Group #21433050 — subgroups

Subgroups — Q₈×#11745664 #21433050

Order 7552.

388 conjugacy classes of subgroups; only the 241 normal subgroups are shown.

normal subgroups only

This group has 388 conjugacy classes of subgroups — too many to draw. Only its 241 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	#16837533	C₂	1	6	yes	maximal, complemented
2	Q₈×C₅₉⋊C₈	C₂	1	2	yes	maximal, complemented
3	#16837426	C₂	1	3	yes	maximal
4	C₄×#11745664	C₂	1	3	yes	maximal
5	C₂×C₄×Q₈×C₅₉	C₂	1	1	yes	maximal, characteristic
6	C₂×#11745664	C₂²	1	3	yes	—
7	C₂×#11745664	C₂²	1	1	yes	characteristic
8	C₄⋊(C₅₉⋊C₈)	C₂²	1	6	yes	—
9	C₄⋊(C₅₉⋊C₈)	C₂²	1	6	yes	—
10	#14273638	C₂²	1	6	yes	—
11	C₄×C₅₉⋊C₈	C₂²	1	6	yes	—
12	C₂²×Q₈×C₅₉	C₄	1	1	yes	characteristic
13	C₄×Q₈×C₅₉	C₄	1	3	yes	—
14	C₄×Q₈×C₅₉	C₄	1	1	yes	characteristic
15	C₄×Q₈×C₅₉	C₂²	1	1	yes	characteristic
16	C₄×Q₈×C₅₉	C₂²	1	3	yes	—
17	C₂×C₄⋊C₄×C₅₉	C₄	1	3	yes	—
18	C₂×C₄²×C₅₉	C₂²	1	3	yes	—
19	C₂×Q₈×C₅₉	—	2	4	no	—
20	C₂×Q₈×C₅₉	C₂×C₄	1	3	yes	—
21	C₂×Q₈×C₅₉	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₄×C₅₉	C₂×C₄	1	3	yes	—
23	C₂²×C₄×C₅₉	C₂×C₄	1	3	yes	—
24	C₂²×C₄×C₅₉	C₂³	1	1	yes	characteristic
25	C₄⋊C₄×C₅₉	C₂×C₄	1	6	yes	—
26	C₄⋊C₄×C₅₉	C₂×C₄	1	3	yes	—
27	C₄⋊C₄×C₅₉	C₂×C₄	1	3	yes	—
28	C₄²×C₅₉	C₂×C₄	1	3	yes	—
29	C₄²×C₅₉	C₂×C₄	1	3	yes	—
30	C₄²×C₅₉	C₂³	1	3	yes	—
31	C₄²×C₅₉	C₂³	1	3	yes	—
32	#11745664	—	2	6	no	—
33	#11745664	Q₈	1	4	yes	direct product factor
34	C₂×C₅₉⋊C₈	C₂³	1	6	yes	—
35	C₂×C₅₉⋊C₈	C₂³	1	2	yes	—
36	C₂³×C₅₉	C₂²×C₄	1	1	yes	characteristic
37	Q₈×C₅₉	—	2	6	no	—
38	Q₈×C₅₉	M₁₆	1	4	yes	complemented
39	C₂×C₄×C₅₉	—	2	6	no	—
40	C₂×C₄×C₅₉	—	4	3	no	—
41	C₂×C₄×C₅₉	C₂²×C₄	1	3	yes	—
42	C₂×C₄×C₅₉	C₂²×C₄	1	3	yes	—
43	C₂×C₄×C₅₉	C₂²×C₄	1	3	yes	—
44	C₂×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₄×C₅₉	C₂²×C₄	1	3	yes	—
46	C₂×C₄×C₅₉	C₂×Q₈	1	2	yes	—
47	C₂×C₄×C₅₉	#41	1	2	yes	—
48	C₂×C₄×C₅₉	C₂⁴	1	1	yes	characteristic
49	C₅₉⋊C₈	—	2	6	no	—
50	C₅₉⋊C₈	C₂×Q₈	1	4	yes	complemented
51	C₂²×C₅₉	—	2	1	no	—
52	C₂²×C₅₉	—	2	1	no	—
53	C₂²×C₅₉	C₄×Q₈	1	2	yes	—
54	C₂²×C₅₉	C₂³×C₄	1	1	yes	characteristic
55	C₄×C₅₉	—	2	3	no	—
56	C₄×C₅₉	—	2	3	no	—
57	C₄×C₅₉	—	2	3	no	—
58	C₄×C₅₉	C₄×Q₈	1	2	yes	—
59	C₄×C₅₉	C₂×M₁₆	1	6	yes	—
60	C₄×C₅₉	C₂²×Q₈	1	1	yes	characteristic
61	C₄×C₅₉	C₂×#41	1	1	yes	characteristic
62	Q₈×M₁₆	—	59	1	no	maximal, Sylow
63	C₂×C₅₉	—	2	2	no	—
64	C₂×C₅₉	C₂×C₄×Q₈	1	1	yes	characteristic
65	C₂×C₅₉	C₂²×M₁₆	1	1	yes	characteristic
66	C₂×C₅₉	#568	1	1	yes	characteristic
67	C₂×C₄×Q₈	D₅₉	1	1	yes	characteristic
68	#446	—	59	6	no	—
69	C₈×Q₈	—	59	2	no	—
70	#423	—	59	3	no	—
71	C₄×M₁₆	—	59	3	no	—
72	C₅₉	Q₈×M₁₆	1	1	yes	characteristic, complemented, Sylow
73	C₂²×Q₈	Dic₅₉	1	1	yes	characteristic
74	C₂×M₁₆	—	59	1	no	—
75	C₂×M₁₆	—	59	3	no	—
76	C₄×Q₈	Dic₅₉	1	3	yes	—
77	C₄×Q₈	Dic₅₉	1	1	yes	characteristic
78	C₄×Q₈	C₂×D₅₉	1	1	yes	characteristic
79	C₄×Q₈	C₂×D₅₉	1	3	yes	—
80	C₂×C₄⋊C₄	Dic₅₉	1	3	yes	—
81	C₂×C₄²	C₂×D₅₉	1	3	yes	—
82	C₄⋊C₈	—	59	6	no	—
83	C₄⋊C₈	—	59	6	no	—
84	#97	—	59	6	no	—
85	C₄×C₈	—	59	6	no	—
86	C₂×Q₈	—	2	4	no	—
87	C₂×Q₈	C₂×Dic₅₉	1	3	yes	—
88	C₂×Q₈	C₂×Dic₅₉	1	1	yes	characteristic
89	C₂²×C₄	C₂×Dic₅₉	1	3	yes	—
90	C₂²×C₄	C₂×Dic₅₉	1	3	yes	—
91	C₂²×C₄	C₂²×D₅₉	1	1	yes	characteristic
92	M₁₆	—	59	4	no	—
93	M₁₆	—	118	6	no	—
94	C₂×C₈	—	59	2	no	—
95	C₂×C₈	—	59	6	no	—
96	C₄⋊C₄	C₂×Dic₅₉	1	6	yes	—
97	C₄⋊C₄	C₂×Dic₅₉	1	3	yes	—
98	C₄⋊C₄	C₂×Dic₅₉	1	3	yes	—
99	C₄²	C₂×Dic₅₉	1	3	yes	—
100	C₄²	C₂×Dic₅₉	1	3	yes	—
101	C₄²	C₂²×D₅₉	1	3	yes	—
102	C₄²	C₂²×D₅₉	1	3	yes	—
103	C₂³	C₂²×Dic₅₉	1	1	yes	characteristic
104	Q₈	—	2	6	no	—
105	Q₈	#11745664	1	4	yes	direct product factor
106	C₂×C₄	—	2	6	no	—
107	C₂×C₄	—	4	3	no	—
108	C₂×C₄	Q₈×D₅₉	1	2	yes	—
109	C₂×C₄	#11745689	1	2	yes	—
110	C₂×C₄	C₂²×Dic₅₉	1	3	yes	—
111	C₂×C₄	C₂²×Dic₅₉	1	3	yes	—
112	C₂×C₄	C₂²×Dic₅₉	1	3	yes	—
113	C₂×C₄	C₂²×Dic₅₉	1	1	yes	characteristic
114	C₂×C₄	C₂²×Dic₅₉	1	3	yes	—
115	C₂×C₄	C₂³×D₅₉	1	1	yes	characteristic
116	C₈	—	59	4	no	—
117	C₈	—	118	6	no	—
118	C₂²	—	2	1	no	—
119	C₂²	—	2	1	no	—
120	C₂²	Q₈×Dic₅₉	1	2	yes	—
121	C₂²	C₂³×Dic₅₉	1	1	yes	characteristic
122	C₄	—	2	3	no	—
123	C₄	—	2	3	no	—
124	C₄	—	2	3	no	—
125	C₄	Q₈×Dic₅₉	1	2	yes	—
126	C₄	C₂×#11745664	1	6	yes	—
127	C₄	C₂×Q₈×D₅₉	1	1	yes	characteristic
128	C₄	C₂×#11745689	1	1	yes	characteristic
129	C₂	—	2	2	no	—
130	C₂	C₂×Q₈×Dic₅₉	1	1	yes	characteristic
131	C₂	C₂²×#11745664	1	1	yes	characteristic
132	C₂	#16838169	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16837533

C₂

yes

maximal, complemented

Q₈×C₅₉⋊C₈

C₂

yes

maximal, complemented

#16837426

C₂

yes

maximal

C₄×#11745664

C₂

yes

maximal

C₂×C₄×Q₈×C₅₉

C₂

yes

maximal, characteristic

C₂×#11745664

C₂²

yes

—

C₂×#11745664

C₂²

yes

characteristic

C₄⋊(C₅₉⋊C₈)

C₂²

yes

—

C₄⋊(C₅₉⋊C₈)

C₂²

yes

—

#14273638

C₂²

yes

—

C₄×C₅₉⋊C₈

C₂²

yes

—

C₂²×Q₈×C₅₉

C₄

yes

characteristic

C₄×Q₈×C₅₉

C₄

yes

—

C₄×Q₈×C₅₉

C₄

yes

characteristic

C₄×Q₈×C₅₉

C₂²

yes

characteristic

C₄×Q₈×C₅₉

C₂²

yes

—

C₂×C₄⋊C₄×C₅₉

C₄

yes

—

C₂×C₄²×C₅₉

C₂²

yes

—

C₂×Q₈×C₅₉

—

C₂×Q₈×C₅₉

C₂×C₄

yes

—

C₂×Q₈×C₅₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₅₉

C₂×C₄

yes

—

C₂²×C₄×C₅₉

C₂×C₄

yes

—

C₂²×C₄×C₅₉

C₂³

yes

characteristic

C₄⋊C₄×C₅₉

C₂×C₄

yes

—

C₄⋊C₄×C₅₉

C₂×C₄

yes

—

C₄⋊C₄×C₅₉

C₂×C₄

yes

—

C₄²×C₅₉

C₂×C₄

yes

—

C₄²×C₅₉

C₂×C₄

yes

—

C₄²×C₅₉

C₂³

yes

—

C₄²×C₅₉

C₂³

yes

—

#11745664

—

#11745664

Q₈

yes

direct product factor

C₂×C₅₉⋊C₈

C₂³

yes

—

C₂×C₅₉⋊C₈

C₂³

yes

—

C₂³×C₅₉

C₂²×C₄

yes

characteristic

Q₈×C₅₉

—

Q₈×C₅₉

M₁₆

yes

complemented

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₂²×C₄

yes

—

C₂×C₄×C₅₉

C₂²×C₄

yes

—

C₂×C₄×C₅₉

C₂²×C₄

yes

—

C₂×C₄×C₅₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₉

C₂²×C₄

yes

—

C₂×C₄×C₅₉

C₂×Q₈

yes

—

C₂×C₄×C₅₉

#41

yes

—

C₂×C₄×C₅₉

C₂⁴

yes

characteristic

C₅₉⋊C₈

—

C₅₉⋊C₈

C₂×Q₈

yes

complemented

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

C₄×Q₈

yes

—

C₂²×C₅₉

C₂³×C₄

yes

characteristic

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

C₄×Q₈

yes

—

C₄×C₅₉

C₂×M₁₆

yes

—

C₄×C₅₉

C₂²×Q₈

yes

characteristic

C₄×C₅₉

C₂×#41

yes

characteristic

Q₈×M₁₆

—

maximal, Sylow

C₂×C₅₉

—

C₂×C₅₉

C₂×C₄×Q₈

yes

characteristic

C₂×C₅₉

C₂²×M₁₆

yes

characteristic

C₂×C₅₉

#568

yes

characteristic

C₂×C₄×Q₈

D₅₉

yes

characteristic

#446

—

C₈×Q₈

—

#423

—

C₄×M₁₆

—

C₅₉

Q₈×M₁₆

yes

characteristic, complemented, Sylow

C₂²×Q₈

Dic₅₉

yes

characteristic

C₂×M₁₆

—

C₂×M₁₆

—

C₄×Q₈

Dic₅₉

yes

—

C₄×Q₈

Dic₅₉

yes

characteristic

C₄×Q₈

C₂×D₅₉

yes

characteristic

C₄×Q₈

C₂×D₅₉

yes

—

C₂×C₄⋊C₄

Dic₅₉

yes

—

C₂×C₄²

C₂×D₅₉

yes

—

C₄⋊C₈

—

C₄⋊C₈

—

#97

—

C₄×C₈

—

C₂×Q₈

—

C₂×Q₈

C₂×Dic₅₉

yes

—

C₂×Q₈

C₂×Dic₅₉

yes

characteristic

C₂²×C₄

C₂×Dic₅₉

yes

—

C₂²×C₄

C₂×Dic₅₉

yes

—

C₂²×C₄

C₂²×D₅₉

yes

characteristic

M₁₆

—

M₁₆

—

118

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

C₂×Dic₅₉

yes

—

C₄⋊C₄

C₂×Dic₅₉

yes

—

C₄⋊C₄

C₂×Dic₅₉

yes

—

C₄²

C₂×Dic₅₉

yes

—

100

C₄²

C₂×Dic₅₉

yes

—

101

C₄²

C₂²×D₅₉

yes

—

102

C₄²

C₂²×D₅₉

yes

—

103

C₂³

C₂²×Dic₅₉

yes

characteristic

104

Q₈

—

105

Q₈

#11745664

yes

direct product factor

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

Q₈×D₅₉

yes

—

109

C₂×C₄

#11745689

yes

—

110

C₂×C₄

C₂²×Dic₅₉

yes

—

111

C₂×C₄

C₂²×Dic₅₉

yes

—

112

C₂×C₄

C₂²×Dic₅₉

yes

—

113

C₂×C₄

C₂²×Dic₅₉

yes

characteristic

114

C₂×C₄

C₂²×Dic₅₉

yes

—

115

C₂×C₄

C₂³×D₅₉

yes

characteristic

116

C₈

—

117

C₈

—

118

—

118

C₂²

—

119

C₂²

—

120

C₂²

Q₈×Dic₅₉

yes

—

121

C₂²

C₂³×Dic₅₉

yes

characteristic

122

C₄

—

123

C₄

—

124

C₄

—

125

C₄

Q₈×Dic₅₉

yes

—

126

C₄

C₂×#11745664

yes

—

127

C₄

C₂×Q₈×D₅₉

yes

characteristic

128

C₄

C₂×#11745689

yes

characteristic

129

C₂

—

130

C₂

C₂×Q₈×Dic₅₉

yes

characteristic

131

C₂

C₂²×#11745664

yes

characteristic

132

C₂

#16838169

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₅₉
commutator, socle	C₂²×C₅₉	C₂³×C₄
Fitting	C₂×C₄×Q₈×C₅₉	C₂
hypercenter	C₂×C₄×Q₈	D₅₉
hypocenter	C₅₉	Q₈×M₁₆
LCS	G ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×Q₈

subgroup

G/H

center, Frattini

C₂×C₄

C₂³×D₅₉

commutator, socle

C₂²×C₅₉

C₂³×C₄

Fitting

C₂×C₄×Q₈×C₅₉

C₂

hypercenter

C₂×C₄×Q₈

D₅₉

hypocenter

C₅₉

Q₈×M₁₆

LCS

G ▷ C₂²×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄×Q₈

N	H ≅ G/N	type
1. #16837533	C₂	semidirect
2. Q₈×C₅₉⋊C₈	C₂	semidirect
33. #11745664	Q₈	direct (N × H)
38. Q₈×C₅₉	M₁₆	semidirect
50. C₅₉⋊C₈	C₂×Q₈	semidirect
72. C₅₉	Q₈×M₁₆	semidirect
105. Q₈	#11745664	direct (N × H)

H ≅ G/N

type

1. #16837533

C₂

semidirect

2. Q₈×C₅₉⋊C₈

C₂

semidirect

33. #11745664

Q₈

direct (N × H)

38. Q₈×C₅₉

M₁₆

semidirect

50. C₅₉⋊C₈

C₂×Q₈

semidirect

72. C₅₉

Q₈×M₁₆

semidirect

105. Q₈

#11745664

direct (N × H)

Subgroups — Q₈×#11745664 #21433050

Conjugacy classes of subgroups

Complements

Subgroups — Q8×#11745664 #21433050

Conjugacy classes of subgroups

Complements

Subgroups — Q₈×#11745664 #21433050