Group #21422936 — subgroups

Subgroups — #21422936

Order 7552.

166 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 166 conjugacy classes of subgroups — too many to draw. Only its 65 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×C₅₉⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄×#11745671	C₂	1	1	yes	maximal, characteristic
3	#16837479	C₂	1	1	yes	maximal, characteristic
4	#16837152	C₂	1	1	yes	maximal, characteristic, complemented
5	#16837131	C₂	1	1	yes	maximal, characteristic
6	#16837124	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₉×Q₈⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#11745671	C₄	1	1	yes	characteristic
9	C₂×#11745659	—	2	1	no	—
10	C₄×D₂₃₆	C₂²	1	1	yes	characteristic
11	#14273658	C₄	1	1	yes	characteristic
12	C₄⋊(C₅₉⋊C₈)	—	2	1	no	—
13	C₄⋊(C₅₉⋊C₈)	C₂²	1	1	yes	characteristic
14	#14273651	C₄	1	1	yes	characteristic
15	#14273647	C₄	1	1	yes	characteristic
16	(C₂×D₅₉)⋊C₈	—	2	1	no	—
17	C₄×C₅₉⋊C₈	—	2	1	no	—
18	C₄×C₅₉⋊C₈	C₂²	1	1	yes	characteristic
19	C₄×C₅₉⋊C₈	C₂²	1	1	yes	characteristic
20	C₄×Q₈×C₅₉	C₂²	1	1	yes	characteristic
21	C₄⋊C₈×C₅₉	C₂²	1	1	yes	characteristic
22	C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
23	C₂×Q₈×C₅₉	C₂×C₄	1	1	yes	characteristic
24	C₂×D₂₃₆	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄×D₅₉	—	2	1	no	—
26	C₂×C₈×C₅₉	—	2	1	no	—
27	C₂×C₈×C₅₉	—	2	1	no	—
28	C₄⋊C₄×C₅₉	—	2	1	no	—
29	C₄⋊C₄×C₅₉	C₂×C₄	1	1	yes	characteristic
30	C₄²×C₅₉	—	2	1	no	—
31	C₄²×C₅₉	C₂³	1	1	yes	characteristic
32	#11745671	—	2	2	no	—
33	(C₂×D₅₉)⋊C₄	—	2	1	no	—
34	C₄⋊Dic₅₉	C₂×C₄	1	1	yes	characteristic
35	C₂×C₅₉⋊C₈	—	2	1	no	—
36	C₂×C₅₉⋊C₈	C₂×C₄	1	1	yes	characteristic
37	C₂×C₅₉⋊C₈	C₂×C₄	1	1	yes	characteristic
38	C₂×C₅₉⋊C₈	D₄	1	1	yes	characteristic
39	C₂×C₅₉⋊C₈	D₄	1	1	yes	characteristic
40	#11745659	—	4	2	no	—
41	C₂²×D₅₉	—	2	1	no	—
42	Q₈×C₅₉	—	2	1	no	—
43	Q₈×C₅₉	M₁₆	1	2	yes	complemented
44	C₂×C₄×C₅₉	—	2	1	no	—
45	C₂×C₄×C₅₉	—	2	1	no	—
46	C₂×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
47	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
48	C₂×C₄×C₅₉	#41	1	1	yes	characteristic
49	C₂×Dic₅₉	—	2	1	no	—
50	D₂₃₆	—	2	1	no	—
51	D₂₃₆	—	2	1	no	—
52	C₄×D₅₉	—	4	1	no	—
53	C₄×D₅₉	—	4	1	no	—
54	C₈×C₅₉	—	4	1	no	—
55	C₈×C₅₉	—	4	1	no	—
56	C₅₉⋊C₈	—	2	1	no	—
57	C₅₉⋊C₈	—	2	1	no	—
58	C₅₉⋊C₈	—	2	1	no	—
59	C₅₉⋊C₈	—	4	1	no	—
60	C₅₉⋊C₈	SD₁₆	1	2	yes	complemented
61	C₂²×C₅₉	C₄×D₄	1	1	yes	characteristic
62	C₂×D₅₉	—	4	1	no	—
63	C₂×D₅₉	—	4	1	no	—
64	C₂×D₅₉	—	4	1	no	—
65	C₄×C₅₉	—	2	2	no	—
66	C₄×C₅₉	—	2	1	no	—
67	C₄×C₅₉	—	4	1	no	—
68	C₄×C₅₉	C₂×M₁₆	1	1	yes	characteristic
69	C₄×C₅₉	#131	1	1	yes	characteristic
70	C₄×C₅₉	C₂×SD₁₆	1	1	yes	characteristic
71	C₄×C₅₉	#135	1	1	yes	characteristic
72	Dic₅₉	—	4	1	no	—
73	#1583	—	59	1	no	maximal, Sylow
74	C₂×C₅₉	#435	1	1	yes	characteristic
75	C₂×C₅₉	C₄×SD₁₆	1	1	yes	characteristic
76	C₂×C₅₉	#444	1	1	yes	characteristic
77	D₅₉	—	8	1	no	—
78	C₈×Q₈	—	59	1	no	—
79	C₄×SD₁₆	—	59	1	no	—
80	#436	—	59	1	no	—
81	#334	—	59	1	no	—
82	Q₈⋊C₈	D₅₉	1	1	yes	characteristic, complemented
83	D₄⋊C₈	—	59	1	no	—
84	#322	—	59	1	no	—
85	C₅₉	#1583	1	1	yes	characteristic, complemented, Sylow
86	C₂×SD₁₆	—	59	1	no	—
87	C₂×M₁₆	—	118	1	no	—
88	C₄×Q₈	C₂×D₅₉	1	1	yes	characteristic
89	C₄×D₄	—	59	1	no	—
90	#106	—	59	1	no	—
91	C₄⋊C₈	—	59	1	no	—
92	C₄⋊C₈	—	118	1	no	—
93	C₄⋊C₈	C₂×D₅₉	1	1	yes	characteristic
94	#103	—	59	1	no	—
95	#102	—	59	1	no	—
96	C₂²⋊C₈	—	118	1	no	—
97	C₄×C₈	—	59	1	no	—
98	C₄×C₈	—	59	1	no	—
99	C₄×C₈	—	118	1	no	—
100	C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
101	C₂×Q₈	C₄×D₅₉	1	1	yes	characteristic
102	C₂×D₄	—	59	1	no	—
103	C₂²×C₄	—	118	1	no	—
104	SD₁₆	—	118	2	no	—
105	M₁₆	—	236	2	no	—
106	C₂×C₈	—	2	1	no	—
107	C₂×C₈	—	2	1	no	—
108	C₂×C₈	—	59	1	no	—
109	C₂×C₈	—	59	1	no	—
110	C₂×C₈	—	59	1	no	—
111	C₂×C₈	—	59	1	no	—
112	C₂×C₈	—	118	1	no	—
113	C₄⋊C₄	—	2	1	no	—
114	C₄⋊C₄	—	59	1	no	—
115	C₄⋊C₄	C₄×D₅₉	1	1	yes	characteristic
116	C₂²⋊C₄	—	118	1	no	—
117	C₄²	—	2	1	no	—
118	C₄²	C₂²×D₅₉	1	1	yes	characteristic
119	C₂³	—	118	1	no	—
120	Q₈	—	2	1	no	—
121	Q₈	#11745659	1	2	yes	complemented
122	D₄	—	118	1	no	—
123	D₄	—	118	1	no	—
124	C₂×C₄	—	2	1	no	—
125	C₂×C₄	—	2	1	no	—
126	C₂×C₄	—	118	1	no	—
127	C₂×C₄	—	236	1	no	—
128	C₂×C₄	—	236	1	no	—
129	C₂×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
130	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
131	C₂×C₄	#11745687	1	1	yes	characteristic
132	C₈	—	4	1	no	—
133	C₈	—	4	1	no	—
134	C₈	—	59	2	no	—
135	C₈	—	118	1	no	—
136	C₈	—	118	1	no	—
137	C₈	—	118	1	no	—
138	C₈	—	236	1	no	—
139	C₂²	—	236	1	no	—
140	C₂²	—	236	1	no	—
141	C₂²	—	236	1	no	—
142	C₂²	(C₂²⋊D₅₉)⋊C₄	1	1	yes	characteristic
143	C₄	—	2	2	no	—
144	C₄	—	2	1	no	—
145	C₄	—	4	1	no	—
146	C₄	—	236	1	no	—
147	C₄	C₂×#11745659	1	1	yes	characteristic
148	C₄	#14273735	1	1	yes	characteristic
149	C₄	SD₁₆×D₅₉	1	1	yes	characteristic
150	C₄	#14273750	1	1	yes	characteristic
151	C₂	—	472	1	no	—
152	C₂	#16837470	1	1	yes	characteristic
153	C₂	#16837487	1	1	yes	characteristic
154	C₂	#16837521	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

Q₈×C₅₉⋊C₈

C₂

yes

maximal, characteristic, complemented

C₄×#11745671

C₂

yes

maximal, characteristic

#16837479

C₂

yes

maximal, characteristic

#16837152

C₂

yes

maximal, characteristic, complemented

#16837131

C₂

yes

maximal, characteristic

#16837124

C₂

yes

maximal, characteristic, complemented

C₅₉×Q₈⋊C₈

C₂

yes

maximal, characteristic, complemented

C₂×#11745671

C₄

yes

characteristic

C₂×#11745659

—

C₄×D₂₃₆

C₂²

yes

characteristic

#14273658

C₄

yes

characteristic

C₄⋊(C₅₉⋊C₈)

—

C₄⋊(C₅₉⋊C₈)

C₂²

yes

characteristic

#14273651

C₄

yes

characteristic

#14273647

C₄

yes

characteristic

(C₂×D₅₉)⋊C₈

—

C₄×C₅₉⋊C₈

—

C₄×C₅₉⋊C₈

C₂²

yes

characteristic

C₄×C₅₉⋊C₈

C₂²

yes

characteristic

C₄×Q₈×C₅₉

C₂²

yes

characteristic

C₄⋊C₈×C₅₉

C₂²

yes

characteristic

C₄×C₈×C₅₉

C₂²

yes

characteristic

C₂×Q₈×C₅₉

C₂×C₄

yes

characteristic

C₂×D₂₃₆

C₂×C₄

yes

characteristic

C₂×C₄×D₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

—

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

C₂×C₄

yes

characteristic

C₄²×C₅₉

—

C₄²×C₅₉

C₂³

yes

characteristic

#11745671

—

(C₂×D₅₉)⋊C₄

—

C₄⋊Dic₅₉

C₂×C₄

yes

characteristic

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

C₂×C₄

yes

characteristic

C₂×C₅₉⋊C₈

C₂×C₄

yes

characteristic

C₂×C₅₉⋊C₈

D₄

yes

characteristic

C₂×C₅₉⋊C₈

D₄

yes

characteristic

#11745659

—

C₂²×D₅₉

—

Q₈×C₅₉

—

Q₈×C₅₉

M₁₆

yes

complemented

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

#41

yes

characteristic

C₂×Dic₅₉

—

D₂₃₆

—

D₂₃₆

—

C₄×D₅₉

—

C₄×D₅₉

—

C₈×C₅₉

—

C₈×C₅₉

—

C₅₉⋊C₈

—

C₅₉⋊C₈

—

C₅₉⋊C₈

—

C₅₉⋊C₈

—

C₅₉⋊C₈

SD₁₆

yes

complemented

C₂²×C₅₉

C₄×D₄

yes

characteristic

C₂×D₅₉

—

C₂×D₅₉

—

C₂×D₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

C₂×M₁₆

yes

characteristic

C₄×C₅₉

#131

yes

characteristic

C₄×C₅₉

C₂×SD₁₆

yes

characteristic

C₄×C₅₉

#135

yes

characteristic

Dic₅₉

—

—

C₂×C₅₉

yes

C₂×C₅₉

C₄×SD₁₆

yes

characteristic

C₂×C₅₉

#444

yes

characteristic

D₅₉

—

C₈×Q₈

—

C₄×SD₁₆

—

#436

—

#334

—

Q₈⋊C₈

D₅₉

yes

characteristic, complemented

D₄⋊C₈

—

#322

—

C₅₉

#1583

yes

characteristic, complemented, Sylow

C₂×SD₁₆

—

C₂×M₁₆

—

118

—

C₄×Q₈

C₂×D₅₉

yes

characteristic

C₄×D₄

—

#106

—

C₄⋊C₈

—

C₄⋊C₈

—

118

—

C₄⋊C₈

C₂×D₅₉

yes

characteristic

#103

—

#102

—

C₂²⋊C₈

—

118

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

118

—

100

C₄×C₈

C₂×D₅₉

yes

characteristic

101

C₂×Q₈

C₄×D₅₉

yes

characteristic

102

C₂×D₄

—

103

C₂²×C₄

—

118

—

104

SD₁₆

—

118

—

105

M₁₆

—

236

—

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂×C₈

—

109

C₂×C₈

—

110

C₂×C₈

—

111

C₂×C₈

—

112

C₂×C₈

—

118

—

113

C₄⋊C₄

—

114

C₄⋊C₄

—

115

C₄⋊C₄

C₄×D₅₉

yes

characteristic

116

C₂²⋊C₄

—

118

—

117

C₄²

—

118

C₄²

C₂²×D₅₉

yes

characteristic

119

C₂³

—

118

—

120

Q₈

—

121

Q₈

#11745659

yes

complemented

122

D₄

—

118

—

123

D₄

—

118

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

118

—

127

C₂×C₄

—

236

—

128

C₂×C₄

—

236

—

129

C₂×C₄

C₂×C₄×D₅₉

yes

characteristic

130

C₂×C₄

D₄×D₅₉

yes

characteristic

131

C₂×C₄

#11745687

yes

characteristic

132

C₈

—

133

C₈

—

134

C₈

—

135

C₈

—

118

—

136

C₈

—

118

—

137

C₈

—

118

—

138

C₈

—

236

—

139

C₂²

—

236

—

140

C₂²

—

236

—

141

C₂²

—

236

—

142

C₂²

(C₂²⋊D₅₉)⋊C₄

yes

characteristic

143

C₄

—

144

C₄

—

145

C₄

—

146

C₄

—

236

—

147

C₄

C₂×#11745659

yes

characteristic

148

C₄

#14273735

yes

characteristic

149

C₄

SD₁₆×D₅₉

yes

characteristic

150

C₄

#14273750

yes

characteristic

151

C₂

—

472

—

152

C₂

#16837470

yes

characteristic

153

C₂

#16837487

yes

characteristic

154

C₂

#16837521

yes

characteristic

	subgroup	G/H
center	C₂×C₄	D₄×D₅₉
commutator	C₂×C₄×C₅₉	C₂²×C₄
Frattini	C₄²	C₂²×D₅₉
Fitting	C₅₉×Q₈⋊C₈	C₂
socle	C₂²×C₅₉	C₄×D₄
hypercenter	Q₈⋊C₈	D₅₉
hypocenter	C₅₉	#1583
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ Q₈⋊C₈

subgroup

G/H

center

C₂×C₄

D₄×D₅₉

commutator

C₂×C₄×C₅₉

C₂²×C₄

Frattini

C₄²

C₂²×D₅₉

Fitting

C₅₉×Q₈⋊C₈

C₂

socle

C₂²×C₅₉

C₄×D₄

hypercenter

Q₈⋊C₈

D₅₉

hypocenter

C₅₉

#1583

LCS

G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂×C₄ ◁ C₄² ◁ Q₈⋊C₈

N	H ≅ G/N	type
1. Q₈×C₅₉⋊C₈	C₂	semidirect
4. #16837152	C₂	semidirect
6. #16837124	C₂	semidirect
7. C₅₉×Q₈⋊C₈	C₂	semidirect
43. Q₈×C₅₉	M₁₆	semidirect
60. C₅₉⋊C₈	SD₁₆	semidirect
82. Q₈⋊C₈	D₅₉	semidirect
85. C₅₉	#1583	semidirect
121. Q₈	#11745659	semidirect

H ≅ G/N

type

1. Q₈×C₅₉⋊C₈

C₂

semidirect

4. #16837152

C₂

semidirect

6. #16837124

C₂

semidirect

7. C₅₉×Q₈⋊C₈

C₂

semidirect

43. Q₈×C₅₉

M₁₆

semidirect

60. C₅₉⋊C₈

SD₁₆

semidirect

82. Q₈⋊C₈

D₅₉

semidirect

85. C₅₉

#1583

semidirect

121. Q₈

#11745659

semidirect

Subgroups — #21422936

Conjugacy classes of subgroups

Complements