Group #22623039 — subgroups

Subgroups — C₂²×#14556534 #22623039

Order 9344.

400 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 149 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	C₂²×D₅₈₄	C₂	1	1	yes	maximal, characteristic
2	C₂×#14556534	C₂	1	12	yes	maximal, direct product factor
3	C₂²×#11924003	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²×Q₁₆×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×D₂₉₂	—	2	1	no	—
6	C₂×D₅₈₄	—	2	3	no	—
7	C₂×D₅₈₄	C₂²	1	6	yes	—
8	#14556534	C₂²	1	16	yes	direct product factor
9	C₂×#11924003	C₂²	1	6	yes	complemented
10	C₂²×Q₈×C₇₃	—	2	1	no	—
11	C₂×Q₁₆×C₇₃	—	2	3	no	—
12	C₂×Q₁₆×C₇₃	C₂²	1	6	yes	complemented
13	C₂²×C₈×C₇₃	C₂²	1	1	yes	characteristic
14	C₂³×D₇₃	—	4	1	no	—
15	C₂×D₂₉₂	—	2	6	no	—
16	C₂×D₂₉₂	—	4	3	no	—
17	D₅₈₄	—	2	6	no	—
18	D₅₈₄	C₂³	1	4	yes	—
19	#11924003	C₂³	1	4	yes	complemented
20	C₂×Q₈×C₇₃	—	2	6	no	—
21	C₂×Q₈×C₇₃	—	4	3	no	—
22	C₂²×C₄×C₇₃	—	4	1	no	—
23	C₂²×C₄×C₇₃	D₄	1	1	yes	characteristic
24	Q₁₆×C₇₃	—	2	6	no	—
25	Q₁₆×C₇₃	C₂³	1	4	yes	complemented
26	C₂×C₈×C₇₃	D₄	1	3	yes	—
27	C₂×C₈×C₇₃	C₂³	1	3	yes	—
28	C₂³×C₇₃	D₈	1	1	yes	characteristic
29	C₂²×D₇₃	—	4	6	no	—
30	C₂²×D₇₃	—	8	4	no	—
31	Q₈×C₇₃	—	2	4	no	—
32	Q₈×C₇₃	—	4	6	no	—
33	C₂×C₄×C₇₃	—	4	6	no	—
34	C₂×C₄×C₇₃	D₈	1	3	yes	—
35	C₂×C₄×C₇₃	C₂×D₄	1	3	yes	—
36	D₂₉₂	—	2	4	no	—
37	D₂₉₂	—	4	6	no	—
38	C₈×C₇₃	C₂×D₄	1	3	yes	—
39	C₈×C₇₃	C₂⁴	1	1	yes	characteristic
40	C₂²×C₇₃	SD₃₂	1	4	yes	complemented
41	C₂²×C₇₃	C₂×D₈	1	3	yes	—
42	C₂×D₇₃	—	4	4	no	—
43	C₂×D₇₃	—	8	12	no	—
44	C₄×C₇₃	—	4	4	no	—
45	C₄×C₇₃	C₂×D₈	1	3	yes	—
46	C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
47	C₂×C₇₃	C₂×SD₃₂	1	6	yes	complemented
48	C₂×C₇₃	C₂²×D₈	1	1	yes	characteristic
49	D₇₃	—	8	4	no	—
50	C₂²×SD₃₂	—	73	1	no	maximal, Sylow
51	C₇₃	C₂²×SD₃₂	1	1	yes	characteristic, complemented, Sylow
52	C₂²×Q₁₆	D₇₃	1	1	yes	characteristic, complemented
53	C₂²×D₈	—	73	1	no	—
54	C₂×SD₃₂	—	73	12	no	—
55	C₂²×C₁₆	—	73	1	no	—
56	C₂²×Q₈	—	2	1	no	—
57	C₂²×D₄	—	146	1	no	—
58	C₂×Q₁₆	—	2	3	no	—
59	C₂×Q₁₆	C₂×D₇₃	1	6	yes	complemented
60	C₂×D₈	—	73	6	no	—
61	C₂×D₈	—	146	3	no	—
62	C₂²×C₈	C₂×D₇₃	1	1	yes	characteristic
63	SD₃₂	—	73	16	no	—
64	C₂×C₁₆	—	73	6	no	—
65	C₂⁴	—	292	1	no	—
66	C₂×Q₈	—	2	6	no	—
67	C₂×Q₈	—	4	3	no	—
68	C₂×D₄	—	292	3	no	—
69	C₂×D₄	—	146	6	no	—
70	C₂²×C₄	—	4	1	no	—
71	C₂²×C₄	C₂²⋊D₇₃	1	1	yes	characteristic
72	Q₁₆	—	2	6	no	—
73	Q₁₆	C₂²×D₇₃	1	4	yes	complemented
74	D₈	—	73	4	no	—
75	D₈	—	146	6	no	—
76	C₂×C₈	C₂²⋊D₇₃	1	3	yes	—
77	C₂×C₈	C₂²×D₇₃	1	3	yes	—
78	C₁₆	—	73	4	no	—
79	C₂³	—	292	6	no	—
80	C₂³	—	584	4	no	—
81	C₂³	#11924013	1	1	yes	characteristic
82	Q₈	—	2	4	no	—
83	Q₈	—	4	6	no	—
84	D₄	—	292	6	no	—
85	D₄	—	146	4	no	—
86	C₂×C₄	—	4	6	no	—
87	C₂×C₄	#11924013	1	3	yes	—
88	C₂×C₄	C₂×C₂²⋊D₇₃	1	3	yes	—
89	C₈	C₂×C₂²⋊D₇₃	1	3	yes	—
90	C₈	C₂³×D₇₃	1	1	yes	characteristic
91	C₂²	—	292	4	no	—
92	C₂²	—	584	12	no	—
93	C₂²	#14556534	1	4	yes	direct product factor
94	C₂²	C₂×#11924013	1	3	yes	—
95	C₄	—	4	4	no	—
96	C₄	C₂×#11924013	1	3	yes	—
97	C₄	C₂²×C₂²⋊D₇₃	1	1	yes	characteristic
98	C₂	—	584	4	no	—
99	C₂	C₂×#14556534	1	6	yes	direct product factor
100	C₂	C₂²×#11924013	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×D₅₈₄

C₂

yes

maximal, characteristic

C₂×#14556534

C₂

yes

maximal, direct product factor

C₂²×#11924003

C₂

yes

maximal, characteristic, complemented

C₂²×Q₁₆×C₇₃

C₂

yes

maximal, characteristic, complemented

C₂²×D₂₉₂

—

C₂×D₅₈₄

—

C₂×D₅₈₄

C₂²

yes

—

#14556534

C₂²

yes

direct product factor

C₂×#11924003

C₂²

yes

complemented

C₂²×Q₈×C₇₃

—

C₂×Q₁₆×C₇₃

—

C₂×Q₁₆×C₇₃

C₂²

yes

complemented

C₂²×C₈×C₇₃

C₂²

yes

characteristic

C₂³×D₇₃

—

C₂×D₂₉₂

—

C₂×D₂₉₂

—

D₅₈₄

—

D₅₈₄

C₂³

yes

—

#11924003

C₂³

yes

complemented

C₂×Q₈×C₇₃

—

C₂×Q₈×C₇₃

—

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

D₄

yes

characteristic

Q₁₆×C₇₃

—

Q₁₆×C₇₃

C₂³

yes

complemented

C₂×C₈×C₇₃

D₄

yes

—

C₂×C₈×C₇₃

C₂³

yes

—

C₂³×C₇₃

D₈

yes

characteristic

C₂²×D₇₃

—

C₂²×D₇₃

—

Q₈×C₇₃

—

Q₈×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

D₈

yes

—

C₂×C₄×C₇₃

C₂×D₄

yes

—

D₂₉₂

—

D₂₉₂

—

C₈×C₇₃

C₂×D₄

yes

—

C₈×C₇₃

C₂⁴

yes

characteristic

C₂²×C₇₃

SD₃₂

yes

complemented

C₂²×C₇₃

C₂×D₈

yes

—

C₂×D₇₃

—

C₂×D₇₃

—

C₄×C₇₃

—

C₄×C₇₃

C₂×D₈

yes

—

C₄×C₇₃

C₂²×D₄

yes

characteristic

C₂×C₇₃

C₂×SD₃₂

yes

complemented

C₂×C₇₃

C₂²×D₈

yes

characteristic

D₇₃

—

C₂²×SD₃₂

—

maximal, Sylow

C₇₃

C₂²×SD₃₂

yes

characteristic, complemented, Sylow

C₂²×Q₁₆

D₇₃

yes

characteristic, complemented

C₂²×D₈

—

C₂×SD₃₂

—

C₂²×C₁₆

—

C₂²×Q₈

—

C₂²×D₄

—

146

—

C₂×Q₁₆

—

C₂×Q₁₆

C₂×D₇₃

yes

complemented

C₂×D₈

—

C₂×D₈

—

146

—

C₂²×C₈

C₂×D₇₃

yes

characteristic

SD₃₂

—

C₂×C₁₆

—

C₂⁴

—

292

—

C₂×Q₈

—

C₂×Q₈

—

C₂×D₄

—

292

—

C₂×D₄

—

146

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊D₇₃

yes

characteristic

Q₁₆

—

Q₁₆

C₂²×D₇₃

yes

complemented

D₈

—

D₈

—

146

—

C₂×C₈

C₂²⋊D₇₃

yes

—

C₂×C₈

C₂²×D₇₃

yes

—

C₁₆

—

C₂³

—

292

—

C₂³

—

584

—

C₂³

#11924013

yes

characteristic

Q₈

—

Q₈

—

D₄

—

292

—

D₄

—

146

—

C₂×C₄

—

C₂×C₄

#11924013

yes

—

C₂×C₄

C₂×C₂²⋊D₇₃

yes

—

C₈

C₂×C₂²⋊D₇₃

yes

—

C₈

C₂³×D₇₃

yes

characteristic

C₂²

—

292

—

C₂²

—

584

—

C₂²

#14556534

yes

direct product factor

C₂²

C₂×#11924013

yes

—

C₄

—

C₄

C₂×#11924013

yes

—

C₄

C₂²×C₂²⋊D₇₃

yes

characteristic

C₂

—

584

—

C₂

C₂×#14556534

yes

direct product factor

100

C₂

C₂²×#11924013

yes

characteristic

	subgroup	G/H
center	C₂³	#11924013
commutator	C₈×C₇₃	C₂⁴
Frattini	C₈	C₂³×D₇₃
Fitting	C₂²×Q₁₆×C₇₃	C₂
socle	C₂³×C₇₃	D₈
hypercenter	C₂²×Q₁₆	D₇₃
hypocenter	C₇₃	C₂²×SD₃₂
LCS	G ▷ C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×Q₁₆

subgroup

G/H

center

C₂³

#11924013

commutator

C₈×C₇₃

C₂⁴

Frattini

C₈

C₂³×D₇₃

Fitting

C₂²×Q₁₆×C₇₃

C₂

socle

C₂³×C₇₃

D₈

hypercenter

C₂²×Q₁₆

D₇₃

hypocenter

C₇₃

C₂²×SD₃₂

LCS

G ▷ C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×Q₁₆

N	H ≅ G/N	type
2. C₂×#14556534	C₂	direct (N × H)
3. C₂²×#11924003	C₂	semidirect
4. C₂²×Q₁₆×C₇₃	C₂	semidirect
8. #14556534	C₂²	direct (N × H)
9. C₂×#11924003	C₂²	semidirect
12. C₂×Q₁₆×C₇₃	C₂²	semidirect
19. #11924003	C₂³	semidirect
25. Q₁₆×C₇₃	C₂³	semidirect
40. C₂²×C₇₃	SD₃₂	semidirect
47. C₂×C₇₃	C₂×SD₃₂	semidirect
51. C₇₃	C₂²×SD₃₂	semidirect
52. C₂²×Q₁₆	D₇₃	semidirect
59. C₂×Q₁₆	C₂×D₇₃	semidirect
73. Q₁₆	C₂²×D₇₃	semidirect
93. C₂²	#14556534	direct (N × H)
99. C₂	C₂×#14556534	direct (N × H)

H ≅ G/N

type

2. C₂×#14556534

C₂

direct (N × H)

3. C₂²×#11924003

C₂

semidirect

4. C₂²×Q₁₆×C₇₃

C₂

semidirect

8. #14556534

C₂²

direct (N × H)

9. C₂×#11924003

C₂²

semidirect

12. C₂×Q₁₆×C₇₃

C₂²

semidirect

19. #11924003

C₂³

semidirect

25. Q₁₆×C₇₃

C₂³

semidirect

40. C₂²×C₇₃

SD₃₂

semidirect

47. C₂×C₇₃

C₂×SD₃₂

semidirect

51. C₇₃

C₂²×SD₃₂

semidirect

52. C₂²×Q₁₆

D₇₃

semidirect

59. C₂×Q₁₆

C₂×D₇₃

semidirect

73. Q₁₆

C₂²×D₇₃

semidirect

93. C₂²

#14556534

direct (N × H)

99. C₂

C₂×#14556534

direct (N × H)

Subgroups — C₂²×#14556534 #22623039

Conjugacy classes of subgroups

Complements

Subgroups — C22×#14556534 #22623039

Conjugacy classes of subgroups

Complements

Subgroups — C₂²×#14556534 #22623039