Subgroups — #33006982

Order 18688.

230 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Dic₁₁₆₈	C₂	1	4	yes	maximal
2	#22612184	C₂	1	2	yes	maximal
3	C₄×C₃₂×C₇₃	C₂	1	1	yes	maximal, characteristic
4	C₂×Dic₅₈₄	—	2	2	no	—
5	C₂×Dic₅₈₄	C₂²	1	4	yes	—
6	#17988967	—	2	2	no	—
7	Dic₁₁₆₈	—	2	8	no	—
8	C₂×C₃₂×C₇₃	C₂²	1	2	yes	—
9	C₄×C₁₆×C₇₃	C₂²	1	1	yes	characteristic
10	C₂×Dic₂₉₂	—	2	4	no	—
11	C₂×Dic₂₉₂	—	4	2	no	—
12	#14556591	—	4	2	no	—
13	Dic₅₈₄	—	2	8	no	—
14	Dic₅₈₄	—	4	4	no	—
15	C₂×C₁₆×C₇₃	D₄	1	1	yes	characteristic
16	C₂×C₁₆×C₇₃	C₂³	1	1	yes	characteristic
17	C₄×C₈×C₇₃	D₄	1	1	yes	characteristic
18	C₃₂×C₇₃	D₄	1	4	yes	—
19	C₂×Dic₁₄₆	—	4	4	no	—
20	Dic₂₉₂	—	4	4	no	—
21	Dic₂₉₂	—	4	4	no	—
22	Dic₂₉₂	—	8	4	no	—
23	C₄⋊Dic₇₃	—	8	4	no	—
24	C₂×C₈×C₇₃	D₈	1	1	yes	characteristic
25	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
26	C₄²×C₇₃	D₈	1	1	yes	characteristic
27	C₁₆×C₇₃	D₈	1	2	yes	—
28	C₁₆×C₇₃	C₂×D₄	1	2	yes	—
29	C₂×C₄×C₇₃	D₁₆	1	2	yes	—
30	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
31	C₂×Dic₇₃	—	8	4	no	—
32	Dic₁₄₆	—	8	4	no	—
33	Dic₁₄₆	—	8	4	no	—
34	C₈×C₇₃	D₁₆	1	2	yes	—
35	C₈×C₇₃	#127	1	1	yes	characteristic
36	C₈×C₇₃	C₂×D₈	1	1	yes	characteristic
37	C₂²×C₇₃	C₂×D₁₆	1	1	yes	characteristic
38	C₄×C₇₃	Q₆₄	1	4	yes	complemented
39	C₄×C₇₃	#493	1	1	yes	characteristic
40	C₄×C₇₃	C₂×D₁₆	1	1	yes	characteristic
41	Dic₇₃	—	16	4	no	—
42	#13726	—	73	1	no	maximal, Sylow
43	C₂×C₇₃	#2246	1	1	yes	characteristic
44	C₂×C₇₃	C₂×Q₆₄	1	2	yes	—
45	C₂×Q₆₄	—	73	4	no	—
46	#2247	—	73	2	no	—
47	C₄×C₃₂	D₇₃	1	1	yes	characteristic
48	C₇₃	#13726	1	1	yes	characteristic, complemented, Sylow
49	C₂×Q₃₂	—	73	4	no	—
50	C₂×Q₃₂	—	146	2	no	—
51	#494	—	146	2	no	—
52	Q₆₄	—	146	8	no	—
53	C₂×C₃₂	C₂×D₇₃	1	2	yes	—
54	C₄×C₁₆	C₂×D₇₃	1	1	yes	characteristic
55	C₂×Q₁₆	—	292	2	no	—
56	C₂×Q₁₆	—	146	4	no	—
57	C₄⋊Q₈	—	292	2	no	—
58	Q₃₂	—	292	4	no	—
59	Q₃₂	—	146	8	no	—
60	C₂×C₁₆	D₂₉₂	1	1	yes	characteristic
61	C₂×C₁₆	C₂²×D₇₃	1	1	yes	characteristic
62	C₄×C₈	D₂₉₂	1	1	yes	characteristic
63	C₃₂	D₂₉₂	1	4	yes	—
64	C₂×Q₈	—	292	4	no	—
65	Q₁₆	—	292	4	no	—
66	Q₁₆	—	292	4	no	—
67	Q₁₆	—	584	4	no	—
68	C₂×C₈	D₅₈₄	1	1	yes	characteristic
69	C₂×C₈	C₂×D₂₉₂	1	1	yes	characteristic
70	C₄⋊C₄	—	584	4	no	—
71	C₄²	D₅₈₄	1	1	yes	characteristic
72	C₁₆	D₅₈₄	1	2	yes	—
73	C₁₆	C₂×D₂₉₂	1	2	yes	—
74	Q₈	—	584	4	no	—
75	Q₈	—	584	4	no	—
76	C₂×C₄	—	584	4	no	—
77	C₂×C₄	D₁₁₆₈	1	2	yes	—
78	C₂×C₄	C₂×D₅₈₄	1	1	yes	characteristic
79	C₈	D₁₁₆₈	1	2	yes	—
80	C₈	#14556588	1	1	yes	characteristic
81	C₈	C₂×D₅₈₄	1	1	yes	characteristic
82	C₂²	C₂×D₁₁₆₈	1	1	yes	characteristic
83	C₄	—	1168	4	no	—
84	C₄	Dic₁₁₆₈	1	4	yes	complemented
85	C₄	#17988964	1	1	yes	characteristic
86	C₄	C₂×D₁₁₆₈	1	1	yes	characteristic
87	C₂	#22612181	1	1	yes	characteristic
88	C₂	C₂×Dic₁₁₆₈	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₁₁₆₈
commutator, cosocle	C₂×C₁₆×C₇₃	C₂³
Frattini	C₂×C₁₆	C₂²×D₇₃
Fitting	C₄×C₃₂×C₇₃	C₂
socle, prime radical	C₂²×C₇₃	C₂×D₁₆
hypercenter	C₄×C₃₂	D₇₃
hypocenter	C₇₃	#13726
LCS	G ▷ C₂×C₁₆×C₇₃ ▷ C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆ ◁ C₄×C₃₂

Complements

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

N	H ≅ G/N	type
38. C₄×C₇₃	Q₆₄	semidirect
48. C₇₃	#13726	semidirect
84. C₄	Dic₁₁₆₈	semidirect

Maximal subgroups

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

subgroup	index
C₂×Dic₁₁₆₈	2
#22612184	2
C₄×C₃₂×C₇₃	2
#13726	73