Group #46811686 — subgroups

Subgroups — #46811686

Order 34432.

332 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 332 conjugacy classes of subgroups — too many to draw. Only its 137 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×#22575453	C₂	1	1	yes	maximal, characteristic
2	C₂×#22575376	C₂	1	4	yes	maximal
3	C₂×C₄×#14524130	C₂	1	1	yes	maximal, characteristic
4	#384×C₂₆₉	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₅₃₈	C₂²	1	2	yes	—
6	C₂²×#14524130	C₂²	1	2	yes	—
7	#22575453	—	2	2	no	—
8	#22575453	C₄	1	4	yes	complemented
9	C₂×C₄⋊Dic₂₆₉	—	2	2	no	—
10	#22575376	—	2	8	no	—
11	C₄×#14524130	—	2	2	no	—
12	C₂×C₄⋊C₄×C₂₆₉	—	2	2	no	—
13	C₂×C₄⋊C₄×C₂₆₉	C₂²	1	2	yes	—
14	C₂×C₄²×C₂₆₉	C₂²	1	1	yes	characteristic
15	C₂×Dic₅₃₈	—	2	2	no	—
16	C₂×Dic₅₃₈	—	2	4	no	—
17	C₂×Dic₅₃₈	—	2	4	no	—
18	C₂×Dic₅₃₈	C₂×C₄	1	4	yes	—
19	C₂²×Dic₂₆₉	—	2	2	no	—
20	C₂×#14524130	—	2	2	no	—
21	C₂×#14524130	—	2	2	no	—
22	C₂×#14524130	D₄	1	4	yes	—
23	C₄⋊Dic₂₆₉	—	2	8	no	—
24	C₂²×C₄×C₂₆₉	—	2	2	no	—
25	C₂²×C₄×C₂₆₉	D₄	1	2	yes	—
26	C₂²×C₄×C₂₆₉	C₂³	1	1	yes	characteristic
27	C₄⋊C₄×C₂₆₉	—	2	4	no	—
28	C₄⋊C₄×C₂₆₉	—	4	4	no	—
29	C₄²×C₂₆₉	C₂×C₄	1	2	yes	—
30	C₄²×C₂₆₉	D₄	1	2	yes	—
31	Dic₅₃₈	—	2	8	no	—
32	Dic₅₃₈	—	4	4	no	—
33	C₂×Dic₂₆₉	—	2	4	no	—
34	C₂×Dic₂₆₉	—	4	2	no	—
35	C₂×Dic₂₆₉	—	4	2	no	—
36	#14524130	—	2	4	no	—
37	C₂³×C₂₆₉	C₂×D₄	1	1	yes	characteristic
38	C₂×C₄×C₂₆₉	—	4	2	no	—
39	C₂×C₄×C₂₆₉	—	4	4	no	—
40	C₂×C₄×C₂₆₉	C₂²⋊C₄	1	4	yes	—
41	C₂×C₄×C₂₆₉	SD₁₆	1	4	yes	—
42	C₂×C₄×C₂₆₉	Q₁₆	1	4	yes	—
43	C₂×C₄×C₂₆₉	C₂²×C₄	1	1	yes	characteristic
44	C₂×C₄×C₂₆₉	C₂×D₄	1	2	yes	—
45	C₂×C₄×C₂₆₉	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×C₂₆₉	#41	1	2	yes	—
47	C₂²×C₂₆₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
48	C₂²×C₂₆₉	#124	1	1	yes	characteristic
49	C₂²×C₂₆₉	#127	1	1	yes	characteristic
50	C₂²×C₂₆₉	C₂×SD₁₆	1	2	yes	—
51	C₂²×C₂₆₉	C₂×Q₁₆	1	2	yes	—
52	C₄×C₂₆₉	—	8	2	no	—
53	C₄×C₂₆₉	#103	1	8	yes	complemented
54	C₄×C₂₆₉	C₄×D₄	1	2	yes	—
55	C₄×C₂₆₉	#121	1	2	yes	—
56	Dic₂₆₉	—	4	4	no	—
57	C₂×C₂₆₉	#390	1	1	yes	characteristic
58	C₂×C₂₆₉	C₂×#103	1	2	yes	—
59	C₂×C₂₆₉	#487	1	2	yes	—
60	C₂×C₂₆₉	#492	1	1	yes	characteristic
61	C₂×C₂₆₉	#494	1	1	yes	characteristic
62	C₂₆₉	#1956	1	1	yes	characteristic, complemented, Sylow
63	#1956	—	269	1	no	maximal, Sylow
64	C₂×C₄⋊Q₈	—	269	1	no	—
65	C₂×#103	—	269	4	no	—
66	C₂×C₄×C₈	—	269	1	no	—
67	#384	D₂₆₉	1	1	yes	characteristic
68	C₂²×Q₈	—	269	2	no	—
69	C₂²×C₈	—	269	2	no	—
70	C₄⋊Q₈	—	269	4	no	—
71	C₄⋊Q₈	—	538	2	no	—
72	C₂×C₄⋊C₄	—	2	2	no	—
73	C₂×C₄⋊C₄	—	538	2	no	—
74	C₂×C₄⋊C₄	C₂×D₂₆₉	1	2	yes	—
75	C₂×C₄²	C₂×D₂₆₉	1	1	yes	characteristic
76	#103	—	538	8	no	—
77	C₄×C₈	—	538	2	no	—
78	C₂×Q₈	—	269	4	no	—
79	C₂×Q₈	—	538	2	no	—
80	C₂×Q₈	—	538	4	no	—
81	C₂×Q₈	—	538	4	no	—
82	C₂²×C₄	—	2	2	no	—
83	C₂²×C₄	—	538	2	no	—
84	C₂²×C₄	C₂²⋊D₂₆₉	1	2	yes	—
85	C₂²×C₄	C₂²×D₂₆₉	1	1	yes	characteristic
86	C₂×C₈	—	269	4	no	—
87	C₂×C₈	—	538	2	no	—
88	C₂×C₈	—	538	2	no	—
89	C₄⋊C₄	—	2	4	no	—
90	C₄⋊C₄	—	4	4	no	—
91	C₄⋊C₄	—	538	8	no	—
92	C₄²	C₄×D₂₆₉	1	2	yes	—
93	C₄²	D₁₀₇₆	1	2	yes	—
94	C₂³	C₂×C₂²⋊D₂₆₉	1	1	yes	characteristic
95	Q₈	—	538	8	no	—
96	Q₈	—	1076	4	no	—
97	C₂×C₄	—	4	2	no	—
98	C₂×C₄	—	4	4	no	—
99	C₂×C₄	—	538	4	no	—
100	C₂×C₄	—	1076	2	no	—
101	C₂×C₄	—	1076	2	no	—
102	C₂×C₄	#16864379	1	4	yes	—
103	C₂×C₄	#16864390	1	4	yes	—
104	C₂×C₄	#16864392	1	4	yes	—
105	C₂×C₄	C₂×C₄×D₂₆₉	1	1	yes	characteristic
106	C₂×C₄	D₄×D₂₆₉	1	2	yes	—
107	C₂×C₄	C₂×D₁₀₇₆	1	1	yes	characteristic
108	C₂×C₄	#16864403	1	2	yes	—
109	C₈	—	538	4	no	—
110	C₂²	C₂×#16864379	1	1	yes	characteristic
111	C₂²	#22575442	1	1	yes	characteristic
112	C₂²	#22575449	1	1	yes	characteristic
113	C₂²	C₂×#16864390	1	2	yes	—
114	C₂²	C₂×#16864392	1	2	yes	—
115	C₄	—	8	2	no	—
116	C₄	—	1076	4	no	—
117	C₄	#22575376	1	8	yes	complemented
118	C₄	#22575415	1	2	yes	—
119	C₄	#22575428	1	2	yes	—
120	C₂	#31770506	1	1	yes	characteristic
121	C₂	C₂×#22575376	1	2	yes	—
122	C₂	#31770957	1	2	yes	—
123	C₂	#31770977	1	1	yes	characteristic
124	C₂	#31770984	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#22575453

C₂

yes

maximal, characteristic

C₂×#22575376

C₂

yes

maximal

C₂×C₄×#14524130

C₂

yes

maximal, characteristic

#384×C₂₆₉

C₂

yes

maximal, characteristic

C₂²×Dic₅₃₈

C₂²

yes

—

C₂²×#14524130

C₂²

yes

—

#22575453

—

#22575453

C₄

yes

complemented

C₂×C₄⋊Dic₂₆₉

—

#22575376

—

C₄×#14524130

—

C₂×C₄⋊C₄×C₂₆₉

—

C₂×C₄⋊C₄×C₂₆₉

C₂²

yes

—

C₂×C₄²×C₂₆₉

C₂²

yes

characteristic

C₂×Dic₅₃₈

—

C₂×Dic₅₃₈

—

C₂×Dic₅₃₈

—

C₂×Dic₅₃₈

C₂×C₄

yes

—

C₂²×Dic₂₆₉

—

C₂×#14524130

—

C₂×#14524130

—

C₂×#14524130

D₄

yes

—

C₄⋊Dic₂₆₉

—

C₂²×C₄×C₂₆₉

—

C₂²×C₄×C₂₆₉

D₄

yes

—

C₂²×C₄×C₂₆₉

C₂³

yes

characteristic

C₄⋊C₄×C₂₆₉

—

C₄⋊C₄×C₂₆₉

—

C₄²×C₂₆₉

C₂×C₄

yes

—

C₄²×C₂₆₉

D₄

yes

—

Dic₅₃₈

—

Dic₅₃₈

—

C₂×Dic₂₆₉

—

C₂×Dic₂₆₉

—

C₂×Dic₂₆₉

—

#14524130

—

C₂³×C₂₆₉

C₂×D₄

yes

characteristic

C₂×C₄×C₂₆₉

—

C₂×C₄×C₂₆₉

—

C₂×C₄×C₂₆₉

C₂²⋊C₄

yes

—

C₂×C₄×C₂₆₉

SD₁₆

yes

—

C₂×C₄×C₂₆₉

Q₁₆

yes

—

C₂×C₄×C₂₆₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₆₉

C₂×D₄

yes

—

C₂×C₄×C₂₆₉

C₂×D₄

yes

characteristic

C₂×C₄×C₂₆₉

#41

yes

—

C₂²×C₂₆₉

C₂×C₂²⋊C₄

yes

C₂²×C₂₆₉

yes

C₂²×C₂₆₉

yes

C₂²×C₂₆₉

C₂×SD₁₆

yes

—

C₂²×C₂₆₉

C₂×Q₁₆

yes

—

C₄×C₂₆₉

—

C₄×C₂₆₉

#103

yes

complemented

C₄×C₂₆₉

C₄×D₄

yes

—

C₄×C₂₆₉

#121

yes

—

Dic₂₆₉

—

C₂×C₂₆₉

#390

yes

characteristic

C₂×C₂₆₉

C₂×#103

yes

—

C₂×C₂₆₉

#487

yes

—

C₂×C₂₆₉

yes

C₂×C₂₆₉

yes

C₂₆₉

yes

characteristic, complemented, Sylow

#1956

—

269

maximal, Sylow

C₂×C₄⋊Q₈

—

269

—

C₂×#103

—

269

—

C₂×C₄×C₈

—

269

—

#384

D₂₆₉

yes

characteristic

C₂²×Q₈

—

269

—

C₂²×C₈

—

269

—

C₄⋊Q₈

—

269

—

C₄⋊Q₈

—

538

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

538

—

C₂×C₄⋊C₄

C₂×D₂₆₉

yes

—

C₂×C₄²

C₂×D₂₆₉

yes

characteristic

#103

—

538

—

C₄×C₈

—

538

—

C₂×Q₈

—

269

—

C₂×Q₈

—

538

—

C₂×Q₈

—

538

—

C₂×Q₈

—

538

—

C₂²×C₄

—

C₂²×C₄

—

538

—

C₂²×C₄

C₂²⋊D₂₆₉

yes

—

C₂²×C₄

C₂²×D₂₆₉

yes

characteristic

C₂×C₈

—

269

—

C₂×C₈

—

538

—

C₂×C₈

—

538

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

538

—

C₄²

C₄×D₂₆₉

yes

—

C₄²

D₁₀₇₆

yes

—

C₂³

C₂×C₂²⋊D₂₆₉

yes

characteristic

Q₈

—

538

—

Q₈

—

1076

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

538

—

100

C₂×C₄

—

1076

—

101

C₂×C₄

—

1076

—

102

C₂×C₄

#16864379

yes

—

103

C₂×C₄

#16864390

yes

—

104

C₂×C₄

#16864392

yes

—

105

C₂×C₄

C₂×C₄×D₂₆₉

yes

characteristic

106

C₂×C₄

D₄×D₂₆₉

yes

—

107

C₂×C₄

C₂×D₁₀₇₆

yes

characteristic

108

C₂×C₄

#16864403

yes

—

109

C₈

—

538

—

110

C₂²

C₂×#16864379

yes

characteristic

111

C₂²

#22575442

yes

characteristic

112

C₂²

#22575449

yes

characteristic

113

C₂²

C₂×#16864390

yes

—

114

C₂²

C₂×#16864392

yes

—

115

C₄

—

116

C₄

—

1076

—

117

C₄

#22575376

yes

complemented

118

C₄

#22575415

yes

—

119

C₄

#22575428

yes

—

120

C₂

#31770506

yes

characteristic

121

C₂

C₂×#22575376

yes

—

122

C₂

#31770957

yes

—

123

C₂

#31770977

yes

characteristic

124

C₂

#31770984

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₂₆₉
commutator	C₂×C₄×C₂₆₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₆₉
Fitting	#384×C₂₆₉	C₂
socle, prime radical	C₂³×C₂₆₉	C₂×D₄
cosocle	C₂²×C₄×C₂₆₉	C₂³
hypercenter	#384	D₂₆₉
hypocenter	C₂₆₉	#1956
LCS	G ▷ C₂×C₄×C₂₆₉ ▷ C₂×C₂₆₉ ▷ C₂₆₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #384

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₂₆₉

commutator

C₂×C₄×C₂₆₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₂₆₉

Fitting

#384×C₂₆₉

C₂

socle, prime radical

C₂³×C₂₆₉

C₂×D₄

cosocle

C₂²×C₄×C₂₆₉

C₂³

D₂₆₉

C₂₆₉

G ▷ C₂×C₄×C₂₆₉ ▷ C₂×C₂₆₉ ▷ C₂₆₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #384

N	H ≅ G/N	type
8. #22575453	C₄	semidirect
53. C₄×C₂₆₉	#103	semidirect
62. C₂₆₉	#1956	semidirect
117. C₄	#22575376	semidirect

H ≅ G/N

type

8. #22575453

C₄

semidirect

53. C₄×C₂₆₉

#103

semidirect

62. C₂₆₉

#1956

semidirect

117. C₄

#22575376

semidirect

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

subgroup	index
C₂×#22575453	2
C₂×#22575376	2
C₂×C₄×#14524130	2
#384×C₂₆₉	2
#1956	269

subgroup

index

#384×C₂₆₉

269

Subgroups — #46811686

Conjugacy classes of subgroups

Complements

Maximal subgroups