Subgroups — #36368696

Order 22144.

110 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26012396	C₂	1	2	yes	maximal
2	#26011865	C₂	1	1	yes	maximal, characteristic
3	#19149738	—	2	2	no	—
4	C₂×#14583813	—	2	2	no	—
5	#19149676	C₂²	1	1	yes	characteristic
6	C₄×#13063436	C₄	1	1	yes	characteristic, complemented
7	C₄⋊Q₈×C₁₇₃	C₄	1	1	yes	characteristic, complemented
8	C₂²⋊#10621924	—	4	2	no	—
9	C₂×D₆₉₂	—	4	1	no	—
10	C₂×D₆₉₂	D₄	1	2	yes	—
11	#14583813	—	4	4	no	—
12	C₂×#13063436	—	2	1	no	—
13	C₂×Q₈×C₁₇₃	—	2	1	no	—
14	C₄⋊C₄×C₁₇₃	—	4	1	no	—
15	C₄²×C₁₇₃	C₂×C₄	1	1	yes	characteristic
16	C₂×#10621924	—	8	2	no	—
17	C₂²×D₁₇₃	—	2	2	no	—
18	D₆₉₂	—	4	2	no	—
19	D₆₉₂	—	8	2	no	—
20	#13063436	—	2	2	no	—
21	Q₈×C₁₇₃	—	4	2	no	—
22	C₂×C₄×C₁₇₃	—	2	1	no	—
23	C₂×C₄×C₁₇₃	—	4	1	no	—
24	C₂×C₄×C₁₇₃	C₂²⋊C₄	1	1	yes	characteristic
25	C₂²×C₁₇₃	#99	1	1	yes	characteristic
26	C₂×D₁₇₃	—	4	2	no	—
27	C₂×D₁₇₃	—	8	2	no	—
28	#10621924	—	16	2	no	—
29	C₄×C₁₇₃	—	2	2	no	—
30	C₄×C₁₇₃	—	2	1	no	—
31	C₄×C₁₇₃	—	8	1	no	—
32	C₂×C₁₇₃	—	2	1	no	—
33	C₂×C₁₇₃	#353	1	1	yes	characteristic
34	D₁₇₃	—	8	2	no	—
35	C₁₇₃	#1409	1	1	yes	characteristic, complemented, Sylow
36	#1409	—	173	1	no	maximal, Sylow
37	#492	—	173	1	no	—
38	#353	—	173	2	no	—
39	C₂×SD₁₆	—	346	2	no	—
40	C₄⋊Q₈	#10621924	1	1	yes	characteristic, complemented
41	#127	—	173	1	no	—
42	#99	—	346	2	no	—
43	C₄×C₈	—	173	1	no	—
44	C₂×Q₈	—	2	1	no	—
45	C₂×D₄	—	173	2	no	—
46	C₂×D₄	—	692	1	no	—
47	SD₁₆	—	692	4	no	—
48	C₂×C₈	—	346	1	no	—
49	C₄⋊C₄	—	4	1	no	—
50	C₂²⋊C₄	—	692	2	no	—
51	C₄²	C₂×#10621924	1	1	yes	characteristic
52	C₂³	—	346	2	no	—
53	Q₈	—	4	2	no	—
54	D₄	—	1384	2	no	—
55	D₄	—	692	2	no	—
56	C₂×C₄	—	2	1	no	—
57	C₂×C₄	—	4	1	no	—
58	C₂×C₄	—	1384	2	no	—
59	C₂×C₄	C₂²⋊#10621924	1	1	yes	characteristic
60	C₈	—	346	2	no	—
61	C₂²	—	1384	2	no	—
62	C₂²	—	692	2	no	—
63	C₂²	#19149737	1	1	yes	characteristic
64	C₄	—	2	2	no	—
65	C₄	—	2	1	no	—
66	C₄	—	8	1	no	—
67	C₄	—	2768	2	no	—
68	C₂	—	2	1	no	—
69	C₂	—	1384	2	no	—
70	C₂	#26012395	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#26012395
commutator	C₄²×C₁₇₃	C₂×C₄
Frattini	C₄²	C₂×#10621924
Fitting	C₄⋊Q₈×C₁₇₃	C₄
socle	C₂×C₁₇₃	#353
prime radical, cosocle	#19149676	C₂²
hypercenter	C₄⋊Q₈	#10621924
hypocenter	C₁₇₃	#1409
LCS	G ▷ C₄²×C₁₇₃ ▷ C₂×C₄×C₁₇₃ ▷ C₂²×C₁₇₃ ▷ C₂×C₁₇₃ ▷ C₁₇₃
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×C₄ ◁ C₄² ◁ C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
6. C₄×#13063436	C₄	semidirect
7. C₄⋊Q₈×C₁₇₃	C₄	semidirect
35. C₁₇₃	#1409	semidirect
40. C₄⋊Q₈	#10621924	semidirect

Maximal subgroups

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

subgroup	index
#26012396	2
#26011865	2
#1409	173