Subgroups — #36368594

Order 22144.

70 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26012354	C₂	1	2	yes	maximal
2	C₁₆×Dic₁₇₃	C₂	1	1	yes	maximal, characteristic
3	C₄⋊#13063443	—	2	2	no	—
4	C₂×#14583801	C₄	1	1	yes	characteristic
5	C₈×Dic₁₇₃	C₂²	1	1	yes	characteristic
6	C₂×C₁₆×C₁₇₃	C₄	1	1	yes	characteristic
7	C₂×#13063443	—	4	2	no	—
8	C₂×#13063436	Q₈	1	1	yes	characteristic
9	C₄×Dic₁₇₃	D₄	1	1	yes	characteristic
10	#14583801	—	2	1	no	—
11	C₂×C₈×C₁₇₃	C₂×C₄	1	1	yes	characteristic
12	C₁₆×C₁₇₃	C₈	1	2	yes	complemented
13	#13063443	—	8	2	no	—
14	C₂×Dic₁₇₃	D₈	1	1	yes	characteristic
15	C₂×Dic₁₇₃	Q₁₆	1	1	yes	characteristic
16	#13063436	—	2	1	no	—
17	C₂×C₄×C₁₇₃	C₄⋊C₄	1	1	yes	characteristic
18	C₈×C₁₇₃	C₂×C₈	1	1	yes	characteristic
19	C₈×C₁₇₃	M₁₆	1	1	yes	characteristic
20	C₂²×C₁₇₃	#107	1	1	yes	characteristic
21	C₄×C₁₇₃	C₄⋊C₈	1	1	yes	characteristic
22	C₄×C₁₇₃	#108	1	1	yes	characteristic
23	Dic₁₇₃	—	2	1	no	—
24	Dic₁₇₃	D₁₆	1	1	yes	characteristic
25	Dic₁₇₃	Q₃₂	1	1	yes	characteristic
26	C₂×C₁₇₃	#335	1	1	yes	characteristic
27	C₂×C₁₇₃	#366	1	1	yes	characteristic
28	C₂×C₁₇₃	#368	1	1	yes	characteristic
29	C₁₇₃	#1371	1	1	yes	characteristic, complemented, Sylow
30	#1371	—	173	1	no	maximal, Sylow
31	C₄×C₁₆	—	173	1	no	—
32	#335	—	173	2	no	—
33	C₂×C₁₆	—	173	1	no	—
34	C₂×C₁₆	#10621924	1	1	yes	characteristic
35	C₄⋊C₈	—	346	2	no	—
36	C₄×C₈	—	173	1	no	—
37	C₂×C₈	—	173	1	no	—
38	C₂×C₈	—	692	2	no	—
39	C₂×C₈	C₂×#10621924	1	1	yes	characteristic
40	C₄²	—	173	1	no	—
41	C₁₆	—	346	1	no	—
42	C₁₆	#13063443	1	2	yes	complemented
43	C₂×C₄	—	173	1	no	—
44	C₂×C₄	—	173	1	no	—
45	C₂×C₄	C₄⋊#10621924	1	1	yes	characteristic
46	C₈	—	346	1	no	—
47	C₈	—	1384	2	no	—
48	C₈	C₂×#13063443	1	1	yes	characteristic
49	C₈	#14583836	1	1	yes	characteristic
50	C₂²	#19149748	1	1	yes	characteristic
51	C₄	—	346	1	no	—
52	C₄	—	173	1	no	—
53	C₄	—	173	1	no	—
54	C₄	C₄⋊#13063443	1	1	yes	characteristic
55	C₄	#19149750	1	1	yes	characteristic
56	C₂	#26012354	1	1	yes	characteristic
57	C₂	#26012417	1	1	yes	characteristic
58	C₂	#26012419	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#19149748
commutator	C₈×C₁₇₃	C₂×C₈
Frattini	C₂×C₈	C₂×#10621924
Fitting	C₂×C₁₆×C₁₇₃	C₄
socle, prime radical	C₂²×C₁₇₃	#107
cosocle	C₈×Dic₁₇₃	C₂²
hypercenter	C₂×C₁₆	#10621924
hypocenter	C₁₇₃	#1371
LCS	G ▷ C₈×C₁₇₃ ▷ C₄×C₁₇₃ ▷ C₂×C₁₇₃ ▷ C₁₇₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆

Complements

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

N	H ≅ G/N	type
12. C₁₆×C₁₇₃	C₈	semidirect
29. C₁₇₃	#1371	semidirect
42. C₁₆	#13063443	semidirect

Maximal subgroups

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

subgroup	index
#26012354	2
C₁₆×Dic₁₇₃	2
#1371	173