Group #21425257 — subgroups

Subgroups — #21425257

Order 7552.

246 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 246 conjugacy classes of subgroups — too many to draw. Only its 69 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	#16838033	C₂	1	1	yes	maximal, characteristic
2	#16837411	C₂	1	2	yes	maximal
3	#16837171	C₂	1	1	yes	maximal, characteristic
4	#16837162	C₂	1	2	yes	maximal
5	C₅₉×#391	C₂	1	1	yes	maximal, characteristic
6	C₂×#11745664	C₂²	1	2	yes	—
7	#14273726	—	2	2	no	—
8	#14273718	—	2	2	no	—
9	#14273706	—	2	2	no	—
10	#14273686	C₂²	1	2	yes	—
11	#14273651	—	2	2	no	—
12	#14273651	—	2	2	no	—
13	C₂²×Q₈×C₅₉	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊C₄×C₅₉	C₂²	1	1	yes	characteristic
15	C₂×C₄²×C₅₉	C₂²	1	1	yes	characteristic
16	#95×C₅₉	—	2	2	no	—
17	C₂×Q₈×C₅₉	—	2	1	no	—
18	C₂×Q₈×C₅₉	—	2	2	no	—
19	C₂×Q₈×C₅₉	—	2	2	no	—
20	C₂×Q₈×C₅₉	Q₈	1	2	yes	complemented
21	C₂²×C₄×C₅₉	—	2	1	no	—
22	C₂²×C₄×C₅₉	—	2	1	no	—
23	C₂²×C₄×C₅₉	D₄	1	2	yes	—
24	C₂²×C₄×C₅₉	C₂³	1	1	yes	characteristic
25	C₂×Dic₁₁₈	—	2	2	no	—
26	C₄⋊C₄×C₅₉	—	2	2	no	—
27	C₄²×C₅₉	—	2	2	no	—
28	C₂²⋊Dic₅₉	—	2	2	no	—
29	C₄⋊Dic₅₉	—	2	2	no	—
30	C₄⋊Dic₅₉	D₄	1	2	yes	—
31	C₄⋊Dic₅₉	D₄	1	2	yes	—
32	Dic₅₉⋊C₄	—	2	4	no	—
33	C₄×Dic₅₉	—	2	2	no	—
34	#11745664	—	2	2	no	—
35	#11745664	—	2	2	no	—
36	C₂×C₅₉⋊C₈	—	2	2	no	—
37	C₂³×C₅₉	C₂×D₄	1	1	yes	characteristic
38	Q₈×C₅₉	—	2	4	no	—
39	Q₈×C₅₉	—	4	2	no	—
40	C₂×C₄×C₅₉	—	2	2	no	—
41	C₂×C₄×C₅₉	—	2	2	no	—
42	C₂×C₄×C₅₉	—	2	2	no	—
43	C₂×C₄×C₅₉	—	4	1	no	—
44	C₂×C₄×C₅₉	—	4	1	no	—
45	C₂×C₄×C₅₉	—	4	1	no	—
46	C₂×C₄×C₅₉	—	4	1	no	—
47	C₂×C₄×C₅₉	—	4	2	no	—
48	C₂×C₄×C₅₉	C₂×D₄	1	2	yes	—
49	C₂×C₄×C₅₉	C₂×Q₈	1	1	yes	characteristic
50	C₂×C₄×C₅₉	#41	1	1	yes	characteristic
51	C₂×C₄×C₅₉	#41	1	2	yes	—
52	C₂×Dic₅₉	—	2	2	no	—
53	C₂×Dic₅₉	—	2	2	no	—
54	Dic₁₁₈	—	8	2	no	—
55	C₅₉⋊C₈	—	4	2	no	—
56	C₂²×C₅₉	—	2	1	no	—
57	C₂²×C₅₉	—	2	1	no	—
58	C₂²×C₅₉	#120	1	1	yes	characteristic
59	C₂²×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
60	C₂²×C₅₉	#124	1	1	yes	characteristic
61	C₄×C₅₉	—	4	2	no	—
62	C₄×C₅₉	—	4	2	no	—
63	C₄×C₅₉	—	4	2	no	—
64	C₄×C₅₉	C₂²⋊Q₈	1	2	yes	—
65	C₄×C₅₉	#124	1	2	yes	—
66	Dic₅₉	—	4	2	no	—
67	Dic₅₉	—	4	2	no	—
68	#2028	—	59	1	no	maximal, Sylow
69	C₂×C₅₉	—	2	1	no	—
70	C₂×C₅₉	—	2	1	no	—
71	C₂×C₅₉	#393	1	1	yes	characteristic
72	C₂×C₅₉	#455	1	1	yes	characteristic
73	C₂×C₅₉	#456	1	1	yes	characteristic
74	#544	—	59	1	no	—
75	#419	—	59	2	no	—
76	#391	D₅₉	1	1	yes	characteristic
77	#343	—	59	1	no	—
78	#339	—	59	2	no	—
79	C₅₉	#2028	1	1	yes	characteristic, complemented, Sylow
80	C₂²×Q₈	C₂×D₅₉	1	1	yes	characteristic
81	C₂×M₁₆	—	59	2	no	—
82	C₄⋊Q₈	—	118	2	no	—
83	#125	—	118	2	no	—
84	C₂²⋊Q₈	—	118	2	no	—
85	#117	—	59	2	no	—
86	C₂×C₄⋊C₄	C₂×D₅₉	1	1	yes	characteristic
87	C₂×C₄²	C₂×D₅₉	1	1	yes	characteristic
88	#103	—	118	2	no	—
89	#103	—	118	2	no	—
90	#95	—	2	2	no	—
91	C₂×Q₈	—	2	1	no	—
92	C₂×Q₈	—	2	2	no	—
93	C₂×Q₈	—	2	2	no	—
94	C₂×Q₈	—	118	2	no	—
95	C₂×Q₈	Dic₁₁₈	1	2	yes	complemented
96	C₂²×C₄	—	2	1	no	—
97	C₂²×C₄	—	2	1	no	—
98	C₂²×C₄	C₂²⋊D₅₉	1	2	yes	—
99	C₂²×C₄	C₂²×D₅₉	1	1	yes	characteristic
100	M₁₆	—	118	2	no	—
101	M₁₆	—	118	2	no	—
102	C₂×C₈	—	118	2	no	—
103	C₄⋊C₄	—	2	2	no	—
104	C₄⋊C₄	—	59	2	no	—
105	C₄⋊C₄	—	59	2	no	—
106	C₄⋊C₄	—	118	2	no	—
107	C₄⋊C₄	—	118	4	no	—
108	C₂²⋊C₄	—	118	2	no	—
109	C₄²	—	2	2	no	—
110	C₄²	—	118	2	no	—
111	C₂³	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
112	Q₈	—	2	4	no	—
113	Q₈	—	4	2	no	—
114	Q₈	—	472	2	no	—
115	C₂×C₄	—	2	2	no	—
116	C₂×C₄	—	2	2	no	—
117	C₂×C₄	—	2	2	no	—
118	C₂×C₄	—	4	1	no	—
119	C₂×C₄	—	4	1	no	—
120	C₂×C₄	—	4	1	no	—
121	C₂×C₄	—	4	1	no	—
122	C₂×C₄	—	4	2	no	—
123	C₂×C₄	—	118	2	no	—
124	C₂×C₄	—	118	2	no	—
125	C₂×C₄	C₂×Dic₁₁₈	1	1	yes	characteristic
126	C₂×C₄	#11745685	1	1	yes	characteristic
127	C₂×C₄	D₄×D₅₉	1	2	yes	—
128	C₂×C₄	#11745687	1	2	yes	—
129	C₈	—	236	2	no	—
130	C₂²	—	2	1	no	—
131	C₂²	—	2	1	no	—
132	C₂²	#14273695	1	1	yes	characteristic
133	C₂²	#14273706	1	1	yes	characteristic
134	C₂²	#14273714	1	1	yes	characteristic
135	C₄	—	4	2	no	—
136	C₄	—	4	2	no	—
137	C₄	—	4	2	no	—
138	C₄	—	236	2	no	—
139	C₄	—	236	2	no	—
140	C₄	#14273704	1	2	yes	—
141	C₄	#14273713	1	2	yes	—
142	C₂	—	2	1	no	—
143	C₂	—	2	1	no	—
144	C₂	#16837314	1	1	yes	characteristic
145	C₂	#16837579	1	1	yes	characteristic
146	C₂	#16837584	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16838033

C₂

yes

maximal, characteristic

#16837411

C₂

yes

maximal

#16837171

C₂

yes

maximal, characteristic

#16837162

C₂

yes

maximal

C₅₉×#391

C₂

yes

maximal, characteristic

C₂×#11745664

C₂²

yes

—

—

—

—

C₂²

yes

—

#14273651

—

#14273651

—

C₂²×Q₈×C₅₉

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₅₉

C₂²

yes

characteristic

C₂×C₄²×C₅₉

C₂²

yes

characteristic

#95×C₅₉

—

C₂×Q₈×C₅₉

—

C₂×Q₈×C₅₉

—

C₂×Q₈×C₅₉

—

C₂×Q₈×C₅₉

Q₈

yes

complemented

C₂²×C₄×C₅₉

—

C₂²×C₄×C₅₉

—

C₂²×C₄×C₅₉

D₄

yes

—

C₂²×C₄×C₅₉

C₂³

yes

characteristic

C₂×Dic₁₁₈

—

C₄⋊C₄×C₅₉

—

C₄²×C₅₉

—

C₂²⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

D₄

yes

—

C₄⋊Dic₅₉

D₄

yes

—

Dic₅₉⋊C₄

—

C₄×Dic₅₉

—

#11745664

—

#11745664

—

C₂×C₅₉⋊C₈

—

C₂³×C₅₉

C₂×D₄

yes

characteristic

Q₈×C₅₉

—

Q₈×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₂×D₄

yes

—

C₂×C₄×C₅₉

C₂×Q₈

yes

C₂×C₄×C₅₉

yes

C₂×C₄×C₅₉

yes

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

Dic₁₁₈

—

C₅₉⋊C₈

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

#120

yes

characteristic

C₂²×C₅₉

C₂²⋊Q₈

yes

characteristic

C₂²×C₅₉

#124

yes

characteristic

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

C₂²⋊Q₈

yes

—

C₄×C₅₉

#124

yes

—

Dic₅₉

—

Dic₅₉

—

#2028

—

maximal, Sylow

C₂×C₅₉

—

C₂×C₅₉

—

C₂×C₅₉

yes

C₂×C₅₉

yes

C₂×C₅₉

yes

—

—

D₅₉

yes

characteristic

#343

—

#339

—

C₅₉

#2028

yes

characteristic, complemented, Sylow

C₂²×Q₈

C₂×D₅₉

yes

characteristic

C₂×M₁₆

—

C₄⋊Q₈

—

118

—

#125

—

118

—

C₂²⋊Q₈

—

118

—

#117

—

C₂×C₄⋊C₄

C₂×D₅₉

yes

characteristic

C₂×C₄²

C₂×D₅₉

yes

characteristic

#103

—

118

—

#103

—

118

—

#95

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

118

—

C₂×Q₈

Dic₁₁₈

yes

complemented

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊D₅₉

yes

—

C₂²×C₄

C₂²×D₅₉

yes

characteristic

100

M₁₆

—

118

—

101

M₁₆

—

118

—

102

C₂×C₈

—

118

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

106

C₄⋊C₄

—

118

—

107

C₄⋊C₄

—

118

—

108

C₂²⋊C₄

—

118

—

109

C₄²

—

110

C₄²

—

118

—

111

C₂³

C₂×C₂²⋊D₅₉

yes

characteristic

112

Q₈

—

113

Q₈

—

114

Q₈

—

472

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

118

—

124

C₂×C₄

—

118

—

125

C₂×C₄

C₂×Dic₁₁₈

yes

characteristic

126

C₂×C₄

#11745685

yes

characteristic

127

C₂×C₄

D₄×D₅₉

yes

—

128

C₂×C₄

#11745687

yes

—

129

C₈

—

236

—

130

C₂²

—

131

C₂²

—

132

C₂²

#14273695

yes

characteristic

133

C₂²

#14273706

yes

characteristic

134

C₂²

#14273714

yes

characteristic

135

C₄

—

136

C₄

—

137

C₄

—

138

C₄

—

236

—

139

C₄

—

236

—

140

C₄

#14273704

yes

—

141

C₄

#14273713

yes

—

142

C₂

—

143

C₂

—

144

C₂

#16837314

yes

characteristic

145

C₂

#16837579

yes

characteristic

146

C₂

#16837584

yes

characteristic

	subgroup	G/H
center	C₂²	#14273695
commutator	C₂²×C₄×C₅₉	C₂³
Frattini	C₂²×C₄	C₂²×D₅₉
Fitting	C₅₉×#391	C₂
socle	C₂²×C₅₉	#120
hypercenter	#391	D₅₉
hypocenter	C₅₉	#2028
LCS	G ▷ C₂²×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #391

subgroup

G/H

center

C₂²

#14273695

commutator

C₂²×C₄×C₅₉

C₂³

Frattini

C₂²×C₄

C₂²×D₅₉

Fitting

C₅₉×#391

C₂

C₂²×C₅₉

D₅₉

C₅₉

G ▷ C₂²×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #391

N	H ≅ G/N	type
20. C₂×Q₈×C₅₉	Q₈	semidirect
79. C₅₉	#2028	semidirect
95. C₂×Q₈	Dic₁₁₈	semidirect

H ≅ G/N

type

20. C₂×Q₈×C₅₉

Q₈

semidirect

79. C₅₉

#2028

semidirect

95. C₂×Q₈

Dic₁₁₈

semidirect

Subgroups — #21425257

Conjugacy classes of subgroups

Complements