Group #28363198 — subgroups

Subgroups — #28363198

Order 13952.

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

normal subgroups only

This group has 256 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	C₂×#16812632	C₂	1	2	yes	maximal
2	C₂×#16812630	C₂	1	2	yes	maximal
3	C₂×#16812610	C₂	1	1	yes	maximal, characteristic
4	#20294892	C₂	1	1	yes	maximal, characteristic
5	#384×C₁₀₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11724784	C₂²	1	2	yes	—
7	C₂×C₄⋊Dic₁₀₉	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊Dic₁₀₉	C₂²	1	1	yes	characteristic
9	#16812632	—	2	4	no	—
10	#16812630	—	2	4	no	—
11	#16812610	C₄	1	4	yes	complemented
12	#16812606	—	2	2	no	—
13	C₂×C₄⋊C₄×C₁₀₉	—	2	2	no	—
14	C₂×C₄⋊C₄×C₁₀₉	C₂²	1	2	yes	—
15	C₂×C₄²×C₁₀₉	C₂²	1	1	yes	characteristic
16	C₂²×Dic₁₀₉	—	2	1	no	—
17	C₂²×Dic₁₀₉	—	2	1	no	—
18	C₂×#11724784	C₂×C₄	1	4	yes	—
19	C₂×#11724784	D₄	1	4	yes	—
20	C₂×#11724784	Q₈	1	4	yes	—
21	C₄⋊Dic₁₀₉	—	2	2	no	—
22	C₄⋊Dic₁₀₉	—	2	2	no	—
23	C₂²×C₄×C₁₀₉	—	2	2	no	—
24	C₂²×C₄×C₁₀₉	D₄	1	2	yes	—
25	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
26	C₄⋊C₄×C₁₀₉	—	2	2	no	—
27	C₄⋊C₄×C₁₀₉	—	2	2	no	—
28	C₄⋊C₄×C₁₀₉	—	4	4	no	—
29	C₄²×C₁₀₉	C₂×C₄	1	2	yes	—
30	C₄²×C₁₀₉	Q₈	1	2	yes	—
31	C₂³×C₁₀₉	C₂×D₄	1	1	yes	characteristic
32	C₂×C₄×C₁₀₉	—	2	2	no	—
33	C₂×C₄×C₁₀₉	—	2	2	no	—
34	C₂×C₄×C₁₀₉	—	4	2	no	—
35	C₂×C₄×C₁₀₉	—	4	4	no	—
36	C₂×C₄×C₁₀₉	C₄⋊C₄	1	4	yes	—
37	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
39	C₂×C₄×C₁₀₉	C₂×Q₈	1	2	yes	—
40	C₂×C₄×C₁₀₉	C₂×Q₈	1	1	yes	characteristic
41	C₂×Dic₁₀₉	—	4	1	no	—
42	C₂×Dic₁₀₉	—	4	1	no	—
43	C₂×Dic₁₀₉	—	4	2	no	—
44	C₂×Dic₁₀₉	—	4	2	no	—
45	#11724784	C₄⋊C₄	1	8	yes	complemented
46	C₂²×C₁₀₉	C₂×C₄⋊C₄	1	1	yes	characteristic
47	C₂²×C₁₀₉	#127	1	1	yes	characteristic
48	C₂²×C₁₀₉	C₄⋊Q₈	1	1	yes	characteristic
49	C₂²×C₁₀₉	AGL(1,ℤ/8)	1	2	yes	—
50	C₂²×C₁₀₉	#137	1	2	yes	—
51	C₄×C₁₀₉	—	2	4	no	—
52	C₄×C₁₀₉	—	8	2	no	—
53	C₄×C₁₀₉	C₂×C₄⋊C₄	1	2	yes	—
54	C₄×C₁₀₉	C₄⋊Q₈	1	2	yes	—
55	Dic₁₀₉	—	8	1	no	—
56	Dic₁₀₉	—	8	1	no	—
57	C₂×C₁₀₉	#384	1	1	yes	characteristic
58	C₂×C₁₀₉	#428	1	2	yes	—
59	C₂×C₁₀₉	#496	1	1	yes	characteristic
60	C₂×C₁₀₉	#497	1	1	yes	characteristic
61	C₂×C₁₀₉	#501	1	2	yes	—
62	#1847	—	109	1	no	maximal, Sylow
63	C₁₀₉	#1847	1	1	yes	characteristic, complemented, Sylow
64	C₂×#107	—	109	2	no	—
65	C₂×#106	—	109	2	no	—
66	C₂×#97	—	109	1	no	—
67	#384	D₁₀₉	1	1	yes	characteristic
68	#382	—	109	1	no	—
69	C₂²×C₈	—	109	2	no	—
70	C₂×C₄⋊C₄	—	2	2	no	—
71	C₂×C₄⋊C₄	—	109	1	no	—
72	C₂×C₄⋊C₄	—	109	1	no	—
73	C₂×C₄⋊C₄	C₂×D₁₀₉	1	2	yes	—
74	C₂×C₄²	C₂×D₁₀₉	1	1	yes	characteristic
75	#107	—	218	4	no	—
76	#106	—	218	4	no	—
77	#97	—	109	4	no	—
78	#95	—	218	2	no	—
79	C₂²×C₄	—	2	2	no	—
80	C₂²×C₄	—	218	1	no	—
81	C₂²×C₄	—	218	1	no	—
82	C₂²×C₄	C₂²⋊D₁₀₉	1	2	yes	—
83	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
84	C₂×C₈	—	109	4	no	—
85	C₂×C₈	—	109	4	no	—
86	C₂×C₈	—	109	4	no	—
87	C₄⋊C₄	—	2	2	no	—
88	C₄⋊C₄	—	2	2	no	—
89	C₄⋊C₄	—	4	4	no	—
90	C₄⋊C₄	—	218	2	no	—
91	C₄⋊C₄	—	218	2	no	—
92	C₄²	Dic₂₁₈	1	2	yes	—
93	C₄²	C₄×D₁₀₉	1	2	yes	—
94	C₂³	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
95	C₂×C₄	—	2	2	no	—
96	C₂×C₄	—	2	2	no	—
97	C₂×C₄	—	4	2	no	—
98	C₂×C₄	—	4	4	no	—
99	C₂×C₄	—	436	1	no	—
100	C₂×C₄	—	436	1	no	—
101	C₂×C₄	—	436	2	no	—
102	C₂×C₄	—	436	2	no	—
103	C₂×C₄	#13182008	1	4	yes	—
104	C₂×C₄	C₂×C₄×D₁₀₉	1	1	yes	characteristic
105	C₂×C₄	D₄×D₁₀₉	1	2	yes	—
106	C₂×C₄	Q₈×D₁₀₉	1	2	yes	—
107	C₂×C₄	C₂×Dic₂₁₈	1	1	yes	characteristic
108	C₈	—	109	8	no	—
109	C₂²	C₂×#13182008	1	1	yes	characteristic
110	C₂²	#16812695	1	1	yes	characteristic
111	C₂²	#16812697	1	1	yes	characteristic
112	C₂²	#16812725	1	2	yes	—
113	C₂²	#16812730	1	2	yes	—
114	C₄	—	2	4	no	—
115	C₄	—	8	2	no	—
116	C₄	—	872	1	no	—
117	C₄	—	872	1	no	—
118	C₄	C₄⋊C₄×D₁₀₉	1	2	yes	—
119	C₄	#16812698	1	2	yes	—
120	C₂	#20294895	1	1	yes	characteristic
121	C₂	#20295066	1	2	yes	—
122	C₂	#20295413	1	1	yes	characteristic
123	C₂	#20295418	1	1	yes	characteristic
124	C₂	#20295432	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16812632

C₂

yes

maximal

C₂×#16812630

C₂

yes

maximal

C₂×#16812610

C₂

yes

maximal, characteristic

#20294892

C₂

yes

maximal, characteristic

#384×C₁₀₉

C₂

yes

maximal, characteristic

C₂²×#11724784

C₂²

yes

—

C₂×C₄⋊Dic₁₀₉

C₂²

yes

characteristic

C₂×C₄⋊Dic₁₀₉

C₂²

yes

—

—

C₄

yes

complemented

#16812606

—

C₂×C₄⋊C₄×C₁₀₉

—

C₂×C₄⋊C₄×C₁₀₉

C₂²

yes

—

C₂×C₄²×C₁₀₉

C₂²

yes

characteristic

C₂²×Dic₁₀₉

—

C₂²×Dic₁₀₉

—

C₂×#11724784

C₂×C₄

yes

—

C₂×#11724784

D₄

yes

—

C₂×#11724784

Q₈

yes

—

C₄⋊Dic₁₀₉

—

C₄⋊Dic₁₀₉

—

C₂²×C₄×C₁₀₉

—

C₂²×C₄×C₁₀₉

D₄

yes

—

C₂²×C₄×C₁₀₉

C₂³

yes

characteristic

C₄⋊C₄×C₁₀₉

—

C₄⋊C₄×C₁₀₉

—

C₄⋊C₄×C₁₀₉

—

C₄²×C₁₀₉

C₂×C₄

yes

—

C₄²×C₁₀₉

Q₈

yes

—

C₂³×C₁₀₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₄⋊C₄

yes

—

C₂×C₄×C₁₀₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂×D₄

yes

—

C₂×C₄×C₁₀₉

C₂×Q₈

yes

—

C₂×C₄×C₁₀₉

C₂×Q₈

yes

characteristic

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

#11724784

C₄⋊C₄

yes

complemented

C₂²×C₁₀₉

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₁₀₉

#127

yes

characteristic

C₂²×C₁₀₉

C₄⋊Q₈

yes

characteristic

C₂²×C₁₀₉

AGL(1,ℤ/8)

yes

—

C₂²×C₁₀₉

#137

yes

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

C₂×C₄⋊C₄

yes

—

C₄×C₁₀₉

C₄⋊Q₈

yes

—

Dic₁₀₉

—

Dic₁₀₉

—

C₂×C₁₀₉

#384

yes

characteristic

C₂×C₁₀₉

#428

yes

—

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

—

#1847

—

109

maximal, Sylow

C₁₀₉

#1847

yes

characteristic, complemented, Sylow

C₂×#107

—

109

—

C₂×#106

—

109

—

C₂×#97

—

109

—

#384

D₁₀₉

yes

characteristic

#382

—

109

—

C₂²×C₈

—

109

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

109

—

C₂×C₄⋊C₄

—

109

—

C₂×C₄⋊C₄

C₂×D₁₀₉

yes

—

C₂×C₄²

C₂×D₁₀₉

yes

characteristic

#107

—

218

—

#106

—

218

—

#97

—

109

—

#95

—

218

—

C₂²×C₄

—

C₂²×C₄

—

218

—

C₂²×C₄

—

218

—

C₂²×C₄

C₂²⋊D₁₀₉

yes

—

C₂²×C₄

C₂²×D₁₀₉

yes

characteristic

C₂×C₈

—

109

—

C₂×C₈

—

109

—

C₂×C₈

—

109

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

218

—

C₄⋊C₄

—

218

—

C₄²

Dic₂₁₈

yes

—

C₄²

C₄×D₁₀₉

yes

—

C₂³

C₂×C₂²⋊D₁₀₉

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

436

—

100

C₂×C₄

—

436

—

101

C₂×C₄

—

436

—

102

C₂×C₄

—

436

—

103

C₂×C₄

#13182008

yes

—

104

C₂×C₄

C₂×C₄×D₁₀₉

yes

characteristic

105

C₂×C₄

D₄×D₁₀₉

yes

—

106

C₂×C₄

Q₈×D₁₀₉

yes

—

107

C₂×C₄

C₂×Dic₂₁₈

yes

characteristic

108

C₈

—

109

—

109

C₂²

C₂×#13182008

yes

characteristic

110

C₂²

#16812695

yes

characteristic

111

C₂²

#16812697

yes

characteristic

112

C₂²

#16812725

yes

—

113

C₂²

#16812730

yes

—

114

C₄

—

115

C₄

—

116

C₄

—

872

—

117

C₄

—

872

—

118

C₄

C₄⋊C₄×D₁₀₉

yes

—

119

C₄

#16812698

yes

—

120

C₂

#20294895

yes

characteristic

121

C₂

#20295066

yes

—

122

C₂

#20295413

yes

characteristic

123

C₂

#20295418

yes

characteristic

124

C₂

#20295432

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₁₀₉
commutator	C₂×C₄×C₁₀₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₀₉
Fitting	#384×C₁₀₉	C₂
socle, prime radical	C₂³×C₁₀₉	C₂×D₄
cosocle	C₂²×C₄×C₁₀₉	C₂³
hypercenter	#384	D₁₀₉
hypocenter	C₁₀₉	#1847
LCS	G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #384

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₁₀₉

commutator

C₂×C₄×C₁₀₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₀₉

Fitting

#384×C₁₀₉

C₂

socle, prime radical

C₂³×C₁₀₉

C₂×D₄

cosocle

C₂²×C₄×C₁₀₉

C₂³

D₁₀₉

C₁₀₉

G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #384

N	H ≅ G/N	type
11. #16812610	C₄	semidirect
45. #11724784	C₄⋊C₄	semidirect
63. C₁₀₉	#1847	semidirect

H ≅ G/N

type

11. #16812610

C₄

semidirect

45. #11724784

C₄⋊C₄

semidirect

63. C₁₀₉

#1847

semidirect

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

subgroup	index
C₂×#16812632	2
C₂×#16812630	2
C₂×#16812610	2
#20294892	2
#384×C₁₀₉	2
#1847	109

subgroup

index

#384×C₁₀₉

109

Subgroups — #28363198

Conjugacy classes of subgroups

Complements

Maximal subgroups