Group #21436952 — subgroups

Subgroups — Q₈×#11745671 #21436952

Order 7552.

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

normal subgroups only

This group has 370 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×D₂₃₆	C₂	1	1	yes	maximal, characteristic
2	#16837798	C₂	1	3	yes	maximal, complemented
3	#16837689	C₂	1	3	yes	maximal
4	#16837687	C₂	1	3	yes	maximal, complemented
5	Q₈×C₅₉⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄×#11745671	C₂	1	3	yes	maximal
7	Q₈²×C₅₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×Q₈×D₅₉	—	2	1	no	—
9	C₂×#11745671	C₂²	1	1	yes	characteristic
10	#14273727	C₂²	1	3	yes	—
11	#14273707	—	2	3	no	—
12	C₄×D₂₃₆	C₂²	1	3	yes	—
13	#14273658	C₂²	1	6	yes	—
14	#14273658	C₂²	1	3	yes	—
15	C₄⋊(C₅₉⋊C₈)	C₂²	1	3	yes	—
16	#14273651	C₂²	1	3	yes	—
17	#14273647	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	Q₈×D₅₉	—	4	4	no	—
28	C₂×D₂₃₆	C₂³	1	1	yes	characteristic
29	C₂×C₄×D₅₉	—	2	3	no	—
30	C₂×Dic₁₁₈	—	2	3	no	—
31	C₄⋊C₄×C₅₉	—	2	3	no	—
32	C₄⋊C₄×C₅₉	—	2	3	no	—
33	C₄⋊C₄×C₅₉	D₄	1	3	yes	—
34	C₄⋊C₄×C₅₉	C₂³	1	3	yes	—
35	C₄²×C₅₉	—	2	3	no	—
36	C₄²×C₅₉	C₂³	1	3	yes	—
37	#11745671	Q₈	1	4	yes	direct product factor
38	(C₂×D₅₉)⋊C₄	—	2	3	no	—
39	C₄⋊Dic₅₉	—	2	3	no	—
40	C₄⋊Dic₅₉	C₂³	1	3	yes	—
41	C₂×C₅₉⋊C₈	C₂³	1	3	yes	—
42	C₂×C₅₉⋊C₈	C₂³	1	1	yes	characteristic
43	C₂²×D₅₉	—	2	1	no	—
44	Q₈×C₅₉	—	2	1	no	—
45	Q₈×C₅₉	—	8	3	no	—
46	Q₈×C₅₉	SD₁₆	1	4	yes	complemented
47	Q₈×C₅₉	C₂×Q₈	1	2	yes	complemented
48	C₂×C₄×C₅₉	—	2	1	no	—
49	C₂×C₄×C₅₉	—	2	3	no	—
50	C₂×C₄×C₅₉	C₂×D₄	1	3	yes	—
51	C₂×C₄×C₅₉	C₂×D₄	1	3	yes	—
52	C₂×C₄×C₅₉	C₂⁴	1	1	yes	characteristic
53	C₂×Dic₅₉	—	2	3	no	—
54	D₂₃₆	—	2	1	no	—
55	D₂₃₆	C₂×Q₈	1	2	yes	—
56	C₄×D₅₉	—	4	6	no	—
57	Dic₁₁₈	—	4	6	no	—
58	C₅₉⋊C₈	—	2	3	no	—
59	C₅₉⋊C₈	C₂×Q₈	1	2	yes	complemented
60	C₂²×C₅₉	C₂²×D₄	1	1	yes	characteristic
61	C₂×D₅₉	—	2	2	no	—
62	C₂×D₅₉	—	4	1	no	—
63	C₂×D₅₉	—	4	1	no	—
64	C₄×C₅₉	—	2	2	no	—
65	C₄×C₅₉	—	2	3	no	—
66	C₄×C₅₉	—	4	3	no	—
67	C₄×C₅₉	C₂×SD₁₆	1	6	yes	—
68	C₄×C₅₉	C₂²×Q₈	1	1	yes	characteristic
69	C₄×C₅₉	2₋¹⁺⁴	1	1	yes	characteristic
70	Dic₅₉	—	4	3	no	—
71	Q₈×SD₁₆	—	59	1	no	maximal, Sylow
72	C₂×C₅₉	D₄×Q₈	1	1	yes	characteristic
73	C₂×C₅₉	C₂²×SD₁₆	1	1	yes	characteristic
74	C₂×C₅₉	#577	1	1	yes	characteristic
75	D₅₉	—	4	2	no	—
76	Q₈²	D₅₉	1	1	yes	characteristic, complemented
77	D₄×Q₈	—	59	1	no	—
78	#498	—	59	3	no	—
79	#476	—	59	3	no	—
80	#475	—	59	3	no	—
81	C₈×Q₈	—	59	1	no	—
82	C₄×SD₁₆	—	59	3	no	—
83	C₅₉	Q₈×SD₁₆	1	1	yes	characteristic, complemented, Sylow
84	C₂²×Q₈	—	118	1	no	—
85	C₂×SD₁₆	—	59	1	no	—
86	C₄⋊Q₈	—	2	3	no	—
87	C₄⋊Q₈	—	59	3	no	—
88	C₄⋊Q₈	C₂×D₅₉	1	3	yes	—
89	C₂²⋊Q₈	—	118	3	no	—
90	C₄×Q₈	—	2	1	no	—
91	C₄×Q₈	C₂×D₅₉	1	3	yes	—
92	C₄×Q₈	C₂×D₅₉	1	1	yes	characteristic
93	C₄×D₄	—	59	3	no	—
94	#106	—	59	3	no	—
95	#106	—	59	6	no	—
96	C₄⋊C₈	—	59	3	no	—
97	#103	—	59	3	no	—
98	#102	—	59	3	no	—
99	C₄×C₈	—	59	3	no	—
100	C₂×Q₈	—	2	3	no	—
101	C₂×Q₈	—	118	3	no	—
102	C₂×Q₈	—	236	4	no	—
103	C₂×Q₈	C₂²⋊D₅₉	1	1	yes	characteristic
104	C₂×Q₈	C₂²×D₅₉	1	1	yes	characteristic
105	C₂×D₄	—	59	1	no	—
106	C₂²×C₄	—	118	3	no	—
107	SD₁₆	—	59	4	no	—
108	C₂×C₈	—	59	1	no	—
109	C₂×C₈	—	59	3	no	—
110	C₄⋊C₄	—	2	3	no	—
111	C₄⋊C₄	—	2	3	no	—
112	C₄⋊C₄	—	59	3	no	—
113	C₄⋊C₄	—	118	3	no	—
114	C₄⋊C₄	C₂²⋊D₅₉	1	3	yes	—
115	C₄⋊C₄	C₂²×D₅₉	1	3	yes	—
116	C₂²⋊C₄	—	118	3	no	—
117	C₄²	—	2	3	no	—
118	C₄²	C₂²×D₅₉	1	3	yes	—
119	C₂³	—	118	1	no	—
120	Q₈	—	2	1	no	—
121	Q₈	—	8	3	no	—
122	Q₈	—	236	6	no	—
123	Q₈	#11745671	1	4	yes	direct product factor
124	Q₈	Q₈×D₅₉	1	2	yes	complemented
125	D₄	—	59	2	no	—
126	D₄	—	118	1	no	—
127	C₂×C₄	—	2	1	no	—
128	C₂×C₄	—	2	3	no	—
129	C₂×C₄	—	118	3	no	—
130	C₂×C₄	—	236	6	no	—
131	C₂×C₄	C₂×C₂²⋊D₅₉	1	3	yes	—
132	C₂×C₄	C₂×C₂²⋊D₅₉	1	3	yes	—
133	C₂×C₄	C₂³×D₅₉	1	1	yes	characteristic
134	C₈	—	59	2	no	—
135	C₈	—	118	3	no	—
136	C₂²	—	118	2	no	—
137	C₂²	—	236	1	no	—
138	C₂²	—	236	1	no	—
139	C₂²	C₂²×C₂²⋊D₅₉	1	1	yes	characteristic
140	C₄	—	2	3	no	—
141	C₄	—	2	2	no	—
142	C₄	—	4	3	no	—
143	C₄	—	236	3	no	—
144	C₄	C₂×#11745671	1	6	yes	—
145	C₄	C₂×Q₈×D₅₉	1	1	yes	characteristic
146	C₄	#14273775	1	1	yes	characteristic
147	C₂	—	236	2	no	—
148	C₂	Q₈×C₂²⋊D₅₉	1	1	yes	characteristic
149	C₂	C₂²×#11745671	1	1	yes	characteristic
150	C₂	#16838213	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

Q₈×D₂₃₆

C₂

yes

maximal, characteristic

#16837798

C₂

yes

maximal, complemented

#16837689

C₂

yes

maximal

#16837687

C₂

yes

maximal, complemented

Q₈×C₅₉⋊C₈

C₂

yes

maximal, characteristic, complemented

C₄×#11745671

C₂

yes

maximal

Q₈²×C₅₉

C₂

yes

maximal, characteristic, complemented

C₂×Q₈×D₅₉

—

C₂×#11745671

C₂²

yes

characteristic

#14273727

C₂²

yes

—

#14273707

—

C₄×D₂₃₆

C₂²

yes

—

#14273658

C₂²

yes

—

#14273658

C₂²

yes

—

C₄⋊(C₅₉⋊C₈)

C₂²

yes

—

#14273651

C₂²

yes

—

#14273647

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

Q₈×D₅₉

—

C₂×D₂₃₆

C₂³

yes

characteristic

C₂×C₄×D₅₉

—

C₂×Dic₁₁₈

—

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

D₄

yes

—

C₄⋊C₄×C₅₉

C₂³

yes

—

C₄²×C₅₉

—

C₄²×C₅₉

C₂³

yes

—

#11745671

Q₈

yes

direct product factor

(C₂×D₅₉)⋊C₄

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

C₂³

yes

—

C₂×C₅₉⋊C₈

C₂³

yes

—

C₂×C₅₉⋊C₈

C₂³

yes

characteristic

C₂²×D₅₉

—

Q₈×C₅₉

—

Q₈×C₅₉

—

Q₈×C₅₉

SD₁₆

yes

complemented

Q₈×C₅₉

C₂×Q₈

yes

complemented

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

—

D₂₃₆

—

D₂₃₆

C₂×Q₈

yes

—

C₄×D₅₉

—

Dic₁₁₈

—

C₅₉⋊C₈

—

C₅₉⋊C₈

C₂×Q₈

yes

complemented

C₂²×C₅₉

C₂²×D₄

yes

characteristic

C₂×D₅₉

—

C₂×D₅₉

—

C₂×D₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

C₂×SD₁₆

yes

—

C₄×C₅₉

C₂²×Q₈

yes

characteristic

C₄×C₅₉

2₋¹⁺⁴

yes

characteristic

Dic₅₉

—

Q₈×SD₁₆

—

maximal, Sylow

C₂×C₅₉

D₄×Q₈

yes

characteristic

C₂×C₅₉

C₂²×SD₁₆

yes

characteristic

C₂×C₅₉

#577

yes

characteristic

D₅₉

—

Q₈²

D₅₉

yes

characteristic, complemented

D₄×Q₈

—

#498

—

#476

—

#475

—

C₈×Q₈

—

C₄×SD₁₆

—

C₅₉

Q₈×SD₁₆

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

118

—

C₂×SD₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₅₉

yes

—

C₂²⋊Q₈

—

118

—

C₄×Q₈

—

C₄×Q₈

C₂×D₅₉

yes

—

C₄×Q₈

C₂×D₅₉

yes

characteristic

C₄×D₄

—

#106

—

#106

—

C₄⋊C₈

—

#103

—

#102

—

C₄×C₈

—

100

C₂×Q₈

—

101

C₂×Q₈

—

118

—

102

C₂×Q₈

—

236

—

103

C₂×Q₈

C₂²⋊D₅₉

yes

characteristic

104

C₂×Q₈

C₂²×D₅₉

yes

characteristic

105

C₂×D₄

—

106

C₂²×C₄

—

118

—

107

SD₁₆

—

108

C₂×C₈

—

109

C₂×C₈

—

110

C₄⋊C₄

—

111

C₄⋊C₄

—

112

C₄⋊C₄

—

113

C₄⋊C₄

—

118

—

114

C₄⋊C₄

C₂²⋊D₅₉

yes

—

115

C₄⋊C₄

C₂²×D₅₉

yes

—

116

C₂²⋊C₄

—

118

—

117

C₄²

—

118

C₄²

C₂²×D₅₉

yes

—

119

C₂³

—

118

—

120

Q₈

—

121

Q₈

—

122

Q₈

—

236

—

123

Q₈

#11745671

yes

direct product factor

124

Q₈

Q₈×D₅₉

yes

complemented

125

D₄

—

126

D₄

—

118

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

118

—

130

C₂×C₄

—

236

—

131

C₂×C₄

C₂×C₂²⋊D₅₉

yes

—

132

C₂×C₄

C₂×C₂²⋊D₅₉

yes

—

133

C₂×C₄

C₂³×D₅₉

yes

characteristic

134

C₈

—

135

C₈

—

118

—

136

C₂²

—

118

—

137

C₂²

—

236

—

138

C₂²

—

236

—

139

C₂²

C₂²×C₂²⋊D₅₉

yes

characteristic

140

C₄

—

141

C₄

—

142

C₄

—

143

C₄

—

236

—

144

C₄

C₂×#11745671

yes

—

145

C₄

C₂×Q₈×D₅₉

yes

characteristic

146

C₄

#14273775

yes

characteristic

147

C₂

—

236

—

148

C₂

Q₈×C₂²⋊D₅₉

yes

characteristic

149

C₂

C₂²×#11745671

yes

characteristic

150

C₂

#16838213

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×SD₁₆
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×SD₁₆

LCS

G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂² ◁ C₄×Q₈ ◁ Q₈²

N	H ≅ G/N	type
2. #16837798	C₂	semidirect
4. #16837687	C₂	semidirect
5. Q₈×C₅₉⋊C₈	C₂	semidirect
7. Q₈²×C₅₉	C₂	semidirect
37. #11745671	Q₈	direct (N × H)
46. Q₈×C₅₉	SD₁₆	semidirect
47. Q₈×C₅₉	C₂×Q₈	semidirect
59. C₅₉⋊C₈	C₂×Q₈	semidirect
76. Q₈²	D₅₉	semidirect
83. C₅₉	Q₈×SD₁₆	semidirect
123. Q₈	#11745671	direct (N × H)
124. Q₈	Q₈×D₅₉	semidirect

H ≅ G/N

type

2. #16837798

C₂

semidirect

4. #16837687

C₂

semidirect

5. Q₈×C₅₉⋊C₈

C₂

semidirect

7. Q₈²×C₅₉

C₂

semidirect

37. #11745671

Q₈

direct (N × H)

46. Q₈×C₅₉

SD₁₆

semidirect

47. Q₈×C₅₉

C₂×Q₈

semidirect

59. C₅₉⋊C₈

C₂×Q₈

semidirect

76. Q₈²

D₅₉

semidirect

83. C₅₉

Q₈×SD₁₆

semidirect

123. Q₈

#11745671

direct (N × H)

124. Q₈

Q₈×D₅₉

semidirect

Subgroups — Q₈×#11745671 #21436952

Conjugacy classes of subgroups

Complements

Subgroups — Q8×#11745671 #21436952

Conjugacy classes of subgroups

Complements

Subgroups — Q₈×#11745671 #21436952