Subgroups — #36351533

Order 22144.

96 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26011267	C₂	1	1	yes	maximal, characteristic
2	#26011242	C₂	1	1	yes	maximal, characteristic
3	C₄×Q₁₆×C₁₇₃	C₂	1	1	yes	maximal, characteristic
4	C₂×#14583801	—	2	1	no	—
5	C₄⋊#13063436	—	2	1	no	—
6	C₂×Q₁₆×C₁₇₃	C₄	1	1	yes	characteristic
7	C₄×Q₈×C₁₇₃	—	2	1	no	—
8	#107×C₁₇₃	C₄	1	1	yes	characteristic
9	#103×C₁₇₃	—	2	1	no	—
10	C₄×C₈×C₁₇₃	C₂²	1	1	yes	characteristic
11	C₂×#13063436	—	4	1	no	—
12	#14583801	—	2	2	no	—
13	C₂×Q₈×C₁₇₃	—	2	1	no	—
14	Q₁₆×C₁₇₃	—	2	1	no	—
15	Q₁₆×C₁₇₃	C₈	1	2	yes	complemented
16	C₂×C₈×C₁₇₃	C₂×C₄	1	1	yes	characteristic
17	C₂×C₈×C₁₇₃	D₄	1	1	yes	characteristic
18	C₄⋊C₄×C₁₇₃	—	2	1	no	—
19	C₄⋊C₄×C₁₇₃	—	4	1	no	—
20	C₄²×C₁₇₃	—	4	1	no	—
21	C₄²×C₁₇₃	D₄	1	1	yes	characteristic
22	#13063436	—	8	1	no	—
23	Q₈×C₁₇₃	—	2	2	no	—
24	Q₈×C₁₇₃	—	4	1	no	—
25	C₂×C₄×C₁₇₃	—	4	1	no	—
26	C₂×C₄×C₁₇₃	—	4	1	no	—
27	C₂×C₄×C₁₇₃	C₂²⋊C₄	1	1	yes	characteristic
28	C₂×C₄×C₁₇₃	D₈	1	1	yes	characteristic
29	C₂×C₄×C₁₇₃	SD₁₆	1	1	yes	characteristic
30	C₈×C₁₇₃	—	2	1	no	—
31	C₈×C₁₇₃	C₂×C₈	1	1	yes	characteristic
32	C₈×C₁₇₃	M₁₆	1	1	yes	characteristic
33	C₂²×C₁₇₃	#102	1	1	yes	characteristic
34	C₄×C₁₇₃	—	2	1	no	—
35	C₄×C₁₇₃	—	2	1	no	—
36	C₄×C₁₇₃	—	4	2	no	—
37	C₄×C₁₇₃	—	8	1	no	—
38	C₄×C₁₇₃	C₂²⋊C₈	1	1	yes	characteristic
39	C₄×C₁₇₃	#104	1	1	yes	characteristic
40	C₂×C₁₇₃	D₄⋊C₈	1	1	yes	characteristic
41	C₂×C₁₇₃	#360	1	1	yes	characteristic
42	C₂×C₁₇₃	#362	1	1	yes	characteristic
43	C₁₇₃	#1334	1	1	yes	characteristic, complemented, Sylow
44	#1334	—	173	1	no	maximal, Sylow
45	C₄×Q₁₆	D₁₇₃	1	1	yes	characteristic
46	#346	—	173	1	no	—
47	#334	—	173	1	no	—
48	C₂×Q₁₆	Dic₁₇₃	1	1	yes	characteristic
49	C₄×Q₈	—	2	1	no	—
50	C₂×C₁₆	—	346	1	no	—
51	#107	Dic₁₇₃	1	1	yes	characteristic
52	C₄⋊C₈	—	346	1	no	—
53	#103	—	2	1	no	—
54	C₄×C₈	C₂×D₁₇₃	1	1	yes	characteristic
55	C₂×Q₈	—	2	1	no	—
56	Q₁₆	—	2	1	no	—
57	Q₁₆	#13063436	1	2	yes	complemented
58	C₂×C₈	—	692	1	no	—
59	C₂×C₈	C₂×Dic₁₇₃	1	1	yes	characteristic
60	C₂×C₈	C₂²⋊D₁₇₃	1	1	yes	characteristic
61	C₄⋊C₄	—	2	1	no	—
62	C₄⋊C₄	—	4	1	no	—
63	C₄²	—	4	1	no	—
64	C₄²	C₂²⋊D₁₇₃	1	1	yes	characteristic
65	C₁₆	—	346	2	no	—
66	Q₈	—	2	2	no	—
67	Q₈	—	4	1	no	—
68	C₂×C₄	—	4	1	no	—
69	C₂×C₄	—	4	1	no	—
70	C₂×C₄	C₂²⋊Dic₁₇₃	1	1	yes	characteristic
71	C₂×C₄	#14583811	1	1	yes	characteristic
72	C₂×C₄	#14583814	1	1	yes	characteristic
73	C₈	—	2	1	no	—
74	C₈	—	1384	1	no	—
75	C₈	C₂×#13063436	1	1	yes	characteristic
76	C₈	#14583810	1	1	yes	characteristic
77	C₂²	#19149600	1	1	yes	characteristic
78	C₄	—	2	1	no	—
79	C₄	—	2	1	no	—
80	C₄	—	4	2	no	—
81	C₄	—	8	1	no	—
82	C₄	C₂²⋊#13063436	1	1	yes	characteristic
83	C₄	#19149606	1	1	yes	characteristic
84	C₂	D₄⋊#13063436	1	1	yes	characteristic
85	C₂	#26011301	1	1	yes	characteristic
86	C₂	#26011307	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#19149600
commutator	C₈×C₁₇₃	C₂×C₈
Frattini	C₄×C₈	C₂×D₁₇₃
Fitting	C₄×Q₁₆×C₁₇₃	C₂
socle, prime radical	C₂²×C₁₇₃	#102
cosocle	C₄×C₈×C₁₇₃	C₂²
hypercenter	C₄×Q₁₆	D₁₇₃
hypocenter	C₁₇₃	#1334
LCS	G ▷ C₈×C₁₇₃ ▷ C₄×C₁₇₃ ▷ C₂×C₁₇₃ ▷ C₁₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×Q₁₆

Complements

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

N	H ≅ G/N	type
15. Q₁₆×C₁₇₃	C₈	semidirect
43. C₁₇₃	#1334	semidirect
57. Q₁₆	#13063436	semidirect

Maximal subgroups

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

subgroup	index
#26011267	2
#26011242	2
C₄×Q₁₆×C₁₇₃	2
#1334	173