Group #21432522 — subgroups

Subgroups — #21432522

Order 7552.

556 conjugacy classes of subgroups; only the 373 normal subgroups are shown.

normal subgroups only

This group has 556 conjugacy classes of subgroups — too many to draw. Only its 373 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×#14273734	C₂	1	1	yes	maximal, characteristic
2	#41×Dic₅₉	C₂	1	1	yes	maximal, characteristic
3	#16837360	C₂	1	3	yes	maximal, complemented
4	#16837357	C₂	1	3	yes	maximal, complemented
5	M₁₆×Dic₅₉	C₂	1	3	yes	maximal, complemented
6	C₂×#14273637	C₂	1	3	yes	maximal, complemented
7	C₂×#131×C₅₉	C₂	1	1	yes	maximal, characteristic
8	#14273734	C₄	1	8	yes	complemented
9	C₂×#11745664	C₂²	1	3	yes	—
10	C₂²×C₅₉⋊C₈	C₂²	1	3	yes	—
11	Q₈×Dic₅₉	C₄	1	2	yes	—
12	D₄×Dic₅₉	C₄	1	6	yes	—
13	#14273686	C₂²	1	3	yes	—
14	C₂×C₄×Dic₅₉	C₂²	1	3	yes	—
15	#14273637	C₂²	1	1	yes	characteristic
16	#14273637	C₂²	1	3	yes	—
17	#14273637	C₂²	1	3	yes	—
18	#14273637	C₂²	1	3	yes	—
19	C₈×Dic₅₉	C₂²	1	6	yes	—
20	C₂×#41×C₅₉	C₂²	1	1	yes	characteristic
21	#131×C₅₉	C₄	1	8	yes	complemented
22	C₂×M₁₆×C₅₉	C₂²	1	3	yes	—
23	C₂²×C₈×C₅₉	C₂²	1	3	yes	—
24	#41×C₅₉	C₂×C₄	1	4	yes	—
25	#41×C₅₉	C₂×C₄	1	4	yes	—
26	C₂×Q₈×C₅₉	C₂×C₄	1	1	yes	characteristic
27	C₂×D₄×C₅₉	C₂×C₄	1	3	yes	—
28	C₂²×C₄×C₅₉	C₂³	1	3	yes	—
29	C₂²×Dic₅₉	C₂×C₄	1	6	yes	—
30	M₁₆×C₅₉	C₂×C₄	1	12	yes	complemented
31	C₂×C₈×C₅₉	C₂×C₄	1	12	yes	complemented
32	C₂×C₈×C₅₉	C₂³	1	1	yes	characteristic
33	C₂×C₈×C₅₉	C₂³	1	3	yes	—
34	C₂²⋊Dic₅₉	C₂×C₄	1	6	yes	—
35	C₄⋊Dic₅₉	C₂×C₄	1	6	yes	—
36	C₄×Dic₅₉	C₂×C₄	1	6	yes	—
37	C₄×Dic₅₉	C₂³	1	3	yes	—
38	C₄×Dic₅₉	C₂³	1	1	yes	characteristic
39	#11745664	C₂×C₄	1	12	yes	complemented
40	C₂×C₅₉⋊C₈	C₂×C₄	1	12	yes	complemented
41	C₂×C₅₉⋊C₈	C₂³	1	3	yes	—
42	C₂×C₅₉⋊C₈	C₂³	1	1	yes	characteristic
43	C₂³×C₅₉	C₂²×C₄	1	3	yes	—
44	Q₈×C₅₉	C₄²	1	4	yes	—
45	D₄×C₅₉	C₄²	1	12	yes	—
46	C₂×C₄×C₅₉	C₂²×C₄	1	3	yes	—
47	C₂×C₄×C₅₉	C₂²×C₄	1	6	yes	—
48	C₂×C₄×C₅₉	C₂²×C₄	1	6	yes	—
49	C₂×C₄×C₅₉	C₂⁴	1	1	yes	characteristic
50	C₂×Dic₅₉	—	4	6	no	—
51	C₂×Dic₅₉	C₂²×C₄	1	2	yes	—
52	C₂×Dic₅₉	C₂²×C₄	1	6	yes	—
53	C₈×C₅₉	C₂²×C₄	1	2	yes	—
54	C₈×C₅₉	C₂²×C₄	1	6	yes	—
55	C₅₉⋊C₈	C₂²×C₄	1	2	yes	—
56	C₅₉⋊C₈	C₂²×C₄	1	6	yes	—
57	C₂²×C₅₉	—	2	6	no	—
58	C₂²×C₅₉	C₂×C₄²	1	6	yes	—
59	C₂²×C₅₉	C₂³×C₄	1	1	yes	characteristic
60	C₄×C₅₉	C₂×C₄²	1	6	yes	—
61	C₄×C₅₉	C₂³×C₄	1	1	yes	characteristic
62	C₄×C₅₉	C₂³×C₄	1	1	yes	characteristic
63	Dic₅₉	—	2	2	no	—
64	Dic₅₉	—	2	6	no	—
65	#2875	—	59	1	no	maximal, Sylow
66	C₂×C₅₉	—	2	6	no	—
67	C₂×C₅₉	C₂²×C₄²	1	1	yes	characteristic
68	C₂×C₅₉	#568	1	2	yes	—
69	C₂×#131	—	59	1	no	—
70	C₂×#131	D₅₉	1	1	yes	characteristic
71	C₄×#41	—	59	1	no	—
72	#405	—	59	3	no	—
73	#405	—	59	3	no	—
74	C₄×M₁₆	—	59	3	no	—
75	C₂×#97	—	59	3	no	—
76	C₅₉	#2875	1	1	yes	characteristic, complemented, Sylow
77	C₂×#41	C₂×D₅₉	1	1	yes	characteristic
78	#131	—	59	8	no	—
79	#131	Dic₅₉	1	8	yes	complemented
80	C₂×M₁₆	—	59	3	no	—
81	C₂×M₁₆	C₂×D₅₉	1	3	yes	—
82	C₂²×C₈	—	59	3	no	—
83	C₂²×C₈	C₂×D₅₉	1	3	yes	—
84	C₄×Q₈	—	59	2	no	—
85	C₄×D₄	—	59	6	no	—
86	#117	—	59	3	no	—
87	C₂×C₄²	—	59	3	no	—
88	#97	—	59	1	no	—
89	#97	—	59	3	no	—
90	#97	—	59	3	no	—
91	#97	—	59	3	no	—
92	C₄×C₈	—	59	6	no	—
93	#41	C₄×D₅₉	1	4	yes	—
94	#41	C₂×Dic₅₉	1	4	yes	—
95	C₂×Q₈	C₄×D₅₉	1	1	yes	characteristic
96	C₂×D₄	C₄×D₅₉	1	3	yes	—
97	C₂²×C₄	—	59	6	no	—
98	C₂²×C₄	C₂²×D₅₉	1	3	yes	—
99	M₁₆	—	59	12	no	—
100	M₁₆	C₂×Dic₅₉	1	12	yes	complemented
101	C₂×C₈	—	59	12	no	—
102	C₂×C₈	—	59	3	no	—
103	C₂×C₈	—	59	1	no	—
104	C₂×C₈	C₂×Dic₅₉	1	12	yes	complemented
105	C₂×C₈	C₂²×D₅₉	1	3	yes	—
106	C₂×C₈	C₂²×D₅₉	1	1	yes	characteristic
107	C₄⋊C₄	—	59	6	no	—
108	C₂²⋊C₄	—	59	6	no	—
109	C₄²	—	59	6	no	—
110	C₄²	—	59	3	no	—
111	C₄²	—	59	1	no	—
112	C₂³	C₂×C₄×D₅₉	1	3	yes	—
113	Q₈	C₄×Dic₅₉	1	4	yes	—
114	D₄	C₄×Dic₅₉	1	12	yes	—
115	C₂×C₄	—	59	2	no	—
116	C₂×C₄	—	59	6	no	—
117	C₂×C₄	—	236	6	no	—
118	C₂×C₄	C₂×C₄×D₅₉	1	3	yes	—
119	C₂×C₄	C₂×C₄×D₅₉	1	6	yes	—
120	C₂×C₄	C₂²×Dic₅₉	1	6	yes	—
121	C₂×C₄	C₂³×D₅₉	1	1	yes	characteristic
122	C₈	—	59	2	no	—
123	C₈	—	59	6	no	—
124	C₈	C₂²×Dic₅₉	1	2	yes	—
125	C₈	C₂²×Dic₅₉	1	6	yes	—
126	C₂²	—	2	6	no	—
127	C₂²	C₂×C₄×Dic₅₉	1	6	yes	—
128	C₂²	C₂²×C₄×D₅₉	1	1	yes	characteristic
129	C₄	—	118	2	no	—
130	C₄	—	118	6	no	—
131	C₄	C₂×C₄×Dic₅₉	1	6	yes	—
132	C₄	C₂²×C₄×D₅₉	1	1	yes	characteristic
133	C₄	C₂³×Dic₅₉	1	1	yes	characteristic
134	C₂	—	2	6	no	—
135	C₂	C₂²×C₄×Dic₅₉	1	1	yes	characteristic
136	C₂	#16838167	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14273734

C₂

yes

maximal, characteristic

#41×Dic₅₉

C₂

yes

maximal, characteristic

#16837360

C₂

yes

maximal, complemented

#16837357

C₂

yes

maximal, complemented

M₁₆×Dic₅₉

C₂

yes

maximal, complemented

C₂×#14273637

C₂

yes

maximal, complemented

C₂×#131×C₅₉

C₂

yes

maximal, characteristic

#14273734

C₄

yes

complemented

C₂×#11745664

C₂²

yes

—

C₂²×C₅₉⋊C₈

C₂²

yes

—

Q₈×Dic₅₉

C₄

yes

—

D₄×Dic₅₉

C₄

yes

—

#14273686

C₂²

yes

—

C₂×C₄×Dic₅₉

C₂²

yes

—

#14273637

C₂²

yes

characteristic

#14273637

C₂²

yes

—

#14273637

C₂²

yes

—

#14273637

C₂²

yes

—

C₈×Dic₅₉

C₂²

yes

—

C₂×#41×C₅₉

C₂²

yes

characteristic

#131×C₅₉

C₄

yes

complemented

C₂×M₁₆×C₅₉

C₂²

yes

—

C₂²×C₈×C₅₉

C₂²

yes

—

#41×C₅₉

C₂×C₄

yes

—

#41×C₅₉

C₂×C₄

yes

—

C₂×Q₈×C₅₉

C₂×C₄

yes

characteristic

C₂×D₄×C₅₉

C₂×C₄

yes

—

C₂²×C₄×C₅₉

C₂³

yes

—

C₂²×Dic₅₉

C₂×C₄

yes

—

M₁₆×C₅₉

C₂×C₄

yes

complemented

C₂×C₈×C₅₉

C₂×C₄

yes

complemented

C₂×C₈×C₅₉

C₂³

yes

characteristic

C₂×C₈×C₅₉

C₂³

yes

—

C₂²⋊Dic₅₉

C₂×C₄

yes

—

C₄⋊Dic₅₉

C₂×C₄

yes

—

C₄×Dic₅₉

C₂×C₄

yes

—

C₄×Dic₅₉

C₂³

yes

—

C₄×Dic₅₉

C₂³

yes

characteristic

#11745664

C₂×C₄

yes

complemented

C₂×C₅₉⋊C₈

C₂×C₄

yes

complemented

C₂×C₅₉⋊C₈

C₂³

yes

—

C₂×C₅₉⋊C₈

C₂³

yes

characteristic

C₂³×C₅₉

C₂²×C₄

yes

—

Q₈×C₅₉

C₄²

yes

—

D₄×C₅₉

C₄²

yes

—

C₂×C₄×C₅₉

C₂²×C₄

yes

—

C₂×C₄×C₅₉

C₂²×C₄

yes

—

C₂×C₄×C₅₉

C₂²×C₄

yes

—

C₂×C₄×C₅₉

C₂⁴

yes

characteristic

C₂×Dic₅₉

—

C₂×Dic₅₉

C₂²×C₄

yes

—

C₂×Dic₅₉

C₂²×C₄

yes

—

C₈×C₅₉

C₂²×C₄

yes

—

C₈×C₅₉

C₂²×C₄

yes

—

C₅₉⋊C₈

C₂²×C₄

yes

—

C₅₉⋊C₈

C₂²×C₄

yes

—

C₂²×C₅₉

—

C₂²×C₅₉

C₂×C₄²

yes

—

C₂²×C₅₉

C₂³×C₄

yes

characteristic

C₄×C₅₉

C₂×C₄²

yes

—

C₄×C₅₉

C₂³×C₄

yes

characteristic

C₄×C₅₉

C₂³×C₄

yes

characteristic

Dic₅₉

—

Dic₅₉

—

#2875

—

maximal, Sylow

C₂×C₅₉

—

C₂×C₅₉

C₂²×C₄²

yes

characteristic

C₂×C₅₉

#568

yes

—

C₂×#131

—

C₂×#131

D₅₉

yes

characteristic

C₄×#41

—

#405

—

#405

—

C₄×M₁₆

—

C₂×#97

—

C₅₉

#2875

yes

characteristic, complemented, Sylow

C₂×#41

C₂×D₅₉

yes

characteristic

#131

—

#131

Dic₅₉

yes

complemented

C₂×M₁₆

—

C₂×M₁₆

C₂×D₅₉

yes

—

C₂²×C₈

—

C₂²×C₈

C₂×D₅₉

yes

—

C₄×Q₈

—

C₄×D₄

—

#117

—

C₂×C₄²

—

—

—

—

—

C₄×C₈

—

#41

C₄×D₅₉

yes

—

#41

C₂×Dic₅₉

yes

—

C₂×Q₈

C₄×D₅₉

yes

characteristic

C₂×D₄

C₄×D₅₉

yes

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₅₉

yes

—

M₁₆

—

100

M₁₆

C₂×Dic₅₉

yes

complemented

101

C₂×C₈

—

102

C₂×C₈

—

103

C₂×C₈

—

104

C₂×C₈

C₂×Dic₅₉

yes

complemented

105

C₂×C₈

C₂²×D₅₉

yes

—

106

C₂×C₈

C₂²×D₅₉

yes

characteristic

107

C₄⋊C₄

—

108

C₂²⋊C₄

—

109

C₄²

—

110

C₄²

—

111

C₄²

—

112

C₂³

C₂×C₄×D₅₉

yes

—

113

Q₈

C₄×Dic₅₉

yes

—

114

D₄

C₄×Dic₅₉

yes

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

236

—

118

C₂×C₄

C₂×C₄×D₅₉

yes

—

119

C₂×C₄

C₂×C₄×D₅₉

yes

—

120

C₂×C₄

C₂²×Dic₅₉

yes

—

121

C₂×C₄

C₂³×D₅₉

yes

characteristic

122

C₈

—

123

C₈

—

124

C₈

C₂²×Dic₅₉

yes

—

125

C₈

C₂²×Dic₅₉

yes

—

126

C₂²

—

127

C₂²

C₂×C₄×Dic₅₉

yes

—

128

C₂²

C₂²×C₄×D₅₉

yes

characteristic

129

C₄

—

118

—

130

C₄

—

118

—

131

C₄

C₂×C₄×Dic₅₉

yes

—

132

C₄

C₂²×C₄×D₅₉

yes

characteristic

133

C₄

C₂³×Dic₅₉

yes

characteristic

134

C₂

—

135

C₂

C₂²×C₄×Dic₅₉

yes

characteristic

136

C₂

#16838167

yes

—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₅₉
commutator	C₂×C₅₉	C₂²×C₄²
Fitting	C₂×#131×C₅₉	C₂
socle	C₂²×C₅₉	C₂³×C₄
hypercenter	C₂×#131	D₅₉
hypocenter	C₅₉	#2875
LCS	G ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#131

subgroup

G/H

center, Frattini

C₂×C₄

C₂³×D₅₉

commutator

C₂×C₅₉

C₂²×C₄²

Fitting

C₂×#131×C₅₉

C₂

socle

C₂²×C₅₉

C₂³×C₄

D₅₉

C₅₉

G ▷ C₂×C₅₉ ▷ C₅₉

C₁ ◁ C₂×C₄ ◁ C₂×#131

N	H ≅ G/N	type
3. #16837360	C₂	semidirect
4. #16837357	C₂	semidirect
5. M₁₆×Dic₅₉	C₂	semidirect
6. C₂×#14273637	C₂	semidirect
8. #14273734	C₄	semidirect
21. #131×C₅₉	C₄	semidirect
30. M₁₆×C₅₉	C₂×C₄	semidirect
31. C₂×C₈×C₅₉	C₂×C₄	semidirect
39. #11745664	C₂×C₄	semidirect
40. C₂×C₅₉⋊C₈	C₂×C₄	semidirect
76. C₅₉	#2875	semidirect
79. #131	Dic₅₉	semidirect
100. M₁₆	C₂×Dic₅₉	semidirect
104. C₂×C₈	C₂×Dic₅₉	semidirect

H ≅ G/N

type

3. #16837360

C₂

semidirect

4. #16837357

C₂

semidirect

5. M₁₆×Dic₅₉

C₂

semidirect

6. C₂×#14273637

C₂

semidirect

8. #14273734

C₄

semidirect

21. #131×C₅₉

C₄

semidirect

30. M₁₆×C₅₉

C₂×C₄

semidirect

31. C₂×C₈×C₅₉

C₂×C₄

semidirect

39. #11745664

C₂×C₄

semidirect

40. C₂×C₅₉⋊C₈

C₂×C₄

semidirect

76. C₅₉

#2875

semidirect

79. #131

Dic₅₉

semidirect

100. M₁₆

C₂×Dic₅₉

semidirect

104. C₂×C₈

C₂×Dic₅₉

semidirect

Subgroups — #21432522

Conjugacy classes of subgroups

Complements