Subgroups — C₂³×#14583835 #36370548

Order 22144.

1720 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14583835	C₂	1	28	yes	maximal, direct product factor
2	C₂³×#13063443	C₂	1	2	yes	maximal, complemented
3	C₂⁴×Dic₁₇₃	C₂	1	1	yes	maximal, characteristic
4	C₂×#14583835	C₂²	1	112	yes	direct product factor
5	C₂²×#13063443	C₂²	1	28	yes	complemented
6	C₂³×Dic₁₇₃	C₄	1	14	yes	—
7	C₂³×Dic₁₇₃	C₄	1	1	yes	characteristic
8	C₂³×Dic₁₇₃	C₂²	1	1	yes	characteristic
9	C₂³×Dic₁₇₃	C₂²	1	14	yes	—
10	C₂⁵×C₁₇₃	C₄	1	1	yes	characteristic
11	#14583835	C₂³	1	64	yes	direct product factor
12	C₂×#13063443	C₂³	1	56	yes	complemented
13	C₂²×Dic₁₇₃	C₂×C₄	1	84	yes	—
14	C₂²×Dic₁₇₃	C₂×C₄	1	7	yes	—
15	C₂²×Dic₁₇₃	C₂×C₄	1	14	yes	—
16	C₂²×Dic₁₇₃	C₂³	1	7	yes	—
17	C₂²×Dic₁₇₃	C₂³	1	28	yes	—
18	C₂⁴×C₁₇₃	—	2	8	no	—
19	C₂⁴×C₁₇₃	C₂×C₄	1	14	yes	—
20	C₂⁴×C₁₇₃	C₂×C₄	1	1	yes	characteristic
21	#13063443	C₂⁴	1	16	yes	complemented
22	C₂×Dic₁₇₃	C₂²×C₄	1	56	yes	—
23	C₂×Dic₁₇₃	C₂²×C₄	1	21	yes	—
24	C₂×Dic₁₇₃	C₂²×C₄	1	28	yes	—
25	C₂×Dic₁₇₃	C₂⁴	1	7	yes	—
26	C₂×Dic₁₇₃	C₂⁴	1	8	yes	—
27	C₂³×C₁₇₃	—	2	56	no	—
28	C₂³×C₁₇₃	M₁₆	1	8	yes	complemented
29	C₂³×C₁₇₃	C₂²×C₄	1	28	yes	—
30	C₂³×C₁₇₃	C₂²×C₄	1	7	yes	—
31	C₂²×C₁₇₃	—	2	56	no	—
32	C₂²×C₁₇₃	C₂×M₁₆	1	28	yes	complemented
33	C₂²×C₁₇₃	C₂³×C₄	1	8	yes	—
34	C₂²×C₁₇₃	C₂³×C₄	1	7	yes	—
35	Dic₁₇₃	C₂³×C₄	1	7	yes	—
36	Dic₁₇₃	C₂³×C₄	1	8	yes	—
37	Dic₁₇₃	C₂⁵	1	1	yes	characteristic
38	C₂×C₁₇₃	—	2	8	no	—
39	C₂×C₁₇₃	C₂²×M₁₆	1	14	yes	complemented
40	C₂×C₁₇₃	C₂⁴×C₄	1	1	yes	characteristic
41	C₁₇₃	C₂³×M₁₆	1	1	yes	characteristic, complemented, Sylow
42	C₂³×M₁₆	—	173	1	no	maximal, Sylow
43	C₂⁴×C₄	—	173	1	no	—
44	C₂²×M₁₆	—	173	28	no	—
45	C₂³×C₈	—	173	2	no	—
46	C₂⁵	#10621924	1	1	yes	characteristic
47	C₂³×C₄	—	173	14	no	—
48	C₂³×C₄	—	173	14	no	—
49	C₂³×C₄	—	173	1	no	—
50	C₂³×C₄	—	173	1	no	—
51	C₂×M₁₆	—	173	112	no	—
52	C₂²×C₈	—	173	28	no	—
53	C₂⁴	—	2	8	no	—
54	C₂⁴	C₂×#10621924	1	14	yes	—
55	C₂⁴	C₂×#10621924	1	1	yes	characteristic
56	C₂²×C₄	—	173	84	no	—
57	C₂²×C₄	—	173	14	no	—
58	C₂²×C₄	—	173	7	no	—
59	C₂²×C₄	—	173	7	no	—
60	C₂²×C₄	—	173	28	no	—
61	M₁₆	—	173	64	no	—
62	C₂×C₈	—	173	56	no	—
63	C₂³	—	2	56	no	—
64	C₂³	#14583835	1	8	yes	direct product factor
65	C₂³	C₂²×#10621924	1	28	yes	—
66	C₂³	C₂²×#10621924	1	7	yes	—
67	C₂×C₄	—	173	56	no	—
68	C₂×C₄	—	173	21	no	—
69	C₂×C₄	—	173	28	no	—
70	C₂×C₄	—	173	7	no	—
71	C₂×C₄	—	173	8	no	—
72	C₈	—	173	16	no	—
73	C₂²	—	2	56	no	—
74	C₂²	C₂×#14583835	1	28	yes	direct product factor
75	C₂²	C₂³×#10621924	1	8	yes	—
76	C₂²	C₂³×#10621924	1	7	yes	—
77	C₄	—	173	1	no	—
78	C₄	—	173	7	no	—
79	C₄	—	173	8	no	—
80	C₂	—	2	8	no	—
81	C₂	C₂²×#14583835	1	14	yes	direct product factor
82	C₂	C₂⁴×#10621924	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	C₂×#10621924
commutator	C₂×C₁₇₃	C₂⁴×C₄
Frattini	C₂	C₂⁴×#10621924
Fitting, prime radical	C₂⁵×C₁₇₃	C₄
socle	C₂⁴×C₁₇₃	C₂×C₄
cosocle	Dic₁₇₃	C₂⁵
hypercenter	C₂⁵	#10621924
hypocenter	C₁₇₃	C₂³×M₁₆
LCS	G ▷ C₂×C₁₇₃ ▷ C₁₇₃
UCS	C₁ ◁ C₂⁴ ◁ C₂⁵

Complements

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

N	H ≅ G/N	type
1. C₂²×#14583835	C₂	direct (N × H)
2. C₂³×#13063443	C₂	semidirect
4. C₂×#14583835	C₂²	direct (N × H)
5. C₂²×#13063443	C₂²	semidirect
11. #14583835	C₂³	direct (N × H)
12. C₂×#13063443	C₂³	semidirect
21. #13063443	C₂⁴	semidirect
28. C₂³×C₁₇₃	M₁₆	semidirect
32. C₂²×C₁₇₃	C₂×M₁₆	semidirect
39. C₂×C₁₇₃	C₂²×M₁₆	semidirect
41. C₁₇₃	C₂³×M₁₆	semidirect
64. C₂³	#14583835	direct (N × H)
74. C₂²	C₂×#14583835	direct (N × H)
81. C₂	C₂²×#14583835	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#14583835	2
C₂³×#13063443	2
C₂⁴×Dic₁₇₃	2
C₂³×M₁₆	173

Subgroups — C23×#14583835 #36370548

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#14583835 #36370548