Group #21425070 — subgroups

Subgroups — #21425070

Order 7552.

276 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 276 conjugacy classes of subgroups — too many to draw. Only its 113 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×#14273638	C₂	1	1	yes	maximal, characteristic
2	#16837268	C₂	1	1	yes	maximal, characteristic
3	#16837164	C₂	1	4	yes	maximal
4	C₂×C₄⋊Q₈×C₅₉	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₅₉⋊C₈	C₂²	1	2	yes	—
6	C₂×C₄⋊Dic₅₉	C₂²	1	2	yes	—
7	#14273638	—	2	2	no	—
8	C₅₉⋊#95	—	2	1	no	—
9	C₅₉⋊#95	—	2	1	no	—
10	C₂²×Q₈×C₅₉	—	2	1	no	—
11	C₄⋊Q₈×C₅₉	—	2	2	no	—
12	C₄⋊Q₈×C₅₉	C₄	1	4	yes	complemented
13	C₂×C₄⋊C₄×C₅₉	—	2	1	no	—
14	C₂×C₄⋊C₄×C₅₉	C₂²	1	2	yes	—
15	C₂×C₄²×C₅₉	C₂²	1	1	yes	characteristic
16	C₂×Q₈×C₅₉	—	2	4	no	—
17	C₂×Q₈×C₅₉	—	4	4	no	—
18	C₂²×C₄×C₅₉	—	2	2	no	—
19	C₂²×C₄×C₅₉	D₄	1	1	yes	characteristic
20	C₂²×C₄×C₅₉	D₄	1	1	yes	characteristic
21	C₂²×C₄×C₅₉	C₂³	1	1	yes	characteristic
22	C₂²×Dic₅₉	—	2	2	no	—
23	C₄⋊C₄×C₅₉	—	2	4	no	—
24	C₄⋊C₄×C₅₉	—	2	2	no	—
25	C₄⋊C₄×C₅₉	C₂×C₄	1	4	yes	—
26	C₄²×C₅₉	C₂×C₄	1	2	yes	—
27	C₄²×C₅₉	D₄	1	2	yes	—
28	C₄⋊Dic₅₉	—	2	4	no	—
29	C₂×C₅₉⋊C₈	—	2	2	no	—
30	C₂×C₅₉⋊C₈	—	2	2	no	—
31	C₂×C₅₉⋊C₈	Q₈	1	4	yes	complemented
32	C₂³×C₅₉	C₂×D₄	1	1	yes	characteristic
33	Q₈×C₅₉	—	4	8	no	—
34	C₂×C₄×C₅₉	—	2	2	no	—
35	C₂×C₄×C₅₉	—	2	2	no	—
36	C₂×C₄×C₅₉	—	2	4	no	—
37	C₂×C₄×C₅₉	—	4	4	no	—
38	C₂×C₄×C₅₉	C₂²⋊C₄	1	2	yes	—
39	C₂×C₄×C₅₉	C₂²⋊C₄	1	2	yes	—
40	C₂×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₅₉	C₂×Q₈	1	2	yes	—
43	C₂×C₄×C₅₉	#41	1	2	yes	—
44	C₂×Dic₅₉	—	4	2	no	—
45	C₂×Dic₅₉	—	4	2	no	—
46	C₂×Dic₅₉	—	4	2	no	—
47	C₅₉⋊C₈	—	2	4	no	—
48	C₂²×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
49	C₂²×C₅₉	#124	1	1	yes	characteristic
50	C₂²×C₅₉	C₄⋊Q₈	1	1	yes	characteristic
51	C₂²×C₅₉	AGL(1,ℤ/8)	1	2	yes	—
52	C₂²×C₅₉	#137	1	2	yes	—
53	C₄×C₅₉	—	2	2	no	—
54	C₄×C₅₉	—	2	2	no	—
55	C₄×C₅₉	—	4	4	no	—
56	C₄×C₅₉	C₄×Q₈	1	2	yes	—
57	C₄×C₅₉	C₂²⋊Q₈	1	2	yes	—
58	Dic₅₉	—	8	2	no	—
59	#1993	—	59	1	no	maximal, Sylow
60	C₂×C₅₉	#391	1	1	yes	characteristic
61	C₂×C₅₉	#418	1	1	yes	characteristic
62	C₂×C₅₉	#419	1	1	yes	characteristic
63	C₂×C₅₉	#489	1	2	yes	—
64	C₂×C₅₉	#501	1	2	yes	—
65	C₂×C₄⋊Q₈	D₅₉	1	1	yes	characteristic
66	C₂×#97	—	59	1	no	—
67	#382	—	59	1	no	—
68	#340	—	59	4	no	—
69	C₅₉	#1993	1	1	yes	characteristic, complemented, Sylow
70	C₂²×Q₈	—	2	1	no	—
71	C₂²×C₈	—	59	2	no	—
72	C₄⋊Q₈	—	2	2	no	—
73	C₄⋊Q₈	Dic₅₉	1	4	yes	complemented
74	C₂×C₄⋊C₄	—	2	1	no	—
75	C₂×C₄⋊C₄	—	59	2	no	—
76	C₂×C₄⋊C₄	C₂×D₅₉	1	2	yes	—
77	C₂×C₄²	C₂×D₅₉	1	1	yes	characteristic
78	#97	—	118	2	no	—
79	#95	—	118	1	no	—
80	#95	—	118	1	no	—
81	C₂×Q₈	—	2	4	no	—
82	C₂×Q₈	—	4	4	no	—
83	C₂²×C₄	—	2	2	no	—
84	C₂²×C₄	—	118	2	no	—
85	C₂²×C₄	C₂²⋊D₅₉	1	1	yes	characteristic
86	C₂²×C₄	C₂²⋊D₅₉	1	1	yes	characteristic
87	C₂²×C₄	C₂²×D₅₉	1	1	yes	characteristic
88	C₂×C₈	—	59	4	no	—
89	C₂×C₈	—	118	2	no	—
90	C₂×C₈	—	118	2	no	—
91	C₄⋊C₄	—	2	4	no	—
92	C₄⋊C₄	—	2	2	no	—
93	C₄⋊C₄	—	118	4	no	—
94	C₄⋊C₄	C₂×Dic₅₉	1	4	yes	—
95	C₄²	C₂×Dic₅₉	1	2	yes	—
96	C₄²	C₂²⋊D₅₉	1	2	yes	—
97	C₂³	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
98	Q₈	—	4	8	no	—
99	C₂×C₄	—	2	2	no	—
100	C₂×C₄	—	2	2	no	—
101	C₂×C₄	—	2	4	no	—
102	C₂×C₄	—	4	4	no	—
103	C₂×C₄	—	236	2	no	—
104	C₂×C₄	—	236	2	no	—
105	C₂×C₄	—	236	2	no	—
106	C₂×C₄	C₂²⋊Dic₅₉	1	2	yes	—
107	C₂×C₄	C₂²⋊Dic₅₉	1	2	yes	—
108	C₂×C₄	Q₈×D₅₉	1	2	yes	—
109	C₂×C₄	#11745689	1	2	yes	—
110	C₂×C₄	C₂²×Dic₅₉	1	1	yes	characteristic
111	C₂×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
112	C₈	—	118	4	no	—
113	C₂²	C₂×C₂²⋊Dic₅₉	1	1	yes	characteristic
114	C₂²	#14273716	1	1	yes	characteristic
115	C₂²	#14273725	1	1	yes	characteristic
116	C₂²	#14273753	1	2	yes	—
117	C₂²	#14273758	1	2	yes	—
118	C₄	—	2	2	no	—
119	C₄	—	2	2	no	—
120	C₄	—	4	4	no	—
121	C₄	—	472	2	no	—
122	C₄	Q₈×Dic₅₉	1	2	yes	—
123	C₄	#14273703	1	2	yes	—
124	C₂	#16837309	1	1	yes	characteristic
125	C₂	#16837407	1	1	yes	characteristic
126	C₂	#16837411	1	1	yes	characteristic
127	C₂	#16837764	1	2	yes	—
128	C₂	#16837808	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14273638

C₂

yes

maximal, characteristic

#16837268

C₂

yes

maximal, characteristic

#16837164

C₂

yes

maximal

C₂×C₄⋊Q₈×C₅₉

C₂

yes

maximal, characteristic

C₂²×C₅₉⋊C₈

C₂²

yes

—

C₂×C₄⋊Dic₅₉

C₂²

yes

—

#14273638

—

C₅₉⋊#95

—

C₅₉⋊#95

—

C₂²×Q₈×C₅₉

—

C₄⋊Q₈×C₅₉

—

C₄⋊Q₈×C₅₉

C₄

yes

complemented

C₂×C₄⋊C₄×C₅₉

—

C₂×C₄⋊C₄×C₅₉

C₂²

yes

—

C₂×C₄²×C₅₉

C₂²

yes

characteristic

C₂×Q₈×C₅₉

—

C₂×Q₈×C₅₉

—

C₂²×C₄×C₅₉

—

C₂²×C₄×C₅₉

D₄

yes

characteristic

C₂²×C₄×C₅₉

D₄

yes

characteristic

C₂²×C₄×C₅₉

C₂³

yes

characteristic

C₂²×Dic₅₉

—

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

C₂×C₄

yes

—

C₄²×C₅₉

C₂×C₄

yes

—

C₄²×C₅₉

D₄

yes

—

C₄⋊Dic₅₉

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

Q₈

yes

complemented

C₂³×C₅₉

C₂×D₄

yes

characteristic

Q₈×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₂²⋊C₄

yes

—

C₂×C₄×C₅₉

C₂²⋊C₄

yes

—

C₂×C₄×C₅₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

C₂×Q₈

yes

—

C₂×C₄×C₅₉

#41

yes

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₅₉⋊C₈

—

C₂²×C₅₉

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₅₉

#124

yes

characteristic

C₂²×C₅₉

C₄⋊Q₈

yes

characteristic

C₂²×C₅₉

AGL(1,ℤ/8)

yes

—

C₂²×C₅₉

#137

yes

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

C₄×Q₈

yes

—

C₄×C₅₉

C₂²⋊Q₈

yes

—

Dic₅₉

—

—

C₂×C₅₉

yes

C₂×C₅₉

yes

C₂×C₅₉

yes

C₂×C₅₉

yes

—

C₂×C₅₉

#501

yes

—

C₂×C₄⋊Q₈

D₅₉

yes

characteristic

C₂×#97

—

#382

—

#340

—

C₅₉

#1993

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₂²×C₈

—

C₄⋊Q₈

—

C₄⋊Q₈

Dic₅₉

yes

complemented

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₅₉

yes

—

C₂×C₄²

C₂×D₅₉

yes

characteristic

#97

—

118

—

#95

—

118

—

#95

—

118

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

118

—

C₂²×C₄

C₂²⋊D₅₉

yes

characteristic

C₂²×C₄

C₂²⋊D₅₉

yes

characteristic

C₂²×C₄

C₂²×D₅₉

yes

characteristic

C₂×C₈

—

C₂×C₈

—

118

—

C₂×C₈

—

118

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

118

—

C₄⋊C₄

C₂×Dic₅₉

yes

—

C₄²

C₂×Dic₅₉

yes

—

C₄²

C₂²⋊D₅₉

yes

—

C₂³

C₂×C₂²⋊D₅₉

yes

characteristic

Q₈

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

236

—

104

C₂×C₄

—

236

—

105

C₂×C₄

—

236

—

106

C₂×C₄

C₂²⋊Dic₅₉

yes

—

107

C₂×C₄

C₂²⋊Dic₅₉

yes

—

108

C₂×C₄

Q₈×D₅₉

yes

—

109

C₂×C₄

#11745689

yes

—

110

C₂×C₄

C₂²×Dic₅₉

yes

characteristic

111

C₂×C₄

C₂×C₂²⋊D₅₉

yes

characteristic

112

C₈

—

118

—

113

C₂²

C₂×C₂²⋊Dic₅₉

yes

characteristic

114

C₂²

#14273716

yes

characteristic

115

C₂²

#14273725

yes

characteristic

116

C₂²

#14273753

yes

—

117

C₂²

#14273758

yes

—

118

C₄

—

119

C₄

—

120

C₄

—

121

C₄

—

472

—

122

C₄

Q₈×Dic₅₉

yes

—

123

C₄

#14273703

yes

—

124

C₂

#16837309

yes

characteristic

125

C₂

#16837407

yes

characteristic

126

C₂

#16837411

yes

characteristic

127

C₂

#16837764

yes

—

128

C₂

#16837808

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₅₉
commutator	C₂×C₄×C₅₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₅₉
Fitting	C₂×C₄⋊Q₈×C₅₉	C₂
socle	C₂³×C₅₉	C₂×D₄
hypercenter	C₂×C₄⋊Q₈	D₅₉
hypocenter	C₅₉	#1993
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₅₉

commutator

C₂×C₄×C₅₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₅₉

Fitting

C₂×C₄⋊Q₈×C₅₉

C₂

socle

C₂³×C₅₉

C₂×D₄

hypercenter

C₂×C₄⋊Q₈

D₅₉

hypocenter

C₅₉

#1993

LCS

G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

N	H ≅ G/N	type
12. C₄⋊Q₈×C₅₉	C₄	semidirect
31. C₂×C₅₉⋊C₈	Q₈	semidirect
69. C₅₉	#1993	semidirect
73. C₄⋊Q₈	Dic₅₉	semidirect

H ≅ G/N

type

12. C₄⋊Q₈×C₅₉

C₄

semidirect

31. C₂×C₅₉⋊C₈

Q₈

semidirect

69. C₅₉

#1993

semidirect

73. C₄⋊Q₈

Dic₅₉

semidirect

Subgroups — #21425070

Conjugacy classes of subgroups

Complements