Group #21436978 — subgroups

Subgroups — Q₈×#11745672 #21436978

Order 7552.

336 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 336 conjugacy classes of subgroups — too many to draw. Only its 149 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	Q₈×Dic₁₁₈	C₂	1	1	yes	maximal, characteristic
2	#16837805	C₂	1	3	yes	maximal
3	#16837697	C₂	1	3	yes	maximal
4	#16837694	C₂	1	3	yes	maximal
5	Q₈×C₅₉⋊C₈	C₂	1	1	yes	maximal, characteristic
6	C₄×#11745672	C₂	1	3	yes	maximal
7	Q₈²×C₅₉	C₂	1	1	yes	maximal, characteristic
8	C₂×#11745672	C₂²	1	1	yes	characteristic
9	#14273727	C₂²	1	3	yes	—
10	#14273726	—	2	3	no	—
11	Q₈×Dic₅₉	—	2	1	no	—
12	C₄×Dic₁₁₈	C₂²	1	3	yes	—
13	#14273660	C₂²	1	6	yes	—
14	#14273660	C₂²	1	3	yes	—
15	C₄⋊(C₅₉⋊C₈)	C₂²	1	3	yes	—
16	#14273651	C₂²	1	3	yes	—
17	#14273650	C₂²	1	3	yes	—
18	C₄×C₅₉⋊C₈	C₂²	1	3	yes	—
19	C₄⋊Q₈×C₅₉	—	2	3	no	—
20	C₄⋊Q₈×C₅₉	C₂²	1	3	yes	—
21	C₄×Q₈×C₅₉	—	2	1	no	—
22	C₄×Q₈×C₅₉	C₂²	1	3	yes	—
23	C₄×Q₈×C₅₉	C₂²	1	1	yes	characteristic
24	C₂×Q₈×C₅₉	—	2	3	no	—
25	C₂×Q₈×C₅₉	D₄	1	1	yes	characteristic
26	C₂×Q₈×C₅₉	C₂³	1	1	yes	characteristic
27	C₂×Dic₁₁₈	—	2	3	no	—
28	C₂×Dic₁₁₈	C₂³	1	1	yes	characteristic
29	C₄⋊C₄×C₅₉	—	2	3	no	—
30	C₄⋊C₄×C₅₉	—	2	3	no	—
31	C₄⋊C₄×C₅₉	D₄	1	3	yes	—
32	C₄⋊C₄×C₅₉	C₂³	1	3	yes	—
33	C₄²×C₅₉	—	2	3	no	—
34	C₄²×C₅₉	C₂³	1	3	yes	—
35	#11745672	Q₈	1	4	yes	direct product factor
36	C₄⋊Dic₅₉	—	2	3	no	—
37	C₄⋊Dic₅₉	C₂³	1	3	yes	—
38	Dic₅₉⋊C₄	—	2	3	no	—
39	C₄×Dic₅₉	—	2	3	no	—
40	C₂×C₅₉⋊C₈	C₂³	1	3	yes	—
41	C₂×C₅₉⋊C₈	C₂³	1	1	yes	characteristic
42	Q₈×C₅₉	—	2	1	no	—
43	Q₈×C₅₉	—	8	3	no	—
44	Q₈×C₅₉	Q₁₆	1	4	yes	complemented
45	Q₈×C₅₉	C₂×Q₈	1	2	yes	—
46	C₂×C₄×C₅₉	—	2	1	no	—
47	C₂×C₄×C₅₉	—	2	3	no	—
48	C₂×C₄×C₅₉	C₂×D₄	1	3	yes	—
49	C₂×C₄×C₅₉	C₂×D₄	1	3	yes	—
50	C₂×C₄×C₅₉	C₂⁴	1	1	yes	characteristic
51	C₂×Dic₅₉	—	2	1	no	—
52	C₂×Dic₅₉	—	2	3	no	—
53	Dic₁₁₈	—	2	1	no	—
54	Dic₁₁₈	—	8	3	no	—
55	Dic₁₁₈	C₂×Q₈	1	2	yes	—
56	C₅₉⋊C₈	—	2	3	no	—
57	C₅₉⋊C₈	C₂×Q₈	1	2	yes	—
58	C₂²×C₅₉	C₂²×D₄	1	1	yes	characteristic
59	C₄×C₅₉	—	2	2	no	—
60	C₄×C₅₉	—	2	3	no	—
61	C₄×C₅₉	—	4	3	no	—
62	C₄×C₅₉	C₂×Q₁₆	1	6	yes	—
63	C₄×C₅₉	C₂²×Q₈	1	1	yes	characteristic
64	C₄×C₅₉	2₋¹⁺⁴	1	1	yes	characteristic
65	Dic₅₉	—	2	2	no	—
66	Dic₅₉	—	4	3	no	—
67	Q₈×Q₁₆	—	59	1	no	maximal, Sylow
68	C₂×C₅₉	D₄×Q₈	1	1	yes	characteristic
69	C₂×C₅₉	C₂²×Q₁₆	1	1	yes	characteristic
70	C₂×C₅₉	#576	1	1	yes	characteristic
71	Q₈²	—	59	1	no	—
72	Q₈²	D₅₉	1	1	yes	characteristic
73	#500	—	59	3	no	—
74	#477	—	59	3	no	—
75	#477	—	59	3	no	—
76	C₈×Q₈	—	59	1	no	—
77	C₄×Q₁₆	—	59	3	no	—
78	C₅₉	Q₈×Q₁₆	1	1	yes	characteristic, complemented, Sylow
79	C₂×Q₁₆	—	59	1	no	—
80	C₄⋊Q₈	—	2	3	no	—
81	C₄⋊Q₈	—	59	3	no	—
82	C₄⋊Q₈	—	118	3	no	—
83	C₄⋊Q₈	C₂×D₅₉	1	3	yes	—
84	C₄×Q₈	—	2	1	no	—
85	C₄×Q₈	—	59	3	no	—
86	C₄×Q₈	—	118	1	no	—
87	C₄×Q₈	C₂×D₅₉	1	3	yes	—
88	C₄×Q₈	C₂×D₅₉	1	1	yes	characteristic
89	#107	—	59	3	no	—
90	#107	—	59	6	no	—
91	C₄⋊C₈	—	59	3	no	—
92	#103	—	59	3	no	—
93	#103	—	59	3	no	—
94	C₄×C₈	—	59	3	no	—
95	C₂×Q₈	—	2	3	no	—
96	C₂×Q₈	—	59	1	no	—
97	C₂×Q₈	—	118	3	no	—
98	C₂×Q₈	C₂²⋊D₅₉	1	1	yes	characteristic
99	C₂×Q₈	C₂²×D₅₉	1	1	yes	characteristic
100	Q₁₆	—	59	4	no	—
101	C₂×C₈	—	59	1	no	—
102	C₂×C₈	—	59	3	no	—
103	C₄⋊C₄	—	2	3	no	—
104	C₄⋊C₄	—	2	3	no	—
105	C₄⋊C₄	—	59	3	no	—
106	C₄⋊C₄	—	118	3	no	—
107	C₄⋊C₄	—	118	3	no	—
108	C₄⋊C₄	C₂²⋊D₅₉	1	3	yes	—
109	C₄⋊C₄	C₂²×D₅₉	1	3	yes	—
110	C₄²	—	2	3	no	—
111	C₄²	—	118	3	no	—
112	C₄²	C₂²×D₅₉	1	3	yes	—
113	Q₈	—	2	1	no	—
114	Q₈	—	8	3	no	—
115	Q₈	—	59	2	no	—
116	Q₈	—	118	1	no	—
117	Q₈	—	472	3	no	—
118	Q₈	#11745672	1	4	yes	direct product factor
119	Q₈	Q₈×D₅₉	1	2	yes	—
120	C₂×C₄	—	2	1	no	—
121	C₂×C₄	—	2	3	no	—
122	C₂×C₄	—	118	1	no	—
123	C₂×C₄	—	118	3	no	—
124	C₂×C₄	C₂×C₂²⋊D₅₉	1	3	yes	—
125	C₂×C₄	C₂×C₂²⋊D₅₉	1	3	yes	—
126	C₂×C₄	C₂³×D₅₉	1	1	yes	characteristic
127	C₈	—	59	2	no	—
128	C₈	—	118	3	no	—
129	C₂²	C₂²×C₂²⋊D₅₉	1	1	yes	characteristic
130	C₄	—	2	3	no	—
131	C₄	—	2	2	no	—
132	C₄	—	4	3	no	—
133	C₄	—	118	2	no	—
134	C₄	—	236	3	no	—
135	C₄	C₂×#11745672	1	6	yes	—
136	C₄	C₂×Q₈×D₅₉	1	1	yes	characteristic
137	C₄	#14273775	1	1	yes	characteristic
138	C₂	Q₈×C₂²⋊D₅₉	1	1	yes	characteristic
139	C₂	C₂²×#11745672	1	1	yes	characteristic
140	C₂	#16838205	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

Q₈×Dic₁₁₈

C₂

yes

maximal, characteristic

#16837805

C₂

yes

maximal

#16837697

C₂

yes

maximal

#16837694

C₂

yes

maximal

Q₈×C₅₉⋊C₈

C₂

yes

maximal, characteristic

C₄×#11745672

C₂

yes

maximal

Q₈²×C₅₉

C₂

yes

maximal, characteristic

C₂×#11745672

C₂²

yes

characteristic

#14273727

C₂²

yes

—

#14273726

—

Q₈×Dic₅₉

—

C₄×Dic₁₁₈

C₂²

yes

—

#14273660

C₂²

yes

—

#14273660

C₂²

yes

—

C₄⋊(C₅₉⋊C₈)

C₂²

yes

—

#14273651

C₂²

yes

—

#14273650

C₂²

yes

—

C₄×C₅₉⋊C₈

C₂²

yes

—

C₄⋊Q₈×C₅₉

—

C₄⋊Q₈×C₅₉

C₂²

yes

—

C₄×Q₈×C₅₉

—

C₄×Q₈×C₅₉

C₂²

yes

—

C₄×Q₈×C₅₉

C₂²

yes

characteristic

C₂×Q₈×C₅₉

—

C₂×Q₈×C₅₉

D₄

yes

characteristic

C₂×Q₈×C₅₉

C₂³

yes

characteristic

C₂×Dic₁₁₈

—

C₂×Dic₁₁₈

C₂³

yes

characteristic

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

D₄

yes

—

C₄⋊C₄×C₅₉

C₂³

yes

—

C₄²×C₅₉

—

C₄²×C₅₉

C₂³

yes

—

#11745672

Q₈

yes

direct product factor

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

C₂³

yes

—

Dic₅₉⋊C₄

—

C₄×Dic₅₉

—

C₂×C₅₉⋊C₈

C₂³

yes

—

C₂×C₅₉⋊C₈

C₂³

yes

characteristic

Q₈×C₅₉

—

Q₈×C₅₉

—

Q₈×C₅₉

Q₁₆

yes

complemented

Q₈×C₅₉

C₂×Q₈

yes

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₂×D₄

yes

—

C₂×C₄×C₅₉

C₂×D₄

yes

—

C₂×C₄×C₅₉

C₂⁴

yes

characteristic

C₂×Dic₅₉

—

C₂×Dic₅₉

—

Dic₁₁₈

—

Dic₁₁₈

—

Dic₁₁₈

C₂×Q₈

yes

—

C₅₉⋊C₈

—

C₅₉⋊C₈

C₂×Q₈

yes

—

C₂²×C₅₉

C₂²×D₄

yes

characteristic

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

C₂×Q₁₆

yes

—

C₄×C₅₉

C₂²×Q₈

yes

characteristic

C₄×C₅₉

2₋¹⁺⁴

yes

characteristic

Dic₅₉

—

Dic₅₉

—

Q₈×Q₁₆

—

maximal, Sylow

C₂×C₅₉

D₄×Q₈

yes

characteristic

C₂×C₅₉

C₂²×Q₁₆

yes

characteristic

C₂×C₅₉

#576

yes

characteristic

Q₈²

—

Q₈²

D₅₉

yes

—

—

—

C₈×Q₈

—

C₄×Q₁₆

—

C₅₉

Q₈×Q₁₆

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

118

—

C₄⋊Q₈

C₂×D₅₉

yes

—

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

—

118

—

C₄×Q₈

C₂×D₅₉

yes

—

C₄×Q₈

C₂×D₅₉

yes

characteristic

#107

—

#107

—

C₄⋊C₈

—

#103

—

#103

—

C₄×C₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

118

—

C₂×Q₈

C₂²⋊D₅₉

yes

characteristic

C₂×Q₈

C₂²×D₅₉

yes

characteristic

100

Q₁₆

—

101

C₂×C₈

—

102

C₂×C₈

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

106

C₄⋊C₄

—

118

—

107

C₄⋊C₄

—

118

—

108

C₄⋊C₄

C₂²⋊D₅₉

yes

—

109

C₄⋊C₄

C₂²×D₅₉

yes

—

110

C₄²

—

111

C₄²

—

118

—

112

C₄²

C₂²×D₅₉

yes

—

113

Q₈

—

114

Q₈

—

115

Q₈

—

116

Q₈

—

118

—

117

Q₈

—

472

—

118

Q₈

#11745672

yes

direct product factor

119

Q₈

Q₈×D₅₉

yes

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

118

—

123

C₂×C₄

—

118

—

124

C₂×C₄

C₂×C₂²⋊D₅₉

yes

—

125

C₂×C₄

C₂×C₂²⋊D₅₉

yes

—

126

C₂×C₄

C₂³×D₅₉

yes

characteristic

127

C₈

—

128

C₈

—

118

—

129

C₂²

C₂²×C₂²⋊D₅₉

yes

characteristic

130

C₄

—

131

C₄

—

132

C₄

—

133

C₄

—

118

—

134

C₄

—

236

—

135

C₄

C₂×#11745672

yes

—

136

C₄

C₂×Q₈×D₅₉

yes

characteristic

137

C₄

#14273775

yes

characteristic

138

C₂

Q₈×C₂²⋊D₅₉

yes

characteristic

139

C₂

C₂²×#11745672

yes

characteristic

140

C₂

#16838205

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₅₉
commutator	C₂×C₄×C₅₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₅₉
Fitting	Q₈²×C₅₉	C₂
socle	C₂²×C₅₉	C₂²×D₄
hypercenter	Q₈²	D₅₉
hypocenter	C₅₉	Q₈×Q₁₆
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ Q₈²

subgroup

G/H

center

C₂²

C₂²×C₂²⋊D₅₉

commutator

C₂×C₄×C₅₉

C₂⁴

Frattini

C₂×C₄

C₂³×D₅₉

Fitting

Q₈²×C₅₉

C₂

socle

C₂²×C₅₉

C₂²×D₄

hypercenter

Q₈²

D₅₉

hypocenter

C₅₉

Q₈×Q₁₆

LCS

G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂² ◁ C₄×Q₈ ◁ Q₈²

N	H ≅ G/N	type
35. #11745672	Q₈	direct (N × H)
44. Q₈×C₅₉	Q₁₆	semidirect
78. C₅₉	Q₈×Q₁₆	semidirect
118. Q₈	#11745672	direct (N × H)

H ≅ G/N

type

35. #11745672

Q₈

direct (N × H)

44. Q₈×C₅₉

Q₁₆

semidirect

78. C₅₉

Q₈×Q₁₆

semidirect

118. Q₈

#11745672

direct (N × H)

Subgroups — Q₈×#11745672 #21436978

Conjugacy classes of subgroups

Complements

Subgroups — Q8×#11745672 #21436978

Conjugacy classes of subgroups

Complements

Subgroups — Q₈×#11745672 #21436978