Subgroups — #32979677

Order 18688.

294 conjugacy classes of subgroups.

The subgroup lattice has 174 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22610103	C₂	1	1	yes	maximal, characteristic
2	#22610095	C₂	1	1	yes	maximal, characteristic
3	#22607790	C₂	1	4	yes	maximal
4	C₇₃×#1450	C₂	1	1	yes	maximal, characteristic
5	C₂×#14556527	—	2	1	no	—
6	C₂×#14556527	—	2	1	no	—
7	C₂×#14556525	—	2	1	no	—
8	C₂×#14556523	—	2	1	no	—
9	#17988611	C₂²	1	2	yes	—
10	#17988453	C₂²	1	2	yes	—
11	#431×C₇₃	C₂²	1	2	yes	—
12	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
13	#322×C₇₃	C₄	1	2	yes	complemented
14	C₈²×C₇₃	C₄	1	2	yes	complemented
15	C₂×#11924012	—	2	2	no	—
16	C₂×C₄⋊Dic₇₃	—	2	2	no	—
17	#14556527	—	4	2	no	—
18	#14556527	—	4	2	no	—
19	#14556525	—	4	2	no	—
20	#14556523	—	4	2	no	—
21	C₄⋊#10596831	—	2	4	no	—
22	#14556498	—	2	4	no	—
23	C₂²×C₈×C₇₃	D₄	1	1	yes	characteristic
24	C₂²×C₈×C₇₃	D₄	1	1	yes	characteristic
25	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
26	C₄⋊C₈×C₇₃	—	2	2	no	—
27	C₂²⋊C₈×C₇₃	—	2	2	no	—
28	C₄×C₈×C₇₃	C₂×C₄	1	4	yes	—
29	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
30	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
31	C₄×C₈×C₇₃	Q₈	1	1	yes	characteristic
32	C₄×C₈×C₇₃	Q₈	1	1	yes	characteristic
33	C₂²×Dic₇₃	—	4	2	no	—
34	#11924012	—	4	2	no	—
35	#11924012	—	4	2	no	—
36	C₂×#10596831	—	4	2	no	—
37	C₄⋊Dic₇₃	—	4	2	no	—
38	C₄⋊Dic₇₃	—	4	2	no	—
39	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
40	C₂²×C₄×C₇₃	#41	1	2	yes	—
41	C₂×C₈×C₇₃	—	2	4	no	—
42	C₂×C₈×C₇₃	C₄⋊C₄	1	1	yes	characteristic
43	C₂×C₈×C₇₃	C₄⋊C₄	1	1	yes	characteristic
44	C₂×C₈×C₇₃	C₄⋊C₄	1	1	yes	characteristic
45	C₂×C₈×C₇₃	C₄⋊C₄	1	1	yes	characteristic
46	C₂×C₈×C₇₃	D₈	1	2	yes	—
47	C₂×C₈×C₇₃	SD₁₆	1	2	yes	—
48	C₂×C₈×C₇₃	SD₁₆	1	2	yes	—
49	C₂×C₈×C₇₃	Q₁₆	1	2	yes	—
50	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
51	C₄²×C₇₃	C₂×Q₈	1	1	yes	characteristic
52	C₄²×C₇₃	#41	1	2	yes	—
53	C₂³×C₇₃	#124	1	1	yes	characteristic
54	C₂×C₄×C₇₃	—	4	2	no	—
55	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
56	C₂×C₄×C₇₃	#117	1	2	yes	—
57	C₂×C₄×C₇₃	#124	1	1	yes	characteristic
58	C₂×C₄×C₇₃	#125	1	2	yes	—
59	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
60	C₂×C₄×C₇₃	C₂×SD₁₆	1	1	yes	characteristic
61	C₂×C₄×C₇₃	C₂×SD₁₆	1	1	yes	characteristic
62	C₂×C₄×C₇₃	C₂×Q₁₆	1	1	yes	characteristic
63	C₂×Dic₇₃	—	8	4	no	—
64	C₂×Dic₇₃	—	8	2	no	—
65	C₈×C₇₃	—	4	4	no	—
66	C₈×C₇₃	#106	1	2	yes	—
67	C₈×C₇₃	#106	1	2	yes	—
68	C₈×C₇₃	#107	1	2	yes	—
69	C₈×C₇₃	#107	1	2	yes	—
70	#10596831	—	8	2	no	—
71	C₂²×C₇₃	—	2	2	no	—
72	C₂²×C₇₃	#382	1	1	yes	characteristic
73	C₂²×C₇₃	#486	1	1	yes	characteristic
74	C₂²×C₇₃	#487	1	1	yes	characteristic
75	C₄×C₇₃	—	2	2	no	—
76	C₄×C₇₃	—	2	2	no	—
77	C₄×C₇₃	C₂×#106	1	1	yes	characteristic
78	C₄×C₇₃	C₂×#107	1	1	yes	characteristic
79	C₄×C₇₃	#486	1	1	yes	characteristic
80	C₄×C₇₃	#487	1	1	yes	characteristic
81	Dic₇₃	—	16	2	no	—
82	#8793	—	73	1	no	maximal, Sylow
83	C₂×C₇₃	—	2	1	no	—
84	C₂×C₇₃	—	2	1	no	—
85	C₂×C₇₃	#1833	1	2	yes	—
86	C₂×C₇₃	#1834	1	1	yes	characteristic
87	#1848	—	73	1	no	—
88	#1846	—	73	1	no	—
89	#1450	D₇₃	1	1	yes	characteristic
90	#1286	—	73	4	no	—
91	C₇₃	#8793	1	1	yes	characteristic, complemented, Sylow
92	#431	C₂×D₇₃	1	2	yes	—
93	C₂×#108	—	146	1	no	—
94	C₂×#108	—	146	1	no	—
95	C₂×#107	—	146	1	no	—
96	C₂×#106	—	146	1	no	—
97	#423	—	73	2	no	—
98	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
99	#382	—	73	2	no	—
100	#322	Dic₇₃	1	2	yes	complemented
101	C₈²	Dic₇₃	1	2	yes	complemented
102	C₂×M₁₆	—	146	2	no	—
103	C₂²×C₈	D₂₉₂	1	1	yes	characteristic
104	C₂²×C₈	D₂₉₂	1	1	yes	characteristic
105	C₂×C₄⋊C₄	—	146	2	no	—
106	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
107	#108	—	292	2	no	—
108	#108	—	292	2	no	—
109	#107	—	292	2	no	—
110	#106	—	292	2	no	—
111	C₄⋊C₈	—	2	2	no	—
112	C₄⋊C₈	—	146	4	no	—
113	C₂²⋊C₈	—	2	2	no	—
114	C₄×C₈	Dic₁₄₆	1	1	yes	characteristic
115	C₄×C₈	Dic₁₄₆	1	1	yes	characteristic
116	C₄×C₈	C₂×Dic₇₃	1	4	yes	—
117	C₄×C₈	C₂×Dic₇₃	1	1	yes	characteristic
118	C₄×C₈	C₂×Dic₇₃	1	1	yes	characteristic
119	#95	—	146	4	no	—
120	C₂²×C₄	—	292	2	no	—
121	C₂²×C₄	C₂×D₂₉₂	1	1	yes	characteristic
122	C₂²×C₄	#11924028	1	2	yes	—
123	M₁₆	—	292	2	no	—
124	M₁₆	—	292	2	no	—
125	C₂×C₈	—	2	4	no	—
126	C₂×C₈	—	292	2	no	—
127	C₂×C₈	C₄⋊Dic₇₃	1	1	yes	characteristic
128	C₂×C₈	C₄⋊Dic₇₃	1	1	yes	characteristic
129	C₂×C₈	C₄⋊Dic₇₃	1	1	yes	characteristic
130	C₂×C₈	C₄⋊Dic₇₃	1	1	yes	characteristic
131	C₂×C₈	D₅₈₄	1	2	yes	—
132	C₂×C₈	#11924017	1	2	yes	—
133	C₂×C₈	#11924017	1	2	yes	—
134	C₂×C₈	Dic₂₉₂	1	2	yes	—
135	C₄⋊C₄	—	292	2	no	—
136	C₄⋊C₄	—	292	2	no	—
137	C₄²	C₂²×Dic₇₃	1	1	yes	characteristic
138	C₄²	C₂×Dic₁₄₆	1	1	yes	characteristic
139	C₄²	#11924028	1	2	yes	—
140	C₂³	#14556579	1	1	yes	characteristic
141	C₂×C₄	—	4	2	no	—
142	C₂×C₄	—	584	4	no	—
143	C₂×C₄	—	584	2	no	—
144	C₂×C₄	C₂×C₄⋊Dic₇₃	1	1	yes	characteristic
145	C₂×C₄	#14556550	1	2	yes	—
146	C₂×C₄	#14556579	1	1	yes	characteristic
147	C₂×C₄	#14556583	1	2	yes	—
148	C₂×C₄	C₂×D₅₈₄	1	1	yes	characteristic
149	C₂×C₄	C₂×#11924017	1	1	yes	characteristic
150	C₂×C₄	C₂×#11924017	1	1	yes	characteristic
151	C₂×C₄	C₂×Dic₂₉₂	1	1	yes	characteristic
152	C₈	—	4	4	no	—
153	C₈	—	584	2	no	—
154	C₈	#14556523	1	2	yes	—
155	C₈	#14556523	1	2	yes	—
156	C₈	#14556525	1	2	yes	—
157	C₈	#14556525	1	2	yes	—
158	C₂²	—	2	2	no	—
159	C₂²	#17988453	1	1	yes	characteristic
160	C₂²	#17988934	1	1	yes	characteristic
161	C₂²	#17988938	1	1	yes	characteristic
162	C₄	—	2	2	no	—
163	C₄	—	2	2	no	—
164	C₄	—	1168	2	no	—
165	C₄	C₂×#14556523	1	1	yes	characteristic
166	C₄	C₂×#14556525	1	1	yes	characteristic
167	C₄	#17988934	1	1	yes	characteristic
168	C₄	#17988938	1	1	yes	characteristic
169	C₂	—	2	1	no	—
170	C₂	—	2	1	no	—
171	C₂	#22610055	1	2	yes	—
172	C₂	#22610059	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#17988453
commutator	C₄²×C₇₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₃
Fitting	C₇₃×#1450	C₂
socle	C₂²×C₇₃	#382
prime radical	C₂³×C₇₃	#124
cosocle	C₂×C₄²×C₇₃	C₂³
hypercenter	#1450	D₇₃
hypocenter	C₇₃	#8793
LCS	G ▷ C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1450

Complements

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

N	H ≅ G/N	type
13. #322×C₇₃	C₄	semidirect
14. C₈²×C₇₃	C₄	semidirect
91. C₇₃	#8793	semidirect
100. #322	Dic₇₃	semidirect
101. C₈²	Dic₇₃	semidirect

Maximal subgroups

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

subgroup	index
#22610103	2
#22610095	2
#22607790	2
C₇₃×#1450	2
#8793	73