Subgroups — #36353165

Order 22144.

190 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26011864	C₂	1	1	yes	maximal, characteristic
2	#26011841	C₂	1	2	yes	maximal, complemented
3	#26011839	C₂	1	2	yes	maximal
4	C₈×#13063436	C₂	1	1	yes	maximal, characteristic, complemented
5	#498×C₁₇₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#14583815	—	2	2	no	—
7	#19149679	C₂²	1	1	yes	characteristic
8	#19149676	C₂²	1	1	yes	characteristic
9	#19149603	—	2	4	no	—
10	#19149599	—	2	4	no	—
11	C₄×#13063436	C₂²	1	2	yes	—
12	C₄⋊Q₈×C₁₇₃	C₂²	1	2	yes	—
13	#106×C₁₇₃	—	2	2	no	—
14	C₄×C₈×C₁₇₃	C₂²	1	1	yes	characteristic
15	C₂×Dic₃₄₆	—	2	1	no	—
16	C₂×D₆₉₂	—	2	2	no	—
17	C₂×D₆₉₂	—	2	1	no	—
18	#14583815	—	4	4	no	—
19	C₂×#13063436	D₄	1	4	yes	—
20	C₄⋊Dic₁₇₃	—	2	2	no	—
21	C₂×Q₈×C₁₇₃	—	2	2	no	—
22	C₂×C₈×C₁₇₃	D₄	1	2	yes	—
23	C₄⋊C₄×C₁₇₃	—	2	2	no	—
24	C₄⋊C₄×C₁₇₃	—	2	2	no	—
25	C₄²×C₁₇₃	C₂³	1	1	yes	characteristic
26	C₂²×D₁₇₃	—	4	1	no	—
27	Dic₃₄₆	—	4	2	no	—
28	D₆₉₂	—	4	2	no	—
29	D₆₉₂	—	4	4	no	—
30	C₂×Dic₁₇₃	—	4	1	no	—
31	#13063436	—	2	4	no	—
32	Q₈×C₁₇₃	—	4	4	no	—
33	C₂×C₄×C₁₇₃	—	4	2	no	—
34	C₂×C₄×C₁₇₃	C₂×D₄	1	2	yes	—
35	C₂×C₄×C₁₇₃	C₂×D₄	1	1	yes	characteristic
36	C₈×C₁₇₃	SD₁₆	1	4	yes	—
37	C₂²×C₁₇₃	#127	1	1	yes	characteristic
38	C₂×D₁₇₃	—	8	1	no	—
39	C₂×D₁₇₃	—	8	2	no	—
40	C₄×C₁₇₃	—	8	2	no	—
41	C₄×C₁₇₃	C₂×SD₁₆	1	2	yes	—
42	C₄×C₁₇₃	#135	1	4	yes	—
43	Dic₁₇₃	—	8	1	no	—
44	C₂×C₁₇₃	#492	1	1	yes	characteristic
45	C₂×C₁₇₃	#495	1	2	yes	—
46	D₁₇₃	—	16	1	no	—
47	C₁₇₃	#1708	1	1	yes	characteristic, complemented, Sylow
48	#1708	—	173	1	no	maximal, Sylow
49	#498	D₁₇₃	1	1	yes	characteristic, complemented
50	#492	—	173	1	no	—
51	#487	—	173	2	no	—
52	#486	—	173	2	no	—
53	C₈²	—	173	1	no	—
54	C₂×SD₁₆	—	346	2	no	—
55	C₄⋊Q₈	—	173	1	no	—
56	C₄⋊Q₈	C₂×D₁₇₃	1	2	yes	—
57	#127	—	173	1	no	—
58	#106	—	2	2	no	—
59	#103	—	346	4	no	—
60	#102	—	346	4	no	—
61	C₄×C₈	—	173	2	no	—
62	C₄×C₈	C₂×D₁₇₃	1	1	yes	characteristic
63	C₂×Q₈	—	2	2	no	—
64	C₂×Q₈	—	346	1	no	—
65	C₂×D₄	—	346	2	no	—
66	C₂×D₄	—	346	1	no	—
67	SD₁₆	—	692	4	no	—
68	C₂×C₈	—	173	4	no	—
69	C₂×C₈	C₂²⋊D₁₇₃	1	2	yes	—
70	C₄⋊C₄	—	2	2	no	—
71	C₄⋊C₄	—	2	2	no	—
72	C₄⋊C₄	—	346	2	no	—
73	C₄²	C₂²×D₁₇₃	1	1	yes	characteristic
74	C₂³	—	692	1	no	—
75	Q₈	—	4	4	no	—
76	Q₈	—	692	2	no	—
77	D₄	—	692	2	no	—
78	D₄	—	692	4	no	—
79	C₂×C₄	—	4	2	no	—
80	C₂×C₄	—	692	1	no	—
81	C₂×C₄	D₄×D₁₇₃	1	2	yes	—
82	C₂×C₄	C₂×C₂²⋊D₁₇₃	1	1	yes	characteristic
83	C₈	—	346	4	no	—
84	C₈	#14583813	1	4	yes	—
85	C₂²	—	1384	1	no	—
86	C₂²	—	1384	2	no	—
87	C₂²	#19149675	1	1	yes	characteristic
88	C₄	—	8	2	no	—
89	C₄	—	1384	1	no	—
90	C₄	C₂×#14583813	1	2	yes	—
91	C₄	#19149700	1	4	yes	—
92	C₂	—	2768	1	no	—
93	C₂	#26011865	1	1	yes	characteristic
94	C₂	#26011872	1	2	yes	—

	subgroup	G/H
center	C₂²	#19149675
commutator, cosocle	C₄²×C₁₇₃	C₂³
Frattini	C₄²	C₂²×D₁₇₃
Fitting	#498×C₁₇₃	C₂
socle	C₂²×C₁₇₃	#127
prime radical	#19149676	C₂²
hypercenter	#498	D₁₇₃
hypocenter	C₁₇₃	#1708
LCS	G ▷ C₄²×C₁₇₃ ▷ C₂²×C₁₇₃ ▷ C₁₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #498

Complements

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

N	H ≅ G/N	type
2. #26011841	C₂	semidirect
4. C₈×#13063436	C₂	semidirect
5. #498×C₁₇₃	C₂	semidirect
47. C₁₇₃	#1708	semidirect
49. #498	D₁₇₃	semidirect

Maximal subgroups

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

subgroup	index
#26011864	2
#26011841	2
#26011839	2
C₈×#13063436	2
#498×C₁₇₃	2
#1708	173