Group #28554312 — subgroups

Subgroups — C₂×#21422395 #28554312

Order 15104.

802 conjugacy classes of subgroups; only the 379 normal subgroups are shown.

normal subgroups only

This group has 802 conjugacy classes of subgroups — too many to draw. Only its 379 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×#16837453	C₂	1	3	yes	maximal
2	#21422395	C₂	1	8	yes	maximal, direct product factor
3	C₂×#16837124	C₂	1	3	yes	maximal, complemented
4	C₂×C₈²×C₅₉	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₄²×D₅₉	C₂²	1	1	yes	characteristic
6	#16837453	C₄	1	12	yes	—
7	#16837453	C₂²	1	12	yes	—
8	C₂×Dic₅₉⋊C₈	C₄	1	6	yes	—
9	C₂×(C₂×D₅₉)⋊C₈	C₄	1	6	yes	—
10	C₂×C₄×C₅₉⋊C₈	C₂²	1	3	yes	—
11	#16837124	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×C₄×Dic₅₉	C₂×C₄	1	3	yes	—
17	C₄²×D₅₉	C₂×C₄	1	6	yes	—
18	C₄²×D₅₉	C₂³	1	2	yes	—
19	Dic₅₉⋊C₈	C₂×C₄	1	24	yes	—
20	(C₂×D₅₉)⋊C₈	C₂×C₄	1	24	yes	—
21	C₄×C₅₉⋊C₈	C₂×C₄	1	6	yes	—
22	C₄×C₅₉⋊C₈	C₂³	1	6	yes	—
23	C₂²×C₈×C₅₉	C₂×C₄	1	6	yes	—
24	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
25	C₄×C₈×C₅₉	C₂×C₄	1	6	yes	—
26	C₄×C₈×C₅₉	C₂³	1	6	yes	—
27	C₂³×D₅₉	C₄²	1	1	yes	characteristic
28	C₂²×C₄×C₅₉	C₂²×C₄	1	3	yes	—
29	C₂²×Dic₅₉	C₄²	1	3	yes	—
30	C₂×C₄×D₅₉	—	2	12	no	—
31	C₂×C₄×D₅₉	C₄²	1	6	yes	—
32	C₂×C₄×D₅₉	C₂²×C₄	1	6	yes	—
33	C₂×C₈×C₅₉	—	2	12	no	—
34	C₂×C₈×C₅₉	C₂²×C₄	1	12	yes	—
35	C₄²×C₅₉	C₂²×C₄	1	3	yes	—
36	C₄²×C₅₉	C₂⁴	1	1	yes	characteristic
37	C₄×Dic₅₉	C₄²	1	6	yes	—
38	C₄×Dic₅₉	C₂²×C₄	1	6	yes	—
39	C₂×C₅₉⋊C₈	—	2	12	no	—
40	C₂×C₅₉⋊C₈	C₂²×C₄	1	12	yes	—
41	C₂³×C₅₉	C₂×C₄²	1	1	yes	characteristic
42	C₂²×D₅₉	—	2	6	no	—
43	C₂²×D₅₉	C₂×C₄²	1	2	yes	—
44	C₂×C₄×C₅₉	C₂×C₄²	1	3	yes	—
45	C₂×C₄×C₅₉	#131	1	12	yes	—
46	C₂×C₄×C₅₉	C₂³×C₄	1	3	yes	—
47	C₂×Dic₅₉	—	2	6	no	—
48	C₂×Dic₅₉	C₂×C₄²	1	6	yes	—
49	C₄×D₅₉	—	2	12	no	—
50	C₄×D₅₉	—	2	12	no	—
51	C₈×C₅₉	—	2	12	no	—
52	C₅₉⋊C₈	—	2	12	no	—
53	C₂×#1454	—	59	1	no	maximal, Sylow
54	C₂²×C₅₉	#405	1	6	yes	—
55	C₂²×C₅₉	C₂²×C₄²	1	1	yes	characteristic
56	C₂×D₅₉	—	2	6	no	—
57	C₂×D₅₉	—	4	4	no	—
58	C₄×C₅₉	#405	1	6	yes	—
59	C₄×C₅₉	C₂×#131	1	6	yes	—
60	Dic₅₉	—	2	6	no	—
61	C₂×#431	—	59	3	no	—
62	#1454	—	59	8	no	—
63	C₂×#322	—	59	3	no	—
64	C₂×C₈²	D₅₉	1	1	yes	characteristic, complemented
65	C₂×C₅₉	#1454	1	4	yes	complemented
66	C₂×C₅₉	C₂×#405	1	3	yes	—
67	D₅₉	—	4	2	no	—
68	C₂²×C₄²	—	59	1	no	—
69	#431	—	59	12	no	—
70	#431	—	59	12	no	—
71	C₂×C₄⋊C₈	—	59	6	no	—
72	C₂×C₂²⋊C₈	—	59	6	no	—
73	C₂×C₄×C₈	—	59	3	no	—
74	C₂×C₄×C₈	C₂×D₅₉	1	3	yes	—
75	#322	—	59	12	no	—
76	C₈²	C₂×D₅₉	1	4	yes	complemented
77	C₅₉	C₂×#1454	1	1	yes	characteristic, complemented, Sylow
78	C₂³×C₄	—	59	3	no	—
79	C₂²×C₈	—	59	6	no	—
80	C₂²×C₈	C₄×D₅₉	1	6	yes	—
81	C₂×C₄²	—	59	3	no	—
82	C₂×C₄²	—	59	6	no	—
83	C₂×C₄²	—	59	2	no	—
84	C₂×C₄²	C₂²×D₅₉	1	1	yes	characteristic
85	C₄⋊C₈	—	59	24	no	—
86	C₂²⋊C₈	—	59	24	no	—
87	C₄×C₈	—	59	6	no	—
88	C₄×C₈	—	59	6	no	—
89	C₄×C₈	C₄×D₅₉	1	6	yes	—
90	C₄×C₈	C₂²×D₅₉	1	6	yes	—
91	C₂⁴	—	59	1	no	—
92	C₂²×C₄	—	59	3	no	—
93	C₂²×C₄	—	59	6	no	—
94	C₂²×C₄	—	59	6	no	—
95	C₂²×C₄	—	118	12	no	—
96	C₂²×C₄	C₂×C₄×D₅₉	1	3	yes	—
97	C₂×C₈	—	2	12	no	—
98	C₂×C₈	—	59	12	no	—
99	C₂×C₈	—	118	12	no	—
100	C₂×C₈	C₂×C₄×D₅₉	1	12	yes	—
101	C₄²	—	59	6	no	—
102	C₄²	—	59	6	no	—
103	C₄²	C₂×C₄×D₅₉	1	3	yes	—
104	C₄²	C₂³×D₅₉	1	1	yes	characteristic
105	C₂³	—	59	2	no	—
106	C₂³	—	118	6	no	—
107	C₂³	C₄²×D₅₉	1	1	yes	characteristic
108	C₂×C₄	—	59	6	no	—
109	C₂×C₄	—	118	6	no	—
110	C₂×C₄	—	118	12	no	—
111	C₂×C₄	—	118	12	no	—
112	C₂×C₄	C₄²×D₅₉	1	3	yes	—
113	C₂×C₄	#14273733	1	12	yes	—
114	C₂×C₄	C₂²×C₄×D₅₉	1	3	yes	—
115	C₈	—	2	12	no	—
116	C₈	—	118	12	no	—
117	C₂²	—	118	6	no	—
118	C₂²	—	236	4	no	—
119	C₂²	#16837356	1	6	yes	—
120	C₂²	C₂×C₄²×D₅₉	1	1	yes	characteristic
121	C₄	—	118	6	no	—
122	C₄	#16837356	1	6	yes	—
123	C₄	C₂×#14273733	1	6	yes	—
124	C₂	—	236	2	no	—
125	C₂	#21422395	1	4	yes	direct product factor
126	C₂	C₂×#16837356	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16837453

C₂

yes

maximal

#21422395

C₂

yes

maximal, direct product factor

C₂×#16837124

C₂

yes

maximal, complemented

C₂×C₈²×C₅₉

C₂

yes

maximal, characteristic, complemented

C₂×C₄²×D₅₉

C₂²

yes

characteristic

#16837453

C₄

yes

—

#16837453

C₂²

yes

—

C₂×Dic₅₉⋊C₈

C₄

yes

—

C₂×(C₂×D₅₉)⋊C₈

C₄

yes

—

C₂×C₄×C₅₉⋊C₈

C₂²

yes

—

#16837124

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×C₄×Dic₅₉

C₂×C₄

yes

—

C₄²×D₅₉

C₂×C₄

yes

—

C₄²×D₅₉

C₂³

yes

—

Dic₅₉⋊C₈

C₂×C₄

yes

—

(C₂×D₅₉)⋊C₈

C₂×C₄

yes

—

C₄×C₅₉⋊C₈

C₂×C₄

yes

—

C₄×C₅₉⋊C₈

C₂³

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

—

C₂×C₄×D₅₉

C₄²

yes

—

C₂×C₄×D₅₉

C₂²×C₄

yes

—

C₂×C₈×C₅₉

—

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×C₅₉⋊C₈

C₂²×C₄

yes

—

C₂³×C₅₉

C₂×C₄²

yes

characteristic

C₂²×D₅₉

—

C₂²×D₅₉

C₂×C₄²

yes

—

C₂×C₄×C₅₉

C₂×C₄²

yes

—

C₂×C₄×C₅₉

#131

yes

—

C₂×C₄×C₅₉

C₂³×C₄

yes

—

C₂×Dic₅₉

—

C₂×Dic₅₉

C₂×C₄²

yes

—

C₄×D₅₉

—

C₄×D₅₉

—

C₈×C₅₉

—

C₅₉⋊C₈

—

C₂×#1454

—

maximal, Sylow

C₂²×C₅₉

#405

yes

—

C₂²×C₅₉

C₂²×C₄²

yes

characteristic

C₂×D₅₉

—

C₂×D₅₉

—

C₄×C₅₉

#405

yes

—

C₄×C₅₉

C₂×#131

yes

—

Dic₅₉

—

C₂×#431

—

#1454

—

C₂×#322

—

C₂×C₈²

D₅₉

yes

characteristic, complemented

C₂×C₅₉

#1454

yes

complemented

C₂×C₅₉

C₂×#405

yes

—

D₅₉

—

C₂²×C₄²

—

#431

—

#431

—

C₂×C₄⋊C₈

—

C₂×C₂²⋊C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂×D₅₉

yes

—

#322

—

C₈²

C₂×D₅₉

yes

complemented

C₅₉

C₂×#1454

yes

characteristic, complemented, Sylow

C₂³×C₄

—

C₂²×C₈

—

C₂²×C₈

C₄×D₅₉

yes

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

C₂²×D₅₉

yes

characteristic

C₄⋊C₈

—

C₂²⋊C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₄×D₅₉

yes

—

C₄×C₈

C₂²×D₅₉

yes

—

C₂⁴

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

118

—

C₂²×C₄

C₂×C₄×D₅₉

yes

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

118

—

100

C₂×C₈

C₂×C₄×D₅₉

yes

—

101

C₄²

—

102

C₄²

—

103

C₄²

C₂×C₄×D₅₉

yes

—

104

C₄²

C₂³×D₅₉

yes

characteristic

105

C₂³

—

106

C₂³

—

118

—

107

C₂³

C₄²×D₅₉

yes

characteristic

108

C₂×C₄

—

109

C₂×C₄

—

118

—

110

C₂×C₄

—

118

—

111

C₂×C₄

—

118

—

112

C₂×C₄

C₄²×D₅₉

yes

—

113

C₂×C₄

#14273733

yes

—

114

C₂×C₄

C₂²×C₄×D₅₉

yes

—

115

C₈

—

116

C₈

—

118

—

117

C₂²

—

118

—

118

C₂²

—

236

—

119

C₂²

#16837356

yes

—

120

C₂²

C₂×C₄²×D₅₉

yes

characteristic

121

C₄

—

118

—

122

C₄

#16837356

yes

—

123

C₄

C₂×#14273733

yes

—

124

C₂

—

236

—

125

C₂

#21422395

yes

direct product factor

126

C₂

C₂×#16837356

yes

—

	subgroup	G/H
center	C₂×C₄²	C₂²×D₅₉
commutator	C₂²×C₅₉	C₂²×C₄²
Frattini	C₄²	C₂³×D₅₉
Fitting	C₂×C₈²×C₅₉	C₂
socle	C₂³×C₅₉	C₂×C₄²
prime radical	C₂³×D₅₉	C₄²
cosocle	C₄²×C₅₉	C₂⁴
hypercenter	C₂×C₈²	D₅₉
hypocenter	C₅₉	C₂×#1454
LCS	G ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₄² ◁ C₂×C₈²

subgroup

G/H

center

C₂×C₄²

C₂²×D₅₉

commutator

C₂²×C₅₉

C₂²×C₄²

Frattini

C₄²

C₂³×D₅₉

Fitting

C₂×C₈²×C₅₉

C₂

socle

C₂³×C₅₉

C₂×C₄²

prime radical

C₂³×D₅₉

C₄²

cosocle

C₄²×C₅₉

C₂⁴

hypercenter

C₂×C₈²

D₅₉

C₅₉

G ▷ C₂²×C₅₉ ▷ C₅₉

C₁ ◁ C₂×C₄² ◁ C₂×C₈²

N	H ≅ G/N	type
2. #21422395	C₂	direct (N × H)
3. C₂×#16837124	C₂	semidirect
4. C₂×C₈²×C₅₉	C₂	semidirect
11. #16837124	C₂²	semidirect
13. C₈²×C₅₉	C₂²	semidirect
64. C₂×C₈²	D₅₉	semidirect
65. C₂×C₅₉	#1454	semidirect
76. C₈²	C₂×D₅₉	semidirect
77. C₅₉	C₂×#1454	semidirect
125. C₂	#21422395	direct (N × H)

H ≅ G/N

type

2. #21422395

C₂

direct (N × H)

3. C₂×#16837124

C₂

semidirect

4. C₂×C₈²×C₅₉

C₂

semidirect

11. #16837124

C₂²

semidirect

13. C₈²×C₅₉

C₂²

semidirect

64. C₂×C₈²

D₅₉

semidirect

65. C₂×C₅₉

#1454

semidirect

76. C₈²

C₂×D₅₉

semidirect

77. C₅₉

C₂×#1454

semidirect

125. C₂

#21422395

direct (N × H)

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

subgroup	index
C₂×#16837453	2
#21422395	2
C₂×#16837124	2
C₂×C₈²×C₅₉	2
C₂×#1454	59

subgroup

index

C₂×C₈²×C₅₉

Subgroups — C₂×#21422395 #28554312

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×#21422395 #28554312

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#21422395 #28554312