Group #28378255 — subgroups

Subgroups — #28378255

Order 13952.

214 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 214 conjugacy classes of subgroups — too many to draw. Only its 92 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	C₄×#13182038	C₂	1	2	yes	maximal
2	#11724786⋊C₈	C₂	1	2	yes	maximal, complemented
3	#20295867	C₂	1	1	yes	maximal, characteristic, complemented
4	C₈×#11724786	C₂	1	1	yes	maximal, characteristic, complemented
5	M₁₆×Dic₁₀₉	C₂	1	1	yes	maximal, characteristic
6	C₂×#13182038	C₄	1	4	yes	—
7	#16812754	C₄	1	4	yes	—
8	C₄×#11724786	C₂²	1	2	yes	—
9	C₄×#11724786	C₂²	1	2	yes	—
10	C₂×#13182013	C₄	1	1	yes	characteristic
11	C₂×C₄×Dic₁₀₉	C₂²	1	1	yes	characteristic
12	#16812609	C₄	1	1	yes	characteristic
13	#16812609	C₄	1	1	yes	characteristic
14	C₈×Dic₁₀₉	C₂²	1	1	yes	characteristic
15	C₈×Dic₁₀₉	C₂²	1	1	yes	characteristic
16	C₂×M₁₆×C₁₀₉	C₄	1	1	yes	characteristic
17	#13182038	—	2	8	no	—
18	C₂×#11724786	C₂×C₄	1	4	yes	—
19	C₂×#11724786	C₂×C₄	1	4	yes	—
20	C₂²×Dic₁₀₉	C₂×C₄	1	2	yes	—
21	#13182013	—	2	2	no	—
22	C₂×#11724784	C₂×C₄	1	1	yes	characteristic
23	C₂×#11724784	C₂×C₄	1	1	yes	characteristic
24	C₄×Dic₁₀₉	C₂×C₄	1	1	yes	characteristic
25	C₄×Dic₁₀₉	C₂×C₄	1	2	yes	—
26	C₄×Dic₁₀₉	C₂³	1	1	yes	characteristic
27	C₂²×C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
28	M₁₆×C₁₀₉	—	2	2	no	—
29	C₂×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
30	C₂×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
31	C₂³×C₁₀₉	C₄²	1	1	yes	characteristic
32	C₂×C₄×C₁₀₉	—	2	1	no	—
33	C₂×C₄×C₁₀₉	—	2	1	no	—
34	C₂×C₄×C₁₀₉	C₄²	1	1	yes	characteristic
35	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
36	C₂×Dic₁₀₉	—	2	4	no	—
37	C₂×Dic₁₀₉	C₄²	1	2	yes	—
38	C₂×Dic₁₀₉	C₂²×C₄	1	2	yes	—
39	#11724786	—	2	4	no	—
40	#11724786	M₁₆	1	8	yes	complemented
41	C₈×C₁₀₉	—	2	1	no	—
42	C₈×C₁₀₉	M₁₆	1	2	yes	complemented
43	#11724784	—	2	1	no	—
44	#11724784	M₁₆	1	2	yes	complemented
45	C₂²×C₁₀₉	—	2	2	no	—
46	C₂²×C₁₀₉	—	2	1	no	—
47	C₂²×C₁₀₉	C₂×C₄²	1	1	yes	characteristic
48	C₄×C₁₀₉	—	2	1	no	—
49	C₄×C₁₀₉	C₂×M₁₆	1	1	yes	characteristic
50	C₄×C₁₀₉	C₂×M₁₆	1	1	yes	characteristic
51	Dic₁₀₉	—	2	2	no	—
52	Dic₁₀₉	C₂×M₁₆	1	4	yes	—
53	C₂×C₁₀₉	—	4	1	no	—
54	C₂×C₁₀₉	C₄×M₁₆	1	2	yes	—
55	C₂×C₁₀₉	C₄×M₁₆	1	1	yes	characteristic
56	#1455	—	109	1	no	maximal, Sylow
57	C₁₀₉	#1455	1	1	yes	characteristic, complemented, Sylow
58	C₄×M₁₆	—	109	2	no	—
59	C₄×M₁₆	—	109	1	no	—
60	#322	—	109	1	no	—
61	#322	—	109	2	no	—
62	C₈²	—	109	1	no	—
63	C₂×M₁₆	—	109	4	no	—
64	C₂×M₁₆	—	109	1	no	—
65	C₂×M₁₆	#89137	1	1	yes	characteristic
66	C₂×C₄²	—	109	1	no	—
67	#97	—	109	4	no	—
68	#97	—	109	1	no	—
69	#97	—	109	1	no	—
70	C₄×C₈	—	109	2	no	—
71	C₄×C₈	—	109	2	no	—
72	C₄×C₈	—	109	1	no	—
73	C₄×C₈	—	109	1	no	—
74	C₂²×C₄	—	109	2	no	—
75	C₂²×C₄	C₂×#89137	1	1	yes	characteristic
76	M₁₆	—	2	2	no	—
77	M₁₆	—	218	2	no	—
78	M₁₆	—	218	8	no	—
79	C₂×C₈	—	109	4	no	—
80	C₂×C₈	—	109	1	no	—
81	C₂×C₈	—	109	4	no	—
82	C₂×C₈	—	109	1	no	—
83	C₂×C₈	C₂×#89137	1	1	yes	characteristic
84	C₂×C₈	C₂×#89137	1	1	yes	characteristic
85	C₄²	—	109	2	no	—
86	C₄²	—	109	1	no	—
87	C₄²	—	109	1	no	—
88	C₂³	C₄×#89137	1	1	yes	characteristic
89	C₂×C₄	—	2	1	no	—
90	C₂×C₄	—	2	1	no	—
91	C₂×C₄	—	109	2	no	—
92	C₂×C₄	—	109	2	no	—
93	C₂×C₄	—	218	4	no	—
94	C₂×C₄	C₄×#89137	1	1	yes	characteristic
95	C₂×C₄	C₂²×#89137	1	1	yes	characteristic
96	C₈	—	2	1	no	—
97	C₈	—	109	8	no	—
98	C₈	—	109	2	no	—
99	C₈	—	218	4	no	—
100	C₈	—	218	1	no	—
101	C₈	#13182038	1	2	yes	complemented
102	C₂²	—	2	2	no	—
103	C₂²	—	2	1	no	—
104	C₂²	C₂×C₄×#89137	1	1	yes	characteristic
105	C₄	—	2	1	no	—
106	C₄	—	109	4	no	—
107	C₄	—	218	2	no	—
108	C₄	C₂×#13182038	1	1	yes	characteristic
109	C₄	D₁₀₉⋊M₁₆	1	1	yes	characteristic
110	C₂	—	4	1	no	—
111	C₂	M₁₆×#89137	1	2	yes	—
112	C₂	C₄×#13182038	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×#13182038

C₂

yes

maximal

#11724786⋊C₈

C₂

yes

maximal, complemented

#20295867

C₂

yes

maximal, characteristic, complemented

C₈×#11724786

C₂

yes

maximal, characteristic, complemented

M₁₆×Dic₁₀₉

C₂

yes

maximal, characteristic

C₂×#13182038

C₄

yes

—

#16812754

C₄

yes

—

C₄×#11724786

C₂²

yes

—

C₄×#11724786

C₂²

yes

—

C₂×#13182013

C₄

yes

characteristic

C₂×C₄×Dic₁₀₉

C₂²

yes

characteristic

#16812609

C₄

yes

characteristic

#16812609

C₄

yes

characteristic

C₈×Dic₁₀₉

C₂²

yes

characteristic

C₈×Dic₁₀₉

C₂²

yes

characteristic

C₂×M₁₆×C₁₀₉

C₄

yes

characteristic

#13182038

—

C₂×#11724786

C₂×C₄

yes

—

C₂×#11724786

C₂×C₄

yes

—

C₂²×Dic₁₀₉

C₂×C₄

yes

—

#13182013

—

C₂×#11724784

C₂×C₄

yes

characteristic

C₂×#11724784

C₂×C₄

yes

characteristic

C₄×Dic₁₀₉

C₂×C₄

yes

characteristic

C₄×Dic₁₀₉

C₂×C₄

yes

—

C₄×Dic₁₀₉

C₂³

yes

characteristic

C₂²×C₄×C₁₀₉

C₂×C₄

yes

characteristic

M₁₆×C₁₀₉

—

C₂×C₈×C₁₀₉

C₂×C₄

yes

characteristic

C₂×C₈×C₁₀₉

C₂×C₄

yes

characteristic

C₂³×C₁₀₉

C₄²

yes

characteristic

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₄²

yes

characteristic

C₂×C₄×C₁₀₉

C₂²×C₄

yes

characteristic

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

C₄²

yes

—

C₂×Dic₁₀₉

C₂²×C₄

yes

—

#11724786

—

#11724786

M₁₆

yes

complemented

C₈×C₁₀₉

—

C₈×C₁₀₉

M₁₆

yes

complemented

#11724784

—

#11724784

M₁₆

yes

complemented

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

C₂×C₄²

yes

characteristic

C₄×C₁₀₉

—

C₄×C₁₀₉

C₂×M₁₆

yes

characteristic

C₄×C₁₀₉

C₂×M₁₆

yes

characteristic

Dic₁₀₉

—

Dic₁₀₉

C₂×M₁₆

yes

—

C₂×C₁₀₉

—

C₂×C₁₀₉

C₄×M₁₆

yes

—

C₂×C₁₀₉

C₄×M₁₆

yes

characteristic

#1455

—

109

maximal, Sylow

C₁₀₉

#1455

yes

characteristic, complemented, Sylow

C₄×M₁₆

—

109

—

C₄×M₁₆

—

109

—

#322

—

109

—

#322

—

109

—

C₈²

—

109

—

C₂×M₁₆

—

109

—

C₂×M₁₆

—

109

—

C₂×M₁₆

#89137

yes

characteristic

C₂×C₄²

—

109

—

#97

—

109

—

#97

—

109

—

#97

—

109

—

C₄×C₈

—

109

—

C₄×C₈

—

109

—

C₄×C₈

—

109

—

C₄×C₈

—

109

—

C₂²×C₄

—

109

—

C₂²×C₄

C₂×#89137

yes

characteristic

M₁₆

—

M₁₆

—

218

—

M₁₆

—

218

—

C₂×C₈

—

109

—

C₂×C₈

—

109

—

C₂×C₈

—

109

—

C₂×C₈

—

109

—

C₂×C₈

yes

C₂×C₈

yes

C₄²

—

109

—

C₄²

—

109

—

C₄²

—

109

—

C₂³

C₄×#89137

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

218

—

C₂×C₄

C₄×#89137

yes

characteristic

C₂×C₄

C₂²×#89137

yes

characteristic

C₈

—

C₈

—

109

—

C₈

—

109

—

C₈

—

218

—

100

C₈

—

218

—

101

C₈

#13182038

yes

complemented

102

C₂²

—

103

C₂²

—

104

C₂²

C₂×C₄×#89137

yes

characteristic

105

C₄

—

106

C₄

—

109

—

107

C₄

—

218

—

108

C₄

C₂×#13182038

yes

characteristic

109

C₄

D₁₀₉⋊M₁₆

yes

characteristic

110

C₂

—

111

C₂

M₁₆×#89137

yes

—

112

C₂

C₄×#13182038

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×#89137
commutator, socle	C₂²×C₁₀₉	C₂×C₄²
Fitting	C₂×M₁₆×C₁₀₉	C₄
prime radical	C₂³×C₁₀₉	C₄²
cosocle	C₄×Dic₁₀₉	C₂³
hypercenter	C₂×M₁₆	#89137
hypocenter	C₁₀₉	#1455
LCS	G ▷ C₂²×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×M₁₆

subgroup

G/H

center, Frattini

C₂×C₄

C₂²×#89137

commutator, socle

C₂²×C₁₀₉

C₂×C₄²

Fitting

C₂×M₁₆×C₁₀₉

C₄

prime radical

C₂³×C₁₀₉

C₄²

cosocle

C₄×Dic₁₀₉

C₂³

C₂×M₁₆

C₁₀₉

G ▷ C₂²×C₁₀₉ ▷ C₁₀₉

C₁ ◁ C₂×C₄ ◁ C₂×M₁₆

N	H ≅ G/N	type
2. #11724786⋊C₈	C₂	semidirect
3. #20295867	C₂	semidirect
4. C₈×#11724786	C₂	semidirect
40. #11724786	M₁₆	semidirect
42. C₈×C₁₀₉	M₁₆	semidirect
44. #11724784	M₁₆	semidirect
57. C₁₀₉	#1455	semidirect
101. C₈	#13182038	semidirect

H ≅ G/N

type

2. #11724786⋊C₈

C₂

semidirect

3. #20295867

C₂

semidirect

4. C₈×#11724786

C₂

semidirect

40. #11724786

M₁₆

semidirect

42. C₈×C₁₀₉

M₁₆

semidirect

44. #11724784

M₁₆

semidirect

57. C₁₀₉

#1455

semidirect

101. C₈

#13182038

semidirect

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

subgroup	index
C₄×#13182038	2
#11724786⋊C₈	2
#20295867	2
C₈×#11724786	2
M₁₆×Dic₁₀₉	2
#1455	109

subgroup

index

M₁₆×Dic₁₀₉

109

Subgroups — #28378255

Conjugacy classes of subgroups

Complements

Maximal subgroups