Group #21436947 — subgroups

Subgroups — Q₈×#11745669 #21436947

Order 7552.

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

normal subgroups only

This group has 404 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×D₂₃₆	C₂	1	1	yes	maximal, characteristic, complemented
2	#16837805	C₂	1	3	yes	maximal, complemented
3	#16837682	C₂	1	3	yes	maximal, complemented
4	#16837679	C₂	1	3	yes	maximal, complemented
5	Q₈×C₅₉⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄×#11745669	C₂	1	3	yes	maximal
7	D₄×Q₈×C₅₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×Q₈×D₅₉	—	2	1	no	—
9	C₂×#11745669	C₂²	1	1	yes	characteristic
10	#14273727	C₂²	1	3	yes	—
11	#14273707	—	2	3	no	—
12	C₄×D₂₃₆	C₂²	1	3	yes	—
13	#14273660	C₂²	1	6	yes	—
14	#14273660	C₂²	1	3	yes	—
15	C₄⋊(C₅₉⋊C₈)	C₂²	1	3	yes	—
16	#14273648	C₂²	1	3	yes	—
17	#14273647	C₂²	1	3	yes	—
18	C₄×C₅₉⋊C₈	C₂²	1	3	yes	—
19	C₂²×Q₈×C₅₉	—	2	1	no	—
20	C₄⋊Q₈×C₅₉	C₂²	1	3	yes	—
21	C₂²⋊Q₈×C₅₉	—	2	3	no	—
22	C₄×Q₈×C₅₉	C₂²	1	1	yes	characteristic
23	C₄×D₄×C₅₉	C₂²	1	3	yes	—
24	C₂×Q₈×C₅₉	—	2	3	no	—
25	C₂×Q₈×C₅₉	—	4	4	no	—
26	C₂×Q₈×C₅₉	D₄	1	1	yes	characteristic
27	C₂×D₄×C₅₉	C₂³	1	1	yes	characteristic
28	C₂²×C₄×C₅₉	—	2	3	no	—
29	Q₈×D₅₉	—	4	4	no	—
30	C₂×D₂₃₆	C₂³	1	1	yes	characteristic
31	C₂×C₄×D₅₉	—	2	3	no	—
32	C₂×Dic₁₁₈	—	2	3	no	—
33	C₄⋊C₄×C₅₉	—	2	3	no	—
34	C₄⋊C₄×C₅₉	D₄	1	3	yes	—
35	C₄⋊C₄×C₅₉	C₂³	1	3	yes	—
36	C₂²⋊C₄×C₅₉	—	2	3	no	—
37	C₄²×C₅₉	C₂³	1	3	yes	—
38	#11745669	Q₈	1	4	yes	direct product factor
39	(C₂×D₅₉)⋊C₄	—	2	3	no	—
40	C₄⋊Dic₅₉	—	2	3	no	—
41	C₄⋊Dic₅₉	C₂³	1	3	yes	—
42	C₂×C₅₉⋊C₈	C₂³	1	3	yes	—
43	C₂×C₅₉⋊C₈	C₂³	1	1	yes	characteristic
44	C₂³×C₅₉	—	2	1	no	—
45	C₂²×D₅₉	—	2	1	no	—
46	Q₈×C₅₉	—	4	6	no	—
47	Q₈×C₅₉	D₈	1	4	yes	complemented
48	D₄×C₅₉	—	2	1	no	—
49	D₄×C₅₉	C₂×Q₈	1	2	yes	complemented
50	C₂×C₄×C₅₉	—	2	3	no	—
51	C₂×C₄×C₅₉	—	4	6	no	—
52	C₂×C₄×C₅₉	C₂×D₄	1	3	yes	—
53	C₂×C₄×C₅₉	C₂×D₄	1	3	yes	—
54	C₂×C₄×C₅₉	C₂⁴	1	1	yes	characteristic
55	C₂×Dic₅₉	—	2	3	no	—
56	D₂₃₆	—	2	1	no	—
57	D₂₃₆	C₂×Q₈	1	2	yes	complemented
58	C₄×D₅₉	—	4	6	no	—
59	Dic₁₁₈	—	4	6	no	—
60	C₅₉⋊C₈	—	2	3	no	—
61	C₅₉⋊C₈	C₂×Q₈	1	2	yes	complemented
62	C₂²×C₅₉	—	2	2	no	—
63	C₂²×C₅₉	—	4	1	no	—
64	C₂²×C₅₉	—	4	1	no	—
65	C₂²×C₅₉	C₂²×D₄	1	1	yes	characteristic
66	C₂×D₅₉	—	2	2	no	—
67	C₂×D₅₉	—	4	1	no	—
68	C₂×D₅₉	—	4	1	no	—
69	C₄×C₅₉	—	2	3	no	—
70	C₄×C₅₉	—	4	3	no	—
71	C₄×C₅₉	C₂×D₈	1	6	yes	—
72	C₄×C₅₉	C₂²×Q₈	1	1	yes	characteristic
73	C₄×C₅₉	2₋¹⁺⁴	1	1	yes	characteristic
74	Dic₅₉	—	4	3	no	—
75	Q₈×D₈	—	59	1	no	maximal, Sylow
76	C₂×C₅₉	—	4	2	no	—
77	C₂×C₅₉	D₄×Q₈	1	1	yes	characteristic
78	C₂×C₅₉	C₂²×D₈	1	1	yes	characteristic
79	C₂×C₅₉	#578	1	1	yes	characteristic
80	D₅₉	—	4	2	no	—
81	D₄×Q₈	—	59	1	no	—
82	D₄×Q₈	D₅₉	1	1	yes	characteristic, complemented
83	#500	—	59	3	no	—
84	#474	—	59	3	no	—
85	#474	—	59	3	no	—
86	C₈×Q₈	—	59	1	no	—
87	C₄×D₈	—	59	3	no	—
88	C₅₉	Q₈×D₈	1	1	yes	characteristic, complemented, Sylow
89	C₂²×Q₈	—	2	1	no	—
90	C₂²×Q₈	—	118	1	no	—
91	C₂×D₈	—	59	1	no	—
92	C₄⋊Q₈	—	59	3	no	—
93	C₄⋊Q₈	C₂×D₅₉	1	3	yes	—
94	C₂²⋊Q₈	—	2	3	no	—
95	C₂²⋊Q₈	—	118	3	no	—
96	C₄×Q₈	C₂×D₅₉	1	1	yes	characteristic
97	C₄×D₄	—	59	3	no	—
98	C₄×D₄	C₂×D₅₉	1	3	yes	—
99	#107	—	59	3	no	—
100	#107	—	59	6	no	—
101	C₄⋊C₈	—	59	3	no	—
102	#102	—	59	3	no	—
103	#102	—	59	3	no	—
104	C₄×C₈	—	59	3	no	—
105	C₂×Q₈	—	2	3	no	—
106	C₂×Q₈	—	4	4	no	—
107	C₂×Q₈	—	118	3	no	—
108	C₂×Q₈	—	236	4	no	—
109	C₂×Q₈	C₂²⋊D₅₉	1	1	yes	characteristic
110	C₂×D₄	—	59	1	no	—
111	C₂×D₄	C₂²×D₅₉	1	1	yes	characteristic
112	C₂²×C₄	—	2	3	no	—
113	C₂²×C₄	—	118	3	no	—
114	D₈	—	59	4	no	—
115	C₂×C₈	—	59	1	no	—
116	C₂×C₈	—	59	3	no	—
117	C₄⋊C₄	—	2	3	no	—
118	C₄⋊C₄	—	59	3	no	—
119	C₄⋊C₄	—	118	3	no	—
120	C₄⋊C₄	C₂²⋊D₅₉	1	3	yes	—
121	C₄⋊C₄	C₂²×D₅₉	1	3	yes	—
122	C₂²⋊C₄	—	2	3	no	—
123	C₂²⋊C₄	—	118	3	no	—
124	C₄²	C₂²×D₅₉	1	3	yes	—
125	C₂³	—	2	1	no	—
126	C₂³	—	118	1	no	—
127	Q₈	—	4	6	no	—
128	Q₈	—	236	6	no	—
129	Q₈	#11745669	1	4	yes	direct product factor
130	D₄	—	2	1	no	—
131	D₄	—	59	2	no	—
132	D₄	—	118	1	no	—
133	D₄	Q₈×D₅₉	1	2	yes	complemented
134	C₂×C₄	—	2	3	no	—
135	C₂×C₄	—	4	6	no	—
136	C₂×C₄	—	118	3	no	—
137	C₂×C₄	—	236	6	no	—
138	C₂×C₄	C₂×C₂²⋊D₅₉	1	3	yes	—
139	C₂×C₄	C₂×C₂²⋊D₅₉	1	3	yes	—
140	C₂×C₄	C₂³×D₅₉	1	1	yes	characteristic
141	C₈	—	59	2	no	—
142	C₈	—	118	3	no	—
143	C₂²	—	2	2	no	—
144	C₂²	—	4	1	no	—
145	C₂²	—	4	1	no	—
146	C₂²	—	118	2	no	—
147	C₂²	—	236	1	no	—
148	C₂²	—	236	1	no	—
149	C₂²	C₂²×C₂²⋊D₅₉	1	1	yes	characteristic
150	C₄	—	2	3	no	—
151	C₄	—	4	3	no	—
152	C₄	—	236	3	no	—
153	C₄	C₂×#11745669	1	6	yes	—
154	C₄	C₂×Q₈×D₅₉	1	1	yes	characteristic
155	C₄	#14273775	1	1	yes	characteristic
156	C₂	—	4	2	no	—
157	C₂	—	236	2	no	—
158	C₂	Q₈×C₂²⋊D₅₉	1	1	yes	characteristic
159	C₂	C₂²×#11745669	1	1	yes	characteristic
160	C₂	#16838217	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

Q₈×D₂₃₆

C₂

yes

maximal, characteristic, complemented

#16837805

C₂

yes

maximal, complemented

#16837682

C₂

yes

maximal, complemented

#16837679

C₂

yes

maximal, complemented

Q₈×C₅₉⋊C₈

C₂

yes

maximal, characteristic, complemented

C₄×#11745669

C₂

yes

maximal

D₄×Q₈×C₅₉

C₂

yes

maximal, characteristic, complemented

C₂×Q₈×D₅₉

—

C₂×#11745669

C₂²

yes

characteristic

#14273727

C₂²

yes

—

#14273707

—

C₄×D₂₃₆

C₂²

yes

—

#14273660

C₂²

yes

—

#14273660

C₂²

yes

—

C₄⋊(C₅₉⋊C₈)

C₂²

yes

—

#14273648

C₂²

yes

—

#14273647

C₂²

yes

—

C₄×C₅₉⋊C₈

C₂²

yes

—

C₂²×Q₈×C₅₉

—

C₄⋊Q₈×C₅₉

C₂²

yes

—

C₂²⋊Q₈×C₅₉

—

C₄×Q₈×C₅₉

C₂²

yes

characteristic

C₄×D₄×C₅₉

C₂²

yes

—

C₂×Q₈×C₅₉

—

C₂×Q₈×C₅₉

—

C₂×Q₈×C₅₉

D₄

yes

characteristic

C₂×D₄×C₅₉

C₂³

yes

characteristic

C₂²×C₄×C₅₉

—

Q₈×D₅₉

—

C₂×D₂₃₆

C₂³

yes

characteristic

C₂×C₄×D₅₉

—

C₂×Dic₁₁₈

—

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

D₄

yes

—

C₄⋊C₄×C₅₉

C₂³

yes

—

C₂²⋊C₄×C₅₉

—

C₄²×C₅₉

C₂³

yes

—

#11745669

Q₈

yes

direct product factor

(C₂×D₅₉)⋊C₄

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

C₂³

yes

—

C₂×C₅₉⋊C₈

C₂³

yes

—

C₂×C₅₉⋊C₈

C₂³

yes

characteristic

C₂³×C₅₉

—

C₂²×D₅₉

—

Q₈×C₅₉

—

Q₈×C₅₉

D₈

yes

complemented

D₄×C₅₉

—

D₄×C₅₉

C₂×Q₈

yes

complemented

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₂×D₄

yes

—

C₂×C₄×C₅₉

C₂×D₄

yes

—

C₂×C₄×C₅₉

C₂⁴

yes

characteristic

C₂×Dic₅₉

—

D₂₃₆

—

D₂₃₆

C₂×Q₈

yes

complemented

C₄×D₅₉

—

Dic₁₁₈

—

C₅₉⋊C₈

—

C₅₉⋊C₈

C₂×Q₈

yes

complemented

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

C₂²×D₄

yes

characteristic

C₂×D₅₉

—

C₂×D₅₉

—

C₂×D₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

C₂×D₈

yes

—

C₄×C₅₉

C₂²×Q₈

yes

characteristic

C₄×C₅₉

2₋¹⁺⁴

yes

characteristic

Dic₅₉

—

Q₈×D₈

—

maximal, Sylow

C₂×C₅₉

—

C₂×C₅₉

D₄×Q₈

yes

characteristic

C₂×C₅₉

C₂²×D₈

yes

characteristic

C₂×C₅₉

#578

yes

characteristic

D₅₉

—

D₄×Q₈

—

D₄×Q₈

D₅₉

yes

characteristic, complemented

#500

—

#474

—

#474

—

C₈×Q₈

—

C₄×D₈

—

C₅₉

Q₈×D₈

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₂²×Q₈

—

118

—

C₂×D₈

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₅₉

yes

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

118

—

C₄×Q₈

C₂×D₅₉

yes

characteristic

C₄×D₄

—

C₄×D₄

C₂×D₅₉

yes

—

#107

—

100

#107

—

101

C₄⋊C₈

—

102

#102

—

103

#102

—

104

C₄×C₈

—

105

C₂×Q₈

—

106

C₂×Q₈

—

107

C₂×Q₈

—

118

—

108

C₂×Q₈

—

236

—

109

C₂×Q₈

C₂²⋊D₅₉

yes

characteristic

110

C₂×D₄

—

111

C₂×D₄

C₂²×D₅₉

yes

characteristic

112

C₂²×C₄

—

113

C₂²×C₄

—

118

—

114

D₈

—

115

C₂×C₈

—

116

C₂×C₈

—

117

C₄⋊C₄

—

118

C₄⋊C₄

—

119

C₄⋊C₄

—

118

—

120

C₄⋊C₄

C₂²⋊D₅₉

yes

—

121

C₄⋊C₄

C₂²×D₅₉

yes

—

122

C₂²⋊C₄

—

123

C₂²⋊C₄

—

118

—

124

C₄²

C₂²×D₅₉

yes

—

125

C₂³

—

126

C₂³

—

118

—

127

Q₈

—

128

Q₈

—

236

—

129

Q₈

#11745669

yes

direct product factor

130

D₄

—

131

D₄

—

132

D₄

—

118

—

133

D₄

Q₈×D₅₉

yes

complemented

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

118

—

137

C₂×C₄

—

236

—

138

C₂×C₄

C₂×C₂²⋊D₅₉

yes

—

139

C₂×C₄

C₂×C₂²⋊D₅₉

yes

—

140

C₂×C₄

C₂³×D₅₉

yes

characteristic

141

C₈

—

142

C₈

—

118

—

143

C₂²

—

144

C₂²

—

145

C₂²

—

146

C₂²

—

118

—

147

C₂²

—

236

—

148

C₂²

—

236

—

149

C₂²

C₂²×C₂²⋊D₅₉

yes

characteristic

150

C₄

—

151

C₄

—

152

C₄

—

236

—

153

C₄

C₂×#11745669

yes

—

154

C₄

C₂×Q₈×D₅₉

yes

characteristic

155

C₄

#14273775

yes

characteristic

156

C₂

—

157

C₂

—

236

—

158

C₂

Q₈×C₂²⋊D₅₉

yes

characteristic

159

C₂

C₂²×#11745669

yes

characteristic

160

C₂

#16838217

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₅₉
commutator	C₂×C₄×C₅₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₅₉
Fitting	D₄×Q₈×C₅₉	C₂
socle	C₂²×C₅₉	C₂²×D₄
hypercenter	D₄×Q₈	D₅₉
hypocenter	C₅₉	Q₈×D₈
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ D₄×Q₈

subgroup

G/H

center

C₂²

C₂²×C₂²⋊D₅₉

commutator

C₂×C₄×C₅₉

C₂⁴

Frattini

C₂×C₄

C₂³×D₅₉

Fitting

D₄×Q₈×C₅₉

C₂

socle

C₂²×C₅₉

C₂²×D₄

hypercenter

D₄×Q₈

D₅₉

hypocenter

C₅₉

Q₈×D₈

LCS

G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂² ◁ C₄×Q₈ ◁ D₄×Q₈

N	H ≅ G/N	type
1. Q₈×D₂₃₆	C₂	semidirect
2. #16837805	C₂	semidirect
3. #16837682	C₂	semidirect
4. #16837679	C₂	semidirect
5. Q₈×C₅₉⋊C₈	C₂	semidirect
7. D₄×Q₈×C₅₉	C₂	semidirect
38. #11745669	Q₈	direct (N × H)
47. Q₈×C₅₉	D₈	semidirect
49. D₄×C₅₉	C₂×Q₈	semidirect
57. D₂₃₆	C₂×Q₈	semidirect
61. C₅₉⋊C₈	C₂×Q₈	semidirect
82. D₄×Q₈	D₅₉	semidirect
88. C₅₉	Q₈×D₈	semidirect
129. Q₈	#11745669	direct (N × H)
133. D₄	Q₈×D₅₉	semidirect

H ≅ G/N

type

1. Q₈×D₂₃₆

C₂

semidirect

2. #16837805

C₂

semidirect

3. #16837682

C₂

semidirect

4. #16837679

C₂

semidirect

5. Q₈×C₅₉⋊C₈

C₂

semidirect

7. D₄×Q₈×C₅₉

C₂

semidirect

38. #11745669

Q₈

direct (N × H)

47. Q₈×C₅₉

D₈

semidirect

49. D₄×C₅₉

C₂×Q₈

semidirect

57. D₂₃₆

C₂×Q₈

semidirect

61. C₅₉⋊C₈

C₂×Q₈

semidirect

82. D₄×Q₈

D₅₉

semidirect

88. C₅₉

Q₈×D₈

semidirect

129. Q₈

#11745669

direct (N × H)

133. D₄

Q₈×D₅₉

semidirect

Subgroups — Q₈×#11745669 #21436947

Conjugacy classes of subgroups

Complements

Subgroups — Q8×#11745669 #21436947

Conjugacy classes of subgroups

Complements

Subgroups — Q₈×#11745669 #21436947