Group #28363876 — subgroups

Subgroups — #28363876

Order 13952.

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

normal subgroups only

This group has 292 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	C₂×#16812699	C₂	1	1	yes	maximal, characteristic
2	#20295050	C₂	1	1	yes	maximal, characteristic
3	C₂×#16812617	C₂	1	4	yes	maximal
4	C₄×M₁₆×C₁₀₉	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₂₁₈	C₂²	1	2	yes	—
6	C₂×#13182013	C₂²	1	2	yes	—
7	#16812699	—	2	2	no	—
8	#16812699	—	2	1	no	—
9	#16812699	—	2	1	no	—
10	C₂×C₄⋊Dic₁₀₉	C₄	1	4	yes	—
11	C₄⋊#11724784	—	2	2	no	—
12	#16812617	—	2	8	no	—
13	C₂×M₁₆×C₁₀₉	C₂²	1	2	yes	—
14	C₂×C₄²×C₁₀₉	C₂²	1	1	yes	characteristic
15	#97×C₁₀₉	—	2	1	no	—
16	C₄×C₈×C₁₀₉	—	2	1	no	—
17	C₂×Dic₂₁₈	—	2	4	no	—
18	C₂×Dic₂₁₈	—	2	2	no	—
19	C₂×Dic₂₁₈	—	2	2	no	—
20	C₂×Dic₂₁₈	—	2	4	no	—
21	C₂²×Dic₁₀₉	C₂×C₄	1	4	yes	—
22	#13182013	—	2	4	no	—
23	C₂×#11724784	—	2	2	no	—
24	C₄⋊Dic₁₀₉	—	2	4	no	—
25	C₄⋊Dic₁₀₉	—	2	4	no	—
26	C₂²×C₄×C₁₀₉	C₂×C₄	1	2	yes	—
27	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
28	M₁₆×C₁₀₉	—	2	2	no	—
29	M₁₆×C₁₀₉	D₄	1	4	yes	—
30	C₂×C₈×C₁₀₉	—	2	2	no	—
31	C₄²×C₁₀₉	D₄	1	2	yes	—
32	C₄²×C₁₀₉	D₄	1	1	yes	characteristic
33	C₄²×C₁₀₉	D₄	1	1	yes	characteristic
34	C₂³×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
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	C₂×D₄	1	1	yes	characteristic
39	C₂×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
40	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
41	C₂×C₄×C₁₀₉	#41	1	2	yes	—
42	C₂×Dic₁₀₉	—	2	4	no	—
43	C₂×Dic₁₀₉	—	2	8	no	—
44	Dic₂₁₈	—	4	4	no	—
45	Dic₂₁₈	—	4	2	no	—
46	Dic₂₁₈	—	4	2	no	—
47	C₈×C₁₀₉	—	2	4	no	—
48	#11724784	—	4	2	no	—
49	C₂²×C₁₀₉	—	2	2	no	—
50	C₂²×C₁₀₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
51	C₂²×C₁₀₉	C₄×D₄	1	2	yes	—
52	C₄×C₁₀₉	—	2	2	no	—
53	C₄×C₁₀₉	#101	1	4	yes	complemented
54	C₄×C₁₀₉	#121	1	2	yes	—
55	C₄×C₁₀₉	#124	1	1	yes	characteristic
56	C₄×C₁₀₉	#127	1	1	yes	characteristic
57	Dic₁₀₉	—	4	4	no	—
58	C₂×C₁₀₉	—	2	2	no	—
59	C₂×C₁₀₉	#390	1	1	yes	characteristic
60	C₂×C₁₀₉	C₂×#101	1	2	yes	—
61	#1966	—	109	1	no	maximal, Sylow
62	C₁₀₉	#1966	1	1	yes	characteristic, complemented, Sylow
63	C₂×C₄⋊Q₈	—	109	1	no	—
64	#423	—	109	1	no	—
65	C₂×#101	—	109	4	no	—
66	C₄×M₁₆	D₁₀₉	1	1	yes	characteristic
67	C₂²×Q₈	—	109	2	no	—
68	C₂×M₁₆	—	109	2	no	—
69	C₂×M₁₆	C₂×D₁₀₉	1	2	yes	—
70	C₄⋊Q₈	—	218	2	no	—
71	C₄⋊Q₈	—	218	1	no	—
72	C₄⋊Q₈	—	218	1	no	—
73	C₂×C₄⋊C₄	—	109	4	no	—
74	C₂×C₄²	C₂×D₁₀₉	1	1	yes	characteristic
75	C₄⋊C₈	—	218	2	no	—
76	#101	—	218	8	no	—
77	#97	—	2	1	no	—
78	C₄×C₈	—	2	1	no	—
79	C₂×Q₈	—	218	2	no	—
80	C₂×Q₈	—	218	4	no	—
81	C₂×Q₈	—	218	2	no	—
82	C₂×Q₈	—	218	4	no	—
83	C₂²×C₄	—	109	4	no	—
84	C₂²×C₄	C₄×D₁₀₉	1	2	yes	—
85	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
86	M₁₆	—	2	2	no	—
87	M₁₆	—	218	4	no	—
88	M₁₆	D₄₃₆	1	4	yes	—
89	C₂×C₈	—	2	2	no	—
90	C₂×C₈	—	218	2	no	—
91	C₄⋊C₄	—	218	4	no	—
92	C₄⋊C₄	—	218	4	no	—
93	C₄²	D₄₃₆	1	2	yes	—
94	C₄²	C₂²⋊D₁₀₉	1	1	yes	characteristic
95	C₄²	C₂²⋊D₁₀₉	1	1	yes	characteristic
96	C₂³	C₂×C₄×D₁₀₉	1	1	yes	characteristic
97	Q₈	—	436	4	no	—
98	Q₈	—	436	2	no	—
99	Q₈	—	436	2	no	—
100	C₂×C₄	—	2	4	no	—
101	C₂×C₄	—	218	4	no	—
102	C₂×C₄	—	218	8	no	—
103	C₂×C₄	#13182006	1	2	yes	—
104	C₂×C₄	#13182006	1	2	yes	—
105	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
106	C₂×C₄	C₂×D₄₃₆	1	2	yes	—
107	C₂×C₄	C₂×D₄₃₆	1	1	yes	characteristic
108	C₂×C₄	#13182029	1	2	yes	—
109	C₈	—	2	4	no	—
110	C₈	—	436	2	no	—
111	C₂²	—	2	2	no	—
112	C₂²	C₂×#13182006	1	1	yes	characteristic
113	C₂²	C₄×D₄₃₆	1	2	yes	—
114	C₄	—	2	2	no	—
115	C₄	—	436	4	no	—
116	C₄	#16812617	1	4	yes	complemented
117	C₄	#16812673	1	2	yes	—
118	C₄	#16812687	1	1	yes	characteristic
119	C₄	#16812696	1	1	yes	characteristic
120	C₂	—	2	2	no	—
121	C₂	#20294928	1	1	yes	characteristic
122	C₂	C₂×#16812617	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16812699

C₂

yes

maximal, characteristic

#20295050

C₂

yes

maximal, characteristic

C₂×#16812617

C₂

yes

maximal

C₄×M₁₆×C₁₀₉

C₂

yes

maximal, characteristic

C₂²×Dic₂₁₈

C₂²

yes

—

C₂×#13182013

C₂²

yes

—

#16812699

—

#16812699

—

#16812699

—

C₂×C₄⋊Dic₁₀₉

C₄

yes

—

C₄⋊#11724784

—

#16812617

—

C₂×M₁₆×C₁₀₉

C₂²

yes

—

C₂×C₄²×C₁₀₉

C₂²

yes

characteristic

#97×C₁₀₉

—

C₄×C₈×C₁₀₉

—

C₂×Dic₂₁₈

—

C₂×Dic₂₁₈

—

C₂×Dic₂₁₈

—

C₂×Dic₂₁₈

—

C₂²×Dic₁₀₉

C₂×C₄

yes

—

#13182013

—

C₂×#11724784

—

C₄⋊Dic₁₀₉

—

C₄⋊Dic₁₀₉

—

C₂²×C₄×C₁₀₉

C₂×C₄

yes

—

C₂²×C₄×C₁₀₉

C₂³

yes

characteristic

M₁₆×C₁₀₉

—

M₁₆×C₁₀₉

D₄

yes

—

C₂×C₈×C₁₀₉

—

C₄²×C₁₀₉

D₄

yes

—

C₄²×C₁₀₉

D₄

yes

characteristic

C₄²×C₁₀₉

D₄

yes

characteristic

C₂³×C₁₀₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₂²⋊C₄

yes

—

C₂×C₄×C₁₀₉

C₂²⋊C₄

yes

—

C₂×C₄×C₁₀₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂×D₄

yes

—

C₂×C₄×C₁₀₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₉

#41

yes

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

Dic₂₁₈

—

Dic₂₁₈

—

Dic₂₁₈

—

C₈×C₁₀₉

—

#11724784

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₁₀₉

C₄×D₄

yes

—

C₄×C₁₀₉

—

C₄×C₁₀₉

#101

yes

complemented

C₄×C₁₀₉

#121

yes

—

C₄×C₁₀₉

yes

C₄×C₁₀₉

yes

Dic₁₀₉

—

C₂×C₁₀₉

—

C₂×C₁₀₉

#390

yes

characteristic

C₂×C₁₀₉

C₂×#101

yes

—

#1966

—

109

maximal, Sylow

C₁₀₉

#1966

yes

characteristic, complemented, Sylow

C₂×C₄⋊Q₈

—

109

—

#423

—

109

—

C₂×#101

—

109

—

C₄×M₁₆

D₁₀₉

yes

characteristic

C₂²×Q₈

—

109

—

C₂×M₁₆

—

109

—

C₂×M₁₆

C₂×D₁₀₉

yes

—

C₄⋊Q₈

—

218

—

C₄⋊Q₈

—

218

—

C₄⋊Q₈

—

218

—

C₂×C₄⋊C₄

—

109

—

C₂×C₄²

C₂×D₁₀₉

yes

characteristic

C₄⋊C₈

—

218

—

#101

—

218

—

#97

—

C₄×C₈

—

C₂×Q₈

—

218

—

C₂×Q₈

—

218

—

C₂×Q₈

—

218

—

C₂×Q₈

—

218

—

C₂²×C₄

—

109

—

C₂²×C₄

C₄×D₁₀₉

yes

—

C₂²×C₄

C₂²×D₁₀₉

yes

characteristic

M₁₆

—

M₁₆

—

218

—

M₁₆

D₄₃₆

yes

—

C₂×C₈

—

C₂×C₈

—

218

—

C₄⋊C₄

—

218

—

C₄⋊C₄

—

218

—

C₄²

D₄₃₆

yes

—

C₄²

C₂²⋊D₁₀₉

yes

characteristic

C₄²

C₂²⋊D₁₀₉

yes

characteristic

C₂³

C₂×C₄×D₁₀₉

yes

characteristic

Q₈

—

436

—

Q₈

—

436

—

Q₈

—

436

—

100

C₂×C₄

—

101

C₂×C₄

—

218

—

102

C₂×C₄

—

218

—

103

C₂×C₄

#13182006

yes

—

104

C₂×C₄

#13182006

yes

—

105

C₂×C₄

C₂×C₂²⋊D₁₀₉

yes

characteristic

106

C₂×C₄

C₂×D₄₃₆

yes

—

107

C₂×C₄

C₂×D₄₃₆

yes

characteristic

108

C₂×C₄

#13182029

yes

—

109

C₈

—

110

C₈

—

436

—

111

C₂²

—

112

C₂²

C₂×#13182006

yes

characteristic

113

C₂²

C₄×D₄₃₆

yes

—

114

C₄

—

115

C₄

—

436

—

116

C₄

#16812617

yes

complemented

117

C₄

#16812673

yes

—

118

C₄

#16812687

yes

characteristic

119

C₄

#16812696

yes

characteristic

120

C₂

—

121

C₂

#20294928

yes

characteristic

122

C₂

C₂×#16812617

yes

—

	subgroup	G/H
center	C₂²	C₂×#13182006
commutator, prime radical	C₂³×C₁₀₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₀₉
Fitting	C₄×M₁₆×C₁₀₉	C₂
socle	C₂²×C₁₀₉	C₂×C₂²⋊C₄
cosocle	C₂²×C₄×C₁₀₉	C₂³
hypercenter	C₄×M₁₆	D₁₀₉
hypocenter	C₁₀₉	#1966
LCS	G ▷ C₂³×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄×M₁₆

subgroup

G/H

center

C₂²

C₂×#13182006

commutator, prime radical

C₂³×C₁₀₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₀₉

Fitting

C₄×M₁₆×C₁₀₉

C₂

socle

C₂²×C₁₀₉

C₂×C₂²⋊C₄

cosocle

C₂²×C₄×C₁₀₉

C₂³

hypercenter

C₄×M₁₆

D₁₀₉

hypocenter

C₁₀₉

#1966

LCS

G ▷ C₂³×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄×M₁₆

N	H ≅ G/N	type
53. C₄×C₁₀₉	#101	semidirect
62. C₁₀₉	#1966	semidirect
116. C₄	#16812617	semidirect

H ≅ G/N

type

53. C₄×C₁₀₉

#101

semidirect

62. C₁₀₉

#1966

semidirect

116. C₄

#16812617

semidirect

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

subgroup	index
C₂×#16812699	2
#20295050	2
C₂×#16812617	2
C₄×M₁₆×C₁₀₉	2
#1966	109

subgroup

index

C₄×M₁₆×C₁₀₉

109

Subgroups — #28363876

Conjugacy classes of subgroups

Complements

Maximal subgroups