Subgroups — #36368582

Order 22144.

86 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26012412	C₂	1	1	yes	maximal, characteristic
2	#26012355	C₂	1	1	yes	maximal, characteristic
3	#26011895	C₂	1	1	yes	maximal, characteristic
4	C₂×#14583829	—	2	1	no	—
5	C₄⋊#13063443	—	2	1	no	—
6	#19149678	—	2	1	no	—
7	#19149613	C₄	1	1	yes	characteristic
8	#19149612	—	2	1	no	—
9	C₈×Dic₁₇₃	C₂²	1	1	yes	characteristic
10	#107×C₁₇₃	C₄	1	1	yes	characteristic
11	C₂×#13063443	—	4	1	no	—
12	#14583829	—	4	1	no	—
13	C₂×Dic₃₄₆	—	4	1	no	—
14	C₂×#13063436	D₄	1	1	yes	characteristic
15	C₄⋊Dic₁₇₃	—	2	1	no	—
16	#14583805	—	4	1	no	—
17	C₄×Dic₁₇₃	D₄	1	1	yes	characteristic
18	C₂×C₈×C₁₇₃	C₂×C₄	1	1	yes	characteristic
19	C₄⋊C₄×C₁₇₃	—	2	1	no	—
20	#13063443	—	8	1	no	—
21	Dic₃₄₆	—	8	2	no	—
22	C₂×Dic₁₇₃	—	4	1	no	—
23	C₂×Dic₁₇₃	D₈	1	1	yes	characteristic
24	C₂×Dic₁₇₃	SD₁₆	1	1	yes	characteristic
25	#13063436	SD₁₆	1	1	yes	characteristic
26	#13063436	Q₁₆	1	1	yes	characteristic
27	C₂×C₄×C₁₇₃	—	4	1	no	—
28	C₂×C₄×C₁₇₃	C₂²⋊C₄	1	1	yes	characteristic
29	C₈×C₁₇₃	—	2	1	no	—
30	C₂²×C₁₇₃	#102	1	1	yes	characteristic
31	C₄×C₁₇₃	—	8	1	no	—
32	C₄×C₁₇₃	#101	1	1	yes	characteristic
33	C₄×C₁₇₃	#103	1	1	yes	characteristic
34	Dic₁₇₃	—	2	1	no	—
35	Dic₁₇₃	—	8	1	no	—
36	Dic₁₇₃	SD₃₂	1	1	yes	characteristic
37	Dic₁₇₃	Q₃₂	1	1	yes	characteristic
38	C₂×C₁₇₃	#332	1	1	yes	characteristic
39	C₂×C₁₇₃	#358	1	1	yes	characteristic
40	C₂×C₁₇₃	#361	1	1	yes	characteristic
41	C₁₇₃	#1365	1	1	yes	characteristic, complemented, Sylow
42	#1365	—	173	1	no	maximal, Sylow
43	#500	—	173	1	no	—
44	C₄⋊C₁₆	—	173	1	no	—
45	#335	—	173	1	no	—
46	C₄⋊Q₈	—	346	1	no	—
47	C₂×C₁₆	—	346	1	no	—
48	#107	—	346	1	no	—
49	#107	—	173	1	no	—
50	#107	#10621924	1	1	yes	characteristic
51	C₄⋊C₈	—	346	1	no	—
52	C₄×C₈	—	173	1	no	—
53	C₂×Q₈	—	692	1	no	—
54	C₂×C₈	—	173	1	no	—
55	C₂×C₈	—	692	1	no	—
56	C₂×C₈	C₂×#10621924	1	1	yes	characteristic
57	C₄⋊C₄	—	2	1	no	—
58	C₄⋊C₄	—	346	1	no	—
59	C₄⋊C₄	—	692	1	no	—
60	C₄²	—	173	1	no	—
61	C₁₆	—	692	1	no	—
62	Q₈	—	1384	2	no	—
63	C₂×C₄	—	4	1	no	—
64	C₂×C₄	—	173	1	no	—
65	C₂×C₄	—	173	1	no	—
66	C₂×C₄	—	692	1	no	—
67	C₂×C₄	C₂²⋊#10621924	1	1	yes	characteristic
68	C₈	—	2	1	no	—
69	C₈	—	1384	1	no	—
70	C₈	—	173	1	no	—
71	C₈	—	173	1	no	—
72	C₂²	#19149741	1	1	yes	characteristic
73	C₄	—	8	1	no	—
74	C₄	—	346	1	no	—
75	C₄	—	1384	1	no	—
76	C₄	—	173	1	no	—
77	C₄	—	173	1	no	—
78	C₄	#19149740	1	1	yes	characteristic
79	C₄	#19149742	1	1	yes	characteristic
80	C₂	#26012350	1	1	yes	characteristic
81	C₂	#26012404	1	1	yes	characteristic
82	C₂	#26012409	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#19149741
commutator	C₂×C₈×C₁₇₃	C₂×C₄
Frattini	C₂×C₈	C₂×#10621924
Fitting	#107×C₁₇₃	C₄
socle, prime radical	C₂²×C₁₇₃	#102
cosocle	C₈×Dic₁₇₃	C₂²
hypercenter	#107	#10621924
hypocenter	C₁₇₃	#1365
LCS	G ▷ C₂×C₈×C₁₇₃ ▷ C₂×C₄×C₁₇₃ ▷ C₂×C₁₇₃ ▷ C₁₇₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #107

Complements

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

N	H ≅ G/N	type
41. C₁₇₃	#1365	semidirect

Maximal subgroups

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

subgroup	index
#26012412	2
#26012355	2
#26011895	2
#1365	173