Group #21434841 — subgroups

Subgroups — #21434841

Order 7552.

358 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 358 conjugacy classes of subgroups — too many to draw. Only its 133 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	#16837959	C₂	1	2	yes	maximal
2	#16837800	C₂	1	1	yes	maximal, characteristic, complemented
3	#16837782	C₂	1	1	yes	maximal, characteristic, complemented
4	#16837753	C₂	1	2	yes	maximal, complemented
5	#16837660	C₂	1	4	yes	maximal
6	#16837510	C₂	1	4	yes	maximal, complemented
7	C₄×M₁₆×C₅₉	C₂	1	1	yes	maximal, characteristic
8	C₂×Dic₂₃₆	C₂²	1	4	yes	—
9	#14273727	C₂²	1	2	yes	—
10	#14273727	C₂²	1	2	yes	—
11	#14273719	—	2	2	no	—
12	#14273706	—	2	2	no	—
13	#14273706	C₂²	1	4	yes	—
14	C₄×Dic₁₁₈	—	2	2	no	—
15	C₄×Dic₁₁₈	C₂²	1	4	yes	—
16	#14273686	—	2	2	no	—
17	#14273659	C₂²	1	4	yes	—
18	#14273652	C₂²	1	8	yes	—
19	C₂×M₁₆×C₅₉	C₂²	1	2	yes	—
20	C₂×C₄²×C₅₉	C₂²	1	1	yes	characteristic
21	#97×C₅₉	C₂²	1	2	yes	—
22	C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
23	C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
24	C₂²×C₄×C₅₉	D₄	1	2	yes	—
25	C₂²×C₄×C₅₉	C₂³	1	1	yes	characteristic
26	C₂×Dic₁₁₈	—	2	2	no	—
27	C₂×Dic₁₁₈	C₂³	1	4	yes	—
28	M₁₆×C₅₉	—	2	4	no	—
29	C₂×C₈×C₅₉	C₂³	1	2	yes	—
30	C₂×C₈×C₅₉	C₂³	1	2	yes	—
31	C₄²×C₅₉	D₄	1	2	yes	—
32	C₄²×C₅₉	C₂³	1	1	yes	characteristic
33	C₄²×C₅₉	C₂³	1	1	yes	characteristic
34	C₂²⋊Dic₅₉	—	2	4	no	—
35	C₄⋊Dic₅₉	—	2	2	no	—
36	C₄⋊Dic₅₉	—	2	2	no	—
37	C₄⋊Dic₅₉	—	2	2	no	—
38	C₄⋊Dic₅₉	C₂³	1	4	yes	—
39	Dic₅₉⋊C₄	—	2	4	no	—
40	Dic₅₉⋊C₄	—	2	4	no	—
41	C₄×Dic₅₉	—	2	4	no	—
42	Dic₂₃₆	—	2	8	no	—
43	C₂³×C₅₉	C₂×D₄	1	1	yes	characteristic
44	C₂×C₄×C₅₉	—	2	4	no	—
45	C₂×C₄×C₅₉	—	2	2	no	—
46	C₂×C₄×C₅₉	C₂×D₄	1	2	yes	—
47	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
48	C₂×C₄×C₅₉	C₂×D₄	1	2	yes	—
49	C₂×C₄×C₅₉	C₂⁴	1	1	yes	characteristic
50	C₂×Dic₅₉	—	2	4	no	—
51	C₂×Dic₅₉	—	2	4	no	—
52	Dic₁₁₈	—	2	8	no	—
53	Dic₁₁₈	—	4	4	no	—
54	C₈×C₅₉	—	2	2	no	—
55	C₈×C₅₉	#41	1	4	yes	—
56	C₂²×C₅₉	—	2	1	no	—
57	C₂²×C₅₉	—	2	2	no	—
58	C₂²×C₅₉	C₂²×D₄	1	1	yes	characteristic
59	C₄×C₅₉	—	2	1	no	—
60	C₄×C₅₉	—	2	2	no	—
61	C₄×C₅₉	#137	1	4	yes	—
62	C₄×C₅₉	C₂×#41	1	2	yes	—
63	Dic₅₉	—	4	4	no	—
64	Dic₅₉	—	4	4	no	—
65	#3185	—	59	1	no	maximal, Sylow
66	C₂×C₅₉	—	4	1	no	—
67	C₂×C₅₉	#532	1	1	yes	characteristic
68	C₂×C₅₉	C₂×#137	1	2	yes	—
69	#533	—	59	2	no	—
70	#498	—	59	1	no	—
71	#494	—	59	1	no	—
72	#487	—	59	2	no	—
73	#470	—	59	4	no	—
74	#441	—	59	4	no	—
75	C₄×M₁₆	D₅₉	1	1	yes	characteristic
76	C₅₉	#3185	1	1	yes	characteristic, complemented, Sylow
77	C₂×Q₁₆	—	59	4	no	—
78	C₂×M₁₆	C₂×D₅₉	1	2	yes	—
79	C₄⋊Q₈	—	59	2	no	—
80	C₄⋊Q₈	—	59	2	no	—
81	#125	—	118	2	no	—
82	C₂²⋊Q₈	—	59	4	no	—
83	C₂²⋊Q₈	—	118	2	no	—
84	C₄×Q₈	—	59	4	no	—
85	C₄×Q₈	—	118	2	no	—
86	#117	—	118	2	no	—
87	C₂×C₄²	C₂×D₅₉	1	1	yes	characteristic
88	#106	—	59	4	no	—
89	#103	—	59	8	no	—
90	#97	C₂×D₅₉	1	2	yes	—
91	C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
92	C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
93	C₂×Q₈	—	59	4	no	—
94	C₂×Q₈	—	118	2	no	—
95	C₂²×C₄	D₂₃₆	1	2	yes	—
96	C₂²×C₄	C₂²×D₅₉	1	1	yes	characteristic
97	Q₁₆	—	118	8	no	—
98	M₁₆	—	2	4	no	—
99	C₂×C₈	C₂²×D₅₉	1	2	yes	—
100	C₂×C₈	C₂²×D₅₉	1	2	yes	—
101	C₄⋊C₄	—	59	4	no	—
102	C₄⋊C₄	—	118	2	no	—
103	C₄⋊C₄	—	118	2	no	—
104	C₄⋊C₄	—	118	4	no	—
105	C₄⋊C₄	—	118	4	no	—
106	C₄⋊C₄	—	118	2	no	—
107	C₂²⋊C₄	—	118	4	no	—
108	C₄²	—	118	4	no	—
109	C₄²	D₂₃₆	1	2	yes	—
110	C₄²	C₂²×D₅₉	1	1	yes	characteristic
111	C₄²	C₂²×D₅₉	1	1	yes	characteristic
112	C₂³	C₂×D₂₃₆	1	1	yes	characteristic
113	Q₈	—	118	8	no	—
114	Q₈	—	236	4	no	—
115	C₂×C₄	—	2	4	no	—
116	C₂×C₄	—	2	2	no	—
117	C₂×C₄	—	118	4	no	—
118	C₂×C₄	—	118	4	no	—
119	C₂×C₄	C₂×D₂₃₆	1	2	yes	—
120	C₂×C₄	C₂×D₂₃₆	1	1	yes	characteristic
121	C₂×C₄	C₂×D₂₃₆	1	2	yes	—
122	C₂×C₄	C₂³×D₅₉	1	1	yes	characteristic
123	C₈	—	2	2	no	—
124	C₈	#11745685	1	4	yes	—
125	C₂²	—	2	2	no	—
126	C₂²	—	2	1	no	—
127	C₂²	C₂²×D₂₃₆	1	1	yes	characteristic
128	C₄	—	2	2	no	—
129	C₄	—	2	1	no	—
130	C₄	—	236	4	no	—
131	C₄	—	236	4	no	—
132	C₄	#14273760	1	4	yes	—
133	C₄	C₂×#11745685	1	2	yes	—
134	C₂	—	4	1	no	—
135	C₂	#16837952	1	1	yes	characteristic
136	C₂	C₂×#14273760	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#16837959

C₂

yes

maximal

#16837800

C₂

yes

maximal, characteristic, complemented

#16837782

C₂

yes

maximal, characteristic, complemented

#16837753

C₂

yes

maximal, complemented

#16837660

C₂

yes

maximal

#16837510

C₂

yes

maximal, complemented

C₄×M₁₆×C₅₉

C₂

yes

maximal, characteristic

C₂×Dic₂₃₆

C₂²

yes

—

#14273727

C₂²

yes

—

#14273727

C₂²

yes

—

#14273719

—

#14273706

—

#14273706

C₂²

yes

—

C₄×Dic₁₁₈

—

C₄×Dic₁₁₈

C₂²

yes

—

#14273686

—

#14273659

C₂²

yes

—

#14273652

C₂²

yes

—

C₂×M₁₆×C₅₉

C₂²

yes

—

C₂×C₄²×C₅₉

C₂²

yes

characteristic

#97×C₅₉

C₂²

yes

—

C₄×C₈×C₅₉

C₂²

yes

characteristic

C₄×C₈×C₅₉

C₂²

yes

characteristic

C₂²×C₄×C₅₉

D₄

yes

—

C₂²×C₄×C₅₉

C₂³

yes

characteristic

C₂×Dic₁₁₈

—

C₂×Dic₁₁₈

C₂³

yes

—

M₁₆×C₅₉

—

C₂×C₈×C₅₉

C₂³

yes

—

C₂×C₈×C₅₉

C₂³

yes

—

C₄²×C₅₉

D₄

yes

—

C₄²×C₅₉

C₂³

yes

characteristic

C₄²×C₅₉

C₂³

yes

characteristic

C₂²⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

C₂³

yes

—

Dic₅₉⋊C₄

—

Dic₅₉⋊C₄

—

C₄×Dic₅₉

—

Dic₂₃₆

—

C₂³×C₅₉

C₂×D₄

yes

characteristic

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

C₂⁴

yes

characteristic

C₂×Dic₅₉

—

C₂×Dic₅₉

—

Dic₁₁₈

—

Dic₁₁₈

—

C₈×C₅₉

—

C₈×C₅₉

#41

yes

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

C₂²×D₄

yes

characteristic

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

#137

yes

—

C₄×C₅₉

C₂×#41

yes

—

Dic₅₉

—

Dic₅₉

—

#3185

—

maximal, Sylow

C₂×C₅₉

—

C₂×C₅₉

#532

yes

characteristic

C₂×C₅₉

C₂×#137

yes

—

—

—

—

—

—

—

C₄×M₁₆

D₅₉

yes

characteristic

C₅₉

#3185

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

C₂×M₁₆

C₂×D₅₉

yes

—

C₄⋊Q₈

—

C₄⋊Q₈

—

#125

—

118

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

118

—

C₄×Q₈

—

C₄×Q₈

—

118

—

#117

—

118

—

C₂×C₄²

C₂×D₅₉

yes

—

—

C₂×D₅₉

yes

—

C₄×C₈

C₂×D₅₉

yes

characteristic

C₄×C₈

C₂×D₅₉

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

118

—

C₂²×C₄

D₂₃₆

yes

—

C₂²×C₄

C₂²×D₅₉

yes

characteristic

Q₁₆

—

118

—

M₁₆

—

C₂×C₈

C₂²×D₅₉

yes

—

100

C₂×C₈

C₂²×D₅₉

yes

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

118

—

103

C₄⋊C₄

—

118

—

104

C₄⋊C₄

—

118

—

105

C₄⋊C₄

—

118

—

106

C₄⋊C₄

—

118

—

107

C₂²⋊C₄

—

118

—

108

C₄²

—

118

—

109

C₄²

D₂₃₆

yes

—

110

C₄²

C₂²×D₅₉

yes

characteristic

111

C₄²

C₂²×D₅₉

yes

characteristic

112

C₂³

C₂×D₂₃₆

yes

characteristic

113

Q₈

—

118

—

114

Q₈

—

236

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

—

118

C₂×C₄

—

118

—

119

C₂×C₄

C₂×D₂₃₆

yes

—

120

C₂×C₄

C₂×D₂₃₆

yes

characteristic

121

C₂×C₄

C₂×D₂₃₆

yes

—

122

C₂×C₄

C₂³×D₅₉

yes

characteristic

123

C₈

—

124

C₈

#11745685

yes

—

125

C₂²

—

126

C₂²

—

127

C₂²

C₂²×D₂₃₆

yes

characteristic

128

C₄

—

129

C₄

—

130

C₄

—

236

—

131

C₄

—

236

—

132

C₄

#14273760

yes

—

133

C₄

C₂×#11745685

yes

—

134

C₂

—

135

C₂

#16837952

yes

characteristic

136

C₂

C₂×#14273760

yes

—

	subgroup	G/H
center	C₂²	C₂²×D₂₃₆
commutator	C₂×C₄×C₅₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₅₉
Fitting	C₄×M₁₆×C₅₉	C₂
socle	C₂²×C₅₉	C₂²×D₄
hypercenter	C₄×M₁₆	D₅₉
hypocenter	C₅₉	#3185
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄×M₁₆

subgroup

G/H

center

C₂²

C₂²×D₂₃₆

commutator

C₂×C₄×C₅₉

C₂⁴

Frattini

C₂×C₄

C₂³×D₅₉

Fitting

C₄×M₁₆×C₅₉

C₂

socle

C₂²×C₅₉

C₂²×D₄

hypercenter

C₄×M₁₆

D₅₉

hypocenter

C₅₉

#3185

LCS

G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄×M₁₆

N	H ≅ G/N	type
2. #16837800	C₂	semidirect
3. #16837782	C₂	semidirect
4. #16837753	C₂	semidirect
6. #16837510	C₂	semidirect
76. C₅₉	#3185	semidirect

H ≅ G/N

type

2. #16837800

C₂

semidirect

3. #16837782

C₂

semidirect

4. #16837753

C₂

semidirect

6. #16837510

C₂

semidirect

76. C₅₉

#3185

semidirect

Subgroups — #21434841

Conjugacy classes of subgroups

Complements