Group #28364009 — subgroups

Subgroups — #28364009

Order 13952.

284 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 284 conjugacy classes of subgroups — too many to draw. Only its 101 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	#20295583	C₂	1	1	yes	maximal, characteristic
2	#20295050	C₂	1	1	yes	maximal, characteristic
3	#20294786	C₂	1	4	yes	maximal
4	C₄³×C₁₀₉	C₂	1	1	yes	maximal, characteristic
5	C₂×#13182013	C₂²	1	2	yes	—
6	#16812699	C₄	1	2	yes	complemented
7	#16812691	C₄	1	2	yes	complemented
8	#16812678	—	2	2	no	—
9	C₄×Dic₂₁₈	—	2	2	no	—
10	#16812658	C₂²	1	2	yes	—
11	C₄⋊#11724784	—	2	2	no	—
12	C₂×C₄²×C₁₀₉	—	2	2	no	—
13	C₂×C₄²×C₁₀₉	C₂²	1	2	yes	—
14	C₂×C₄²×C₁₀₉	C₂²	1	1	yes	characteristic
15	C₂×Dic₂₁₈	—	2	2	no	—
16	#13182013	—	2	4	no	—
17	C₂×#11724784	—	2	2	no	—
18	C₄⋊Dic₁₀₉	—	2	2	no	—
19	C₄⋊Dic₁₀₉	C₂×C₄	1	4	yes	—
20	#13182008	—	2	4	no	—
21	C₂²⋊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	C₄²×C₁₀₉	—	2	8	no	—
27	C₄²×C₁₀₉	—	2	2	no	—
28	C₄²×C₁₀₉	C₂×C₄	1	1	yes	characteristic
29	C₄²×C₁₀₉	C₂×C₄	1	1	yes	characteristic
30	C₄²×C₁₀₉	D₄	1	2	yes	—
31	C₄²×C₁₀₉	Q₈	1	4	yes	complemented
32	C₂³×C₁₀₉	C₂×D₄	1	1	yes	characteristic
33	C₂×C₄×C₁₀₉	—	2	8	no	—
34	C₂×C₄×C₁₀₉	—	2	4	no	—
35	C₂×C₄×C₁₀₉	—	2	4	no	—
36	C₂×C₄×C₁₀₉	C₂²⋊C₄	1	2	yes	—
37	C₂×C₄×C₁₀₉	C₂²⋊C₄	1	2	yes	—
38	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
39	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
40	C₂×C₄×C₁₀₉	C₂×Q₈	1	2	yes	—
41	C₂×C₄×C₁₀₉	#41	1	2	yes	—
42	C₂×Dic₁₀₉	—	2	4	no	—
43	Dic₂₁₈	—	4	4	no	—
44	#11724784	—	4	2	no	—
45	C₂²×C₁₀₉	—	2	2	no	—
46	C₂²×C₁₀₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
47	C₂²×C₁₀₉	C₂²⋊Q₈	1	2	yes	—
48	C₄×C₁₀₉	—	2	8	no	—
49	C₄×C₁₀₉	—	2	2	no	—
50	C₄×C₁₀₉	#104	1	4	yes	—
51	C₄×C₁₀₉	C₄×Q₈	1	2	yes	—
52	C₄×C₁₀₉	#124	1	1	yes	characteristic
53	C₄×C₁₀₉	C₄⋊Q₈	1	1	yes	characteristic
54	Dic₁₀₉	—	4	4	no	—
55	C₂×C₁₀₉	—	2	2	no	—
56	C₂×C₁₀₉	#391	1	1	yes	characteristic
57	C₂×C₁₀₉	C₂×#104	1	2	yes	—
58	#1994	—	109	1	no	maximal, Sylow
59	C₁₀₉	#1994	1	1	yes	characteristic, complemented, Sylow
60	#533	—	109	1	no	—
61	#423	—	109	1	no	—
62	C₄³	D₁₀₉	1	1	yes	characteristic
63	#339	—	109	4	no	—
64	C₂×M₁₆	—	109	2	no	—
65	C₄⋊Q₈	—	109	2	no	—
66	#125	—	109	2	no	—
67	C₂²⋊Q₈	—	218	2	no	—
68	C₄×Q₈	—	218	2	no	—
69	#117	—	109	2	no	—
70	C₂×C₄²	—	2	2	no	—
71	C₂×C₄²	C₂×D₁₀₉	1	2	yes	—
72	C₂×C₄²	C₂×D₁₀₉	1	1	yes	characteristic
73	C₄⋊C₈	—	218	2	no	—
74	C₂×Q₈	—	218	2	no	—
75	C₂²×C₄	—	2	2	no	—
76	C₂²×C₄	C₂²⋊D₁₀₉	1	2	yes	—
77	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
78	M₁₆	—	218	4	no	—
79	C₂×C₈	—	218	2	no	—
80	C₄⋊C₄	—	109	4	no	—
81	C₄⋊C₄	—	218	4	no	—
82	C₄⋊C₄	—	218	2	no	—
83	C₂²⋊C₄	—	218	2	no	—
84	C₄²	—	2	8	no	—
85	C₄²	—	2	2	no	—
86	C₄²	—	218	2	no	—
87	C₄²	Dic₂₁₈	1	4	yes	complemented
88	C₄²	C₄×D₁₀₉	1	1	yes	characteristic
89	C₄²	C₄×D₁₀₉	1	1	yes	characteristic
90	C₄²	D₄₃₆	1	2	yes	—
91	C₂³	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
92	Q₈	—	436	4	no	—
93	C₂×C₄	—	2	8	no	—
94	C₂×C₄	—	2	4	no	—
95	C₂×C₄	—	2	4	no	—
96	C₂×C₄	—	218	4	no	—
97	C₂×C₄	#13182006	1	2	yes	—
98	C₂×C₄	#13182006	1	2	yes	—
99	C₂×C₄	C₂×C₄×D₁₀₉	1	1	yes	characteristic
100	C₂×C₄	C₂×D₄₃₆	1	1	yes	characteristic
101	C₂×C₄	C₂×Dic₂₁₈	1	2	yes	—
102	C₂×C₄	#13182029	1	2	yes	—
103	C₈	—	436	2	no	—
104	C₂²	—	2	2	no	—
105	C₂²	C₂×#13182006	1	1	yes	characteristic
106	C₂²	#16812678	1	2	yes	—
107	C₄	—	2	8	no	—
108	C₄	—	2	2	no	—
109	C₄	—	436	4	no	—
110	C₄	#16812627	1	4	yes	—
111	C₄	C₄×Dic₂₁₈	1	2	yes	—
112	C₄	#16812687	1	1	yes	characteristic
113	C₄	#16812699	1	1	yes	characteristic
114	C₂	—	2	2	no	—
115	C₂	#20294932	1	1	yes	characteristic
116	C₂	C₂×#16812627	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20295583

C₂

yes

maximal, characteristic

#20295050

C₂

yes

maximal, characteristic

#20294786

C₂

yes

maximal

C₄³×C₁₀₉

C₂

yes

maximal, characteristic

C₂×#13182013

C₂²

yes

—

#16812699

C₄

yes

complemented

#16812691

C₄

yes

complemented

#16812678

—

C₄×Dic₂₁₈

—

#16812658

C₂²

yes

—

C₄⋊#11724784

—

C₂×C₄²×C₁₀₉

—

C₂×C₄²×C₁₀₉

C₂²

yes

—

C₂×C₄²×C₁₀₉

C₂²

yes

characteristic

C₂×Dic₂₁₈

—

#13182013

—

C₂×#11724784

—

C₄⋊Dic₁₀₉

—

C₄⋊Dic₁₀₉

C₂×C₄

yes

—

#13182008

—

C₂²⋊Dic₁₀₉

—

C₄×Dic₁₀₉

—

C₂²×C₄×C₁₀₉

—

C₂²×C₄×C₁₀₉

D₄

yes

—

C₂²×C₄×C₁₀₉

C₂³

yes

characteristic

C₄²×C₁₀₉

—

C₄²×C₁₀₉

—

C₄²×C₁₀₉

C₂×C₄

yes

characteristic

C₄²×C₁₀₉

C₂×C₄

yes

characteristic

C₄²×C₁₀₉

D₄

yes

—

C₄²×C₁₀₉

Q₈

yes

complemented

C₂³×C₁₀₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₂²⋊C₄

yes

—

C₂×C₄×C₁₀₉

C₂²⋊C₄

yes

—

C₂×C₄×C₁₀₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂×Q₈

yes

—

C₂×C₄×C₁₀₉

#41

yes

—

C₂×Dic₁₀₉

—

Dic₂₁₈

—

#11724784

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₁₀₉

C₂²⋊Q₈

yes

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

#104

yes

—

C₄×C₁₀₉

C₄×Q₈

yes

—

C₄×C₁₀₉

#124

yes

characteristic

C₄×C₁₀₉

C₄⋊Q₈

yes

characteristic

Dic₁₀₉

—

C₂×C₁₀₉

—

C₂×C₁₀₉

#391

yes

characteristic

C₂×C₁₀₉

C₂×#104

yes

—

#1994

—

109

maximal, Sylow

C₁₀₉

#1994

yes

characteristic, complemented, Sylow

#533

—

109

—

#423

—

109

—

C₄³

D₁₀₉

yes

characteristic

#339

—

109

—

C₂×M₁₆

—

109

—

C₄⋊Q₈

—

109

—

#125

—

109

—

C₂²⋊Q₈

—

218

—

C₄×Q₈

—

218

—

#117

—

109

—

C₂×C₄²

—

C₂×C₄²

C₂×D₁₀₉

yes

—

C₂×C₄²

C₂×D₁₀₉

yes

characteristic

C₄⋊C₈

—

218

—

C₂×Q₈

—

218

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊D₁₀₉

yes

—

C₂²×C₄

C₂²×D₁₀₉

yes

characteristic

M₁₆

—

218

—

C₂×C₈

—

218

—

C₄⋊C₄

—

109

—

C₄⋊C₄

—

218

—

C₄⋊C₄

—

218

—

C₂²⋊C₄

—

218

—

C₄²

—

C₄²

—

C₄²

—

218

—

C₄²

Dic₂₁₈

yes

complemented

C₄²

C₄×D₁₀₉

yes

characteristic

C₄²

C₄×D₁₀₉

yes

characteristic

C₄²

D₄₃₆

yes

—

C₂³

C₂×C₂²⋊D₁₀₉

yes

characteristic

Q₈

—

436

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

218

—

C₂×C₄

#13182006

yes

—

C₂×C₄

#13182006

yes

—

C₂×C₄

C₂×C₄×D₁₀₉

yes

characteristic

100

C₂×C₄

C₂×D₄₃₆

yes

characteristic

101

C₂×C₄

C₂×Dic₂₁₈

yes

—

102

C₂×C₄

#13182029

yes

—

103

C₈

—

436

—

104

C₂²

—

105

C₂²

C₂×#13182006

yes

characteristic

106

C₂²

#16812678

yes

—

107

C₄

—

108

C₄

—

109

C₄

—

436

—

110

C₄

#16812627

yes

—

111

C₄

C₄×Dic₂₁₈

yes

—

112

C₄

#16812687

yes

characteristic

113

C₄

#16812699

yes

characteristic

114

C₂

—

115

C₂

#20294932

yes

characteristic

116

C₂

C₂×#16812627

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×D₄₃₆
commutator	C₂×C₄×C₁₀₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₀₉
Fitting	C₄³×C₁₀₉	C₂
socle	C₂²×C₁₀₉	C₂×C₂²⋊C₄
prime radical	C₂³×C₁₀₉	C₂×D₄
cosocle	C₂²×C₄×C₁₀₉	C₂³
hypercenter	C₄³	D₁₀₉
hypocenter	C₁₀₉	#1994
LCS	G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄³

subgroup

G/H

center

C₂×C₄

C₂×D₄₃₆

commutator

C₂×C₄×C₁₀₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₀₉

Fitting

C₄³×C₁₀₉

C₂

socle

C₂²×C₁₀₉

C₂×C₂²⋊C₄

prime radical

C₂³×C₁₀₉

C₂×D₄

cosocle

C₂²×C₄×C₁₀₉

C₂³

hypercenter

C₄³

D₁₀₉

hypocenter

C₁₀₉

#1994

LCS

G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄³

N	H ≅ G/N	type
6. #16812699	C₄	semidirect
7. #16812691	C₄	semidirect
31. C₄²×C₁₀₉	Q₈	semidirect
59. C₁₀₉	#1994	semidirect
87. C₄²	Dic₂₁₈	semidirect

H ≅ G/N

type

6. #16812699

C₄

semidirect

7. #16812691

C₄

semidirect

31. C₄²×C₁₀₉

Q₈

semidirect

59. C₁₀₉

#1994

semidirect

87. C₄²

Dic₂₁₈

semidirect

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

subgroup	index
#20295583	2
#20295050	2
#20294786	2
C₄³×C₁₀₉	2
#1994	109

subgroup

index

C₄³×C₁₀₉

109

Subgroups — #28364009

Conjugacy classes of subgroups

Complements

Maximal subgroups