Group #20294925 — subgroups

Subgroups — #20294925

Order 6976.

170 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 170 conjugacy classes of subgroups — too many to draw. Only its 91 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×C₄⋊Dic₁₀₉	C₂	1	2	yes	maximal
2	C₂×C₄⋊Dic₁₀₉	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×Dic₁₀₉	C₂	1	1	yes	maximal, characteristic
4	#16812606	C₂	1	2	yes	maximal
5	C₂×C₄⋊C₄×C₁₀₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₁₀₉	C₂²	1	2	yes	—
7	C₂²×Dic₁₀₉	C₂²	1	1	yes	characteristic
8	C₂²×Dic₁₀₉	C₂²	1	1	yes	characteristic
9	C₄⋊Dic₁₀₉	—	2	2	no	—
10	C₄⋊Dic₁₀₉	—	2	4	no	—
11	C₄×Dic₁₀₉	—	2	2	no	—
12	C₂²×C₄×C₁₀₉	C₂²	1	2	yes	—
13	C₂²×C₄×C₁₀₉	C₂²	1	1	yes	characteristic
14	C₄⋊C₄×C₁₀₉	C₄	1	4	yes	complemented
15	C₂³×C₁₀₉	C₂³	1	1	yes	characteristic
16	C₂×C₄×C₁₀₉	—	2	2	no	—
17	C₂×C₄×C₁₀₉	—	2	2	no	—
18	C₂×C₄×C₁₀₉	C₂×C₄	1	4	yes	—
19	C₂×C₄×C₁₀₉	C₂×C₄	1	2	yes	—
20	C₂×C₄×C₁₀₉	D₄	1	2	yes	—
21	C₂×C₄×C₁₀₉	Q₈	1	2	yes	—
22	C₂×Dic₁₀₉	—	2	1	no	—
23	C₂×Dic₁₀₉	—	2	1	no	—
24	C₂×Dic₁₀₉	—	2	1	no	—
25	C₂×Dic₁₀₉	—	2	1	no	—
26	C₂×Dic₁₀₉	—	2	2	no	—
27	C₂×Dic₁₀₉	—	2	2	no	—
28	C₂×Dic₁₀₉	—	2	2	no	—
29	C₂×Dic₁₀₉	D₄	1	2	yes	—
30	C₂×Dic₁₀₉	Q₈	1	2	yes	—
31	C₂²×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
32	C₂²×C₁₀₉	C₂×D₄	1	1	yes	characteristic
33	C₂²×C₁₀₉	C₂×D₄	1	1	yes	characteristic
34	C₂²×C₁₀₉	C₂×Q₈	1	1	yes	characteristic
35	C₂²×C₁₀₉	C₂×Q₈	1	1	yes	characteristic
36	C₂²×C₁₀₉	#41	1	1	yes	characteristic
37	C₂²×C₁₀₉	#41	1	1	yes	characteristic
38	C₄×C₁₀₉	—	2	4	no	—
39	C₄×C₁₀₉	C₄⋊C₄	1	4	yes	complemented
40	Dic₁₀₉	—	2	2	no	—
41	Dic₁₀₉	—	2	2	no	—
42	Dic₁₀₉	—	4	2	no	—
43	C₂×C₁₀₉	C₂×C₄⋊C₄	1	1	yes	characteristic
44	C₂×C₁₀₉	C₄×D₄	1	1	yes	characteristic
45	C₂×C₁₀₉	C₄×Q₈	1	1	yes	characteristic
46	C₂×C₁₀₉	#121	1	1	yes	characteristic
47	C₂×C₁₀₉	C₂²⋊Q₈	1	1	yes	characteristic
48	C₂×C₁₀₉	#125	1	1	yes	characteristic
49	C₂×C₁₀₉	C₄⋊Q₈	1	1	yes	characteristic
50	C₁₀₉	#389	1	1	yes	characteristic, complemented, Sylow
51	#389	—	109	1	no	maximal, Sylow
52	C₂×C₄⋊C₄	—	109	2	no	—
53	C₂×C₄⋊C₄	—	109	1	no	—
54	C₂×C₄⋊C₄	D₁₀₉	1	1	yes	characteristic
55	C₂×C₄²	—	109	1	no	—
56	#95	—	109	2	no	—
57	C₂²×C₄	—	109	1	no	—
58	C₂²×C₄	—	109	1	no	—
59	C₂²×C₄	—	109	2	no	—
60	C₂²×C₄	C₂×D₁₀₉	1	2	yes	—
61	C₂²×C₄	C₂×D₁₀₉	1	1	yes	characteristic
62	C₄⋊C₄	—	218	4	no	—
63	C₄⋊C₄	—	218	2	no	—
64	C₄⋊C₄	Dic₁₀₉	1	4	yes	complemented
65	C₄²	—	218	2	no	—
66	C₂³	C₂²×D₁₀₉	1	1	yes	characteristic
67	C₂×C₄	—	2	2	no	—
68	C₂×C₄	—	2	2	no	—
69	C₂×C₄	—	109	2	no	—
70	C₂×C₄	—	109	2	no	—
71	C₂×C₄	—	218	2	no	—
72	C₂×C₄	—	218	2	no	—
73	C₂×C₄	—	218	2	no	—
74	C₂×C₄	—	218	1	no	—
75	C₂×C₄	—	218	1	no	—
76	C₂×C₄	—	218	1	no	—
77	C₂×C₄	—	218	1	no	—
78	C₂×C₄	Dic₂₁₈	1	2	yes	—
79	C₂×C₄	D₄₃₆	1	2	yes	—
80	C₂×C₄	C₂×Dic₁₀₉	1	4	yes	—
81	C₂×C₄	C₂×Dic₁₀₉	1	2	yes	—
82	C₂²	C₂²×Dic₁₀₉	1	1	yes	characteristic
83	C₂²	D₄×D₁₀₉	1	1	yes	characteristic
84	C₂²	C₂×D₄₃₆	1	1	yes	characteristic
85	C₂²	Q₈×D₁₀₉	1	1	yes	characteristic
86	C₂²	C₂×Dic₂₁₈	1	1	yes	characteristic
87	C₂²	#13182028	1	1	yes	characteristic
88	C₂²	#13182030	1	1	yes	characteristic
89	C₄	—	2	4	no	—
90	C₄	—	436	2	no	—
91	C₄	—	218	2	no	—
92	C₄	—	218	2	no	—
93	C₄	C₄⋊Dic₁₀₉	1	4	yes	complemented
94	C₂	C₂×C₄⋊Dic₁₀₉	1	1	yes	characteristic
95	C₂	D₄×Dic₁₀₉	1	1	yes	characteristic
96	C₂	Q₈×Dic₁₀₉	1	1	yes	characteristic
97	C₂	#16812674	1	1	yes	characteristic
98	C₂	#16812679	1	1	yes	characteristic
99	C₂	#16812690	1	1	yes	characteristic
100	C₂	#16812698	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊Dic₁₀₉

C₂

yes

maximal

C₂×C₄⋊Dic₁₀₉

C₂

yes

maximal, characteristic

C₂×C₄×Dic₁₀₉

C₂

yes

maximal, characteristic

#16812606

C₂

yes

maximal

C₂×C₄⋊C₄×C₁₀₉

C₂

yes

maximal, characteristic

C₂²×Dic₁₀₉

C₂²

yes

—

C₂²×Dic₁₀₉

C₂²

yes

characteristic

C₂²×Dic₁₀₉

C₂²

yes

characteristic

C₄⋊Dic₁₀₉

—

C₄⋊Dic₁₀₉

—

C₄×Dic₁₀₉

—

C₂²×C₄×C₁₀₉

C₂²

yes

—

C₂²×C₄×C₁₀₉

C₂²

yes

characteristic

C₄⋊C₄×C₁₀₉

C₄

yes

complemented

C₂³×C₁₀₉

C₂³

yes

characteristic

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₂×C₄

yes

—

C₂×C₄×C₁₀₉

C₂×C₄

yes

—

C₂×C₄×C₁₀₉

D₄

yes

—

C₂×C₄×C₁₀₉

Q₈

yes

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

D₄

yes

—

C₂×Dic₁₀₉

Q₈

yes

—

C₂²×C₁₀₉

C₂²×C₄

yes

characteristic

C₂²×C₁₀₉

C₂×D₄

yes

characteristic

C₂²×C₁₀₉

C₂×D₄

yes

characteristic

C₂²×C₁₀₉

C₂×Q₈

yes

characteristic

C₂²×C₁₀₉

C₂×Q₈

yes

C₂²×C₁₀₉

yes

C₂²×C₁₀₉

yes

C₄×C₁₀₉

—

C₄×C₁₀₉

C₄⋊C₄

yes

complemented

Dic₁₀₉

—

Dic₁₀₉

—

Dic₁₀₉

—

C₂×C₁₀₉

C₂×C₄⋊C₄

yes

characteristic

C₂×C₁₀₉

C₄×D₄

yes

characteristic

C₂×C₁₀₉

C₄×Q₈

yes

characteristic

C₂×C₁₀₉

#121

yes

characteristic

C₂×C₁₀₉

C₂²⋊Q₈

yes

characteristic

C₂×C₁₀₉

#125

yes

characteristic

C₂×C₁₀₉

C₄⋊Q₈

yes

characteristic

C₁₀₉

#389

yes

characteristic, complemented, Sylow

#389

—

109

maximal, Sylow

C₂×C₄⋊C₄

—

109

—

C₂×C₄⋊C₄

—

109

—

C₂×C₄⋊C₄

D₁₀₉

yes

characteristic

C₂×C₄²

—

109

—

#95

—

109

—

C₂²×C₄

—

109

—

C₂²×C₄

—

109

—

C₂²×C₄

—

109

—

C₂²×C₄

C₂×D₁₀₉

yes

—

C₂²×C₄

C₂×D₁₀₉

yes

characteristic

C₄⋊C₄

—

218

—

C₄⋊C₄

—

218

—

C₄⋊C₄

Dic₁₀₉

yes

complemented

C₄²

—

218

—

C₂³

C₂²×D₁₀₉

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

218

—

C₂×C₄

—

218

—

C₂×C₄

—

218

—

C₂×C₄

—

218

—

C₂×C₄

—

218

—

C₂×C₄

—

218

—

C₂×C₄

—

218

—

C₂×C₄

Dic₂₁₈

yes

—

C₂×C₄

D₄₃₆

yes

—

C₂×C₄

C₂×Dic₁₀₉

yes

—

C₂×C₄

C₂×Dic₁₀₉

yes

—

C₂²

C₂²×Dic₁₀₉

yes

characteristic

C₂²

D₄×D₁₀₉

yes

characteristic

C₂²

C₂×D₄₃₆

yes

characteristic

C₂²

Q₈×D₁₀₉

yes

characteristic

C₂²

C₂×Dic₂₁₈

yes

C₂²

yes

C₂²

yes

C₄

—

C₄

—

436

—

C₄

—

218

—

C₄

—

218

—

C₄

C₄⋊Dic₁₀₉

yes

complemented

C₂

C₂×C₄⋊Dic₁₀₉

yes

characteristic

C₂

D₄×Dic₁₀₉

yes

characteristic

C₂

Q₈×Dic₁₀₉

yes

C₂

yes

C₂

yes

C₂

yes

100

C₂

#16812698

yes

characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂²×D₁₀₉
commutator	C₂²×C₁₀₉	C₂²×C₄
Fitting	C₂×C₄⋊C₄×C₁₀₉	C₂
socle	C₂³×C₁₀₉	C₂³
hypercenter	C₂×C₄⋊C₄	D₁₀₉
hypocenter	C₁₀₉	#389
LCS	G ▷ C₂²×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄⋊C₄

subgroup

G/H

center, Frattini

C₂³

C₂²×D₁₀₉

commutator

C₂²×C₁₀₉

C₂²×C₄

Fitting

C₂×C₄⋊C₄×C₁₀₉

C₂

socle

C₂³×C₁₀₉

C₂³

hypercenter

C₂×C₄⋊C₄

D₁₀₉

C₁₀₉

G ▷ C₂²×C₁₀₉ ▷ C₁₀₉

C₁ ◁ C₂³ ◁ C₂×C₄⋊C₄

N	H ≅ G/N	type
14. C₄⋊C₄×C₁₀₉	C₄	semidirect
39. C₄×C₁₀₉	C₄⋊C₄	semidirect
50. C₁₀₉	#389	semidirect
64. C₄⋊C₄	Dic₁₀₉	semidirect
93. C₄	C₄⋊Dic₁₀₉	semidirect

H ≅ G/N

type

14. C₄⋊C₄×C₁₀₉

C₄

semidirect

39. C₄×C₁₀₉

C₄⋊C₄

semidirect

50. C₁₀₉

#389

semidirect

64. C₄⋊C₄

Dic₁₀₉

semidirect

93. C₄

C₄⋊Dic₁₀₉

semidirect

Subgroups — #20294925

Conjugacy classes of subgroups

Complements