Group #46823552 — subgroups

Subgroups — #46823552

Order 34432.

392 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 392 conjugacy classes of subgroups — too many to draw. Only its 139 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×Dic₅₃₈	C₂	1	1	yes	maximal, characteristic, complemented
2	#31770951	C₂	1	3	yes	maximal, complemented
3	#31770895	C₂	1	3	yes	maximal, complemented
4	#31770802	C₂	1	3	yes	maximal, complemented
5	Q₈×#14524130	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄×#16864390	C₂	1	3	yes	maximal, complemented
7	#550×C₂₆₉	C₂	1	1	yes	maximal, characteristic
8	C₂×#16864390	C₂²	1	3	yes	—
9	#22575453	C₂²	1	3	yes	—
10	#22575452	—	2	3	no	—
11	Q₈×Dic₂₆₉	—	2	1	no	—
12	C₄×Dic₅₃₈	C₂²	1	3	yes	—
13	#22575384	C₂²	1	3	yes	—
14	C₄⋊#14524130	C₂²	1	3	yes	—
15	#22575376	C₂²	1	1	yes	characteristic
16	#22575374	C₂²	1	6	yes	—
17	#22575374	C₂²	1	3	yes	—
18	C₄×#14524130	C₂²	1	3	yes	—
19	C₂×#41×C₂₆₉	—	2	1	no	—
20	#127×C₂₆₉	C₂²	1	3	yes	—
21	#121×C₂₆₉	—	2	3	no	—
22	C₄×Q₈×C₂₆₉	C₂²	1	1	yes	characteristic
23	C₄×D₄×C₂₆₉	C₂²	1	3	yes	—
24	C₂×Dic₅₃₈	—	2	3	no	—
25	C₂×Dic₅₃₈	C₂³	1	1	yes	characteristic
26	#16864390	—	2	6	no	—
27	C₂×#14524130	C₂³	1	1	yes	characteristic
28	C₂×#14524130	C₂³	1	3	yes	—
29	C₄⋊Dic₂₆₉	—	2	3	no	—
30	C₄⋊Dic₂₆₉	C₂³	1	3	yes	—
31	#16864381	—	2	3	no	—
32	C₄×Dic₂₆₉	—	2	3	no	—
33	#41×C₂₆₉	—	4	4	no	—
34	C₂×Q₈×C₂₆₉	D₄	1	1	yes	characteristic
35	C₂×D₄×C₂₆₉	—	2	3	no	—
36	C₂×D₄×C₂₆₉	—	2	3	no	—
37	C₂×D₄×C₂₆₉	C₂³	1	3	yes	—
38	C₂²×C₄×C₂₆₉	—	2	3	no	—
39	C₄⋊C₄×C₂₆₉	D₄	1	3	yes	—
40	C₄⋊C₄×C₂₆₉	C₂³	1	1	yes	characteristic
41	C₂²⋊C₄×C₂₆₉	—	2	3	no	—
42	C₄²×C₂₆₉	C₂³	1	3	yes	—
43	Dic₅₃₈	—	2	1	no	—
44	Dic₅₃₈	—	8	3	no	—
45	Dic₅₃₈	#41	1	2	yes	complemented
46	C₂×Dic₂₆₉	—	2	1	no	—
47	C₂×Dic₂₆₉	—	2	3	no	—
48	#14524130	—	2	3	no	—
49	#14524130	—	2	1	no	—
50	C₂³×C₂₆₉	—	2	3	no	—
51	Q₈×C₂₆₉	SD₁₆	1	4	yes	complemented
52	D₄×C₂₆₉	—	2	3	no	—
53	D₄×C₂₆₉	—	2	3	no	—
54	D₄×C₂₆₉	—	4	6	no	—
55	D₄×C₂₆₉	—	8	3	no	—
56	C₂×C₄×C₂₆₉	—	2	1	no	—
57	C₂×C₄×C₂₆₉	—	4	6	no	—
58	C₂×C₄×C₂₆₉	C₂×D₄	1	3	yes	—
59	C₂×C₄×C₂₆₉	C₂×D₄	1	3	yes	—
60	C₂×C₄×C₂₆₉	C₂⁴	1	1	yes	characteristic
61	C₂²×C₂₆₉	—	4	3	no	—
62	C₂²×C₂₆₉	—	4	3	no	—
63	C₂²×C₂₆₉	—	4	3	no	—
64	C₂²×C₂₆₉	C₂²×D₄	1	1	yes	characteristic
65	C₄×C₂₆₉	—	2	3	no	—
66	C₄×C₂₆₉	—	4	1	no	—
67	C₄×C₂₆₉	C₂×SD₁₆	1	6	yes	—
68	C₄×C₂₆₉	C₂×#41	1	1	yes	characteristic
69	C₄×C₂₆₉	2₋¹⁺⁴	1	1	yes	characteristic
70	Dic₂₆₉	—	2	2	no	—
71	Dic₂₆₉	—	4	3	no	—
72	C₂×C₂₆₉	—	8	3	no	—
73	C₂×C₂₆₉	#548	1	1	yes	characteristic
74	C₂×C₂₆₉	C₂²×SD₁₆	1	1	yes	characteristic
75	C₂×C₂₆₉	#576	1	1	yes	characteristic
76	C₂₆₉	#3362	1	1	yes	characteristic, complemented, Sylow
77	#3362	—	269	1	no	maximal, Sylow
78	Q₈²	—	269	1	no	—
79	#550	D₂₆₉	1	1	yes	characteristic
80	#486	—	269	3	no	—
81	#476	—	269	3	no	—
82	#460	—	269	3	no	—
83	C₈×Q₈	—	269	1	no	—
84	C₄×SD₁₆	—	269	3	no	—
85	C₂×#41	—	2	1	no	—
86	C₂×SD₁₆	—	269	3	no	—
87	C₄⋊Q₈	—	269	3	no	—
88	C₄⋊Q₈	—	538	3	no	—
89	#127	C₂×D₂₆₉	1	3	yes	—
90	#121	—	2	3	no	—
91	C₄×Q₈	—	269	3	no	—
92	C₄×Q₈	—	538	1	no	—
93	C₄×Q₈	C₂×D₂₆₉	1	1	yes	characteristic
94	C₄×D₄	C₂×D₂₆₉	1	3	yes	—
95	#106	—	269	3	no	—
96	C₄⋊C₈	—	269	3	no	—
97	#103	—	269	1	no	—
98	#102	—	269	3	no	—
99	#102	—	269	6	no	—
100	C₄×C₈	—	269	3	no	—
101	#41	—	4	4	no	—
102	C₂×Q₈	—	269	1	no	—
103	C₂×Q₈	—	538	3	no	—
104	C₂×Q₈	C₂²⋊D₂₆₉	1	1	yes	characteristic
105	C₂×D₄	—	2	3	no	—
106	C₂×D₄	—	2	3	no	—
107	C₂×D₄	C₂²×D₂₆₉	1	3	yes	—
108	C₂²×C₄	—	2	3	no	—
109	SD₁₆	—	538	6	no	—
110	C₂×C₈	—	269	1	no	—
111	C₂×C₈	—	269	3	no	—
112	C₄⋊C₄	—	269	3	no	—
113	C₄⋊C₄	—	538	3	no	—
114	C₄⋊C₄	—	538	3	no	—
115	C₄⋊C₄	C₂²⋊D₂₆₉	1	3	yes	—
116	C₄⋊C₄	C₂²×D₂₆₉	1	1	yes	characteristic
117	C₂²⋊C₄	—	2	3	no	—
118	C₄²	—	538	3	no	—
119	C₄²	C₂²×D₂₆₉	1	3	yes	—
120	C₂³	—	2	3	no	—
121	Q₈	—	269	2	no	—
122	Q₈	—	538	1	no	—
123	Q₈	—	2152	3	no	—
124	Q₈	#16864390	1	4	yes	complemented
125	D₄	—	2	3	no	—
126	D₄	—	2	3	no	—
127	D₄	—	4	6	no	—
128	D₄	—	8	3	no	—
129	C₂×C₄	—	2	1	no	—
130	C₂×C₄	—	4	6	no	—
131	C₂×C₄	—	538	1	no	—
132	C₂×C₄	—	538	3	no	—
133	C₂×C₄	C₂×C₂²⋊D₂₆₉	1	3	yes	—
134	C₂×C₄	C₂×C₂²⋊D₂₆₉	1	3	yes	—
135	C₂×C₄	C₂³×D₂₆₉	1	1	yes	characteristic
136	C₈	—	538	3	no	—
137	C₈	—	538	1	no	—
138	C₂²	—	4	3	no	—
139	C₂²	—	4	3	no	—
140	C₂²	—	4	3	no	—
141	C₂²	C₂²×C₂²⋊D₂₆₉	1	1	yes	characteristic
142	C₄	—	2	3	no	—
143	C₄	—	4	1	no	—
144	C₄	—	538	2	no	—
145	C₄	—	1076	3	no	—
146	C₄	C₂×#16864390	1	6	yes	—
147	C₄	#41×D₂₆₉	1	1	yes	characteristic
148	C₄	#22575500	1	1	yes	characteristic
149	C₂	—	8	3	no	—
150	C₂	#31771262	1	1	yes	characteristic
151	C₂	C₂²×#16864390	1	1	yes	characteristic
152	C₂	#31771408	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

Q₈×Dic₅₃₈

C₂

yes

maximal, characteristic, complemented

#31770951

C₂

yes

maximal, complemented

#31770895

C₂

yes

maximal, complemented

#31770802

C₂

yes

maximal, complemented

Q₈×#14524130

C₂

yes

maximal, characteristic, complemented

C₄×#16864390

C₂

yes

maximal, complemented

#550×C₂₆₉

C₂

yes

maximal, characteristic

C₂×#16864390

C₂²

yes

—

#22575453

C₂²

yes

—

#22575452

—

Q₈×Dic₂₆₉

—

C₄×Dic₅₃₈

C₂²

yes

—

#22575384

C₂²

yes

—

C₄⋊#14524130

C₂²

yes

—

#22575376

C₂²

yes

characteristic

#22575374

C₂²

yes

—

#22575374

C₂²

yes

—

C₄×#14524130

C₂²

yes

—

C₂×#41×C₂₆₉

—

#127×C₂₆₉

C₂²

yes

—

#121×C₂₆₉

—

C₄×Q₈×C₂₆₉

C₂²

yes

characteristic

C₄×D₄×C₂₆₉

C₂²

yes

—

C₂×Dic₅₃₈

—

C₂×Dic₅₃₈

C₂³

yes

characteristic

#16864390

—

C₂×#14524130

C₂³

yes

characteristic

C₂×#14524130

C₂³

yes

—

C₄⋊Dic₂₆₉

—

C₄⋊Dic₂₆₉

C₂³

yes

—

#16864381

—

C₄×Dic₂₆₉

—

#41×C₂₆₉

—

C₂×Q₈×C₂₆₉

D₄

yes

characteristic

C₂×D₄×C₂₆₉

—

C₂×D₄×C₂₆₉

—

C₂×D₄×C₂₆₉

C₂³

yes

—

C₂²×C₄×C₂₆₉

—

C₄⋊C₄×C₂₆₉

D₄

yes

—

C₄⋊C₄×C₂₆₉

C₂³

yes

characteristic

C₂²⋊C₄×C₂₆₉

—

C₄²×C₂₆₉

C₂³

yes

—

Dic₅₃₈

—

Dic₅₃₈

—

Dic₅₃₈

#41

yes

complemented

C₂×Dic₂₆₉

—

C₂×Dic₂₆₉

—

#14524130

—

#14524130

—

C₂³×C₂₆₉

—

Q₈×C₂₆₉

SD₁₆

yes

complemented

D₄×C₂₆₉

—

D₄×C₂₆₉

—

D₄×C₂₆₉

—

D₄×C₂₆₉

—

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×C₂₆₉

—

C₂²×C₂₆₉

—

C₂²×C₂₆₉

C₂²×D₄

yes

characteristic

C₄×C₂₆₉

—

C₄×C₂₆₉

—

C₄×C₂₆₉

C₂×SD₁₆

yes

—

C₄×C₂₆₉

C₂×#41

yes

characteristic

C₄×C₂₆₉

2₋¹⁺⁴

yes

characteristic

Dic₂₆₉

—

Dic₂₆₉

—

C₂×C₂₆₉

—

C₂×C₂₆₉

#548

yes

characteristic

C₂×C₂₆₉

C₂²×SD₁₆

yes

C₂×C₂₆₉

yes

C₂₆₉

yes

characteristic, complemented, Sylow

#3362

—

269

maximal, Sylow

Q₈²

—

269

—

#550

D₂₆₉

yes

characteristic

#486

—

269

—

#476

—

269

—

#460

—

269

—

C₈×Q₈

—

269

—

C₄×SD₁₆

—

269

—

C₂×#41

—

C₂×SD₁₆

—

269

—

C₄⋊Q₈

—

269

—

C₄⋊Q₈

—

538

—

#127

C₂×D₂₆₉

yes

—

#121

—

C₄×Q₈

—

269

—

C₄×Q₈

—

538

—

C₄×Q₈

C₂×D₂₆₉

yes

characteristic

C₄×D₄

C₂×D₂₆₉

yes

—

#106

—

269

—

C₄⋊C₈

—

269

—

#103

—

269

—

#102

—

269

—

#102

—

269

—

100

C₄×C₈

—

269

—

101

#41

—

102

C₂×Q₈

—

269

—

103

C₂×Q₈

—

538

—

104

C₂×Q₈

C₂²⋊D₂₆₉

yes

characteristic

105

C₂×D₄

—

106

C₂×D₄

—

107

C₂×D₄

C₂²×D₂₆₉

yes

—

108

C₂²×C₄

—

109

SD₁₆

—

538

—

110

C₂×C₈

—

269

—

111

C₂×C₈

—

269

—

112

C₄⋊C₄

—

269

—

113

C₄⋊C₄

—

538

—

114

C₄⋊C₄

—

538

—

115

C₄⋊C₄

C₂²⋊D₂₆₉

yes

—

116

C₄⋊C₄

C₂²×D₂₆₉

yes

characteristic

117

C₂²⋊C₄

—

118

C₄²

—

538

—

119

C₄²

C₂²×D₂₆₉

yes

—

120

C₂³

—

121

Q₈

—

269

—

122

Q₈

—

538

—

123

Q₈

—

2152

—

124

Q₈

#16864390

yes

complemented

125

D₄

—

126

D₄

—

127

D₄

—

128

D₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

538

—

132

C₂×C₄

—

538

—

133

C₂×C₄

C₂×C₂²⋊D₂₆₉

yes

—

134

C₂×C₄

C₂×C₂²⋊D₂₆₉

yes

—

135

C₂×C₄

C₂³×D₂₆₉

yes

characteristic

136

C₈

—

538

—

137

C₈

—

538

—

138

C₂²

—

139

C₂²

—

140

C₂²

—

141

C₂²

C₂²×C₂²⋊D₂₆₉

yes

characteristic

142

C₄

—

143

C₄

—

144

C₄

—

538

—

145

C₄

—

1076

—

146

C₄

C₂×#16864390

yes

—

147

C₄

#41×D₂₆₉

yes

characteristic

148

C₄

#22575500

yes

characteristic

149

C₂

—

150

C₂

#31771262

yes

characteristic

151

C₂

C₂²×#16864390

yes

characteristic

152

C₂

#31771408

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₂₆₉
commutator, cosocle	C₂×C₄×C₂₆₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₂₆₉
Fitting, prime radical	#550×C₂₆₉	C₂
socle	C₂²×C₂₆₉	C₂²×D₄
hypercenter	#550	D₂₆₉
hypocenter	C₂₆₉	#3362
LCS	G ▷ C₂×C₄×C₂₆₉ ▷ C₂×C₂₆₉ ▷ C₂₆₉
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ #550

subgroup

G/H

center

C₂²

C₂²×C₂²⋊D₂₆₉

commutator, cosocle

C₂×C₄×C₂₆₉

C₂⁴

Frattini

C₂×C₄

C₂³×D₂₆₉

Fitting, prime radical

#550×C₂₆₉

C₂

socle

C₂²×C₂₆₉

C₂²×D₄

D₂₆₉

C₂₆₉

G ▷ C₂×C₄×C₂₆₉ ▷ C₂×C₂₆₉ ▷ C₂₆₉

UCS

C₁ ◁ C₂² ◁ C₄×Q₈ ◁ #550

N	H ≅ G/N	type
1. Q₈×Dic₅₃₈	C₂	semidirect
2. #31770951	C₂	semidirect
3. #31770895	C₂	semidirect
4. #31770802	C₂	semidirect
5. Q₈×#14524130	C₂	semidirect
6. C₄×#16864390	C₂	semidirect
45. Dic₅₃₈	#41	semidirect
51. Q₈×C₂₆₉	SD₁₆	semidirect
76. C₂₆₉	#3362	semidirect
124. Q₈	#16864390	semidirect

H ≅ G/N

type

1. Q₈×Dic₅₃₈

C₂

semidirect

2. #31770951

C₂

semidirect

3. #31770895

C₂

semidirect

4. #31770802

C₂

semidirect

5. Q₈×#14524130

C₂

semidirect

6. C₄×#16864390

C₂

semidirect

45. Dic₅₃₈

#41

semidirect

51. Q₈×C₂₆₉

SD₁₆

semidirect

76. C₂₆₉

#3362

semidirect

124. Q₈

#16864390

semidirect

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

subgroup	index
Q₈×Dic₅₃₈	2
#31770951	2
#31770895	2
#31770802	2
Q₈×#14524130	2
C₄×#16864390	2
#550×C₂₆₉	2
#3362	269

subgroup

index

Q₈×Dic₅₃₈

#550×C₂₆₉

269

Subgroups — #46823552

Conjugacy classes of subgroups

Complements

Maximal subgroups