄

—

114

C₄⋊C₄

—

115

C₄⋊C₄

D₁₇₂

yes

—

116

C₂²⋊C₄

—

117

C₄²

—

118

C₄²

C₂²×D₄₃

yes

—

119

C₂³

—

120

Q₈

—

121

Q₈

—

122

Q₈

—

344

—

123

Q₈

—

172

—

124

Q₈

#10621873

yes

direct product factor

125

D₄

—

126

D₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

172

—

131

C₂×C₄

C₂×D₁₇₂

yes

—

132

C₂×C₄

C₂×D₁₇₂

yes

—

133

C₂×C₄

C₂³×D₄₃

yes

characteristic

134

C₈

—

135

C₈

Q₈×D₄₃

yes

complemented

136

C₂²

—

137

C₂²

—

172

—

138

C₂²

—

172

—

139

C₂²

C₂²×D₁₇₂

yes

characteristic

140

C₄

—

141

C₄

—

142

C₄

—

172

—

143

C₄

—

172

—

144

C₄

C₂×#10621873

yes

—

145

C₄

C₂×Q₈×D₄₃

yes

characteristic

146

C₄

#13063372

yes

characteristic

147

C₂

—

172

—

148

C₂

Q₈×D₁₇₂

yes

characteristic

149

C₂

C₂²×#10621873

yes

characteristic

150

C₂

#14583711

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²×D₁₇₂
commutator	C₂×C₄×C₄₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₄₃
Fitting	C₈×Q₈×C₄₃	C₂
socle	C₂²×C₄₃	C₂²×D₄
hypercenter	C₈×Q₈	D₄₃
hypocenter	C₄₃	Q₈×SD₁₆
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ C₈×Q₈

subgroup

G/H

center

C₂²

C₂²×D₁₇₂

commutator

C₂×C₄×C₄₃

C₂⁴

Frattini

C₂×C₄

C₂³×D₄₃

Fitting

C₈×Q₈×C₄₃

C₂

socle

C₂²×C₄₃

C₂²×D₄

hypercenter

C₈×Q₈

D₄₃

hypocenter

C₄₃

Q₈×SD₁₆

LCS

G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂² ◁ C₄×Q₈ ◁ C₈×Q₈

N	H ≅ G/N	type
1. Q₈×Dic₈₆	C₂	semidirect
3. #14583297	C₂	semidirect
5. #14583185	C₂	semidirect
7. C₈×Q₈×C₄₃	C₂	semidirect
42. #10621873	Q₈	direct (N × H)
44. Q₈×C₄₃	SD₁₆	semidirect
57. Dic₈₆	C₂×Q₈	semidirect
59. C₈×C₄₃	C₂×Q₈	semidirect
81. C₈×Q₈	D₄₃	semidirect
83. C₄₃	Q₈×SD₁₆	semidirect
124. Q₈	#10621873	direct (N × H)
135. C₈	Q₈×D₄₃	semidirect

H ≅ G/N

type

1. Q₈×Dic₈₆

C₂

semidirect

3. #14583297

C₂

semidirect

5. #14583185

C₂

semidirect

7. C₈×Q₈×C₄₃

C₂

semidirect

42. #10621873

Q₈

direct (N × H)

44. Q₈×C₄₃

SD₁₆

semidirect

57. Dic₈₆

C₂×Q₈

semidirect

59. C₈×C₄₃

C₂×Q₈

semidirect

81. C₈×Q₈

D₄₃

semidirect

83. C₄₃

Q₈×SD₁₆

semidirect

124. Q₈

#10621873

direct (N × H)

135. C₈

Q₈×D₄₃

semidirect

Subgroups — Q₈×#10621873 #19147456

Conjugacy classes of subgroups

Complements

Subgroups — Q8×#10621873 #19147456

Conjugacy classes of subgroups

Complements

Subgroups — Q₈×#10621873 #19147456