Group #28374836 — subgroups

Subgroups — #28374836

Order 13952.

444 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 117 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	#20295839	C₂	1	4	yes	maximal, complemented
2	#20295298	C₂	1	4	yes	maximal, complemented
3	#20295208	C₂	1	4	yes	maximal, complemented
4	#20295144	C₂	1	2	yes	maximal, complemented
5	#494×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
6	#16812747	C₂²	1	4	yes	—
7	#16812731	—	2	8	no	—
8	#16812730	C₂²	1	8	yes	—
9	#16812722	—	2	4	no	—
10	#16812721	C₂²	1	2	yes	—
11	Q₁₆×D₁₀₉	—	2	4	no	—
12	#16812705	C₂²	1	4	yes	—
13	#16812635	C₂²	1	2	yes	—
14	#16812627	C₂²	1	4	yes	—
15	#16812618	C₂²	1	4	yes	—
16	C₂×Q₁₆×C₁₀₉	C₂²	1	4	yes	—
17	C₄⋊Q₈×C₁₀₉	C₂²	1	2	yes	—
18	C₄×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
19	#13182030	—	2	8	no	—
20	#13182029	—	4	4	no	—
21	#13182029	C₂³	1	4	yes	—
22	Q₈×D₁₀₉	—	2	8	no	—
23	Dic₄₃₆	D₄	1	2	yes	—
24	#13182019	—	2	8	no	—
25	#13182018	D₄	1	4	yes	—
26	#13182016	—	2	8	no	—
27	D₈₇₂	D₄	1	2	yes	—
28	#13182013	C₂³	1	4	yes	—
29	#13182012	—	2	4	no	—
30	C₈×D₁₀₉	—	2	4	no	—
31	C₂×Q₈×C₁₀₉	C₂³	1	4	yes	—
32	Q₁₆×C₁₀₉	—	2	4	no	—
33	Q₁₆×C₁₀₉	—	2	4	no	—
34	C₂×C₈×C₁₀₉	C₂³	1	2	yes	—
35	C₄⋊C₄×C₁₀₉	—	4	2	no	—
36	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
37	Q₈×C₁₀₉	—	2	8	no	—
38	C₂×C₄×C₁₀₉	—	2	1	no	—
39	C₂×C₄×C₁₀₉	—	2	4	no	—
40	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
41	C₂²⋊D₁₀₉	—	2	4	no	—
42	D₄₃₆	—	4	4	no	—
43	D₄₃₆	C₂×D₄	1	4	yes	—
44	C₄×D₁₀₉	—	2	4	no	—
45	C₄×D₁₀₉	—	4	8	no	—
46	Dic₂₁₈	—	4	4	no	—
47	Dic₂₁₈	C₂×D₄	1	4	yes	—
48	C₈×C₁₀₉	C₂×D₄	1	2	yes	—
49	C₈×C₁₀₉	C₂×D₄	1	2	yes	—
50	#11724784	—	2	4	no	—
51	C₂²×C₁₀₉	H(5,𝔽₂)	1	1	yes	characteristic
52	C₂×D₁₀₉	—	2	4	no	—
53	C₄×C₁₀₉	—	2	2	no	—
54	C₄×C₁₀₉	—	4	4	no	—
55	C₄×C₁₀₉	C₂²×D₄	1	2	yes	—
56	Dic₁₀₉	—	2	4	no	—
57	C₂×C₁₀₉	—	2	1	no	—
58	C₂×C₁₀₉	D₄²	1	1	yes	characteristic
59	D₁₀₉	—	4	4	no	—
60	#3293	—	109	1	no	maximal, Sylow
61	C₁₀₉	#3293	1	1	yes	characteristic, complemented, Sylow
62	#578	—	109	4	no	—
63	#494	D₁₀₉	1	1	yes	characteristic, complemented
64	#473	—	109	4	no	—
65	#456	—	109	4	no	—
66	#443	—	109	2	no	—
67	2₋¹⁺⁴	—	109	4	no	—
68	#137	—	109	8	no	—
69	#137	—	218	8	no	—
70	#135	—	109	2	no	—
71	#135	—	218	4	no	—
72	C₂×Q₁₆	—	218	4	no	—
73	C₂×Q₁₆	C₂×D₁₀₉	1	4	yes	—
74	#131	—	109	4	no	—
75	C₄⋊Q₈	C₂×D₁₀₉	1	2	yes	—
76	#108	—	109	2	no	—
77	#104	—	109	4	no	—
78	#101	—	109	4	no	—
79	C₄×C₈	C₂×D₁₀₉	1	1	yes	characteristic
80	#41	—	109	4	no	—
81	#41	—	436	4	no	—
82	#41	—	218	8	no	—
83	C₂×Q₈	—	218	8	no	—
84	C₂×Q₈	C₂²×D₁₀₉	1	4	yes	—
85	Q₁₆	—	2	4	no	—
86	Q₁₆	—	2	4	no	—
87	Q₁₆	—	109	2	no	—
88	Q₁₆	—	218	8	no	—
89	SD₁₆	—	109	4	no	—
90	SD₁₆	—	218	8	no	—
91	D₈	—	109	2	no	—
92	M₁₆	—	109	4	no	—
93	M₁₆	—	218	4	no	—
94	C₂×C₈	—	218	4	no	—
95	C₂×C₈	C₂²×D₁₀₉	1	2	yes	—
96	C₄⋊C₄	—	4	2	no	—
97	C₄²	C₂²×D₁₀₉	1	1	yes	characteristic
98	Q₈	—	2	8	no	—
99	Q₈	—	109	4	no	—
100	Q₈	—	436	4	no	—
101	D₄	—	109	4	no	—
102	D₄	—	436	4	no	—
103	D₄	—	218	4	no	—
104	C₂×C₄	—	2	1	no	—
105	C₂×C₄	—	2	4	no	—
106	C₂×C₄	—	436	8	no	—
107	C₂×C₄	—	218	4	no	—
108	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
109	C₈	—	218	4	no	—
110	C₈	D₄×D₁₀₉	1	2	yes	—
111	C₈	D₄×D₁₀₉	1	2	yes	—
112	C₂²	—	218	4	no	—
113	C₂²	#16812744	1	1	yes	characteristic
114	C₄	—	2	2	no	—
115	C₄	—	4	4	no	—
116	C₄	—	218	4	no	—
117	C₄	C₂×D₄×D₁₀₉	1	2	yes	—
118	C₂	—	2	1	no	—
119	C₂	—	436	4	no	—
120	C₂	#20295661	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20295839

C₂

yes

maximal, complemented

#20295298

C₂

yes

maximal, complemented

#20295208

C₂

yes

maximal, complemented

#20295144

C₂

yes

maximal, complemented

#494×C₁₀₉

C₂

yes

maximal, characteristic, complemented

#16812747

C₂²

yes

—

#16812731

—

#16812730

C₂²

yes

—

#16812722

—

#16812721

C₂²

yes

—

Q₁₆×D₁₀₉

—

#16812705

C₂²

yes

—

#16812635

C₂²

yes

—

#16812627

C₂²

yes

—

#16812618

C₂²

yes

—

C₂×Q₁₆×C₁₀₉

C₂²

yes

—

C₄⋊Q₈×C₁₀₉

C₂²

yes

—

C₄×C₈×C₁₀₉

C₂²

yes

—

—

C₂³

yes

—

Q₈×D₁₀₉

—

Dic₄₃₆

D₄

yes

—

#13182019

—

#13182018

D₄

yes

—

#13182016

—

D₈₇₂

D₄

yes

—

#13182013

C₂³

yes

—

#13182012

—

C₈×D₁₀₉

—

C₂×Q₈×C₁₀₉

C₂³

yes

—

Q₁₆×C₁₀₉

—

Q₁₆×C₁₀₉

—

C₂×C₈×C₁₀₉

C₂³

yes

—

C₄⋊C₄×C₁₀₉

—

C₄²×C₁₀₉

C₂³

yes

characteristic

Q₈×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₂⁴

yes

characteristic

C₂²⋊D₁₀₉

—

D₄₃₆

—

D₄₃₆

C₂×D₄

yes

—

C₄×D₁₀₉

—

C₄×D₁₀₉

—

Dic₂₁₈

—

Dic₂₁₈

C₂×D₄

yes

—

C₈×C₁₀₉

C₂×D₄

yes

—

C₈×C₁₀₉

C₂×D₄

yes

—

#11724784

—

C₂²×C₁₀₉

H(5,𝔽₂)

yes

characteristic

C₂×D₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

C₂²×D₄

yes

—

Dic₁₀₉

—

C₂×C₁₀₉

—

C₂×C₁₀₉

D₄²

yes

characteristic

D₁₀₉

—

#3293

—

109

maximal, Sylow

C₁₀₉

#3293

yes

characteristic, complemented, Sylow

#578

—

109

—

#494

D₁₀₉

yes

characteristic, complemented

#473

—

109

—

#456

—

109

—

#443

—

109

—

2₋¹⁺⁴

—

109

—

#137

—

109

—

#137

—

218

—

#135

—

109

—

#135

—

218

—

C₂×Q₁₆

—

218

—

C₂×Q₁₆

C₂×D₁₀₉

yes

—

#131

—

109

—

C₄⋊Q₈

C₂×D₁₀₉

yes

—

#108

—

109

—

#104

—

109

—

#101

—

109

—

C₄×C₈

C₂×D₁₀₉

yes

characteristic

#41

—

109

—

#41

—

436

—

#41

—

218

—

C₂×Q₈

—

218

—

C₂×Q₈

C₂²×D₁₀₉

yes

—

Q₁₆

—

Q₁₆

—

Q₁₆

—

109

—

Q₁₆

—

218

—

SD₁₆

—

109

—

SD₁₆

—

218

—

D₈

—

109

—

M₁₆

—

109

—

M₁₆

—

218

—

C₂×C₈

—

218

—

C₂×C₈

C₂²×D₁₀₉

yes

—

C₄⋊C₄

—

C₄²

C₂²×D₁₀₉

yes

characteristic

Q₈

—

Q₈

—

109

—

100

Q₈

—

436

—

101

D₄

—

109

—

102

D₄

—

436

—

103

D₄

—

218

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

436

—

107

C₂×C₄

—

218

—

108

C₂×C₄

C₂³×D₁₀₉

yes

characteristic

109

C₈

—

218

—

110

C₈

D₄×D₁₀₉

yes

—

111

C₈

D₄×D₁₀₉

yes

—

112

C₂²

—

218

—

113

C₂²

#16812744

yes

characteristic

114

C₄

—

115

C₄

—

116

C₄

—

218

—

117

C₄

C₂×D₄×D₁₀₉

yes

—

118

C₂

—

119

C₂

—

436

—

120

C₂

#20295661

yes

characteristic

	subgroup	G/H
center	C₂	#20295661
commutator, cosocle	C₂×C₄×C₁₀₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₀₉
Fitting	#494×C₁₀₉	C₂
socle	C₂×C₁₀₉	D₄²
hypercenter	#494	D₁₀₉
hypocenter	C₁₀₉	#3293
LCS	G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #494

subgroup

G/H

center

C₂

#20295661

commutator, cosocle

C₂×C₄×C₁₀₉

C₂⁴

Frattini

C₂×C₄

C₂³×D₁₀₉

Fitting

#494×C₁₀₉

C₂

socle

C₂×C₁₀₉

D₄²

D₁₀₉

C₁₀₉

G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ #494

N	H ≅ G/N	type
1. #20295839	C₂	semidirect
2. #20295298	C₂	semidirect
3. #20295208	C₂	semidirect
4. #20295144	C₂	semidirect
5. #494×C₁₀₉	C₂	semidirect
61. C₁₀₉	#3293	semidirect
63. #494	D₁₀₉	semidirect

H ≅ G/N

type

1. #20295839

C₂

semidirect

2. #20295298

C₂

semidirect

3. #20295208

C₂

semidirect

4. #20295144

C₂

semidirect

5. #494×C₁₀₉

C₂

semidirect

61. C₁₀₉

#3293

semidirect

63. #494

D₁₀₉

semidirect

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

subgroup	index
#20295839	2
#20295298	2
#20295208	2
#20295144	2
#494×C₁₀₉	2
#3293	109

subgroup

index

#494×C₁₀₉

109

Subgroups — #28374836

Conjugacy classes of subgroups

Complements

Maximal subgroups