Subgroups — #2241×D₇₃ #33241270

Order 18688.

622 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#500×D₇₃	C₂	1	1	yes	maximal, characteristic
2	#494×D₇₃	C₂	1	1	yes	maximal, characteristic
3	#22612160	C₂	1	1	yes	maximal, characteristic, complemented
4	#22612159	C₂	1	1	yes	maximal, characteristic, complemented
5	#22612158	C₂	1	1	yes	maximal, characteristic, complemented
6	#22612157	C₂	1	4	yes	maximal, complemented
7	#358×D₇₃	C₂	1	4	yes	maximal
8	C₄×C₁₆×D₇₃	C₂	1	1	yes	maximal, characteristic
9	C₇₃×#2241	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×Q₁₆×D₇₃	—	2	1	no	—
11	C₂×Q₁₆×D₇₃	C₂²	1	2	yes	—
12	C₄⋊Q₈×D₇₃	—	2	1	no	—
13	C₄⋊Q₈×D₇₃	—	2	1	no	—
14	C₂×C₁₆×D₇₃	C₂²	1	2	yes	—
15	#17988992	C₂²	1	1	yes	characteristic
16	#17988991	—	2	1	no	—
17	#17988991	C₂²	1	2	yes	—
18	#17988990	—	2	1	no	—
19	#17988967	C₂²	1	1	yes	characteristic
20	#17988966	—	2	1	no	—
21	#17988965	—	2	1	no	—
22	#17988965	C₂²	1	2	yes	—
23	#107×D₇₃	—	2	1	no	—
24	#107×D₇₃	C₂²	1	2	yes	—
25	C₄×C₈×D₇₃	C₂²	1	1	yes	characteristic
26	#17988392	C₂²	1	4	yes	—
27	#17988391	C₂²	1	4	yes	—
28	#17988390	C₂²	1	4	yes	—
29	C₄×#11924003	C₂²	1	1	yes	characteristic
30	C₁₆×Dic₇₃	C₂²	1	2	yes	—
31	#500×C₇₃	C₂²	1	1	yes	characteristic
32	#494×C₇₃	C₂²	1	1	yes	characteristic
33	#358×C₇₃	C₂²	1	4	yes	—
34	C₄×C₁₆×C₇₃	C₂²	1	1	yes	characteristic
35	C₂×Q₈×D₇₃	—	2	2	no	—
36	C₂×Q₈×D₇₃	—	4	1	no	—
37	C₂×Dic₂₉₂	—	2	1	no	—
38	C₂×Dic₂₉₂	C₂³	1	2	yes	—
39	C₂×#11924018	—	2	2	no	—
40	C₂×#11924018	—	2	2	no	—
41	Q₁₆×D₇₃	—	2	8	no	—
42	Q₁₆×D₇₃	—	4	4	no	—
43	C₂×C₈×D₇₃	D₄	1	1	yes	characteristic
44	C₂×C₈×D₇₃	C₂³	1	1	yes	characteristic
45	#14556591	—	2	1	no	—
46	#14556591	—	2	1	no	—
47	#14556590	—	2	2	no	—
48	#14556590	—	4	1	no	—
49	#14556590	—	4	1	no	—
50	#14556590	—	4	1	no	—
51	#14556589	—	2	2	no	—
52	#14556589	—	4	1	no	—
53	C₄⋊C₄×D₇₃	—	2	2	no	—
54	C₄⋊C₄×D₇₃	—	4	1	no	—
55	C₄⋊C₄×D₇₃	—	4	1	no	—
56	C₄⋊C₄×D₇₃	—	4	1	no	—
57	C₄²×D₇₃	D₄	1	1	yes	characteristic
58	C₂×#11924003	C₂³	1	2	yes	—
59	C₁₆×D₇₃	—	2	4	no	—
60	#14556525	—	2	1	no	—
61	#14556525	C₂³	1	2	yes	—
62	#14556524	—	2	2	no	—
63	#14556524	—	2	2	no	—
64	C₄×#10596831	D₄	1	1	yes	characteristic
65	C₈×Dic₇₃	D₄	1	1	yes	characteristic
66	C₈×Dic₇₃	C₂³	1	1	yes	characteristic
67	C₂×Q₁₆×C₇₃	—	2	1	no	—
68	C₂×Q₁₆×C₇₃	C₂³	1	2	yes	—
69	C₄⋊Q₈×C₇₃	—	2	1	no	—
70	C₄⋊Q₈×C₇₃	—	2	1	no	—
71	C₂×C₁₆×C₇₃	C₂³	1	2	yes	—
72	#107×C₇₃	—	2	1	no	—
73	#107×C₇₃	C₂³	1	2	yes	—
74	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
75	C₂×Dic₁₄₆	—	2	2	no	—
76	C₂×Dic₁₄₆	—	4	1	no	—
77	C₂×Dic₁₄₆	—	4	2	no	—
78	C₂×Dic₁₄₆	—	4	2	no	—
79	Q₈×D₇₃	—	4	4	no	—
80	Q₈×D₇₃	—	4	4	no	—
81	Q₈×D₇₃	—	8	4	no	—
82	C₂×C₄×D₇₃	—	4	2	no	—
83	C₂×C₄×D₇₃	—	4	2	no	—
84	C₂×C₄×D₇₃	D₈	1	1	yes	characteristic
85	C₂×C₄×D₇₃	D₈	1	1	yes	characteristic
86	C₂×C₄×D₇₃	C₂×D₄	1	1	yes	characteristic
87	Dic₂₉₂	—	2	4	no	—
88	Dic₂₉₂	—	4	2	no	—
89	#11924018	—	4	4	no	—
90	#11924018	—	4	4	no	—
91	C₂×#10596831	C₂×D₄	1	1	yes	characteristic
92	C₂×#10596831	C₂×D₄	1	1	yes	characteristic
93	C₈×D₇₃	SD₁₆	1	4	yes	—
94	C₈×D₇₃	#41	1	4	yes	—
95	C₄⋊Dic₇₃	—	2	2	no	—
96	C₄⋊Dic₇₃	—	4	1	no	—
97	C₄⋊Dic₇₃	—	4	1	no	—
98	C₄⋊Dic₇₃	—	4	1	no	—
99	#11924007	—	4	2	no	—
100	#11924007	—	4	2	no	—
101	#11924007	—	4	2	no	—
102	#11924007	—	4	2	no	—
103	C₄×Dic₇₃	D₈	1	1	yes	characteristic
104	C₄×Dic₇₃	D₈	1	1	yes	characteristic
105	C₄×Dic₇₃	C₂×D₄	1	1	yes	characteristic
106	#11924003	—	2	2	no	—
107	C₂×Q₈×C₇₃	—	2	2	no	—
108	C₂×Q₈×C₇₃	—	4	1	no	—
109	Q₁₆×C₇₃	—	2	4	no	—
110	Q₁₆×C₇₃	—	4	2	no	—
111	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
112	C₂×C₈×C₇₃	C₂⁴	1	1	yes	characteristic
113	C₄⋊C₄×C₇₃	—	2	2	no	—
114	C₄⋊C₄×C₇₃	—	4	1	no	—
115	C₄⋊C₄×C₇₃	—	4	1	no	—
116	C₄⋊C₄×C₇₃	—	4	1	no	—
117	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
118	C₁₆×C₇₃	—	2	2	no	—
119	C₂²×D₇₃	C₂×D₈	1	1	yes	characteristic
120	Q₈×C₇₃	—	4	2	no	—
121	Q₈×C₇₃	—	4	2	no	—
122	Q₈×C₇₃	—	8	2	no	—
123	C₂×C₄×C₇₃	—	4	2	no	—
124	C₂×C₄×C₇₃	—	4	2	no	—
125	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
126	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
127	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
128	C₂×Dic₇₃	—	4	2	no	—
129	C₂×Dic₇₃	—	4	2	no	—
130	C₂×Dic₇₃	C₂×D₈	1	1	yes	characteristic
131	C₂×Dic₇₃	C₂×D₈	1	1	yes	characteristic
132	C₂×Dic₇₃	C₂×D₈	1	1	yes	characteristic
133	C₄×D₇₃	—	8	4	no	—
134	C₄×D₇₃	—	8	4	no	—
135	C₄×D₇₃	SD₃₂	1	4	yes	—
136	C₄×D₇₃	Q₃₂	1	4	yes	—
137	C₄×D₇₃	#124	1	2	yes	—
138	C₄×D₇₃	C₂×SD₁₆	1	2	yes	—
139	Dic₁₄₆	—	4	2	no	—
140	Dic₁₄₆	—	4	2	no	—
141	Dic₁₄₆	—	8	2	no	—
142	Dic₁₄₆	—	8	4	no	—
143	Dic₁₄₆	—	8	4	no	—
144	C₈×C₇₃	C₂×SD₁₆	1	2	yes	—
145	C₈×C₇₃	C₂×#41	1	2	yes	—
146	#10596831	#124	1	2	yes	—
147	#10596831	C₂×SD₁₆	1	2	yes	—
148	C₂²×C₇₃	C₂²×D₈	1	1	yes	characteristic
149	C₂×D₇₃	#486	1	2	yes	—
150	C₂×D₇₃	C₂×SD₃₂	1	2	yes	—
151	C₂×D₇₃	C₂×Q₃₂	1	2	yes	—
152	C₄×C₇₃	—	8	2	no	—
153	C₄×C₇₃	—	8	2	no	—
154	C₄×C₇₃	C₂×SD₃₂	1	2	yes	—
155	C₄×C₇₃	C₂×Q₃₂	1	2	yes	—
156	C₄×C₇₃	C₂×#124	1	1	yes	characteristic
157	C₄×C₇₃	C₂²×SD₁₆	1	1	yes	characteristic
158	Dic₇₃	—	8	2	no	—
159	Dic₇₃	—	8	2	no	—
160	Dic₇₃	#486	1	2	yes	—
161	Dic₇₃	C₂×SD₃₂	1	2	yes	—
162	Dic₇₃	C₂×Q₃₂	1	2	yes	—
163	C₂×#2241	—	73	1	no	maximal, Sylow
164	C₂×C₇₃	C₂×#486	1	1	yes	characteristic
165	C₂×C₇₃	C₂²×SD₃₂	1	1	yes	characteristic
166	C₂×C₇₃	C₂²×Q₃₂	1	1	yes	characteristic
167	D₇₃	#2241	1	4	yes	direct product factor
168	C₂×#500	—	73	1	no	—
169	C₂×#494	—	73	1	no	—
170	#2241	—	73	1	no	—
171	#2241	—	73	1	no	—
172	#2241	—	73	4	no	—
173	#2241	—	73	1	no	—
174	#2241	D₇₃	1	1	yes	characteristic, direct product factor
175	C₂×#358	—	73	4	no	—
176	C₂×C₄×C₁₆	—	73	1	no	—
177	C₇₃	C₂×#2241	1	1	yes	characteristic, complemented, Sylow
178	C₂²×Q₁₆	—	73	2	no	—
179	C₂²×Q₁₆	—	146	1	no	—
180	C₂×C₄⋊Q₈	—	146	1	no	—
181	C₂×C₄⋊Q₈	—	146	1	no	—
182	C₂²×C₁₆	—	73	2	no	—
183	#500	—	73	2	no	—
184	#500	—	73	1	no	—
185	#500	—	146	1	no	—
186	#500	—	146	1	no	—
187	#500	C₂×D₇₃	1	1	yes	characteristic
188	#494	—	73	2	no	—
189	#494	—	73	1	no	—
190	#494	—	146	1	no	—
191	#494	—	146	1	no	—
192	#494	C₂×D₇₃	1	1	yes	characteristic
193	C₂×#107	—	73	2	no	—
194	C₂×#107	—	146	1	no	—
195	C₂×C₄×C₈	—	73	1	no	—
196	#358	—	73	4	no	—
197	#358	—	73	4	no	—
198	#358	—	73	4	no	—
199	#358	C₂×D₇₃	1	4	yes	—
200	C₄×C₁₆	—	73	2	no	—
201	C₄×C₁₆	—	73	1	no	—
202	C₄×C₁₆	C₂×D₇₃	1	1	yes	characteristic
203	C₂²×Q₈	—	292	1	no	—
204	C₂²×Q₈	—	146	2	no	—
205	C₂×Q₁₆	—	2	1	no	—
206	C₂×Q₁₆	—	292	4	no	—
207	C₂×Q₁₆	—	73	2	no	—
208	C₂×Q₁₆	—	146	8	no	—
209	C₂×Q₁₆	—	146	2	no	—
210	C₂×Q₁₆	—	146	2	no	—
211	C₂×Q₁₆	—	146	1	no	—
212	C₂×Q₁₆	C₂²×D₇₃	1	2	yes	—
213	C₂²×C₈	—	73	1	no	—
214	C₂²×C₈	—	73	1	no	—
215	C₄⋊Q₈	—	2	1	no	—
216	C₄⋊Q₈	—	2	1	no	—
217	C₄⋊Q₈	—	292	1	no	—
218	C₄⋊Q₈	—	292	1	no	—
219	C₄⋊Q₈	—	292	1	no	—
220	C₄⋊Q₈	—	292	1	no	—
221	C₄⋊Q₈	—	146	2	no	—
222	C₄⋊Q₈	—	146	2	no	—
223	C₄⋊Q₈	—	146	1	no	—
224	C₄⋊Q₈	—	146	1	no	—
225	C₂×C₄⋊C₄	—	292	1	no	—
226	C₂×C₄⋊C₄	—	292	1	no	—
227	C₂×C₄⋊C₄	—	292	1	no	—
228	C₂×C₄⋊C₄	—	146	2	no	—
229	C₂×C₄²	—	73	1	no	—
230	C₂×C₁₆	—	73	2	no	—
231	C₂×C₁₆	—	146	4	no	—
232	C₂×C₁₆	C₂²×D₇₃	1	2	yes	—
233	#107	—	2	1	no	—
234	#107	—	73	2	no	—
235	#107	—	146	2	no	—
236	#107	—	146	2	no	—
237	#107	—	146	1	no	—
238	#107	C₂²×D₇₃	1	2	yes	—
239	C₄×C₈	—	73	1	no	—
240	C₄×C₈	—	73	1	no	—
241	C₄×C₈	—	73	1	no	—
242	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
243	C₂×Q₈	—	2	2	no	—
244	C₂×Q₈	—	4	1	no	—
245	C₂×Q₈	—	292	2	no	—
246	C₂×Q₈	—	292	2	no	—
247	C₂×Q₈	—	292	1	no	—
248	C₂×Q₈	—	292	4	no	—
249	C₂×Q₈	—	292	4	no	—
250	C₂×Q₈	—	584	4	no	—
251	C₂×Q₈	—	146	2	no	—
252	C₂²×C₄	—	292	2	no	—
253	C₂²×C₄	—	292	2	no	—
254	C₂²×C₄	—	73	1	no	—
255	C₂²×C₄	—	73	1	no	—
256	C₂²×C₄	—	73	1	no	—
257	Q₁₆	—	2	4	no	—
258	Q₁₆	—	4	2	no	—
259	Q₁₆	—	292	4	no	—
260	Q₁₆	—	292	4	no	—
261	Q₁₆	—	292	2	no	—
262	Q₁₆	—	146	4	no	—
263	C₂×C₈	—	73	1	no	—
264	C₂×C₈	—	73	1	no	—
265	C₂×C₈	—	73	4	no	—
266	C₂×C₈	—	73	4	no	—
267	C₂×C₈	D₄×D₇₃	1	1	yes	characteristic
268	C₂×C₈	C₂³×D₇₃	1	1	yes	characteristic
269	C₄⋊C₄	—	2	2	no	—
270	C₄⋊C₄	—	4	1	no	—
271	C₄⋊C₄	—	4	1	no	—
272	C₄⋊C₄	—	4	1	no	—
273	C₄⋊C₄	—	292	2	no	—
274	C₄⋊C₄	—	292	2	no	—
275	C₄⋊C₄	—	292	2	no	—
276	C₄⋊C₄	—	292	2	no	—
277	C₄⋊C₄	—	292	1	no	—
278	C₄⋊C₄	—	292	1	no	—
279	C₄⋊C₄	—	292	1	no	—
280	C₄⋊C₄	—	146	2	no	—
281	C₄²	—	73	1	no	—
282	C₄²	—	73	1	no	—
283	C₄²	—	73	1	no	—
284	C₄²	D₄×D₇₃	1	1	yes	characteristic
285	C₁₆	—	2	2	no	—
286	C₁₆	—	146	2	no	—
287	C₂³	—	73	1	no	—
288	Q₈	—	4	2	no	—
289	Q₈	—	4	2	no	—
290	Q₈	—	8	2	no	—
291	Q₈	—	292	2	no	—
292	Q₈	—	292	2	no	—
293	Q₈	—	584	4	no	—
294	Q₈	—	584	4	no	—
295	Q₈	—	584	2	no	—
296	C₂×C₄	—	4	2	no	—
297	C₂×C₄	—	4	2	no	—
298	C₂×C₄	—	292	2	no	—
299	C₂×C₄	—	292	2	no	—
300	C₂×C₄	—	584	4	no	—
301	C₂×C₄	—	584	4	no	—
302	C₂×C₄	—	73	1	no	—
303	C₂×C₄	—	73	1	no	—
304	C₂×C₄	—	73	1	no	—
305	C₂×C₄	—	73	4	no	—
306	C₂×C₄	—	73	4	no	—
307	C₂×C₄	—	73	2	no	—
308	C₂×C₄	—	73	2	no	—
309	C₂×C₄	D₈×D₇₃	1	1	yes	characteristic
310	C₂×C₄	D₈×D₇₃	1	1	yes	characteristic
311	C₂×C₄	C₂×D₄×D₇₃	1	1	yes	characteristic
312	C₈	—	73	2	no	—
313	C₈	—	73	2	no	—
314	C₈	SD₁₆×D₇₃	1	2	yes	—
315	C₈	#41×D₇₃	1	2	yes	—
316	C₂²	—	73	2	no	—
317	C₂²	—	73	2	no	—
318	C₂²	—	73	2	no	—
319	C₂²	C₂×D₈×D₇₃	1	1	yes	characteristic
320	C₄	—	8	2	no	—
321	C₄	—	8	2	no	—
322	C₄	—	584	2	no	—
323	C₄	—	584	2	no	—
324	C₄	—	73	2	no	—
325	C₄	—	73	2	no	—
326	C₄	—	73	2	no	—
327	C₄	SD₃₂×D₇₃	1	2	yes	—
328	C₄	Q₃₂×D₇₃	1	2	yes	—
329	C₄	#124×D₇₃	1	1	yes	characteristic
330	C₄	C₂×SD₁₆×D₇₃	1	1	yes	characteristic
331	C₂	—	73	4	no	—
332	C₂	#486×D₇₃	1	1	yes	characteristic
333	C₂	C₂×SD₃₂×D₇₃	1	1	yes	characteristic
334	C₂	C₂×Q₃₂×D₇₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₈×D₇₃
commutator, cosocle	C₂×C₈×C₇₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₇₃
Fitting	C₇₃×#2241	C₂
socle	C₂²×C₇₃	C₂²×D₈
prime radical	C₂²×D₇₃	C₂×D₈
hypercenter	#2241	D₇₃
hypocenter	C₇₃	C₂×#2241
LCS	G ▷ C₂×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2241

Complements

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

N	H ≅ G/N	type
3. #22612160	C₂	semidirect
4. #22612159	C₂	semidirect
5. #22612158	C₂	semidirect
6. #22612157	C₂	semidirect
9. C₇₃×#2241	C₂	semidirect
167. D₇₃	#2241	direct (N × H)
174. #2241	D₇₃	direct (N × H)
177. C₇₃	C₂×#2241	semidirect

Maximal subgroups

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

subgroup	index
#500×D₇₃	2
#494×D₇₃	2
#22612160	2
#22612159	2
#22612158	2
#22612157	2
#358×D₇₃	2
C₄×C₁₆×D₇₃	2
C₇₃×#2241	2
C₂×#2241	73

Subgroups — #2241×D73 #33241270

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #2241×D₇₃ #33241270