Group #46809966 — subgroups

Subgroups — #46809966

Order 34432.

240 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 53 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	#31771321	C₂	1	1	yes	maximal, characteristic, complemented
2	#31770797	C₂	1	2	yes	maximal, complemented
3	#31770354	C₂	1	1	yes	maximal, characteristic, complemented
4	#31770334	C₂	1	2	yes	maximal, complemented
5	#492×C₂₆₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#16864387	—	2	2	no	—
7	#22575442	—	2	2	no	—
8	#22575428	—	2	2	no	—
9	C₄×D₁₀₇₆	C₂²	1	2	yes	—
10	#22575411	—	2	2	no	—
11	C₄⋊#14524130	C₂²	1	2	yes	—
12	#22575374	—	2	2	no	—
13	C₂×SD₁₆×C₂₆₉	—	2	2	no	—
14	C₄⋊Q₈×C₂₆₉	C₂²	1	1	yes	characteristic
15	#127×C₂₆₉	C₂²	1	1	yes	characteristic
16	C₄×C₈×C₂₆₉	C₂²	1	1	yes	characteristic
17	C₂×D₁₀₇₆	—	2	2	no	—
18	C₂×D₁₀₇₆	D₄	1	2	yes	—
19	C₂×C₄×D₂₆₉	—	2	2	no	—
20	#16864387	—	4	4	no	—
21	C₂×#14524130	—	2	2	no	—
22	C₄⋊Dic₂₆₉	D₄	1	2	yes	—
23	#16864379	—	2	4	no	—
24	#16864379	—	2	2	no	—
25	C₄×Dic₂₆₉	—	2	2	no	—
26	C₂×Q₈×C₂₆₉	D₄	1	2	yes	—
27	C₂×D₄×C₂₆₉	—	2	2	no	—
28	C₂×D₄×C₂₆₉	—	2	1	no	—
29	SD₁₆×C₂₆₉	—	4	4	no	—
30	C₂×C₈×C₂₆₉	—	2	1	no	—
31	C₄⋊C₄×C₂₆₉	—	2	2	no	—
32	C₄²×C₂₆₉	C₂³	1	1	yes	characteristic
33	C₂²×D₂₆₉	—	2	2	no	—
34	D₁₀₇₆	—	2	2	no	—
35	D₁₀₇₆	—	2	2	no	—
36	D₁₀₇₆	—	8	2	no	—
37	C₂×Dic₂₆₉	—	2	2	no	—
38	C₄×D₂₆₉	—	4	2	no	—
39	C₄×D₂₆₉	—	4	2	no	—
40	#14524130	—	4	2	no	—
41	C₂³×C₂₆₉	—	4	1	no	—
42	Q₈×C₂₆₉	—	4	2	no	—
43	D₄×C₂₆₉	—	4	2	no	—
44	D₄×C₂₆₉	—	4	2	no	—
45	D₄×C₂₆₉	—	4	2	no	—
46	C₂×C₄×C₂₆₉	—	2	2	no	—
47	C₂×C₄×C₂₆₉	C₂×D₄	1	2	yes	—
48	C₂×C₄×C₂₆₉	C₂×D₄	1	1	yes	characteristic
49	C₈×C₂₆₉	—	2	2	no	—
50	C₂²×C₂₆₉	—	8	2	no	—
51	C₂²×C₂₆₉	—	8	1	no	—
52	C₂²×C₂₆₉	#120	1	1	yes	characteristic
53	C₂×D₂₆₉	—	4	2	no	—
54	C₂×D₂₆₉	—	4	2	no	—
55	C₂×D₂₆₉	—	4	2	no	—
56	C₄×C₂₆₉	—	2	1	no	—
57	C₄×C₂₆₉	—	4	2	no	—
58	C₄×C₂₆₉	#135	1	2	yes	—
59	C₄×C₂₆₉	AGL(1,ℤ/8)	1	2	yes	—
60	Dic₂₆₉	—	4	2	no	—
61	C₂×C₂₆₉	—	16	1	no	—
62	C₂×C₂₆₉	#449	1	2	yes	—
63	C₂×C₂₆₉	#453	1	1	yes	characteristic
64	D₂₆₉	—	8	2	no	—
65	C₂₆₉	#1624	1	1	yes	characteristic, complemented, Sylow
66	#1624	—	269	1	no	maximal, Sylow
67	#555	—	269	1	no	—
68	#492	D₂₆₉	1	1	yes	characteristic, complemented
69	#459	—	269	2	no	—
70	#333	—	269	1	no	—
71	D₄⋊C₈	—	269	2	no	—
72	C₂×SD₁₆	—	2	2	no	—
73	C₂×D₈	—	538	2	no	—
74	C₄⋊Q₈	C₂×D₂₆₉	1	1	yes	characteristic
75	#127	C₂×D₂₆₉	1	1	yes	characteristic
76	#124	—	538	2	no	—
77	#121	—	538	2	no	—
78	C₄×D₄	—	269	2	no	—
79	#117	—	538	2	no	—
80	C₄⋊C₈	—	269	2	no	—
81	#102	—	538	2	no	—
82	C₄×C₈	C₂×D₂₆₉	1	1	yes	characteristic
83	C₂×Q₈	C₂²⋊D₂₆₉	1	2	yes	—
84	C₂×D₄	—	2	2	no	—
85	C₂×D₄	—	2	1	no	—
86	C₂×D₄	—	269	2	no	—
87	C₂×D₄	—	538	2	no	—
88	C₂²×C₄	—	538	2	no	—
89	SD₁₆	—	4	4	no	—
90	D₈	—	1076	4	no	—
91	C₂×C₈	—	2	1	no	—
92	C₂×C₈	—	538	2	no	—
93	C₄⋊C₄	—	2	2	no	—
94	C₄⋊C₄	—	269	2	no	—
95	C₂²⋊C₄	—	538	4	no	—
96	C₂²⋊C₄	—	538	2	no	—
97	C₄²	—	538	2	no	—
98	C₄²	C₂²×D₂₆₉	1	1	yes	characteristic
99	C₂³	—	4	1	no	—
100	C₂³	—	538	2	no	—
101	Q₈	—	4	2	no	—
102	D₄	—	4	2	no	—
103	D₄	—	4	2	no	—
104	D₄	—	4	2	no	—
105	D₄	—	538	2	no	—
106	D₄	—	538	2	no	—
107	D₄	—	2152	2	no	—
108	C₂×C₄	—	2	2	no	—
109	C₂×C₄	—	538	2	no	—
110	C₂×C₄	—	1076	2	no	—
111	C₂×C₄	—	1076	2	no	—
112	C₂×C₄	D₄×D₂₆₉	1	2	yes	—
113	C₂×C₄	C₂×C₂²⋊D₂₆₉	1	1	yes	characteristic
114	C₈	—	2	2	no	—
115	C₈	—	1076	2	no	—
116	C₂²	—	8	2	no	—
117	C₂²	—	8	1	no	—
118	C₂²	—	1076	2	no	—
119	C₂²	—	1076	2	no	—
120	C₂²	—	1076	2	no	—
121	C₂²	#22575421	1	1	yes	characteristic
122	C₄	—	2	1	no	—
123	C₄	—	4	2	no	—
124	C₄	—	1076	2	no	—
125	C₄	#22575474	1	2	yes	—
126	C₄	#22575480	1	2	yes	—
127	C₂	—	16	1	no	—
128	C₂	—	2152	2	no	—
129	C₂	#31770749	1	2	yes	—
130	C₂	#31770772	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#31771321

C₂

yes

maximal, characteristic, complemented

#31770797

C₂

yes

maximal, complemented

#31770354

C₂

yes

maximal, characteristic, complemented

#31770334

C₂

yes

maximal, complemented

#492×C₂₆₉

C₂

yes

maximal, characteristic, complemented

C₂×#16864387

—

#22575442

—

#22575428

—

C₄×D₁₀₇₆

C₂²

yes

—

#22575411

—

C₄⋊#14524130

C₂²

yes

—

#22575374

—

C₂×SD₁₆×C₂₆₉

—

C₄⋊Q₈×C₂₆₉

C₂²

yes

characteristic

#127×C₂₆₉

C₂²

yes

characteristic

C₄×C₈×C₂₆₉

C₂²

yes

characteristic

C₂×D₁₀₇₆

—

C₂×D₁₀₇₆

D₄

yes

—

C₂×C₄×D₂₆₉

—

#16864387

—

C₂×#14524130

—

C₄⋊Dic₂₆₉

D₄

yes

—

#16864379

—

#16864379

—

C₄×Dic₂₆₉

—

C₂×Q₈×C₂₆₉

D₄

yes

—

C₂×D₄×C₂₆₉

—

C₂×D₄×C₂₆₉

—

SD₁₆×C₂₆₉

—

C₂×C₈×C₂₆₉

—

C₄⋊C₄×C₂₆₉

—

C₄²×C₂₆₉

C₂³

yes

characteristic

C₂²×D₂₆₉

—

D₁₀₇₆

—

D₁₀₇₆

—

D₁₀₇₆

—

C₂×Dic₂₆₉

—

C₄×D₂₆₉

—

C₄×D₂₆₉

—

#14524130

—

C₂³×C₂₆₉

—

Q₈×C₂₆₉

—

D₄×C₂₆₉

—

D₄×C₂₆₉

—

D₄×C₂₆₉

—

C₂×C₄×C₂₆₉

—

C₂×C₄×C₂₆₉

C₂×D₄

yes

—

C₂×C₄×C₂₆₉

C₂×D₄

yes

characteristic

C₈×C₂₆₉

—

C₂²×C₂₆₉

—

C₂²×C₂₆₉

—

C₂²×C₂₆₉

#120

yes

characteristic

C₂×D₂₆₉

—

C₂×D₂₆₉

—

C₂×D₂₆₉

—

C₄×C₂₆₉

—

C₄×C₂₆₉

—

C₄×C₂₆₉

#135

yes

—

C₄×C₂₆₉

AGL(1,ℤ/8)

yes

—

Dic₂₆₉

—

C₂×C₂₆₉

—

C₂×C₂₆₉

#449

yes

—

C₂×C₂₆₉

#453

yes

characteristic

D₂₆₉

—

C₂₆₉

#1624

yes

characteristic, complemented, Sylow

#1624

—

269

maximal, Sylow

#555

—

269

—

#492

D₂₆₉

yes

characteristic, complemented

#459

—

269

—

#333

—

269

—

D₄⋊C₈

—

269

—

C₂×SD₁₆

—

C₂×D₈

—

538

—

C₄⋊Q₈

C₂×D₂₆₉

yes

characteristic

#127

C₂×D₂₆₉

yes

characteristic

#124

—

538

—

#121

—

538

—

C₄×D₄

—

269

—

#117

—

538

—

C₄⋊C₈

—

269

—

#102

—

538

—

C₄×C₈

C₂×D₂₆₉

yes

characteristic

C₂×Q₈

C₂²⋊D₂₆₉

yes

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

269

—

C₂×D₄

—

538

—

C₂²×C₄

—

538

—

SD₁₆

—

D₈

—

1076

—

C₂×C₈

—

C₂×C₈

—

538

—

C₄⋊C₄

—

C₄⋊C₄

—

269

—

C₂²⋊C₄

—

538

—

C₂²⋊C₄

—

538

—

C₄²

—

538

—

C₄²

C₂²×D₂₆₉

yes

characteristic

C₂³

—

100

C₂³

—

538

—

101

Q₈

—

102

D₄

—

103

D₄

—

104

D₄

—

105

D₄

—

538

—

106

D₄

—

538

—

107

D₄

—

2152

—

108

C₂×C₄

—

109

C₂×C₄

—

538

—

110

C₂×C₄

—

1076

—

111

C₂×C₄

—

1076

—

112

C₂×C₄

D₄×D₂₆₉

yes

—

113

C₂×C₄

C₂×C₂²⋊D₂₆₉

yes

characteristic

114

C₈

—

115

C₈

—

1076

—

116

C₂²

—

117

C₂²

—

118

C₂²

—

1076

—

119

C₂²

—

1076

—

120

C₂²

—

1076

—

121

C₂²

#22575421

yes

characteristic

122

C₄

—

123

C₄

—

124

C₄

—

1076

—

125

C₄

#22575474

yes

—

126

C₄

#22575480

yes

—

127

C₂

—

128

C₂

—

2152

—

129

C₂

#31770749

yes

—

130

C₂

#31770772

yes

characteristic

	subgroup	G/H
center	C₂²	#22575421
commutator, cosocle	C₄²×C₂₆₉	C₂³
Frattini	C₄²	C₂²×D₂₆₉
Fitting	#492×C₂₆₉	C₂
socle	C₂²×C₂₆₉	#120
hypercenter	#492	D₂₆₉
hypocenter	C₂₆₉	#1624
LCS	G ▷ C₄²×C₂₆₉ ▷ C₂²×C₂₆₉ ▷ C₂₆₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #492

subgroup

G/H

center

C₂²

#22575421

commutator, cosocle

C₄²×C₂₆₉

C₂³

Frattini

C₄²

C₂²×D₂₆₉

Fitting

#492×C₂₆₉

C₂

C₂²×C₂₆₉

D₂₆₉

C₂₆₉

G ▷ C₄²×C₂₆₉ ▷ C₂²×C₂₆₉ ▷ C₂₆₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #492

N	H ≅ G/N	type
1. #31771321	C₂	semidirect
2. #31770797	C₂	semidirect
3. #31770354	C₂	semidirect
4. #31770334	C₂	semidirect
5. #492×C₂₆₉	C₂	semidirect
65. C₂₆₉	#1624	semidirect
68. #492	D₂₆₉	semidirect

H ≅ G/N

type

1. #31771321

C₂

semidirect

2. #31770797

C₂

semidirect

3. #31770354

C₂

semidirect

4. #31770334

C₂

semidirect

5. #492×C₂₆₉

C₂

semidirect

65. C₂₆₉

#1624

semidirect

68. #492

D₂₆₉

semidirect

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

subgroup	index
#31771321	2
#31770797	2
#31770354	2
#31770334	2
#492×C₂₆₉	2
#1624	269

subgroup

index

#492×C₂₆₉

269

Subgroups — #46809966

Conjugacy classes of subgroups

Complements

Maximal subgroups