Group #20295299 — subgroups

Subgroups — #20295299

Order 6976.

100 conjugacy classes of subgroups; only the 37 normal subgroups are shown.

normal subgroups only

This group has 100 conjugacy classes of subgroups — too many to draw. Only its 37 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	#16812729	C₂	1	1	yes	maximal, characteristic
2	C₂×#13182019	C₂	1	1	yes	maximal, characteristic, complemented
3	#16812706	C₂	1	1	yes	maximal, characteristic
4	#16812634	C₂	1	1	yes	maximal, characteristic, complemented
5	#16812618	C₂	1	1	yes	maximal, characteristic, complemented
6	#16812617	C₂	1	1	yes	maximal, characteristic, complemented
7	#137×C₁₀₉	C₂	1	1	yes	maximal, characteristic
8	C₂×Dic₂₁₈	C₂²	1	1	yes	characteristic
9	#13182019	—	2	1	no	—
10	#13182019	—	2	1	no	—
11	#13182019	—	2	1	no	—
12	#13182017	—	2	1	no	—
13	#13182013	—	2	1	no	—
14	#13182013	C₂²	1	1	yes	characteristic
15	#13182013	C₂²	1	1	yes	characteristic
16	C₂×#11724784	—	2	1	no	—
17	C₂×#11724784	C₂²	1	1	yes	characteristic
18	#41×C₁₀₉	C₂²	1	1	yes	characteristic
19	C₂×Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
20	Q₁₆×C₁₀₉	—	2	1	no	—
21	SD₁₆×C₁₀₉	—	2	1	no	—
22	M₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
23	Q₈×C₁₀₉	—	2	1	no	—
24	Q₈×C₁₀₉	—	2	1	no	—
25	Q₈×C₁₀₉	D₄	1	1	yes	characteristic
26	D₄×C₁₀₉	—	2	1	no	—
27	D₄×C₁₀₉	D₄	1	1	yes	characteristic
28	C₂×C₄×C₁₀₉	—	2	1	no	—
29	C₂×C₄×C₁₀₉	—	2	1	no	—
30	C₂×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
31	C₂×Dic₁₀₉	—	2	1	no	—
32	Dic₂₁₈	—	2	1	no	—
33	Dic₂₁₈	—	2	1	no	—
34	C₈×C₁₀₉	—	2	1	no	—
35	#11724784	—	2	1	no	—
36	#11724784	—	2	1	no	—
37	#11724784	D₄	1	1	yes	characteristic
38	#11724784	D₄	1	1	yes	characteristic
39	C₂²×C₁₀₉	—	2	1	no	—
40	C₂²×C₁₀₉	#41	1	1	yes	characteristic
41	C₄×C₁₀₉	—	2	1	no	—
42	C₄×C₁₀₉	—	4	1	no	—
43	C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
44	C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
45	Dic₁₀₉	—	4	1	no	—
46	C₂×C₁₀₉	—	2	1	no	—
47	C₂×C₁₀₉	—	4	1	no	—
48	C₂×C₁₀₉	#121	1	1	yes	characteristic
49	C₁₀₉	#473	1	1	yes	characteristic, complemented, Sylow
50	#473	—	109	1	no	maximal, Sylow
51	#137	—	109	1	no	—
52	#137	D₁₀₉	1	1	yes	characteristic
53	C₂×Q₁₆	—	109	1	no	—
54	#131	—	109	1	no	—
55	#108	—	109	1	no	—
56	#101	—	109	1	no	—
57	#101	—	109	1	no	—
58	#41	C₂×D₁₀₉	1	1	yes	characteristic
59	C₂×Q₈	—	109	1	no	—
60	C₂×Q₈	C₂×D₁₀₉	1	1	yes	characteristic
61	Q₁₆	—	2	1	no	—
62	Q₁₆	—	218	1	no	—
63	Q₁₆	—	218	1	no	—
64	Q₁₆	—	218	1	no	—
65	SD₁₆	—	2	1	no	—
66	SD₁₆	—	218	1	no	—
67	M₁₆	—	109	1	no	—
68	M₁₆	—	109	1	no	—
69	M₁₆	—	218	1	no	—
70	M₁₆	C₂×D₁₀₉	1	1	yes	characteristic
71	C₂×C₈	—	109	1	no	—
72	C₂×C₈	—	218	1	no	—
73	Q₈	—	2	1	no	—
74	Q₈	—	2	1	no	—
75	Q₈	—	218	1	no	—
76	Q₈	—	218	1	no	—
77	Q₈	C₂²⋊D₁₀₉	1	1	yes	characteristic
78	D₄	—	2	1	no	—
79	D₄	C₂²⋊D₁₀₉	1	1	yes	characteristic
80	C₂×C₄	—	2	1	no	—
81	C₂×C₄	—	2	1	no	—
82	C₂×C₄	—	218	1	no	—
83	C₂×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
84	C₈	—	2	1	no	—
85	C₈	—	109	1	no	—
86	C₈	—	109	1	no	—
87	C₈	—	218	1	no	—
88	C₈	—	218	1	no	—
89	C₂²	—	2	1	no	—
90	C₂²	#13182028	1	1	yes	characteristic
91	C₄	—	2	1	no	—
92	C₄	—	4	1	no	—
93	C₄	—	436	1	no	—
94	C₄	D₄×D₁₀₉	1	1	yes	characteristic
95	C₄	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
96	C₂	—	2	1	no	—
97	C₂	—	4	1	no	—
98	C₂	#16812671	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16812729

C₂

yes

maximal, characteristic

C₂×#13182019

C₂

yes

maximal, characteristic, complemented

#16812706

C₂

yes

maximal, characteristic

#16812634

C₂

yes

maximal, characteristic, complemented

#16812618

C₂

yes

maximal, characteristic, complemented

#16812617

C₂

yes

maximal, characteristic, complemented

#137×C₁₀₉

C₂

yes

maximal, characteristic

C₂×Dic₂₁₈

C₂²

yes

—

—

—

—

—

C₂²

yes

characteristic

#13182013

C₂²

yes

characteristic

C₂×#11724784

—

C₂×#11724784

C₂²

yes

characteristic

#41×C₁₀₉

C₂²

yes

characteristic

C₂×Q₈×C₁₀₉

C₂²

yes

characteristic

Q₁₆×C₁₀₉

—

SD₁₆×C₁₀₉

—

M₁₆×C₁₀₉

C₂²

yes

characteristic

Q₈×C₁₀₉

—

Q₈×C₁₀₉

—

Q₈×C₁₀₉

D₄

yes

characteristic

D₄×C₁₀₉

—

D₄×C₁₀₉

D₄

yes

characteristic

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₂³

yes

characteristic

C₂×Dic₁₀₉

—

Dic₂₁₈

—

Dic₂₁₈

—

C₈×C₁₀₉

—

#11724784

—

#11724784

—

#11724784

D₄

yes

characteristic

#11724784

D₄

yes

characteristic

C₂²×C₁₀₉

—

C₂²×C₁₀₉

#41

yes

characteristic

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

C₂×D₄

yes

characteristic

C₄×C₁₀₉

C₂×D₄

yes

characteristic

Dic₁₀₉

—

C₂×C₁₀₉

—

C₂×C₁₀₉

—

C₂×C₁₀₉

#121

yes

characteristic

C₁₀₉

#473

yes

characteristic, complemented, Sylow

#473

—

109

maximal, Sylow

#137

—

109

—

#137

D₁₀₉

yes

characteristic

C₂×Q₁₆

—

109

—

#131

—

109

—

#108

—

109

—

#101

—

109

—

#101

—

109

—

#41

C₂×D₁₀₉

yes

characteristic

C₂×Q₈

—

109

—

C₂×Q₈

C₂×D₁₀₉

yes

characteristic

Q₁₆

—

Q₁₆

—

218

—

Q₁₆

—

218

—

Q₁₆

—

218

—

SD₁₆

—

SD₁₆

—

218

—

M₁₆

—

109

—

M₁₆

—

109

—

M₁₆

—

218

—

M₁₆

C₂×D₁₀₉

yes

characteristic

C₂×C₈

—

109

—

C₂×C₈

—

218

—

Q₈

—

Q₈

—

Q₈

—

218

—

Q₈

—

218

—

Q₈

C₂²⋊D₁₀₉

yes

characteristic

D₄

—

D₄

C₂²⋊D₁₀₉

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

218

—

C₂×C₄

C₂²×D₁₀₉

yes

characteristic

C₈

—

C₈

—

109

—

C₈

—

109

—

C₈

—

218

—

C₈

—

218

—

C₂²

—

C₂²

#13182028

yes

characteristic

C₄

—

C₄

—

C₄

—

436

—

C₄

D₄×D₁₀₉

yes

characteristic

C₄

C₂×C₂²⋊D₁₀₉

yes

characteristic

C₂

—

C₂

—

C₂

#16812671

yes

characteristic

	subgroup	G/H
center	C₂	#16812671
commutator	C₂×C₄×C₁₀₉	C₂³
Frattini	C₂×C₄	C₂²×D₁₀₉
Fitting	#137×C₁₀₉	C₂
socle	C₂×C₁₀₉	#121
hypercenter	#137	D₁₀₉
hypocenter	C₁₀₉	#473
LCS	G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #137

subgroup

G/H

center

C₂

#16812671

commutator

C₂×C₄×C₁₀₉

C₂³

Frattini

C₂×C₄

C₂²×D₁₀₉

Fitting

#137×C₁₀₉

C₂

C₂×C₁₀₉

D₁₀₉

C₁₀₉

G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ #137

N	H ≅ G/N	type
2. C₂×#13182019	C₂	semidirect
4. #16812634	C₂	semidirect
5. #16812618	C₂	semidirect
6. #16812617	C₂	semidirect
49. C₁₀₉	#473	semidirect

H ≅ G/N

type

2. C₂×#13182019

C₂

semidirect

4. #16812634

C₂

semidirect

5. #16812618

C₂

semidirect

6. #16812617

C₂

semidirect

49. C₁₀₉

#473

semidirect

Subgroups — #20295299

Conjugacy classes of subgroups

Complements