Subgroups — #36368481

Order 22144.

80 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26012379	C₂	1	1	yes	maximal, characteristic
2	#26012379	C₂	1	1	yes	maximal, characteristic
3	#26011213	C₂	1	1	yes	maximal, characteristic
4	C₂×#14583829	C₄	1	2	yes	—
5	C₂×#14583829	C₄	1	2	yes	—
6	C₄×#13063436	C₄	1	1	yes	characteristic
7	C₄×#13063436	C₂²	1	1	yes	characteristic
8	C₄×C₈×C₁₇₃	C₄	1	1	yes	characteristic
9	#14583829	—	2	2	no	—
10	#14583829	C₈	1	4	yes	complemented
11	C₂×#13063436	—	2	1	no	—
12	C₂×#13063436	C₂×C₄	1	1	yes	characteristic
13	C₂×#13063436	C₂×C₄	1	1	yes	characteristic
14	C₂×C₈×C₁₇₃	—	2	1	no	—
15	C₄²×C₁₇₃	C₂×C₄	1	1	yes	characteristic
16	#13063436	—	4	1	no	—
17	#13063436	C₂×C₈	1	2	yes	—
18	#13063436	M₁₆	1	2	yes	—
19	C₂×C₄×C₁₇₃	C₄²	1	1	yes	characteristic
20	C₂×C₄×C₁₇₃	M₁₆	1	1	yes	characteristic
21	C₂×C₄×C₁₇₃	M₁₆	1	1	yes	characteristic
22	C₈×C₁₇₃	—	4	1	no	—
23	C₂²×C₁₇₃	#97	1	1	yes	characteristic
24	C₄×C₁₇₃	—	2	1	no	—
25	C₄×C₁₇₃	—	2	1	no	—
26	C₄×C₁₇₃	C₄×C₈	1	1	yes	characteristic
27	C₄×C₁₇₃	#97	1	1	yes	characteristic
28	C₂×C₁₇₃	#322	1	1	yes	characteristic
29	C₂×C₁₇₃	#347	1	1	yes	characteristic
30	C₂×C₁₇₃	#347	1	1	yes	characteristic
31	C₁₇₃	#1313	1	1	yes	characteristic, complemented, Sylow
32	#1313	—	173	1	no	maximal, Sylow
33	#346	—	173	1	no	—
34	#346	—	173	1	no	—
35	#322	—	173	1	no	—
36	C₂×C₁₆	—	173	2	no	—
37	C₂×C₁₆	—	173	2	no	—
38	C₄×C₈	—	173	1	no	—
39	C₄×C₈	—	173	1	no	—
40	C₄×C₈	#10621924	1	1	yes	characteristic
41	C₂×C₈	—	2	1	no	—
42	C₂×C₈	—	346	1	no	—
43	C₂×C₈	—	173	1	no	—
44	C₂×C₈	—	173	1	no	—
45	C₄²	C₂×#10621924	1	1	yes	characteristic
46	C₁₆	—	346	2	no	—
47	C₁₆	—	173	4	no	—
48	C₂×C₄	C₄×#10621924	1	1	yes	characteristic
49	C₂×C₄	#14583835	1	1	yes	characteristic
50	C₂×C₄	#14583836	1	1	yes	characteristic
51	C₈	—	4	1	no	—
52	C₈	—	173	2	no	—
53	C₈	—	173	2	no	—
54	C₈	—	692	1	no	—
55	C₂²	#19149734	1	1	yes	characteristic
56	C₄	—	2	1	no	—
57	C₄	—	2	1	no	—
58	C₄	C₈×#10621924	1	1	yes	characteristic
59	C₄	#19149733	1	1	yes	characteristic
60	C₂	#13063443⋊C₈	1	1	yes	characteristic
61	C₂	#26012383	1	1	yes	characteristic
62	C₂	#26012384	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₄×#10621924
commutator	C₄×C₁₇₃	C₄×C₈
Frattini	C₄²	C₂×#10621924
Fitting	C₄×C₈×C₁₇₃	C₄
socle, prime radical	C₂²×C₁₇₃	#97
cosocle	C₄×#13063436	C₂²
hypercenter	C₄×C₈	#10621924
hypocenter	C₁₇₃	#1313
LCS	G ▷ C₄×C₁₇₃ ▷ C₁₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈

Complements

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

N	H ≅ G/N	type
10. #14583829	C₈	semidirect
31. C₁₇₃	#1313	semidirect

Maximal subgroups

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

subgroup	index
#26012379	2
#26011213	2
#1313	173