Group #21422813 — subgroups

Subgroups — #21422813

Order 7552.

228 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 85 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	#16837459	C₂	1	1	yes	maximal, characteristic
2	#16837459	C₂	1	1	yes	maximal, characteristic
3	#16837148	C₂	1	4	yes	maximal, complemented
4	C₂×C₄⋊Q₈×C₅₉	C₂	1	1	yes	maximal, characteristic
5	C₄⋊(C₅₉⋊C₈)	C₂²	1	2	yes	—
6	C₄⋊(C₅₉⋊C₈)	C₂²	1	2	yes	—
7	C₂²⋊(C₅₉⋊C₈)	—	2	1	no	—
8	C₂²⋊(C₅₉⋊C₈)	—	2	1	no	—
9	#14273638	—	2	1	no	—
10	#14273638	—	2	1	no	—
11	C₂²×Q₈×C₅₉	—	2	1	no	—
12	C₄⋊Q₈×C₅₉	—	2	1	no	—
13	C₄⋊Q₈×C₅₉	—	2	1	no	—
14	C₄⋊Q₈×C₅₉	C₄	1	2	yes	—
15	C₄⋊Q₈×C₅₉	C₂²	1	2	yes	—
16	C₂×C₄⋊C₄×C₅₉	—	2	1	no	—
17	C₂×C₄⋊C₄×C₅₉	C₄	1	2	yes	—
18	C₂×C₄²×C₅₉	C₂²	1	1	yes	characteristic
19	C₂×Q₈×C₅₉	—	2	2	no	—
20	C₂×Q₈×C₅₉	—	2	2	no	—
21	C₂×Q₈×C₅₉	—	4	4	no	—
22	C₂²×C₄×C₅₉	—	2	2	no	—
23	C₂²×C₄×C₅₉	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₄×C₅₉	D₄	1	1	yes	characteristic
25	C₂²×C₄×C₅₉	D₄	1	1	yes	characteristic
26	C₄⋊C₄×C₅₉	—	2	2	no	—
27	C₄⋊C₄×C₅₉	—	2	2	no	—
28	C₄⋊C₄×C₅₉	—	2	2	no	—
29	C₄⋊C₄×C₅₉	C₂×C₄	1	4	yes	—
30	C₄²×C₅₉	C₂×C₄	1	1	yes	characteristic
31	C₄²×C₅₉	D₄	1	1	yes	characteristic
32	C₄²×C₅₉	D₄	1	1	yes	characteristic
33	C₄²×C₅₉	C₂³	1	1	yes	characteristic
34	C₂×C₅₉⋊C₈	—	2	2	no	—
35	C₂×C₅₉⋊C₈	—	2	2	no	—
36	C₂³×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
37	Q₈×C₅₉	—	4	4	no	—
38	Q₈×C₅₉	—	4	4	no	—
39	C₂×C₄×C₅₉	—	2	2	no	—
40	C₂×C₄×C₅₉	—	2	2	no	—
41	C₂×C₄×C₅₉	—	2	2	no	—
42	C₂×C₄×C₅₉	—	2	4	no	—
43	C₂×C₄×C₅₉	—	4	4	no	—
44	C₂×C₄×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
45	C₂×C₄×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
46	C₂×C₄×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
47	C₂×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
48	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
50	C₅₉⋊C₈	—	4	2	no	—
51	C₅₉⋊C₈	—	4	2	no	—
52	C₂²×C₅₉	—	2	1	no	—
53	C₂²×C₅₉	—	2	1	no	—
54	C₂²×C₅₉	#101	1	2	yes	—
55	C₂²×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
56	C₄×C₅₉	—	2	1	no	—
57	C₄×C₅₉	—	2	1	no	—
58	C₄×C₅₉	—	2	1	no	—
59	C₄×C₅₉	—	2	1	no	—
60	C₄×C₅₉	—	4	4	no	—
61	C₄×C₅₉	AGL(1,ℤ/8)	1	2	yes	—
62	C₄×C₅₉	#137	1	2	yes	—
63	#1556	—	59	1	no	maximal, Sylow
64	C₂×C₅₉	—	2	2	no	—
65	C₂×C₅₉	C₂×#101	1	1	yes	characteristic
66	C₂×C₅₉	#418	1	1	yes	characteristic
67	C₂×C₅₉	#419	1	1	yes	characteristic
68	C₂×C₄⋊Q₈	D₅₉	1	1	yes	characteristic
69	#432	—	59	1	no	—
70	#432	—	59	1	no	—
71	#332	—	59	4	no	—
72	C₅₉	#1556	1	1	yes	characteristic, complemented, Sylow
73	C₂²×Q₈	—	2	1	no	—
74	C₄⋊Q₈	—	2	1	no	—
75	C₄⋊Q₈	—	2	1	no	—
76	C₄⋊Q₈	Dic₅₉	1	2	yes	—
77	C₄⋊Q₈	C₂×D₅₉	1	2	yes	—
78	C₂×C₄⋊C₄	—	2	1	no	—
79	C₂×C₄⋊C₄	Dic₅₉	1	2	yes	—
80	C₂×C₄²	C₂×D₅₉	1	1	yes	characteristic
81	C₄⋊C₈	—	59	2	no	—
82	C₄⋊C₈	—	59	2	no	—
83	C₂²⋊C₈	—	118	1	no	—
84	C₂²⋊C₈	—	118	1	no	—
85	#97	—	118	1	no	—
86	#97	—	118	1	no	—
87	C₂×Q₈	—	2	2	no	—
88	C₂×Q₈	—	2	2	no	—
89	C₂×Q₈	—	4	4	no	—
90	C₂²×C₄	—	2	2	no	—
91	C₂²×C₄	C₂×Dic₅₉	1	1	yes	characteristic
92	C₂²×C₄	C₂²⋊D₅₉	1	1	yes	characteristic
93	C₂²×C₄	C₂²⋊D₅₉	1	1	yes	characteristic
94	C₂×C₈	—	118	2	no	—
95	C₂×C₈	—	118	2	no	—
96	C₄⋊C₄	—	2	2	no	—
97	C₄⋊C₄	—	2	2	no	—
98	C₄⋊C₄	—	2	2	no	—
99	C₄⋊C₄	C₂×Dic₅₉	1	4	yes	—
100	C₄²	C₂×Dic₅₉	1	1	yes	characteristic
101	C₄²	C₂²⋊D₅₉	1	1	yes	characteristic
102	C₄²	C₂²⋊D₅₉	1	1	yes	characteristic
103	C₄²	C₂²×D₅₉	1	1	yes	characteristic
104	C₂³	C₂²⋊Dic₅₉	1	1	yes	characteristic
105	Q₈	—	4	4	no	—
106	Q₈	—	4	4	no	—
107	C₂×C₄	—	2	2	no	—
108	C₂×C₄	—	2	2	no	—
109	C₂×C₄	—	2	2	no	—
110	C₂×C₄	—	2	4	no	—
111	C₂×C₄	—	4	4	no	—
112	C₂×C₄	C₂²⋊Dic₅₉	1	1	yes	characteristic
113	C₂×C₄	C₂²⋊Dic₅₉	1	1	yes	characteristic
114	C₂×C₄	C₂²⋊Dic₅₉	1	1	yes	characteristic
115	C₂×C₄	C₂²×Dic₅₉	1	1	yes	characteristic
116	C₂×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
117	C₂×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
118	C₈	—	236	2	no	—
119	C₈	—	236	2	no	—
120	C₂²	—	2	1	no	—
121	C₂²	—	2	1	no	—
122	C₂²	#14273646	1	2	yes	—
123	C₂²	C₂×C₂²⋊Dic₅₉	1	1	yes	characteristic
124	C₄	—	2	1	no	—
125	C₄	—	2	1	no	—
126	C₄	—	2	1	no	—
127	C₄	—	2	1	no	—
128	C₄	—	4	4	no	—
129	C₄	#14273753	1	2	yes	—
130	C₄	#14273758	1	2	yes	—
131	C₂	—	2	2	no	—
132	C₂	C₂×#14273646	1	1	yes	characteristic
133	C₂	#16837407	1	1	yes	characteristic
134	C₂	#16837411	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16837459

C₂

yes

maximal, characteristic

#16837459

C₂

yes

maximal, characteristic

#16837148

C₂

yes

maximal, complemented

C₂×C₄⋊Q₈×C₅₉

C₂

yes

maximal, characteristic

C₄⋊(C₅₉⋊C₈)

C₂²

yes

—

C₄⋊(C₅₉⋊C₈)

C₂²

yes

—

C₂²⋊(C₅₉⋊C₈)

—

C₂²⋊(C₅₉⋊C₈)

—

#14273638

—

#14273638

—

C₂²×Q₈×C₅₉

—

C₄⋊Q₈×C₅₉

—

C₄⋊Q₈×C₅₉

—

C₄⋊Q₈×C₅₉

C₄

yes

—

C₄⋊Q₈×C₅₉

C₂²

yes

—

C₂×C₄⋊C₄×C₅₉

—

C₂×C₄⋊C₄×C₅₉

C₄

yes

—

C₂×C₄²×C₅₉

C₂²

yes

characteristic

C₂×Q₈×C₅₉

—

C₂×Q₈×C₅₉

—

C₂×Q₈×C₅₉

—

C₂²×C₄×C₅₉

—

C₂²×C₄×C₅₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₅₉

D₄

yes

characteristic

C₂²×C₄×C₅₉

D₄

yes

characteristic

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

C₂×C₄

yes

—

C₄²×C₅₉

C₂×C₄

yes

characteristic

C₄²×C₅₉

D₄

yes

characteristic

C₄²×C₅₉

D₄

yes

characteristic

C₄²×C₅₉

C₂³

yes

characteristic

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂³×C₅₉

C₂²⋊C₄

yes

characteristic

Q₈×C₅₉

—

Q₈×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₉

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₉

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₅₉⋊C₈

—

C₅₉⋊C₈

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

#101

yes

—

C₂²×C₅₉

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

AGL(1,ℤ/8)

yes

—

C₄×C₅₉

#137

yes

—

#1556

—

maximal, Sylow

C₂×C₅₉

—

C₂×C₅₉

yes

C₂×C₅₉

yes

C₂×C₅₉

yes

C₂×C₄⋊Q₈

D₅₉

yes

—

—

—

C₅₉

#1556

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

Dic₅₉

yes

—

C₄⋊Q₈

C₂×D₅₉

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

Dic₅₉

yes

—

C₂×C₄²

C₂×D₅₉

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

118

—

C₂²⋊C₈

—

118

—

#97

—

118

—

#97

—

118

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

C₂×Dic₅₉

yes

characteristic

C₂²×C₄

C₂²⋊D₅₉

yes

characteristic

C₂²×C₄

C₂²⋊D₅₉

yes

characteristic

C₂×C₈

—

118

—

C₂×C₈

—

118

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂×Dic₅₉

yes

—

100

C₄²

C₂×Dic₅₉

yes

characteristic

101

C₄²

C₂²⋊D₅₉

yes

characteristic

102

C₄²

C₂²⋊D₅₉

yes

characteristic

103

C₄²

C₂²×D₅₉

yes

characteristic

104

C₂³

C₂²⋊Dic₅₉

yes

characteristic

105

Q₈

—

106

Q₈

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

C₂²⋊Dic₅₉

yes

characteristic

113

C₂×C₄

C₂²⋊Dic₅₉

yes

characteristic

114

C₂×C₄

C₂²⋊Dic₅₉

yes

characteristic

115

C₂×C₄

C₂²×Dic₅₉

yes

characteristic

116

C₂×C₄

C₂×C₂²⋊D₅₉

yes

characteristic

117

C₂×C₄

C₂×C₂²⋊D₅₉

yes

characteristic

118

C₈

—

236

—

119

C₈

—

236

—

120

C₂²

—

121

C₂²

—

122

C₂²

#14273646

yes

—

123

C₂²

C₂×C₂²⋊Dic₅₉

yes

characteristic

124

C₄

—

125

C₄

—

126

C₄

—

127

C₄

—

128

C₄

—

129

C₄

#14273753

yes

—

130

C₄

#14273758

yes

—

131

C₂

—

132

C₂

C₂×#14273646

yes

characteristic

133

C₂

#16837407

yes

characteristic

134

C₂

#16837411

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊Dic₅₉
commutator	C₂×C₄×C₅₉	C₂²×C₄
Frattini	C₄²	C₂²×D₅₉
Fitting	C₂×C₄⋊Q₈×C₅₉	C₂
socle	C₂²×C₅₉	C₂×C₂²⋊C₄
hypercenter	C₂×C₄⋊Q₈	D₅₉
hypocenter	C₅₉	#1556
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

subgroup

G/H

center

C₂²

C₂×C₂²⋊Dic₅₉

commutator

C₂×C₄×C₅₉

C₂²×C₄

Frattini

C₄²

C₂²×D₅₉

Fitting

C₂×C₄⋊Q₈×C₅₉

C₂

socle

C₂²×C₅₉

C₂×C₂²⋊C₄

hypercenter

C₂×C₄⋊Q₈

D₅₉

hypocenter

C₅₉

#1556

LCS

G ▷ C₂×C₄×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

N	H ≅ G/N	type
3. #16837148	C₂	semidirect
72. C₅₉	#1556	semidirect

H ≅ G/N

type

3. #16837148

C₂

semidirect

72. C₅₉

#1556

semidirect

Subgroups — #21422813

Conjugacy classes of subgroups

Complements