Subgroups — #36355843

Order 22144.

128 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26011896	C₂	1	1	yes	maximal, characteristic
2	#26011318	C₂	1	4	yes	maximal
3	C₄×#14583801	C₂	1	1	yes	maximal, characteristic
4	#500×C₁₇₃	C₂	1	1	yes	maximal, characteristic
5	#19149679	—	2	1	no	—
6	C₂×#14583801	C₂²	1	2	yes	—
7	#19149613	—	2	1	no	—
8	#19149613	C₂²	1	2	yes	—
9	C₄⋊Q₈×C₁₇₃	—	2	1	no	—
10	#107×C₁₇₃	—	2	1	no	—
11	#107×C₁₇₃	C₂²	1	2	yes	—
12	C₄×C₈×C₁₇₃	C₂²	1	1	yes	characteristic
13	C₂×Dic₃₄₆	—	4	1	no	—
14	C₄⋊Dic₁₇₃	—	2	2	no	—
15	C₄⋊Dic₁₇₃	—	4	1	no	—
16	#14583801	Q₈	1	4	yes	complemented
17	C₂×Q₈×C₁₇₃	—	4	1	no	—
18	C₂×C₈×C₁₇₃	D₄	1	1	yes	characteristic
19	C₂×C₈×C₁₇₃	C₂³	1	1	yes	characteristic
20	C₄⋊C₄×C₁₇₃	—	2	2	no	—
21	C₄⋊C₄×C₁₇₃	—	4	1	no	—
22	C₄²×C₁₇₃	D₄	1	1	yes	characteristic
23	Dic₃₄₆	—	8	2	no	—
24	C₂×Dic₁₇₃	—	4	2	no	—
25	Q₈×C₁₇₃	—	8	2	no	—
26	C₂×C₄×C₁₇₃	—	4	2	no	—
27	C₂×C₄×C₁₇₃	D₈	1	1	yes	characteristic
28	C₂×C₄×C₁₇₃	D₈	1	1	yes	characteristic
29	C₂×C₄×C₁₇₃	C₂×D₄	1	1	yes	characteristic
30	C₈×C₁₇₃	Q₁₆	1	2	yes	—
31	C₈×C₁₇₃	C₂×Q₈	1	2	yes	—
32	C₂²×C₁₇₃	C₂×D₈	1	1	yes	characteristic
33	C₄×C₁₇₃	—	8	2	no	—
34	C₄×C₁₇₃	SD₃₂	1	2	yes	—
35	C₄×C₁₇₃	SD₃₂	1	2	yes	—
36	C₄×C₁₇₃	C₄⋊Q₈	1	1	yes	characteristic
37	C₄×C₁₇₃	C₂×Q₁₆	1	1	yes	characteristic
38	Dic₁₇₃	—	8	2	no	—
39	C₂×C₁₇₃	#500	1	1	yes	characteristic
40	C₂×C₁₇₃	C₂×SD₃₂	1	1	yes	characteristic
41	C₂×C₁₇₃	C₂×SD₃₂	1	1	yes	characteristic
42	C₁₇₃	#2254	1	1	yes	characteristic, complemented, Sylow
43	#2254	—	173	1	no	maximal, Sylow
44	#500	—	173	1	no	—
45	#500	D₁₇₃	1	1	yes	characteristic
46	#367	—	173	4	no	—
47	C₄×C₁₆	—	173	1	no	—
48	C₄⋊Q₈	—	2	1	no	—
49	C₄⋊Q₈	—	346	1	no	—
50	C₂×C₁₆	—	173	2	no	—
51	#107	—	2	1	no	—
52	#107	—	346	1	no	—
53	#107	—	173	2	no	—
54	#107	C₂×D₁₇₃	1	2	yes	—
55	C₄×C₈	C₂×D₁₇₃	1	1	yes	characteristic
56	C₂×Q₈	—	4	1	no	—
57	C₂×Q₈	—	692	1	no	—
58	C₂×C₈	C₂²⋊D₁₇₃	1	1	yes	characteristic
59	C₂×C₈	C₂²×D₁₇₃	1	1	yes	characteristic
60	C₄⋊C₄	—	2	2	no	—
61	C₄⋊C₄	—	4	1	no	—
62	C₄⋊C₄	—	346	2	no	—
63	C₄⋊C₄	—	692	1	no	—
64	C₄²	C₂²⋊D₁₇₃	1	1	yes	characteristic
65	C₁₆	—	173	4	no	—
66	Q₈	—	8	2	no	—
67	Q₈	—	1384	2	no	—
68	C₂×C₄	—	4	2	no	—
69	C₂×C₄	—	692	2	no	—
70	C₂×C₄	#14583811	1	1	yes	characteristic
71	C₂×C₄	#14583811	1	1	yes	characteristic
72	C₂×C₄	C₂×C₂²⋊D₁₇₃	1	1	yes	characteristic
73	C₈	#14583816	1	2	yes	—
74	C₈	Q₈×D₁₇₃	1	2	yes	—
75	C₂²	C₂×#14583811	1	1	yes	characteristic
76	C₄	—	8	2	no	—
77	C₄	—	1384	2	no	—
78	C₄	#19149622	1	2	yes	—
79	C₄	#19149623	1	2	yes	—
80	C₄	#19149677	1	1	yes	characteristic
81	C₄	C₂×#14583816	1	1	yes	characteristic
82	C₂	#26011894	1	1	yes	characteristic
83	C₂	C₂×#19149622	1	1	yes	characteristic
84	C₂	C₂×#19149623	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#14583811
commutator, cosocle	C₂×C₈×C₁₇₃	C₂³
Frattini	C₂×C₈	C₂²×D₁₇₃
Fitting	#500×C₁₇₃	C₂
socle, prime radical	C₂²×C₁₇₃	C₂×D₈
hypercenter	#500	D₁₇₃
hypocenter	C₁₇₃	#2254
LCS	G ▷ C₂×C₈×C₁₇₃ ▷ C₄×C₁₇₃ ▷ C₂×C₁₇₃ ▷ C₁₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #500

Complements

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

N	H ≅ G/N	type
16. #14583801	Q₈	semidirect
42. C₁₇₃	#2254	semidirect

Maximal subgroups

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

subgroup	index
#26011896	2
#26011318	2
C₄×#14583801	2
#500×C₁₇₃	2
#2254	173