Group #28362175 — subgroups

Subgroups — #28362175

Order 13952.

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

normal subgroups only

This group has 218 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	#20295742	C₂	1	1	yes	maximal, characteristic
2	#20295313	C₂	1	2	yes	maximal, complemented
3	#20294775	C₂	1	1	yes	maximal, characteristic, complemented
4	#20294755	C₂	1	2	yes	maximal, complemented
5	#494×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
6	#16812688	—	2	2	no	—
7	#16812674	—	2	2	no	—
8	C₄×D₄₃₆	C₂²	1	2	yes	—
9	#16812657	—	2	2	no	—
10	#16812630	—	2	2	no	—
11	C₄⋊#11724784	C₂²	1	2	yes	—
12	#16812619	—	2	2	no	—
13	C₂×Q₁₆×C₁₀₉	—	2	2	no	—
14	C₄⋊Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
15	C₄⋊Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
16	C₄×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
17	C₂×D₄₃₆	—	2	2	no	—
18	C₂×D₄₃₆	D₄	1	2	yes	—
19	C₂×C₄×D₁₀₉	—	2	2	no	—
20	C₂×#11724784	—	2	2	no	—
21	C₄⋊Dic₁₀₉	D₄	1	2	yes	—
22	#13182006	—	2	4	no	—
23	#13182006	—	2	2	no	—
24	C₄×Dic₁₀₉	—	2	2	no	—
25	C₂×Q₈×C₁₀₉	—	2	1	no	—
26	C₂×Q₈×C₁₀₉	D₄	1	2	yes	—
27	Q₁₆×C₁₀₉	—	4	4	no	—
28	C₂×C₈×C₁₀₉	—	2	1	no	—
29	C₄⋊C₄×C₁₀₉	—	2	2	no	—
30	C₄⋊C₄×C₁₀₉	—	2	2	no	—
31	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
32	C₂²×D₁₀₉	—	2	2	no	—
33	Q₈×C₁₀₉	—	4	2	no	—
34	Q₈×C₁₀₉	—	4	2	no	—
35	C₂×C₄×C₁₀₉	—	2	2	no	—
36	C₂×C₄×C₁₀₉	—	4	1	no	—
37	C₂×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
38	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
39	C₂×Dic₁₀₉	—	2	2	no	—
40	D₄₃₆	—	2	2	no	—
41	D₄₃₆	—	2	2	no	—
42	D₄₃₆	—	8	2	no	—
43	C₄×D₁₀₉	—	4	2	no	—
44	C₄×D₁₀₉	—	4	2	no	—
45	C₈×C₁₀₉	—	2	2	no	—
46	#11724784	—	4	2	no	—
47	C₂²×C₁₀₉	#120	1	1	yes	characteristic
48	C₂×D₁₀₉	—	4	2	no	—
49	C₂×D₁₀₉	—	4	2	no	—
50	C₂×D₁₀₉	—	4	2	no	—
51	C₄×C₁₀₉	—	2	1	no	—
52	C₄×C₁₀₉	—	4	2	no	—
53	C₄×C₁₀₉	—	8	1	no	—
54	C₄×C₁₀₉	#135	1	2	yes	—
55	C₄×C₁₀₉	#137	1	2	yes	—
56	Dic₁₀₉	—	4	2	no	—
57	C₂×C₁₀₉	#452	1	2	yes	—
58	C₂×C₁₀₉	#453	1	1	yes	characteristic
59	D₁₀₉	—	8	2	no	—
60	#1634	—	109	1	no	maximal, Sylow
61	C₁₀₉	#1634	1	1	yes	characteristic, complemented, Sylow
62	#555	—	109	1	no	—
63	#494	D₁₀₉	1	1	yes	characteristic, complemented
64	#476	—	109	2	no	—
65	#333	—	109	1	no	—
66	D₄⋊C₈	—	109	2	no	—
67	C₂×Q₁₆	—	2	2	no	—
68	C₄⋊Q₈	C₂×D₁₀₉	1	1	yes	characteristic
69	C₄⋊Q₈	C₂×D₁₀₉	1	1	yes	characteristic
70	#124	—	218	2	no	—
71	#121	—	218	2	no	—
72	C₄×D₄	—	109	2	no	—
73	#117	—	218	2	no	—
74	#106	—	218	2	no	—
75	C₄⋊C₈	—	109	2	no	—
76	#102	—	218	2	no	—
77	C₄×C₈	C₂×D₁₀₉	1	1	yes	characteristic
78	C₂×Q₈	—	2	1	no	—
79	C₂×Q₈	C₂²⋊D₁₀₉	1	2	yes	—
80	C₂×D₄	—	109	2	no	—
81	C₂×D₄	—	218	2	no	—
82	C₂²×C₄	—	218	2	no	—
83	Q₁₆	—	4	4	no	—
84	C₂×C₈	—	2	1	no	—
85	C₂×C₈	—	218	2	no	—
86	C₄⋊C₄	—	2	2	no	—
87	C₄⋊C₄	—	2	2	no	—
88	C₄⋊C₄	—	109	2	no	—
89	C₂²⋊C₄	—	218	4	no	—
90	C₂²⋊C₄	—	218	2	no	—
91	C₄²	—	218	2	no	—
92	C₄²	C₂²×D₁₀₉	1	1	yes	characteristic
93	C₂³	—	218	2	no	—
94	Q₈	—	4	2	no	—
95	Q₈	—	4	2	no	—
96	D₄	—	872	2	no	—
97	D₄	—	218	2	no	—
98	D₄	—	218	2	no	—
99	C₂×C₄	—	2	2	no	—
100	C₂×C₄	—	4	1	no	—
101	C₂×C₄	—	436	2	no	—
102	C₂×C₄	—	436	2	no	—
103	C₂×C₄	—	218	2	no	—
104	C₂×C₄	D₄×D₁₀₉	1	2	yes	—
105	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
106	C₈	—	2	2	no	—
107	C₈	—	436	2	no	—
108	C₂²	—	436	2	no	—
109	C₂²	—	436	2	no	—
110	C₂²	—	436	2	no	—
111	C₂²	#16812667	1	1	yes	characteristic
112	C₄	—	2	1	no	—
113	C₄	—	4	2	no	—
114	C₄	—	8	1	no	—
115	C₄	—	436	2	no	—
116	C₄	#16812722	1	2	yes	—
117	C₄	#16812731	1	2	yes	—
118	C₂	—	872	2	no	—
119	C₂	#20295187	1	2	yes	—
120	C₂	#20295193	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20295742

C₂

yes

maximal, characteristic

#20295313

C₂

yes

maximal, complemented

#20294775

C₂

yes

maximal, characteristic, complemented

#20294755

C₂

yes

maximal, complemented

#494×C₁₀₉

C₂

yes

maximal, characteristic, complemented

#16812688

—

#16812674

—

C₄×D₄₃₆

C₂²

yes

—

#16812657

—

#16812630

—

C₄⋊#11724784

C₂²

yes

—

#16812619

—

C₂×Q₁₆×C₁₀₉

—

C₄⋊Q₈×C₁₀₉

C₂²

yes

characteristic

C₄⋊Q₈×C₁₀₉

C₂²

yes

characteristic

C₄×C₈×C₁₀₉

C₂²

yes

characteristic

C₂×D₄₃₆

—

C₂×D₄₃₆

D₄

yes

—

C₂×C₄×D₁₀₉

—

C₂×#11724784

—

C₄⋊Dic₁₀₉

D₄

yes

—

#13182006

—

#13182006

—

C₄×Dic₁₀₉

—

C₂×Q₈×C₁₀₉

—

C₂×Q₈×C₁₀₉

D₄

yes

—

Q₁₆×C₁₀₉

—

C₂×C₈×C₁₀₉

—

C₄⋊C₄×C₁₀₉

—

C₄⋊C₄×C₁₀₉

—

C₄²×C₁₀₉

C₂³

yes

characteristic

C₂²×D₁₀₉

—

Q₈×C₁₀₉

—

Q₈×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₂×D₄

yes

—

C₂×C₄×C₁₀₉

C₂×D₄

yes

characteristic

C₂×Dic₁₀₉

—

D₄₃₆

—

D₄₃₆

—

D₄₃₆

—

C₄×D₁₀₉

—

C₄×D₁₀₉

—

C₈×C₁₀₉

—

#11724784

—

C₂²×C₁₀₉

#120

yes

characteristic

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

#135

yes

—

C₄×C₁₀₉

#137

yes

—

Dic₁₀₉

—

C₂×C₁₀₉

#452

yes

—

C₂×C₁₀₉

#453

yes

characteristic

D₁₀₉

—

#1634

—

109

maximal, Sylow

C₁₀₉

#1634

yes

characteristic, complemented, Sylow

#555

—

109

—

#494

D₁₀₉

yes

characteristic, complemented

#476

—

109

—

#333

—

109

—

D₄⋊C₈

—

109

—

C₂×Q₁₆

—

C₄⋊Q₈

C₂×D₁₀₉

yes

characteristic

C₄⋊Q₈

C₂×D₁₀₉

yes

characteristic

#124

—

218

—

#121

—

218

—

C₄×D₄

—

109

—

#117

—

218

—

#106

—

218

—

C₄⋊C₈

—

109

—

#102

—

218

—

C₄×C₈

C₂×D₁₀₉

yes

characteristic

C₂×Q₈

—

C₂×Q₈

C₂²⋊D₁₀₉

yes

—

C₂×D₄

—

109

—

C₂×D₄

—

218

—

C₂²×C₄

—

218

—

Q₁₆

—

C₂×C₈

—

C₂×C₈

—

218

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

109

—

C₂²⋊C₄

—

218

—

C₂²⋊C₄

—

218

—

C₄²

—

218

—

C₄²

C₂²×D₁₀₉

yes

characteristic

C₂³

—

218

—

Q₈

—

Q₈

—

D₄

—

872

—

D₄

—

218

—

D₄

—

218

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

436

—

102

C₂×C₄

—

436

—

103

C₂×C₄

—

218

—

104

C₂×C₄

D₄×D₁₀₉

yes

—

105

C₂×C₄

C₂×C₂²⋊D₁₀₉

yes

characteristic

106

C₈

—

107

C₈

—

436

—

108

C₂²

—

436

—

109

C₂²

—

436

—

110

C₂²

—

436

—

111

C₂²

#16812667

yes

characteristic

112

C₄

—

113

C₄

—

114

C₄

—

115

C₄

—

436

—

116

C₄

#16812722

yes

—

117

C₄

#16812731

yes

—

118

C₂

—

872

—

119

C₂

#20295187

yes

—

120

C₂

#20295193

yes

characteristic

	subgroup	G/H
center	C₂²	#16812667
commutator, cosocle	C₄²×C₁₀₉	C₂³
Frattini	C₄²	C₂²×D₁₀₉
Fitting	#494×C₁₀₉	C₂
socle	C₂²×C₁₀₉	#120
prime radical	#20295742	C₂
hypercenter	#494	D₁₀₉
hypocenter	C₁₀₉	#1634
LCS	G ▷ C₄²×C₁₀₉ ▷ C₂²×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #494

subgroup

G/H

center

C₂²

#16812667

commutator, cosocle

C₄²×C₁₀₉

C₂³

Frattini

C₄²

C₂²×D₁₀₉

Fitting

#494×C₁₀₉

C₂

C₂²×C₁₀₉

C₂

D₁₀₉

C₁₀₉

G ▷ C₄²×C₁₀₉ ▷ C₂²×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #494

N	H ≅ G/N	type
2. #20295313	C₂	semidirect
3. #20294775	C₂	semidirect
4. #20294755	C₂	semidirect
5. #494×C₁₀₉	C₂	semidirect
61. C₁₀₉	#1634	semidirect
63. #494	D₁₀₉	semidirect

H ≅ G/N

type

2. #20295313

C₂

semidirect

3. #20294775

C₂

semidirect

4. #20294755

C₂

semidirect

5. #494×C₁₀₉

C₂

semidirect

61. C₁₀₉

#1634

semidirect

63. #494

D₁₀₉

semidirect

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

subgroup	index
#20295742	2
#20295313	2
#20294775	2
#20294755	2
#494×C₁₀₉	2
#1634	109

subgroup

index

#494×C₁₀₉

109

Subgroups — #28362175

Conjugacy classes of subgroups

Complements

Maximal subgroups