Subgroups — #36351704

Order 22144.

72 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26011907	C₂	1	1	yes	maximal, characteristic
2	#26011325	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃₂×C₁₇₃	C₂	1	1	yes	maximal, characteristic, complemented
4	#19149685	C₄	1	1	yes	characteristic
5	#19149619	C₄	1	1	yes	characteristic
6	#19149618	—	2	1	no	—
7	C₁₆×D₁₇₃	—	2	1	no	—
8	#19149585	—	2	1	no	—
9	C₂×C₁₆×C₁₇₃	C₂²	1	1	yes	characteristic
10	C₃₂×C₁₇₃	—	2	1	no	—
11	#14583826	C₈	1	1	yes	characteristic
12	#14583810	C₈	1	1	yes	characteristic
13	#14583809	—	2	1	no	—
14	C₈×D₁₇₃	—	2	1	no	—
15	#14583801	—	2	1	no	—
16	C₂×C₈×C₁₇₃	C₂×C₄	1	1	yes	characteristic
17	C₁₆×C₁₇₃	D₄	1	1	yes	characteristic
18	C₁₆×C₁₇₃	D₄	1	1	yes	characteristic
19	Dic₃₄₆	C₁₆	1	1	yes	characteristic
20	D₆₉₂	C₁₆	1	1	yes	characteristic
21	C₂²⋊D₁₇₃	—	2	1	no	—
22	C₄×D₁₇₃	—	2	1	no	—
23	#13063436	—	2	1	no	—
24	C₂×C₄×C₁₇₃	C₂×C₈	1	1	yes	characteristic
25	C₈×C₁₇₃	C₂²⋊C₄	1	1	yes	characteristic
26	C₈×C₁₇₃	M₁₆	1	1	yes	characteristic
27	C₂²×C₁₇₃	M₃₂	1	1	yes	characteristic
28	C₂×D₁₇₃	—	2	1	no	—
29	C₄×C₁₇₃	C₂²⋊C₈	1	1	yes	characteristic
30	C₄×C₁₇₃	C₂×C₁₆	1	1	yes	characteristic
31	Dic₁₇₃	—	2	1	no	—
32	C₂×C₁₇₃	—	2	1	no	—
33	C₂×C₁₇₃	C₂²⋊C₁₆	1	1	yes	characteristic
34	D₁₇₃	—	4	1	no	—
35	C₁₇₃	#1401	1	1	yes	characteristic, complemented, Sylow
36	#1401	—	173	1	no	maximal, Sylow
37	#504	—	173	1	no	—
38	M₆₄	—	173	1	no	—
39	C₂×C₃₂	D₁₇₃	1	1	yes	characteristic, complemented
40	#131	—	173	1	no	—
41	M₃₂	—	346	1	no	—
42	M₃₂	—	173	1	no	—
43	C₂×C₁₆	—	346	1	no	—
44	C₂×C₁₆	C₂×D₁₇₃	1	1	yes	characteristic
45	C₃₂	—	2	1	no	—
46	C₃₂	—	346	1	no	—
47	#41	—	173	1	no	—
48	M₁₆	—	346	1	no	—
49	M₁₆	—	173	1	no	—
50	C₂×C₈	—	346	1	no	—
51	C₂×C₈	C₄×D₁₇₃	1	1	yes	characteristic
52	C₁₆	—	346	1	no	—
53	C₁₆	C₂²⋊D₁₇₃	1	1	yes	characteristic
54	C₁₆	D₆₉₂	1	1	yes	characteristic
55	Q₈	—	173	1	no	—
56	D₄	—	346	1	no	—
57	D₄	—	173	1	no	—
58	C₂×C₄	—	346	1	no	—
59	C₂×C₄	C₈×D₁₇₃	1	1	yes	characteristic
60	C₈	—	346	1	no	—
61	C₈	#14583803	1	1	yes	characteristic
62	C₈	#14583809	1	1	yes	characteristic
63	C₂²	—	346	1	no	—
64	C₂²	#19149618	1	1	yes	characteristic
65	C₄	—	346	1	no	—
66	C₄	#19149591	1	1	yes	characteristic
67	C₄	C₁₆×D₁₇₃	1	1	yes	characteristic
68	C₂	—	2	1	no	—
69	C₂	—	692	1	no	—
70	C₂	#26011270	1	1	yes	characteristic

	subgroup	G/H
center	C₁₆	D₆₉₂
commutator	C₄×C₁₇₃	C₂×C₁₆
Frattini	C₂×C₁₆	C₂×D₁₇₃
Fitting	C₂×C₃₂×C₁₇₃	C₂
socle	C₂×C₁₇₃	C₂²⋊C₁₆
prime radical	#14583826	C₈
cosocle	C₂×C₁₆×C₁₇₃	C₂²
hypercenter	C₂×C₃₂	D₁₇₃
hypocenter	C₁₇₃	#1401
LCS	G ▷ C₄×C₁₇₃ ▷ C₂×C₁₇₃ ▷ C₁₇₃
UCS	C₁ ◁ C₁₆ ◁ C₂×C₁₆ ◁ C₂×C₃₂

Complements

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

N	H ≅ G/N	type
2. #26011325	C₂	semidirect
3. C₂×C₃₂×C₁₇₃	C₂	semidirect
35. C₁₇₃	#1401	semidirect
39. C₂×C₃₂	D₁₇₃	semidirect

Maximal subgroups

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

subgroup	index
#26011907	2
#26011325	2
C₂×C₃₂×C₁₇₃	2
#1401	173