Group #21425638 — subgroups

Subgroups — #21425638

Order 7552.

254 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 254 conjugacy classes of subgroups — too many to draw. Only its 95 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×#14273659	C₂	1	2	yes	maximal
2	C₂×#14273652	C₂	1	2	yes	maximal
3	#16837308	C₂	1	1	yes	maximal, characteristic
4	#16837272	C₂	1	1	yes	maximal, characteristic
5	C₅₉×#336	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₁₁₈	C₂²	1	1	yes	characteristic
7	C₂×C₄⋊Dic₅₉	—	2	2	no	—
8	C₂×C₄⋊Dic₅₉	C₂²	1	2	yes	—
9	C₂×C₄⋊Dic₅₉	C₂²	1	1	yes	characteristic
10	#14273659	—	2	4	no	—
11	#14273652	—	2	4	no	—
12	C₅₉⋊#95	—	2	2	no	—
13	C₂²×C₈×C₅₉	C₂²	1	2	yes	—
14	C₂×C₄²×C₅₉	C₂²	1	1	yes	characteristic
15	C₂²×C₄×C₅₉	D₄	1	2	yes	—
16	C₂²×C₄×C₅₉	C₂³	1	1	yes	characteristic
17	C₂²×Dic₅₉	—	2	1	no	—
18	C₂²×Dic₅₉	—	2	1	no	—
19	C₂²×Dic₅₉	—	2	2	no	—
20	C₂×Dic₁₁₈	—	2	2	no	—
21	C₂×Dic₁₁₈	—	2	2	no	—
22	C₂×Dic₁₁₈	—	2	1	no	—
23	C₂×Dic₁₁₈	Q₈	1	2	yes	—
24	C₂×C₈×C₅₉	—	2	2	no	—
25	C₂×C₈×C₅₉	—	2	2	no	—
26	C₂×C₈×C₅₉	D₄	1	4	yes	—
27	C₄²×C₅₉	—	2	2	no	—
28	C₄⋊Dic₅₉	—	2	2	no	—
29	C₄⋊Dic₅₉	—	2	2	no	—
30	C₄⋊Dic₅₉	—	2	2	no	—
31	C₄⋊Dic₅₉	—	4	4	no	—
32	C₂³×C₅₉	C₂×D₄	1	1	yes	characteristic
33	C₂×C₄×C₅₉	—	2	2	no	—
34	C₂×C₄×C₅₉	—	2	2	no	—
35	C₂×C₄×C₅₉	SD₁₆	1	4	yes	—
36	C₂×C₄×C₅₉	C₂×D₄	1	2	yes	—
37	C₂×C₄×C₅₉	C₂×Q₈	1	1	yes	characteristic
38	C₂×C₄×C₅₉	#41	1	2	yes	—
39	C₂×C₄×C₅₉	#41	1	1	yes	characteristic
40	C₂×Dic₅₉	—	2	2	no	—
41	C₂×Dic₅₉	—	4	1	no	—
42	C₂×Dic₅₉	—	4	1	no	—
43	C₂×Dic₅₉	—	4	2	no	—
44	C₂×Dic₅₉	—	4	1	no	—
45	C₂×Dic₅₉	—	4	4	no	—
46	C₂×Dic₅₉	—	4	2	no	—
47	Dic₁₁₈	—	2	4	no	—
48	Dic₁₁₈	—	4	2	no	—
49	C₈×C₅₉	—	2	4	no	—
50	C₂²×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
51	C₂²×C₅₉	#123	1	1	yes	characteristic
52	C₂²×C₅₉	#127	1	1	yes	characteristic
53	C₂²×C₅₉	C₂×SD₁₆	1	2	yes	—
54	C₂²×C₅₉	#137	1	2	yes	—
55	C₄×C₅₉	—	2	4	no	—
56	C₄×C₅₉	C₂²⋊Q₈	1	2	yes	—
57	C₄×C₅₉	#123	1	2	yes	—
58	Dic₅₉	—	4	2	no	—
59	Dic₅₉	—	8	2	no	—
60	Dic₅₉	—	8	1	no	—
61	#2096	—	59	1	no	maximal, Sylow
62	C₂×C₅₉	#399	1	1	yes	characteristic
63	C₂×C₅₉	#475	1	2	yes	—
64	C₂×C₅₉	#483	1	2	yes	—
65	C₂×C₅₉	#492	1	1	yes	characteristic
66	C₂×C₅₉	#497	1	1	yes	characteristic
67	C₂×#106	—	59	2	no	—
68	C₂×#103	—	59	2	no	—
69	#391	—	59	1	no	—
70	#384	—	59	1	no	—
71	#336	D₅₉	1	1	yes	characteristic
72	C₅₉	#2096	1	1	yes	characteristic, complemented, Sylow
73	C₂²×Q₈	—	59	1	no	—
74	C₂²×C₈	C₂×D₅₉	1	2	yes	—
75	C₂×C₄⋊C₄	—	59	2	no	—
76	C₂×C₄⋊C₄	—	59	1	no	—
77	C₂×C₄⋊C₄	—	118	2	no	—
78	C₂×C₄²	C₂×D₅₉	1	1	yes	characteristic
79	#106	—	118	4	no	—
80	#103	—	118	4	no	—
81	#95	—	118	2	no	—
82	C₂×Q₈	—	59	2	no	—
83	C₂×Q₈	—	118	1	no	—
84	C₂×Q₈	—	118	2	no	—
85	C₂×Q₈	—	118	2	no	—
86	C₂²×C₄	—	118	1	no	—
87	C₂²×C₄	—	118	1	no	—
88	C₂²×C₄	—	118	2	no	—
89	C₂²×C₄	D₂₃₆	1	2	yes	—
90	C₂²×C₄	C₂²×D₅₉	1	1	yes	characteristic
91	C₂×C₈	—	2	2	no	—
92	C₂×C₈	—	2	2	no	—
93	C₂×C₈	D₂₃₆	1	4	yes	—
94	C₄⋊C₄	—	118	2	no	—
95	C₄⋊C₄	—	118	2	no	—
96	C₄⋊C₄	—	118	2	no	—
97	C₄⋊C₄	—	236	4	no	—
98	C₄²	—	2	2	no	—
99	C₂³	C₂×D₂₃₆	1	1	yes	characteristic
100	Q₈	—	118	4	no	—
101	Q₈	—	236	2	no	—
102	C₂×C₄	—	2	2	no	—
103	C₂×C₄	—	2	2	no	—
104	C₂×C₄	—	118	2	no	—
105	C₂×C₄	—	236	2	no	—
106	C₂×C₄	—	236	1	no	—
107	C₂×C₄	—	236	4	no	—
108	C₂×C₄	—	236	2	no	—
109	C₂×C₄	—	236	1	no	—
110	C₂×C₄	—	236	1	no	—
111	C₂×C₄	#11745660	1	4	yes	—
112	C₂×C₄	C₂×D₂₃₆	1	2	yes	—
113	C₂×C₄	#11745687	1	2	yes	—
114	C₂×C₄	#11745687	1	1	yes	characteristic
115	C₂×C₄	Q₈×D₅₉	1	1	yes	characteristic
116	C₈	—	2	4	no	—
117	C₂²	#14273707	1	1	yes	characteristic
118	C₂²	#14273711	1	1	yes	characteristic
119	C₂²	#14273724	1	1	yes	characteristic
120	C₂²	C₂×#11745660	1	2	yes	—
121	C₂²	#14273760	1	2	yes	—
122	C₄	—	2	4	no	—
123	C₄	—	472	2	no	—
124	C₄	—	472	1	no	—
125	C₄	—	236	2	no	—
126	C₄	#14273707	1	2	yes	—
127	C₄	#14273711	1	2	yes	—
128	C₂	#16837339	1	1	yes	characteristic
129	C₂	#16837688	1	2	yes	—
130	C₂	#16837733	1	2	yes	—
131	C₂	#16837775	1	1	yes	characteristic
132	C₂	#16837796	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14273659

C₂

yes

maximal

C₂×#14273652

C₂

yes

maximal

#16837308

C₂

yes

maximal, characteristic

#16837272

C₂

yes

maximal, characteristic

C₅₉×#336

C₂

yes

maximal, characteristic

C₂²×Dic₁₁₈

C₂²

yes

characteristic

C₂×C₄⋊Dic₅₉

—

C₂×C₄⋊Dic₅₉

C₂²

yes

—

C₂×C₄⋊Dic₅₉

C₂²

yes

characteristic

#14273659

—

#14273652

—

C₅₉⋊#95

—

C₂²×C₈×C₅₉

C₂²

yes

—

C₂×C₄²×C₅₉

C₂²

yes

characteristic

C₂²×C₄×C₅₉

D₄

yes

—

C₂²×C₄×C₅₉

C₂³

yes

characteristic

C₂²×Dic₅₉

—

C₂²×Dic₅₉

—

C₂²×Dic₅₉

—

C₂×Dic₁₁₈

—

C₂×Dic₁₁₈

—

C₂×Dic₁₁₈

—

C₂×Dic₁₁₈

Q₈

yes

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

D₄

yes

—

C₄²×C₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₂³×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

SD₁₆

yes

—

C₂×C₄×C₅₉

C₂×D₄

yes

—

C₂×C₄×C₅₉

C₂×Q₈

yes

characteristic

C₂×C₄×C₅₉

#41

yes

—

C₂×C₄×C₅₉

#41

yes

characteristic

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

Dic₁₁₈

—

Dic₁₁₈

—

C₈×C₅₉

—

C₂²×C₅₉

C₂²⋊Q₈

yes

C₂²×C₅₉

yes

C₂²×C₅₉

yes

C₂²×C₅₉

C₂×SD₁₆

yes

—

C₂²×C₅₉

#137

yes

—

C₄×C₅₉

—

C₄×C₅₉

C₂²⋊Q₈

yes

—

C₄×C₅₉

#123

yes

—

Dic₅₉

—

Dic₅₉

—

Dic₅₉

—

—

C₂×C₅₉

yes

C₂×C₅₉

yes

—

C₂×C₅₉

#483

yes

—

C₂×C₅₉

yes

C₂×C₅₉

yes

—

—

—

—

D₅₉

yes

characteristic

C₅₉

#2096

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₂²×C₈

C₂×D₅₉

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

118

—

C₂×C₄²

C₂×D₅₉

yes

characteristic

#106

—

118

—

#103

—

118

—

#95

—

118

—

C₂×Q₈

—

C₂×Q₈

—

118

—

C₂×Q₈

—

118

—

C₂×Q₈

—

118

—

C₂²×C₄

—

118

—

C₂²×C₄

—

118

—

C₂²×C₄

—

118

—

C₂²×C₄

D₂₃₆

yes

—

C₂²×C₄

C₂²×D₅₉

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

D₂₃₆

yes

—

C₄⋊C₄

—

118

—

C₄⋊C₄

—

118

—

C₄⋊C₄

—

118

—

C₄⋊C₄

—

236

—

C₄²

—

C₂³

C₂×D₂₃₆

yes

characteristic

100

Q₈

—

118

—

101

Q₈

—

236

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

118

—

105

C₂×C₄

—

236

—

106

C₂×C₄

—

236

—

107

C₂×C₄

—

236

—

108

C₂×C₄

—

236

—

109

C₂×C₄

—

236

—

110

C₂×C₄

—

236

—

111

C₂×C₄

#11745660

yes

—

112

C₂×C₄

C₂×D₂₃₆

yes

—

113

C₂×C₄

#11745687

yes

—

114

C₂×C₄

#11745687

yes

characteristic

115

C₂×C₄

Q₈×D₅₉

yes

characteristic

116

C₈

—

117

C₂²

#14273707

yes

characteristic

118

C₂²

#14273711

yes

characteristic

119

C₂²

#14273724

yes

characteristic

120

C₂²

C₂×#11745660

yes

—

121

C₂²

#14273760

yes

—

122

C₄

—

123

C₄

—

472

—

124

C₄

—

472

—

125

C₄

—

236

—

126

C₄

#14273707

yes

—

127

C₄

#14273711

yes

—

128

C₂

#16837339

yes

characteristic

129

C₂

#16837688

yes

—

130

C₂

#16837733

yes

—

131

C₂

#16837775

yes

characteristic

132

C₂

#16837796

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×D₂₃₆
commutator	C₂²×C₄×C₅₉	C₂³
Frattini	C₂²×C₄	C₂²×D₅₉
Fitting	C₅₉×#336	C₂
socle	C₂³×C₅₉	C₂×D₄
hypercenter	#336	D₅₉
hypocenter	C₅₉	#2096
LCS	G ▷ C₂²×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #336

subgroup

G/H

center

C₂³

C₂×D₂₃₆

commutator

C₂²×C₄×C₅₉

C₂³

Frattini

C₂²×C₄

C₂²×D₅₉

Fitting

C₅₉×#336

C₂

socle

C₂³×C₅₉

C₂×D₄

D₅₉

C₅₉

G ▷ C₂²×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #336

N	H ≅ G/N	type
72. C₅₉	#2096	semidirect

H ≅ G/N

type

72. C₅₉

#2096

semidirect

Subgroups — #21425638

Conjugacy classes of subgroups

Complements