Group #46811666 — subgroups

Subgroups — #46811666

Order 34432.

196 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 196 conjugacy classes of subgroups — too many to draw. Only its 97 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	#31770647	C₂	1	2	yes	maximal
2	C₂×#22575385	C₂	1	1	yes	maximal, characteristic
3	#31770560	C₂	1	1	yes	maximal, characteristic
4	#31770373	C₂	1	2	yes	maximal
5	C₂×#108×C₂₆₉	C₂	1	1	yes	maximal, characteristic
6	C₂×#16864386	C₂²	1	1	yes	characteristic
7	#22575412	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊Dic₂₆₉	C₂²	1	2	yes	—
9	#22575387	—	2	2	no	—
10	#22575385	—	2	2	no	—
11	#22575385	—	2	1	no	—
12	#22575385	—	2	1	no	—
13	#22575363	—	2	1	no	—
14	C₈×Dic₂₆₉	—	2	1	no	—
15	C₂×M₁₆×C₂₆₉	C₂²	1	2	yes	—
16	C₂²×C₈×C₂₆₉	C₂²	1	1	yes	characteristic
17	#108×C₂₆₉	C₄	1	4	yes	complemented
18	C₂²×Dic₂₆₉	—	2	2	no	—
19	#16864386	—	2	1	no	—
20	#16864386	Q₈	1	2	yes	—
21	C₂×#14524130	—	2	1	no	—
22	C₄⋊Dic₂₆₉	—	2	2	no	—
23	C₄⋊Dic₂₆₉	—	2	2	no	—
24	C₄⋊Dic₂₆₉	D₄	1	2	yes	—
25	C₂²⋊Dic₂₆₉	—	2	1	no	—
26	C₄×Dic₂₆₉	—	2	1	no	—
27	C₂²×C₄×C₂₆₉	C₂³	1	1	yes	characteristic
28	M₁₆×C₂₆₉	—	2	2	no	—
29	M₁₆×C₂₆₉	C₂×C₄	1	4	yes	—
30	C₂×C₈×C₂₆₉	—	2	2	no	—
31	C₂×C₈×C₂₆₉	C₂×C₄	1	2	yes	—
32	C₂×C₈×C₂₆₉	D₄	1	1	yes	characteristic
33	C₂×C₈×C₂₆₉	D₄	1	1	yes	characteristic
34	C₂×C₈×C₂₆₉	Q₈	1	2	yes	—
35	C₂×Dic₂₆₉	—	2	2	no	—
36	C₂×Dic₂₆₉	—	4	2	no	—
37	C₂×Dic₂₆₉	—	8	2	no	—
38	#14524130	—	2	2	no	—
39	C₂³×C₂₆₉	#41	1	1	yes	characteristic
40	C₂×C₄×C₂₆₉	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₄×C₂₆₉	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₂₆₉	C₂×D₄	1	1	yes	characteristic
43	C₂×C₄×C₂₆₉	C₂×Q₈	1	1	yes	characteristic
44	C₂×C₄×C₂₆₉	C₂×Q₈	1	1	yes	characteristic
45	C₂×C₄×C₂₆₉	#41	1	1	yes	characteristic
46	C₈×C₂₆₉	—	2	4	no	—
47	C₈×C₂₆₉	C₄⋊C₄	1	2	yes	—
48	C₈×C₂₆₉	C₄⋊C₄	1	2	yes	—
49	C₂²×C₂₆₉	—	2	2	no	—
50	C₂²×C₂₆₉	C₄×Q₈	1	1	yes	characteristic
51	C₂²×C₂₆₉	#121	1	1	yes	characteristic
52	C₂²×C₂₆₉	#125	1	1	yes	characteristic
53	C₄×C₂₆₉	C₂×C₄⋊C₄	1	1	yes	characteristic
54	C₄×C₂₆₉	C₄×D₄	1	1	yes	characteristic
55	C₄×C₂₆₉	C₂²⋊Q₈	1	1	yes	characteristic
56	C₄×C₂₆₉	C₄⋊Q₈	1	1	yes	characteristic
57	Dic₂₆₉	—	4	2	no	—
58	Dic₂₆₉	—	8	2	no	—
59	C₂×C₂₆₉	—	2	1	no	—
60	C₂×C₂₆₉	—	2	1	no	—
61	C₂×C₂₆₉	#389	1	1	yes	characteristic
62	C₂×C₂₆₉	#471	1	2	yes	—
63	C₂₆₉	#1951	1	1	yes	characteristic, complemented, Sylow
64	#1951	—	269	1	no	maximal, Sylow
65	C₂×#108	D₂₆₉	1	1	yes	characteristic
66	#428	—	269	2	no	—
67	C₂×#106	—	269	1	no	—
68	#405	—	269	1	no	—
69	#343	—	269	2	no	—
70	C₂×M₁₆	—	269	1	no	—
71	C₂×M₁₆	C₂×D₂₆₉	1	2	yes	—
72	C₂²×C₈	C₂×D₂₆₉	1	1	yes	characteristic
73	#117	—	269	1	no	—
74	C₂×C₄⋊C₄	—	269	2	no	—
75	#108	Dic₂₆₉	1	4	yes	complemented
76	#107	—	538	2	no	—
77	#106	—	538	2	no	—
78	#106	—	538	1	no	—
79	#106	—	538	1	no	—
80	#97	—	538	1	no	—
81	C₄×C₈	—	538	1	no	—
82	C₂²×C₄	—	538	2	no	—
83	C₂²×C₄	C₂²×D₂₆₉	1	1	yes	characteristic
84	M₁₆	—	2	2	no	—
85	M₁₆	—	269	2	no	—
86	M₁₆	—	538	1	no	—
87	M₁₆	C₂×Dic₂₆₉	1	4	yes	—
88	C₂×C₈	—	2	2	no	—
89	C₂×C₈	—	538	1	no	—
90	C₂×C₈	C₂×Dic₂₆₉	1	2	yes	—
91	C₂×C₈	D₁₀₇₆	1	1	yes	characteristic
92	C₂×C₈	D₁₀₇₆	1	1	yes	characteristic
93	C₂×C₈	Dic₅₃₈	1	2	yes	—
94	C₄⋊C₄	—	269	2	no	—
95	C₄⋊C₄	—	538	2	no	—
96	C₄⋊C₄	—	538	2	no	—
97	C₂²⋊C₄	—	538	1	no	—
98	C₄²	—	538	1	no	—
99	C₂³	#16864403	1	1	yes	characteristic
100	C₂×C₄	—	538	2	no	—
101	C₂×C₄	—	1076	2	no	—
102	C₂×C₄	—	2152	2	no	—
103	C₂×C₄	C₂²×Dic₂₆₉	1	1	yes	characteristic
104	C₂×C₄	D₄×D₂₆₉	1	1	yes	characteristic
105	C₂×C₄	C₂×D₁₀₇₆	1	1	yes	characteristic
106	C₂×C₄	Q₈×D₂₆₉	1	1	yes	characteristic
107	C₂×C₄	C₂×Dic₅₃₈	1	1	yes	characteristic
108	C₂×C₄	#16864401	1	1	yes	characteristic
109	C₈	—	2	4	no	—
110	C₈	—	538	2	no	—
111	C₈	C₄⋊Dic₂₆₉	1	2	yes	—
112	C₈	C₄⋊Dic₂₆₉	1	2	yes	—
113	C₂²	—	2	2	no	—
114	C₂²	Q₈×Dic₂₆₉	1	1	yes	characteristic
115	C₂²	#22575428	1	1	yes	characteristic
116	C₂²	#22575444	1	1	yes	characteristic
117	C₄	—	1076	2	no	—
118	C₄	—	2152	2	no	—
119	C₄	C₂×C₄⋊Dic₂₆₉	1	1	yes	characteristic
120	C₄	D₄×Dic₂₆₉	1	1	yes	characteristic
121	C₄	#22575433	1	1	yes	characteristic
122	C₄	#22575452	1	1	yes	characteristic
123	C₂	—	2	1	no	—
124	C₂	—	2	1	no	—
125	C₂	#31770504	1	1	yes	characteristic
126	C₂	#31770869	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#31770647

C₂

yes

maximal

C₂×#22575385

C₂

yes

maximal, characteristic

#31770560

C₂

yes

maximal, characteristic

#31770373

C₂

yes

maximal

C₂×#108×C₂₆₉

C₂

yes

maximal, characteristic

C₂×#16864386

C₂²

yes

characteristic

#22575412

C₂²

yes

characteristic

C₂×C₄⋊Dic₂₆₉

C₂²

yes

—

—

—

—

—

—

C₈×Dic₂₆₉

—

C₂×M₁₆×C₂₆₉

C₂²

yes

—

C₂²×C₈×C₂₆₉

C₂²

yes

characteristic

#108×C₂₆₉

C₄

yes

complemented

C₂²×Dic₂₆₉

—

#16864386

—

#16864386

Q₈

yes

—

C₂×#14524130

—

C₄⋊Dic₂₆₉

—

C₄⋊Dic₂₆₉

—

C₄⋊Dic₂₆₉

D₄

yes

—

C₂²⋊Dic₂₆₉

—

C₄×Dic₂₆₉

—

C₂²×C₄×C₂₆₉

C₂³

yes

characteristic

M₁₆×C₂₆₉

—

M₁₆×C₂₆₉

C₂×C₄

yes

—

C₂×C₈×C₂₆₉

—

C₂×C₈×C₂₆₉

C₂×C₄

yes

—

C₂×C₈×C₂₆₉

D₄

yes

characteristic

C₂×C₈×C₂₆₉

D₄

yes

characteristic

C₂×C₈×C₂₆₉

Q₈

yes

—

C₂×Dic₂₆₉

—

C₂×Dic₂₆₉

—

C₂×Dic₂₆₉

—

#14524130

—

C₂³×C₂₆₉

#41

yes

characteristic

C₂×C₄×C₂₆₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₆₉

C₂×D₄

yes

characteristic

C₂×C₄×C₂₆₉

C₂×D₄

yes

characteristic

C₂×C₄×C₂₆₉

C₂×Q₈

yes

characteristic

C₂×C₄×C₂₆₉

C₂×Q₈

yes

characteristic

C₂×C₄×C₂₆₉

#41

yes

characteristic

C₈×C₂₆₉

—

C₈×C₂₆₉

C₄⋊C₄

yes

—

C₈×C₂₆₉

C₄⋊C₄

yes

—

C₂²×C₂₆₉

—

C₂²×C₂₆₉

C₄×Q₈

yes

C₂²×C₂₆₉

yes

C₂²×C₂₆₉

yes

C₄×C₂₆₉

C₂×C₄⋊C₄

yes

characteristic

C₄×C₂₆₉

C₄×D₄

yes

characteristic

C₄×C₂₆₉

C₂²⋊Q₈

yes

characteristic

C₄×C₂₆₉

C₄⋊Q₈

yes

characteristic

Dic₂₆₉

—

Dic₂₆₉

—

C₂×C₂₆₉

—

C₂×C₂₆₉

—

C₂×C₂₆₉

#389

yes

characteristic

C₂×C₂₆₉

#471

yes

—

C₂₆₉

#1951

yes

characteristic, complemented, Sylow

#1951

—

269

maximal, Sylow

C₂×#108

D₂₆₉

yes

characteristic

#428

—

269

—

C₂×#106

—

269

—

#405

—

269

—

#343

—

269

—

C₂×M₁₆

—

269

—

C₂×M₁₆

C₂×D₂₆₉

yes

—

C₂²×C₈

C₂×D₂₆₉

yes

characteristic

#117

—

269

—

C₂×C₄⋊C₄

—

269

—

#108

Dic₂₆₉

yes

complemented

#107

—

538

—

#106

—

538

—

#106

—

538

—

#106

—

538

—

#97

—

538

—

C₄×C₈

—

538

—

C₂²×C₄

—

538

—

C₂²×C₄

C₂²×D₂₆₉

yes

characteristic

M₁₆

—

M₁₆

—

269

—

M₁₆

—

538

—

M₁₆

C₂×Dic₂₆₉

yes

—

C₂×C₈

—

C₂×C₈

—

538

—

C₂×C₈

C₂×Dic₂₆₉

yes

—

C₂×C₈

D₁₀₇₆

yes

characteristic

C₂×C₈

D₁₀₇₆

yes

characteristic

C₂×C₈

Dic₅₃₈

yes

—

C₄⋊C₄

—

269

—

C₄⋊C₄

—

538

—

C₄⋊C₄

—

538

—

C₂²⋊C₄

—

538

—

C₄²

—

538

—

C₂³

#16864403

yes

characteristic

100

C₂×C₄

—

538

—

101

C₂×C₄

—

1076

—

102

C₂×C₄

—

2152

—

103

C₂×C₄

C₂²×Dic₂₆₉

yes

characteristic

104

C₂×C₄

D₄×D₂₆₉

yes

characteristic

105

C₂×C₄

C₂×D₁₀₇₆

yes

characteristic

106

C₂×C₄

Q₈×D₂₆₉

yes

characteristic

107

C₂×C₄

C₂×Dic₅₃₈

yes

characteristic

108

C₂×C₄

#16864401

yes

characteristic

109

C₈

—

110

C₈

—

538

—

111

C₈

C₄⋊Dic₂₆₉

yes

—

112

C₈

C₄⋊Dic₂₆₉

yes

—

113

C₂²

—

114

C₂²

Q₈×Dic₂₆₉

yes

characteristic

115

C₂²

#22575428

yes

characteristic

116

C₂²

#22575444

yes

characteristic

117

C₄

—

1076

—

118

C₄

—

2152

—

119

C₄

C₂×C₄⋊Dic₂₆₉

yes

characteristic

120

C₄

D₄×Dic₂₆₉

yes

characteristic

121

C₄

#22575433

yes

characteristic

122

C₄

#22575452

yes

characteristic

123

C₂

—

124

C₂

—

125

C₂

#31770504

yes

characteristic

126

C₂

#31770869

yes

—

	subgroup	G/H
center	C₂²	#22575428
commutator	C₂×C₄×C₂₆₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₆₉
Fitting	C₂×#108×C₂₆₉	C₂
socle	C₂²×C₂₆₉	#121
prime radical	C₂³×C₂₆₉	#41
cosocle	C₂²×C₄×C₂₆₉	C₂³
hypercenter	C₂×#108	D₂₆₉
hypocenter	C₂₆₉	#1951
LCS	G ▷ C₂×C₄×C₂₆₉ ▷ C₂×C₂₆₉ ▷ C₂₆₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#108

subgroup

G/H

center

C₂²

#22575428

commutator

C₂×C₄×C₂₆₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₂₆₉

Fitting

C₂×#108×C₂₆₉

C₂

C₂²×C₂₆₉

C₂³×C₂₆₉

C₂²×C₄×C₂₆₉

C₂³

D₂₆₉

C₂₆₉

G ▷ C₂×C₄×C₂₆₉ ▷ C₂×C₂₆₉ ▷ C₂₆₉

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#108

N	H ≅ G/N	type
17. #108×C₂₆₉	C₄	semidirect
63. C₂₆₉	#1951	semidirect
75. #108	Dic₂₆₉	semidirect

H ≅ G/N

type

17. #108×C₂₆₉

C₄

semidirect

63. C₂₆₉

#1951

semidirect

75. #108

Dic₂₆₉

semidirect

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

subgroup	index
#31770647	2
C₂×#22575385	2
#31770560	2
#31770373	2
C₂×#108×C₂₆₉	2
#1951	269

subgroup

index

C₂×#108×C₂₆₉

269

Subgroups — #46811666

Conjugacy classes of subgroups

Complements

Maximal subgroups