Group #28379362 — subgroups

Subgroups — D₁₀₉⋊(C₄⋊C₁₆) #28379362

Order 13952.

184 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 184 conjugacy classes of subgroups — too many to draw. Only its 94 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×D₁₀₉⋊C₁₆	C₂	1	2	yes	maximal
2	#20295947	C₂	1	2	yes	maximal, complemented
3	C₄⋊#13182032	C₂	1	2	yes	maximal, complemented
4	C₄×C₈×D₁₀₉	C₂	1	1	yes	maximal, characteristic
5	D₁₀₉⋊C₁₆	—	2	4	no	—
6	C₂×#13182032	C₂²	1	4	yes	—
7	C₂×C₈×D₁₀₉	C₄	1	1	yes	characteristic
8	C₂×C₈×D₁₀₉	C₂²	1	1	yes	characteristic
9	C₄²×D₁₀₉	C₄	1	1	yes	characteristic
10	C₄×#11724784	C₂²	1	1	yes	characteristic
11	C₈×Dic₁₀₉	C₄	1	1	yes	characteristic
12	C₈×Dic₁₀₉	C₂²	1	1	yes	characteristic
13	C₄×C₈×C₁₀₉	C₄	1	1	yes	characteristic
14	#13182032	—	2	4	no	—
15	C₂×C₄×D₁₀₉	C₈	1	1	yes	characteristic
16	C₂×C₄×D₁₀₉	C₈	1	1	yes	characteristic
17	C₂×C₄×D₁₀₉	C₂×C₄	1	1	yes	characteristic
18	C₂×#11724784	C₂×C₄	1	1	yes	characteristic
19	C₂×#11724784	C₂³	1	1	yes	characteristic
20	C₈×D₁₀₉	—	2	2	no	—
21	C₈×D₁₀₉	D₄	1	2	yes	—
22	C₈×D₁₀₉	Q₈	1	2	yes	—
23	C₄×Dic₁₀₉	C₈	1	1	yes	characteristic
24	C₄×Dic₁₀₉	C₈	1	1	yes	characteristic
25	C₄×Dic₁₀₉	C₂×C₄	1	1	yes	characteristic
26	C₂×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
27	C₂×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
28	C₄²×C₁₀₉	C₂×C₄	1	1	yes	characteristic
29	C₂²×D₁₀₉	C₂×C₈	1	1	yes	characteristic
30	C₂×C₄×C₁₀₉	C₂×C₈	1	1	yes	characteristic
31	C₂×C₄×C₁₀₉	C₂×C₈	1	1	yes	characteristic
32	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
33	C₂×Dic₁₀₉	C₂×C₈	1	1	yes	characteristic
34	C₂×Dic₁₀₉	C₂×C₈	1	1	yes	characteristic
35	C₂×Dic₁₀₉	C₂×C₈	1	1	yes	characteristic
36	C₄×D₁₀₉	—	2	2	no	—
37	C₄×D₁₀₉	C₁₆	1	4	yes	complemented
38	C₄×D₁₀₉	C₄⋊C₄	1	2	yes	—
39	C₄×D₁₀₉	M₁₆	1	2	yes	—
40	C₈×C₁₀₉	—	2	1	no	—
41	C₈×C₁₀₉	C₄⋊C₄	1	2	yes	—
42	#11724784	—	2	1	no	—
43	#11724784	C₂×D₄	1	1	yes	characteristic
44	#11724784	C₂×Q₈	1	1	yes	characteristic
45	C₂²×C₁₀₉	C₂²×C₈	1	1	yes	characteristic
46	C₂×D₁₀₉	C₄⋊C₈	1	2	yes	—
47	C₂×D₁₀₉	C₂×C₁₆	1	2	yes	—
48	C₂×D₁₀₉	M₃₂	1	2	yes	—
49	C₄×C₁₀₉	—	2	1	no	—
50	C₄×C₁₀₉	C₂×C₁₆	1	2	yes	complemented
51	C₄×C₁₀₉	C₂×C₄⋊C₄	1	1	yes	characteristic
52	C₄×C₁₀₉	C₂×M₁₆	1	1	yes	characteristic
53	Dic₁₀₉	—	2	1	no	—
54	Dic₁₀₉	C₄⋊C₈	1	2	yes	—
55	Dic₁₀₉	C₂×C₁₆	1	2	yes	complemented
56	C₂×C₁₀₉	C₂×C₄⋊C₈	1	1	yes	characteristic
57	C₂×C₁₀₉	C₂²×C₁₆	1	1	yes	characteristic
58	C₂×C₁₀₉	C₂×M₃₂	1	1	yes	characteristic
59	D₁₀₉	C₄⋊C₁₆	1	4	yes	complemented
60	C₂×C₄⋊C₁₆	—	109	1	no	maximal, Sylow
61	C₁₀₉	C₂×C₄⋊C₁₆	1	1	yes	characteristic, complemented, Sylow
62	C₂²×C₁₆	—	109	2	no	—
63	C₂×C₄×C₈	—	109	1	no	—
64	C₄⋊C₁₆	—	109	2	no	—
65	C₄⋊C₁₆	—	109	2	no	—
66	C₂²×C₈	—	109	1	no	—
67	C₂²×C₈	—	109	1	no	—
68	C₂×C₄²	—	109	1	no	—
69	C₂×C₁₆	—	109	4	no	—
70	C₂×C₁₆	—	218	4	no	—
71	C₄×C₈	—	109	1	no	—
72	C₄×C₈	—	109	1	no	—
73	C₄×C₈	—	109	1	no	—
74	C₄×C₈	#89137	1	1	yes	characteristic
75	C₂²×C₄	—	109	1	no	—
76	C₂²×C₄	—	109	1	no	—
77	C₂²×C₄	—	109	1	no	—
78	C₂×C₈	—	109	1	no	—
79	C₂×C₈	—	109	2	no	—
80	C₂×C₈	—	109	2	no	—
81	C₂×C₈	—	109	1	no	—
82	C₂×C₈	—	218	2	no	—
83	C₂×C₈	C₂×#89137	1	1	yes	characteristic
84	C₂×C₈	C₂×#89137	1	1	yes	characteristic
85	C₄²	—	109	1	no	—
86	C₄²	—	109	1	no	—
87	C₄²	—	109	1	no	—
88	C₄²	C₂×#89137	1	1	yes	characteristic
89	C₁₆	—	218	4	no	—
90	C₂³	—	109	1	no	—
91	C₂×C₄	—	109	1	no	—
92	C₂×C₄	—	109	1	no	—
93	C₂×C₄	—	109	4	no	—
94	C₂×C₄	—	109	2	no	—
95	C₂×C₄	—	109	2	no	—
96	C₂×C₄	—	109	1	no	—
97	C₂×C₄	—	218	2	no	—
98	C₂×C₄	D₁₀₉⋊C₈	1	1	yes	characteristic
99	C₂×C₄	D₁₀₉⋊C₈	1	1	yes	characteristic
100	C₂×C₄	C₂²×#89137	1	1	yes	characteristic
101	C₈	—	2	1	no	—
102	C₈	—	109	1	no	—
103	C₈	—	109	1	no	—
104	C₈	—	218	1	no	—
105	C₈	C₄⋊#89137	1	2	yes	—
106	C₂²	—	109	2	no	—
107	C₂²	—	109	2	no	—
108	C₂²	—	109	2	no	—
109	C₂²	C₂×D₁₀₉⋊C₈	1	1	yes	characteristic
110	C₄	—	2	1	no	—
111	C₄	—	109	2	no	—
112	C₄	—	109	2	no	—
113	C₄	—	218	1	no	—
114	C₄	D₁₀₉⋊C₁₆	1	2	yes	complemented
115	C₄	C₂×C₄⋊#89137	1	1	yes	characteristic
116	C₄	D₁₀₉⋊M₁₆	1	1	yes	characteristic
117	C₂	—	109	4	no	—
118	C₂	D₁₀₉⋊(C₄⋊C₈)	1	1	yes	characteristic
119	C₂	C₂×D₁₀₉⋊C₁₆	1	1	yes	characteristic
120	C₂	D₁₀₉⋊M₃₂	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×D₁₀₉⋊C₁₆

C₂

yes

maximal

#20295947

C₂

yes

maximal, complemented

C₄⋊#13182032

C₂

yes

maximal, complemented

C₄×C₈×D₁₀₉

C₂

yes

maximal, characteristic

D₁₀₉⋊C₁₆

—

C₂×#13182032

C₂²

yes

—

C₂×C₈×D₁₀₉

C₄

yes

characteristic

C₂×C₈×D₁₀₉

C₂²

yes

characteristic

C₄²×D₁₀₉

C₄

yes

characteristic

C₄×#11724784

C₂²

yes

characteristic

C₈×Dic₁₀₉

C₄

yes

characteristic

C₈×Dic₁₀₉

C₂²

yes

characteristic

C₄×C₈×C₁₀₉

C₄

yes

characteristic

#13182032

—

C₂×C₄×D₁₀₉

C₈

yes

characteristic

C₂×C₄×D₁₀₉

C₈

yes

characteristic

C₂×C₄×D₁₀₉

C₂×C₄

yes

characteristic

C₂×#11724784

C₂×C₄

yes

characteristic

C₂×#11724784

C₂³

yes

characteristic

C₈×D₁₀₉

—

C₈×D₁₀₉

D₄

yes

—

C₈×D₁₀₉

Q₈

yes

—

C₄×Dic₁₀₉

C₈

yes

characteristic

C₄×Dic₁₀₉

C₈

yes

characteristic

C₄×Dic₁₀₉

C₂×C₄

yes

characteristic

C₂×C₈×C₁₀₉

C₂×C₄

yes

characteristic

C₂×C₈×C₁₀₉

C₂×C₄

yes

characteristic

C₄²×C₁₀₉

C₂×C₄

yes

characteristic

C₂²×D₁₀₉

C₂×C₈

yes

characteristic

C₂×C₄×C₁₀₉

C₂×C₈

yes

characteristic

C₂×C₄×C₁₀₉

C₂×C₈

yes

characteristic

C₂×C₄×C₁₀₉

C₂²×C₄

yes

characteristic

C₂×Dic₁₀₉

C₂×C₈

yes

characteristic

C₂×Dic₁₀₉

C₂×C₈

yes

characteristic

C₂×Dic₁₀₉

C₂×C₈

yes

characteristic

C₄×D₁₀₉

—

C₄×D₁₀₉

C₁₆

yes

complemented

C₄×D₁₀₉

C₄⋊C₄

yes

—

C₄×D₁₀₉

M₁₆

yes

—

C₈×C₁₀₉

—

C₈×C₁₀₉

C₄⋊C₄

yes

—

#11724784

—

#11724784

C₂×D₄

yes

characteristic

#11724784

C₂×Q₈

yes

characteristic

C₂²×C₁₀₉

C₂²×C₈

yes

characteristic

C₂×D₁₀₉

C₄⋊C₈

yes

—

C₂×D₁₀₉

C₂×C₁₆

yes

—

C₂×D₁₀₉

M₃₂

yes

—

C₄×C₁₀₉

—

C₄×C₁₀₉

C₂×C₁₆

yes

complemented

C₄×C₁₀₉

C₂×C₄⋊C₄

yes

characteristic

C₄×C₁₀₉

C₂×M₁₆

yes

characteristic

Dic₁₀₉

—

Dic₁₀₉

C₄⋊C₈

yes

—

Dic₁₀₉

C₂×C₁₆

yes

complemented

C₂×C₁₀₉

C₂×C₄⋊C₈

yes

characteristic

C₂×C₁₀₉

C₂²×C₁₆

yes

characteristic

C₂×C₁₀₉

C₂×M₃₂

yes

characteristic

D₁₀₉

C₄⋊C₁₆

yes

complemented

C₂×C₄⋊C₁₆

—

109

maximal, Sylow

C₁₀₉

C₂×C₄⋊C₁₆

yes

characteristic, complemented, Sylow

C₂²×C₁₆

—

109

—

C₂×C₄×C₈

—

109

—

C₄⋊C₁₆

—

109

—

C₄⋊C₁₆

—

109

—

C₂²×C₈

—

109

—

C₂²×C₈

—

109

—

C₂×C₄²

—

109

—

C₂×C₁₆

—

109

—

C₂×C₁₆

—

218

—

C₄×C₈

—

109

—

C₄×C₈

—

109

—

C₄×C₈

—

109

—

C₄×C₈

#89137

yes

characteristic

C₂²×C₄

—

109

—

C₂²×C₄

—

109

—

C₂²×C₄

—

109

—

C₂×C₈

—

109

—

C₂×C₈

—

109

—

C₂×C₈

—

109

—

C₂×C₈

—

109

—

C₂×C₈

—

218

—

C₂×C₈

yes

C₂×C₈

yes

C₄²

—

109

—

C₄²

—

109

—

C₄²

—

109

—

C₄²

C₂×#89137

yes

characteristic

C₁₆

—

218

—

C₂³

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

218

—

C₂×C₄

D₁₀₉⋊C₈

yes

characteristic

C₂×C₄

D₁₀₉⋊C₈

yes

characteristic

100

C₂×C₄

C₂²×#89137

yes

characteristic

101

C₈

—

102

C₈

—

109

—

103

C₈

—

109

—

104

C₈

—

218

—

105

C₈

C₄⋊#89137

yes

—

106

C₂²

—

109

—

107

C₂²

—

109

—

108

C₂²

—

109

—

109

C₂²

C₂×D₁₀₉⋊C₈

yes

characteristic

110

C₄

—

111

C₄

—

109

—

112

C₄

—

109

—

113

C₄

—

218

—

114

C₄

D₁₀₉⋊C₁₆

yes

complemented

115

C₄

C₂×C₄⋊#89137

yes

characteristic

116

C₄

D₁₀₉⋊M₁₆

yes

characteristic

117

C₂

—

109

—

118

C₂

D₁₀₉⋊(C₄⋊C₈)

yes

characteristic

119

C₂

C₂×D₁₀₉⋊C₁₆

yes

characteristic

120

C₂

D₁₀₉⋊M₃₂

yes

characteristic

	subgroup	G/H
center	C₂×C₈	C₂×#89137
commutator	C₂×C₁₀₉	C₂²×C₁₆
Frattini	C₂×C₄	C₂²×#89137
Fitting	C₄×C₈×C₁₀₉	C₄
socle	C₂²×C₁₀₉	C₂²×C₈
prime radical	C₂²×D₁₀₉	C₂×C₈
cosocle	C₂×#11724784	C₂³
hypercenter	C₄×C₈	#89137
hypocenter	C₁₀₉	C₂×C₄⋊C₁₆
LCS	G ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂×C₈ ◁ C₄×C₈

subgroup

G/H

center

C₂×C₈

C₂×#89137

commutator

C₂×C₁₀₉

C₂²×C₁₆

Frattini

C₂×C₄

C₂²×#89137

Fitting

C₄×C₈×C₁₀₉

C₄

socle

C₂²×C₁₀₉

C₂²×C₈

prime radical

C₂²×D₁₀₉

C₂×C₈

C₂³

C₄×C₈

C₁₀₉

C₂×C₄⋊C₁₆

LCS

G ▷ C₂×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂×C₈ ◁ C₄×C₈

N	H ≅ G/N	type
2. #20295947	C₂	semidirect
3. C₄⋊#13182032	C₂	semidirect
37. C₄×D₁₀₉	C₁₆	semidirect
50. C₄×C₁₀₉	C₂×C₁₆	semidirect
55. Dic₁₀₉	C₂×C₁₆	semidirect
59. D₁₀₉	C₄⋊C₁₆	semidirect
61. C₁₀₉	C₂×C₄⋊C₁₆	semidirect
114. C₄	D₁₀₉⋊C₁₆	semidirect

H ≅ G/N

type

2. #20295947

C₂

semidirect

3. C₄⋊#13182032

C₂

semidirect

37. C₄×D₁₀₉

C₁₆

semidirect

50. C₄×C₁₀₉

C₂×C₁₆

semidirect

55. Dic₁₀₉

C₂×C₁₆

semidirect

59. D₁₀₉

C₄⋊C₁₆

semidirect

61. C₁₀₉

C₂×C₄⋊C₁₆

semidirect

114. C₄

D₁₀₉⋊C₁₆

semidirect

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

subgroup	index
C₂×D₁₀₉⋊C₁₆	2
#20295947	2
C₄⋊#13182032	2
C₄×C₈×D₁₀₉	2
C₂×C₄⋊C₁₆	109

subgroup

index

C₂×D₁₀₉⋊C₁₆

#20295947

C₄⋊#13182032

C₄×C₈×D₁₀₉

C₂×C₄⋊C₁₆

109

Subgroups — D₁₀₉⋊(C₄⋊C₁₆) #28379362

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D109⋊(C4⋊C16) #28379362

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₁₀₉⋊(C₄⋊C₁₆) #28379362