Subgroups — #26012531

Order 11072.

108 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×#10621924	C₂	1	2	yes	maximal
2	#19149747	C₂	1	1	yes	maximal, characteristic
3	#19149742	C₂	1	2	yes	maximal
4	#19149733	C₂	1	1	yes	maximal, characteristic
5	Q₁₆×D₁₇₃	C₂	1	1	yes	maximal, characteristic
6	D₁₇₃⋊C₈	C₂²	1	1	yes	characteristic
7	C₄⋊#10621924	—	2	2	no	—
8	C₄⋊#10621924	C₂²	1	2	yes	—
9	C₄×#10621924	—	2	2	no	—
10	C₄×#10621924	C₂²	1	1	yes	characteristic
11	Q₈×D₁₇₃	C₂²	1	2	yes	—
12	Dic₆₉₂	C₄	1	1	yes	characteristic, complemented
13	#14583816	C₄	1	2	yes	complemented
14	C₈×D₁₇₃	C₂²	1	1	yes	characteristic
15	Q₁₆×C₁₇₃	C₄	1	1	yes	characteristic, complemented
16	C₂×#10621924	—	2	2	no	—
17	C₂×#10621924	D₄	1	2	yes	—
18	#13063443	—	2	1	no	—
19	Dic₃₄₆	—	2	2	no	—
20	Dic₃₄₆	C₂×C₄	1	2	yes	—
21	C₄×D₁₇₃	—	2	2	no	—
22	C₄×D₁₇₃	C₂³	1	1	yes	characteristic
23	#13063436	C₂×C₄	1	1	yes	characteristic
24	Q₈×C₁₇₃	C₂×C₄	1	2	yes	—
25	C₈×C₁₇₃	C₂×C₄	1	1	yes	characteristic
26	C₂×D₁₇₃	C₂×D₄	1	1	yes	characteristic
27	#10621924	—	2	2	no	—
28	#10621924	—	4	2	no	—
29	C₄×C₁₇₃	—	2	2	no	—
30	C₄×C₁₇₃	C₂²×C₄	1	1	yes	characteristic
31	Dic₁₇₃	—	2	2	no	—
32	Dic₁₇₃	#41	1	1	yes	characteristic
33	C₂×C₁₇₃	C₄×D₄	1	1	yes	characteristic
34	D₁₇₃	#137	1	2	yes	—
35	C₁₇₃	#441	1	1	yes	characteristic, complemented, Sylow
36	#441	—	173	1	no	maximal, Sylow
37	C₂×Q₁₆	—	173	1	no	—
38	C₄×Q₈	—	173	2	no	—
39	#106	—	173	1	no	—
40	#103	—	173	2	no	—
41	#97	—	173	1	no	—
42	C₂×Q₈	—	173	2	no	—
43	Q₁₆	—	173	2	no	—
44	Q₁₆	—	173	1	no	—
45	Q₁₆	#10621924	1	1	yes	characteristic, complemented
46	C₂×C₈	—	173	1	no	—
47	C₂×C₈	—	173	1	no	—
48	C₄⋊C₄	—	346	2	no	—
49	C₄⋊C₄	—	173	2	no	—
50	C₄²	—	346	2	no	—
51	C₄²	—	173	1	no	—
52	Q₈	—	346	2	no	—
53	Q₈	—	173	2	no	—
54	Q₈	C₂×#10621924	1	2	yes	—
55	C₂×C₄	—	346	2	no	—
56	C₂×C₄	—	346	2	no	—
57	C₂×C₄	—	173	2	no	—
58	C₂×C₄	—	173	1	no	—
59	C₈	—	346	1	no	—
60	C₈	—	173	1	no	—
61	C₈	C₂×#10621924	1	1	yes	characteristic
62	C₂²	—	173	1	no	—
63	C₄	—	2	2	no	—
64	C₄	—	346	2	no	—
65	C₄	—	346	2	no	—
66	C₄	—	173	1	no	—
67	C₄	—	692	2	no	—
68	C₄	C₂²×#10621924	1	1	yes	characteristic
69	C₂	—	173	2	no	—
70	C₂	D₄×#10621924	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₄×#10621924
commutator	C₄×C₁₇₃	C₂²×C₄
Frattini	C₄	C₂²×#10621924
Fitting	Q₁₆×C₁₇₃	C₄
socle	C₂×C₁₇₃	C₄×D₄
prime radical	C₂×D₁₇₃	C₂×D₄
cosocle	C₄×D₁₇₃	C₂³
hypercenter	Q₁₆	#10621924
hypocenter	C₁₇₃	#441
LCS	G ▷ C₄×C₁₇₃ ▷ C₂×C₁₇₃ ▷ C₁₇₃
UCS	C₁ ◁ C₂ ◁ C₄ ◁ Q₁₆

Complements

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

N	H ≅ G/N	type
12. Dic₆₉₂	C₄	semidirect
13. #14583816	C₄	semidirect
15. Q₁₆×C₁₇₃	C₄	semidirect
35. C₁₇₃	#441	semidirect
45. Q₁₆	#10621924	semidirect

Maximal subgroups

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

subgroup	index
Q₈×#10621924	2
#19149747	2
#19149742	2
#19149733	2
Q₁₆×D₁₇₃	2
#441	173