Group #28378618 — subgroups

Subgroups — Q₁₆×#11724786 #28378618

Order 13952.

164 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 164 conjugacy classes of subgroups — too many to draw. Only its 74 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	Q₈×#11724786	C₂	1	2	yes	maximal
2	#20295889	C₂	1	1	yes	maximal, characteristic
3	Q₈⋊#11724786	C₂	1	2	yes	maximal
4	C₈×#11724786	C₂	1	1	yes	maximal, characteristic
5	Q₁₆×Dic₁₀₉	C₂	1	1	yes	maximal, characteristic
6	C₄⋊#11724786	—	2	2	no	—
7	C₄⋊#11724786	C₂²	1	2	yes	—
8	C₄×#11724786	—	2	2	no	—
9	C₄×#11724786	C₂²	1	1	yes	characteristic
10	C₄×#11724786	C₂²	1	1	yes	characteristic
11	Q₈×Dic₁₀₉	C₂²	1	2	yes	—
12	#16812633	C₄	1	1	yes	characteristic
13	#16812623	C₄	1	2	yes	—
14	C₈×Dic₁₀₉	C₂²	1	1	yes	characteristic
15	C₂×Q₁₆×C₁₀₉	C₄	1	1	yes	characteristic
16	C₂×#11724786	—	2	1	no	—
17	C₂×#11724786	—	2	2	no	—
18	C₂×#11724786	D₄	1	1	yes	characteristic
19	C₂×#11724786	D₄	1	1	yes	characteristic
20	C₂×#11724784	C₂×C₄	1	1	yes	characteristic
21	C₄⋊Dic₁₀₉	—	2	2	no	—
22	C₄⋊Dic₁₀₉	C₂×C₄	1	2	yes	—
23	C₄×Dic₁₀₉	—	2	2	no	—
24	C₄×Dic₁₀₉	C₂³	1	1	yes	characteristic
25	C₂×Q₈×C₁₀₉	C₂×C₄	1	2	yes	—
26	Q₁₆×C₁₀₉	C₈	1	4	yes	complemented
27	C₂×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
28	Q₈×C₁₀₉	—	2	2	no	—
29	Q₈×C₁₀₉	C₂×C₈	1	4	yes	—
30	C₂×C₄×C₁₀₉	—	2	2	no	—
31	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
32	C₂×Dic₁₀₉	—	2	2	no	—
33	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
34	C₂×Dic₁₀₉	#41	1	1	yes	characteristic
35	#11724786	—	2	2	no	—
36	#11724786	—	2	1	no	—
37	#11724786	—	4	2	no	—
38	#11724786	Q₁₆	1	2	yes	direct product factor
39	C₈×C₁₀₉	C₂×C₈	1	2	yes	—
40	#11724784	—	2	1	no	—
41	C₂²×C₁₀₉	C₄×D₄	1	1	yes	characteristic
42	C₄×C₁₀₉	—	2	4	no	—
43	C₄×C₁₀₉	C₂²×C₈	1	1	yes	characteristic
44	C₄×C₁₀₉	#131	1	1	yes	characteristic
45	Dic₁₀₉	—	2	1	no	—
46	Dic₁₀₉	—	4	2	no	—
47	Dic₁₀₉	C₂×Q₁₆	1	1	yes	characteristic
48	Dic₁₀₉	#135	1	1	yes	characteristic
49	C₂×C₁₀₉	C₈×D₄	1	1	yes	characteristic
50	C₂×C₁₀₉	C₄×Q₁₆	1	1	yes	characteristic
51	C₂×C₁₀₉	#443	1	1	yes	characteristic
52	C₈×Q₁₆	—	109	1	no	maximal, Sylow
53	C₁₀₉	C₈×Q₁₆	1	1	yes	characteristic, complemented, Sylow
54	C₈×Q₈	—	109	2	no	—
55	C₄×Q₁₆	—	109	1	no	—
56	#335	—	109	1	no	—
57	Q₈⋊C₈	—	109	2	no	—
58	C₈²	—	109	1	no	—
59	C₂×Q₁₆	#89137	1	1	yes	characteristic
60	C₄×Q₈	—	109	2	no	—
61	#107	—	109	1	no	—
62	C₄⋊C₈	—	109	2	no	—
63	C₄⋊C₈	—	218	2	no	—
64	#103	—	109	2	no	—
65	C₄×C₈	—	109	1	no	—
66	C₄×C₈	—	109	1	no	—
67	C₄×C₈	—	109	1	no	—
68	C₄×C₈	—	218	2	no	—
69	C₂×Q₈	C₂×#89137	1	2	yes	—
70	Q₁₆	#11724786	1	4	yes	direct product factor
71	C₂×C₈	—	109	1	no	—
72	C₂×C₈	—	109	1	no	—
73	C₂×C₈	—	109	1	no	—
74	C₂×C₈	—	218	2	no	—
75	C₂×C₈	—	218	1	no	—
76	C₂×C₈	C₂×#89137	1	1	yes	characteristic
77	C₄⋊C₄	—	109	2	no	—
78	C₄⋊C₄	—	218	2	no	—
79	C₄²	—	109	1	no	—
80	C₄²	—	218	2	no	—
81	Q₈	—	2	2	no	—
82	Q₈	C₂×#11724786	1	4	yes	—
83	C₂×C₄	—	2	2	no	—
84	C₂×C₄	—	109	1	no	—
85	C₂×C₄	—	109	1	no	—
86	C₂×C₄	—	218	2	no	—
87	C₂×C₄	C₂²×#89137	1	1	yes	characteristic
88	C₈	—	109	2	no	—
89	C₈	—	436	2	no	—
90	C₈	—	218	1	no	—
91	C₈	—	218	2	no	—
92	C₈	—	218	1	no	—
93	C₈	C₂×#11724786	1	2	yes	—
94	C₂²	D₄×#89137	1	1	yes	characteristic
95	C₄	—	2	4	no	—
96	C₄	—	109	1	no	—
97	C₄	—	109	1	no	—
98	C₄	—	436	2	no	—
99	C₄	—	218	1	no	—
100	C₄	C₂²×#11724786	1	1	yes	characteristic
101	C₄	#16812786	1	1	yes	characteristic
102	C₂	D₄×#11724786	1	1	yes	characteristic
103	C₂	Q₁₆×#89137	1	1	yes	characteristic
104	C₂	#20296070	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

Q₈×#11724786

C₂

yes

maximal

#20295889

C₂

yes

maximal, characteristic

Q₈⋊#11724786

C₂

yes

maximal

C₈×#11724786

C₂

yes

maximal, characteristic

Q₁₆×Dic₁₀₉

C₂

yes

maximal, characteristic

C₄⋊#11724786

—

C₄⋊#11724786

C₂²

yes

—

C₄×#11724786

—

C₄×#11724786

C₂²

yes

characteristic

C₄×#11724786

C₂²

yes

characteristic

Q₈×Dic₁₀₉

C₂²

yes

—

#16812633

C₄

yes

characteristic

#16812623

C₄

yes

—

C₈×Dic₁₀₉

C₂²

yes

characteristic

C₂×Q₁₆×C₁₀₉

C₄

yes

—

—

D₄

yes

characteristic

C₂×#11724786

D₄

yes

characteristic

C₂×#11724784

C₂×C₄

yes

characteristic

C₄⋊Dic₁₀₉

—

C₄⋊Dic₁₀₉

C₂×C₄

yes

—

C₄×Dic₁₀₉

—

C₄×Dic₁₀₉

C₂³

yes

characteristic

C₂×Q₈×C₁₀₉

C₂×C₄

yes

—

Q₁₆×C₁₀₉

C₈

yes

complemented

C₂×C₈×C₁₀₉

C₂×C₄

yes

characteristic

Q₈×C₁₀₉

—

Q₈×C₁₀₉

C₂×C₈

yes

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₂²×C₄

yes

characteristic

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

C₂×D₄

yes

C₂×Dic₁₀₉

yes

—

—

—

Q₁₆

yes

direct product factor

C₈×C₁₀₉

C₂×C₈

yes

—

#11724784

—

C₂²×C₁₀₉

C₄×D₄

yes

characteristic

C₄×C₁₀₉

—

C₄×C₁₀₉

C₂²×C₈

yes

characteristic

C₄×C₁₀₉

#131

yes

characteristic

Dic₁₀₉

—

Dic₁₀₉

—

Dic₁₀₉

C₂×Q₁₆

yes

characteristic

Dic₁₀₉

#135

yes

characteristic

C₂×C₁₀₉

C₈×D₄

yes

characteristic

C₂×C₁₀₉

C₄×Q₁₆

yes

characteristic

C₂×C₁₀₉

#443

yes

characteristic

C₈×Q₁₆

—

109

maximal, Sylow

C₁₀₉

C₈×Q₁₆

yes

characteristic, complemented, Sylow

C₈×Q₈

—

109

—

C₄×Q₁₆

—

109

—

#335

—

109

—

Q₈⋊C₈

—

109

—

C₈²

—

109

—

C₂×Q₁₆

#89137

yes

characteristic

C₄×Q₈

—

109

—

#107

—

109

—

C₄⋊C₈

—

109

—

C₄⋊C₈

—

218

—

#103

—

109

—

C₄×C₈

—

109

—

C₄×C₈

—

109

—

C₄×C₈

—

109

—

C₄×C₈

—

218

—

C₂×Q₈

C₂×#89137

yes

—

Q₁₆

#11724786

yes

direct product factor

C₂×C₈

—

109

—

C₂×C₈

—

109

—

C₂×C₈

—

109

—

C₂×C₈

—

218

—

C₂×C₈

—

218

—

C₂×C₈

C₂×#89137

yes

characteristic

C₄⋊C₄

—

109

—

C₄⋊C₄

—

218

—

C₄²

—

109

—

C₄²

—

218

—

Q₈

—

Q₈

C₂×#11724786

yes

—

C₂×C₄

—

C₂×C₄

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

218

—

C₂×C₄

C₂²×#89137

yes

characteristic

C₈

—

109

—

C₈

—

436

—

C₈

—

218

—

C₈

—

218

—

C₈

—

218

—

C₈

C₂×#11724786

yes

—

C₂²

D₄×#89137

yes

characteristic

C₄

—

C₄

—

109

—

C₄

—

109

—

C₄

—

436

—

C₄

—

218

—

100

C₄

C₂²×#11724786

yes

characteristic

101

C₄

#16812786

yes

characteristic

102

C₂

D₄×#11724786

yes

characteristic

103

C₂

Q₁₆×#89137

yes

characteristic

104

C₂

#20296070

yes

characteristic

	subgroup	G/H
center	C₂²	D₄×#89137
commutator	C₄×C₁₀₉	C₂²×C₈
Frattini	C₂×C₄	C₂²×#89137
Fitting	C₂×Q₁₆×C₁₀₉	C₄
socle, prime radical	C₂²×C₁₀₉	C₄×D₄
cosocle	C₄×Dic₁₀₉	C₂³
hypercenter	C₂×Q₁₆	#89137
hypocenter	C₁₀₉	C₈×Q₁₆
LCS	G ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×Q₁₆

subgroup

G/H

center

C₂²

D₄×#89137

commutator

C₄×C₁₀₉

C₂²×C₈

Frattini

C₂×C₄

C₂²×#89137

Fitting

C₂×Q₁₆×C₁₀₉

C₄

socle, prime radical

C₂²×C₁₀₉

C₄×D₄

cosocle

C₄×Dic₁₀₉

C₂³

hypercenter

C₂×Q₁₆

#89137

hypocenter

C₁₀₉

C₈×Q₁₆

LCS

G ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×Q₁₆

N	H ≅ G/N	type
26. Q₁₆×C₁₀₉	C₈	semidirect
38. #11724786	Q₁₆	direct (N × H)
53. C₁₀₉	C₈×Q₁₆	semidirect
70. Q₁₆	#11724786	direct (N × H)

H ≅ G/N

type

26. Q₁₆×C₁₀₉

C₈

semidirect

38. #11724786

Q₁₆

direct (N × H)

53. C₁₀₉

C₈×Q₁₆

semidirect

70. Q₁₆

#11724786

direct (N × H)

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

subgroup	index
Q₈×#11724786	2
#20295889	2
Q₈⋊#11724786	2
C₈×#11724786	2
Q₁₆×Dic₁₀₉	2
C₈×Q₁₆	109

subgroup

index

Q₁₆×Dic₁₀₉

C₈×Q₁₆

109

Subgroups — Q₁₆×#11724786 #28378618

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q16×#11724786 #28378618

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₁₆×#11724786 #28378618