Subgroups — #36351540

Order 22144.

96 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26011609	C₂	1	1	yes	maximal, characteristic
2	#26011312	C₂	1	1	yes	maximal, characteristic, complemented
3	#322×C₁₇₃	C₂	1	1	yes	maximal, characteristic, complemented
4	#19149701	C₄	1	1	yes	characteristic
5	#19149685	—	2	1	no	—
6	#19149619	—	2	1	no	—
7	#19149615	C₄	1	1	yes	characteristic
8	#19149607	—	2	1	no	—
9	C₄×C₈×C₁₇₃	—	2	1	no	—
10	C₄×C₈×C₁₇₃	C₂²	1	1	yes	characteristic
11	#14583826	—	2	1	no	—
12	Dic₆₉₂	C₈	1	1	yes	characteristic, complemented
13	#14583815	—	2	1	no	—
14	D₁₃₈₄	C₈	1	1	yes	characteristic, complemented
15	#14583810	—	2	1	no	—
16	#14583809	—	4	1	no	—
17	C₈×D₁₇₃	—	4	1	no	—
18	#14583801	—	4	1	no	—
19	C₂×C₈×C₁₇₃	—	2	2	no	—
20	C₂×C₈×C₁₇₃	C₂×C₄	1	1	yes	characteristic
21	C₂×C₈×C₁₇₃	D₄	1	1	yes	characteristic
22	C₄²×C₁₇₃	D₄	1	1	yes	characteristic
23	Dic₃₄₆	—	2	1	no	—
24	D₆₉₂	—	2	1	no	—
25	C₂²⋊D₁₇₃	—	4	1	no	—
26	C₄×D₁₇₃	—	4	1	no	—
27	#13063436	—	4	1	no	—
28	C₂×C₄×C₁₇₃	—	2	1	no	—
29	C₂×C₄×C₁₇₃	C₂²⋊C₄	1	1	yes	characteristic
30	C₈×C₁₇₃	—	4	2	no	—
31	C₈×C₁₇₃	C₂×C₈	1	1	yes	characteristic
32	C₈×C₁₇₃	M₁₆	1	1	yes	characteristic
33	C₈×C₁₇₃	D₈	1	1	yes	characteristic
34	C₈×C₁₇₃	SD₁₆	1	1	yes	characteristic
35	C₂²×C₁₇₃	#104	1	1	yes	characteristic
36	C₂×D₁₇₃	—	4	1	no	—
37	C₄×C₁₇₃	—	2	2	no	—
38	C₄×C₁₇₃	C₂²⋊C₈	1	1	yes	characteristic
39	C₄×C₁₇₃	#102	1	1	yes	characteristic
40	Dic₁₇₃	—	4	1	no	—
41	C₂×C₁₇₃	—	2	1	no	—
42	C₂×C₁₇₃	D₄⋊C₈	1	1	yes	characteristic
43	D₁₇₃	—	8	1	no	—
44	C₁₇₃	#1336	1	1	yes	characteristic, complemented, Sylow
45	#1336	—	173	1	no	maximal, Sylow
46	#443	—	173	1	no	—
47	#364	—	173	1	no	—
48	#322	D₁₇₃	1	1	yes	characteristic, complemented
49	#135	—	173	1	no	—
50	#131	—	346	1	no	—
51	M₃₂	—	346	1	no	—
52	#108	—	173	1	no	—
53	#104	—	346	1	no	—
54	C₄×C₈	—	2	1	no	—
55	C₄×C₈	C₂×D₁₇₃	1	1	yes	characteristic
56	#41	—	346	1	no	—
57	Q₁₆	—	173	1	no	—
58	SD₁₆	—	346	1	no	—
59	D₈	—	173	1	no	—
60	M₁₆	—	346	1	no	—
61	M₁₆	—	692	1	no	—
62	C₂×C₈	—	2	2	no	—
63	C₂×C₈	—	692	1	no	—
64	C₂×C₈	C₄×D₁₇₃	1	1	yes	characteristic
65	C₂×C₈	D₆₉₂	1	1	yes	characteristic
66	C₄²	C₂²⋊D₁₇₃	1	1	yes	characteristic
67	C₁₆	—	692	1	no	—
68	Q₈	—	346	1	no	—
69	D₄	—	346	1	no	—
70	D₄	—	692	1	no	—
71	C₂×C₄	—	2	1	no	—
72	C₂×C₄	—	692	1	no	—
73	C₂×C₄	#14583803	1	1	yes	characteristic
74	C₈	—	4	2	no	—
75	C₈	—	692	1	no	—
76	C₈	C₈×D₁₇₃	1	1	yes	characteristic
77	C₈	#14583809	1	1	yes	characteristic
78	C₈	D₁₃₈₄	1	1	yes	characteristic
79	C₈	#14583815	1	1	yes	characteristic
80	C₂²	—	692	1	no	—
81	C₂²	#19149607	1	1	yes	characteristic
82	C₄	—	2	2	no	—
83	C₄	—	692	1	no	—
84	C₄	#19149591	1	1	yes	characteristic
85	C₄	#19149601	1	1	yes	characteristic
86	C₂	—	2	1	no	—
87	C₂	—	1384	1	no	—
88	C₂	#26011220	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#19149601
commutator	C₈×C₁₇₃	C₂×C₈
Frattini	C₄×C₈	C₂×D₁₇₃
Fitting	#322×C₁₇₃	C₂
socle	C₂×C₁₇₃	D₄⋊C₈
prime radical	#19149701	C₄
cosocle	C₄×C₈×C₁₇₃	C₂²
hypercenter	#322	D₁₇₃
hypocenter	C₁₇₃	#1336
LCS	G ▷ C₈×C₁₇₃ ▷ C₄×C₁₇₃ ▷ C₂×C₁₇₃ ▷ C₁₇₃
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ C₄×C₈ ◁ #322

Complements

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

N	H ≅ G/N	type
2. #26011312	C₂	semidirect
3. #322×C₁₇₃	C₂	semidirect
12. Dic₆₉₂	C₈	semidirect
14. D₁₃₈₄	C₈	semidirect
44. C₁₇₃	#1336	semidirect
48. #322	D₁₇₃	semidirect

Maximal subgroups

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

subgroup	index
#26011609	2
#26011312	2
#322×C₁₇₃	2
#1336	173