Group #28378069 — subgroups

Subgroups — (C₂×#89137)⋊C₁₆ #28378069

Order 13952.

160 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 160 conjugacy classes of subgroups — too many to draw. Only its 68 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	1	yes	maximal, characteristic
2	C₂×C₈×#89137	C₂	1	1	yes	maximal, characteristic
3	C₂×C₁₆×D₁₀₉	C₂	1	1	yes	maximal, characteristic
4	C₂×D₁₀₉⋊C₈	C₄	1	1	yes	characteristic
5	C₂×C₄×#89137	C₄	1	1	yes	characteristic
6	D₁₀₉⋊C₁₆	—	2	1	no	—
7	D₁₀₉⋊C₁₆	—	2	1	no	—
8	C₂×#13182032	C₄	1	2	yes	complemented
9	C₈×#89137	—	2	1	no	—
10	C₈×#89137	—	2	1	no	—
11	C₂×C₈×D₁₀₉	C₂²	1	1	yes	characteristic
12	C₂×#13182004	C₄	1	1	yes	characteristic, complemented
13	C₁₆×D₁₀₉	—	2	1	no	—
14	C₁₆×D₁₀₉	—	2	1	no	—
15	C₂×C₁₆×C₁₀₉	C₄	1	1	yes	characteristic, complemented
16	C₂²×#89137	C₈	1	2	yes	—
17	D₁₀₉⋊C₈	—	2	1	no	—
18	D₁₀₉⋊C₈	—	2	1	no	—
19	C₂×#11724786	C₈	1	2	yes	—
20	C₄×#89137	—	2	1	no	—
21	C₄×#89137	—	2	1	no	—
22	#13182032	—	2	2	no	—
23	C₂×C₄×D₁₀₉	C₂×C₄	1	1	yes	characteristic
24	C₂×#11724784	C₂×C₄	1	1	yes	characteristic
25	C₈×D₁₀₉	D₄	1	1	yes	characteristic
26	C₈×D₁₀₉	D₄	1	1	yes	characteristic
27	C₈×D₁₀₉	D₄	1	1	yes	characteristic
28	C₈×D₁₀₉	Q₈	1	1	yes	characteristic
29	#13182004	—	2	1	no	—
30	C₂×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
31	C₁₆×C₁₀₉	—	2	1	no	—
32	C₂²×D₁₀₉	C₂×C₈	1	1	yes	characteristic
33	C₂×#89137	—	2	2	no	—
34	C₂×#89137	—	2	2	no	—
35	C₂×#89137	C₁₆	1	4	yes	complemented
36	C₂×C₄×C₁₀₉	C₄²	1	1	yes	characteristic
37	C₂×Dic₁₀₉	C₂×C₈	1	1	yes	characteristic
38	C₄×D₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
39	C₄×D₁₀₉	C₄⋊C₄	1	1	yes	characteristic
40	C₄×D₁₀₉	M₁₆	1	1	yes	characteristic
41	C₄×D₁₀₉	M₁₆	1	1	yes	characteristic
42	#11724786	—	2	2	no	—
43	C₈×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
44	C₈×C₁₀₉	C₄⋊C₄	1	1	yes	characteristic
45	#11724784	C₂²⋊C₄	1	1	yes	characteristic
46	#11724784	C₄⋊C₄	1	1	yes	characteristic
47	C₂²×C₁₀₉	C₄×C₈	1	1	yes	characteristic
48	C₂×D₁₀₉	C₂²⋊C₈	1	1	yes	characteristic
49	C₂×D₁₀₉	C₄⋊C₈	1	1	yes	characteristic
50	C₂×D₁₀₉	C₂×C₁₆	1	2	yes	—
51	C₂×D₁₀₉	M₃₂	1	2	yes	—
52	#89137	—	2	4	no	—
53	C₄×C₁₀₉	#95	1	1	yes	characteristic
54	C₄×C₁₀₉	#97	1	1	yes	characteristic
55	Dic₁₀₉	C₂²⋊C₈	1	1	yes	characteristic
56	Dic₁₀₉	C₄⋊C₈	1	1	yes	characteristic
57	C₂×C₁₀₉	#336	1	1	yes	characteristic
58	C₂×C₁₀₉	C₄×C₁₆	1	1	yes	characteristic
59	C₂×C₁₀₉	#346	1	1	yes	characteristic
60	D₁₀₉	C₂²⋊C₁₆	1	2	yes	—
61	D₁₀₉	C₄⋊C₁₆	1	2	yes	—
62	#1374	—	109	1	no	maximal, Sylow
63	C₁₀₉	#1374	1	1	yes	characteristic, complemented, Sylow
64	C₂²×C₁₆	—	109	1	no	—
65	C₂²×C₁₆	—	109	1	no	—
66	C₂×C₄×C₈	—	109	1	no	—
67	C₂²×C₈	—	109	1	no	—
68	C₂²×C₈	—	109	1	no	—
69	C₂×C₄²	—	109	1	no	—
70	C₂×C₁₆	—	109	2	no	—
71	C₂×C₁₆	—	109	1	no	—
72	C₂×C₁₆	—	218	1	no	—
73	C₂×C₁₆	—	218	1	no	—
74	C₂×C₁₆	—	218	1	no	—
75	C₂×C₁₆	—	218	1	no	—
76	C₂×C₁₆	#89137	1	1	yes	characteristic, complemented
77	C₄×C₈	—	218	1	no	—
78	C₄×C₈	—	218	1	no	—
79	C₂²×C₄	—	109	1	no	—
80	C₂²×C₄	—	109	2	no	—
81	C₂×C₈	—	109	1	no	—
82	C₂×C₈	—	109	1	no	—
83	C₂×C₈	—	109	1	no	—
84	C₂×C₈	—	109	1	no	—
85	C₂×C₈	—	109	1	no	—
86	C₂×C₈	—	109	2	no	—
87	C₂×C₈	—	218	1	no	—
88	C₂×C₈	—	218	1	no	—
89	C₂×C₈	C₂×#89137	1	1	yes	characteristic
90	C₄²	—	218	1	no	—
91	C₄²	—	218	1	no	—
92	C₁₆	—	2	1	no	—
93	C₁₆	—	218	2	no	—
94	C₁₆	—	218	1	no	—
95	C₂³	—	109	1	no	—
96	C₂×C₄	—	109	1	no	—
97	C₂×C₄	—	109	1	no	—
98	C₂×C₄	—	109	1	no	—
99	C₂×C₄	—	109	1	no	—
100	C₂×C₄	—	109	1	no	—
101	C₂×C₄	—	109	4	no	—
102	C₂×C₄	—	218	2	no	—
103	C₂×C₄	—	218	2	no	—
104	C₂×C₄	C₄×#89137	1	1	yes	characteristic
105	C₈	—	109	1	no	—
106	C₈	—	109	1	no	—
107	C₈	—	218	2	no	—
108	C₈	C₂²⋊#89137	1	1	yes	characteristic
109	C₈	C₄⋊#89137	1	1	yes	characteristic
110	C₂²	—	109	1	no	—
111	C₂²	—	109	1	no	—
112	C₂²	—	109	2	no	—
113	C₂²	—	109	2	no	—
114	C₂²	C₈×#89137	1	1	yes	characteristic
115	C₄	—	109	1	no	—
116	C₄	—	109	1	no	—
117	C₄	—	218	4	no	—
118	C₄	(C₂×#89137)⋊C₄	1	1	yes	characteristic
119	C₄	#16812753	1	1	yes	characteristic
120	C₂	—	109	2	no	—
121	C₂	—	109	2	no	—
122	C₂	(C₂×#89137)⋊C₈	1	1	yes	characteristic
123	C₂	C₁₆×#89137	1	1	yes	characteristic
124	C₂	#20295913	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×D₁₀₉⋊C₁₆

C₂

yes

maximal, characteristic

C₂×C₈×#89137

C₂

yes

maximal, characteristic

C₂×C₁₆×D₁₀₉

C₂

yes

maximal, characteristic

C₂×D₁₀₉⋊C₈

C₄

yes

characteristic

C₂×C₄×#89137

C₄

yes

characteristic

D₁₀₉⋊C₁₆

—

D₁₀₉⋊C₁₆

—

C₂×#13182032

C₄

yes

complemented

C₈×#89137

—

C₈×#89137

—

C₂×C₈×D₁₀₉

C₂²

yes

characteristic

C₂×#13182004

C₄

yes

characteristic, complemented

C₁₆×D₁₀₉

—

C₁₆×D₁₀₉

—

C₂×C₁₆×C₁₀₉

C₄

yes

characteristic, complemented

C₂²×#89137

C₈

yes

—

D₁₀₉⋊C₈

—

D₁₀₉⋊C₈

—

C₂×#11724786

C₈

yes

—

C₄×#89137

—

C₄×#89137

—

#13182032

—

C₂×C₄×D₁₀₉

C₂×C₄

yes

characteristic

C₂×#11724784

C₂×C₄

yes

characteristic

C₈×D₁₀₉

D₄

yes

characteristic

C₈×D₁₀₉

D₄

yes

characteristic

C₈×D₁₀₉

D₄

yes

characteristic

C₈×D₁₀₉

Q₈

yes

characteristic

#13182004

—

C₂×C₈×C₁₀₉

C₂×C₄

yes

characteristic

C₁₆×C₁₀₉

—

C₂²×D₁₀₉

C₂×C₈

yes

—

—

C₁₆

yes

complemented

C₂×C₄×C₁₀₉

C₄²

yes

characteristic

C₂×Dic₁₀₉

C₂×C₈

yes

characteristic

C₄×D₁₀₉

C₂²⋊C₄

yes

characteristic

C₄×D₁₀₉

C₄⋊C₄

yes

characteristic

C₄×D₁₀₉

M₁₆

yes

characteristic

C₄×D₁₀₉

M₁₆

yes

characteristic

#11724786

—

C₈×C₁₀₉

C₂²⋊C₄

yes

characteristic

C₈×C₁₀₉

C₄⋊C₄

yes

characteristic

#11724784

C₂²⋊C₄

yes

characteristic

#11724784

C₄⋊C₄

yes

characteristic

C₂²×C₁₀₉

C₄×C₈

yes

characteristic

C₂×D₁₀₉

C₂²⋊C₈

yes

characteristic

C₂×D₁₀₉

C₄⋊C₈

yes

characteristic

C₂×D₁₀₉

C₂×C₁₆

yes

—

C₂×D₁₀₉

M₃₂

yes

—

#89137

—

C₄×C₁₀₉

yes

C₄×C₁₀₉

yes

Dic₁₀₉

C₂²⋊C₈

yes

characteristic

Dic₁₀₉

C₄⋊C₈

yes

characteristic

C₂×C₁₀₉

#336

yes

characteristic

C₂×C₁₀₉

C₄×C₁₆

yes

characteristic

C₂×C₁₀₉

#346

yes

characteristic

D₁₀₉

C₂²⋊C₁₆

yes

—

D₁₀₉

C₄⋊C₁₆

yes

—

#1374

—

109

maximal, Sylow

C₁₀₉

#1374

yes

characteristic, complemented, Sylow

C₂²×C₁₆

—

109

—

C₂²×C₁₆

—

109

—

C₂×C₄×C₈

—

109

—

C₂²×C₈

—

109

—

C₂²×C₈

—

109

—

C₂×C₄²

—

109

—

C₂×C₁₆

—

109

—

C₂×C₁₆

—

109

—

C₂×C₁₆

—

218

—

C₂×C₁₆

—

218

—

C₂×C₁₆

—

218

—

C₂×C₁₆

—

218

—

C₂×C₁₆

#89137

yes

characteristic, complemented

C₄×C₈

—

218

—

C₄×C₈

—

218

—

C₂²×C₄

—

109

—

C₂²×C₄

—

109

—

C₂×C₈

—

109

—

C₂×C₈

—

109

—

C₂×C₈

—

109

—

C₂×C₈

—

109

—

C₂×C₈

—

109

—

C₂×C₈

—

109

—

C₂×C₈

—

218

—

C₂×C₈

—

218

—

C₂×C₈

C₂×#89137

yes

characteristic

C₄²

—

218

—

C₄²

—

218

—

C₁₆

—

C₁₆

—

218

—

C₁₆

—

218

—

C₂³

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

109

—

100

C₂×C₄

—

109

—

101

C₂×C₄

—

109

—

102

C₂×C₄

—

218

—

103

C₂×C₄

—

218

—

104

C₂×C₄

C₄×#89137

yes

characteristic

105

C₈

—

109

—

106

C₈

—

109

—

107

C₈

—

218

—

108

C₈

C₂²⋊#89137

yes

characteristic

109

C₈

C₄⋊#89137

yes

characteristic

110

C₂²

—

109

—

111

C₂²

—

109

—

112

C₂²

—

109

—

113

C₂²

—

109

—

114

C₂²

C₈×#89137

yes

characteristic

115

C₄

—

109

—

116

C₄

—

109

—

117

C₄

—

218

—

118

C₄

(C₂×#89137)⋊C₄

yes

characteristic

119

C₄

#16812753

yes

characteristic

120

C₂

—

109

—

121

C₂

—

109

—

122

C₂

(C₂×#89137)⋊C₈

yes

characteristic

123

C₂

C₁₆×#89137

yes

characteristic

124

C₂

#20295913

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂×#89137
commutator	C₂×C₁₀₉	C₄×C₁₆
Fitting	C₂×C₁₆×C₁₀₉	C₄
socle	C₂²×C₁₀₉	C₄×C₈
prime radical	C₂²×D₁₀₉	C₂×C₈
cosocle	C₂×C₈×D₁₀₉	C₂²
hypercenter	C₂×C₁₆	#89137
hypocenter	C₁₀₉	#1374
LCS	G ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₁₆

subgroup

G/H

center, Frattini

C₂×C₈

C₂×#89137

commutator

C₂×C₁₀₉

C₄×C₁₆

Fitting

C₂×C₁₆×C₁₀₉

C₄

socle

C₂²×C₁₀₉

C₄×C₈

prime radical

C₂²×D₁₀₉

C₂×C₈

cosocle

C₂×C₈×D₁₀₉

C₂²

C₂×C₁₆

C₁₀₉

G ▷ C₂×C₁₀₉ ▷ C₁₀₉

C₁ ◁ C₂×C₈ ◁ C₂×C₁₆

N	H ≅ G/N	type
8. C₂×#13182032	C₄	semidirect
12. C₂×#13182004	C₄	semidirect
15. C₂×C₁₆×C₁₀₉	C₄	semidirect
35. C₂×#89137	C₁₆	semidirect
63. C₁₀₉	#1374	semidirect
76. C₂×C₁₆	#89137	semidirect

H ≅ G/N

type

8. C₂×#13182032

C₄

semidirect

12. C₂×#13182004

C₄

semidirect

15. C₂×C₁₆×C₁₀₉

C₄

semidirect

35. C₂×#89137

C₁₆

semidirect

63. C₁₀₉

#1374

semidirect

76. C₂×C₁₆

#89137

semidirect

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

subgroup	index
C₂×D₁₀₉⋊C₁₆	2
C₂×C₈×#89137	2
C₂×C₁₆×D₁₀₉	2
#1374	109

subgroup

index

C₂×D₁₀₉⋊C₁₆

C₂×C₈×#89137

C₂×C₁₆×D₁₀₉

#1374

109

Subgroups — (C₂×#89137)⋊C₁₆ #28378069

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — (C2×#89137)⋊C16 #28378069

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — (C₂×#89137)⋊C₁₆ #28378069