Subgroups — #33000809

Order 18688.

478 conjugacy classes of subgroups.

The subgroup lattice has 164 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17988362	C₂	1	2	yes	maximal
2	C₂×C₄×#11924003	C₂	1	1	yes	maximal, characteristic
3	C₂×C₁₆×Dic₇₃	C₂	1	2	yes	maximal
4	#22609617	C₂	1	1	yes	maximal, characteristic
5	C₂×#346×C₇₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11924003	C₂²	1	1	yes	characteristic
7	C₂²×#11924003	C₂²	1	1	yes	characteristic
8	C₂×#14556502	C₄	1	1	yes	characteristic
9	C₂×#14556501	C₄	1	2	yes	—
10	C₂×C₈×Dic₇₃	C₂²	1	2	yes	—
11	#17988433	C₄	1	1	yes	characteristic
12	#17988362	C₄	1	8	yes	complemented
13	C₄×#11924003	C₄	1	4	yes	complemented
14	C₁₆×Dic₇₃	C₄	1	8	yes	complemented
15	C₂²×C₁₆×C₇₃	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
17	#346×C₇₃	C₄	1	4	yes	complemented
18	C₂²×#10596831	C₂×C₄	1	2	yes	—
19	C₂×C₄×Dic₇₃	C₂×C₄	1	2	yes	—
20	C₂×#11924003	C₂×C₄	1	4	yes	—
21	C₂×#11924003	C₂×C₄	1	2	yes	—
22	C₂×#11924003	C₂×C₄	1	2	yes	—
23	C₂×#11924003	C₂×C₄	1	4	yes	—
24	#14556502	C₈	1	4	yes	—
25	#14556501	C₈	1	8	yes	—
26	C₈×Dic₇₃	C₂×C₄	1	4	yes	—
27	C₈×Dic₇₃	C₂×C₄	1	4	yes	—
28	#14556498	C₈	1	4	yes	—
29	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
31	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
32	C₂×C₁₆×C₇₃	C₂×C₄	1	4	yes	—
33	C₂×C₁₆×C₇₃	C₂×C₄	1	4	yes	—
34	C₂×C₁₆×C₇₃	C₂×C₄	1	4	yes	—
35	C₄×C₈×C₇₃	C₂×C₄	1	2	yes	—
36	C₄×C₈×C₇₃	C₂×C₄	1	2	yes	—
37	C₂²×Dic₇₃	C₂×C₈	1	4	yes	—
38	C₂×#10596831	C₄²	1	4	yes	—
39	C₂×#10596831	C₂×C₈	1	4	yes	—
40	C₂×#10596831	C₂×C₈	1	4	yes	—
41	C₄×Dic₇₃	C₄²	1	4	yes	—
42	C₄×Dic₇₃	C₂×C₈	1	4	yes	—
43	#11924003	C₄²	1	4	yes	—
44	#11924003	C₄²	1	4	yes	—
45	C₂²×C₄×C₇₃	C₂×C₈	1	2	yes	—
46	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
47	C₂×C₈×C₇₃	C₄²	1	2	yes	—
48	C₂×C₈×C₇₃	C₂×C₈	1	4	yes	—
49	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
50	C₂×C₈×C₇₃	C₂²×C₄	1	2	yes	—
51	C₂×C₈×C₇₃	C₂²×C₄	1	2	yes	—
52	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
53	C₄²×C₇₃	C₄²	1	2	yes	—
54	C₄²×C₇₃	C₂×C₈	1	2	yes	—
55	C₁₆×C₇₃	C₄²	1	8	yes	—
56	C₂³×C₇₃	C₂²×C₈	1	1	yes	characteristic
57	C₂×C₄×C₇₃	—	2	4	no	—
58	C₂×C₄×C₇₃	C₄×C₈	1	4	yes	—
59	C₂×C₄×C₇₃	C₂×C₄²	1	2	yes	—
60	C₂×C₄×C₇₃	C₂×C₄²	1	1	yes	characteristic
61	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
62	C₂×C₄×C₇₃	C₂²×C₈	1	2	yes	—
63	C₂×Dic₇₃	—	2	8	no	—
64	C₂×Dic₇₃	C₄×C₈	1	8	yes	—
65	C₈×C₇₃	C₄×C₈	1	4	yes	—
66	C₈×C₇₃	C₂×C₄²	1	1	yes	characteristic
67	C₈×C₇₃	C₂×C₄²	1	1	yes	characteristic
68	C₈×C₇₃	C₂×C₄²	1	2	yes	—
69	#10596831	C₄×C₈	1	8	yes	—
70	C₂²×C₇₃	C₂×C₄×C₈	1	2	yes	—
71	C₂²×C₇₃	C₂×C₄×C₈	1	1	yes	characteristic
72	C₂²×C₇₃	#504	1	4	yes	—
73	C₄×C₇₃	—	2	4	no	—
74	C₄×C₇₃	C₄³	1	1	yes	characteristic
75	C₄×C₇₃	C₂×C₄×C₈	1	2	yes	—
76	C₄×C₇₃	C₂×C₄×C₈	1	1	yes	characteristic
77	Dic₇₃	—	2	8	no	—
78	#12539	—	73	1	no	maximal, Sylow
79	C₂×C₇₃	C₄²×C₈	1	1	yes	characteristic
80	C₂×C₇₃	#2108	1	4	yes	—
81	C₂×C₇₃	#2108	1	2	yes	—
82	C₂×#346	—	73	2	no	—
83	C₂×#346	D₇₃	1	1	yes	characteristic
84	C₂×C₄×C₁₆	—	73	2	no	—
85	C₂×C₄×C₁₆	—	73	1	no	—
86	#1726	—	73	1	no	—
87	C₇₃	#12539	1	1	yes	characteristic, complemented, Sylow
88	C₂²×C₁₆	—	73	1	no	—
89	C₂²×C₁₆	—	73	1	no	—
90	C₂²×C₁₆	C₂×D₇₃	1	2	yes	—
91	C₂×#97	—	73	2	no	—
92	C₂×#97	—	73	1	no	—
93	C₂×C₄×C₈	—	73	2	no	—
94	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
95	#376	—	73	1	no	—
96	#346	—	73	8	no	—
97	#346	Dic₇₃	1	4	yes	complemented
98	C₄×C₁₆	—	73	8	no	—
99	C₄×C₁₆	—	73	4	no	—
100	C₂²×C₈	—	73	2	no	—
101	C₂²×C₈	C₄×D₇₃	1	1	yes	characteristic
102	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
103	C₂×C₄²	—	73	2	no	—
104	C₂×C₄²	C₄×D₇₃	1	1	yes	characteristic
105	C₂×C₁₆	—	73	2	no	—
106	C₂×C₁₆	—	73	2	no	—
107	C₂×C₁₆	—	73	4	no	—
108	C₂×C₁₆	—	73	4	no	—
109	C₂×C₁₆	C₄×D₇₃	1	4	yes	—
110	C₂×C₁₆	C₄×D₇₃	1	4	yes	—
111	C₂×C₁₆	C₂×Dic₇₃	1	4	yes	—
112	#97	—	73	8	no	—
113	#97	—	73	4	no	—
114	C₄×C₈	—	73	4	no	—
115	C₄×C₈	—	73	4	no	—
116	C₄×C₈	C₄×D₇₃	1	2	yes	—
117	C₄×C₈	C₂×Dic₇₃	1	2	yes	—
118	#95	—	73	4	no	—
119	C₂²×C₄	—	73	4	no	—
120	C₂²×C₄	C₈×D₇₃	1	2	yes	—
121	C₂²×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
122	C₂×C₈	—	73	4	no	—
123	C₂×C₈	—	73	4	no	—
124	C₂×C₈	—	73	4	no	—
125	C₂×C₈	C₄×Dic₇₃	1	2	yes	—
126	C₂×C₈	C₈×D₇₃	1	4	yes	—
127	C₂×C₈	C₂×C₄×D₇₃	1	1	yes	characteristic
128	C₂×C₈	C₂×C₄×D₇₃	1	2	yes	—
129	C₂×C₈	C₂×C₄×D₇₃	1	2	yes	—
130	C₂×C₈	C₂²×Dic₇₃	1	1	yes	characteristic
131	C₄²	—	73	4	no	—
132	C₄²	—	73	4	no	—
133	C₄²	C₄×Dic₇₃	1	2	yes	—
134	C₄²	C₈×D₇₃	1	2	yes	—
135	C₁₆	—	73	4	no	—
136	C₁₆	—	73	4	no	—
137	C₁₆	C₄×Dic₇₃	1	8	yes	—
138	C₂³	C₂×C₈×D₇₃	1	1	yes	characteristic
139	C₂×C₄	—	2	4	no	—
140	C₂×C₄	—	73	8	no	—
141	C₂×C₄	—	146	8	no	—
142	C₂×C₄	C₈×Dic₇₃	1	4	yes	—
143	C₂×C₄	C₄²×D₇₃	1	2	yes	—
144	C₂×C₄	C₂×C₄×Dic₇₃	1	1	yes	characteristic
145	C₂×C₄	C₂×C₈×D₇₃	1	1	yes	characteristic
146	C₂×C₄	C₂×C₈×D₇₃	1	2	yes	—
147	C₈	—	73	8	no	—
148	C₈	C₈×Dic₇₃	1	4	yes	—
149	C₈	C₄²×D₇₃	1	1	yes	characteristic
150	C₈	C₄²×D₇₃	1	1	yes	characteristic
151	C₈	C₂×C₄×Dic₇₃	1	2	yes	—
152	C₂²	C₄×C₈×D₇₃	1	2	yes	—
153	C₂²	C₂×C₈×Dic₇₃	1	1	yes	characteristic
154	C₂²	#17989005	1	4	yes	—
155	C₄	—	2	4	no	—
156	C₄	—	146	8	no	—
157	C₄	C₄²×Dic₇₃	1	1	yes	characteristic
158	C₄	C₄×C₈×D₇₃	1	2	yes	—
159	C₄	C₂×C₈×Dic₇₃	1	1	yes	characteristic
160	C₂	C₄×C₈×Dic₇₃	1	1	yes	characteristic
161	C₂	#22611529	1	4	yes	—
162	C₂	#22611530	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₇₃
commutator	C₂×C₇₃	C₄²×C₈
Fitting	C₂×#346×C₇₃	C₂
socle, prime radical	C₂³×C₇₃	C₂²×C₈
cosocle	C₂²×C₈×C₇₃	C₂³
hypercenter	C₂×#346	D₇₃
hypocenter	C₇₃	#12539
LCS	G ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×#346

Complements

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

N	H ≅ G/N	type
12. #17988362	C₄	semidirect
13. C₄×#11924003	C₄	semidirect
14. C₁₆×Dic₇₃	C₄	semidirect
17. #346×C₇₃	C₄	semidirect
87. C₇₃	#12539	semidirect
97. #346	Dic₇₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×#17988362	2
C₂×C₄×#11924003	2
C₂×C₁₆×Dic₇₃	2
#22609617	2
C₂×#346×C₇₃	2
#12539	73