Group #28364586 — subgroups

Subgroups — #28364586

Order 13952.

210 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 210 conjugacy classes of subgroups — too many to draw. Only its 67 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	#20295075	C₂	1	1	yes	maximal, characteristic, complemented
2	#20295053	C₂	1	2	yes	maximal, complemented
3	#20295044	C₂	1	2	yes	maximal
4	#20294994	C₂	1	1	yes	maximal, characteristic
5	H(3,ℤ/4)×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#13182029	C₂²	1	1	yes	characteristic
7	C₂×#13182013	C₂²	1	2	yes	—
8	C₂²×#11724784	C₂²	1	1	yes	characteristic
9	#16812635	—	2	2	no	—
10	C₄⋊#11724784	—	2	4	no	—
11	#16812627	—	2	4	no	—
12	#16812611	—	2	2	no	—
13	C₂×M₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
14	#117×C₁₀₉	C₂²	1	2	yes	—
15	#13182029	—	2	2	no	—
16	#13182029	—	2	2	no	—
17	C₂×Dic₂₁₈	Q₈	1	1	yes	characteristic
18	C₂×D₄₃₆	Q₈	1	1	yes	characteristic
19	C₂×C₂²⋊D₁₀₉	—	2	1	no	—
20	C₂×C₄×D₁₀₉	—	2	1	no	—
21	#13182013	—	2	2	no	—
22	#13182013	—	2	2	no	—
23	C₂×#11724784	—	2	1	no	—
24	C₂×#11724784	—	2	2	no	—
25	C₂×#11724784	—	2	2	no	—
26	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
27	M₁₆×C₁₀₉	—	2	2	no	—
28	C₂×C₈×C₁₀₉	D₄	1	2	yes	—
29	C₄⋊C₄×C₁₀₉	—	2	2	no	—
30	C₂²⋊C₄×C₁₀₉	—	2	2	no	—
31	C₄²×C₁₀₉	D₄	1	4	yes	—
32	C₂³×C₁₀₉	#41	1	1	yes	characteristic
33	C₂²×D₁₀₉	—	2	1	no	—
34	C₂×C₄×C₁₀₉	—	2	4	no	—
35	C₂×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
36	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
37	C₂×C₄×C₁₀₉	C₂×Q₈	1	1	yes	characteristic
38	C₂×C₄×C₁₀₉	#41	1	2	yes	—
39	C₂²⋊D₁₀₉	—	4	2	no	—
40	C₂×Dic₁₀₉	—	2	1	no	—
41	D₄₃₆	—	2	2	no	—
42	C₄×D₁₀₉	—	4	1	no	—
43	C₄×D₁₀₉	—	4	1	no	—
44	Dic₂₁₈	—	2	2	no	—
45	C₈×C₁₀₉	—	2	2	no	—
46	#11724784	—	4	1	no	—
47	#11724784	—	4	2	no	—
48	C₂²×C₁₀₉	—	4	1	no	—
49	C₂²×C₁₀₉	C₂²⋊Q₈	1	1	yes	characteristic
50	C₂²×C₁₀₉	#123	1	2	yes	—
51	C₂×D₁₀₉	—	4	2	no	—
52	C₂×D₁₀₉	—	4	1	no	—
53	C₄×C₁₀₉	—	4	4	no	—
54	C₄×C₁₀₉	C₂²⋊Q₈	1	2	yes	—
55	C₄×C₁₀₉	#123	1	1	yes	characteristic
56	C₄×C₁₀₉	#127	1	1	yes	characteristic
57	Dic₁₀₉	—	4	1	no	—
58	C₂×C₁₀₉	—	2	2	no	—
59	C₂×C₁₀₉	—	2	1	no	—
60	C₂×C₁₀₉	#399	1	1	yes	characteristic
61	D₁₀₉	—	8	1	no	—
62	#2098	—	109	1	no	maximal, Sylow
63	C₁₀₉	#2098	1	1	yes	characteristic, complemented, Sylow
64	#430	—	109	1	no	—
65	#424	—	109	2	no	—
66	#421	—	109	2	no	—
67	#408	—	109	1	no	—
68	H(3,ℤ/4)	D₁₀₉	1	1	yes	characteristic, complemented
69	C₂×#41	—	109	1	no	—
70	C₂×M₁₆	—	109	2	no	—
71	C₂×M₁₆	C₂×D₁₀₉	1	1	yes	characteristic
72	C₂²×C₈	—	109	1	no	—
73	#117	C₂×D₁₀₉	1	2	yes	—
74	#108	—	218	2	no	—
75	C₄⋊C₈	—	218	4	no	—
76	#104	—	218	4	no	—
77	C₂²⋊C₈	—	218	2	no	—
78	#41	—	218	2	no	—
79	#41	—	218	2	no	—
80	C₂×Q₈	—	109	1	no	—
81	C₂×D₄	—	109	1	no	—
82	C₂×D₄	—	218	1	no	—
83	C₂²×C₄	—	218	1	no	—
84	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
85	M₁₆	—	2	2	no	—
86	M₁₆	—	218	2	no	—
87	M₁₆	—	218	2	no	—
88	C₂×C₈	—	218	1	no	—
89	C₂×C₈	—	218	2	no	—
90	C₂×C₈	—	218	2	no	—
91	C₂×C₈	D₄₃₆	1	2	yes	—
92	C₄⋊C₄	—	2	2	no	—
93	C₂²⋊C₄	—	2	2	no	—
94	C₄²	D₄₃₆	1	4	yes	—
95	C₂³	—	218	1	no	—
96	C₂³	#13182028	1	1	yes	characteristic
97	Q₈	—	218	2	no	—
98	D₄	—	436	2	no	—
99	D₄	—	218	2	no	—
100	C₂×C₄	—	2	4	no	—
101	C₂×C₄	—	436	1	no	—
102	C₂×C₄	—	436	1	no	—
103	C₂×C₄	—	218	1	no	—
104	C₂×C₄	C₂×D₄₃₆	1	2	yes	—
105	C₂×C₄	C₂×D₄₃₆	1	1	yes	characteristic
106	C₂×C₄	Q₈×D₁₀₉	1	1	yes	characteristic
107	C₂×C₄	#13182028	1	2	yes	—
108	C₈	—	2	2	no	—
109	C₈	—	436	1	no	—
110	C₈	—	436	2	no	—
111	C₂²	—	4	1	no	—
112	C₂²	—	436	2	no	—
113	C₂²	—	436	1	no	—
114	C₂²	#16812679	1	1	yes	characteristic
115	C₂²	#16812683	1	2	yes	—
116	C₄	—	4	4	no	—
117	C₄	—	436	1	no	—
118	C₄	#16812679	1	2	yes	—
119	C₄	#16812683	1	1	yes	characteristic
120	C₄	#16812696	1	1	yes	characteristic
121	C₂	—	2	2	no	—
122	C₂	—	2	1	no	—
123	C₂	—	872	1	no	—
124	C₂	#20294963	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20295075

C₂

yes

maximal, characteristic, complemented

#20295053

C₂

yes

maximal, complemented

#20295044

C₂

yes

maximal

#20294994

C₂

yes

maximal, characteristic

H(3,ℤ/4)×C₁₀₉

C₂

yes

maximal, characteristic, complemented

C₂×#13182029

C₂²

yes

characteristic

C₂×#13182013

C₂²

yes

—

C₂²×#11724784

C₂²

yes

—

—

—

—

C₂×M₁₆×C₁₀₉

C₂²

yes

characteristic

#117×C₁₀₉

C₂²

yes

—

#13182029

—

#13182029

—

C₂×Dic₂₁₈

Q₈

yes

characteristic

C₂×D₄₃₆

Q₈

yes

characteristic

C₂×C₂²⋊D₁₀₉

—

C₂×C₄×D₁₀₉

—

—

—

—

—

—

C₂²×C₄×C₁₀₉

C₂³

yes

characteristic

M₁₆×C₁₀₉

—

C₂×C₈×C₁₀₉

D₄

yes

—

C₄⋊C₄×C₁₀₉

—

C₂²⋊C₄×C₁₀₉

—

C₄²×C₁₀₉

D₄

yes

—

C₂³×C₁₀₉

#41

yes

characteristic

C₂²×D₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₂×D₄

yes

—

C₂×C₄×C₁₀₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₀₉

#41

yes

—

C₂²⋊D₁₀₉

—

C₂×Dic₁₀₉

—

D₄₃₆

—

C₄×D₁₀₉

—

C₄×D₁₀₉

—

Dic₂₁₈

—

C₈×C₁₀₉

—

#11724784

—

#11724784

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

C₂²⋊Q₈

yes

characteristic

C₂²×C₁₀₉

#123

yes

—

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

C₂²⋊Q₈

yes

—

C₄×C₁₀₉

yes

C₄×C₁₀₉

yes

Dic₁₀₉

—

C₂×C₁₀₉

—

C₂×C₁₀₉

—

C₂×C₁₀₉

#399

yes

characteristic

D₁₀₉

—

#2098

—

109

maximal, Sylow

C₁₀₉

#2098

yes

characteristic, complemented, Sylow

#430

—

109

—

#424

—

109

—

#421

—

109

—

#408

—

109

—

H(3,ℤ/4)

D₁₀₉

yes

characteristic, complemented

C₂×#41

—

109

—

C₂×M₁₆

—

109

—

C₂×M₁₆

C₂×D₁₀₉

yes

characteristic

C₂²×C₈

—

109

—

#117

C₂×D₁₀₉

yes

—

#108

—

218

—

C₄⋊C₈

—

218

—

#104

—

218

—

C₂²⋊C₈

—

218

—

#41

—

218

—

#41

—

218

—

C₂×Q₈

—

109

—

C₂×D₄

—

109

—

C₂×D₄

—

218

—

C₂²×C₄

—

218

—

C₂²×C₄

C₂²×D₁₀₉

yes

characteristic

M₁₆

—

M₁₆

—

218

—

M₁₆

—

218

—

C₂×C₈

—

218

—

C₂×C₈

—

218

—

C₂×C₈

—

218

—

C₂×C₈

D₄₃₆

yes

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

D₄₃₆

yes

—

C₂³

—

218

—

C₂³

#13182028

yes

characteristic

Q₈

—

218

—

D₄

—

436

—

D₄

—

218

—

100

C₂×C₄

—

101

C₂×C₄

—

436

—

102

C₂×C₄

—

436

—

103

C₂×C₄

—

218

—

104

C₂×C₄

C₂×D₄₃₆

yes

—

105

C₂×C₄

C₂×D₄₃₆

yes

characteristic

106

C₂×C₄

Q₈×D₁₀₉

yes

characteristic

107

C₂×C₄

#13182028

yes

—

108

C₈

—

109

C₈

—

436

—

110

C₈

—

436

—

111

C₂²

—

112

C₂²

—

436

—

113

C₂²

—

436

—

114

C₂²

#16812679

yes

characteristic

115

C₂²

#16812683

yes

—

116

C₄

—

117

C₄

—

436

—

118

C₄

#16812679

yes

—

119

C₄

#16812683

yes

characteristic

120

C₄

#16812696

yes

characteristic

121

C₂

—

122

C₂

—

123

C₂

—

872

—

124

C₂

#20294963

yes

characteristic

	subgroup	G/H
center	C₄	#16812696
commutator, cosocle	C₂²×C₄×C₁₀₉	C₂³
Frattini	C₂²×C₄	C₂²×D₁₀₉
Fitting	H(3,ℤ/4)×C₁₀₉	C₂
socle	C₂×C₁₀₉	#399
prime radical	C₂×#13182029	C₂²
hypercenter	H(3,ℤ/4)	D₁₀₉
hypocenter	C₁₀₉	#2098
LCS	G ▷ C₂²×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ H(3,ℤ/4)

subgroup

G/H

center

C₄

#16812696

commutator, cosocle

C₂²×C₄×C₁₀₉

C₂³

Frattini

C₂²×C₄

C₂²×D₁₀₉

Fitting

H(3,ℤ/4)×C₁₀₉

C₂

C₂×C₁₀₉

C₂²

D₁₀₉

C₁₀₉

G ▷ C₂²×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ H(3,ℤ/4)

N	H ≅ G/N	type
1. #20295075	C₂	semidirect
2. #20295053	C₂	semidirect
5. H(3,ℤ/4)×C₁₀₉	C₂	semidirect
63. C₁₀₉	#2098	semidirect
68. H(3,ℤ/4)	D₁₀₉	semidirect

H ≅ G/N

type

1. #20295075

C₂

semidirect

2. #20295053

C₂

semidirect

5. H(3,ℤ/4)×C₁₀₉

C₂

semidirect

63. C₁₀₉

#2098

semidirect

68. H(3,ℤ/4)

D₁₀₉

semidirect

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

subgroup	index
#20295075	2
#20295053	2
#20295044	2
#20294994	2
H(3,ℤ/4)×C₁₀₉	2
#2098	109

subgroup

index

H(3,ℤ/4)×C₁₀₉

109

Subgroups — #28364586

Conjugacy classes of subgroups

Complements

Maximal subgroups