Group #46823563 — subgroups

Subgroups — #46823563

Order 34432.

336 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 336 conjugacy classes of subgroups — too many to draw. Only its 141 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	#31771334	C₂	1	1	yes	maximal, characteristic
2	#31770957	C₂	1	3	yes	maximal
3	#31770897	C₂	1	3	yes	maximal
4	#31770815	C₂	1	3	yes	maximal
5	Q₈×#14524130	C₂	1	1	yes	maximal, characteristic
6	C₄×#16864392	C₂	1	3	yes	maximal
7	Q₈²×C₂₆₉	C₂	1	1	yes	maximal, characteristic
8	C₂×#16864392	C₂²	1	3	yes	—
9	#22575453	C₂²	1	3	yes	—
10	#22575444	—	2	3	no	—
11	Q₈×Dic₂₆₉	—	2	1	no	—
12	C₄×Dic₅₃₈	C₂²	1	3	yes	—
13	#22575386	C₂²	1	3	yes	—
14	C₄⋊#14524130	C₂²	1	3	yes	—
15	#22575377	C₂²	1	1	yes	characteristic
16	#22575376	C₂²	1	6	yes	—
17	#22575376	C₂²	1	3	yes	—
18	C₄×#14524130	C₂²	1	3	yes	—
19	C₄⋊Q₈×C₂₆₉	—	2	3	no	—
20	C₄⋊Q₈×C₂₆₉	C₂²	1	3	yes	—
21	C₄×Q₈×C₂₆₉	—	2	1	no	—
22	C₄×Q₈×C₂₆₉	C₂²	1	3	yes	—
23	C₄×Q₈×C₂₆₉	C₂²	1	1	yes	characteristic
24	C₂×Dic₅₃₈	C₂³	1	3	yes	—
25	#16864392	—	2	6	no	—
26	C₂×#14524130	C₂³	1	1	yes	characteristic
27	C₂×#14524130	C₂³	1	3	yes	—
28	C₄⋊Dic₂₆₉	—	2	3	no	—
29	C₄⋊Dic₂₆₉	—	2	3	no	—
30	C₄⋊Dic₂₆₉	C₂³	1	1	yes	characteristic
31	#16864381	—	2	3	no	—
32	C₄×Dic₂₆₉	—	2	3	no	—
33	C₂×Q₈×C₂₆₉	—	2	3	no	—
34	C₂×Q₈×C₂₆₉	D₄	1	1	yes	characteristic
35	C₂×Q₈×C₂₆₉	C₂³	1	1	yes	characteristic
36	C₄⋊C₄×C₂₆₉	—	2	3	no	—
37	C₄⋊C₄×C₂₆₉	—	2	3	no	—
38	C₄⋊C₄×C₂₆₉	D₄	1	3	yes	—
39	C₄⋊C₄×C₂₆₉	C₂³	1	3	yes	—
40	C₄²×C₂₆₉	—	2	3	no	—
41	C₄²×C₂₆₉	C₂³	1	3	yes	—
42	Dic₅₃₈	—	2	3	no	—
43	Dic₅₃₈	—	2	3	no	—
44	C₂×Dic₂₆₉	—	2	1	no	—
45	C₂×Dic₂₆₉	—	2	3	no	—
46	#14524130	—	2	3	no	—
47	#14524130	—	2	1	no	—
48	Q₈×C₂₆₉	—	2	1	no	—
49	Q₈×C₂₆₉	—	8	3	no	—
50	Q₈×C₂₆₉	Q₁₆	1	4	yes	complemented
51	Q₈×C₂₆₉	#41	1	2	yes	—
52	C₂×C₄×C₂₆₉	—	2	1	no	—
53	C₂×C₄×C₂₆₉	—	2	3	no	—
54	C₂×C₄×C₂₆₉	C₂×D₄	1	3	yes	—
55	C₂×C₄×C₂₆₉	C₂×D₄	1	3	yes	—
56	C₂×C₄×C₂₆₉	C₂⁴	1	1	yes	characteristic
57	C₂²×C₂₆₉	C₂²×D₄	1	1	yes	characteristic
58	C₄×C₂₆₉	—	2	2	no	—
59	C₄×C₂₆₉	—	2	3	no	—
60	C₄×C₂₆₉	—	4	3	no	—
61	C₄×C₂₆₉	C₂×Q₁₆	1	6	yes	—
62	C₄×C₂₆₉	C₂×#41	1	1	yes	characteristic
63	C₄×C₂₆₉	2₋¹⁺⁴	1	1	yes	characteristic
64	Dic₂₆₉	—	4	3	no	—
65	Dic₂₆₉	—	4	1	no	—
66	C₂×C₂₆₉	#548	1	1	yes	characteristic
67	C₂×C₂₆₉	C₂²×Q₁₆	1	1	yes	characteristic
68	C₂×C₂₆₉	#577	1	1	yes	characteristic
69	C₂₆₉	#3363	1	1	yes	characteristic, complemented, Sylow
70	#3363	—	269	1	no	maximal, Sylow
71	Q₈²	D₂₆₉	1	1	yes	characteristic
72	#557	—	269	1	no	—
73	#487	—	269	3	no	—
74	#477	—	269	3	no	—
75	#462	—	269	3	no	—
76	C₈×Q₈	—	269	1	no	—
77	C₄×Q₁₆	—	269	3	no	—
78	C₂×Q₁₆	—	269	3	no	—
79	C₄⋊Q₈	—	2	3	no	—
80	C₄⋊Q₈	—	269	3	no	—
81	C₄⋊Q₈	C₂×D₂₆₉	1	3	yes	—
82	#125	—	538	3	no	—
83	C₄×Q₈	—	2	1	no	—
84	C₄×Q₈	—	269	3	no	—
85	C₄×Q₈	—	538	1	no	—
86	C₄×Q₈	C₂×D₂₆₉	1	3	yes	—
87	C₄×Q₈	C₂×D₂₆₉	1	1	yes	characteristic
88	#107	—	269	3	no	—
89	C₄⋊C₈	—	269	3	no	—
90	#103	—	269	1	no	—
91	#103	—	269	3	no	—
92	#103	—	269	6	no	—
93	C₄×C₈	—	269	3	no	—
94	C₂×Q₈	—	2	3	no	—
95	C₂×Q₈	—	269	3	no	—
96	C₂×Q₈	C₂²⋊D₂₆₉	1	1	yes	characteristic
97	C₂×Q₈	C₂²×D₂₆₉	1	1	yes	characteristic
98	Q₁₆	—	538	6	no	—
99	C₂×C₈	—	269	1	no	—
100	C₂×C₈	—	269	3	no	—
101	C₄⋊C₄	—	2	3	no	—
102	C₄⋊C₄	—	2	3	no	—
103	C₄⋊C₄	—	269	1	no	—
104	C₄⋊C₄	—	538	3	no	—
105	C₄⋊C₄	—	538	3	no	—
106	C₄⋊C₄	—	538	3	no	—
107	C₄⋊C₄	C₂²⋊D₂₆₉	1	3	yes	—
108	C₄⋊C₄	C₂²×D₂₆₉	1	3	yes	—
109	C₄²	—	2	3	no	—
110	C₄²	—	538	3	no	—
111	C₄²	C₂²×D₂₆₉	1	3	yes	—
112	Q₈	—	2	1	no	—
113	Q₈	—	8	3	no	—
114	Q₈	—	538	3	no	—
115	Q₈	—	538	3	no	—
116	Q₈	#16864392	1	4	yes	complemented
117	Q₈	#16864403	1	2	yes	—
118	C₂×C₄	—	2	1	no	—
119	C₂×C₄	—	2	3	no	—
120	C₂×C₄	—	538	1	no	—
121	C₂×C₄	—	538	3	no	—
122	C₂×C₄	C₂×C₂²⋊D₂₆₉	1	3	yes	—
123	C₂×C₄	C₂×C₂²⋊D₂₆₉	1	3	yes	—
124	C₂×C₄	C₂³×D₂₆₉	1	1	yes	characteristic
125	C₈	—	538	3	no	—
126	C₈	—	538	1	no	—
127	C₂²	C₂²×C₂²⋊D₂₆₉	1	1	yes	characteristic
128	C₄	—	2	3	no	—
129	C₄	—	2	2	no	—
130	C₄	—	4	3	no	—
131	C₄	—	1076	1	no	—
132	C₄	—	1076	3	no	—
133	C₄	C₂×#16864392	1	6	yes	—
134	C₄	C₂×#16864403	1	1	yes	characteristic
135	C₄	#22575501	1	1	yes	characteristic
136	C₂	#31771266	1	1	yes	characteristic
137	C₂	C₂²×#16864392	1	1	yes	characteristic
138	C₂	#31771416	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#31771334

C₂

yes

maximal, characteristic

#31770957

C₂

yes

maximal

#31770897

C₂

yes

maximal

#31770815

C₂

yes

maximal

Q₈×#14524130

C₂

yes

maximal, characteristic

C₄×#16864392

C₂

yes

maximal

Q₈²×C₂₆₉

C₂

yes

maximal, characteristic

C₂×#16864392

C₂²

yes

—

#22575453

C₂²

yes

—

#22575444

—

Q₈×Dic₂₆₉

—

C₄×Dic₅₃₈

C₂²

yes

—

#22575386

C₂²

yes

—

C₄⋊#14524130

C₂²

yes

—

#22575377

C₂²

yes

characteristic

#22575376

C₂²

yes

—

#22575376

C₂²

yes

—

C₄×#14524130

C₂²

yes

—

C₄⋊Q₈×C₂₆₉

—

C₄⋊Q₈×C₂₆₉

C₂²

yes

—

C₄×Q₈×C₂₆₉

—

C₄×Q₈×C₂₆₉

C₂²

yes

—

C₄×Q₈×C₂₆₉

C₂²

yes

characteristic

C₂×Dic₅₃₈

C₂³

yes

—

#16864392

—

C₂×#14524130

C₂³

yes

characteristic

C₂×#14524130

C₂³

yes

—

C₄⋊Dic₂₆₉

—

C₄⋊Dic₂₆₉

—

C₄⋊Dic₂₆₉

C₂³

yes

characteristic

#16864381

—

C₄×Dic₂₆₉

—

C₂×Q₈×C₂₆₉

—

C₂×Q₈×C₂₆₉

D₄

yes

characteristic

C₂×Q₈×C₂₆₉

C₂³

yes

characteristic

C₄⋊C₄×C₂₆₉

—

C₄⋊C₄×C₂₆₉

—

C₄⋊C₄×C₂₆₉

D₄

yes

—

C₄⋊C₄×C₂₆₉

C₂³

yes

—

C₄²×C₂₆₉

—

C₄²×C₂₆₉

C₂³

yes

—

Dic₅₃₈

—

Dic₅₃₈

—

C₂×Dic₂₆₉

—

C₂×Dic₂₆₉

—

#14524130

—

#14524130

—

Q₈×C₂₆₉

—

Q₈×C₂₆₉

—

Q₈×C₂₆₉

Q₁₆

yes

complemented

Q₈×C₂₆₉

#41

yes

—

C₂×C₄×C₂₆₉

—

C₂×C₄×C₂₆₉

—

C₂×C₄×C₂₆₉

C₂×D₄

yes

—

C₂×C₄×C₂₆₉

C₂×D₄

yes

—

C₂×C₄×C₂₆₉

C₂⁴

yes

characteristic

C₂²×C₂₆₉

C₂²×D₄

yes

characteristic

C₄×C₂₆₉

—

C₄×C₂₆₉

—

C₄×C₂₆₉

—

C₄×C₂₆₉

C₂×Q₁₆

yes

—

C₄×C₂₆₉

C₂×#41

yes

characteristic

C₄×C₂₆₉

2₋¹⁺⁴

yes

characteristic

Dic₂₆₉

—

Dic₂₆₉

—

C₂×C₂₆₉

#548

yes

characteristic

C₂×C₂₆₉

C₂²×Q₁₆

yes

C₂×C₂₆₉

yes

C₂₆₉

yes

characteristic, complemented, Sylow

#3363

—

269

maximal, Sylow

Q₈²

D₂₆₉

yes

characteristic

#557

—

269

—

#487

—

269

—

#477

—

269

—

#462

—

269

—

C₈×Q₈

—

269

—

C₄×Q₁₆

—

269

—

C₂×Q₁₆

—

269

—

C₄⋊Q₈

—

C₄⋊Q₈

—

269

—

C₄⋊Q₈

C₂×D₂₆₉

yes

—

#125

—

538

—

C₄×Q₈

—

C₄×Q₈

—

269

—

C₄×Q₈

—

538

—

C₄×Q₈

C₂×D₂₆₉

yes

—

C₄×Q₈

C₂×D₂₆₉

yes

characteristic

#107

—

269

—

C₄⋊C₈

—

269

—

#103

—

269

—

#103

—

269

—

#103

—

269

—

C₄×C₈

—

269

—

C₂×Q₈

—

C₂×Q₈

—

269

—

C₂×Q₈

C₂²⋊D₂₆₉

yes

characteristic

C₂×Q₈

C₂²×D₂₆₉

yes

characteristic

Q₁₆

—

538

—

C₂×C₈

—

269

—

100

C₂×C₈

—

269

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

269

—

104

C₄⋊C₄

—

538

—

105

C₄⋊C₄

—

538

—

106

C₄⋊C₄

—

538

—

107

C₄⋊C₄

C₂²⋊D₂₆₉

yes

—

108

C₄⋊C₄

C₂²×D₂₆₉

yes

—

109

C₄²

—

110

C₄²

—

538

—

111

C₄²

C₂²×D₂₆₉

yes

—

112

Q₈

—

113

Q₈

—

114

Q₈

—

538

—

115

Q₈

—

538

—

116

Q₈

#16864392

yes

complemented

117

Q₈

#16864403

yes

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

538

—

121

C₂×C₄

—

538

—

122

C₂×C₄

C₂×C₂²⋊D₂₆₉

yes

—

123

C₂×C₄

C₂×C₂²⋊D₂₆₉

yes

—

124

C₂×C₄

C₂³×D₂₆₉

yes

characteristic

125

C₈

—

538

—

126

C₈

—

538

—

127

C₂²

C₂²×C₂²⋊D₂₆₉

yes

characteristic

128

C₄

—

129

C₄

—

130

C₄

—

131

C₄

—

1076

—

132

C₄

—

1076

—

133

C₄

C₂×#16864392

yes

—

134

C₄

C₂×#16864403

yes

characteristic

135

C₄

#22575501

yes

characteristic

136

C₂

#31771266

yes

characteristic

137

C₂

C₂²×#16864392

yes

characteristic

138

C₂

#31771416

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₂₆₉
commutator, cosocle	C₂×C₄×C₂₆₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₂₆₉
Fitting	Q₈²×C₂₆₉	C₂
socle, prime radical	C₂²×C₂₆₉	C₂²×D₄
hypercenter	Q₈²	D₂₆₉
hypocenter	C₂₆₉	#3363
LCS	G ▷ C₂×C₄×C₂₆₉ ▷ C₂×C₂₆₉ ▷ C₂₆₉
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ Q₈²

subgroup

G/H

center

C₂²

C₂²×C₂²⋊D₂₆₉

commutator, cosocle

C₂×C₄×C₂₆₉

C₂⁴

Frattini

C₂×C₄

C₂³×D₂₆₉

Fitting

Q₈²×C₂₆₉

C₂

socle, prime radical

C₂²×C₂₆₉

C₂²×D₄

hypercenter

Q₈²

D₂₆₉

hypocenter

C₂₆₉

#3363

LCS

G ▷ C₂×C₄×C₂₆₉ ▷ C₂×C₂₆₉ ▷ C₂₆₉

UCS

C₁ ◁ C₂² ◁ C₄×Q₈ ◁ Q₈²

N	H ≅ G/N	type
50. Q₈×C₂₆₉	Q₁₆	semidirect
69. C₂₆₉	#3363	semidirect
116. Q₈	#16864392	semidirect

H ≅ G/N

type

50. Q₈×C₂₆₉

Q₁₆

semidirect

69. C₂₆₉

#3363

semidirect

116. Q₈

#16864392

semidirect

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

subgroup	index
#31771334	2
#31770957	2
#31770897	2
#31770815	2
Q₈×#14524130	2
C₄×#16864392	2
Q₈²×C₂₆₉	2
#3363	269

subgroup

index

Q₈²×C₂₆₉

269

Subgroups — #46823563

Conjugacy classes of subgroups

Complements

Maximal subgroups