Group #20295015 — subgroups

Subgroups — #103×D₁₀₉ #20295015

Order 6976.

162 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 162 conjugacy classes of subgroups — too many to draw. Only its 63 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×Q₈×D₁₀₉	C₂	1	1	yes	maximal, characteristic
2	C₂×C₈×D₁₀₉	C₂	1	1	yes	maximal, characteristic
3	C₄⋊C₄×D₁₀₉	C₂	1	1	yes	maximal, characteristic
4	#16812624	C₂	1	1	yes	maximal, characteristic, complemented
5	#16812623	C₂	1	1	yes	maximal, characteristic, complemented
6	#16812622	C₂	1	1	yes	maximal, characteristic, complemented
7	#103×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×Dic₂₁₈	—	2	1	no	—
9	C₂×Dic₂₁₈	C₂²	1	1	yes	characteristic
10	Q₈×D₁₀₉	—	2	2	no	—
11	Q₈×D₁₀₉	C₄	1	4	yes	complemented
12	C₂×C₄×D₁₀₉	—	2	1	no	—
13	C₂×C₄×D₁₀₉	—	2	1	no	—
14	C₂×C₄×D₁₀₉	C₂²	1	1	yes	characteristic
15	C₂×#11724784	C₂²	1	1	yes	characteristic
16	C₈×D₁₀₉	—	2	2	no	—
17	C₄⋊Dic₁₀₉	C₂²	1	1	yes	characteristic
18	#13182008	—	2	1	no	—
19	C₂×Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
20	C₂×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
21	C₄⋊C₄×C₁₀₉	C₂²	1	1	yes	characteristic
22	C₂²×D₁₀₉	D₄	1	1	yes	characteristic
23	Q₈×C₁₀₉	—	2	1	no	—
24	Q₈×C₁₀₉	C₂×C₄	1	2	yes	complemented
25	C₂×C₄×C₁₀₉	—	2	1	no	—
26	C₂×C₄×C₁₀₉	—	2	1	no	—
27	C₂×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
28	C₂×Dic₁₀₉	—	2	1	no	—
29	C₂×Dic₁₀₉	—	2	1	no	—
30	C₂×Dic₁₀₉	D₄	1	1	yes	characteristic
31	C₄×D₁₀₉	—	2	4	no	—
32	C₄×D₁₀₉	—	4	2	no	—
33	C₄×D₁₀₉	C₂×C₄	1	2	yes	—
34	C₄×D₁₀₉	D₄	1	2	yes	—
35	Dic₂₁₈	—	2	1	no	—
36	Dic₂₁₈	—	2	4	no	—
37	Dic₂₁₈	C₂×C₄	1	2	yes	complemented
38	C₈×C₁₀₉	—	2	1	no	—
39	#11724784	—	2	1	no	—
40	C₂²×C₁₀₉	C₂×D₄	1	1	yes	characteristic
41	C₂×D₁₀₉	C₂²⋊C₄	1	2	yes	—
42	C₂×D₁₀₉	SD₁₆	1	2	yes	—
43	C₂×D₁₀₉	Q₁₆	1	2	yes	—
44	C₄×C₁₀₉	—	2	2	no	—
45	C₄×C₁₀₉	—	4	1	no	—
46	C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
47	C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
48	Dic₁₀₉	—	2	2	no	—
49	Dic₁₀₉	—	4	1	no	—
50	Dic₁₀₉	C₂²⋊C₄	1	2	yes	—
51	C₂×C₁₀₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
52	C₂×C₁₀₉	C₂×SD₁₆	1	1	yes	characteristic
53	C₂×C₁₀₉	C₂×Q₁₆	1	1	yes	characteristic
54	D₁₀₉	#103	1	4	yes	direct product factor
55	C₁₀₉	C₂×#103	1	1	yes	characteristic, complemented, Sylow
56	C₂×#103	—	109	1	no	maximal, Sylow
57	C₂²×Q₈	—	109	1	no	—
58	C₂²×C₈	—	109	1	no	—
59	C₂×C₄⋊C₄	—	109	1	no	—
60	#103	—	109	1	no	—
61	#103	—	109	1	no	—
62	#103	—	109	1	no	—
63	#103	D₁₀₉	1	1	yes	characteristic, direct product factor
64	C₂×Q₈	—	109	4	no	—
65	C₂×Q₈	—	109	1	no	—
66	C₂×Q₈	—	218	1	no	—
67	C₂×Q₈	—	218	2	no	—
68	C₂×Q₈	C₂×D₁₀₉	1	1	yes	characteristic
69	C₂²×C₄	—	109	1	no	—
70	C₂²×C₄	—	218	1	no	—
71	C₂²×C₄	—	218	1	no	—
72	C₂×C₈	—	109	1	no	—
73	C₂×C₈	—	218	2	no	—
74	C₂×C₈	C₂×D₁₀₉	1	1	yes	characteristic
75	C₄⋊C₄	—	109	1	no	—
76	C₄⋊C₄	—	218	1	no	—
77	C₄⋊C₄	C₂×D₁₀₉	1	1	yes	characteristic
78	C₂³	—	109	1	no	—
79	Q₈	—	2	1	no	—
80	Q₈	—	109	2	no	—
81	Q₈	—	218	4	no	—
82	Q₈	—	218	1	no	—
83	Q₈	C₄×D₁₀₉	1	2	yes	complemented
84	C₂×C₄	—	2	1	no	—
85	C₂×C₄	—	2	1	no	—
86	C₂×C₄	—	109	1	no	—
87	C₂×C₄	—	109	2	no	—
88	C₂×C₄	—	109	2	no	—
89	C₂×C₄	—	436	2	no	—
90	C₂×C₄	—	218	4	no	—
91	C₂×C₄	—	218	1	no	—
92	C₂×C₄	—	218	1	no	—
93	C₂×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
94	C₈	—	2	1	no	—
95	C₈	—	218	1	no	—
96	C₂²	—	109	2	no	—
97	C₂²	—	109	2	no	—
98	C₂²	—	109	2	no	—
99	C₂²	D₄×D₁₀₉	1	1	yes	characteristic
100	C₄	—	2	2	no	—
101	C₄	—	4	1	no	—
102	C₄	—	109	2	no	—
103	C₄	—	436	1	no	—
104	C₄	—	218	2	no	—
105	C₄	C₂×C₄×D₁₀₉	1	1	yes	characteristic
106	C₄	D₄×D₁₀₉	1	1	yes	characteristic
107	C₂	—	109	4	no	—
108	C₂	C₂²⋊C₄×D₁₀₉	1	1	yes	characteristic
109	C₂	SD₁₆×D₁₀₉	1	1	yes	characteristic
110	C₂	Q₁₆×D₁₀₉	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×Q₈×D₁₀₉

C₂

yes

maximal, characteristic

C₂×C₈×D₁₀₉

C₂

yes

maximal, characteristic

C₄⋊C₄×D₁₀₉

C₂

yes

maximal, characteristic

#16812624

C₂

yes

maximal, characteristic, complemented

#16812623

C₂

yes

maximal, characteristic, complemented

#16812622

C₂

yes

maximal, characteristic, complemented

#103×C₁₀₉

C₂

yes

maximal, characteristic, complemented

C₂×Dic₂₁₈

—

C₂×Dic₂₁₈

C₂²

yes

characteristic

Q₈×D₁₀₉

—

Q₈×D₁₀₉

C₄

yes

complemented

C₂×C₄×D₁₀₉

—

C₂×C₄×D₁₀₉

—

C₂×C₄×D₁₀₉

C₂²

yes

characteristic

C₂×#11724784

C₂²

yes

characteristic

C₈×D₁₀₉

—

C₄⋊Dic₁₀₉

C₂²

yes

characteristic

#13182008

—

C₂×Q₈×C₁₀₉

C₂²

yes

characteristic

C₂×C₈×C₁₀₉

C₂²

yes

characteristic

C₄⋊C₄×C₁₀₉

C₂²

yes

characteristic

C₂²×D₁₀₉

D₄

yes

characteristic

Q₈×C₁₀₉

—

Q₈×C₁₀₉

C₂×C₄

yes

complemented

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₂³

yes

characteristic

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

D₄

yes

characteristic

C₄×D₁₀₉

—

C₄×D₁₀₉

—

C₄×D₁₀₉

C₂×C₄

yes

—

C₄×D₁₀₉

D₄

yes

—

Dic₂₁₈

—

Dic₂₁₈

—

Dic₂₁₈

C₂×C₄

yes

complemented

C₈×C₁₀₉

—

#11724784

—

C₂²×C₁₀₉

C₂×D₄

yes

characteristic

C₂×D₁₀₉

C₂²⋊C₄

yes

—

C₂×D₁₀₉

SD₁₆

yes

—

C₂×D₁₀₉

Q₁₆

yes

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

C₂²×C₄

yes

characteristic

C₄×C₁₀₉

C₂×D₄

yes

characteristic

Dic₁₀₉

—

Dic₁₀₉

—

Dic₁₀₉

C₂²⋊C₄

yes

—

C₂×C₁₀₉

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₁₀₉

C₂×SD₁₆

yes

characteristic

C₂×C₁₀₉

C₂×Q₁₆

yes

characteristic

D₁₀₉

#103

yes

direct product factor

C₁₀₉

C₂×#103

yes

characteristic, complemented, Sylow

C₂×#103

—

109

maximal, Sylow

C₂²×Q₈

—

109

—

C₂²×C₈

—

109

—

C₂×C₄⋊C₄

—

109

—

#103

—

109

—

#103

—

109

—

#103

—

109

—

#103

D₁₀₉

yes

characteristic, direct product factor

C₂×Q₈

—

109

—

C₂×Q₈

—

109

—

C₂×Q₈

—

218

—

C₂×Q₈

—

218

—

C₂×Q₈

C₂×D₁₀₉

yes

characteristic

C₂²×C₄

—

109

—

C₂²×C₄

—

218

—

C₂²×C₄

—

218

—

C₂×C₈

—

109

—

C₂×C₈

—

218

—

C₂×C₈

C₂×D₁₀₉

yes

characteristic

C₄⋊C₄

—

109

—

C₄⋊C₄

—

218

—

C₄⋊C₄

C₂×D₁₀₉

yes

characteristic

C₂³

—

109

—

Q₈

—

Q₈

—

109

—

Q₈

—

218

—

Q₈

—

218

—

Q₈

C₄×D₁₀₉

yes

complemented

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

436

—

C₂×C₄

—

218

—

C₂×C₄

—

218

—

C₂×C₄

—

218

—

C₂×C₄

C₂²×D₁₀₉

yes

characteristic

C₈

—

C₈

—

218

—

C₂²

—

109

—

C₂²

—

109

—

C₂²

—

109

—

C₂²

D₄×D₁₀₉

yes

characteristic

100

C₄

—

101

C₄

—

102

C₄

—

109

—

103

C₄

—

436

—

104

C₄

—

218

—

105

C₄

C₂×C₄×D₁₀₉

yes

characteristic

106

C₄

D₄×D₁₀₉

yes

characteristic

107

C₂

—

109

—

108

C₂

C₂²⋊C₄×D₁₀₉

yes

characteristic

109

C₂

SD₁₆×D₁₀₉

yes

characteristic

110

C₂

Q₁₆×D₁₀₉

yes

characteristic

	subgroup	G/H
center	C₂²	D₄×D₁₀₉
commutator	C₄×C₁₀₉	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₁₀₉
Fitting	#103×C₁₀₉	C₂
socle	C₂²×C₁₀₉	C₂×D₄
hypercenter	#103	D₁₀₉
hypocenter	C₁₀₉	C₂×#103
LCS	G ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #103

subgroup

G/H

center

C₂²

D₄×D₁₀₉

commutator

C₄×C₁₀₉

C₂²×C₄

Frattini

C₂×C₄

C₂²×D₁₀₉

Fitting

#103×C₁₀₉

C₂

socle

C₂²×C₁₀₉

C₂×D₄

D₁₀₉

C₁₀₉

G ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ #103

N	H ≅ G/N	type
4. #16812624	C₂	semidirect
5. #16812623	C₂	semidirect
6. #16812622	C₂	semidirect
7. #103×C₁₀₉	C₂	semidirect
11. Q₈×D₁₀₉	C₄	semidirect
24. Q₈×C₁₀₉	C₂×C₄	semidirect
37. Dic₂₁₈	C₂×C₄	semidirect
54. D₁₀₉	#103	direct (N × H)
55. C₁₀₉	C₂×#103	semidirect
63. #103	D₁₀₉	direct (N × H)
83. Q₈	C₄×D₁₀₉	semidirect

H ≅ G/N

type

4. #16812624

C₂

semidirect

5. #16812623

C₂

semidirect

6. #16812622

C₂

semidirect

7. #103×C₁₀₉

C₂

semidirect

11. Q₈×D₁₀₉

C₄

semidirect

24. Q₈×C₁₀₉

C₂×C₄

semidirect

37. Dic₂₁₈

C₂×C₄

semidirect

54. D₁₀₉

#103

direct (N × H)

55. C₁₀₉

C₂×#103

semidirect

63. #103

D₁₀₉

direct (N × H)

83. Q₈

C₄×D₁₀₉

semidirect

Subgroups — #103×D₁₀₉ #20295015

Conjugacy classes of subgroups

Complements

Subgroups — #103×D109 #20295015

Conjugacy classes of subgroups

Complements

Subgroups — #103×D₁₀₉ #20295015