Subgroups — #322×Dic₇₃ #32975331

Order 18688.

654 conjugacy classes of subgroups.

The subgroup lattice has 146 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₈×Dic₇₃	C₂	1	1	yes	maximal, characteristic
2	C₄×C₈×Dic₇₃	C₂	1	2	yes	maximal
3	C₂×#17988309	C₂	1	2	yes	maximal
4	C₂×#17988308	C₂	1	1	yes	maximal, characteristic
5	C₂×#322×C₇₃	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄×#10596831	C₂²	1	2	yes	—
7	C₂×C₄×#10596831	C₂²	1	1	yes	characteristic
8	C₂×C₈×Dic₇₃	C₄	1	8	yes	—
9	C₂×C₈×Dic₇₃	C₄	1	4	yes	—
10	C₄²×Dic₇₃	C₂²	1	1	yes	characteristic
11	#17988309	C₄	1	8	yes	complemented
12	#17988308	C₄	1	4	yes	complemented
13	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₈×C₇₃	C₂²	1	2	yes	—
15	#322×C₇₃	C₄	1	4	yes	complemented
16	C₂²×#10596831	C₂×C₄	1	4	yes	—
17	C₂²×#10596831	C₂×C₄	1	2	yes	—
18	C₂×C₄×Dic₇₃	C₂×C₄	1	2	yes	—
19	C₂×C₄×Dic₇₃	C₂×C₄	1	4	yes	—
20	C₄×#10596831	C₂×C₄	1	4	yes	—
21	C₄×#10596831	C₂×C₄	1	4	yes	—
22	C₄×#10596831	C₂×C₄	1	4	yes	—
23	C₈×Dic₇₃	—	2	16	no	—
24	C₈×Dic₇₃	C₈	1	16	yes	complemented
25	C₂²×C₈×C₇₃	C₂×C₄	1	4	yes	—
26	C₂²×C₈×C₇₃	C₂×C₄	1	2	yes	—
27	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
28	C₄×C₈×C₇₃	C₂×C₄	1	2	yes	—
29	C₄×C₈×C₇₃	C₂×C₄	1	4	yes	—
30	C₄×C₈×C₇₃	C₂×C₄	1	4	yes	—
31	C₄×C₈×C₇₃	C₂×C₄	1	2	yes	—
32	C₂²×Dic₇₃	C₄²	1	4	yes	—
33	C₂×#10596831	—	2	8	no	—
34	C₂×#10596831	C₄²	1	8	yes	—
35	C₂×#10596831	C₄²	1	4	yes	—
36	C₂×#10596831	C₂×C₈	1	8	yes	—
37	C₄×Dic₇₃	C₂×C₈	1	8	yes	—
38	C₄×Dic₇₃	M₁₆	1	16	yes	—
39	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
40	C₂²×C₄×C₇₃	C₂²×C₄	1	2	yes	—
41	C₂×C₈×C₇₃	—	2	8	no	—
42	C₂×C₈×C₇₃	C₄²	1	8	yes	—
43	C₂×C₈×C₇₃	C₄²	1	4	yes	—
44	C₂×C₈×C₇₃	C₂×C₈	1	8	yes	—
45	C₄²×C₇₃	C₂²×C₄	1	2	yes	—
46	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
47	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
48	C₂³×C₇₃	C₂×C₄²	1	1	yes	characteristic
49	C₂×C₄×C₇₃	C₂×C₄²	1	1	yes	characteristic
50	C₂×C₄×C₇₃	C₂×C₄²	1	2	yes	—
51	C₂×C₄×C₇₃	C₂×C₄²	1	1	yes	characteristic
52	C₂×C₄×C₇₃	C₂×C₄²	1	2	yes	—
53	C₂×C₄×C₇₃	C₂²×C₈	1	4	yes	—
54	C₂×C₄×C₇₃	C₂×M₁₆	1	8	yes	—
55	C₂×Dic₇₃	C₄×C₈	1	8	yes	—
56	C₂×Dic₇₃	#97	1	16	yes	—
57	C₈×C₇₃	—	2	8	no	—
58	C₈×C₇₃	C₄×C₈	1	8	yes	complemented
59	#10596831	—	2	8	no	—
60	#10596831	C₄×C₈	1	8	yes	complemented
61	C₂²×C₇₃	C₄³	1	1	yes	characteristic
62	C₂²×C₇₃	C₂×C₄×C₈	1	2	yes	—
63	C₂²×C₇₃	C₂×#97	1	4	yes	—
64	C₄×C₇₃	C₂×C₄×C₈	1	2	yes	—
65	C₄×C₇₃	C₂×C₄×C₈	1	2	yes	—
66	C₄×C₇₃	C₄×M₁₆	1	4	yes	—
67	C₄×C₇₃	C₄×M₁₆	1	4	yes	—
68	Dic₇₃	#322	1	16	yes	direct product factor
69	C₄×#322	—	73	1	no	maximal, Sylow
70	C₂×C₇₃	C₂×#322	1	4	yes	—
71	C₂×C₇₃	C₄²×C₈	1	1	yes	characteristic
72	C₂×C₇₃	C₄×#97	1	2	yes	—
73	C₄²×C₈	—	73	1	no	—
74	C₄²×C₈	—	73	2	no	—
75	C₂×#322	—	73	1	no	—
76	C₂×#322	—	73	2	no	—
77	C₂×#322	D₇₃	1	1	yes	characteristic
78	C₇₃	C₄×#322	1	1	yes	characteristic, complemented, Sylow
79	C₂×C₄×C₈	—	73	4	no	—
80	C₂×C₄×C₈	—	73	1	no	—
81	C₂×C₄×C₈	—	73	2	no	—
82	C₂×C₄×C₈	—	73	8	no	—
83	C₂×C₄×C₈	C₂×D₇₃	1	2	yes	—
84	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
85	C₄³	—	73	1	no	—
86	#322	—	73	4	no	—
87	#322	—	73	8	no	—
88	#322	Dic₇₃	1	4	yes	direct product factor
89	C₂²×C₈	—	73	2	no	—
90	C₂²×C₈	—	73	4	no	—
91	C₂²×C₈	C₄×D₇₃	1	4	yes	—
92	C₂²×C₈	C₄×D₇₃	1	2	yes	—
93	C₂×C₄²	—	73	2	no	—
94	C₂×C₄²	—	73	4	no	—
95	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
96	C₄×C₈	—	73	16	no	—
97	C₄×C₈	—	73	4	no	—
98	C₄×C₈	—	73	4	no	—
99	C₄×C₈	—	73	4	no	—
100	C₄×C₈	—	146	16	no	—
101	C₄×C₈	C₄×D₇₃	1	4	yes	—
102	C₄×C₈	C₄×D₇₃	1	2	yes	—
103	C₄×C₈	C₂×Dic₇₃	1	4	yes	—
104	C₄×C₈	C₂×Dic₇₃	1	2	yes	—
105	C₂²×C₄	—	73	4	no	—
106	C₂²×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
107	C₂²×C₄	C₂×C₄×D₇₃	1	2	yes	—
108	C₂×C₈	—	2	8	no	—
109	C₂×C₈	—	73	4	no	—
110	C₂×C₈	—	73	8	no	—
111	C₂×C₈	—	73	8	no	—
112	C₂×C₈	—	146	8	no	—
113	C₂×C₈	C₄×Dic₇₃	1	8	yes	—
114	C₂×C₈	C₄×Dic₇₃	1	4	yes	—
115	C₂×C₈	C₈×D₇₃	1	8	yes	—
116	C₄²	—	73	8	no	—
117	C₄²	—	73	16	no	—
118	C₄²	C₂×C₄×D₇₃	1	2	yes	—
119	C₄²	C₂×C₄×D₇₃	1	1	yes	characteristic
120	C₄²	C₂²×Dic₇₃	1	1	yes	characteristic
121	C₂³	C₄²×D₇₃	1	1	yes	characteristic
122	C₂×C₄	—	73	8	no	—
123	C₂×C₄	—	73	16	no	—
124	C₂×C₄	C₄²×D₇₃	1	1	yes	characteristic
125	C₂×C₄	C₄²×D₇₃	1	2	yes	—
126	C₂×C₄	C₂×C₄×Dic₇₃	1	1	yes	characteristic
127	C₂×C₄	C₂×C₄×Dic₇₃	1	2	yes	—
128	C₂×C₄	C₂×C₈×D₇₃	1	4	yes	—
129	C₂×C₄	M₁₆×D₇₃	1	8	yes	—
130	C₈	—	2	8	no	—
131	C₈	—	73	8	no	—
132	C₈	—	146	8	no	—
133	C₈	C₈×Dic₇₃	1	8	yes	complemented
134	C₂²	C₄²×Dic₇₃	1	1	yes	characteristic
135	C₂²	C₄×C₈×D₇₃	1	2	yes	—
136	C₂²	#97×D₇₃	1	4	yes	—
137	C₄	—	73	16	no	—
138	C₄	C₄×C₈×D₇₃	1	2	yes	—
139	C₄	C₂×C₈×Dic₇₃	1	2	yes	—
140	C₄	M₁₆×Dic₇₃	1	4	yes	—
141	C₄	#17988539	1	4	yes	—
142	C₂	#322×D₇₃	1	4	yes	—
143	C₂	C₄×C₈×Dic₇₃	1	1	yes	characteristic
144	C₂	#97×Dic₇₃	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄²	C₂²×D₇₃
commutator	C₂×C₇₃	C₄²×C₈
Fitting	C₂×#322×C₇₃	C₂
socle, prime radical	C₂³×C₇₃	C₂×C₄²
cosocle	C₂×C₄²×C₇₃	C₂³
hypercenter	C₂×#322	D₇₃
hypocenter	C₇₃	C₄×#322
LCS	G ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄² ◁ C₂×#322

Complements

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

N	H ≅ G/N	type
11. #17988309	C₄	semidirect
12. #17988308	C₄	semidirect
15. #322×C₇₃	C₄	semidirect
24. C₈×Dic₇₃	C₈	semidirect
58. C₈×C₇₃	C₄×C₈	semidirect
60. #10596831	C₄×C₈	semidirect
68. Dic₇₃	#322	direct (N × H)
78. C₇₃	C₄×#322	semidirect
88. #322	Dic₇₃	direct (N × H)
133. C₈	C₈×Dic₇₃	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₈×Dic₇₃	2
C₂×#17988309	2
C₂×#17988308	2
C₂×#322×C₇₃	2
C₄×#322	73

Subgroups — #322×Dic73 #32975331

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #322×Dic₇₃ #32975331