Group #46813721 — subgroups

Subgroups — Q₈×C₂²⋊Dic₂₆₉ #46813721

Order 34432.

820 conjugacy classes of subgroups; only the 333 normal subgroups are shown.

normal subgroups only

This group has 820 conjugacy classes of subgroups — too many to draw. Only its 333 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×Q₈×Dic₂₆₉	C₂	1	2	yes	maximal, complemented
2	#31770512	C₂	1	6	yes	maximal, complemented
3	#31770465	C₂	1	3	yes	maximal
4	C₄×C₂²⋊Dic₂₆₉	C₂	1	3	yes	maximal
5	C₂³×Q₈×C₂₆₉	C₂	1	1	yes	maximal, characteristic
6	Q₈×Dic₂₆₉	—	2	8	no	—
7	C₂×C₄⋊Dic₂₆₉	C₂²	1	6	yes	—
8	C₂×C₂²⋊Dic₂₆₉	C₂²	1	3	yes	—
9	C₂×C₂²⋊Dic₂₆₉	C₂²	1	1	yes	characteristic
10	C₂×C₄×Dic₂₆₉	C₂²	1	6	yes	—
11	#22575360	C₂²	1	6	yes	—
12	#22575360	C₂²	1	6	yes	—
13	C₂²×Q₈×C₂₆₉	—	2	8	no	—
14	C₂²×Q₈×C₂₆₉	C₄	1	8	yes	complemented
15	C₂²×Q₈×C₂₆₉	C₂²	1	3	yes	—
16	C₂²×Q₈×C₂₆₉	C₂²	1	1	yes	characteristic
17	C₂³×C₄×C₂₆₉	C₂²	1	3	yes	—
18	C₂²×Dic₂₆₉	C₂³	1	6	yes	—
19	C₂²×Dic₂₆₉	C₂³	1	2	yes	—
20	C₄⋊Dic₂₆₉	—	2	12	no	—
21	C₂²⋊Dic₂₆₉	—	2	6	no	—
22	C₂²⋊Dic₂₆₉	Q₈	1	4	yes	direct product factor
23	C₄×Dic₂₆₉	—	2	12	no	—
24	C₂⁴×C₂₆₉	C₂³	1	1	yes	characteristic
25	C₂×Q₈×C₂₆₉	—	2	32	no	—
26	C₂×Q₈×C₂₆₉	—	2	12	no	—
27	C₂×Q₈×C₂₆₉	C₂×C₄	1	12	yes	—
28	C₂×Q₈×C₂₆₉	C₂×C₄	1	4	yes	—
29	C₂×Q₈×C₂₆₉	D₄	1	8	yes	—
30	C₂²×C₄×C₂₆₉	—	2	12	no	—
31	C₂²×C₄×C₂₆₉	C₂×C₄	1	12	yes	—
32	C₂²×C₄×C₂₆₉	C₂³	1	3	yes	—
33	C₂²×C₄×C₂₆₉	C₂³	1	3	yes	—
34	C₂×Dic₂₆₉	—	2	2	no	—
35	C₂×Dic₂₆₉	—	2	2	no	—
36	C₂×Dic₂₆₉	—	2	6	no	—
37	C₂×Dic₂₆₉	—	2	6	no	—
38	C₂×Dic₂₆₉	—	2	6	no	—
39	C₂×Dic₂₆₉	C₂×Q₈	1	4	yes	complemented
40	C₂³×C₂₆₉	—	2	2	no	—
41	C₂³×C₂₆₉	—	2	2	no	—
42	C₂³×C₂₆₉	C₂²×C₄	1	2	yes	—
43	C₂³×C₂₆₉	C₂×Q₈	1	2	yes	—
44	C₂³×C₂₆₉	#41	1	2	yes	—
45	C₂³×C₂₆₉	C₂⁴	1	1	yes	characteristic
46	Q₈×C₂₆₉	—	2	24	no	—
47	Q₈×C₂₆₉	C₂²⋊C₄	1	16	yes	complemented
48	C₂×C₄×C₂₆₉	—	2	6	no	—
49	C₂×C₄×C₂₆₉	—	2	24	no	—
50	C₂×C₄×C₂₆₉	C₂²×C₄	1	6	yes	—
51	C₂×C₄×C₂₆₉	C₂²×C₄	1	6	yes	—
52	C₂×C₄×C₂₆₉	C₂×D₄	1	12	yes	—
53	C₂²×C₂₆₉	—	2	2	no	—
54	C₂²×C₂₆₉	—	2	2	no	—
55	C₂²×C₂₆₉	—	2	4	no	—
56	C₂²×C₂₆₉	—	2	4	no	—
57	C₂²×C₂₆₉	C₄×Q₈	1	4	yes	complemented
58	C₂²×C₂₆₉	C₂³×C₄	1	1	yes	characteristic
59	C₂²×C₂₆₉	C₂²×D₄	1	2	yes	—
60	C₂²×C₂₆₉	C₂²×Q₈	1	1	yes	characteristic
61	C₂²×C₂₆₉	C₂×#41	1	1	yes	characteristic
62	C₂²×C₂₆₉	2₋¹⁺⁴	1	2	yes	—
63	C₄×C₂₆₉	—	2	6	no	—
64	C₄×C₂₆₉	C₂×C₂²⋊C₄	1	12	yes	—
65	Dic₂₆₉	—	2	4	no	—
66	Dic₂₆₉	—	4	6	no	—
67	C₂×C₂₆₉	—	2	4	no	—
68	C₂×C₂₆₉	C₂²×C₂²⋊C₄	1	1	yes	characteristic
69	C₂×C₂₆₉	C₂×C₄×Q₈	1	1	yes	characteristic
70	C₂×C₂₆₉	#519	1	1	yes	characteristic
71	C₂×C₂₆₉	#548	1	2	yes	—
72	C₂×C₂₆₉	D₄×Q₈	1	2	yes	—
73	C₂₆₉	Q₈×C₂²⋊C₄	1	1	yes	characteristic, complemented, Sylow
74	Q₈×C₂²⋊C₄	—	269	1	no	maximal, Sylow
75	C₂³×Q₈	D₂₆₉	1	1	yes	characteristic
76	C₂×C₄×Q₈	—	269	2	no	—
77	#391	—	269	6	no	—
78	#380	—	269	3	no	—
79	C₄×C₂²⋊C₄	—	269	3	no	—
80	C₂²×Q₈	—	2	8	no	—
81	C₂²×Q₈	Dic₂₆₉	1	8	yes	complemented
82	C₂²×Q₈	C₂×D₂₆₉	1	3	yes	—
83	C₂²×Q₈	C₂×D₂₆₉	1	1	yes	characteristic
84	C₂³×C₄	C₂×D₂₆₉	1	3	yes	—
85	C₄×Q₈	—	538	8	no	—
86	C₂×C₄⋊C₄	—	269	6	no	—
87	C₂×C₂²⋊C₄	—	269	1	no	—
88	C₂×C₂²⋊C₄	—	269	3	no	—
89	C₂×C₄²	—	269	6	no	—
90	#95	—	269	6	no	—
91	#95	—	269	6	no	—
92	C₂⁴	C₂²×D₂₆₉	1	1	yes	characteristic
93	C₂×Q₈	—	2	32	no	—
94	C₂×Q₈	—	2	12	no	—
95	C₂×Q₈	C₂×Dic₂₆₉	1	12	yes	—
96	C₂×Q₈	C₂×Dic₂₆₉	1	4	yes	—
97	C₂×Q₈	C₂²⋊D₂₆₉	1	8	yes	—
98	C₂²×C₄	—	2	12	no	—
99	C₂²×C₄	—	269	2	no	—
100	C₂²×C₄	—	269	6	no	—
101	C₂²×C₄	C₂×Dic₂₆₉	1	12	yes	—
102	C₂²×C₄	C₂²×D₂₆₉	1	3	yes	—
103	C₂²×C₄	C₂²×D₂₆₉	1	3	yes	—
104	C₄⋊C₄	—	538	12	no	—
105	C₂²⋊C₄	—	269	4	no	—
106	C₂²⋊C₄	—	538	6	no	—
107	C₄²	—	538	12	no	—
108	C₂³	—	2	2	no	—
109	C₂³	—	2	2	no	—
110	C₂³	C₂²×Dic₂₆₉	1	2	yes	—
111	C₂³	Q₈×D₂₆₉	1	2	yes	—
112	C₂³	#16864403	1	2	yes	—
113	C₂³	C₂³×D₂₆₉	1	1	yes	characteristic
114	Q₈	—	2	24	no	—
115	Q₈	C₂²⋊Dic₂₆₉	1	16	yes	direct product factor
116	C₂×C₄	—	2	6	no	—
117	C₂×C₄	—	2	24	no	—
118	C₂×C₄	—	269	4	no	—
119	C₂×C₄	—	538	2	no	—
120	C₂×C₄	—	538	2	no	—
121	C₂×C₄	—	538	6	no	—
122	C₂×C₄	—	538	6	no	—
123	C₂×C₄	—	538	6	no	—
124	C₂×C₄	C₂²×Dic₂₆₉	1	6	yes	—
125	C₂×C₄	C₂²×Dic₂₆₉	1	6	yes	—
126	C₂×C₄	C₂×C₂²⋊D₂₆₉	1	12	yes	—
127	C₂²	—	2	4	no	—
128	C₂²	—	2	4	no	—
129	C₂²	—	2	2	no	—
130	C₂²	—	2	2	no	—
131	C₂²	Q₈×Dic₂₆₉	1	4	yes	complemented
132	C₂²	C₂³×Dic₂₆₉	1	1	yes	characteristic
133	C₂²	C₂²×C₂²⋊D₂₆₉	1	2	yes	—
134	C₂²	C₂×Q₈×D₂₆₉	1	1	yes	characteristic
135	C₂²	C₂×#16864403	1	1	yes	characteristic
136	C₂²	#22575501	1	2	yes	—
137	C₄	—	2	6	no	—
138	C₄	—	538	4	no	—
139	C₄	—	1076	6	no	—
140	C₄	C₂×C₂²⋊Dic₂₆₉	1	12	yes	—
141	C₂	—	2	4	no	—
142	C₂	C₂²×C₂²⋊Dic₂₆₉	1	1	yes	characteristic
143	C₂	C₂×Q₈×Dic₂₆₉	1	1	yes	characteristic
144	C₂	#31771083	1	1	yes	characteristic
145	C₂	#31771266	1	2	yes	—
146	C₂	Q₈×C₂²⋊D₂₆₉	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×Q₈×Dic₂₆₉

C₂

yes

maximal, complemented

#31770512

C₂

yes

maximal, complemented

#31770465

C₂

yes

maximal

C₄×C₂²⋊Dic₂₆₉

C₂

yes

maximal

C₂³×Q₈×C₂₆₉

C₂

yes

maximal, characteristic

Q₈×Dic₂₆₉

—

C₂×C₄⋊Dic₂₆₉

C₂²

yes

—

C₂×C₂²⋊Dic₂₆₉

C₂²

yes

—

C₂×C₂²⋊Dic₂₆₉

C₂²

yes

characteristic

C₂×C₄×Dic₂₆₉

C₂²

yes

—

#22575360

C₂²

yes

—

#22575360

C₂²

yes

—

C₂²×Q₈×C₂₆₉

—

C₂²×Q₈×C₂₆₉

C₄

yes

complemented

C₂²×Q₈×C₂₆₉

C₂²

yes

—

C₂²×Q₈×C₂₆₉

C₂²

yes

characteristic

C₂³×C₄×C₂₆₉

C₂²

yes

—

C₂²×Dic₂₆₉

C₂³

yes

—

C₂²×Dic₂₆₉

C₂³

yes

—

C₄⋊Dic₂₆₉

—

C₂²⋊Dic₂₆₉

—

C₂²⋊Dic₂₆₉

Q₈

yes

direct product factor

C₄×Dic₂₆₉

—

C₂⁴×C₂₆₉

C₂³

yes

characteristic

C₂×Q₈×C₂₆₉

—

C₂×Q₈×C₂₆₉

—

C₂×Q₈×C₂₆₉

C₂×C₄

yes

—

C₂×Q₈×C₂₆₉

C₂×C₄

yes

—

C₂×Q₈×C₂₆₉

D₄

yes

—

C₂²×C₄×C₂₆₉

—

C₂²×C₄×C₂₆₉

C₂×C₄

yes

—

C₂²×C₄×C₂₆₉

C₂³

yes

—

C₂²×C₄×C₂₆₉

C₂³

yes

—

C₂×Dic₂₆₉

—

C₂×Dic₂₆₉

—

C₂×Dic₂₆₉

—

C₂×Dic₂₆₉

—

C₂×Dic₂₆₉

—

C₂×Dic₂₆₉

C₂×Q₈

yes

complemented

C₂³×C₂₆₉

—

C₂³×C₂₆₉

—

C₂³×C₂₆₉

C₂²×C₄

yes

—

C₂³×C₂₆₉

C₂×Q₈

yes

—

C₂³×C₂₆₉

#41

yes

—

C₂³×C₂₆₉

C₂⁴

yes

characteristic

Q₈×C₂₆₉

—

Q₈×C₂₆₉

C₂²⋊C₄

yes

complemented

C₂×C₄×C₂₆₉

—

C₂×C₄×C₂₆₉

—

C₂×C₄×C₂₆₉

C₂²×C₄

yes

—

C₂×C₄×C₂₆₉

C₂²×C₄

yes

—

C₂×C₄×C₂₆₉

C₂×D₄

yes

—

C₂²×C₂₆₉

—

C₂²×C₂₆₉

—

C₂²×C₂₆₉

—

C₂²×C₂₆₉

—

C₂²×C₂₆₉

C₄×Q₈

yes

complemented

C₂²×C₂₆₉

C₂³×C₄

yes

characteristic

C₂²×C₂₆₉

C₂²×D₄

yes

—

C₂²×C₂₆₉

C₂²×Q₈

yes

characteristic

C₂²×C₂₆₉

C₂×#41

yes

characteristic

C₂²×C₂₆₉

2₋¹⁺⁴

yes

—

C₄×C₂₆₉

—

C₄×C₂₆₉

C₂×C₂²⋊C₄

yes

—

Dic₂₆₉

—

Dic₂₆₉

—

C₂×C₂₆₉

—

C₂×C₂₆₉

C₂²×C₂²⋊C₄

yes

characteristic

C₂×C₂₆₉

C₂×C₄×Q₈

yes

C₂×C₂₆₉

yes

C₂×C₂₆₉

yes

—

C₂×C₂₆₉

D₄×Q₈

yes

—

C₂₆₉

Q₈×C₂²⋊C₄

yes

characteristic, complemented, Sylow

Q₈×C₂²⋊C₄

—

269

maximal, Sylow

C₂³×Q₈

D₂₆₉

yes

characteristic

C₂×C₄×Q₈

—

269

—

#391

—

269

—

#380

—

269

—

C₄×C₂²⋊C₄

—

269

—

C₂²×Q₈

—

C₂²×Q₈

Dic₂₆₉

yes

complemented

C₂²×Q₈

C₂×D₂₆₉

yes

—

C₂²×Q₈

C₂×D₂₆₉

yes

characteristic

C₂³×C₄

C₂×D₂₆₉

yes

—

C₄×Q₈

—

538

—

C₂×C₄⋊C₄

—

269

—

C₂×C₂²⋊C₄

—

269

—

C₂×C₂²⋊C₄

—

269

—

C₂×C₄²

—

269

—

#95

—

269

—

#95

—

269

—

C₂⁴

C₂²×D₂₆₉

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂×Dic₂₆₉

yes

—

C₂×Q₈

C₂×Dic₂₆₉

yes

—

C₂×Q₈

C₂²⋊D₂₆₉

yes

—

C₂²×C₄

—

C₂²×C₄

—

269

—

100

C₂²×C₄

—

269

—

101

C₂²×C₄

C₂×Dic₂₆₉

yes

—

102

C₂²×C₄

C₂²×D₂₆₉

yes

—

103

C₂²×C₄

C₂²×D₂₆₉

yes

—

104

C₄⋊C₄

—

538

—

105

C₂²⋊C₄

—

269

—

106

C₂²⋊C₄

—

538

—

107

C₄²

—

538

—

108

C₂³

—

109

C₂³

—

110

C₂³

C₂²×Dic₂₆₉

yes

—

111

C₂³

Q₈×D₂₆₉

yes

—

112

C₂³

#16864403

yes

—

113

C₂³

C₂³×D₂₆₉

yes

characteristic

114

Q₈

—

115

Q₈

C₂²⋊Dic₂₆₉

yes

direct product factor

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

269

—

119

C₂×C₄

—

538

—

120

C₂×C₄

—

538

—

121

C₂×C₄

—

538

—

122

C₂×C₄

—

538

—

123

C₂×C₄

—

538

—

124

C₂×C₄

C₂²×Dic₂₆₉

yes

—

125

C₂×C₄

C₂²×Dic₂₆₉

yes

—

126

C₂×C₄

C₂×C₂²⋊D₂₆₉

yes

—

127

C₂²

—

128

C₂²

—

129

C₂²

—

130

C₂²

—

131

C₂²

Q₈×Dic₂₆₉

yes

complemented

132

C₂²

C₂³×Dic₂₆₉

yes

characteristic

133

C₂²

C₂²×C₂²⋊D₂₆₉

yes

—

134

C₂²

C₂×Q₈×D₂₆₉

yes

characteristic

135

C₂²

C₂×#16864403

yes

characteristic

136

C₂²

#22575501

yes

—

137

C₄

—

138

C₄

—

538

—

139

C₄

—

1076

—

140

C₄

C₂×C₂²⋊Dic₂₆₉

yes

—

141

C₂

—

142

C₂

C₂²×C₂²⋊Dic₂₆₉

yes

characteristic

143

C₂

C₂×Q₈×Dic₂₆₉

yes

characteristic

144

C₂

#31771083

yes

characteristic

145

C₂

#31771266

yes

—

146

C₂

Q₈×C₂²⋊D₂₆₉

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₂₆₉
commutator	C₂²×C₂₆₉	C₂³×C₄
Fitting	C₂³×Q₈×C₂₆₉	C₂
socle, cosocle	C₂³×C₂₆₉	C₂⁴
prime radical	C₂⁴×C₂₆₉	C₂³
hypercenter	C₂³×Q₈	D₂₆₉
hypocenter	C₂₆₉	Q₈×C₂²⋊C₄
LCS	G ▷ C₂²×C₂₆₉ ▷ C₂₆₉
UCS	C₁ ◁ C₂³ ◁ C₂³×Q₈

subgroup

G/H

center, Frattini

C₂³

C₂³×D₂₆₉

commutator

C₂²×C₂₆₉

C₂³×C₄

Fitting

C₂³×Q₈×C₂₆₉

C₂

socle, cosocle

C₂³×C₂₆₉

C₂⁴

prime radical

C₂⁴×C₂₆₉

C₂³

hypercenter

C₂³×Q₈

D₂₆₉

hypocenter

C₂₆₉

Q₈×C₂²⋊C₄

LCS

G ▷ C₂²×C₂₆₉ ▷ C₂₆₉

UCS

C₁ ◁ C₂³ ◁ C₂³×Q₈

N	H ≅ G/N	type
1. C₂×Q₈×Dic₂₆₉	C₂	semidirect
2. #31770512	C₂	semidirect
14. C₂²×Q₈×C₂₆₉	C₄	semidirect
22. C₂²⋊Dic₂₆₉	Q₈	direct (N × H)
39. C₂×Dic₂₆₉	C₂×Q₈	semidirect
47. Q₈×C₂₆₉	C₂²⋊C₄	semidirect
57. C₂²×C₂₆₉	C₄×Q₈	semidirect
73. C₂₆₉	Q₈×C₂²⋊C₄	semidirect
81. C₂²×Q₈	Dic₂₆₉	semidirect
115. Q₈	C₂²⋊Dic₂₆₉	direct (N × H)
131. C₂²	Q₈×Dic₂₆₉	semidirect

H ≅ G/N

type

1. C₂×Q₈×Dic₂₆₉

C₂

semidirect

2. #31770512

C₂

semidirect

14. C₂²×Q₈×C₂₆₉

C₄

semidirect

22. C₂²⋊Dic₂₆₉

Q₈

direct (N × H)

39. C₂×Dic₂₆₉

C₂×Q₈

semidirect

47. Q₈×C₂₆₉

C₂²⋊C₄

semidirect

57. C₂²×C₂₆₉

C₄×Q₈

semidirect

73. C₂₆₉

Q₈×C₂²⋊C₄

semidirect

81. C₂²×Q₈

Dic₂₆₉

semidirect

115. Q₈

C₂²⋊Dic₂₆₉

direct (N × H)

131. C₂²

Q₈×Dic₂₆₉

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×Q₈×Dic₂₆₉	2
#31770512	2
#31770465	2
C₄×C₂²⋊Dic₂₆₉	2
C₂³×Q₈×C₂₆₉	2
Q₈×C₂²⋊C₄	269

subgroup

index

C₂×Q₈×Dic₂₆₉

#31770512

#31770465

C₄×C₂²⋊Dic₂₆₉

C₂³×Q₈×C₂₆₉

Q₈×C₂²⋊C₄

269

Subgroups — Q₈×C₂²⋊Dic₂₆₉ #46813721

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q8×C22⋊Dic269 #46813721

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×C₂²⋊Dic₂₆₉ #46813721