Subgroups — #36353169

Order 22144.

212 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26011871	C₂	1	1	yes	maximal, characteristic, complemented
2	#26011868	C₂	1	1	yes	maximal, characteristic
3	#26011838	C₂	1	4	yes	maximal, complemented
4	C₈²×C₁₇₃	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×Dic₆₉₂	—	2	2	no	—
6	C₂×D₁₃₈₄	—	2	2	no	—
7	#19149679	C₂²	1	2	yes	—
8	#19149676	C₂²	1	2	yes	—
9	#19149601	—	2	8	no	—
10	C₄×C₈×C₁₇₃	C₂²	1	2	yes	—
11	C₄×C₈×C₁₇₃	C₂²	1	1	yes	characteristic
12	C₂×Dic₃₄₆	—	2	2	no	—
13	C₂×D₆₉₂	—	2	4	no	—
14	C₂×D₆₉₂	—	2	2	no	—
15	Dic₆₉₂	—	4	4	no	—
16	D₁₃₈₄	—	4	4	no	—
17	C₄⋊Dic₁₇₃	—	2	4	no	—
18	C₂×C₈×C₁₇₃	D₄	1	4	yes	—
19	C₂×C₈×C₁₇₃	D₄	1	2	yes	—
20	C₄²×C₁₇₃	C₂³	1	1	yes	characteristic
21	C₂²×D₁₇₃	—	4	2	no	—
22	Dic₃₄₆	—	4	4	no	—
23	D₆₉₂	—	4	4	no	—
24	D₆₉₂	—	4	8	no	—
25	C₂×Dic₁₇₃	—	4	2	no	—
26	C₂×C₄×C₁₇₃	C₂×D₄	1	1	yes	characteristic
27	C₂×C₄×C₁₇₃	C₂×D₄	1	2	yes	—
28	C₈×C₁₇₃	—	2	4	no	—
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	C₂×D₈	1	2	yes	—
34	C₄×C₁₇₃	#135	1	4	yes	—
35	Dic₁₇₃	—	8	2	no	—
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	—	173	1	no	—
42	#493	—	173	1	no	—
43	#486	—	173	4	no	—
44	C₈²	D₁₇₃	1	1	yes	characteristic, complemented
45	C₂×Q₁₆	—	346	2	no	—
46	C₂×D₈	—	346	2	no	—
47	C₄⋊Q₈	—	173	2	no	—
48	#127	—	173	2	no	—
49	#102	—	346	8	no	—
50	C₄×C₈	C₂×D₁₇₃	1	2	yes	—
51	C₄×C₈	C₂×D₁₇₃	1	1	yes	characteristic
52	C₂×Q₈	—	346	2	no	—
53	C₂×D₄	—	346	4	no	—
54	C₂×D₄	—	346	2	no	—
55	Q₁₆	—	692	4	no	—
56	D₈	—	692	4	no	—
57	C₂×C₈	D₆₉₂	1	4	yes	—
58	C₂×C₈	D₆₉₂	1	2	yes	—
59	C₄⋊C₄	—	346	4	no	—
60	C₄²	C₂²×D₁₇₃	1	1	yes	characteristic
61	C₂³	—	692	2	no	—
62	Q₈	—	692	4	no	—
63	D₄	—	692	4	no	—
64	D₄	—	692	8	no	—
65	C₂×C₄	—	692	2	no	—
66	C₂×C₄	C₂×D₆₉₂	1	1	yes	characteristic
67	C₂×C₄	C₂×D₆₉₂	1	2	yes	—
68	C₈	—	2	4	no	—
69	C₈	D₁₃₈₄	1	4	yes	—
70	C₂²	—	1384	2	no	—
71	C₂²	—	1384	4	no	—
72	C₂²	#19149676	1	1	yes	characteristic
73	C₄	—	1384	2	no	—
74	C₄	C₂×D₁₃₈₄	1	2	yes	—
75	C₄	#19149701	1	4	yes	—
76	C₂	—	2768	2	no	—
77	C₂	#26011868	1	1	yes	characteristic
78	C₂	#26011875	1	2	yes	—

	subgroup	G/H
center	C₂²	#19149676
commutator, cosocle	C₄²×C₁₇₃	C₂³
Frattini	C₄²	C₂²×D₁₇₃
Fitting	C₈²×C₁₇₃	C₂
socle	C₂²×C₁₇₃	#127
prime radical	#26011868	C₂
hypercenter	C₈²	D₁₇₃
hypocenter	C₁₇₃	#1709
LCS	G ▷ C₄²×C₁₇₃ ▷ C₂²×C₁₇₃ ▷ C₁₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈²

Complements

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

N	H ≅ G/N	type
1. #26011871	C₂	semidirect
3. #26011838	C₂	semidirect
4. C₈²×C₁₇₃	C₂	semidirect
39. C₁₇₃	#1709	semidirect
44. C₈²	D₁₇₃	semidirect

Maximal subgroups

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

subgroup	index
#26011871	2
#26011868	2
#26011838	2
C₈²×C₁₇₃	2
#1709	173