Group #28554254 — subgroups

Subgroups — C₂×C₈²×D₅₉ #28554254

Order 15104.

1342 conjugacy classes of subgroups; only the 811 normal subgroups are shown.

normal subgroups only

This group has 1342 conjugacy classes of subgroups — too many to draw. Only its 811 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×C₄×C₈×D₅₉	C₂	1	3	yes	maximal
2	C₂×C₈×C₅₉⋊C₈	C₂	1	3	yes	maximal, complemented
3	C₈²×D₅₉	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₈²×C₅₉	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×C₈×D₅₉	C₄	1	6	yes	—
6	C₂×C₄²×D₅₉	C₂²	1	1	yes	characteristic
7	C₂×C₄×C₅₉⋊C₈	C₂²	1	3	yes	—
8	C₂×C₈×Dic₅₉	C₄	1	6	yes	—
9	C₄×C₈×D₅₉	C₄	1	12	yes	—
10	C₄×C₈×D₅₉	C₂²	1	12	yes	—
11	C₈×C₅₉⋊C₈	C₂²	1	12	yes	complemented
12	C₂×C₄×C₈×C₅₉	C₂²	1	3	yes	—
13	C₈²×C₅₉	C₂²	1	4	yes	complemented
14	C₂²×C₄×D₅₉	C₂×C₄	1	3	yes	—
15	C₂²×C₅₉⋊C₈	C₂×C₄	1	6	yes	—
16	C₂×C₈×D₅₉	C₈	1	48	yes	direct product factor
17	C₂×C₈×D₅₉	C₂×C₄	1	24	yes	—
18	C₂×C₄×Dic₅₉	C₂×C₄	1	3	yes	—
19	C₄²×D₅₉	C₂×C₄	1	6	yes	—
20	C₄²×D₅₉	C₂³	1	2	yes	—
21	C₄×C₅₉⋊C₈	C₂×C₄	1	6	yes	—
22	C₄×C₅₉⋊C₈	C₂³	1	6	yes	—
23	C₈×Dic₅₉	C₂×C₄	1	24	yes	—
24	C₂²×C₈×C₅₉	C₂×C₄	1	6	yes	—
25	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
26	C₄×C₈×C₅₉	C₂×C₄	1	6	yes	—
27	C₄×C₈×C₅₉	C₂³	1	6	yes	—
28	C₂³×D₅₉	C₄²	1	1	yes	characteristic
29	C₂²×C₄×C₅₉	C₂²×C₄	1	3	yes	—
30	C₂²×Dic₅₉	C₄²	1	3	yes	—
31	C₂×C₄×D₅₉	C₄²	1	6	yes	—
32	C₂×C₄×D₅₉	C₂×C₈	1	24	yes	—
33	C₂×C₄×D₅₉	C₂²×C₄	1	6	yes	—
34	C₂×C₈×C₅₉	C₂×C₈	1	24	yes	complemented
35	C₂×C₈×C₅₉	C₂²×C₄	1	12	yes	—
36	C₄²×C₅₉	C₂²×C₄	1	3	yes	—
37	C₄²×C₅₉	C₂⁴	1	1	yes	characteristic
38	C₄×Dic₅₉	C₄²	1	6	yes	—
39	C₄×Dic₅₉	C₂²×C₄	1	6	yes	—
40	C₂×C₅₉⋊C₈	C₂×C₈	1	24	yes	complemented
41	C₂×C₅₉⋊C₈	C₂²×C₄	1	12	yes	—
42	C₈×D₅₉	C₂×C₈	1	96	yes	direct product factor
43	C₂³×C₅₉	C₂×C₄²	1	1	yes	characteristic
44	C₂²×D₅₉	C₄×C₈	1	12	yes	—
45	C₂²×D₅₉	C₂×C₄²	1	2	yes	—
46	C₂×C₄×C₅₉	C₂×C₄²	1	3	yes	—
47	C₂×C₄×C₅₉	C₂²×C₈	1	12	yes	—
48	C₂×C₄×C₅₉	C₂³×C₄	1	3	yes	—
49	C₂×Dic₅₉	C₄×C₈	1	12	yes	—
50	C₂×Dic₅₉	C₂×C₄²	1	6	yes	—
51	C₄×D₅₉	C₄×C₈	1	24	yes	—
52	C₄×D₅₉	C₂²×C₈	1	24	yes	—
53	C₈×C₅₉	C₂²×C₈	1	24	yes	complemented
54	C₅₉⋊C₈	C₂²×C₈	1	24	yes	complemented
55	C₂²×C₈²	—	59	1	no	maximal, Sylow
56	C₂²×C₅₉	C₂×C₄×C₈	1	6	yes	—
57	C₂²×C₅₉	C₂²×C₄²	1	1	yes	characteristic
58	C₂×D₅₉	C₈²	1	16	yes	direct product factor
59	C₂×D₅₉	C₂×C₄×C₈	1	12	yes	—
60	C₄×C₅₉	C₂×C₄×C₈	1	6	yes	—
61	C₄×C₅₉	C₂³×C₈	1	6	yes	—
62	Dic₅₉	C₂×C₄×C₈	1	12	yes	—
63	C₂²×C₄×C₈	—	59	3	no	—
64	C₂×C₈²	—	59	3	no	—
65	C₂×C₈²	—	59	8	no	—
66	C₂×C₈²	D₅₉	1	1	yes	characteristic, direct product factor
67	C₂×C₅₉	C₂×C₈²	1	4	yes	complemented
68	C₂×C₅₉	C₂²×C₄×C₈	1	3	yes	—
69	D₅₉	C₂×C₈²	1	8	yes	direct product factor
70	C₂³×C₈	—	59	6	no	—
71	C₂²×C₄²	—	59	1	no	—
72	C₂×C₄×C₈	—	59	6	no	—
73	C₂×C₄×C₈	—	59	3	no	—
74	C₂×C₄×C₈	—	59	12	no	—
75	C₂×C₄×C₈	—	59	12	no	—
76	C₂×C₄×C₈	C₂×D₅₉	1	3	yes	—
77	C₈²	—	59	12	no	—
78	C₈²	C₂×D₅₉	1	4	yes	direct product factor
79	C₅₉	C₂²×C₈²	1	1	yes	characteristic, complemented, Sylow
80	C₂³×C₄	—	59	3	no	—
81	C₂²×C₈	—	59	6	no	—
82	C₂²×C₈	—	59	48	no	—
83	C₂²×C₈	—	59	24	no	—
84	C₂²×C₈	C₄×D₅₉	1	6	yes	—
85	C₂×C₄²	—	59	3	no	—
86	C₂×C₄²	—	59	6	no	—
87	C₂×C₄²	—	59	2	no	—
88	C₂×C₄²	C₂²×D₅₉	1	1	yes	characteristic
89	C₄×C₈	—	59	24	no	—
90	C₄×C₈	—	59	6	no	—
91	C₄×C₈	—	59	6	no	—
92	C₄×C₈	C₄×D₅₉	1	6	yes	—
93	C₄×C₈	C₂²×D₅₉	1	6	yes	—
94	C₂⁴	—	59	1	no	—
95	C₂²×C₄	—	59	3	no	—
96	C₂²×C₄	—	59	6	no	—
97	C₂²×C₄	—	59	24	no	—
98	C₂²×C₄	—	59	6	no	—
99	C₂²×C₄	C₂×C₄×D₅₉	1	3	yes	—
100	C₂×C₈	—	59	24	no	—
101	C₂×C₈	—	59	12	no	—
102	C₂×C₈	—	59	96	no	—
103	C₂×C₈	C₈×D₅₉	1	24	yes	direct product factor
104	C₂×C₈	C₂×C₄×D₅₉	1	12	yes	—
105	C₄²	—	59	6	no	—
106	C₄²	—	59	6	no	—
107	C₄²	C₂×C₄×D₅₉	1	3	yes	—
108	C₄²	C₂³×D₅₉	1	1	yes	characteristic
109	C₂³	—	59	2	no	—
110	C₂³	—	59	12	no	—
111	C₂³	C₄²×D₅₉	1	1	yes	characteristic
112	C₂×C₄	—	59	6	no	—
113	C₂×C₄	—	59	12	no	—
114	C₂×C₄	—	59	24	no	—
115	C₂×C₄	—	59	24	no	—
116	C₂×C₄	C₄²×D₅₉	1	3	yes	—
117	C₂×C₄	C₂×C₈×D₅₉	1	12	yes	—
118	C₂×C₄	C₂²×C₄×D₅₉	1	3	yes	—
119	C₈	—	59	24	no	—
120	C₈	C₂×C₈×D₅₉	1	24	yes	direct product factor
121	C₂²	—	59	16	no	—
122	C₂²	—	59	12	no	—
123	C₂²	C₄×C₈×D₅₉	1	6	yes	—
124	C₂²	C₂×C₄²×D₅₉	1	1	yes	characteristic
125	C₄	—	59	12	no	—
126	C₄	C₄×C₈×D₅₉	1	6	yes	—
127	C₄	C₂²×C₈×D₅₉	1	6	yes	—
128	C₂	—	59	8	no	—
129	C₂	C₈²×D₅₉	1	4	yes	direct product factor
130	C₂	C₂×C₄×C₈×D₅₉	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄×C₈×D₅₉

C₂

yes

maximal

C₂×C₈×C₅₉⋊C₈

C₂

yes

maximal, complemented

C₈²×D₅₉

C₂

yes

maximal, direct product factor

C₂×C₈²×C₅₉

C₂

yes

maximal, characteristic, complemented

C₂²×C₈×D₅₉

C₄

yes

—

C₂×C₄²×D₅₉

C₂²

yes

characteristic

C₂×C₄×C₅₉⋊C₈

C₂²

yes

—

C₂×C₈×Dic₅₉

C₄

yes

—

C₄×C₈×D₅₉

C₄

yes

—

C₄×C₈×D₅₉

C₂²

yes

—

C₈×C₅₉⋊C₈

C₂²

yes

complemented

C₂×C₄×C₈×C₅₉

C₂²

yes

—

C₈²×C₅₉

C₂²

yes

complemented

C₂²×C₄×D₅₉

C₂×C₄

yes

—

C₂²×C₅₉⋊C₈

C₂×C₄

yes

—

C₂×C₈×D₅₉

C₈

yes

direct product factor

C₂×C₈×D₅₉

C₂×C₄

yes

—

C₂×C₄×Dic₅₉

C₂×C₄

yes

—

C₄²×D₅₉

C₂×C₄

yes

—

C₄²×D₅₉

C₂³

yes

—

C₄×C₅₉⋊C₈

C₂×C₄

yes

—

C₄×C₅₉⋊C₈

C₂³

yes

—

C₈×Dic₅₉

C₂×C₄

yes

—

C₂²×C₈×C₅₉

C₂×C₄

yes

—

C₂×C₄²×C₅₉

C₂³

yes

characteristic

C₄×C₈×C₅₉

C₂×C₄

yes

—

C₄×C₈×C₅₉

C₂³

yes

—

C₂³×D₅₉

C₄²

yes

characteristic

C₂²×C₄×C₅₉

C₂²×C₄

yes

—

C₂²×Dic₅₉

C₄²

yes

—

C₂×C₄×D₅₉

C₄²

yes

—

C₂×C₄×D₅₉

C₂×C₈

yes

—

C₂×C₄×D₅₉

C₂²×C₄

yes

—

C₂×C₈×C₅₉

C₂×C₈

yes

complemented

C₂×C₈×C₅₉

C₂²×C₄

yes

—

C₄²×C₅₉

C₂²×C₄

yes

—

C₄²×C₅₉

C₂⁴

yes

characteristic

C₄×Dic₅₉

C₄²

yes

—

C₄×Dic₅₉

C₂²×C₄

yes

—

C₂×C₅₉⋊C₈

C₂×C₈

yes

complemented

C₂×C₅₉⋊C₈

C₂²×C₄

yes

—

C₈×D₅₉

C₂×C₈

yes

direct product factor

C₂³×C₅₉

C₂×C₄²

yes

characteristic

C₂²×D₅₉

C₄×C₈

yes

—

C₂²×D₅₉

C₂×C₄²

yes

—

C₂×C₄×C₅₉

C₂×C₄²

yes

—

C₂×C₄×C₅₉

C₂²×C₈

yes

—

C₂×C₄×C₅₉

C₂³×C₄

yes

—

C₂×Dic₅₉

C₄×C₈

yes

—

C₂×Dic₅₉

C₂×C₄²

yes

—

C₄×D₅₉

C₄×C₈

yes

—

C₄×D₅₉

C₂²×C₈

yes

—

C₈×C₅₉

C₂²×C₈

yes

complemented

C₅₉⋊C₈

C₂²×C₈

yes

complemented

C₂²×C₈²

—

maximal, Sylow

C₂²×C₅₉

C₂×C₄×C₈

yes

—

C₂²×C₅₉

C₂²×C₄²

yes

characteristic

C₂×D₅₉

C₈²

yes

direct product factor

C₂×D₅₉

C₂×C₄×C₈

yes

—

C₄×C₅₉

C₂×C₄×C₈

yes

—

C₄×C₅₉

C₂³×C₈

yes

—

Dic₅₉

C₂×C₄×C₈

yes

—

C₂²×C₄×C₈

—

C₂×C₈²

—

C₂×C₈²

—

C₂×C₈²

D₅₉

yes

characteristic, direct product factor

C₂×C₅₉

C₂×C₈²

yes

complemented

C₂×C₅₉

C₂²×C₄×C₈

yes

—

D₅₉

C₂×C₈²

yes

direct product factor

C₂³×C₈

—

C₂²×C₄²

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂×D₅₉

yes

—

C₈²

—

C₈²

C₂×D₅₉

yes

direct product factor

C₅₉

C₂²×C₈²

yes

characteristic, complemented, Sylow

C₂³×C₄

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

C₄×D₅₉

yes

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

C₂²×D₅₉

yes

characteristic

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₄×D₅₉

yes

—

C₄×C₈

C₂²×D₅₉

yes

—

C₂⁴

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×C₄×D₅₉

yes

—

100

C₂×C₈

—

101

C₂×C₈

—

102

C₂×C₈

—

103

C₂×C₈

C₈×D₅₉

yes

direct product factor

104

C₂×C₈

C₂×C₄×D₅₉

yes

—

105

C₄²

—

106

C₄²

—

107

C₄²

C₂×C₄×D₅₉

yes

—

108

C₄²

C₂³×D₅₉

yes

characteristic

109

C₂³

—

110

C₂³

—

111

C₂³

C₄²×D₅₉

yes

characteristic

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

C₄²×D₅₉

yes

—

117

C₂×C₄

C₂×C₈×D₅₉

yes

—

118

C₂×C₄

C₂²×C₄×D₅₉

yes

—

119

C₈

—

120

C₈

C₂×C₈×D₅₉

yes

direct product factor

121

C₂²

—

122

C₂²

—

123

C₂²

C₄×C₈×D₅₉

yes

—

124

C₂²

C₂×C₄²×D₅₉

yes

characteristic

125

C₄

—

126

C₄

C₄×C₈×D₅₉

yes

—

127

C₄

C₂²×C₈×D₅₉

yes

—

128

C₂

—

129

C₂

C₈²×D₅₉

yes

direct product factor

130

C₂

C₂×C₄×C₈×D₅₉

yes

—

	subgroup	G/H
center, hypercenter	C₂×C₈²	D₅₉
commutator, hypocenter	C₅₉	C₂²×C₈²
Frattini	C₄²	C₂³×D₅₉
Fitting	C₂×C₈²×C₅₉	C₂
socle	C₂³×C₅₉	C₂×C₄²
prime radical	C₂³×D₅₉	C₄²
cosocle	C₄²×C₅₉	C₂⁴
LCS	G ▷ C₅₉
UCS	C₁ ◁ C₂×C₈²

subgroup

G/H

center, hypercenter

C₂×C₈²

D₅₉

commutator, hypocenter

C₅₉

C₂²×C₈²

Frattini

C₄²

C₂³×D₅₉

Fitting

C₂×C₈²×C₅₉

C₂

socle

C₂³×C₅₉

C₂×C₄²

prime radical

C₂³×D₅₉

C₄²

cosocle

C₄²×C₅₉

C₂⁴

LCS

G ▷ C₅₉

UCS

C₁ ◁ C₂×C₈²

N	H ≅ G/N	type
2. C₂×C₈×C₅₉⋊C₈	C₂	semidirect
3. C₈²×D₅₉	C₂	direct (N × H)
4. C₂×C₈²×C₅₉	C₂	semidirect
11. C₈×C₅₉⋊C₈	C₂²	semidirect
13. C₈²×C₅₉	C₂²	semidirect
16. C₂×C₈×D₅₉	C₈	direct (N × H)
34. C₂×C₈×C₅₉	C₂×C₈	semidirect
40. C₂×C₅₉⋊C₈	C₂×C₈	semidirect
42. C₈×D₅₉	C₂×C₈	direct (N × H)
53. C₈×C₅₉	C₂²×C₈	semidirect
54. C₅₉⋊C₈	C₂²×C₈	semidirect
58. C₂×D₅₉	C₈²	direct (N × H)
66. C₂×C₈²	D₅₉	direct (N × H)
67. C₂×C₅₉	C₂×C₈²	semidirect
69. D₅₉	C₂×C₈²	direct (N × H)
78. C₈²	C₂×D₅₉	direct (N × H)
79. C₅₉	C₂²×C₈²	semidirect
103. C₂×C₈	C₈×D₅₉	direct (N × H)
120. C₈	C₂×C₈×D₅₉	direct (N × H)
129. C₂	C₈²×D₅₉	direct (N × H)

H ≅ G/N

type

2. C₂×C₈×C₅₉⋊C₈

C₂

semidirect

3. C₈²×D₅₉

C₂

direct (N × H)

4. C₂×C₈²×C₅₉

C₂

semidirect

11. C₈×C₅₉⋊C₈

C₂²

semidirect

13. C₈²×C₅₉

C₂²

semidirect

16. C₂×C₈×D₅₉

C₈

direct (N × H)

34. C₂×C₈×C₅₉

C₂×C₈

semidirect

40. C₂×C₅₉⋊C₈

C₂×C₈

semidirect

42. C₈×D₅₉

C₂×C₈

direct (N × H)

53. C₈×C₅₉

C₂²×C₈

semidirect

54. C₅₉⋊C₈

C₂²×C₈

semidirect

58. C₂×D₅₉

C₈²

direct (N × H)

66. C₂×C₈²

D₅₉

direct (N × H)

67. C₂×C₅₉

C₂×C₈²

semidirect

69. D₅₉

C₂×C₈²

direct (N × H)

78. C₈²

C₂×D₅₉

direct (N × H)

79. C₅₉

C₂²×C₈²

semidirect

103. C₂×C₈

C₈×D₅₉

direct (N × H)

120. C₈

C₂×C₈×D₅₉

direct (N × H)

129. C₂

C₈²×D₅₉

direct (N × H)

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₄×C₈×D₅₉	2
C₂×C₈×C₅₉⋊C₈	2
C₈²×D₅₉	2
C₂×C₈²×C₅₉	2
C₂²×C₈²	59

subgroup

index

C₂×C₄×C₈×D₅₉

C₂×C₈×C₅₉⋊C₈

C₈²×D₅₉

C₂×C₈²×C₅₉

C₂²×C₈²

Subgroups — C₂×C₈²×D₅₉ #28554254

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×C82×D59 #28554254

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₈²×D₅₉ #28554254