Group #21423721 — subgroups

Subgroups — #21423721

Order 7552.

198 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 198 conjugacy classes of subgroups — too many to draw. Only its 61 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	#16837805	C₂	1	1	yes	maximal, characteristic, complemented
2	#16837776	C₂	1	1	yes	maximal, characteristic
3	#16837776	C₂	1	1	yes	maximal, characteristic
4	#16837754	C₂	1	1	yes	maximal, characteristic, complemented
5	#16837749	C₂	1	1	yes	maximal, characteristic
6	#16837124	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₉×#498	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#11745671	—	2	2	no	—
9	C₂×#11745671	—	2	2	no	—
10	#14273727	C₂²	1	1	yes	characteristic
11	#14273724	C₂²	1	1	yes	characteristic
12	#14273660	—	2	2	no	—
13	#14273650	—	2	2	no	—
14	#14273649	—	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	#106×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×Dic₁₁₈	—	2	1	no	—
27	C₂×C₈×C₅₉	D₄	1	2	yes	—
28	C₄⋊C₄×C₅₉	—	2	1	no	—
29	C₄⋊C₄×C₅₉	—	2	1	no	—
30	C₄⋊C₄×C₅₉	—	2	1	no	—
31	C₄⋊C₄×C₅₉	—	2	1	no	—
32	C₄²×C₅₉	C₂³	1	1	yes	characteristic
33	#11745671	—	4	4	no	—
34	#11745671	—	4	4	no	—
35	C₄⋊Dic₅₉	—	2	1	no	—
36	C₄⋊Dic₅₉	—	2	1	no	—
37	C₂×C₅₉⋊C₈	D₄	1	2	yes	—
38	C₂×C₅₉⋊C₈	D₄	1	2	yes	—
39	C₂²×D₅₉	—	4	1	no	—
40	Q₈×C₅₉	—	4	2	no	—
41	Q₈×C₅₉	—	4	2	no	—
42	C₂×C₄×C₅₉	—	4	1	no	—
43	C₂×C₄×C₅₉	—	4	1	no	—
44	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
47	C₂×Dic₅₉	—	4	1	no	—
48	D₂₃₆	—	4	2	no	—
49	D₂₃₆	—	4	2	no	—
50	D₂₃₆	—	4	2	no	—
51	Dic₁₁₈	—	4	2	no	—
52	C₈×C₅₉	—	2	2	no	—
53	C₅₉⋊C₈	—	2	2	no	—
54	C₅₉⋊C₈	SD₁₆	1	4	yes	complemented
55	C₂²×C₅₉	#127	1	1	yes	characteristic
56	C₂×D₅₉	—	8	1	no	—
57	C₂×D₅₉	—	8	1	no	—
58	C₂×D₅₉	—	8	1	no	—
59	C₄×C₅₉	—	8	1	no	—
60	C₄×C₅₉	—	8	1	no	—
61	C₄×C₅₉	C₂×SD₁₆	1	2	yes	—
62	C₄×C₅₉	AGL(1,ℤ/8)	1	2	yes	—
63	C₄×C₅₉	#137	1	2	yes	—
64	Dic₅₉	—	8	1	no	—
65	#1714	—	59	1	no	maximal, Sylow
66	C₂×C₅₉	#492	1	1	yes	characteristic
67	C₂×C₅₉	#496	1	1	yes	characteristic
68	C₂×C₅₉	#497	1	1	yes	characteristic
69	D₅₉	—	16	1	no	—
70	#500	—	59	1	no	—
71	#498	D₅₉	1	1	yes	characteristic, complemented
72	#492	—	59	1	no	—
73	#492	—	59	1	no	—
74	#487	—	59	1	no	—
75	#486	—	59	1	no	—
76	#322	—	59	1	no	—
77	C₅₉	#1714	1	1	yes	characteristic, complemented, Sylow
78	C₂×SD₁₆	—	118	2	no	—
79	C₂×SD₁₆	—	118	2	no	—
80	C₄⋊Q₈	—	59	1	no	—
81	C₄⋊Q₈	C₂×D₅₉	1	1	yes	characteristic
82	C₄⋊Q₈	C₂×D₅₉	1	1	yes	characteristic
83	#127	—	59	1	no	—
84	#107	—	118	2	no	—
85	#106	—	2	2	no	—
86	#103	—	118	2	no	—
87	#102	—	118	2	no	—
88	C₄×C₈	—	59	1	no	—
89	C₄×C₈	—	59	1	no	—
90	C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
91	C₂×Q₈	—	2	1	no	—
92	C₂×Q₈	—	2	1	no	—
93	C₂×Q₈	—	118	1	no	—
94	C₂×D₄	—	118	1	no	—
95	C₂×D₄	—	118	1	no	—
96	C₂×D₄	—	118	1	no	—
97	SD₁₆	—	236	4	no	—
98	SD₁₆	—	236	4	no	—
99	C₂×C₈	—	59	2	no	—
100	C₂×C₈	—	59	2	no	—
101	C₂×C₈	C₂²⋊D₅₉	1	2	yes	—
102	C₄⋊C₄	—	2	1	no	—
103	C₄⋊C₄	—	2	1	no	—
104	C₄⋊C₄	—	2	1	no	—
105	C₄⋊C₄	—	2	1	no	—
106	C₄⋊C₄	—	118	1	no	—
107	C₄⋊C₄	—	118	1	no	—
108	C₄²	C₂²×D₅₉	1	1	yes	characteristic
109	C₂³	—	236	1	no	—
110	Q₈	—	4	2	no	—
111	Q₈	—	4	2	no	—
112	Q₈	—	236	2	no	—
113	D₄	—	236	2	no	—
114	D₄	—	236	2	no	—
115	D₄	—	236	2	no	—
116	C₂×C₄	—	4	1	no	—
117	C₂×C₄	—	4	1	no	—
118	C₂×C₄	—	236	1	no	—
119	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
120	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
121	C₂×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
122	C₈	—	2	2	no	—
123	C₈	—	59	4	no	—
124	C₈	—	118	2	no	—
125	C₂²	—	472	1	no	—
126	C₂²	—	472	1	no	—
127	C₂²	—	472	1	no	—
128	C₂²	#14273723	1	1	yes	characteristic
129	C₄	—	8	1	no	—
130	C₄	—	8	1	no	—
131	C₄	—	472	1	no	—
132	C₄	SD₁₆×D₅₉	1	2	yes	—
133	C₄	#14273754	1	2	yes	—
134	C₄	#14273758	1	2	yes	—
135	C₂	—	944	1	no	—
136	C₂	#16837773	1	1	yes	characteristic
137	C₂	#16837788	1	1	yes	characteristic
138	C₂	#16837794	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16837805

C₂

yes

maximal, characteristic, complemented

#16837776

C₂

yes

maximal, characteristic

#16837776

C₂

yes

maximal, characteristic

#16837754

C₂

yes

maximal, characteristic, complemented

#16837749

C₂

yes

maximal, characteristic

#16837124

C₂

yes

maximal, characteristic, complemented

C₅₉×#498

C₂

yes

maximal, characteristic, complemented

C₂×#11745671

—

C₂×#11745671

—

#14273727

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

#106×C₅₉

—

C₄×C₈×C₅₉

C₂²

yes

characteristic

C₂×Q₈×C₅₉

—

C₂×Q₈×C₅₉

—

C₂×D₂₃₆

—

C₂×D₂₃₆

—

C₂×D₂₃₆

—

C₂×Dic₁₁₈

—

C₂×C₈×C₅₉

D₄

yes

—

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

—

C₄²×C₅₉

C₂³

yes

characteristic

#11745671

—

#11745671

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₂×C₅₉⋊C₈

D₄

yes

—

C₂×C₅₉⋊C₈

D₄

yes

—

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

C₂×Dic₅₉

—

D₂₃₆

—

D₂₃₆

—

D₂₃₆

—

Dic₁₁₈

—

C₈×C₅₉

—

C₅₉⋊C₈

—

C₅₉⋊C₈

SD₁₆

yes

complemented

C₂²×C₅₉

#127

yes

characteristic

C₂×D₅₉

—

C₂×D₅₉

—

C₂×D₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

C₂×SD₁₆

yes

—

C₄×C₅₉

AGL(1,ℤ/8)

yes

—

C₄×C₅₉

#137

yes

—

Dic₅₉

—

—

C₂×C₅₉

yes

C₂×C₅₉

yes

C₂×C₅₉

yes

D₅₉

—

#500

—

#498

D₅₉

yes

characteristic, complemented

—

—

—

—

—

C₅₉

#1714

yes

characteristic, complemented, Sylow

C₂×SD₁₆

—

118

—

C₂×SD₁₆

—

118

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₅₉

yes

characteristic

C₄⋊Q₈

C₂×D₅₉

yes

characteristic

#127

—

#107

—

118

—

#106

—

#103

—

118

—

#102

—

118

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×D₅₉

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

118

—

C₂×D₄

—

118

—

C₂×D₄

—

118

—

C₂×D₄

—

118

—

SD₁₆

—

236

—

SD₁₆

—

236

—

C₂×C₈

—

100

C₂×C₈

—

101

C₂×C₈

C₂²⋊D₅₉

yes

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

106

C₄⋊C₄

—

118

—

107

C₄⋊C₄

—

118

—

108

C₄²

C₂²×D₅₉

yes

characteristic

109

C₂³

—

236

—

110

Q₈

—

111

Q₈

—

112

Q₈

—

236

—

113

D₄

—

236

—

114

D₄

—

236

—

115

D₄

—

236

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

236

—

119

C₂×C₄

D₄×D₅₉

yes

characteristic

120

C₂×C₄

D₄×D₅₉

yes

characteristic

121

C₂×C₄

C₂×C₂²⋊D₅₉

yes

characteristic

122

C₈

—

123

C₈

—

124

C₈

—

118

—

125

C₂²

—

472

—

126

C₂²

—

472

—

127

C₂²

—

472

—

128

C₂²

#14273723

yes

characteristic

129

C₄

—

130

C₄

—

131

C₄

—

472

—

132

C₄

SD₁₆×D₅₉

yes

—

133

C₄

#14273754

yes

—

134

C₄

#14273758

yes

—

135

C₂

—

944

—

136

C₂

#16837773

yes

characteristic

137

C₂

#16837788

yes

characteristic

138

C₂

#16837794

yes

characteristic

	subgroup	G/H
center	C₂²	#14273723
commutator	C₄²×C₅₉	C₂³
Frattini	C₄²	C₂²×D₅₉
Fitting	C₅₉×#498	C₂
socle	C₂²×C₅₉	#127
hypercenter	#498	D₅₉
hypocenter	C₅₉	#1714
LCS	G ▷ C₄²×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #498

subgroup

G/H

center

C₂²

#14273723

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
1. #16837805	C₂	semidirect
4. #16837754	C₂	semidirect
6. #16837124	C₂	semidirect
7. C₅₉×#498	C₂	semidirect
54. C₅₉⋊C₈	SD₁₆	semidirect
71. #498	D₅₉	semidirect
77. C₅₉	#1714	semidirect

H ≅ G/N

type

1. #16837805

C₂

semidirect

4. #16837754

C₂

semidirect

6. #16837124

C₂

semidirect

7. C₅₉×#498

C₂

semidirect

54. C₅₉⋊C₈

SD₁₆

semidirect

71. #498

D₅₉

semidirect

77. C₅₉

#1714

semidirect

Subgroups — #21423721

Conjugacy classes of subgroups

Complements