Group #21423738 — subgroups

Subgroups — #21423738

Order 7552.

208 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 208 conjugacy classes of subgroups — too many to draw. Only its 57 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	#16837779	C₂	1	1	yes	maximal, characteristic
2	#16837776	C₂	1	1	yes	maximal, characteristic, complemented
3	#16837750	C₂	1	1	yes	maximal, characteristic, complemented
4	#16837750	C₂	1	1	yes	maximal, characteristic, complemented
5	#16837750	C₂	1	1	yes	maximal, characteristic, complemented
6	#16837124	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₉×#487	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#11745671	—	2	2	no	—
9	C₂×D₄₇₂	—	2	2	no	—
10	#14273724	C₂²	1	1	yes	characteristic
11	#14273724	C₂²	1	1	yes	characteristic
12	#14273647	—	2	2	no	—
13	#14273647	—	2	2	no	—
14	#14273647	—	2	2	no	—
15	C₄×C₅₉⋊C₈	C₂²	1	1	yes	characteristic
16	C₄×C₅₉⋊C₈	C₂²	1	1	yes	characteristic
17	C₄⋊Q₈×C₅₉	C₂²	1	1	yes	characteristic
18	C₄⋊Q₈×C₅₉	C₂²	1	1	yes	characteristic
19	#103×C₅₉	—	2	2	no	—
20	C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
21	C₂×Q₈×C₅₉	—	2	1	no	—
22	C₂×Q₈×C₅₉	—	2	1	no	—
23	C₂×D₂₃₆	—	2	1	no	—
24	C₂×D₂₃₆	—	2	1	no	—
25	C₂×D₂₃₆	—	2	1	no	—
26	C₂×D₂₃₆	—	2	1	no	—
27	C₂×D₂₃₆	—	2	1	no	—
28	C₂×D₂₃₆	—	2	1	no	—
29	C₂×C₈×C₅₉	D₄	1	2	yes	—
30	C₄⋊C₄×C₅₉	—	2	1	no	—
31	C₄⋊C₄×C₅₉	—	2	1	no	—
32	C₄⋊C₄×C₅₉	—	2	1	no	—
33	C₄⋊C₄×C₅₉	—	2	1	no	—
34	C₄²×C₅₉	C₂³	1	1	yes	characteristic
35	#11745671	—	4	4	no	—
36	C₂×C₅₉⋊C₈	D₄	1	2	yes	—
37	C₂×C₅₉⋊C₈	D₄	1	2	yes	—
38	D₄₇₂	—	4	4	no	—
39	C₂²×D₅₉	—	4	1	no	—
40	C₂²×D₅₉	—	4	1	no	—
41	Q₈×C₅₉	—	4	2	no	—
42	Q₈×C₅₉	—	4	2	no	—
43	C₂×C₄×C₅₉	—	4	1	no	—
44	C₂×C₄×C₅₉	—	4	1	no	—
45	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
47	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
48	D₂₃₆	—	4	2	no	—
49	D₂₃₆	—	4	2	no	—
50	D₂₃₆	—	4	2	no	—
51	D₂₃₆	—	4	2	no	—
52	D₂₃₆	—	4	2	no	—
53	D₂₃₆	—	4	2	no	—
54	C₈×C₅₉	—	2	2	no	—
55	C₅₉⋊C₈	—	2	2	no	—
56	C₅₉⋊C₈	—	2	2	no	—
57	C₂²×C₅₉	#127	1	1	yes	characteristic
58	C₂×D₅₉	—	8	1	no	—
59	C₂×D₅₉	—	8	1	no	—
60	C₂×D₅₉	—	8	1	no	—
61	C₂×D₅₉	—	8	1	no	—
62	C₂×D₅₉	—	8	1	no	—
63	C₂×D₅₉	—	8	1	no	—
64	C₄×C₅₉	—	8	1	no	—
65	C₄×C₅₉	—	8	1	no	—
66	C₄×C₅₉	#135	1	2	yes	—
67	C₄×C₅₉	AGL(1,ℤ/8)	1	2	yes	—
68	C₄×C₅₉	AGL(1,ℤ/8)	1	2	yes	—
69	#1717	—	59	1	no	maximal, Sylow
70	C₂×C₅₉	#495	1	1	yes	characteristic
71	C₂×C₅₉	#496	1	1	yes	characteristic
72	C₂×C₅₉	#496	1	1	yes	characteristic
73	D₅₉	—	16	1	no	—
74	D₅₉	—	16	1	no	—
75	#493	—	59	1	no	—
76	#492	—	59	1	no	—
77	#487	D₅₉	1	1	yes	characteristic, complemented
78	#486	—	59	1	no	—
79	#486	—	59	1	no	—
80	#486	—	59	1	no	—
81	#322	—	59	1	no	—
82	C₅₉	#1717	1	1	yes	characteristic, complemented, Sylow
83	C₂×SD₁₆	—	118	2	no	—
84	C₂×D₈	—	118	2	no	—
85	C₄⋊Q₈	C₂×D₅₉	1	1	yes	characteristic
86	C₄⋊Q₈	C₂×D₅₉	1	1	yes	characteristic
87	#127	—	59	1	no	—
88	#127	—	59	1	no	—
89	#103	—	2	2	no	—
90	#102	—	118	2	no	—
91	#102	—	118	2	no	—
92	#102	—	118	2	no	—
93	C₄×C₈	—	59	1	no	—
94	C₄×C₈	—	59	1	no	—
95	C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
96	C₂×Q₈	—	2	1	no	—
97	C₂×Q₈	—	2	1	no	—
98	C₂×D₄	—	118	1	no	—
99	C₂×D₄	—	118	1	no	—
100	C₂×D₄	—	118	1	no	—
101	C₂×D₄	—	118	1	no	—
102	C₂×D₄	—	118	1	no	—
103	C₂×D₄	—	118	1	no	—
104	SD₁₆	—	236	4	no	—
105	D₈	—	236	4	no	—
106	C₂×C₈	—	59	2	no	—
107	C₂×C₈	—	59	2	no	—
108	C₂×C₈	C₂²⋊D₅₉	1	2	yes	—
109	C₄⋊C₄	—	2	1	no	—
110	C₄⋊C₄	—	2	1	no	—
111	C₄⋊C₄	—	2	1	no	—
112	C₄⋊C₄	—	2	1	no	—
113	C₄²	C₂²×D₅₉	1	1	yes	characteristic
114	C₂³	—	236	1	no	—
115	C₂³	—	236	1	no	—
116	Q₈	—	4	2	no	—
117	Q₈	—	4	2	no	—
118	D₄	—	236	2	no	—
119	D₄	—	236	2	no	—
120	D₄	—	236	2	no	—
121	D₄	—	236	2	no	—
122	D₄	—	236	2	no	—
123	D₄	—	236	2	no	—
124	C₂×C₄	—	4	1	no	—
125	C₂×C₄	—	4	1	no	—
126	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
127	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
128	C₂×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
129	C₈	—	2	2	no	—
130	C₈	—	118	2	no	—
131	C₈	—	118	2	no	—
132	C₂²	—	472	1	no	—
133	C₂²	—	472	1	no	—
134	C₂²	—	472	1	no	—
135	C₂²	—	472	1	no	—
136	C₂²	—	472	1	no	—
137	C₂²	—	472	1	no	—
138	C₂²	#14273723	1	1	yes	characteristic
139	C₄	—	8	1	no	—
140	C₄	—	8	1	no	—
141	C₄	#14273750	1	2	yes	—
142	C₄	#14273752	1	2	yes	—
143	C₄	#14273754	1	2	yes	—
144	C₂	—	944	1	no	—
145	C₂	—	944	1	no	—
146	C₂	#16837787	1	1	yes	characteristic
147	C₂	#16837788	1	1	yes	characteristic
148	C₂	#16837792	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16837779

C₂

yes

maximal, characteristic

#16837776

C₂

yes

maximal, characteristic, complemented

#16837750

C₂

yes

maximal, characteristic, complemented

#16837750

C₂

yes

maximal, characteristic, complemented

#16837750

C₂

yes

maximal, characteristic, complemented

#16837124

C₂

yes

maximal, characteristic, complemented

C₅₉×#487

C₂

yes

maximal, characteristic, complemented

C₂×#11745671

—

C₂×D₄₇₂

—

#14273724

C₂²

yes

characteristic

#14273724

C₂²

yes

—

—

—

C₄×C₅₉⋊C₈

C₂²

yes

characteristic

C₄×C₅₉⋊C₈

C₂²

yes

characteristic

C₄⋊Q₈×C₅₉

C₂²

yes

characteristic

C₄⋊Q₈×C₅₉

C₂²

yes

characteristic

#103×C₅₉

—

C₄×C₈×C₅₉

C₂²

yes

characteristic

C₂×Q₈×C₅₉

—

C₂×Q₈×C₅₉

—

C₂×D₂₃₆

—

C₂×D₂₃₆

—

C₂×D₂₃₆

—

C₂×D₂₃₆

—

C₂×D₂₃₆

—

C₂×D₂₃₆

—

C₂×C₈×C₅₉

D₄

yes

—

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

—

C₄²×C₅₉

C₂³

yes

characteristic

#11745671

—

C₂×C₅₉⋊C₈

D₄

yes

—

C₂×C₅₉⋊C₈

D₄

yes

—

D₄₇₂

—

C₂²×D₅₉

—

C₂²×D₅₉

—

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

C₂×D₄

yes

characteristic

D₂₃₆

—

D₂₃₆

—

D₂₃₆

—

D₂₃₆

—

D₂₃₆

—

D₂₃₆

—

C₈×C₅₉

—

C₅₉⋊C₈

—

C₅₉⋊C₈

—

C₂²×C₅₉

#127

yes

characteristic

C₂×D₅₉

—

C₂×D₅₉

—

C₂×D₅₉

—

C₂×D₅₉

—

C₂×D₅₉

—

C₂×D₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

#135

yes

—

C₄×C₅₉

AGL(1,ℤ/8)

yes

—

C₄×C₅₉

AGL(1,ℤ/8)

yes

—

—

C₂×C₅₉

yes

C₂×C₅₉

yes

C₂×C₅₉

yes

D₅₉

—

D₅₉

—

#493

—

#492

—

#487

D₅₉

yes

characteristic, complemented

—

—

—

—

C₅₉

#1717

yes

characteristic, complemented, Sylow

C₂×SD₁₆

—

118

—

C₂×D₈

—

118

—

C₄⋊Q₈

C₂×D₅₉

yes

characteristic

C₄⋊Q₈

C₂×D₅₉

yes

—

—

—

—

118

—

#102

—

118

—

#102

—

118

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×D₅₉

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

C₂×D₄

—

118

—

C₂×D₄

—

118

—

100

C₂×D₄

—

118

—

101

C₂×D₄

—

118

—

102

C₂×D₄

—

118

—

103

C₂×D₄

—

118

—

104

SD₁₆

—

236

—

105

D₈

—

236

—

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂×C₈

C₂²⋊D₅₉

yes

—

109

C₄⋊C₄

—

110

C₄⋊C₄

—

111

C₄⋊C₄

—

112

C₄⋊C₄

—

113

C₄²

C₂²×D₅₉

yes

characteristic

114

C₂³

—

236

—

115

C₂³

—

236

—

116

Q₈

—

117

Q₈

—

118

D₄

—

236

—

119

D₄

—

236

—

120

D₄

—

236

—

121

D₄

—

236

—

122

D₄

—

236

—

123

D₄

—

236

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

D₄×D₅₉

yes

characteristic

127

C₂×C₄

D₄×D₅₉

yes

characteristic

128

C₂×C₄

C₂×C₂²⋊D₅₉

yes

characteristic

129

C₈

—

130

C₈

—

118

—

131

C₈

—

118

—

132

C₂²

—

472

—

133

C₂²

—

472

—

134

C₂²

—

472

—

135

C₂²

—

472

—

136

C₂²

—

472

—

137

C₂²

—

472

—

138

C₂²

#14273723

yes

characteristic

139

C₄

—

140

C₄

—

141

C₄

#14273750

yes

—

142

C₄

#14273752

yes

—

143

C₄

#14273754

yes

—

144

C₂

—

944

—

145

C₂

—

944

—

146

C₂

#16837787

yes

characteristic

147

C₂

#16837788

yes

characteristic

148

C₂

#16837792

yes

characteristic

	subgroup	G/H
center	C₂²	#14273723
commutator	C₄²×C₅₉	C₂³
Frattini	C₄²	C₂²×D₅₉
Fitting	C₅₉×#487	C₂
socle	C₂²×C₅₉	#127
hypercenter	#487	D₅₉
hypocenter	C₅₉	#1717
LCS	G ▷ C₄²×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #487

subgroup

G/H

center

C₂²

#14273723

commutator

C₄²×C₅₉

C₂³

Frattini

C₄²

C₂²×D₅₉

Fitting

C₅₉×#487

C₂

C₂²×C₅₉

D₅₉

C₅₉

G ▷ C₄²×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #487

N	H ≅ G/N	type
2. #16837776	C₂	semidirect
3. #16837750	C₂	semidirect
4. #16837750	C₂	semidirect
5. #16837750	C₂	semidirect
6. #16837124	C₂	semidirect
7. C₅₉×#487	C₂	semidirect
77. #487	D₅₉	semidirect
82. C₅₉	#1717	semidirect

H ≅ G/N

type

2. #16837776

C₂

semidirect

3. #16837750

C₂

semidirect

4. #16837750

C₂

semidirect

5. #16837750

C₂

semidirect

6. #16837124

C₂

semidirect

7. C₅₉×#487

C₂

semidirect

77. #487

D₅₉

semidirect

82. C₅₉

#1717

semidirect

Subgroups — #21423738

Conjugacy classes of subgroups

Complements