Group #28379244 — subgroups

Subgroups — #28379244

Order 13952.

184 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 184 conjugacy classes of subgroups — too many to draw. Only its 96 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	#20295989	C₂	1	1	yes	maximal, characteristic
3	#20295913	C₂	1	2	yes	maximal, complemented
4	C₁₆×#89137	C₂	1	2	yes	maximal, complemented
5	M₃₂×D₁₀₉	C₂	1	1	yes	maximal, characteristic
6	D₁₀₉⋊M₁₆	C₄	1	1	yes	characteristic
7	#16812778	C₄	1	1	yes	characteristic
8	D₁₀₉⋊C₁₆	C₄	1	2	yes	complemented
9	D₁₀₉⋊C₁₆	C₂²	1	1	yes	characteristic
10	D₁₀₉⋊C₁₆	C₂²	1	1	yes	characteristic
11	C₂×#13182032	C₄	1	2	yes	complemented
12	#16812753	C₄	1	2	yes	—
13	C₈×#89137	C₂²	1	2	yes	—
14	C₂×C₈×D₁₀₉	C₂²	1	1	yes	characteristic
15	#16812639	C₄	1	1	yes	characteristic, complemented
16	#16812638	C₄	1	2	yes	complemented
17	C₁₆×D₁₀₉	C₂²	1	2	yes	—
18	M₃₂×C₁₀₉	C₄	1	1	yes	characteristic, complemented
19	#13182039	C₈	1	2	yes	—
20	#13182038	C₈	1	2	yes	—
21	D₁₀₉⋊C₈	C₂×C₄	1	2	yes	—
22	C₄⋊#89137	C₈	1	2	yes	—
23	C₂²⋊#89137	C₈	1	2	yes	—
24	C₄×#89137	C₂×C₄	1	2	yes	—
25	#13182032	C₂×C₄	1	2	yes	—
26	#13182032	C₂×C₄	1	2	yes	—
27	C₂×C₄×D₁₀₉	C₂×C₄	1	1	yes	characteristic
28	C₂×#11724784	C₂×C₄	1	1	yes	characteristic
29	C₈×D₁₀₉	C₂×C₄	1	1	yes	characteristic
30	C₈×D₁₀₉	C₂×C₄	1	1	yes	characteristic
31	C₈×D₁₀₉	C₂×C₄	1	1	yes	characteristic
32	C₈×D₁₀₉	C₂³	1	1	yes	characteristic
33	#13182004	C₂×C₄	1	2	yes	—
34	C₂×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
35	C₁₆×C₁₀₉	C₂×C₄	1	2	yes	—
36	C₂²×D₁₀₉	C₂×C₈	1	1	yes	characteristic
37	C₂×#89137	C₂×C₈	1	4	yes	—
38	C₂×C₄×C₁₀₉	C₄²	1	1	yes	characteristic
39	C₂×Dic₁₀₉	C₂×C₈	1	1	yes	characteristic
40	C₄×D₁₀₉	C₄²	1	1	yes	characteristic
41	C₄×D₁₀₉	C₂×C₈	1	1	yes	characteristic
42	C₄×D₁₀₉	C₂×C₈	1	1	yes	characteristic
43	C₄×D₁₀₉	C₂²×C₄	1	1	yes	characteristic
44	#11724786	C₂×C₈	1	4	yes	—
45	C₈×C₁₀₉	C₄²	1	1	yes	characteristic
46	C₈×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
47	#11724784	C₄²	1	1	yes	characteristic
48	#11724784	C₂²×C₄	1	1	yes	characteristic
49	C₂²×C₁₀₉	C₄×C₈	1	1	yes	characteristic
50	C₂×D₁₀₉	—	2	2	no	—
51	C₂×D₁₀₉	C₄×C₈	1	1	yes	characteristic
52	C₂×D₁₀₉	C₂²×C₈	1	1	yes	characteristic
53	#89137	—	2	4	no	—
54	C₄×C₁₀₉	C₄×C₈	1	1	yes	characteristic
55	C₄×C₁₀₉	C₂×C₄²	1	1	yes	characteristic
56	Dic₁₀₉	C₄×C₈	1	1	yes	characteristic
57	Dic₁₀₉	C₂²×C₈	1	1	yes	characteristic
58	C₂×C₁₀₉	—	2	1	no	—
59	C₂×C₁₀₉	C₂×C₄×C₈	1	1	yes	characteristic
60	D₁₀₉	—	2	1	no	—
61	D₁₀₉	#504	1	2	yes	—
62	#2108	—	109	1	no	maximal, Sylow
63	C₁₀₉	#2108	1	1	yes	characteristic, complemented, Sylow
64	C₂×M₃₂	—	109	1	no	—
65	C₂²×C₁₆	—	109	1	no	—
66	#405	—	109	1	no	—
67	#346	—	109	2	no	—
68	C₄×C₁₆	—	109	2	no	—
69	C₂×M₁₆	—	109	1	no	—
70	C₂²×C₈	—	109	1	no	—
71	#117	—	109	1	no	—
72	M₃₂	—	109	1	no	—
73	M₃₂	—	109	2	no	—
74	M₃₂	#89137	1	1	yes	characteristic, complemented
75	C₂×C₁₆	—	109	1	no	—
76	C₂×C₁₆	—	109	1	no	—
77	C₂×C₁₆	—	109	2	no	—
78	C₂×C₁₆	—	109	2	no	—
79	C₂×C₁₆	—	109	2	no	—
80	#97	—	109	2	no	—
81	C₄×C₈	—	109	2	no	—
82	C₂²×C₄	—	109	1	no	—
83	M₁₆	—	109	2	no	—
84	M₁₆	—	109	2	no	—
85	C₂×C₈	—	109	2	no	—
86	C₂×C₈	—	109	1	no	—
87	C₂×C₈	—	109	1	no	—
88	C₂×C₈	—	109	1	no	—
89	C₂×C₈	—	109	1	no	—
90	C₂×C₈	—	109	1	no	—
91	C₂×C₈	C₂×#89137	1	1	yes	characteristic
92	C₄⋊C₄	—	109	2	no	—
93	C₂²⋊C₄	—	109	2	no	—
94	C₄²	—	109	2	no	—
95	C₁₆	—	109	2	no	—
96	C₁₆	—	109	2	no	—
97	C₁₆	—	109	2	no	—
98	C₁₆	C₂×#89137	1	2	yes	—
99	C₂³	—	109	1	no	—
100	C₂×C₄	—	109	1	no	—
101	C₂×C₄	—	109	1	no	—
102	C₂×C₄	—	109	1	no	—
103	C₂×C₄	—	109	1	no	—
104	C₂×C₄	—	109	1	no	—
105	C₂×C₄	—	109	4	no	—
106	C₂×C₄	C₄×#89137	1	1	yes	characteristic
107	C₈	—	109	4	no	—
108	C₈	—	109	1	no	—
109	C₈	—	109	1	no	—
110	C₈	C₄×#89137	1	1	yes	characteristic
111	C₈	C₂²×#89137	1	1	yes	characteristic
112	C₂²	—	109	1	no	—
113	C₂²	—	109	1	no	—
114	C₂²	—	218	2	no	—
115	C₂²	C₈×#89137	1	1	yes	characteristic
116	C₄	—	109	1	no	—
117	C₄	—	109	1	no	—
118	C₄	—	218	4	no	—
119	C₄	C₈×#89137	1	1	yes	characteristic
120	C₄	C₂×C₄×#89137	1	1	yes	characteristic
121	C₂	—	2	1	no	—
122	C₂	—	109	2	no	—
123	C₂	—	218	1	no	—
124	C₂	C₂×C₈×#89137	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×D₁₀₉⋊C₁₆

C₂

yes

maximal, characteristic

#20295989

C₂

yes

maximal, characteristic

#20295913

C₂

yes

maximal, complemented

C₁₆×#89137

C₂

yes

maximal, complemented

M₃₂×D₁₀₉

C₂

yes

maximal, characteristic

D₁₀₉⋊M₁₆

C₄

yes

characteristic

#16812778

C₄

yes

characteristic

D₁₀₉⋊C₁₆

C₄

yes

complemented

D₁₀₉⋊C₁₆

C₂²

yes

characteristic

D₁₀₉⋊C₁₆

C₂²

yes

characteristic

C₂×#13182032

C₄

yes

complemented

#16812753

C₄

yes

—

C₈×#89137

C₂²

yes

—

C₂×C₈×D₁₀₉

C₂²

yes

characteristic

#16812639

C₄

yes

characteristic, complemented

#16812638

C₄

yes

complemented

C₁₆×D₁₀₉

C₂²

yes

—

M₃₂×C₁₀₉

C₄

yes

characteristic, complemented

#13182039

C₈

yes

—

#13182038

C₈

yes

—

D₁₀₉⋊C₈

C₂×C₄

yes

—

C₄⋊#89137

C₈

yes

—

C₂²⋊#89137

C₈

yes

—

C₄×#89137

C₂×C₄

yes

—

#13182032

C₂×C₄

yes

—

#13182032

C₂×C₄

yes

—

C₂×C₄×D₁₀₉

C₂×C₄

yes

characteristic

C₂×#11724784

C₂×C₄

yes

characteristic

C₈×D₁₀₉

C₂×C₄

yes

characteristic

C₈×D₁₀₉

C₂×C₄

yes

characteristic

C₈×D₁₀₉

C₂×C₄

yes

characteristic

C₈×D₁₀₉

C₂³

yes

characteristic

#13182004

C₂×C₄

yes

—

C₂×C₈×C₁₀₉

C₂×C₄

yes

characteristic

C₁₆×C₁₀₉

C₂×C₄

yes

—

C₂²×D₁₀₉

C₂×C₈

yes

characteristic

C₂×#89137

C₂×C₈

yes

—

C₂×C₄×C₁₀₉

C₄²

yes

characteristic

C₂×Dic₁₀₉

C₂×C₈

yes

characteristic

C₄×D₁₀₉

C₄²

yes

characteristic

C₄×D₁₀₉

C₂×C₈

yes

characteristic

C₄×D₁₀₉

C₂×C₈

yes

characteristic

C₄×D₁₀₉

C₂²×C₄

yes

characteristic

#11724786

C₂×C₈

yes

—

C₈×C₁₀₉

C₄²

yes

characteristic

C₈×C₁₀₉

C₂²×C₄

yes

characteristic

#11724784

C₄²

yes

characteristic

#11724784

C₂²×C₄

yes

characteristic

C₂²×C₁₀₉

C₄×C₈

yes

characteristic

C₂×D₁₀₉

—

C₂×D₁₀₉

C₄×C₈

yes

characteristic

C₂×D₁₀₉

C₂²×C₈

yes

characteristic

#89137

—

C₄×C₁₀₉

C₄×C₈

yes

characteristic

C₄×C₁₀₉

C₂×C₄²

yes

characteristic

Dic₁₀₉

C₄×C₈

yes

characteristic

Dic₁₀₉

C₂²×C₈

yes

characteristic

C₂×C₁₀₉

—

C₂×C₁₀₉

C₂×C₄×C₈

yes

characteristic

D₁₀₉

—

D₁₀₉

#504

yes

—

#2108

—

109

maximal, Sylow

C₁₀₉

#2108

yes

characteristic, complemented, Sylow

C₂×M₃₂

—

109

—

C₂²×C₁₆

—

109

—

#405

—

109

—

#346

—

109

—

C₄×C₁₆

—

109

—

C₂×M₁₆

—

109

—

C₂²×C₈

—

109

—

#117

—

109

—

M₃₂

—

109

—

M₃₂

—

109

—

M₃₂

#89137

yes

characteristic, complemented

C₂×C₁₆

—

109

—

C₂×C₁₆

—

109

—

C₂×C₁₆

—

109

—

C₂×C₁₆

—

109

—

C₂×C₁₆

—

109

—

#97

—

109

—

C₄×C₈

—

109

—

C₂²×C₄

—

109

—

M₁₆

—

109

—

M₁₆

—

109

—

C₂×C₈

—

109

—

C₂×C₈

—

109

—

C₂×C₈

—

109

—

C₂×C₈

—

109

—

C₂×C₈

—

109

—

C₂×C₈

—

109

—

C₂×C₈

C₂×#89137

yes

characteristic

C₄⋊C₄

—

109

—

C₂²⋊C₄

—

109

—

C₄²

—

109

—

C₁₆

—

109

—

C₁₆

—

109

—

C₁₆

—

109

—

C₁₆

C₂×#89137

yes

—

C₂³

—

109

—

100

C₂×C₄

—

109

—

101

C₂×C₄

—

109

—

102

C₂×C₄

—

109

—

103

C₂×C₄

—

109

—

104

C₂×C₄

—

109

—

105

C₂×C₄

—

109

—

106

C₂×C₄

C₄×#89137

yes

characteristic

107

C₈

—

109

—

108

C₈

—

109

—

109

C₈

—

109

—

110

C₈

C₄×#89137

yes

characteristic

111

C₈

C₂²×#89137

yes

characteristic

112

C₂²

—

109

—

113

C₂²

—

109

—

114

C₂²

—

218

—

115

C₂²

C₈×#89137

yes

characteristic

116

C₄

—

109

—

117

C₄

—

109

—

118

C₄

—

218

—

119

C₄

C₈×#89137

yes

characteristic

120

C₄

C₂×C₄×#89137

yes

characteristic

121

C₂

—

122

C₂

—

109

—

123

C₂

—

218

—

124

C₂

C₂×C₈×#89137

yes

characteristic

	subgroup	G/H
center, Frattini	C₈	C₂²×#89137
commutator, socle	C₂×C₁₀₉	C₂×C₄×C₈
Fitting	M₃₂×C₁₀₉	C₄
prime radical	C₂²×D₁₀₉	C₂×C₈
cosocle	C₈×D₁₀₉	C₂³
hypercenter	M₃₂	#89137
hypocenter	C₁₀₉	#2108
LCS	G ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₈ ◁ M₃₂

subgroup

G/H

center, Frattini

C₈

C₂²×#89137

commutator, socle

C₂×C₁₀₉

C₂×C₄×C₈

Fitting

M₃₂×C₁₀₉

C₄

prime radical

C₂²×D₁₀₉

C₂×C₈

cosocle

C₈×D₁₀₉

C₂³

M₃₂

C₁₀₉

G ▷ C₂×C₁₀₉ ▷ C₁₀₉

C₁ ◁ C₈ ◁ M₃₂

N	H ≅ G/N	type
3. #20295913	C₂	semidirect
4. C₁₆×#89137	C₂	semidirect
8. D₁₀₉⋊C₁₆	C₄	semidirect
11. C₂×#13182032	C₄	semidirect
15. #16812639	C₄	semidirect
16. #16812638	C₄	semidirect
18. M₃₂×C₁₀₉	C₄	semidirect
63. C₁₀₉	#2108	semidirect
74. M₃₂	#89137	semidirect

H ≅ G/N

type

3. #20295913

C₂

semidirect

4. C₁₆×#89137

C₂

semidirect

8. D₁₀₉⋊C₁₆

C₄

semidirect

11. C₂×#13182032

C₄

semidirect

15. #16812639

C₄

semidirect

16. #16812638

C₄

semidirect

18. M₃₂×C₁₀₉

C₄

semidirect

63. C₁₀₉

#2108

semidirect

74. M₃₂

#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
#20295989	2
#20295913	2
C₁₆×#89137	2
M₃₂×D₁₀₉	2
#2108	109

subgroup

index

C₂×D₁₀₉⋊C₁₆

#20295989

#20295913

C₁₆×#89137

M₃₂×D₁₀₉

#2108

109

Subgroups — #28379244

Conjugacy classes of subgroups

Complements

Maximal subgroups