Group #21423552 — subgroups

Subgroups — #21423552

Order 7552.

144 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 144 conjugacy classes of subgroups — too many to draw. Only its 55 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	#16837752	C₂	1	1	yes	maximal, characteristic
2	#16837697	C₂	1	1	yes	maximal, characteristic
3	Q₈×C₅₉⋊C₈	C₂	1	1	yes	maximal, characteristic
4	#16837154	C₂	1	1	yes	maximal, characteristic
5	#16837150	C₂	1	1	yes	maximal, characteristic
6	#16837148	C₂	1	1	yes	maximal, characteristic
7	C₅₉×#475	C₂	1	1	yes	maximal, characteristic
8	#14273727	C₂²	1	1	yes	characteristic
9	#14273660	—	2	1	no	—
10	C₄⋊(C₅₉⋊C₈)	—	2	1	no	—
11	C₄⋊(C₅₉⋊C₈)	C₂²	1	1	yes	characteristic
12	C₄⋊(C₅₉⋊C₈)	C₂²	1	1	yes	characteristic
13	#14273651	—	2	1	no	—
14	#14273651	—	2	1	no	—
15	#14273651	—	2	1	no	—
16	C₄×C₅₉⋊C₈	—	2	1	no	—
17	C₄×C₅₉⋊C₈	C₂²	1	1	yes	characteristic
18	C₄⋊Q₈×C₅₉	C₂²	1	1	yes	characteristic
19	C₄×Q₈×C₅₉	C₂²	1	1	yes	characteristic
20	#106×C₅₉	—	2	1	no	—
21	C₄⋊C₈×C₅₉	C₂²	1	1	yes	characteristic
22	#103×C₅₉	—	2	1	no	—
23	C₂×Q₈×C₅₉	—	2	1	no	—
24	C₂×Q₈×C₅₉	D₄	1	1	yes	characteristic
25	C₂×Dic₁₁₈	—	2	1	no	—
26	C₂×C₈×C₅₉	—	2	1	no	—
27	C₄⋊C₄×C₅₉	—	2	1	no	—
28	C₄⋊C₄×C₅₉	—	2	1	no	—
29	C₄⋊C₄×C₅₉	—	2	1	no	—
30	C₄⋊C₄×C₅₉	D₄	1	1	yes	characteristic
31	C₄²×C₅₉	—	2	1	no	—
32	C₄²×C₅₉	C₂³	1	1	yes	characteristic
33	C₄⋊Dic₅₉	—	2	1	no	—
34	C₄⋊Dic₅₉	—	2	1	no	—
35	C₂×C₅₉⋊C₈	—	2	1	no	—
36	C₂×C₅₉⋊C₈	—	2	1	no	—
37	C₂×C₅₉⋊C₈	D₄	1	1	yes	characteristic
38	C₂×C₅₉⋊C₈	D₄	1	1	yes	characteristic
39	Q₈×C₅₉	—	2	1	no	—
40	Q₈×C₅₉	—	4	1	no	—
41	Q₈×C₅₉	—	4	1	no	—
42	Q₈×C₅₉	Q₁₆	1	2	yes	—
43	C₂×C₄×C₅₉	—	2	1	no	—
44	C₂×C₄×C₅₉	—	2	1	no	—
45	C₂×C₄×C₅₉	—	4	1	no	—
46	C₂×C₄×C₅₉	C₂×D₄	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	—	4	1	no	—
50	Dic₁₁₈	—	8	1	no	—
51	C₈×C₅₉	—	4	1	no	—
52	C₅₉⋊C₈	—	2	1	no	—
53	C₅₉⋊C₈	—	2	1	no	—
54	C₅₉⋊C₈	—	4	1	no	—
55	C₅₉⋊C₈	—	4	1	no	—
56	C₂²×C₅₉	#121	1	1	yes	characteristic
57	C₄×C₅₉	—	2	2	no	—
58	C₄×C₅₉	—	2	1	no	—
59	C₄×C₅₉	—	4	1	no	—
60	C₄×C₅₉	—	8	1	no	—
61	C₄×C₅₉	C₂×Q₁₆	1	1	yes	characteristic
62	C₄×C₅₉	#135	1	1	yes	characteristic
63	C₄×C₅₉	#135	1	1	yes	characteristic
64	C₄×C₅₉	#137	1	1	yes	characteristic
65	Dic₅₉	—	8	1	no	—
66	#1684	—	59	1	no	maximal, Sylow
67	C₂×C₅₉	#464	1	1	yes	characteristic
68	C₂×C₅₉	#467	1	1	yes	characteristic
69	C₂×C₅₉	#471	1	1	yes	characteristic
70	#487	—	59	1	no	—
71	#477	—	59	1	no	—
72	#475	D₅₉	1	1	yes	characteristic
73	C₈×Q₈	—	59	1	no	—
74	#335	—	59	1	no	—
75	#333	—	59	1	no	—
76	#332	—	59	1	no	—
77	C₅₉	#1684	1	1	yes	characteristic, complemented, Sylow
78	C₄⋊Q₈	—	59	1	no	—
79	C₄⋊Q₈	C₂×D₅₉	1	1	yes	characteristic
80	C₄×Q₈	C₂×D₅₉	1	1	yes	characteristic
81	#107	—	118	1	no	—
82	#106	—	2	1	no	—
83	C₄⋊C₈	—	59	1	no	—
84	C₄⋊C₈	—	59	1	no	—
85	C₄⋊C₈	—	118	1	no	—
86	C₄⋊C₈	C₂×D₅₉	1	1	yes	characteristic
87	#103	—	2	1	no	—
88	#103	—	118	1	no	—
89	#103	—	118	1	no	—
90	#103	—	118	1	no	—
91	C₄×C₈	—	59	1	no	—
92	C₄×C₈	—	118	1	no	—
93	C₂×Q₈	—	2	1	no	—
94	C₂×Q₈	—	118	1	no	—
95	C₂×Q₈	C₂²⋊D₅₉	1	1	yes	characteristic
96	C₂×C₈	—	2	1	no	—
97	C₂×C₈	—	59	1	no	—
98	C₂×C₈	—	59	1	no	—
99	C₂×C₈	—	118	1	no	—
100	C₂×C₈	—	118	1	no	—
101	C₄⋊C₄	—	2	1	no	—
102	C₄⋊C₄	—	2	1	no	—
103	C₄⋊C₄	—	2	1	no	—
104	C₄⋊C₄	—	118	1	no	—
105	C₄⋊C₄	—	118	1	no	—
106	C₄⋊C₄	C₂²⋊D₅₉	1	1	yes	characteristic
107	C₄²	—	2	1	no	—
108	C₄²	C₂²×D₅₉	1	1	yes	characteristic
109	Q₈	—	2	1	no	—
110	Q₈	—	4	1	no	—
111	Q₈	—	4	1	no	—
112	Q₈	—	472	1	no	—
113	Q₈	#11745672	1	2	yes	—
114	C₂×C₄	—	2	1	no	—
115	C₂×C₄	—	2	1	no	—
116	C₂×C₄	—	4	1	no	—
117	C₂×C₄	—	236	1	no	—
118	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
119	C₂×C₄	#11745687	1	1	yes	characteristic
120	C₂×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
121	C₈	—	4	1	no	—
122	C₈	—	118	1	no	—
123	C₈	—	118	1	no	—
124	C₈	—	236	1	no	—
125	C₈	—	236	1	no	—
126	C₂²	#14273699	1	1	yes	characteristic
127	C₄	—	2	2	no	—
128	C₄	—	2	1	no	—
129	C₄	—	4	1	no	—
130	C₄	—	8	1	no	—
131	C₄	—	472	1	no	—
132	C₄	C₂×#11745672	1	1	yes	characteristic
133	C₄	#14273747	1	1	yes	characteristic
134	C₄	#14273748	1	1	yes	characteristic
135	C₄	#14273756	1	1	yes	characteristic
136	C₂	#16837622	1	1	yes	characteristic
137	C₂	#16837641	1	1	yes	characteristic
138	C₂	#16837667	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16837752

C₂

yes

maximal, characteristic

#16837697

C₂

yes

maximal, characteristic

Q₈×C₅₉⋊C₈

C₂

yes

maximal, characteristic

#16837154

C₂

yes

maximal, characteristic

#16837150

C₂

yes

maximal, characteristic

#16837148

C₂

yes

maximal, characteristic

C₅₉×#475

C₂

yes

maximal, characteristic

#14273727

C₂²

yes

characteristic

#14273660

—

C₄⋊(C₅₉⋊C₈)

—

C₄⋊(C₅₉⋊C₈)

C₂²

yes

characteristic

C₄⋊(C₅₉⋊C₈)

C₂²

yes

—

—

—

C₄×C₅₉⋊C₈

—

C₄×C₅₉⋊C₈

C₂²

yes

characteristic

C₄⋊Q₈×C₅₉

C₂²

yes

characteristic

C₄×Q₈×C₅₉

C₂²

yes

characteristic

#106×C₅₉

—

C₄⋊C₈×C₅₉

C₂²

yes

characteristic

#103×C₅₉

—

C₂×Q₈×C₅₉

—

C₂×Q₈×C₅₉

D₄

yes

characteristic

C₂×Dic₁₁₈

—

C₂×C₈×C₅₉

—

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

D₄

yes

characteristic

C₄²×C₅₉

—

C₄²×C₅₉

C₂³

yes

characteristic

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

D₄

yes

characteristic

C₂×C₅₉⋊C₈

D₄

yes

characteristic

Q₈×C₅₉

—

Q₈×C₅₉

—

Q₈×C₅₉

—

Q₈×C₅₉

Q₁₆

yes

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

#41

yes

characteristic

C₂×Dic₅₉

—

Dic₁₁₈

—

C₈×C₅₉

—

C₅₉⋊C₈

—

C₅₉⋊C₈

—

C₅₉⋊C₈

—

C₅₉⋊C₈

—

C₂²×C₅₉

#121

yes

characteristic

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

C₂×Q₁₆

yes

C₄×C₅₉

yes

C₄×C₅₉

yes

C₄×C₅₉

yes

Dic₅₉

—

—

C₂×C₅₉

yes

C₂×C₅₉

yes

C₂×C₅₉

yes

—

—

D₅₉

yes

characteristic

C₈×Q₈

—

#335

—

#333

—

#332

—

C₅₉

#1684

yes

characteristic, complemented, Sylow

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₅₉

yes

characteristic

C₄×Q₈

C₂×D₅₉

yes

characteristic

#107

—

118

—

#106

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

118

—

C₄⋊C₈

C₂×D₅₉

yes

characteristic

#103

—

#103

—

118

—

#103

—

118

—

#103

—

118

—

C₄×C₈

—

C₄×C₈

—

118

—

C₂×Q₈

—

C₂×Q₈

—

118

—

C₂×Q₈

C₂²⋊D₅₉

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

118

—

100

C₂×C₈

—

118

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

118

—

105

C₄⋊C₄

—

118

—

106

C₄⋊C₄

C₂²⋊D₅₉

yes

characteristic

107

C₄²

—

108

C₄²

C₂²×D₅₉

yes

characteristic

109

Q₈

—

110

Q₈

—

111

Q₈

—

112

Q₈

—

472

—

113

Q₈

#11745672

yes

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

236

—

118

C₂×C₄

D₄×D₅₉

yes

characteristic

119

C₂×C₄

#11745687

yes

characteristic

120

C₂×C₄

C₂×C₂²⋊D₅₉

yes

characteristic

121

C₈

—

122

C₈

—

118

—

123

C₈

—

118

—

124

C₈

—

236

—

125

C₈

—

236

—

126

C₂²

#14273699

yes

characteristic

127

C₄

—

128

C₄

—

129

C₄

—

130

C₄

—

131

C₄

—

472

—

132

C₄

C₂×#11745672

yes

characteristic

133

C₄

#14273747

yes

characteristic

134

C₄

#14273748

yes

characteristic

135

C₄

#14273756

yes

characteristic

136

C₂

#16837622

yes

characteristic

137

C₂

#16837641

yes

characteristic

138

C₂

#16837667

yes

characteristic

	subgroup	G/H
center	C₂²	#14273699
commutator	C₄²×C₅₉	C₂³
Frattini	C₄²	C₂²×D₅₉
Fitting	C₅₉×#475	C₂
socle	C₂²×C₅₉	#121
hypercenter	#475	D₅₉
hypocenter	C₅₉	#1684
LCS	G ▷ C₄²×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #475

subgroup

G/H

center

C₂²

#14273699

commutator

C₄²×C₅₉

C₂³

Frattini

C₄²

C₂²×D₅₉

Fitting

C₅₉×#475

C₂

C₂²×C₅₉

D₅₉

C₅₉

G ▷ C₄²×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #475

N	H ≅ G/N	type
77. C₅₉	#1684	semidirect

H ≅ G/N

type

77. C₅₉

#1684

semidirect

Subgroups — #21423552

Conjugacy classes of subgroups

Complements