Subgroups — #36355816

Order 22144.

176 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Dic₁₃₈₄	C₂	1	4	yes	maximal
2	#26011871	C₂	1	2	yes	maximal
3	C₄×C₁₆×C₁₇₃	C₂	1	1	yes	maximal, characteristic
4	C₂×Dic₆₉₂	—	2	2	no	—
5	C₂×Dic₆₉₂	C₂²	1	4	yes	—
6	#19149679	—	2	2	no	—
7	Dic₁₃₈₄	—	2	8	no	—
8	C₂×C₁₆×C₁₇₃	C₂²	1	2	yes	—
9	C₄×C₈×C₁₇₃	C₂²	1	1	yes	characteristic
10	C₂×Dic₃₄₆	—	2	4	no	—
11	Dic₆₉₂	—	2	8	no	—
12	Dic₆₉₂	—	4	4	no	—
13	C₄⋊Dic₁₇₃	—	4	4	no	—
14	C₂×C₈×C₁₇₃	D₄	1	1	yes	characteristic
15	C₂×C₈×C₁₇₃	C₂³	1	1	yes	characteristic
16	C₄²×C₁₇₃	D₄	1	1	yes	characteristic
17	C₁₆×C₁₇₃	D₄	1	4	yes	—
18	Dic₃₄₆	—	4	4	no	—
19	Dic₃₄₆	—	4	4	no	—
20	C₂×Dic₁₇₃	—	4	4	no	—
21	C₂×C₄×C₁₇₃	D₈	1	2	yes	—
22	C₂×C₄×C₁₇₃	C₂×D₄	1	1	yes	characteristic
23	C₈×C₁₇₃	D₈	1	2	yes	—
24	C₈×C₁₇₃	C₂×D₄	1	2	yes	—
25	C₂²×C₁₇₃	C₂×D₈	1	1	yes	characteristic
26	C₄×C₁₇₃	Q₃₂	1	4	yes	complemented
27	C₄×C₁₇₃	#127	1	1	yes	characteristic
28	C₄×C₁₇₃	C₂×D₈	1	1	yes	characteristic
29	Dic₁₇₃	—	8	4	no	—
30	C₂×C₁₇₃	#493	1	1	yes	characteristic
31	C₂×C₁₇₃	C₂×Q₃₂	1	2	yes	—
32	C₁₇₃	#2247	1	1	yes	characteristic, complemented, Sylow
33	#2247	—	173	1	no	maximal, Sylow
34	C₂×Q₃₂	—	173	4	no	—
35	#494	—	173	2	no	—
36	C₄×C₁₆	D₁₇₃	1	1	yes	characteristic
37	C₂×Q₁₆	—	346	2	no	—
38	C₂×Q₁₆	—	173	4	no	—
39	C₄⋊Q₈	—	346	2	no	—
40	Q₃₂	—	346	8	no	—
41	C₂×C₁₆	C₂×D₁₇₃	1	2	yes	—
42	C₄×C₈	C₂×D₁₇₃	1	1	yes	characteristic
43	C₂×Q₈	—	346	4	no	—
44	Q₁₆	—	346	8	no	—
45	Q₁₆	—	692	4	no	—
46	C₂×C₈	D₆₉₂	1	1	yes	characteristic
47	C₂×C₈	C₂²×D₁₇₃	1	1	yes	characteristic
48	C₄⋊C₄	—	692	4	no	—
49	C₄²	D₆₉₂	1	1	yes	characteristic
50	C₁₆	D₆₉₂	1	4	yes	—
51	Q₈	—	692	4	no	—
52	Q₈	—	692	4	no	—
53	C₂×C₄	—	692	4	no	—
54	C₂×C₄	D₁₃₈₄	1	2	yes	—
55	C₂×C₄	C₂×D₆₉₂	1	1	yes	characteristic
56	C₈	D₁₃₈₄	1	2	yes	—
57	C₈	C₂×D₆₉₂	1	2	yes	—
58	C₂²	C₂×D₁₃₈₄	1	1	yes	characteristic
59	C₄	—	1384	4	no	—
60	C₄	Dic₁₃₈₄	1	4	yes	complemented
61	C₄	#19149676	1	1	yes	characteristic
62	C₄	C₂×D₁₃₈₄	1	1	yes	characteristic
63	C₂	#26011868	1	1	yes	characteristic
64	C₂	C₂×Dic₁₃₈₄	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₁₃₈₄
commutator, cosocle	C₂×C₈×C₁₇₃	C₂³
Frattini	C₂×C₈	C₂²×D₁₇₃
Fitting	C₄×C₁₆×C₁₇₃	C₂
socle, prime radical	C₂²×C₁₇₃	C₂×D₈
hypercenter	C₄×C₁₆	D₁₇₃
hypocenter	C₁₇₃	#2247
LCS	G ▷ C₂×C₈×C₁₇₃ ▷ C₄×C₁₇₃ ▷ C₂×C₁₇₃ ▷ C₁₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆

Complements

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

N	H ≅ G/N	type
26. C₄×C₁₇₃	Q₃₂	semidirect
32. C₁₇₃	#2247	semidirect
60. C₄	Dic₁₃₈₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×Dic₁₃₈₄	2
#26011871	2
C₄×C₁₆×C₁₇₃	2
#2247	173