Group #21425809 — subgroups

Subgroups — #21425809

Order 7552.

232 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 232 conjugacy classes of subgroups — too many to draw. Only its 85 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	C₂×#14273749	C₂	1	1	yes	maximal, characteristic
2	C₂×#14273661	C₂	1	1	yes	maximal, characteristic, complemented
3	#16837207	C₂	1	2	yes	maximal, complemented
4	#16837204	C₂	1	2	yes	maximal, complemented
5	C₂×M₃₂×C₅₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#11745685	—	2	1	no	—
7	#14273749	—	2	2	no	—
8	#14273749	C₄	1	4	yes	complemented
9	C₂×Dic₂₃₆	C₂²	1	1	yes	characteristic
10	C₂×#11745660	—	2	1	no	—
11	C₂×D₄₇₂	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊Dic₅₉	—	2	1	no	—
13	#14273661	—	2	1	no	—
14	#14273661	C₂²	1	2	yes	—
15	C₂²×C₈×C₅₉	C₂²	1	1	yes	characteristic
16	M₃₂×C₅₉	—	2	2	no	—
17	C₂×C₁₆×C₅₉	C₂²	1	2	yes	—
18	C₂²×C₄×C₅₉	D₄	1	1	yes	characteristic
19	C₂×C₂²⋊D₅₉	—	4	1	no	—
20	C₂²×Dic₅₉	—	4	1	no	—
21	#11745685	—	2	4	no	—
22	#11745685	—	4	2	no	—
23	C₂×D₂₃₆	—	2	1	no	—
24	C₂×C₄×D₅₉	—	4	1	no	—
25	C₂×Dic₁₁₈	—	2	1	no	—
26	C₂×C₈×C₅₉	C₂×C₄	1	2	yes	—
27	C₂×C₈×C₅₉	D₄	1	1	yes	characteristic
28	C₂×C₈×C₅₉	D₄	1	2	yes	—
29	C₂×C₈×C₅₉	C₂³	1	1	yes	characteristic
30	C₄⋊Dic₅₉	—	2	2	no	—
31	C₄⋊Dic₅₉	—	4	1	no	—
32	Dic₂₃₆	—	2	1	no	—
33	Dic₂₃₆	C₂×C₄	1	2	yes	complemented
34	D₄₇₂	—	2	1	no	—
35	D₄₇₂	C₂×C₄	1	2	yes	complemented
36	#11745660	—	2	4	no	—
37	C₁₆×C₅₉	—	2	2	no	—
38	C₂³×C₅₉	D₈	1	1	yes	characteristic
39	C₂²×D₅₉	—	4	1	no	—
40	C₂×C₄×C₅₉	C₂²⋊C₄	1	2	yes	—
41	C₂×C₄×C₅₉	D₈	1	1	yes	characteristic
42	C₂×C₄×C₅₉	SD₁₆	1	2	yes	—
43	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
44	C₂²⋊D₅₉	—	4	4	no	—
45	C₂×Dic₅₉	—	4	1	no	—
46	C₂×Dic₅₉	—	4	2	no	—
47	C₂×Dic₅₉	—	8	2	no	—
48	D₂₃₆	—	2	2	no	—
49	D₂₃₆	—	4	1	no	—
50	C₄×D₅₉	—	4	4	no	—
51	Dic₁₁₈	—	2	2	no	—
52	Dic₁₁₈	—	4	1	no	—
53	C₈×C₅₉	C₂²⋊C₄	1	2	yes	—
54	C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
55	C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
56	C₂²×C₅₉	—	2	1	no	—
57	C₂²×C₅₉	—	2	1	no	—
58	C₂²×C₅₉	#102	1	2	yes	—
59	C₂²×C₅₉	C₂×D₈	1	1	yes	characteristic
60	C₂×D₅₉	—	4	2	no	—
61	C₂×D₅₉	—	8	1	no	—
62	C₂×D₅₉	—	8	1	no	—
63	C₄×C₅₉	#102	1	2	yes	—
64	C₄×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
65	C₄×C₅₉	C₂×SD₁₆	1	1	yes	characteristic
66	Dic₅₉	—	4	2	no	—
67	Dic₅₉	—	8	2	no	—
68	#2139	—	59	1	no	maximal, Sylow
69	C₂×C₅₉	—	2	2	no	—
70	C₂×C₅₉	C₂×#102	1	1	yes	characteristic
71	C₂×C₅₉	#509	1	1	yes	characteristic
72	C₂×C₅₉	#510	1	1	yes	characteristic
73	D₅₉	—	8	2	no	—
74	C₂×#135	—	59	1	no	—
75	C₂×M₃₂	D₅₉	1	1	yes	characteristic, complemented
76	C₂×#107	—	59	1	no	—
77	#358	—	59	2	no	—
78	#357	—	59	2	no	—
79	C₅₉	#2139	1	1	yes	characteristic, complemented, Sylow
80	C₂×#41	—	118	1	no	—
81	#135	—	59	4	no	—
82	#135	—	118	2	no	—
83	C₂×Q₁₆	—	59	1	no	—
84	C₂×SD₁₆	—	118	1	no	—
85	C₂×D₈	—	59	1	no	—
86	C₂²×C₈	C₂×D₅₉	1	1	yes	characteristic
87	C₂×C₄⋊C₄	—	118	1	no	—
88	M₃₂	—	2	2	no	—
89	C₂×C₁₆	C₂×D₅₉	1	2	yes	—
90	#107	—	59	2	no	—
91	#107	—	118	1	no	—
92	#41	—	118	4	no	—
93	#41	—	236	2	no	—
94	C₂×Q₈	—	118	1	no	—
95	C₂×D₄	—	118	1	no	—
96	C₂×D₄	—	236	1	no	—
97	C₂²×C₄	—	236	1	no	—
98	C₂²×C₄	—	236	1	no	—
99	C₂²×C₄	D₂₃₆	1	1	yes	characteristic
100	Q₁₆	—	59	2	no	—
101	Q₁₆	—	118	1	no	—
102	SD₁₆	—	118	4	no	—
103	D₈	—	59	2	no	—
104	D₈	—	118	1	no	—
105	C₂×C₈	C₄×D₅₉	1	2	yes	—
106	C₂×C₈	D₂₃₆	1	1	yes	characteristic
107	C₂×C₈	C₂²⋊D₅₉	1	2	yes	—
108	C₂×C₈	C₂²×D₅₉	1	1	yes	characteristic
109	C₄⋊C₄	—	118	2	no	—
110	C₄⋊C₄	—	236	1	no	—
111	C₁₆	—	2	2	no	—
112	C₂³	—	236	1	no	—
113	C₂³	D₄₇₂	1	1	yes	characteristic
114	Q₈	—	118	2	no	—
115	Q₈	—	236	1	no	—
116	D₄	—	118	2	no	—
117	D₄	—	236	4	no	—
118	D₄	—	236	1	no	—
119	C₂×C₄	—	472	2	no	—
120	C₂×C₄	—	236	2	no	—
121	C₂×C₄	—	236	1	no	—
122	C₂×C₄	—	236	4	no	—
123	C₂×C₄	#11745660	1	2	yes	—
124	C₂×C₄	D₄₇₂	1	1	yes	characteristic
125	C₂×C₄	(C₂×D₅₉)⋊C₄	1	2	yes	—
126	C₂×C₄	C₂×D₂₃₆	1	1	yes	characteristic
127	C₈	(C₂×D₅₉)⋊C₄	1	2	yes	—
128	C₈	C₂×C₄×D₅₉	1	1	yes	characteristic
129	C₈	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
130	C₂²	—	2	1	no	—
131	C₂²	—	2	1	no	—
132	C₂²	—	472	1	no	—
133	C₂²	—	472	1	no	—
134	C₂²	—	236	2	no	—
135	C₂²	#14273649	1	2	yes	—
136	C₂²	C₂×D₄₇₂	1	1	yes	characteristic
137	C₄	—	472	2	no	—
138	C₄	—	236	2	no	—
139	C₄	#14273649	1	2	yes	—
140	C₄	C₂×(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
141	C₄	C₂×#11745660	1	1	yes	characteristic
142	C₂	—	2	2	no	—
143	C₂	—	472	2	no	—
144	C₂	C₂×#14273649	1	1	yes	characteristic
145	C₂	#16837840	1	1	yes	characteristic
146	C₂	#16837845	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14273749

C₂

yes

maximal, characteristic

C₂×#14273661

C₂

yes

maximal, characteristic, complemented

#16837207

C₂

yes

maximal, complemented

#16837204

C₂

yes

maximal, complemented

C₂×M₃₂×C₅₉

C₂

yes

maximal, characteristic, complemented

C₂×#11745685

—

#14273749

—

#14273749

C₄

yes

complemented

C₂×Dic₂₃₆

C₂²

yes

characteristic

C₂×#11745660

—

C₂×D₄₇₂

C₂²

yes

characteristic

C₂×C₄⋊Dic₅₉

—

#14273661

—

#14273661

C₂²

yes

—

C₂²×C₈×C₅₉

C₂²

yes

characteristic

M₃₂×C₅₉

—

C₂×C₁₆×C₅₉

C₂²

yes

—

C₂²×C₄×C₅₉

D₄

yes

characteristic

C₂×C₂²⋊D₅₉

—

C₂²×Dic₅₉

—

#11745685

—

#11745685

—

C₂×D₂₃₆

—

C₂×C₄×D₅₉

—

C₂×Dic₁₁₈

—

C₂×C₈×C₅₉

C₂×C₄

yes

—

C₂×C₈×C₅₉

D₄

yes

characteristic

C₂×C₈×C₅₉

D₄

yes

—

C₂×C₈×C₅₉

C₂³

yes

characteristic

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

Dic₂₃₆

—

Dic₂₃₆

C₂×C₄

yes

complemented

D₄₇₂

—

D₄₇₂

C₂×C₄

yes

complemented

#11745660

—

C₁₆×C₅₉

—

C₂³×C₅₉

D₈

yes

characteristic

C₂²×D₅₉

—

C₂×C₄×C₅₉

C₂²⋊C₄

yes

—

C₂×C₄×C₅₉

D₈

yes

characteristic

C₂×C₄×C₅₉

SD₁₆

yes

—

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂²⋊D₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

D₂₃₆

—

D₂₃₆

—

C₄×D₅₉

—

Dic₁₁₈

—

Dic₁₁₈

—

C₈×C₅₉

C₂²⋊C₄

yes

—

C₈×C₅₉

C₂²×C₄

yes

characteristic

C₈×C₅₉

C₂×D₄

yes

characteristic

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

#102

yes

—

C₂²×C₅₉

C₂×D₈

yes

characteristic

C₂×D₅₉

—

C₂×D₅₉

—

C₂×D₅₉

—

C₄×C₅₉

#102

yes

—

C₄×C₅₉

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₅₉

C₂×SD₁₆

yes

characteristic

Dic₅₉

—

Dic₅₉

—

#2139

—

maximal, Sylow

C₂×C₅₉

—

C₂×C₅₉

yes

C₂×C₅₉

yes

C₂×C₅₉

yes

D₅₉

—

C₂×#135

—

C₂×M₃₂

D₅₉

yes

characteristic, complemented

C₂×#107

—

#358

—

#357

—

C₅₉

#2139

yes

characteristic, complemented, Sylow

C₂×#41

—

118

—

#135

—

#135

—

118

—

C₂×Q₁₆

—

C₂×SD₁₆

—

118

—

C₂×D₈

—

C₂²×C₈

C₂×D₅₉

yes

characteristic

C₂×C₄⋊C₄

—

118

—

M₃₂

—

C₂×C₁₆

C₂×D₅₉

yes

—

#107

—

#107

—

118

—

#41

—

118

—

#41

—

236

—

C₂×Q₈

—

118

—

C₂×D₄

—

118

—

C₂×D₄

—

236

—

C₂²×C₄

—

236

—

C₂²×C₄

—

236

—

C₂²×C₄

D₂₃₆

yes

characteristic

100

Q₁₆

—

101

Q₁₆

—

118

—

102

SD₁₆

—

118

—

103

D₈

—

104

D₈

—

118

—

105

C₂×C₈

C₄×D₅₉

yes

—

106

C₂×C₈

D₂₃₆

yes

characteristic

107

C₂×C₈

C₂²⋊D₅₉

yes

—

108

C₂×C₈

C₂²×D₅₉

yes

characteristic

109

C₄⋊C₄

—

118

—

110

C₄⋊C₄

—

236

—

111

C₁₆

—

112

C₂³

—

236

—

113

C₂³

D₄₇₂

yes

characteristic

114

Q₈

—

118

—

115

Q₈

—

236

—

116

D₄

—

118

—

117

D₄

—

236

—

118

D₄

—

236

—

119

C₂×C₄

—

472

—

120

C₂×C₄

—

236

—

121

C₂×C₄

—

236

—

122

C₂×C₄

—

236

—

123

C₂×C₄

#11745660

yes

—

124

C₂×C₄

D₄₇₂

yes

characteristic

125

C₂×C₄

(C₂×D₅₉)⋊C₄

yes

—

126

C₂×C₄

C₂×D₂₃₆

yes

characteristic

127

C₈

(C₂×D₅₉)⋊C₄

yes

—

128

C₈

C₂×C₄×D₅₉

yes

characteristic

129

C₈

C₂×C₂²⋊D₅₉

yes

characteristic

130

C₂²

—

131

C₂²

—

132

C₂²

—

472

—

133

C₂²

—

472

—

134

C₂²

—

236

—

135

C₂²

#14273649

yes

—

136

C₂²

C₂×D₄₇₂

yes

characteristic

137

C₄

—

472

—

138

C₄

—

236

—

139

C₄

#14273649

yes

—

140

C₄

C₂×(C₂×D₅₉)⋊C₄

yes

characteristic

141

C₄

C₂×#11745660

yes

characteristic

142

C₂

—

143

C₂

—

472

—

144

C₂

C₂×#14273649

yes

characteristic

145

C₂

#16837840

yes

characteristic

146

C₂

#16837845

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×D₄₇₂
commutator	C₈×C₅₉	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₅₉
Fitting	C₂×M₃₂×C₅₉	C₂
socle	C₂²×C₅₉	C₂×D₈
hypercenter	C₂×M₃₂	D₅₉
hypocenter	C₅₉	#2139
LCS	G ▷ C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×M₃₂

subgroup

G/H

center

C₂²

C₂×D₄₇₂

commutator

C₈×C₅₉

C₂²×C₄

Frattini

C₂×C₈

C₂²×D₅₉

Fitting

C₂×M₃₂×C₅₉

C₂

socle

C₂²×C₅₉

C₂×D₈

hypercenter

C₂×M₃₂

D₅₉

hypocenter

C₅₉

#2139

LCS

G ▷ C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×M₃₂

N	H ≅ G/N	type
2. C₂×#14273661	C₂	semidirect
3. #16837207	C₂	semidirect
4. #16837204	C₂	semidirect
5. C₂×M₃₂×C₅₉	C₂	semidirect
8. #14273749	C₄	semidirect
33. Dic₂₃₆	C₂×C₄	semidirect
35. D₄₇₂	C₂×C₄	semidirect
75. C₂×M₃₂	D₅₉	semidirect
79. C₅₉	#2139	semidirect

H ≅ G/N

type

2. C₂×#14273661

C₂

semidirect

3. #16837207

C₂

semidirect

4. #16837204

C₂

semidirect

5. C₂×M₃₂×C₅₉

C₂

semidirect

8. #14273749

C₄

semidirect

33. Dic₂₃₆

C₂×C₄

semidirect

35. D₄₇₂

C₂×C₄

semidirect

75. C₂×M₃₂

D₅₉

semidirect

79. C₅₉

#2139

semidirect

Subgroups — #21425809

Conjugacy classes of subgroups

Complements