Subgroups — #36353172

Order 22144.

168 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26011896	C₂	1	1	yes	maximal, characteristic
2	#26011843	C₂	1	2	yes	maximal
3	#26011841	C₂	1	2	yes	maximal
4	C₈×#13063436	C₂	1	1	yes	maximal, characteristic
5	#500×C₁₇₃	C₂	1	1	yes	maximal, characteristic
6	#19149679	C₂²	1	2	yes	—
7	#19149613	—	2	2	no	—
8	#19149603	—	2	4	no	—
9	#19149602	—	2	4	no	—
10	C₄×#13063436	C₂²	1	1	yes	characteristic
11	C₄×#13063436	C₂²	1	1	yes	characteristic
12	C₄⋊Q₈×C₁₇₃	C₂²	1	2	yes	—
13	#107×C₁₇₃	—	2	2	no	—
14	C₄×C₈×C₁₇₃	C₂²	1	1	yes	characteristic
15	C₂×Dic₃₄₆	—	2	2	no	—
16	C₂×#13063436	D₄	1	2	yes	—
17	C₂×#13063436	D₄	1	2	yes	—
18	C₄⋊Dic₁₇₃	—	2	2	no	—
19	C₄⋊Dic₁₇₃	—	2	2	no	—
20	C₂×Q₈×C₁₇₃	—	2	2	no	—
21	C₂×C₈×C₁₇₃	D₄	1	2	yes	—
22	C₄⋊C₄×C₁₇₃	—	2	2	no	—
23	C₄⋊C₄×C₁₇₃	—	2	2	no	—
24	C₄²×C₁₇₃	C₂³	1	1	yes	characteristic
25	Dic₃₄₆	—	4	4	no	—
26	C₂×Dic₁₇₃	—	4	2	no	—
27	#13063436	—	2	2	no	—
28	#13063436	—	2	2	no	—
29	Q₈×C₁₇₃	—	4	4	no	—
30	C₂×C₄×C₁₇₃	—	4	2	no	—
31	C₂×C₄×C₁₇₃	C₂×D₄	1	1	yes	characteristic
32	C₂×C₄×C₁₇₃	C₂×D₄	1	1	yes	characteristic
33	C₂×C₄×C₁₇₃	C₂×D₄	1	1	yes	characteristic
34	C₈×C₁₇₃	Q₁₆	1	4	yes	—
35	C₂²×C₁₇₃	#127	1	1	yes	characteristic
36	C₄×C₁₇₃	—	8	2	no	—
37	C₄×C₁₇₃	C₂×Q₁₆	1	2	yes	—
38	C₄×C₁₇₃	#135	1	2	yes	—
39	C₄×C₁₇₃	#135	1	2	yes	—
40	Dic₁₇₃	—	8	2	no	—
41	C₂×C₁₇₃	#494	1	1	yes	characteristic
42	C₂×C₁₇₃	#495	1	1	yes	characteristic
43	C₂×C₁₇₃	#495	1	1	yes	characteristic
44	C₁₇₃	#1710	1	1	yes	characteristic, complemented, Sylow
45	#1710	—	173	1	no	maximal, Sylow
46	#500	—	173	1	no	—
47	#500	D₁₇₃	1	1	yes	characteristic
48	#487	—	173	2	no	—
49	#487	—	173	2	no	—
50	C₈²	—	173	1	no	—
51	C₄⋊Q₈	—	173	2	no	—
52	C₄⋊Q₈	C₂×D₁₇₃	1	2	yes	—
53	#107	—	2	2	no	—
54	#107	—	346	2	no	—
55	#103	—	346	4	no	—
56	#103	—	346	4	no	—
57	C₄×C₈	—	173	1	no	—
58	C₄×C₈	—	173	1	no	—
59	C₄×C₈	C₂×D₁₇₃	1	1	yes	characteristic
60	C₂×Q₈	—	2	2	no	—
61	C₂×Q₈	—	346	2	no	—
62	C₂×C₈	—	173	2	no	—
63	C₂×C₈	—	173	2	no	—
64	C₂×C₈	C₂²⋊D₁₇₃	1	2	yes	—
65	C₄⋊C₄	—	2	2	no	—
66	C₄⋊C₄	—	2	2	no	—
67	C₄⋊C₄	—	346	2	no	—
68	C₄⋊C₄	—	346	2	no	—
69	C₄²	C₂²×D₁₇₃	1	1	yes	characteristic
70	Q₈	—	4	4	no	—
71	Q₈	—	692	4	no	—
72	C₂×C₄	—	4	2	no	—
73	C₂×C₄	—	692	2	no	—
74	C₂×C₄	D₄×D₁₇₃	1	1	yes	characteristic
75	C₂×C₄	D₄×D₁₇₃	1	1	yes	characteristic
76	C₂×C₄	C₂×C₂²⋊D₁₇₃	1	1	yes	characteristic
77	C₈	—	346	2	no	—
78	C₈	—	346	2	no	—
79	C₈	#14583816	1	4	yes	—
80	C₂²	#19149675	1	1	yes	characteristic
81	C₄	—	8	2	no	—
82	C₄	—	1384	2	no	—
83	C₄	C₂×#14583816	1	2	yes	—
84	C₄	#19149698	1	2	yes	—
85	C₄	#19149702	1	2	yes	—
86	C₂	#26011870	1	1	yes	characteristic
87	C₂	#26011874	1	1	yes	characteristic
88	C₂	#26011876	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#19149675
commutator, cosocle	C₄²×C₁₇₃	C₂³
Frattini	C₄²	C₂²×D₁₇₃
Fitting	#500×C₁₇₃	C₂
socle, prime radical	C₂²×C₁₇₃	#127
hypercenter	#500	D₁₇₃
hypocenter	C₁₇₃	#1710
LCS	G ▷ C₄²×C₁₇₃ ▷ C₂²×C₁₇₃ ▷ C₁₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #500

Complements

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

N	H ≅ G/N	type
44. C₁₇₃	#1710	semidirect

Maximal subgroups

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

subgroup	index
#26011896	2
#26011843	2
#26011841	2
C₈×#13063436	2
#500×C₁₇₃	2
#1710	173