Subgroups — #33068522

Order 18688.

662 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22619935	C₂	1	2	yes	maximal
2	C₂×#17988666	C₂	1	1	yes	maximal, characteristic
3	#22609451	C₂	1	2	yes	maximal, complemented
4	#22609441	C₂	1	2	yes	maximal, complemented
5	#22609427	C₂	1	4	yes	maximal, complemented
6	#22608679	C₂	1	1	yes	maximal, characteristic, complemented
7	#22608588	C₂	1	2	yes	maximal, complemented
8	C₇₃×#3160	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×#11924011	—	2	1	no	—
10	C₂×C₄×D₂₉₂	C₂²	1	1	yes	characteristic
11	#17988915	—	2	2	no	—
12	#17988874	C₂²	1	4	yes	—
13	#17988789	C₂²	1	4	yes	—
14	#17988737	—	2	2	no	—
15	C₄×#11924015	—	2	4	no	—
16	#17988666	—	2	1	no	—
17	#17988666	—	2	1	no	—
18	#17988666	C₂²	1	2	yes	—
19	#17988666	C₂²	1	2	yes	—
20	#17988640	C₂²	1	2	yes	—
21	C₂×C₄⋊#10596831	C₂²	1	1	yes	characteristic
22	C₂×#14556503	—	2	1	no	—
23	#17988340	C₂²	1	2	yes	complemented
24	#17988338	C₂²	1	2	yes	complemented
25	#17988336	C₂²	1	4	yes	complemented
26	#17988333	C₂²	1	4	yes	complemented
27	#533×C₇₃	C₂²	1	2	yes	—
28	#500×C₇₃	C₂²	1	1	yes	characteristic, complemented
29	#500×C₇₃	C₂²	1	1	yes	characteristic, complemented
30	#499×C₇₃	—	2	1	no	—
31	#499×C₇₃	—	2	1	no	—
32	#498×C₇₃	C₂²	1	2	yes	complemented
33	#427×C₇₃	—	2	1	no	—
34	#427×C₇₃	—	2	1	no	—
35	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
36	C₂²×D₂₉₂	D₄	1	1	yes	characteristic
37	C₂²×C₄×D₇₃	—	2	1	no	—
38	C₂×#11924015	—	2	4	no	—
39	C₂×#11924011	—	2	2	no	—
40	C₂×#11924011	—	2	2	no	—
41	C₂×#11924011	—	4	4	no	—
42	C₂²×#10596831	—	2	1	no	—
43	C₄×D₂₉₂	—	2	1	no	—
44	C₄×D₂₉₂	—	2	1	no	—
45	C₄×D₂₉₂	D₄	1	2	yes	—
46	C₄×D₂₉₂	C₂³	1	2	yes	—
47	C₂×C₄⋊Dic₇₃	D₄	1	1	yes	characteristic
48	C₂×#11924005	—	2	1	no	—
49	#14556522	—	2	4	no	—
50	C₄⋊#10596831	—	2	1	no	—
51	C₄⋊#10596831	C₂³	1	2	yes	—
52	C₄⋊#10596831	C₂³	1	4	yes	—
53	#14556515	—	2	4	no	—
54	#14556511	—	2	4	no	—
55	C₂²⋊#10596831	—	2	2	no	—
56	#14556503	—	2	2	no	—
57	#14556503	—	2	2	no	—
58	C₄×#10596831	—	2	2	no	—
59	C₂²×C₈×C₇₃	D₄	1	1	yes	characteristic
60	C₂²×C₈×C₇₃	D₄	1	1	yes	characteristic
61	C₄⋊Q₈×C₇₃	C₂³	1	2	yes	—
62	C₄⋊Q₈×C₇₃	C₂³	1	2	yes	—
63	#125×C₇₃	—	2	2	no	—
64	C₂²⋊Q₈×C₇₃	—	2	2	no	—
65	C₂²⋊Q₈×C₇₃	—	2	2	no	—
66	C₄×Q₈×C₇₃	—	2	2	no	—
67	C₄×Q₈×C₇₃	—	2	2	no	—
68	#117×C₇₃	—	2	2	no	—
69	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
70	#107×C₇₃	—	2	1	no	—
71	#107×C₇₃	—	2	1	no	—
72	#107×C₇₃	—	2	1	no	—
73	#107×C₇₃	—	2	1	no	—
74	#106×C₇₃	—	2	2	no	—
75	#106×C₇₃	—	2	2	no	—
76	C₄×C₈×C₇₃	D₄	1	1	yes	characteristic
77	C₄×C₈×C₇₃	D₄	1	1	yes	characteristic
78	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
79	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
80	C₂³×D₇₃	—	2	1	no	—
81	C₂×D₂₉₂	—	2	1	no	—
82	C₂×D₂₉₂	—	4	2	no	—
83	C₂×D₂₉₂	C₂×D₄	1	2	yes	—
84	C₂²×Dic₇₃	—	2	1	no	—
85	C₂×C₄×D₇₃	—	2	2	no	—
86	C₂×C₄×D₇₃	—	2	2	no	—
87	C₂×C₄×D₇₃	—	4	2	no	—
88	C₂×C₄×D₇₃	—	4	2	no	—
89	#11924015	—	4	8	no	—
90	#11924011	—	4	4	no	—
91	#11924011	—	4	4	no	—
92	C₂×#10596831	—	2	4	no	—
93	C₂×#10596831	—	2	2	no	—
94	C₂×#10596831	—	4	2	no	—
95	C₄⋊Dic₇₃	—	2	1	no	—
96	C₄⋊Dic₇₃	C₂×D₄	1	2	yes	—
97	#11924005	—	2	2	no	—
98	#11924005	—	2	2	no	—
99	C₂×Q₈×C₇₃	—	2	2	no	—
100	C₂×Q₈×C₇₃	—	2	2	no	—
101	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
102	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
103	C₂²×C₄×C₇₃	#41	1	1	yes	characteristic
104	C₂×C₈×C₇₃	—	2	2	no	—
105	C₂×C₈×C₇₃	SD₁₆	1	4	yes	complemented
106	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
107	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
108	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
109	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
110	C₄⋊C₄×C₇₃	—	2	2	no	—
111	C₄⋊C₄×C₇₃	—	2	2	no	—
112	C₄⋊C₄×C₇₃	—	2	2	no	—
113	C₄⋊C₄×C₇₃	—	2	2	no	—
114	C₄⋊C₄×C₇₃	—	4	2	no	—
115	C₄⋊C₄×C₇₃	—	4	2	no	—
116	C₂²⋊C₄×C₇₃	—	4	2	no	—
117	C₄²×C₇₃	—	4	2	no	—
118	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
119	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
120	C₄²×C₇₃	#41	1	1	yes	characteristic
121	C₄²×C₇₃	C₂⁴	1	1	yes	characteristic
122	C₂³×C₇₃	#121	1	1	yes	characteristic
123	C₂²×D₇₃	—	2	2	no	—
124	C₂²×D₇₃	—	4	2	no	—
125	C₂²×D₇₃	—	4	2	no	—
126	C₂²×D₇₃	—	4	2	no	—
127	Q₈×C₇₃	—	4	4	no	—
128	Q₈×C₇₃	—	8	2	no	—
129	C₂×C₄×C₇₃	—	2	2	no	—
130	C₂×C₄×C₇₃	—	4	1	no	—
131	C₂×C₄×C₇₃	—	4	2	no	—
132	C₂×C₄×C₇₃	—	4	2	no	—
133	C₂×C₄×C₇₃	#121	1	1	yes	characteristic
134	C₂×C₄×C₇₃	#121	1	1	yes	characteristic
135	C₂×C₄×C₇₃	#121	1	1	yes	characteristic
136	C₂×C₄×C₇₃	C₂×SD₁₆	1	2	yes	—
137	C₂×C₄×C₇₃	#137	1	2	yes	—
138	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
139	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
140	C₂×C₄×C₇₃	C₂×#41	1	1	yes	characteristic
141	C₂×Dic₇₃	—	2	2	no	—
142	C₂×Dic₇₃	—	4	2	no	—
143	D₂₉₂	—	2	4	no	—
144	D₂₉₂	—	4	2	no	—
145	C₄×D₇₃	—	4	2	no	—
146	C₄×D₇₃	—	4	2	no	—
147	C₄×D₇₃	—	4	4	no	—
148	C₈×C₇₃	—	2	1	no	—
149	C₈×C₇₃	—	2	1	no	—
150	C₈×C₇₃	C₂×SD₁₆	1	2	yes	—
151	C₈×C₇₃	C₂×SD₁₆	1	2	yes	—
152	#10596831	—	4	2	no	—
153	#10596831	—	4	4	no	—
154	C₂²×C₇₃	—	2	1	no	—
155	C₂²×C₇₃	—	2	1	no	—
156	C₂²×C₇₃	#461	1	2	yes	—
157	C₂²×C₇₃	C₂×#121	1	1	yes	characteristic
158	C₂×D₇₃	—	4	2	no	—
159	C₂×D₇₃	—	4	2	no	—
160	C₂×D₇₃	—	4	2	no	—
161	C₂×D₇₃	—	8	4	no	—
162	C₄×C₇₃	—	2	1	no	—
163	C₄×C₇₃	—	2	1	no	—
164	C₄×C₇₃	—	2	1	no	—
165	C₄×C₇₃	—	8	2	no	—
166	C₄×C₇₃	—	8	2	no	—
167	C₄×C₇₃	#461	1	2	yes	—
168	C₄×C₇₃	C₂²×SD₁₆	1	1	yes	characteristic
169	C₄×C₇₃	C₂×#137	1	1	yes	characteristic
170	C₄×C₇₃	#575	1	2	yes	—
171	Dic₇₃	—	4	2	no	—
172	#19137	—	73	1	no	maximal, Sylow
173	C₂×C₇₃	—	2	2	no	—
174	C₂×C₇₃	C₂×#461	1	1	yes	characteristic
175	C₂×C₇₃	#3054	1	1	yes	characteristic
176	C₂×C₇₃	#3067	1	1	yes	characteristic
177	D₇₃	—	8	2	no	—
178	#3160	D₇₃	1	1	yes	characteristic, complemented
179	#3101	—	73	2	no	—
180	C₂×#436	—	73	1	no	—
181	#1695	—	73	2	no	—
182	#1693	—	73	2	no	—
183	#1691	—	73	4	no	—
184	#1564	—	73	1	no	—
185	#1542	—	73	2	no	—
186	C₇₃	#19137	1	1	yes	characteristic, complemented, Sylow
187	C₂²×M₁₆	—	146	1	no	—
188	#533	C₂×D₇₃	1	2	yes	—
189	C₂×C₄×D₄	—	73	1	no	—
190	#500	C₂×D₇₃	1	1	yes	characteristic, complemented
191	#500	C₂×D₇₃	1	1	yes	characteristic, complemented
192	#499	—	2	1	no	—
193	#499	—	2	1	no	—
194	#498	C₂×D₇₃	1	2	yes	complemented
195	#483	—	146	2	no	—
196	#476	—	73	4	no	—
197	#461	—	73	4	no	—
198	#450	—	146	2	no	—
199	C₄×SD₁₆	—	146	4	no	—
200	#436	—	73	2	no	—
201	#436	—	73	2	no	—
202	#436	—	146	1	no	—
203	#436	—	146	1	no	—
204	#431	—	73	2	no	—
205	#427	—	2	1	no	—
206	#427	—	2	1	no	—
207	C₂×C₄⋊C₈	—	73	1	no	—
208	C₂×C₂²⋊C₈	—	146	1	no	—
209	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
210	#335	—	73	2	no	—
211	#334	—	73	2	no	—
212	#333	—	73	4	no	—
213	#332	—	73	4	no	—
214	C₂²×D₄	—	73	1	no	—
215	C₂³×C₄	—	146	1	no	—
216	C₂×SD₁₆	—	146	4	no	—
217	C₂×M₁₆	—	292	4	no	—
218	C₂×M₁₆	—	146	2	no	—
219	C₂×M₁₆	—	146	2	no	—
220	C₂²×C₈	—	146	1	no	—
221	C₂²×C₈	C₂²⋊D₇₃	1	1	yes	characteristic
222	C₂²×C₈	C₂²⋊D₇₃	1	1	yes	characteristic
223	C₄⋊Q₈	C₂²×D₇₃	1	2	yes	—
224	C₄⋊Q₈	C₂²×D₇₃	1	2	yes	—
225	#125	—	2	2	no	—
226	C₂²⋊Q₈	—	2	2	no	—
227	C₂²⋊Q₈	—	2	2	no	—
228	C₄×Q₈	—	2	2	no	—
229	C₄×Q₈	—	2	2	no	—
230	C₄×D₄	—	73	2	no	—
231	C₄×D₄	—	73	2	no	—
232	C₄×D₄	—	146	1	no	—
233	C₄×D₄	—	146	1	no	—
234	#117	—	2	2	no	—
235	C₂×C₄⋊C₄	—	73	1	no	—
236	C₂×C₂²⋊C₄	—	146	1	no	—
237	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
238	#107	—	2	1	no	—
239	#107	—	2	1	no	—
240	#107	—	2	1	no	—
241	#107	—	2	1	no	—
242	#106	—	2	2	no	—
243	#106	—	2	2	no	—
244	#106	—	146	4	no	—
245	C₄⋊C₈	—	73	2	no	—
246	C₄⋊C₈	—	73	4	no	—
247	C₄⋊C₈	—	146	1	no	—
248	#103	—	146	4	no	—
249	#102	—	146	4	no	—
250	C₂²⋊C₈	—	146	2	no	—
251	C₂²⋊C₈	—	146	2	no	—
252	C₂²⋊C₈	—	146	2	no	—
253	C₄×C₈	—	146	2	no	—
254	C₄×C₈	C₂²⋊D₇₃	1	1	yes	characteristic
255	C₄×C₈	C₂²⋊D₇₃	1	1	yes	characteristic
256	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
257	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
258	C₂⁴	—	146	1	no	—
259	C₂×Q₈	—	2	2	no	—
260	C₂×Q₈	—	2	2	no	—
261	C₂×D₄	—	292	2	no	—
262	C₂×D₄	—	73	2	no	—
263	C₂×D₄	—	146	1	no	—
264	C₂²×C₄	—	292	2	no	—
265	C₂²×C₄	—	292	2	no	—
266	C₂²×C₄	—	146	1	no	—
267	C₂²×C₄	—	146	2	no	—
268	C₂²×C₄	—	146	2	no	—
269	C₂²×C₄	D₄×D₇₃	1	1	yes	characteristic
270	C₂²×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
271	C₂²×C₄	#11924027	1	1	yes	characteristic
272	SD₁₆	—	292	8	no	—
273	M₁₆	—	292	4	no	—
274	M₁₆	—	292	4	no	—
275	C₂×C₈	—	2	2	no	—
276	C₂×C₈	—	292	2	no	—
277	C₂×C₈	—	146	4	no	—
278	C₂×C₈	—	146	2	no	—
279	C₂×C₈	#11924015	1	4	yes	complemented
280	C₂×C₈	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
281	C₂×C₈	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
282	C₂×C₈	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
283	C₂×C₈	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
284	C₄⋊C₄	—	2	2	no	—
285	C₄⋊C₄	—	2	2	no	—
286	C₄⋊C₄	—	2	2	no	—
287	C₄⋊C₄	—	2	2	no	—
288	C₄⋊C₄	—	4	2	no	—
289	C₄⋊C₄	—	4	2	no	—
290	C₄⋊C₄	—	73	2	no	—
291	C₄⋊C₄	—	146	1	no	—
292	C₂²⋊C₄	—	4	2	no	—
293	C₂²⋊C₄	—	146	2	no	—
294	C₂²⋊C₄	—	146	2	no	—
295	C₄²	—	4	2	no	—
296	C₄²	D₄×D₇₃	1	1	yes	characteristic
297	C₄²	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
298	C₄²	#11924027	1	1	yes	characteristic
299	C₄²	C₂³×D₇₃	1	1	yes	characteristic
300	C₂³	—	292	2	no	—
301	C₂³	—	292	2	no	—
302	C₂³	—	292	2	no	—
303	C₂³	—	146	2	no	—
304	C₂³	#14556562	1	1	yes	characteristic
305	Q₈	—	4	4	no	—
306	Q₈	—	8	2	no	—
307	D₄	—	292	2	no	—
308	D₄	—	146	4	no	—
309	C₂×C₄	—	2	2	no	—
310	C₂×C₄	—	4	1	no	—
311	C₂×C₄	—	4	2	no	—
312	C₂×C₄	—	4	2	no	—
313	C₂×C₄	—	292	2	no	—
314	C₂×C₄	—	292	2	no	—
315	C₂×C₄	—	292	4	no	—
316	C₂×C₄	—	292	2	no	—
317	C₂×C₄	—	146	2	no	—
318	C₂×C₄	#14556562	1	1	yes	characteristic
319	C₂×C₄	#14556562	1	1	yes	characteristic
320	C₂×C₄	#14556562	1	1	yes	characteristic
321	C₂×C₄	C₂×#11924015	1	2	yes	—
322	C₂×C₄	#14556622	1	2	yes	—
323	C₂×C₄	C₂×D₄×D₇₃	1	1	yes	characteristic
324	C₂×C₄	C₂²×C₂²⋊D₇₃	1	1	yes	characteristic
325	C₂×C₄	C₂×#11924027	1	1	yes	characteristic
326	C₈	—	2	1	no	—
327	C₈	—	2	1	no	—
328	C₈	—	292	4	no	—
329	C₈	—	292	2	no	—
330	C₈	C₂×#11924015	1	2	yes	—
331	C₈	C₂×#11924015	1	2	yes	—
332	C₂²	—	2	1	no	—
333	C₂²	—	2	1	no	—
334	C₂²	—	292	2	no	—
335	C₂²	—	292	2	no	—
336	C₂²	—	292	2	no	—
337	C₂²	—	584	4	no	—
338	C₂²	#17988789	1	2	yes	—
339	C₂²	C₂×#14556562	1	1	yes	characteristic
340	C₄	—	2	1	no	—
341	C₄	—	2	1	no	—
342	C₄	—	2	1	no	—
343	C₄	—	8	2	no	—
344	C₄	—	8	2	no	—
345	C₄	—	292	2	no	—
346	C₄	#17988789	1	2	yes	—
347	C₄	C₂²×#11924015	1	1	yes	characteristic
348	C₄	C₂×#14556622	1	1	yes	characteristic
349	C₄	#17989384	1	2	yes	—
350	C₂	—	2	2	no	—
351	C₂	—	584	2	no	—
352	C₂	C₂×#17988789	1	1	yes	characteristic
353	C₂	#22619536	1	1	yes	characteristic
354	C₂	#22619647	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#14556562
commutator, cosocle	C₄²×C₇₃	C₂⁴
Frattini	C₄²	C₂³×D₇₃
Fitting	C₇₃×#3160	C₂
socle	C₂²×C₇₃	C₂×#121
prime radical	C₂²×D₂₉₂	D₄
hypercenter	#3160	D₇₃
hypocenter	C₇₃	#19137
LCS	G ▷ C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3160

Complements

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

N	H ≅ G/N	type
3. #22609451	C₂	semidirect
4. #22609441	C₂	semidirect
5. #22609427	C₂	semidirect
6. #22608679	C₂	semidirect
7. #22608588	C₂	semidirect
8. C₇₃×#3160	C₂	semidirect
23. #17988340	C₂²	semidirect
24. #17988338	C₂²	semidirect
25. #17988336	C₂²	semidirect
26. #17988333	C₂²	semidirect
28. #500×C₇₃	C₂²	semidirect
29. #500×C₇₃	C₂²	semidirect
32. #498×C₇₃	C₂²	semidirect
105. C₂×C₈×C₇₃	SD₁₆	semidirect
178. #3160	D₇₃	semidirect
186. C₇₃	#19137	semidirect
190. #500	C₂×D₇₃	semidirect
191. #500	C₂×D₇₃	semidirect
194. #498	C₂×D₇₃	semidirect
279. C₂×C₈	#11924015	semidirect

Maximal subgroups

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

subgroup	index
#22619935	2
C₂×#17988666	2
#22609451	2
#22609441	2
#22609427	2
#22608679	2
#22608588	2
C₇₃×#3160	2
#19137	73