Group #21423592 — subgroups

Subgroups — #21423592

Order 7552.

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

normal subgroups only

This group has 160 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	#16837689	C₂	1	1	yes	maximal, characteristic
2	#16837604	C₂	1	1	yes	maximal, characteristic
3	#16837481	C₂	1	1	yes	maximal, characteristic
4	#16837155	C₂	1	1	yes	maximal, characteristic, complemented
5	#16837151	C₂	1	1	yes	maximal, characteristic, complemented
6	#16837148	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₉×#498	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#11745671	—	2	1	no	—
9	C₂×#11745659	—	2	1	no	—
10	C₄×D₂₃₆	C₂²	1	1	yes	characteristic
11	#14273658	—	2	1	no	—
12	C₄⋊(C₅₉⋊C₈)	C₂²	1	1	yes	characteristic
13	C₄⋊(C₅₉⋊C₈)	C₂²	1	1	yes	characteristic
14	C₄⋊(C₅₉⋊C₈)	C₂²	1	1	yes	characteristic
15	#14273651	—	2	1	no	—
16	#14273647	—	2	1	no	—
17	(C₂×D₅₉)⋊C₈	—	2	1	no	—
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	#106×C₅₉	—	2	1	no	—
22	C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
23	C₂×Q₈×C₅₉	—	2	1	no	—
24	C₂×Q₈×C₅₉	—	2	1	no	—
25	C₂×D₂₃₆	D₄	1	1	yes	characteristic
26	C₂×C₄×D₅₉	—	2	1	no	—
27	C₂×C₈×C₅₉	D₄	1	1	yes	characteristic
28	C₂×C₈×C₅₉	D₄	1	1	yes	characteristic
29	C₄⋊C₄×C₅₉	—	2	1	no	—
30	C₄⋊C₄×C₅₉	—	2	1	no	—
31	C₄⋊C₄×C₅₉	—	2	1	no	—
32	C₄⋊C₄×C₅₉	—	2	1	no	—
33	C₄²×C₅₉	C₂³	1	1	yes	characteristic
34	#11745671	—	4	2	no	—
35	(C₂×D₅₉)⋊C₄	—	2	1	no	—
36	C₄⋊Dic₅₉	D₄	1	1	yes	characteristic
37	C₂×C₅₉⋊C₈	—	2	1	no	—
38	C₂×C₅₉⋊C₈	—	2	1	no	—
39	C₂×C₅₉⋊C₈	—	2	1	no	—
40	#11745659	—	4	2	no	—
41	C₂²×D₅₉	—	2	1	no	—
42	Q₈×C₅₉	—	4	1	no	—
43	Q₈×C₅₉	—	4	1	no	—
44	Q₈×C₅₉	—	8	1	no	—
45	C₂×C₄×C₅₉	—	4	1	no	—
46	C₂×C₄×C₅₉	—	4	1	no	—
47	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
48	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₅₉	#41	1	1	yes	characteristic
50	C₂×Dic₅₉	—	2	1	no	—
51	D₂₃₆	—	2	1	no	—
52	D₂₃₆	—	2	1	no	—
53	C₄×D₅₉	—	4	1	no	—
54	C₄×D₅₉	—	4	1	no	—
55	C₈×C₅₉	—	2	1	no	—
56	C₈×C₅₉	SD₁₆	1	2	yes	complemented
57	C₅₉⋊C₈	—	4	1	no	—
58	C₅₉⋊C₈	—	4	1	no	—
59	C₅₉⋊C₈	—	4	1	no	—
60	C₂²×C₅₉	#121	1	1	yes	characteristic
61	C₂×D₅₉	—	4	1	no	—
62	C₂×D₅₉	—	4	1	no	—
63	C₂×D₅₉	—	4	1	no	—
64	C₄×C₅₉	—	2	1	no	—
65	C₄×C₅₉	—	8	1	no	—
66	C₄×C₅₉	—	8	1	no	—
67	C₄×C₅₉	C₂×SD₁₆	1	1	yes	characteristic
68	C₄×C₅₉	AGL(1,ℤ/8)	1	1	yes	characteristic
69	C₄×C₅₉	#137	1	1	yes	characteristic
70	C₄×C₅₉	#137	1	1	yes	characteristic
71	Dic₅₉	—	4	1	no	—
72	#1691	—	59	1	no	maximal, Sylow
73	C₂×C₅₉	#461	1	1	yes	characteristic
74	C₂×C₅₉	#468	1	1	yes	characteristic
75	C₂×C₅₉	#471	1	1	yes	characteristic
76	D₅₉	—	8	1	no	—
77	#498	D₅₉	1	1	yes	characteristic, complemented
78	#476	—	59	1	no	—
79	#461	—	59	1	no	—
80	#436	—	59	1	no	—
81	#335	—	59	1	no	—
82	#333	—	59	1	no	—
83	#332	—	59	1	no	—
84	C₅₉	#1691	1	1	yes	characteristic, complemented, Sylow
85	C₂×SD₁₆	—	118	1	no	—
86	C₂×M₁₆	—	118	1	no	—
87	C₄⋊Q₈	C₂×D₅₉	1	1	yes	characteristic
88	C₄⋊Q₈	C₂×D₅₉	1	1	yes	characteristic
89	C₄×D₄	—	59	1	no	—
90	#106	—	2	1	no	—
91	#106	—	2	1	no	—
92	#106	—	118	1	no	—
93	C₄⋊C₈	—	59	1	no	—
94	C₄⋊C₈	—	59	1	no	—
95	C₄⋊C₈	—	59	1	no	—
96	#103	—	118	1	no	—
97	#102	—	118	1	no	—
98	C₂²⋊C₈	—	118	1	no	—
99	C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
100	C₂×Q₈	—	2	1	no	—
101	C₂×Q₈	—	2	1	no	—
102	C₂×D₄	—	59	1	no	—
103	C₂²×C₄	—	118	1	no	—
104	SD₁₆	—	236	2	no	—
105	M₁₆	—	236	2	no	—
106	C₂×C₈	—	118	1	no	—
107	C₂×C₈	—	118	1	no	—
108	C₂×C₈	—	118	1	no	—
109	C₂×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
110	C₂×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
111	C₄⋊C₄	—	2	1	no	—
112	C₄⋊C₄	—	2	1	no	—
113	C₄⋊C₄	—	2	1	no	—
114	C₄⋊C₄	—	2	1	no	—
115	C₄⋊C₄	—	59	1	no	—
116	C₂²⋊C₄	—	118	1	no	—
117	C₄²	C₂²×D₅₉	1	1	yes	characteristic
118	C₂³	—	118	1	no	—
119	Q₈	—	4	1	no	—
120	Q₈	—	4	1	no	—
121	Q₈	—	8	1	no	—
122	D₄	—	118	1	no	—
123	D₄	—	118	1	no	—
124	C₂×C₄	—	4	1	no	—
125	C₂×C₄	—	4	1	no	—
126	C₂×C₄	—	118	1	no	—
127	C₂×C₄	—	236	1	no	—
128	C₂×C₄	—	236	1	no	—
129	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
130	C₂×C₄	#11745687	1	1	yes	characteristic
131	C₂×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
132	C₈	—	2	1	no	—
133	C₈	—	236	1	no	—
134	C₈	—	236	1	no	—
135	C₈	—	236	1	no	—
136	C₈	#11745671	1	2	yes	complemented
137	C₂²	—	236	1	no	—
138	C₂²	—	236	1	no	—
139	C₂²	—	236	1	no	—
140	C₂²	#14273698	1	1	yes	characteristic
141	C₄	—	2	1	no	—
142	C₄	—	8	1	no	—
143	C₄	—	8	1	no	—
144	C₄	—	236	1	no	—
145	C₄	C₂×#11745671	1	1	yes	characteristic
146	C₄	#14273754	1	1	yes	characteristic
147	C₄	#14273756	1	1	yes	characteristic
148	C₄	#14273758	1	1	yes	characteristic
149	C₂	—	472	1	no	—
150	C₂	#16837604	1	1	yes	characteristic
151	C₂	#16837645	1	1	yes	characteristic
152	C₂	#16837662	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16837689

C₂

yes

maximal, characteristic

#16837604

C₂

yes

maximal, characteristic

#16837481

C₂

yes

maximal, characteristic

#16837155

C₂

yes

maximal, characteristic, complemented

#16837151

C₂

yes

maximal, characteristic, complemented

#16837148

C₂

yes

maximal, characteristic, complemented

C₅₉×#498

C₂

yes

maximal, characteristic, complemented

C₂×#11745671

—

C₂×#11745659

—

C₄×D₂₃₆

C₂²

yes

characteristic

#14273658

—

C₄⋊(C₅₉⋊C₈)

C₂²

yes

characteristic

C₄⋊(C₅₉⋊C₈)

C₂²

yes

characteristic

C₄⋊(C₅₉⋊C₈)

C₂²

yes

characteristic

#14273651

—

#14273647

—

(C₂×D₅₉)⋊C₈

—

C₄⋊Q₈×C₅₉

C₂²

yes

characteristic

C₄⋊Q₈×C₅₉

C₂²

yes

characteristic

#106×C₅₉

—

#106×C₅₉

—

C₄×C₈×C₅₉

C₂²

yes

characteristic

C₂×Q₈×C₅₉

—

C₂×Q₈×C₅₉

—

C₂×D₂₃₆

D₄

yes

characteristic

C₂×C₄×D₅₉

—

C₂×C₈×C₅₉

D₄

yes

characteristic

C₂×C₈×C₅₉

D₄

yes

characteristic

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

—

C₄²×C₅₉

C₂³

yes

characteristic

#11745671

—

(C₂×D₅₉)⋊C₄

—

C₄⋊Dic₅₉

D₄

yes

characteristic

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

#11745659

—

C₂²×D₅₉

—

Q₈×C₅₉

—

Q₈×C₅₉

—

Q₈×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

—

D₂₃₆

—

D₂₃₆

—

C₄×D₅₉

—

C₄×D₅₉

—

C₈×C₅₉

—

C₈×C₅₉

SD₁₆

yes

complemented

C₅₉⋊C₈

—

C₅₉⋊C₈

—

C₅₉⋊C₈

—

C₂²×C₅₉

#121

yes

characteristic

C₂×D₅₉

—

C₂×D₅₉

—

C₂×D₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

C₂×SD₁₆

yes

C₄×C₅₉

yes

C₄×C₅₉

yes

C₄×C₅₉

yes

Dic₅₉

—

—

C₂×C₅₉

yes

C₂×C₅₉

yes

C₂×C₅₉

yes

D₅₉

—

#498

D₅₉

yes

characteristic, complemented

—

—

—

—

—

—

C₅₉

#1691

yes

characteristic, complemented, Sylow

C₂×SD₁₆

—

118

—

C₂×M₁₆

—

118

—

C₄⋊Q₈

C₂×D₅₉

yes

characteristic

C₄⋊Q₈

C₂×D₅₉

yes

characteristic

C₄×D₄

—

#106

—

#106

—

#106

—

118

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

#103

—

118

—

#102

—

118

—

C₂²⋊C₈

—

118

—

C₄×C₈

C₂×D₅₉

yes

characteristic

100

C₂×Q₈

—

101

C₂×Q₈

—

102

C₂×D₄

—

103

C₂²×C₄

—

118

—

104

SD₁₆

—

236

—

105

M₁₆

—

236

—

106

C₂×C₈

—

118

—

107

C₂×C₈

—

118

—

108

C₂×C₈

—

118

—

109

C₂×C₈

C₂²⋊D₅₉

yes

characteristic

110

C₂×C₈

C₂²⋊D₅₉

yes

characteristic

111

C₄⋊C₄

—

112

C₄⋊C₄

—

113

C₄⋊C₄

—

114

C₄⋊C₄

—

115

C₄⋊C₄

—

116

C₂²⋊C₄

—

118

—

117

C₄²

C₂²×D₅₉

yes

characteristic

118

C₂³

—

118

—

119

Q₈

—

120

Q₈

—

121

Q₈

—

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

D₄×D₅₉

yes

characteristic

130

C₂×C₄

#11745687

yes

characteristic

131

C₂×C₄

C₂×C₂²⋊D₅₉

yes

characteristic

132

C₈

—

133

C₈

—

236

—

134

C₈

—

236

—

135

C₈

—

236

—

136

C₈

#11745671

yes

complemented

137

C₂²

—

236

—

138

C₂²

—

236

—

139

C₂²

—

236

—

140

C₂²

#14273698

yes

characteristic

141

C₄

—

142

C₄

—

143

C₄

—

144

C₄

—

236

—

145

C₄

C₂×#11745671

yes

characteristic

146

C₄

#14273754

yes

characteristic

147

C₄

#14273756

yes

characteristic

148

C₄

#14273758

yes

characteristic

149

C₂

—

472

—

150

C₂

#16837604

yes

characteristic

151

C₂

#16837645

yes

characteristic

152

C₂

#16837662

yes

characteristic

	subgroup	G/H
center	C₂²	#14273698
commutator	C₄²×C₅₉	C₂³
Frattini	C₄²	C₂²×D₅₉
Fitting	C₅₉×#498	C₂
socle	C₂²×C₅₉	#121
hypercenter	#498	D₅₉
hypocenter	C₅₉	#1691
LCS	G ▷ C₄²×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #498

subgroup

G/H

center

C₂²

#14273698

commutator

C₄²×C₅₉

C₂³

Frattini

C₄²

C₂²×D₅₉

Fitting

C₅₉×#498

C₂

C₂²×C₅₉

D₅₉

C₅₉

G ▷ C₄²×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #498

N	H ≅ G/N	type
4. #16837155	C₂	semidirect
5. #16837151	C₂	semidirect
6. #16837148	C₂	semidirect
7. C₅₉×#498	C₂	semidirect
56. C₈×C₅₉	SD₁₆	semidirect
77. #498	D₅₉	semidirect
84. C₅₉	#1691	semidirect
136. C₈	#11745671	semidirect

H ≅ G/N

type

4. #16837155

C₂

semidirect

5. #16837151

C₂

semidirect

6. #16837148

C₂

semidirect

7. C₅₉×#498

C₂

semidirect

56. C₈×C₅₉

SD₁₆

semidirect

77. #498

D₅₉

semidirect

84. C₅₉

#1691

semidirect

136. C₈

#11745671

semidirect

Subgroups — #21423592

Conjugacy classes of subgroups

Complements