Subgroups — C₂²×#19149729 #36369981

Order 22144.

120 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19149729	C₂	1	6	yes	maximal, direct product factor
2	C₂²×#14583801	C₂	1	1	yes	maximal, characteristic
3	#19149729	C₂²	1	4	yes	direct product factor
4	C₂×#14583801	C₄	1	3	yes	—
5	C₂×#14583801	C₂²	1	3	yes	—
6	C₂²×C₈×C₁₇₃	C₄	1	1	yes	characteristic
7	#14583801	C₂×C₄	1	3	yes	—
8	#14583801	C₂³	1	1	yes	characteristic
9	C₂²×C₄×C₁₇₃	C₈	1	1	yes	characteristic
10	C₂×C₈×C₁₇₃	C₈	1	3	yes	—
11	C₂×C₈×C₁₇₃	C₂×C₄	1	3	yes	—
12	C₂³×C₁₇₃	C₁₆	1	1	yes	characteristic
13	C₂×C₄×C₁₇₃	C₁₆	1	3	yes	—
14	C₂×C₄×C₁₇₃	C₂×C₈	1	3	yes	—
15	C₈×C₁₇₃	C₂×C₈	1	3	yes	—
16	C₈×C₁₇₃	C₂²×C₄	1	1	yes	characteristic
17	C₂²×C₁₇₃	C₃₂	1	4	yes	complemented
18	C₂²×C₁₇₃	C₂×C₁₆	1	3	yes	—
19	C₄×C₁₇₃	C₂×C₁₆	1	3	yes	—
20	C₄×C₁₇₃	C₂²×C₈	1	1	yes	characteristic
21	C₂×C₁₇₃	C₂×C₃₂	1	6	yes	complemented
22	C₂×C₁₇₃	C₂²×C₁₆	1	1	yes	characteristic
23	C₁₇₃	C₂²×C₃₂	1	1	yes	characteristic, complemented, Sylow
24	C₂²×C₃₂	—	173	1	no	maximal, Sylow
25	C₂²×C₁₆	—	173	1	no	—
26	C₂×C₃₂	—	173	6	no	—
27	C₂²×C₈	#10621924	1	1	yes	characteristic
28	C₂×C₁₆	—	173	3	no	—
29	C₂×C₁₆	—	173	3	no	—
30	C₃₂	—	173	4	no	—
31	C₂²×C₄	#13063443	1	1	yes	characteristic
32	C₂×C₈	#13063443	1	3	yes	—
33	C₂×C₈	C₂×#10621924	1	3	yes	—
34	C₁₆	—	173	1	no	—
35	C₁₆	—	173	3	no	—
36	C₂³	#14583829	1	1	yes	characteristic
37	C₂×C₄	#14583829	1	3	yes	—
38	C₂×C₄	C₂×#13063443	1	3	yes	—
39	C₈	C₂×#13063443	1	3	yes	—
40	C₈	C₂²×#10621924	1	1	yes	characteristic
41	C₂²	#19149729	1	4	yes	direct product factor
42	C₂²	C₂×#14583829	1	3	yes	—
43	C₄	C₂×#14583829	1	3	yes	—
44	C₄	C₂²×#13063443	1	1	yes	characteristic
45	C₂	C₂×#19149729	1	6	yes	direct product factor
46	C₂	C₂²×#14583829	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂²×C₈	#10621924
commutator, hypocenter	C₁₇₃	C₂²×C₃₂
Frattini	C₈	C₂²×#10621924
Fitting	C₂²×C₈×C₁₇₃	C₄
socle, prime radical	C₂³×C₁₇₃	C₁₆
cosocle	#14583801	C₂³
LCS	G ▷ C₁₇₃
UCS	C₁ ◁ C₂²×C₈

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₂×#19149729	C₂	direct (N × H)
3. #19149729	C₂²	direct (N × H)
17. C₂²×C₁₇₃	C₃₂	semidirect
21. C₂×C₁₇₃	C₂×C₃₂	semidirect
23. C₁₇₃	C₂²×C₃₂	semidirect
41. C₂²	#19149729	direct (N × H)
45. C₂	C₂×#19149729	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#19149729	2
C₂²×#14583801	2
C₂²×C₃₂	173

Subgroups — C22×#19149729 #36369981

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#19149729 #36369981