Subgroups — #36353170

Order 22144.

212 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26011868	C₂	1	1	yes	maximal, characteristic
2	#26011839	C₂	1	4	yes	maximal, complemented
3	C₈×#13063436	C₂	1	1	yes	maximal, characteristic, complemented
4	#494×C₁₇₃	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×D₁₃₈₄	—	2	2	no	—
6	#19149676	C₂²	1	2	yes	—
7	#19149599	—	2	8	no	—
8	C₄×#13063436	C₂²	1	2	yes	—
9	C₂×Q₁₆×C₁₇₃	—	2	2	no	—
10	C₄⋊Q₈×C₁₇₃	C₂²	1	2	yes	—
11	C₄×C₈×C₁₇₃	C₂²	1	1	yes	characteristic
12	C₂×D₆₉₂	—	2	4	no	—
13	C₂×D₆₉₂	—	2	2	no	—
14	D₁₃₈₄	—	4	4	no	—
15	C₂×#13063436	D₄	1	4	yes	—
16	C₂×Q₈×C₁₇₃	—	2	2	no	—
17	Q₁₆×C₁₇₃	—	4	4	no	—
18	C₂×C₈×C₁₇₃	D₄	1	2	yes	—
19	C₄⋊C₄×C₁₇₃	—	2	4	no	—
20	C₄²×C₁₇₃	C₂³	1	1	yes	characteristic
21	C₂²×D₁₇₃	—	4	2	no	—
22	D₆₉₂	—	4	4	no	—
23	D₆₉₂	—	4	8	no	—
24	#13063436	—	2	4	no	—
25	Q₈×C₁₇₃	—	4	4	no	—
26	C₂×C₄×C₁₇₃	—	4	2	no	—
27	C₂×C₄×C₁₇₃	C₂×D₄	1	2	yes	—
28	C₂×C₄×C₁₇₃	C₂×D₄	1	1	yes	characteristic
29	C₈×C₁₇₃	D₈	1	4	yes	—
30	C₂²×C₁₇₃	#127	1	1	yes	characteristic
31	C₂×D₁₇₃	—	8	2	no	—
32	C₂×D₁₇₃	—	8	4	no	—
33	C₄×C₁₇₃	—	8	2	no	—
34	C₄×C₁₇₃	C₂×D₈	1	2	yes	—
35	C₄×C₁₇₃	#135	1	4	yes	—
36	C₂×C₁₇₃	#493	1	1	yes	characteristic
37	C₂×C₁₇₃	#495	1	2	yes	—
38	D₁₇₃	—	16	2	no	—
39	C₁₇₃	#1709	1	1	yes	characteristic, complemented, Sylow
40	#1709	—	173	1	no	maximal, Sylow
41	#494	D₁₇₃	1	1	yes	characteristic, complemented
42	#493	—	173	1	no	—
43	#486	—	173	4	no	—
44	C₈²	—	173	1	no	—
45	C₂×Q₁₆	—	2	2	no	—
46	C₂×D₈	—	346	2	no	—
47	C₄⋊Q₈	C₂×D₁₇₃	1	2	yes	—
48	#127	—	173	2	no	—
49	#102	—	346	8	no	—
50	C₄×C₈	—	173	2	no	—
51	C₄×C₈	C₂×D₁₇₃	1	1	yes	characteristic
52	C₂×Q₈	—	2	2	no	—
53	C₂×D₄	—	346	4	no	—
54	C₂×D₄	—	346	2	no	—
55	Q₁₆	—	4	4	no	—
56	D₈	—	692	4	no	—
57	C₂×C₈	—	173	4	no	—
58	C₂×C₈	C₂²⋊D₁₇₃	1	2	yes	—
59	C₄⋊C₄	—	2	4	no	—
60	C₄²	C₂²×D₁₇₃	1	1	yes	characteristic
61	C₂³	—	692	2	no	—
62	Q₈	—	4	4	no	—
63	D₄	—	692	4	no	—
64	D₄	—	692	8	no	—
65	C₂×C₄	—	4	2	no	—
66	C₂×C₄	D₄×D₁₇₃	1	2	yes	—
67	C₂×C₄	C₂×C₂²⋊D₁₇₃	1	1	yes	characteristic
68	C₈	—	346	4	no	—
69	C₈	#14583811	1	4	yes	—
70	C₂²	—	1384	2	no	—
71	C₂²	—	1384	4	no	—
72	C₂²	#19149675	1	1	yes	characteristic
73	C₄	—	8	2	no	—
74	C₄	C₂×#14583811	1	2	yes	—
75	C₄	#19149702	1	4	yes	—
76	C₂	—	2768	2	no	—
77	C₂	#26011867	1	1	yes	characteristic
78	C₂	#26011876	1	2	yes	—

	subgroup	G/H
center	C₂²	#19149675
commutator, cosocle	C₄²×C₁₇₃	C₂³
Frattini	C₄²	C₂²×D₁₇₃
Fitting	#494×C₁₇₃	C₂
socle	C₂²×C₁₇₃	#127
prime radical	#26011868	C₂
hypercenter	#494	D₁₇₃
hypocenter	C₁₇₃	#1709
LCS	G ▷ C₄²×C₁₇₃ ▷ C₂²×C₁₇₃ ▷ C₁₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #494

Complements

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

N	H ≅ G/N	type
2. #26011839	C₂	semidirect
3. C₈×#13063436	C₂	semidirect
4. #494×C₁₇₃	C₂	semidirect
39. C₁₇₃	#1709	semidirect
41. #494	D₁₇₃	semidirect

Maximal subgroups

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

subgroup	index
#26011868	2
#26011839	2
C₈×#13063436	2
#494×C₁₇₃	2
#1709	173