Group #28361821 — subgroups

Subgroups — #28361821

Order 13952.

158 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 158 conjugacy classes of subgroups — too many to draw. Only its 85 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	#20295079	C₂	1	1	yes	maximal, characteristic
2	#20295050	C₂	1	1	yes	maximal, characteristic
3	#20294779	C₂	1	2	yes	maximal, complemented
4	#20294777	C₂	1	2	yes	maximal, complemented
5	#431×C₁₀₉	C₂	1	1	yes	maximal, characteristic
6	C₂×#13182013	—	2	1	no	—
7	C₄⋊#11724784	—	2	1	no	—
8	C₄⋊#11724784	C₂²	1	1	yes	characteristic
9	C₄⋊#11724784	C₂²	1	2	yes	—
10	C₄⋊#11724784	C₂²	1	1	yes	characteristic
11	C₂²⋊#11724784	—	2	1	no	—
12	C₄×#11724784	—	2	1	no	—
13	C₂×C₄²×C₁₀₉	C₂²	1	1	yes	characteristic
14	C₄⋊C₈×C₁₀₉	C₄	1	2	yes	—
15	C₂²⋊C₈×C₁₀₉	C₄	1	2	yes	—
16	C₄×C₈×C₁₀₉	C₂²	1	2	yes	—
17	#13182013	—	4	2	no	—
18	C₂×#11724784	—	2	1	no	—
19	C₂×#11724784	—	2	1	no	—
20	C₂×#11724784	—	2	2	no	—
21	C₂²×C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₄×C₁₀₉	D₄	1	1	yes	characteristic
23	C₂²×C₄×C₁₀₉	Q₈	1	1	yes	characteristic
24	C₂×C₈×C₁₀₉	C₂×C₄	1	4	yes	—
25	C₄²×C₁₀₉	C₂×C₄	1	1	yes	characteristic
26	C₄²×C₁₀₉	D₄	1	1	yes	characteristic
27	C₄²×C₁₀₉	Q₈	1	1	yes	characteristic
28	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
29	C₂³×C₁₀₉	C₄⋊C₄	1	1	yes	characteristic
30	C₂×C₄×C₁₀₉	—	2	2	no	—
31	C₂×C₄×C₁₀₉	—	2	2	no	—
32	C₂×C₄×C₁₀₉	—	2	1	no	—
33	C₂×C₄×C₁₀₉	—	2	1	no	—
34	C₂×C₄×C₁₀₉	C₄⋊C₄	1	1	yes	characteristic
35	C₂×C₄×C₁₀₉	C₄⋊C₄	1	1	yes	characteristic
36	C₂×C₄×C₁₀₉	C₄⋊C₄	1	1	yes	characteristic
37	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
39	C₂×C₄×C₁₀₉	C₂×Q₈	1	1	yes	characteristic
40	C₈×C₁₀₉	—	2	4	no	—
41	#11724784	—	4	1	no	—
42	#11724784	—	4	2	no	—
43	#11724784	—	4	1	no	—
44	C₂²×C₁₀₉	—	2	1	no	—
45	C₂²×C₁₀₉	—	2	1	no	—
46	C₂²×C₁₀₉	—	2	1	no	—
47	C₂²×C₁₀₉	C₂×C₄⋊C₄	1	1	yes	characteristic
48	C₄×C₁₀₉	—	2	1	no	—
49	C₄×C₁₀₉	—	2	1	no	—
50	C₄×C₁₀₉	—	2	1	no	—
51	C₄×C₁₀₉	#108	1	2	yes	—
52	C₄×C₁₀₉	#131	1	2	yes	—
53	C₄×C₁₀₉	AGL(1,ℤ/8)	1	1	yes	characteristic
54	C₄×C₁₀₉	#137	1	1	yes	characteristic
55	C₂×C₁₀₉	—	4	1	no	—
56	C₂×C₁₀₉	#424	1	1	yes	characteristic
57	C₂×C₁₀₉	#428	1	1	yes	characteristic
58	C₂×C₁₀₉	C₂×#108	1	1	yes	characteristic
59	#1574	—	109	1	no	maximal, Sylow
60	C₁₀₉	#1574	1	1	yes	characteristic, complemented, Sylow
61	#431	—	109	1	no	—
62	#431	D₁₀₉	1	1	yes	characteristic
63	#423	—	109	1	no	—
64	#335	—	109	2	no	—
65	#334	—	109	2	no	—
66	C₂×M₁₆	—	218	1	no	—
67	C₂×C₄²	C₂×D₁₀₉	1	1	yes	characteristic
68	C₄⋊C₈	—	109	1	no	—
69	C₄⋊C₈	—	109	2	no	—
70	C₄⋊C₈	—	109	1	no	—
71	C₄⋊C₈	—	218	1	no	—
72	C₄⋊C₈	Dic₁₀₉	1	2	yes	—
73	C₂²⋊C₈	—	218	1	no	—
74	C₂²⋊C₈	Dic₁₀₉	1	2	yes	—
75	C₄×C₈	—	218	1	no	—
76	C₄×C₈	C₂×D₁₀₉	1	2	yes	—
77	C₂²×C₄	Dic₂₁₈	1	1	yes	characteristic
78	C₂²×C₄	D₄₃₆	1	1	yes	characteristic
79	C₂²×C₄	C₂×Dic₁₀₉	1	1	yes	characteristic
80	M₁₆	—	436	2	no	—
81	C₂×C₈	—	218	1	no	—
82	C₂×C₈	—	218	1	no	—
83	C₂×C₈	—	218	2	no	—
84	C₂×C₈	C₂×Dic₁₀₉	1	4	yes	—
85	C₄²	Dic₂₁₈	1	1	yes	characteristic
86	C₄²	D₄₃₆	1	1	yes	characteristic
87	C₄²	C₂×Dic₁₀₉	1	1	yes	characteristic
88	C₄²	C₂²×D₁₀₉	1	1	yes	characteristic
89	C₂³	C₄⋊Dic₁₀₉	1	1	yes	characteristic
90	C₂×C₄	—	2	2	no	—
91	C₂×C₄	—	2	2	no	—
92	C₂×C₄	—	2	1	no	—
93	C₂×C₄	—	2	1	no	—
94	C₂×C₄	C₄⋊Dic₁₀₉	1	1	yes	characteristic
95	C₂×C₄	C₄⋊Dic₁₀₉	1	1	yes	characteristic
96	C₂×C₄	C₄⋊Dic₁₀₉	1	1	yes	characteristic
97	C₂×C₄	C₂²×Dic₁₀₉	1	1	yes	characteristic
98	C₂×C₄	C₂×D₄₃₆	1	1	yes	characteristic
99	C₂×C₄	C₂×Dic₂₁₈	1	1	yes	characteristic
100	C₈	—	2	4	no	—
101	C₈	—	436	2	no	—
102	C₈	—	436	1	no	—
103	C₈	—	436	1	no	—
104	C₂²	—	2	1	no	—
105	C₂²	—	2	1	no	—
106	C₂²	—	2	1	no	—
107	C₂²	C₂×C₄⋊Dic₁₀₉	1	1	yes	characteristic
108	C₄	—	2	1	no	—
109	C₄	—	2	1	no	—
110	C₄	—	2	1	no	—
111	C₄	#16812635	1	2	yes	—
112	C₄	#16812706	1	2	yes	—
113	C₄	#16812727	1	1	yes	characteristic
114	C₄	#16812732	1	1	yes	characteristic
115	C₂	—	4	1	no	—
116	C₂	#20295053	1	1	yes	characteristic
117	C₂	#20295068	1	1	yes	characteristic
118	C₂	C₂×#16812635	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20295079

C₂

yes

maximal, characteristic

#20295050

C₂

yes

maximal, characteristic

#20294779

C₂

yes

maximal, complemented

#20294777

C₂

yes

maximal, complemented

#431×C₁₀₉

C₂

yes

maximal, characteristic

C₂×#13182013

—

C₄⋊#11724784

—

C₄⋊#11724784

C₂²

yes

characteristic

C₄⋊#11724784

C₂²

yes

—

C₄⋊#11724784

C₂²

yes

characteristic

C₂²⋊#11724784

—

C₄×#11724784

—

C₂×C₄²×C₁₀₉

C₂²

yes

characteristic

C₄⋊C₈×C₁₀₉

C₄

yes

—

C₂²⋊C₈×C₁₀₉

C₄

yes

—

C₄×C₈×C₁₀₉

C₂²

yes

—

—

—

—

—

C₂²×C₄×C₁₀₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₀₉

D₄

yes

characteristic

C₂²×C₄×C₁₀₉

Q₈

yes

characteristic

C₂×C₈×C₁₀₉

C₂×C₄

yes

—

C₄²×C₁₀₉

C₂×C₄

yes

characteristic

C₄²×C₁₀₉

D₄

yes

characteristic

C₄²×C₁₀₉

Q₈

yes

characteristic

C₄²×C₁₀₉

C₂³

yes

characteristic

C₂³×C₁₀₉

C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₀₉

C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₀₉

C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂×Q₈

yes

characteristic

C₈×C₁₀₉

—

#11724784

—

#11724784

—

#11724784

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

C₂×C₄⋊C₄

yes

characteristic

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

#108

yes

—

C₄×C₁₀₉

#131

yes

—

C₄×C₁₀₉

yes

C₄×C₁₀₉

yes

C₂×C₁₀₉

—

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

—

109

maximal, Sylow

C₁₀₉

#1574

yes

characteristic, complemented, Sylow

#431

—

109

—

#431

D₁₀₉

yes

characteristic

#423

—

109

—

#335

—

109

—

#334

—

109

—

C₂×M₁₆

—

218

—

C₂×C₄²

C₂×D₁₀₉

yes

characteristic

C₄⋊C₈

—

109

—

C₄⋊C₈

—

109

—

C₄⋊C₈

—

109

—

C₄⋊C₈

—

218

—

C₄⋊C₈

Dic₁₀₉

yes

—

C₂²⋊C₈

—

218

—

C₂²⋊C₈

Dic₁₀₉

yes

—

C₄×C₈

—

218

—

C₄×C₈

C₂×D₁₀₉

yes

—

C₂²×C₄

Dic₂₁₈

yes

characteristic

C₂²×C₄

D₄₃₆

yes

characteristic

C₂²×C₄

C₂×Dic₁₀₉

yes

characteristic

M₁₆

—

436

—

C₂×C₈

—

218

—

C₂×C₈

—

218

—

C₂×C₈

—

218

—

C₂×C₈

C₂×Dic₁₀₉

yes

—

C₄²

Dic₂₁₈

yes

characteristic

C₄²

D₄₃₆

yes

characteristic

C₄²

C₂×Dic₁₀₉

yes

characteristic

C₄²

C₂²×D₁₀₉

yes

characteristic

C₂³

C₄⋊Dic₁₀₉

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₄⋊Dic₁₀₉

yes

characteristic

C₂×C₄

C₄⋊Dic₁₀₉

yes

characteristic

C₂×C₄

C₄⋊Dic₁₀₉

yes

characteristic

C₂×C₄

C₂²×Dic₁₀₉

yes

characteristic

C₂×C₄

C₂×D₄₃₆

yes

characteristic

C₂×C₄

C₂×Dic₂₁₈

yes

characteristic

100

C₈

—

101

C₈

—

436

—

102

C₈

—

436

—

103

C₈

—

436

—

104

C₂²

—

105

C₂²

—

106

C₂²

—

107

C₂²

C₂×C₄⋊Dic₁₀₉

yes

characteristic

108

C₄

—

109

C₄

—

110

C₄

—

111

C₄

#16812635

yes

—

112

C₄

#16812706

yes

—

113

C₄

#16812727

yes

characteristic

114

C₄

#16812732

yes

characteristic

115

C₂

—

116

C₂

#20295053

yes

characteristic

117

C₂

#20295068

yes

characteristic

118

C₂

C₂×#16812635

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄₃₆
commutator	C₂×C₄×C₁₀₉	C₂²×C₄
Frattini	C₄²	C₂²×D₁₀₉
Fitting	#431×C₁₀₉	C₂
socle	C₂²×C₁₀₉	C₂×C₄⋊C₄
prime radical	C₂³×C₁₀₉	C₄⋊C₄
cosocle	C₄²×C₁₀₉	C₂³
hypercenter	#431	D₁₀₉
hypocenter	C₁₀₉	#1574
LCS	G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #431

subgroup

G/H

center

C₂×C₄

C₂×D₄₃₆

commutator

C₂×C₄×C₁₀₉

C₂²×C₄

Frattini

C₄²

C₂²×D₁₀₉

Fitting

#431×C₁₀₉

C₂

socle

C₂²×C₁₀₉

C₂×C₄⋊C₄

prime radical

C₂³×C₁₀₉

C₄⋊C₄

cosocle

C₄²×C₁₀₉

C₂³

D₁₀₉

C₁₀₉

G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #431

N	H ≅ G/N	type
3. #20294779	C₂	semidirect
4. #20294777	C₂	semidirect
60. C₁₀₉	#1574	semidirect

H ≅ G/N

type

3. #20294779

C₂

semidirect

4. #20294777

C₂

semidirect

60. C₁₀₉

#1574

semidirect

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

subgroup	index
#20295079	2
#20295050	2
#20294779	2
#20294777	2
#431×C₁₀₉	2
#1574	109

subgroup

index

#431×C₁₀₉

109

Subgroups — #28361821

Conjugacy classes of subgroups

Complements

Maximal subgroups