Subgroups — #26011585

Order 11072.

110 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Dic₆₉₂	C₂	1	1	yes	maximal, characteristic
2	#19149644	C₂	1	2	yes	maximal
3	#19149602	C₂	1	2	yes	maximal
4	C₈×Dic₁₇₃	C₂	1	1	yes	maximal, characteristic
5	#107×C₁₇₃	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₃₄₆	C₂²	1	2	yes	—
7	Dic₆₉₂	C₄	1	4	yes	complemented
8	C₂×#13063436	C₂²	1	1	yes	characteristic
9	#14583805	—	2	2	no	—
10	C₄×Dic₁₇₃	—	2	2	no	—
11	C₄×Dic₁₇₃	C₂²	1	1	yes	characteristic
12	C₂×C₈×C₁₇₃	C₂²	1	1	yes	characteristic
13	C₄⋊C₄×C₁₇₃	C₂²	1	2	yes	—
14	Dic₃₄₆	—	2	2	no	—
15	Dic₃₄₆	C₂×C₄	1	4	yes	—
16	C₂×Dic₁₇₃	—	2	2	no	—
17	C₂×Dic₁₇₃	D₄	1	1	yes	characteristic
18	C₂×Dic₁₇₃	D₄	1	1	yes	characteristic
19	#13063436	—	2	1	no	—
20	C₂×C₄×C₁₇₃	—	2	2	no	—
21	C₂×C₄×C₁₇₃	C₂³	1	1	yes	characteristic
22	C₈×C₁₇₃	C₂×C₄	1	2	yes	—
23	C₂²×C₁₇₃	C₂×D₄	1	1	yes	characteristic
24	C₄×C₁₇₃	—	4	2	no	—
25	C₄×C₁₇₃	C₂²×C₄	1	1	yes	characteristic
26	C₄×C₁₇₃	#41	1	1	yes	characteristic
27	Dic₁₇₃	—	2	4	no	—
28	Dic₁₇₃	—	2	1	no	—
29	Dic₁₇₃	Q₁₆	1	2	yes	complemented
30	C₂×C₁₇₃	C₄×D₄	1	1	yes	characteristic
31	C₂×C₁₇₃	C₂×Q₁₆	1	1	yes	characteristic
32	C₂×C₁₇₃	#135	1	1	yes	characteristic
33	C₁₇₃	C₄×Q₁₆	1	1	yes	characteristic, complemented, Sylow
34	C₄×Q₁₆	—	173	1	no	maximal, Sylow
35	C₂×Q₁₆	—	173	1	no	—
36	C₄×Q₈	—	173	2	no	—
37	#107	D₁₇₃	1	1	yes	characteristic
38	#103	—	173	2	no	—
39	C₄×C₈	—	173	1	no	—
40	C₂×Q₈	—	173	2	no	—
41	Q₁₆	—	173	4	no	—
42	C₂×C₈	—	173	1	no	—
43	C₂×C₈	C₂×D₁₇₃	1	1	yes	characteristic
44	C₄⋊C₄	—	346	2	no	—
45	C₄⋊C₄	C₂×D₁₇₃	1	2	yes	—
46	C₄²	—	346	2	no	—
47	C₄²	—	173	1	no	—
48	Q₈	—	346	2	no	—
49	Q₈	—	173	4	no	—
50	C₂×C₄	—	2	2	no	—
51	C₂×C₄	—	346	2	no	—
52	C₂×C₄	—	173	1	no	—
53	C₂×C₄	—	173	1	no	—
54	C₂×C₄	C₂²×D₁₇₃	1	1	yes	characteristic
55	C₈	—	346	1	no	—
56	C₈	C₄×D₁₇₃	1	2	yes	—
57	C₂²	D₄×D₁₇₃	1	1	yes	characteristic
58	C₄	—	4	2	no	—
59	C₄	—	346	4	no	—
60	C₄	—	346	1	no	—
61	C₄	—	173	2	no	—
62	C₄	C₂×C₄×D₁₇₃	1	1	yes	characteristic
63	C₄	#14583827	1	1	yes	characteristic
64	C₂	#19149641	1	1	yes	characteristic
65	C₂	Q₁₆×D₁₇₃	1	1	yes	characteristic
66	C₂	#19149698	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₄×D₁₇₃
commutator	C₄×C₁₇₃	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₁₇₃
Fitting	#107×C₁₇₃	C₂
socle, prime radical	C₂²×C₁₇₃	C₂×D₄
cosocle	C₂×C₄×C₁₇₃	C₂³
hypercenter	#107	D₁₇₃
hypocenter	C₁₇₃	C₄×Q₁₆
LCS	G ▷ C₄×C₁₇₃ ▷ C₂×C₁₇₃ ▷ C₁₇₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #107

Complements

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

N	H ≅ G/N	type
7. Dic₆₉₂	C₄	semidirect
29. Dic₁₇₃	Q₁₆	semidirect
33. C₁₇₃	C₄×Q₁₆	semidirect

Maximal subgroups

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

subgroup	index
C₂×Dic₆₉₂	2
#19149644	2
#19149602	2
C₈×Dic₁₇₃	2
#107×C₁₇₃	2
C₄×Q₁₆	173