Subgroups — #2248×D₇₃ #33241374

Order 18688.

918 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×SD₃₂×D₇₃	C₂	1	4	yes	maximal, complemented
2	#494×D₇₃	C₂	1	1	yes	maximal, characteristic, complemented
3	#493×D₇₃	C₂	1	1	yes	maximal, characteristic
4	#22612188	C₂	1	1	yes	maximal, characteristic, complemented
5	#22612187	C₂	1	1	yes	maximal, characteristic, complemented
6	#22612186	C₂	1	1	yes	maximal, characteristic, complemented
7	#22612185	C₂	1	4	yes	maximal, complemented
8	C₄×C₁₆×D₇₃	C₂	1	1	yes	maximal, characteristic, complemented
9	C₇₃×#2248	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×Q₁₆×D₇₃	—	2	1	no	—
11	C₂×Q₁₆×D₇₃	C₂²	1	2	yes	—
12	C₂×D₈×D₇₃	—	2	1	no	—
13	C₂×D₈×D₇₃	C₂²	1	2	yes	—
14	C₄⋊Q₈×D₇₃	—	2	1	no	—
15	#127×D₇₃	—	2	1	no	—
16	C₂×#14556536	C₂²	1	4	yes	complemented
17	C₂×#14556535	C₂²	1	4	yes	complemented
18	C₂×#14556534	C₂²	1	4	yes	complemented
19	SD₃₂×D₇₃	—	2	16	no	—
20	C₂×C₁₆×D₇₃	C₂²	1	2	yes	—
21	#17988967	C₂²	1	1	yes	characteristic, complemented
22	#17988966	—	2	1	no	—
23	#17988965	—	2	1	no	—
24	#17988965	C₂²	1	2	yes	complemented
25	#17988964	C₂²	1	1	yes	characteristic
26	#17988963	—	2	1	no	—
27	#17988962	—	2	1	no	—
28	#17988962	C₂²	1	2	yes	—
29	C₄×C₈×D₇₃	C₂²	1	1	yes	characteristic
30	C₄×#11924003	C₂²	1	1	yes	characteristic, complemented
31	C₁₆×Dic₇₃	C₂²	1	2	yes	complemented
32	C₂×SD₃₂×C₇₃	C₂²	1	4	yes	complemented
33	#494×C₇₃	C₂²	1	1	yes	characteristic, complemented
34	#493×C₇₃	C₂²	1	1	yes	characteristic
35	C₄×C₁₆×C₇₃	C₂²	1	1	yes	characteristic, complemented
36	C₂×Q₈×D₇₃	—	2	2	no	—
37	C₂×D₄×D₇₃	—	2	2	no	—
38	C₂×D₄×D₇₃	—	4	2	no	—
39	C₂×Dic₂₉₂	—	2	1	no	—
40	C₂×Dic₂₉₂	C₂³	1	2	yes	—
41	C₂×#11924018	—	2	2	no	—
42	C₂×#11924018	—	2	2	no	—
43	Q₁₆×D₇₃	—	2	8	no	—
44	Q₁₆×D₇₃	—	4	4	no	—
45	C₂×D₅₈₄	—	2	1	no	—
46	C₂×D₅₈₄	C₂³	1	2	yes	—
47	C₂×#11924013	—	2	2	no	—
48	C₂×#11924013	—	2	2	no	—
49	D₈×D₇₃	—	2	8	no	—
50	D₈×D₇₃	—	4	4	no	—
51	C₂×C₈×D₇₃	D₄	1	1	yes	characteristic
52	C₂×C₈×D₇₃	C₂³	1	1	yes	characteristic
53	#14556591	—	2	1	no	—
54	#14556590	—	4	2	no	—
55	#14556589	—	2	2	no	—
56	#14556588	—	2	1	no	—
57	#14556587	—	2	2	no	—
58	#14556587	—	4	2	no	—
59	C₄⋊C₄×D₇₃	—	4	2	no	—
60	C₄²×D₇₃	D₄	1	1	yes	characteristic
61	#14556536	—	2	8	no	—
62	#14556535	—	2	8	no	—
63	#14556534	—	2	8	no	—
64	C₂×#11924003	C₂³	1	2	yes	—
65	C₁₆×D₇₃	D₄	1	8	yes	complemented
66	C₄×#10596831	D₄	1	1	yes	characteristic
67	C₈×Dic₇₃	D₄	1	1	yes	characteristic
68	C₈×Dic₇₃	C₂³	1	1	yes	characteristic
69	C₂×Q₁₆×C₇₃	—	2	1	no	—
70	C₂×Q₁₆×C₇₃	C₂³	1	2	yes	—
71	C₂×D₈×C₇₃	—	2	1	no	—
72	C₂×D₈×C₇₃	C₂³	1	2	yes	—
73	C₄⋊Q₈×C₇₃	—	2	1	no	—
74	#127×C₇₃	—	2	1	no	—
75	SD₃₂×C₇₃	—	2	8	no	—
76	C₂×C₁₆×C₇₃	C₂³	1	2	yes	—
77	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
78	C₂³×D₇₃	—	4	2	no	—
79	C₂×Dic₁₄₆	—	2	2	no	—
80	C₂×Dic₁₄₆	—	4	2	no	—
81	Q₈×D₇₃	—	4	4	no	—
82	Q₈×D₇₃	—	4	4	no	—
83	C₂×D₂₉₂	—	2	2	no	—
84	C₂×D₂₉₂	—	4	2	no	—
85	C₂×C₂²⋊D₇₃	—	4	4	no	—
86	C₂×C₂²⋊D₇₃	—	4	2	no	—
87	D₄×D₇₃	—	4	4	no	—
88	D₄×D₇₃	—	4	4	no	—
89	D₄×D₇₃	—	8	8	no	—
90	C₂×C₄×D₇₃	—	4	2	no	—
91	C₂×C₄×D₇₃	D₈	1	2	yes	—
92	C₂×C₄×D₇₃	C₂×D₄	1	1	yes	characteristic
93	Dic₂₉₂	—	2	4	no	—
94	Dic₂₉₂	—	4	2	no	—
95	#11924018	—	4	4	no	—
96	#11924018	—	4	4	no	—
97	D₅₈₄	—	2	4	no	—
98	D₅₈₄	—	4	2	no	—
99	#11924013	—	4	4	no	—
100	#11924013	—	4	4	no	—
101	C₂×#10596831	C₂×D₄	1	1	yes	characteristic
102	C₂×#10596831	C₂×D₄	1	1	yes	characteristic
103	C₈×D₇₃	D₈	1	4	yes	—
104	C₈×D₇₃	C₂×D₄	1	4	yes	—
105	C₄⋊Dic₇₃	—	4	2	no	—
106	#11924007	—	4	4	no	—
107	C₄×Dic₇₃	D₈	1	2	yes	—
108	C₄×Dic₇₃	C₂×D₄	1	1	yes	characteristic
109	#11924003	C₂×D₄	1	4	yes	complemented
110	C₂×Q₈×C₇₃	—	2	2	no	—
111	C₂×D₄×C₇₃	—	2	2	no	—
112	C₂×D₄×C₇₃	—	4	2	no	—
113	Q₁₆×C₇₃	—	2	4	no	—
114	Q₁₆×C₇₃	—	4	2	no	—
115	D₈×C₇₃	—	2	4	no	—
116	D₈×C₇₃	—	4	2	no	—
117	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
118	C₂×C₈×C₇₃	C₂⁴	1	1	yes	characteristic
119	C₄⋊C₄×C₇₃	—	4	2	no	—
120	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
121	C₁₆×C₇₃	C₂×D₄	1	4	yes	complemented
122	C₂³×C₇₃	—	4	2	no	—
123	C₂²×D₇₃	—	4	2	no	—
124	C₂²×D₇₃	—	8	4	no	—
125	C₂²×D₇₃	—	8	8	no	—
126	C₂²×D₇₃	C₂×D₈	1	1	yes	characteristic
127	Q₈×C₇₃	—	4	2	no	—
128	Q₈×C₇₃	—	4	2	no	—
129	D₄×C₇₃	—	4	2	no	—
130	D₄×C₇₃	—	4	2	no	—
131	D₄×C₇₃	—	8	4	no	—
132	C₂×C₄×C₇₃	—	4	2	no	—
133	C₂×C₄×C₇₃	C₂×D₈	1	2	yes	—
134	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
135	C₂²⋊D₇₃	—	8	4	no	—
136	C₂²⋊D₇₃	—	8	8	no	—
137	C₂×Dic₇₃	—	4	2	no	—
138	C₂×Dic₇₃	C₂×D₈	1	1	yes	characteristic
139	C₂×Dic₇₃	C₂×D₈	1	2	yes	—
140	D₂₉₂	—	4	2	no	—
141	D₂₉₂	—	4	2	no	—
142	D₂₉₂	—	8	4	no	—
143	C₄×D₇₃	—	8	4	no	—
144	C₄×D₇₃	SD₃₂	1	8	yes	complemented
145	C₄×D₇₃	#127	1	2	yes	—
146	C₄×D₇₃	C₂×D₈	1	2	yes	—
147	Dic₁₄₆	—	4	2	no	—
148	Dic₁₄₆	—	4	2	no	—
149	Dic₁₄₆	—	8	4	no	—
150	C₈×C₇₃	C₂×D₈	1	2	yes	—
151	C₈×C₇₃	C₂²×D₄	1	2	yes	—
152	#10596831	#127	1	2	yes	—
153	#10596831	C₂×D₈	1	2	yes	—
154	C₂²×C₇₃	—	8	2	no	—
155	C₂²×C₇₃	—	8	4	no	—
156	C₂²×C₇₃	C₂²×D₈	1	1	yes	characteristic
157	C₂×D₇₃	—	8	2	no	—
158	C₂×D₇₃	—	8	4	no	—
159	C₂×D₇₃	—	16	8	no	—
160	C₂×D₇₃	#493	1	2	yes	—
161	C₂×D₇₃	C₂×SD₃₂	1	4	yes	—
162	C₄×C₇₃	—	8	2	no	—
163	C₄×C₇₃	C₂×SD₃₂	1	4	yes	complemented
164	C₄×C₇₃	C₂×#127	1	1	yes	characteristic
165	C₄×C₇₃	C₂²×D₈	1	1	yes	characteristic
166	Dic₇₃	—	8	2	no	—
167	Dic₇₃	#493	1	2	yes	—
168	Dic₇₃	C₂×SD₃₂	1	4	yes	complemented
169	C₂×#2248	—	73	1	no	maximal, Sylow
170	C₂×C₇₃	—	16	2	no	—
171	C₂×C₇₃	C₂×#493	1	1	yes	characteristic
172	C₂×C₇₃	C₂²×SD₃₂	1	2	yes	—
173	D₇₃	—	16	2	no	—
174	D₇₃	#2248	1	4	yes	direct product factor
175	C₂²×SD₃₂	—	73	4	no	—
176	C₂×#494	—	73	1	no	—
177	C₂×#493	—	73	1	no	—
178	#2248	—	73	4	no	—
179	#2248	—	73	1	no	—
180	#2248	—	73	1	no	—
181	#2248	—	73	1	no	—
182	#2248	D₇₃	1	1	yes	characteristic, direct product factor
183	C₂×C₄×C₁₆	—	73	1	no	—
184	C₇₃	C₂×#2248	1	1	yes	characteristic, complemented, Sylow
185	C₂²×Q₁₆	—	73	2	no	—
186	C₂²×Q₁₆	—	146	1	no	—
187	C₂²×D₈	—	73	2	no	—
188	C₂²×D₈	—	146	1	no	—
189	C₂×C₄⋊Q₈	—	146	1	no	—
190	C₂×#127	—	146	1	no	—
191	C₂×SD₃₂	—	73	4	no	—
192	C₂×SD₃₂	—	73	4	no	—
193	C₂×SD₃₂	—	73	4	no	—
194	C₂×SD₃₂	—	146	16	no	—
195	C₂×SD₃₂	C₂×D₇₃	1	4	yes	complemented
196	C₂²×C₁₆	—	73	2	no	—
197	#494	—	73	2	no	—
198	#494	—	73	1	no	—
199	#494	—	146	1	no	—
200	#494	—	146	1	no	—
201	#494	C₂×D₇₃	1	1	yes	characteristic, complemented
202	#493	—	73	2	no	—
203	#493	—	73	1	no	—
204	#493	—	146	1	no	—
205	#493	—	146	1	no	—
206	#493	C₂×D₇₃	1	1	yes	characteristic
207	C₂×C₄×C₈	—	73	1	no	—
208	C₄×C₁₆	—	73	2	no	—
209	C₄×C₁₆	—	73	1	no	—
210	C₄×C₁₆	C₂×D₇₃	1	1	yes	characteristic, complemented
211	C₂²×Q₈	—	146	2	no	—
212	C₂²×D₄	—	292	2	no	—
213	C₂²×D₄	—	146	2	no	—
214	C₂×Q₁₆	—	2	1	no	—
215	C₂×Q₁₆	—	292	4	no	—
216	C₂×Q₁₆	—	73	2	no	—
217	C₂×Q₁₆	—	146	8	no	—
218	C₂×Q₁₆	—	146	2	no	—
219	C₂×Q₁₆	—	146	2	no	—
220	C₂×Q₁₆	—	146	1	no	—
221	C₂×Q₁₆	C₂²×D₇₃	1	2	yes	—
222	C₂×D₈	—	2	1	no	—
223	C₂×D₈	—	292	4	no	—
224	C₂×D₈	—	73	2	no	—
225	C₂×D₈	—	146	8	no	—
226	C₂×D₈	—	146	2	no	—
227	C₂×D₈	—	146	2	no	—
228	C₂×D₈	—	146	1	no	—
229	C₂×D₈	C₂²×D₇₃	1	2	yes	—
230	C₂²×C₈	—	73	1	no	—
231	C₂²×C₈	—	73	1	no	—
232	C₄⋊Q₈	—	2	1	no	—
233	C₄⋊Q₈	—	292	2	no	—
234	C₄⋊Q₈	—	146	2	no	—
235	C₄⋊Q₈	—	146	1	no	—
236	#127	—	2	1	no	—
237	#127	—	292	2	no	—
238	#127	—	146	2	no	—
239	#127	—	146	1	no	—
240	C₂×C₄⋊C₄	—	292	2	no	—
241	C₂×C₄²	—	73	1	no	—
242	SD₃₂	—	2	8	no	—
243	SD₃₂	—	146	8	no	—
244	SD₃₂	—	146	8	no	—
245	SD₃₂	—	146	8	no	—
246	C₂×C₁₆	—	73	2	no	—
247	C₂×C₁₆	—	73	8	no	—
248	C₂×C₁₆	C₂²×D₇₃	1	2	yes	—
249	C₄×C₈	—	73	1	no	—
250	C₄×C₈	—	73	1	no	—
251	C₄×C₈	—	73	1	no	—
252	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
253	C₂⁴	—	292	2	no	—
254	C₂×Q₈	—	2	2	no	—
255	C₂×Q₈	—	292	2	no	—
256	C₂×Q₈	—	292	4	no	—
257	C₂×Q₈	—	292	4	no	—
258	C₂×Q₈	—	146	2	no	—
259	C₂×D₄	—	2	2	no	—
260	C₂×D₄	—	4	2	no	—
261	C₂×D₄	—	292	4	no	—
262	C₂×D₄	—	292	4	no	—
263	C₂×D₄	—	292	2	no	—
264	C₂×D₄	—	292	4	no	—
265	C₂×D₄	—	292	2	no	—
266	C₂×D₄	—	584	8	no	—
267	C₂×D₄	—	146	2	no	—
268	C₂²×C₄	—	292	2	no	—
269	C₂²×C₄	—	73	2	no	—
270	C₂²×C₄	—	73	1	no	—
271	Q₁₆	—	2	4	no	—
272	Q₁₆	—	4	2	no	—
273	Q₁₆	—	292	4	no	—
274	Q₁₆	—	292	4	no	—
275	Q₁₆	—	292	2	no	—
276	Q₁₆	—	146	4	no	—
277	D₈	—	2	4	no	—
278	D₈	—	4	2	no	—
279	D₈	—	292	4	no	—
280	D₈	—	292	4	no	—
281	D₈	—	292	2	no	—
282	D₈	—	146	4	no	—
283	C₂×C₈	—	73	1	no	—
284	C₂×C₈	—	73	1	no	—
285	C₂×C₈	—	73	4	no	—
286	C₂×C₈	—	73	4	no	—
287	C₂×C₈	D₄×D₇₃	1	1	yes	characteristic
288	C₂×C₈	C₂³×D₇₃	1	1	yes	characteristic
289	C₄⋊C₄	—	4	2	no	—
290	C₄⋊C₄	—	292	4	no	—
291	C₄⋊C₄	—	292	2	no	—
292	C₄²	—	73	2	no	—
293	C₄²	—	73	1	no	—
294	C₄²	D₄×D₇₃	1	1	yes	characteristic
295	C₁₆	—	73	4	no	—
296	C₁₆	D₄×D₇₃	1	4	yes	complemented
297	C₂³	—	4	2	no	—
298	C₂³	—	292	2	no	—
299	C₂³	—	584	4	no	—
300	C₂³	—	584	8	no	—
301	C₂³	—	73	1	no	—
302	Q₈	—	4	2	no	—
303	Q₈	—	4	2	no	—
304	Q₈	—	292	2	no	—
305	Q₈	—	292	2	no	—
306	Q₈	—	584	4	no	—
307	D₄	—	4	2	no	—
308	D₄	—	4	2	no	—
309	D₄	—	8	4	no	—
310	D₄	—	292	2	no	—
311	D₄	—	292	2	no	—
312	D₄	—	584	4	no	—
313	D₄	—	584	8	no	—
314	D₄	—	584	4	no	—
315	C₂×C₄	—	4	2	no	—
316	C₂×C₄	—	292	2	no	—
317	C₂×C₄	—	584	4	no	—
318	C₂×C₄	—	73	2	no	—
319	C₂×C₄	—	73	1	no	—
320	C₂×C₄	—	73	2	no	—
321	C₂×C₄	—	73	8	no	—
322	C₂×C₄	—	73	2	no	—
323	C₂×C₄	D₈×D₇₃	1	2	yes	—
324	C₂×C₄	C₂×D₄×D₇₃	1	1	yes	characteristic
325	C₈	—	73	2	no	—
326	C₈	—	73	2	no	—
327	C₈	D₈×D₇₃	1	2	yes	—
328	C₈	C₂×D₄×D₇₃	1	2	yes	—
329	C₂²	—	8	4	no	—
330	C₂²	—	8	2	no	—
331	C₂²	—	584	4	no	—
332	C₂²	—	584	2	no	—
333	C₂²	—	73	4	no	—
334	C₂²	—	73	2	no	—
335	C₂²	—	1168	8	no	—
336	C₂²	C₂×D₈×D₇₃	1	1	yes	characteristic
337	C₄	—	8	2	no	—
338	C₄	—	584	2	no	—
339	C₄	—	73	4	no	—
340	C₄	—	73	2	no	—
341	C₄	SD₃₂×D₇₃	1	4	yes	complemented
342	C₄	#127×D₇₃	1	1	yes	characteristic
343	C₄	C₂×D₈×D₇₃	1	1	yes	characteristic
344	C₂	—	16	2	no	—
345	C₂	—	73	4	no	—
346	C₂	—	1168	2	no	—
347	C₂	#493×D₇₃	1	1	yes	characteristic
348	C₂	C₂×SD₃₂×D₇₃	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₈×D₇₃
commutator, cosocle	C₂×C₈×C₇₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₇₃
Fitting	C₇₃×#2248	C₂
socle	C₂²×C₇₃	C₂²×D₈
prime radical	#493×D₇₃	C₂
hypercenter	#2248	D₇₃
hypocenter	C₇₃	C₂×#2248
LCS	G ▷ C₂×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2248

Complements

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

N	H ≅ G/N	type
1. C₂×SD₃₂×D₇₃	C₂	semidirect
2. #494×D₇₃	C₂	semidirect
4. #22612188	C₂	semidirect
5. #22612187	C₂	semidirect
6. #22612186	C₂	semidirect
7. #22612185	C₂	semidirect
8. C₄×C₁₆×D₇₃	C₂	semidirect
9. C₇₃×#2248	C₂	semidirect
16. C₂×#14556536	C₂²	semidirect
17. C₂×#14556535	C₂²	semidirect
18. C₂×#14556534	C₂²	semidirect
21. #17988967	C₂²	semidirect
24. #17988965	C₂²	semidirect
30. C₄×#11924003	C₂²	semidirect
31. C₁₆×Dic₇₃	C₂²	semidirect
32. C₂×SD₃₂×C₇₃	C₂²	semidirect
33. #494×C₇₃	C₂²	semidirect
35. C₄×C₁₆×C₇₃	C₂²	semidirect
65. C₁₆×D₇₃	D₄	semidirect
109. #11924003	C₂×D₄	semidirect
121. C₁₆×C₇₃	C₂×D₄	semidirect
144. C₄×D₇₃	SD₃₂	semidirect
163. C₄×C₇₃	C₂×SD₃₂	semidirect
168. Dic₇₃	C₂×SD₃₂	semidirect
174. D₇₃	#2248	direct (N × H)
182. #2248	D₇₃	direct (N × H)
184. C₇₃	C₂×#2248	semidirect
195. C₂×SD₃₂	C₂×D₇₃	semidirect
201. #494	C₂×D₇₃	semidirect
210. C₄×C₁₆	C₂×D₇₃	semidirect
296. C₁₆	D₄×D₇₃	semidirect
341. C₄	SD₃₂×D₇₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×SD₃₂×D₇₃	2
#494×D₇₃	2
#493×D₇₃	2
#22612188	2
#22612187	2
#22612186	2
#22612185	2
C₄×C₁₆×D₇₃	2
C₇₃×#2248	2
C₂×#2248	73

Subgroups — #2248×D73 #33241374

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #2248×D₇₃ #33241374