Group #21434478 — subgroups

Subgroups — #21434478

Order 7552.

380 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 141 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×#14273727	C₂	1	1	yes	maximal, characteristic
2	#16837772	C₂	1	4	yes	maximal, complemented
3	#16837754	C₂	1	4	yes	maximal, complemented
4	#16837410	C₂	1	4	yes	maximal
5	C₄×#11745664	C₂	1	1	yes	maximal, characteristic
6	C₅₉×#533	C₂	1	1	yes	maximal, characteristic
7	C₂²×Dic₁₁₈	C₂²	1	2	yes	—
8	C₂×#11745664	C₂²	1	2	yes	—
9	#14273727	—	2	2	no	—
10	#14273727	C₂²	1	2	yes	—
11	#14273727	C₂²	1	2	yes	—
12	C₂×C₄⋊Dic₅₉	—	2	2	no	—
13	#14273650	C₂²	1	16	yes	—
14	#14273638	C₂²	1	2	yes	—
15	C₄×C₅₉⋊C₈	C₂²	1	2	yes	—
16	C₄⋊Q₈×C₅₉	C₂²	1	2	yes	—
17	#125×C₅₉	C₂²	1	2	yes	—
18	C₂²⋊Q₈×C₅₉	—	2	2	no	—
19	C₄×Q₈×C₅₉	—	2	2	no	—
20	#117×C₅₉	C₂²	1	2	yes	—
21	C₂×C₄²×C₅₉	C₂²	1	1	yes	characteristic
22	C₂×Q₈×C₅₉	—	2	2	no	—
23	C₂²×C₄×C₅₉	D₄	1	2	yes	—
24	C₂²×C₄×C₅₉	C₂³	1	1	yes	characteristic
25	C₂²×Dic₅₉	—	2	2	no	—
26	C₂×Dic₁₁₈	—	2	8	no	—
27	C₂×Dic₁₁₈	—	2	2	no	—
28	C₂×Dic₁₁₈	C₂³	1	4	yes	—
29	C₄⋊C₄×C₅₉	—	2	4	no	—
30	C₄⋊C₄×C₅₉	—	2	2	no	—
31	C₄⋊C₄×C₅₉	C₂³	1	4	yes	—
32	C₂²⋊C₄×C₅₉	—	2	2	no	—
33	C₄²×C₅₉	—	2	2	no	—
34	C₄²×C₅₉	D₄	1	2	yes	—
35	C₄²×C₅₉	C₂³	1	1	yes	characteristic
36	C₄²×C₅₉	C₂³	1	1	yes	characteristic
37	C₄⋊Dic₅₉	—	2	4	no	—
38	C₄⋊Dic₅₉	—	2	4	no	—
39	#11745664	—	2	4	no	—
40	C₂×C₅₉⋊C₈	C₂³	1	4	yes	—
41	C₂³×C₅₉	C₂×D₄	1	1	yes	characteristic
42	Q₈×C₅₉	—	4	4	no	—
43	C₂×C₄×C₅₉	—	2	4	no	—
44	C₂×C₄×C₅₉	—	2	4	no	—
45	C₂×C₄×C₅₉	C₂×D₄	1	2	yes	—
46	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
47	C₂×C₄×C₅₉	C₂×D₄	1	2	yes	—
48	C₂×C₄×C₅₉	#41	1	4	yes	—
49	C₂×C₄×C₅₉	C₂⁴	1	1	yes	characteristic
50	C₂×Dic₅₉	—	2	4	no	—
51	C₂×Dic₅₉	—	4	4	no	—
52	Dic₁₁₈	—	2	8	no	—
53	Dic₁₁₈	—	4	4	no	—
54	C₅₉⋊C₈	—	2	4	no	—
55	C₂²×C₅₉	—	2	2	no	—
56	C₂²×C₅₉	#124	1	2	yes	—
57	C₂²×C₅₉	C₂²×D₄	1	1	yes	characteristic
58	C₄×C₅₉	—	2	2	no	—
59	C₄×C₅₉	—	4	4	no	—
60	C₄×C₅₉	#124	1	2	yes	—
61	C₄×C₅₉	#137	1	4	yes	—
62	C₄×C₅₉	C₂×#41	1	2	yes	—
63	Dic₅₉	—	4	4	no	—
64	#3139	—	59	1	no	maximal, Sylow
65	C₂×C₅₉	—	2	2	no	—
66	C₂×C₅₉	C₂×#124	1	1	yes	characteristic
67	C₂×C₅₉	C₂×#137	1	2	yes	—
68	#533	D₅₉	1	1	yes	characteristic
69	C₂×C₄⋊Q₈	—	59	1	no	—
70	#491	—	59	4	no	—
71	#487	—	59	4	no	—
72	#419	—	59	4	no	—
73	C₄×M₁₆	—	59	1	no	—
74	C₅₉	#3139	1	1	yes	characteristic, complemented, Sylow
75	C₂²×Q₈	—	59	2	no	—
76	C₂×M₁₆	—	59	2	no	—
77	C₄⋊Q₈	—	59	2	no	—
78	C₄⋊Q₈	—	59	2	no	—
79	C₄⋊Q₈	—	118	2	no	—
80	C₄⋊Q₈	C₂×D₅₉	1	2	yes	—
81	#125	C₂×D₅₉	1	2	yes	—
82	C₂²⋊Q₈	—	2	2	no	—
83	C₄×Q₈	—	2	2	no	—
84	#117	C₂×D₅₉	1	2	yes	—
85	C₂×C₄⋊C₄	—	118	2	no	—
86	C₂×C₄²	C₂×D₅₉	1	1	yes	characteristic
87	#103	—	59	16	no	—
88	#97	—	59	2	no	—
89	C₄×C₈	—	59	2	no	—
90	C₂×Q₈	—	2	2	no	—
91	C₂×Q₈	—	59	4	no	—
92	C₂×Q₈	—	118	2	no	—
93	C₂×Q₈	—	118	8	no	—
94	C₂²×C₄	—	118	2	no	—
95	C₂²×C₄	C₂²⋊D₅₉	1	2	yes	—
96	C₂²×C₄	C₂²×D₅₉	1	1	yes	characteristic
97	M₁₆	—	118	4	no	—
98	C₂×C₈	—	59	4	no	—
99	C₄⋊C₄	—	2	4	no	—
100	C₄⋊C₄	—	2	2	no	—
101	C₄⋊C₄	—	118	4	no	—
102	C₄⋊C₄	—	118	4	no	—
103	C₄⋊C₄	C₂²×D₅₉	1	4	yes	—
104	C₂²⋊C₄	—	2	2	no	—
105	C₄²	—	2	2	no	—
106	C₄²	C₂²⋊D₅₉	1	2	yes	—
107	C₄²	C₂²×D₅₉	1	1	yes	characteristic
108	C₄²	C₂²×D₅₉	1	1	yes	characteristic
109	C₂³	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
110	Q₈	—	4	4	no	—
111	Q₈	—	118	8	no	—
112	Q₈	—	236	4	no	—
113	C₂×C₄	—	2	4	no	—
114	C₂×C₄	—	2	4	no	—
115	C₂×C₄	—	118	4	no	—
116	C₂×C₄	—	236	4	no	—
117	C₂×C₄	#11745689	1	4	yes	—
118	C₂×C₄	C₂×C₂²⋊D₅₉	1	2	yes	—
119	C₂×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
120	C₂×C₄	C₂×C₂²⋊D₅₉	1	2	yes	—
121	C₂×C₄	C₂³×D₅₉	1	1	yes	characteristic
122	C₈	—	118	4	no	—
123	C₂²	—	2	2	no	—
124	C₂²	#14273716	1	2	yes	—
125	C₂²	C₂²×C₂²⋊D₅₉	1	1	yes	characteristic
126	C₄	—	2	2	no	—
127	C₄	—	4	4	no	—
128	C₄	—	236	4	no	—
129	C₄	#14273716	1	2	yes	—
130	C₄	#14273757	1	4	yes	—
131	C₄	C₂×#11745689	1	2	yes	—
132	C₂	—	2	2	no	—
133	C₂	C₂×#14273716	1	1	yes	characteristic
134	C₂	C₂×#14273757	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14273727

C₂

yes

maximal, characteristic

#16837772

C₂

yes

maximal, complemented

#16837754

C₂

yes

maximal, complemented

#16837410

C₂

yes

maximal

C₄×#11745664

C₂

yes

maximal, characteristic

C₅₉×#533

C₂

yes

maximal, characteristic

C₂²×Dic₁₁₈

C₂²

yes

—

C₂×#11745664

C₂²

yes

—

#14273727

—

#14273727

C₂²

yes

—

#14273727

C₂²

yes

—

C₂×C₄⋊Dic₅₉

—

#14273650

C₂²

yes

—

#14273638

C₂²

yes

—

C₄×C₅₉⋊C₈

C₂²

yes

—

C₄⋊Q₈×C₅₉

C₂²

yes

—

#125×C₅₉

C₂²

yes

—

C₂²⋊Q₈×C₅₉

—

C₄×Q₈×C₅₉

—

#117×C₅₉

C₂²

yes

—

C₂×C₄²×C₅₉

C₂²

yes

characteristic

C₂×Q₈×C₅₉

—

C₂²×C₄×C₅₉

D₄

yes

—

C₂²×C₄×C₅₉

C₂³

yes

characteristic

C₂²×Dic₅₉

—

C₂×Dic₁₁₈

—

C₂×Dic₁₁₈

—

C₂×Dic₁₁₈

C₂³

yes

—

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

C₂³

yes

—

C₂²⋊C₄×C₅₉

—

C₄²×C₅₉

—

C₄²×C₅₉

D₄

yes

—

C₄²×C₅₉

C₂³

yes

characteristic

C₄²×C₅₉

C₂³

yes

characteristic

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

#11745664

—

C₂×C₅₉⋊C₈

C₂³

yes

—

C₂³×C₅₉

C₂×D₄

yes

characteristic

Q₈×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₂×D₄

yes

—

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

C₂×D₄

yes

—

C₂×C₄×C₅₉

#41

yes

—

C₂×C₄×C₅₉

C₂⁴

yes

characteristic

C₂×Dic₅₉

—

C₂×Dic₅₉

—

Dic₁₁₈

—

Dic₁₁₈

—

C₅₉⋊C₈

—

C₂²×C₅₉

—

C₂²×C₅₉

#124

yes

—

C₂²×C₅₉

C₂²×D₄

yes

characteristic

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

#124

yes

—

C₄×C₅₉

#137

yes

—

C₄×C₅₉

C₂×#41

yes

—

Dic₅₉

—

#3139

—

maximal, Sylow

C₂×C₅₉

—

C₂×C₅₉

C₂×#124

yes

characteristic

C₂×C₅₉

C₂×#137

yes

—

#533

D₅₉

yes

characteristic

C₂×C₄⋊Q₈

—

#491

—

#487

—

#419

—

C₄×M₁₆

—

C₅₉

#3139

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₂×M₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

118

—

C₄⋊Q₈

C₂×D₅₉

yes

—

#125

C₂×D₅₉

yes

—

C₂²⋊Q₈

—

C₄×Q₈

—

#117

C₂×D₅₉

yes

—

C₂×C₄⋊C₄

—

118

—

C₂×C₄²

C₂×D₅₉

yes

characteristic

#103

—

#97

—

C₄×C₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

118

—

C₂×Q₈

—

118

—

C₂²×C₄

—

118

—

C₂²×C₄

C₂²⋊D₅₉

yes

—

C₂²×C₄

C₂²×D₅₉

yes

characteristic

M₁₆

—

118

—

C₂×C₈

—

C₄⋊C₄

—

100

C₄⋊C₄

—

101

C₄⋊C₄

—

118

—

102

C₄⋊C₄

—

118

—

103

C₄⋊C₄

C₂²×D₅₉

yes

—

104

C₂²⋊C₄

—

105

C₄²

—

106

C₄²

C₂²⋊D₅₉

yes

—

107

C₄²

C₂²×D₅₉

yes

characteristic

108

C₄²

C₂²×D₅₉

yes

characteristic

109

C₂³

C₂×C₂²⋊D₅₉

yes

characteristic

110

Q₈

—

111

Q₈

—

118

—

112

Q₈

—

236

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

118

—

116

C₂×C₄

—

236

—

117

C₂×C₄

#11745689

yes

—

118

C₂×C₄

C₂×C₂²⋊D₅₉

yes

—

119

C₂×C₄

C₂×C₂²⋊D₅₉

yes

characteristic

120

C₂×C₄

C₂×C₂²⋊D₅₉

yes

—

121

C₂×C₄

C₂³×D₅₉

yes

characteristic

122

C₈

—

118

—

123

C₂²

—

124

C₂²

#14273716

yes

—

125

C₂²

C₂²×C₂²⋊D₅₉

yes

characteristic

126

C₄

—

127

C₄

—

128

C₄

—

236

—

129

C₄

#14273716

yes

—

130

C₄

#14273757

yes

—

131

C₄

C₂×#11745689

yes

—

132

C₂

—

133

C₂

C₂×#14273716

yes

characteristic

134

C₂

C₂×#14273757

yes

—

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₅₉
commutator	C₂×C₄×C₅₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₅₉
Fitting	C₅₉×#533	C₂
socle	C₂²×C₅₉	C₂²×D₄
hypercenter	#533	D₅₉
hypocenter	C₅₉	#3139
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #533

subgroup

G/H

center

C₂²

C₂²×C₂²⋊D₅₉

commutator

C₂×C₄×C₅₉

C₂⁴

Frattini

C₂×C₄

C₂³×D₅₉

Fitting

C₅₉×#533

C₂

socle

C₂²×C₅₉

C₂²×D₄

D₅₉

C₅₉

G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #533

N	H ≅ G/N	type
2. #16837772	C₂	semidirect
3. #16837754	C₂	semidirect
74. C₅₉	#3139	semidirect

H ≅ G/N

type

2. #16837772

C₂

semidirect

3. #16837754

C₂

semidirect

74. C₅₉

#3139

semidirect

Subgroups — #21434478

Conjugacy classes of subgroups

Complements