Group #28378086 — subgroups

Subgroups — #28378086

Order 13952.

136 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 136 conjugacy classes of subgroups — too many to draw. Only its 66 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×#16812773	C₂	1	1	yes	maximal, characteristic
2	C₂²×#13182032	C₂	1	1	yes	maximal, characteristic
3	#20294982	C₂	1	1	yes	maximal, characteristic
4	#16812773	—	2	1	no	—
5	#16812773	C₄	1	2	yes	complemented
6	C₂×#13182032	—	2	1	no	—
7	C₂×#13182032	—	2	1	no	—
8	C₂×#13182032	—	2	1	no	—
9	C₂×#13182032	C₄	1	2	yes	complemented
10	C₂×#13182013	C₄	1	1	yes	characteristic
11	C₂²×#11724784	C₂²	1	1	yes	characteristic
12	#16812610	—	2	1	no	—
13	C₄×#11724784	—	2	1	no	—
14	#117×C₁₀₉	C₄	1	1	yes	characteristic
15	#13182032	—	2	2	no	—
16	#13182032	—	2	2	no	—
17	#13182013	—	2	1	no	—
18	#13182013	C₈	1	2	yes	—
19	C₂×#11724784	—	2	1	no	—
20	C₂×#11724784	C₂×C₄	1	1	yes	characteristic
21	C₂×#11724784	C₂×C₄	1	1	yes	characteristic
22	C₂×#11724784	D₄	1	1	yes	characteristic
23	C₂×#11724784	D₄	1	1	yes	characteristic
24	C₂×#11724784	D₄	1	1	yes	characteristic
25	C₂×#11724784	Q₈	1	1	yes	characteristic
26	C₂²×C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
27	C₄⋊C₄×C₁₀₉	C₈	1	2	yes	—
28	C₂²⋊C₄×C₁₀₉	—	2	1	no	—
29	C₄²×C₁₀₉	—	2	1	no	—
30	C₂³×C₁₀₉	M₁₆	1	1	yes	characteristic
31	C₂×C₄×C₁₀₉	—	2	2	no	—
32	C₂×C₄×C₁₀₉	C₄²	1	1	yes	characteristic
33	C₂×C₄×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
34	C₂×C₄×C₁₀₉	C₄⋊C₄	1	1	yes	characteristic
35	C₂×C₄×C₁₀₉	C₂×C₈	1	1	yes	characteristic
36	C₂×C₄×C₁₀₉	C₂×C₈	1	1	yes	characteristic
37	C₂×C₄×C₁₀₉	M₁₆	1	1	yes	characteristic
38	#11724784	—	2	2	no	—
39	#11724784	C₂²⋊C₄	1	1	yes	characteristic
40	#11724784	C₂²⋊C₄	1	1	yes	characteristic
41	#11724784	C₄⋊C₄	1	1	yes	characteristic
42	#11724784	C₄⋊C₄	1	1	yes	characteristic
43	C₂²×C₁₀₉	—	2	2	no	—
44	C₂²×C₁₀₉	#97	1	1	yes	characteristic
45	C₂²×C₁₀₉	C₂²⋊C₈	1	1	yes	characteristic
46	C₂²×C₁₀₉	C₄⋊C₈	1	1	yes	characteristic
47	C₄×C₁₀₉	—	4	2	no	—
48	C₄×C₁₀₉	#95	1	1	yes	characteristic
49	C₄×C₁₀₉	C₄×C₈	1	1	yes	characteristic
50	C₄×C₁₀₉	C₂²⋊C₈	1	1	yes	characteristic
51	C₄×C₁₀₉	C₄⋊C₈	1	1	yes	characteristic
52	C₂×C₁₀₉	—	2	1	no	—
53	C₂×C₁₀₉	—	2	1	no	—
54	C₂×C₁₀₉	#336	1	1	yes	characteristic
55	C₂×C₁₀₉	#350	1	2	yes	—
56	#1379	—	109	1	no	maximal, Sylow
57	C₁₀₉	#1379	1	1	yes	characteristic, complemented, Sylow
58	C₂×M₃₂	—	109	1	no	—
59	C₂²×C₁₆	—	109	1	no	—
60	#405	—	109	1	no	—
61	C₂×M₁₆	—	109	1	no	—
62	C₂²×C₈	—	109	1	no	—
63	#117	#89137	1	1	yes	characteristic
64	M₃₂	—	109	2	no	—
65	M₃₂	—	218	1	no	—
66	C₂×C₁₆	—	109	2	no	—
67	C₂×C₁₆	—	218	1	no	—
68	C₂×C₁₆	—	218	1	no	—
69	C₂×C₁₆	—	218	1	no	—
70	#97	—	218	1	no	—
71	C₄×C₈	—	218	1	no	—
72	C₂²×C₄	C₂×#89137	1	1	yes	characteristic
73	M₁₆	—	109	2	no	—
74	M₁₆	—	218	1	no	—
75	C₂×C₈	—	109	1	no	—
76	C₂×C₈	—	109	1	no	—
77	C₂×C₈	—	109	1	no	—
78	C₂×C₈	—	109	1	no	—
79	C₂×C₈	—	109	1	no	—
80	C₂×C₈	—	109	1	no	—
81	C₂×C₈	—	218	1	no	—
82	C₄⋊C₄	#11724786	1	2	yes	—
83	C₂²⋊C₄	—	2	1	no	—
84	C₄²	—	2	1	no	—
85	C₁₆	—	218	2	no	—
86	C₁₆	—	218	2	no	—
87	C₂³	#13182038	1	1	yes	characteristic
88	C₂×C₄	—	2	2	no	—
89	C₂×C₄	C₄×#89137	1	1	yes	characteristic
90	C₂×C₄	C₂²⋊#89137	1	1	yes	characteristic
91	C₂×C₄	C₄⋊#89137	1	1	yes	characteristic
92	C₂×C₄	C₂×#11724786	1	1	yes	characteristic
93	C₂×C₄	D₁₀₉⋊C₈	1	1	yes	characteristic
94	C₂×C₄	#13182039	1	1	yes	characteristic
95	C₈	—	109	1	no	—
96	C₈	—	109	1	no	—
97	C₈	—	109	1	no	—
98	C₈	—	109	1	no	—
99	C₈	—	218	2	no	—
100	C₂²	—	2	2	no	—
101	C₂²	#16812754	1	1	yes	characteristic
102	C₂²	C₂²⋊#11724786	1	1	yes	characteristic
103	C₂²	#16812766	1	1	yes	characteristic
104	C₄	—	4	2	no	—
105	C₄	(C₂×#89137)⋊C₄	1	1	yes	characteristic
106	C₄	C₄×#11724786	1	1	yes	characteristic
107	C₄	#16812756	1	1	yes	characteristic
108	C₄	C₄⋊#11724786	1	1	yes	characteristic
109	C₂	—	2	1	no	—
110	C₂	—	2	1	no	—
111	C₂	#20295892	1	1	yes	characteristic
112	C₂	#20295924	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16812773

C₂

yes

maximal, characteristic

C₂²×#13182032

C₂

yes

maximal, characteristic

#20294982

C₂

yes

maximal, characteristic

#16812773

—

#16812773

C₄

yes

—

—

—

C₄

yes

complemented

C₂×#13182013

C₄

yes

characteristic

C₂²×#11724784

C₂²

yes

characteristic

#16812610

—

C₄×#11724784

—

#117×C₁₀₉

C₄

yes

—

—

—

C₈

yes

—

C₂×#11724784

—

C₂×#11724784

C₂×C₄

yes

characteristic

C₂×#11724784

C₂×C₄

yes

characteristic

C₂×#11724784

D₄

yes

characteristic

C₂×#11724784

D₄

yes

characteristic

C₂×#11724784

D₄

yes

characteristic

C₂×#11724784

Q₈

yes

characteristic

C₂²×C₄×C₁₀₉

C₂×C₄

yes

characteristic

C₄⋊C₄×C₁₀₉

C₈

yes

—

C₂²⋊C₄×C₁₀₉

—

C₄²×C₁₀₉

—

C₂³×C₁₀₉

M₁₆

yes

characteristic

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₄²

yes

characteristic

C₂×C₄×C₁₀₉

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₀₉

C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂×C₈

yes

characteristic

C₂×C₄×C₁₀₉

C₂×C₈

yes

characteristic

C₂×C₄×C₁₀₉

M₁₆

yes

characteristic

#11724784

—

#11724784

C₂²⋊C₄

yes

characteristic

#11724784

C₂²⋊C₄

yes

characteristic

#11724784

C₄⋊C₄

yes

characteristic

#11724784

C₄⋊C₄

yes

characteristic

C₂²×C₁₀₉

—

C₂²×C₁₀₉

#97

yes

characteristic

C₂²×C₁₀₉

C₂²⋊C₈

yes

characteristic

C₂²×C₁₀₉

C₄⋊C₈

yes

characteristic

C₄×C₁₀₉

—

C₄×C₁₀₉

#95

yes

characteristic

C₄×C₁₀₉

C₄×C₈

yes

characteristic

C₄×C₁₀₉

C₂²⋊C₈

yes

characteristic

C₄×C₁₀₉

C₄⋊C₈

yes

characteristic

C₂×C₁₀₉

—

C₂×C₁₀₉

—

C₂×C₁₀₉

#336

yes

characteristic

C₂×C₁₀₉

#350

yes

—

#1379

—

109

maximal, Sylow

C₁₀₉

#1379

yes

characteristic, complemented, Sylow

C₂×M₃₂

—

109

—

C₂²×C₁₆

—

109

—

#405

—

109

—

C₂×M₁₆

—

109

—

C₂²×C₈

—

109

—

#117

#89137

yes

characteristic

M₃₂

—

109

—

M₃₂

—

218

—

C₂×C₁₆

—

109

—

C₂×C₁₆

—

218

—

C₂×C₁₆

—

218

—

C₂×C₁₆

—

218

—

#97

—

218

—

C₄×C₈

—

218

—

C₂²×C₄

C₂×#89137

yes

characteristic

M₁₆

—

109

—

M₁₆

—

218

—

C₂×C₈

—

109

—

C₂×C₈

—

109

—

C₂×C₈

—

109

—

C₂×C₈

—

109

—

C₂×C₈

—

109

—

C₂×C₈

—

109

—

C₂×C₈

—

218

—

C₄⋊C₄

#11724786

yes

—

C₂²⋊C₄

—

C₄²

—

C₁₆

—

218

—

C₁₆

—

218

—

C₂³

#13182038

yes

characteristic

C₂×C₄

—

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

D₁₀₉⋊C₈

yes

characteristic

C₂×C₄

#13182039

yes

characteristic

C₈

—

109

—

C₈

—

109

—

C₈

—

109

—

C₈

—

109

—

C₈

—

218

—

100

C₂²

—

101

C₂²

#16812754

yes

characteristic

102

C₂²

C₂²⋊#11724786

yes

characteristic

103

C₂²

#16812766

yes

characteristic

104

C₄

—

105

C₄

(C₂×#89137)⋊C₄

yes

characteristic

106

C₄

C₄×#11724786

yes

characteristic

107

C₄

#16812756

yes

characteristic

108

C₄

C₄⋊#11724786

yes

characteristic

109

C₂

—

110

C₂

—

111

C₂

#20295892

yes

characteristic

112

C₂

#20295924

yes

—

	subgroup	G/H
center	C₂×C₄	C₂²⋊#89137
commutator	C₄×C₁₀₉	C₄×C₈
Frattini	C₂²×C₄	C₂×#89137
Fitting	#117×C₁₀₉	C₄
socle	C₂²×C₁₀₉	C₂²⋊C₈
prime radical	C₂³×C₁₀₉	M₁₆
cosocle	C₂²×#11724784	C₂²
hypercenter	#117	#89137
hypocenter	C₁₀₉	#1379
LCS	G ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #117

subgroup

G/H

center

C₂×C₄

C₂²⋊#89137

commutator

C₄×C₁₀₉

C₄×C₈

Frattini

C₂²×C₄

C₂×#89137

Fitting

#117×C₁₀₉

C₄

socle

C₂²×C₁₀₉

C₂²⋊C₈

prime radical

C₂³×C₁₀₉

M₁₆

C₂²

C₁₀₉

G ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #117

N	H ≅ G/N	type
5. #16812773	C₄	semidirect
9. C₂×#13182032	C₄	semidirect
57. C₁₀₉	#1379	semidirect

H ≅ G/N

type

5. #16812773

C₄

semidirect

9. C₂×#13182032

C₄

semidirect

57. C₁₀₉

#1379

semidirect

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

subgroup	index
C₂×#16812773	2
C₂²×#13182032	2
#20294982	2
#1379	109

subgroup

index

109

Subgroups — #28378086

Conjugacy classes of subgroups

Complements

Maximal subgroups