Group #28475348 — subgroups

Subgroups — #28475348

Order 15104.

140 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 140 conjugacy classes of subgroups — too many to draw. Only its 57 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	C₈×Dic₂₃₆	C₂	1	1	yes	maximal, characteristic
2	#21422160	C₂	1	1	yes	maximal, characteristic
3	C₅₉×#1312	C₂	1	1	yes	maximal, characteristic
4	C₈×Dic₁₁₈	—	2	1	no	—
5	C₄×Dic₂₃₆	C₄	1	1	yes	characteristic
6	C₄⋊(C₅₉⋊C₁₆)	—	2	1	no	—
7	#16837155	C₄	1	1	yes	characteristic
8	#16837134	—	2	1	no	—
9	C₄×C₁₆×C₅₉	—	2	1	no	—
10	C₈²×C₅₉	C₂²	1	1	yes	characteristic
11	C₂×Dic₂₃₆	C₈	1	1	yes	characteristic
12	C₄×Dic₁₁₈	—	2	1	no	—
13	C₂×C₅₉⋊C₁₆	—	4	1	no	—
14	#14273661	C₈	1	1	yes	characteristic
15	C₄⋊(C₅₉⋊C₈)	—	2	1	no	—
16	Dic₅₉⋊C₈	—	4	1	no	—
17	#14273652	—	2	1	no	—
18	C₈×Dic₅₉	—	4	1	no	—
19	C₂×C₁₆×C₅₉	—	2	2	no	—
20	C₄×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
21	C₄×C₈×C₅₉	D₄	1	1	yes	characteristic
22	C₄×C₈×C₅₉	D₄	1	1	yes	characteristic
23	C₂×Dic₁₁₈	—	2	1	no	—
24	C₂×C₈×C₅₉	—	2	1	no	—
25	C₂×C₈×C₅₉	C₂×C₈	1	1	yes	characteristic
26	C₂×C₈×C₅₉	M₁₆	1	1	yes	characteristic
27	C₂×C₈×C₅₉	D₈	1	1	yes	characteristic
28	C₂×C₈×C₅₉	SD₁₆	1	1	yes	characteristic
29	C₄²×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
30	C₄⋊Dic₅₉	—	2	1	no	—
31	Dic₅₉⋊C₄	—	4	1	no	—
32	C₄×Dic₅₉	—	4	1	no	—
33	C₂×C₅₉⋊C₈	—	4	1	no	—
34	Dic₂₃₆	—	2	1	no	—
35	Dic₂₃₆	C₁₆	1	2	yes	complemented
36	C₁₆×C₅₉	—	4	2	no	—
37	C₅₉⋊C₁₆	—	8	1	no	—
38	C₂×C₄×C₅₉	C₂²⋊C₈	1	1	yes	characteristic
39	C₂×C₄×C₅₉	#102	1	1	yes	characteristic
40	C₂×C₄×C₅₉	#104	1	1	yes	characteristic
41	C₂×Dic₅₉	—	4	1	no	—
42	C₂×Dic₅₉	—	4	1	no	—
43	Dic₁₁₈	—	2	2	no	—
44	Dic₁₁₈	—	4	1	no	—
45	C₈×C₅₉	—	2	2	no	—
46	C₈×C₅₉	—	2	1	no	—
47	C₈×C₅₉	—	2	1	no	—
48	C₈×C₅₉	—	2	1	no	—
49	C₈×C₅₉	C₂×C₁₆	1	1	yes	characteristic
50	C₈×C₅₉	M₃₂	1	1	yes	characteristic
51	C₅₉⋊C₈	—	8	1	no	—
52	#7071	—	59	1	no	maximal, Sylow
53	C₂²×C₅₉	D₄⋊C₈	1	1	yes	characteristic
54	C₄×C₅₉	—	2	1	no	—
55	C₄×C₅₉	C₂²⋊C₁₆	1	1	yes	characteristic
56	C₄×C₅₉	#350	1	1	yes	characteristic
57	C₄×C₅₉	#360	1	1	yes	characteristic
58	C₄×C₅₉	#362	1	1	yes	characteristic
59	Dic₅₉	—	4	2	no	—
60	Dic₅₉	—	8	1	no	—
61	C₈×Q₁₆	—	59	1	no	—
62	#1367	—	59	1	no	—
63	#1312	D₅₉	1	1	yes	characteristic
64	C₂×C₅₉	D₄⋊C₁₆	1	1	yes	characteristic
65	C₂×C₅₉	#1334	1	1	yes	characteristic
66	C₂×C₅₉	#1336	1	1	yes	characteristic
67	C₈×Q₈	—	118	1	no	—
68	C₄×Q₁₆	—	59	1	no	—
69	C₄⋊C₁₆	—	118	1	no	—
70	C₄×C₁₆	—	2	1	no	—
71	#335	—	59	1	no	—
72	Q₈⋊C₈	—	118	1	no	—
73	C₈²	C₂×D₅₉	1	1	yes	characteristic
74	C₅₉	#7071	1	1	yes	characteristic, complemented, Sylow
75	C₂×Q₁₆	—	59	1	no	—
76	C₄×Q₈	—	118	1	no	—
77	C₂×C₁₆	—	2	2	no	—
78	C₂×C₁₆	—	236	1	no	—
79	#107	—	59	1	no	—
80	C₄⋊C₈	—	118	1	no	—
81	C₄⋊C₈	—	236	1	no	—
82	#103	—	118	1	no	—
83	C₄×C₈	—	236	1	no	—
84	C₄×C₈	C₄×D₅₉	1	1	yes	characteristic
85	C₄×C₈	D₂₃₆	1	1	yes	characteristic
86	C₄×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
87	C₂×Q₈	—	118	1	no	—
88	Q₁₆	—	59	2	no	—
89	Q₁₆	—	118	1	no	—
90	C₂×C₈	—	2	1	no	—
91	C₂×C₈	—	236	1	no	—
92	C₂×C₈	C₈×D₅₉	1	1	yes	characteristic
93	C₂×C₈	#11745659	1	1	yes	characteristic
94	C₂×C₈	#11745660	1	1	yes	characteristic
95	C₂×C₈	D₄₇₂	1	1	yes	characteristic
96	C₄⋊C₄	—	118	1	no	—
97	C₄⋊C₄	—	236	1	no	—
98	C₄²	—	236	1	no	—
99	C₄²	(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
100	C₁₆	—	4	2	no	—
101	C₁₆	—	472	1	no	—
102	Q₈	—	118	2	no	—
103	Q₈	—	236	1	no	—
104	C₂×C₄	—	236	1	no	—
105	C₂×C₄	—	236	1	no	—
106	C₂×C₄	(C₂×D₅₉)⋊C₈	1	1	yes	characteristic
107	C₂×C₄	#14273649	1	1	yes	characteristic
108	C₂×C₄	#14273655	1	1	yes	characteristic
109	C₈	—	2	2	no	—
110	C₈	—	2	1	no	—
111	C₈	—	2	1	no	—
112	C₈	—	2	1	no	—
113	C₈	—	472	1	no	—
114	C₈	C₁₆×D₅₉	1	1	yes	characteristic
115	C₈	#14273666	1	1	yes	characteristic
116	C₂²	#16837131	1	1	yes	characteristic
117	C₄	—	2	1	no	—
118	C₄	—	472	1	no	—
119	C₄	—	236	2	no	—
120	C₄	(C₂×D₅₉)⋊C₁₆	1	1	yes	characteristic
121	C₄	#16837185	1	1	yes	characteristic
122	C₄	#16837213	1	1	yes	characteristic
123	C₄	#16837219	1	1	yes	characteristic
124	C₂	#21422052	1	1	yes	characteristic
125	C₂	#21422067	1	1	yes	characteristic
126	C₂	#21422073	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₈×Dic₂₃₆

C₂

yes

maximal, characteristic

#21422160

C₂

yes

maximal, characteristic

C₅₉×#1312

C₂

yes

maximal, characteristic

C₈×Dic₁₁₈

—

C₄×Dic₂₃₆

C₄

yes

characteristic

C₄⋊(C₅₉⋊C₁₆)

—

#16837155

C₄

yes

characteristic

#16837134

—

C₄×C₁₆×C₅₉

—

C₈²×C₅₉

C₂²

yes

characteristic

C₂×Dic₂₃₆

C₈

yes

characteristic

C₄×Dic₁₁₈

—

C₂×C₅₉⋊C₁₆

—

#14273661

C₈

yes

characteristic

C₄⋊(C₅₉⋊C₈)

—

Dic₅₉⋊C₈

—

#14273652

—

C₈×Dic₅₉

—

C₂×C₁₆×C₅₉

—

C₄×C₈×C₅₉

C₂×C₄

yes

characteristic

C₄×C₈×C₅₉

D₄

yes

characteristic

C₄×C₈×C₅₉

D₄

yes

characteristic

C₂×Dic₁₁₈

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

C₂×C₈

yes

characteristic

C₂×C₈×C₅₉

M₁₆

yes

characteristic

C₂×C₈×C₅₉

D₈

yes

characteristic

C₂×C₈×C₅₉

SD₁₆

yes

characteristic

C₄²×C₅₉

C₂²⋊C₄

yes

characteristic

C₄⋊Dic₅₉

—

Dic₅₉⋊C₄

—

C₄×Dic₅₉

—

C₂×C₅₉⋊C₈

—

Dic₂₃₆

—

Dic₂₃₆

C₁₆

yes

complemented

C₁₆×C₅₉

—

C₅₉⋊C₁₆

—

C₂×C₄×C₅₉

C₂²⋊C₈

yes

C₂×C₄×C₅₉

yes

C₂×C₄×C₅₉

yes

C₂×Dic₅₉

—

C₂×Dic₅₉

—

Dic₁₁₈

—

Dic₁₁₈

—

C₈×C₅₉

—

C₈×C₅₉

—

C₈×C₅₉

—

C₈×C₅₉

—

C₈×C₅₉

C₂×C₁₆

yes

characteristic

C₈×C₅₉

M₃₂

yes

characteristic

C₅₉⋊C₈

—

#7071

—

maximal, Sylow

C₂²×C₅₉

D₄⋊C₈

yes

characteristic

C₄×C₅₉

—

C₄×C₅₉

C₂²⋊C₁₆

yes

C₄×C₅₉

yes

C₄×C₅₉

yes

C₄×C₅₉

yes

Dic₅₉

—

Dic₅₉

—

C₈×Q₁₆

—

#1367

—

#1312

D₅₉

yes

characteristic

C₂×C₅₉

D₄⋊C₁₆

yes

C₂×C₅₉

yes

C₂×C₅₉

yes

C₈×Q₈

—

118

—

C₄×Q₁₆

—

C₄⋊C₁₆

—

118

—

C₄×C₁₆

—

#335

—

Q₈⋊C₈

—

118

—

C₈²

C₂×D₅₉

yes

characteristic

C₅₉

#7071

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

C₄×Q₈

—

118

—

C₂×C₁₆

—

C₂×C₁₆

—

236

—

#107

—

C₄⋊C₈

—

118

—

C₄⋊C₈

—

236

—

#103

—

118

—

C₄×C₈

—

236

—

C₄×C₈

C₄×D₅₉

yes

characteristic

C₄×C₈

D₂₃₆

yes

characteristic

C₄×C₈

C₂²⋊D₅₉

yes

characteristic

C₂×Q₈

—

118

—

Q₁₆

—

Q₁₆

—

118

—

C₂×C₈

—

C₂×C₈

—

236

—

C₂×C₈

C₈×D₅₉

yes

C₂×C₈

yes

C₂×C₈

yes

C₂×C₈

D₄₇₂

yes

characteristic

C₄⋊C₄

—

118

—

C₄⋊C₄

—

236

—

C₄²

—

236

—

C₄²

(C₂×D₅₉)⋊C₄

yes

characteristic

100

C₁₆

—

101

C₁₆

—

472

—

102

Q₈

—

118

—

103

Q₈

—

236

—

104

C₂×C₄

—

236

—

105

C₂×C₄

—

236

—

106

C₂×C₄

(C₂×D₅₉)⋊C₈

yes

characteristic

107

C₂×C₄

#14273649

yes

characteristic

108

C₂×C₄

#14273655

yes

characteristic

109

C₈

—

110

C₈

—

111

C₈

—

112

C₈

—

113

C₈

—

472

—

114

C₈

C₁₆×D₅₉

yes

characteristic

115

C₈

#14273666

yes

characteristic

116

C₂²

#16837131

yes

characteristic

117

C₄

—

118

C₄

—

472

—

119

C₄

—

236

—

120

C₄

(C₂×D₅₉)⋊C₁₆

yes

characteristic

121

C₄

#16837185

yes

characteristic

122

C₄

#16837213

yes

characteristic

123

C₄

#16837219

yes

characteristic

124

C₂

#21422052

yes

characteristic

125

C₂

#21422067

yes

characteristic

126

C₂

#21422073

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#14273649
commutator	C₈×C₅₉	C₂×C₁₆
Frattini	C₈²	C₂×D₅₉
Fitting	C₅₉×#1312	C₂
socle, prime radical	C₂²×C₅₉	D₄⋊C₈
cosocle	C₈²×C₅₉	C₂²
hypercenter	#1312	D₅₉
hypocenter	C₅₉	#7071
LCS	G ▷ C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ C₈² ◁ #1312

subgroup

G/H

center

C₂×C₄

#14273649

commutator

C₈×C₅₉

C₂×C₁₆

Frattini

C₈²

C₂×D₅₉

Fitting

C₅₉×#1312

C₂

socle, prime radical

C₂²×C₅₉

D₄⋊C₈

cosocle

C₈²×C₅₉

C₂²

D₅₉

C₅₉

G ▷ C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ C₈² ◁ #1312

N	H ≅ G/N	type
35. Dic₂₃₆	C₁₆	semidirect
74. C₅₉	#7071	semidirect

H ≅ G/N

type

35. Dic₂₃₆

C₁₆

semidirect

74. C₅₉

#7071

semidirect

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

subgroup	index
C₈×Dic₂₃₆	2
#21422160	2
C₅₉×#1312	2
#7071	59

subgroup

index

C₈×Dic₂₃₆

#21422160

C₅₉×#1312

#7071

Subgroups — #28475348

Conjugacy classes of subgroups

Complements

Maximal subgroups