Subgroups — #33240479

Order 18688.

714 conjugacy classes of subgroups.

The subgroup lattice has 402 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₁₆×D₇₃	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#17989005	C₂	1	1	yes	maximal, characteristic, complemented
3	#22619574	C₂	1	2	yes	maximal, complemented
4	#22612035	C₂	1	1	yes	maximal, characteristic, complemented
5	#22612034	C₂	1	1	yes	maximal, characteristic, complemented
6	#22612031	C₂	1	1	yes	maximal, characteristic, complemented
7	#22612016	C₂	1	2	yes	maximal, complemented
8	#22612015	C₂	1	2	yes	maximal, complemented
9	#22611714	C₂	1	1	yes	maximal, characteristic, complemented
10	#22611645	C₂	1	2	yes	maximal, complemented
11	C₇₃×#2218	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×D₈×D₇₃	C₂²	1	2	yes	—
13	C₂×#14556597	C₂²	1	1	yes	characteristic
14	#17989153	—	2	2	no	—
15	C₂×D₁₁₆₈	C₂²	1	1	yes	characteristic
16	C₂×#14556532	C₂²	1	2	yes	—
17	D₁₆×D₇₃	—	2	4	no	—
18	#17989005	—	2	4	no	—
19	C₂×#14556530	C₂²	1	1	yes	characteristic
20	C₂×#14556530	C₂²	1	1	yes	characteristic, complemented
21	C₂²×#11924003	C₂²	1	1	yes	characteristic, complemented
22	C₂×C₁₆×D₇₃	C₂²	1	1	yes	characteristic
23	C₂×C₁₆×D₇₃	C₂²	1	1	yes	characteristic, complemented
24	#17988839	—	2	2	no	—
25	#17988838	—	2	2	no	—
26	#17988824	C₂²	1	2	yes	complemented
27	#17988823	C₂²	1	2	yes	complemented
28	#17988820	C₂²	1	2	yes	complemented
29	#17988624	C₂²	1	2	yes	—
30	#17988590	—	2	2	no	—
31	#17988415	C₂²	1	1	yes	characteristic, complemented
32	#17988412	C₂²	1	2	yes	complemented
33	#17988389	C₂²	1	2	yes	complemented
34	#17988388	C₂²	1	2	yes	complemented
35	#17988387	C₂²	1	2	yes	complemented
36	C₂×D₁₆×C₇₃	C₂²	1	1	yes	characteristic
37	C₂×M₃₂×C₇₃	C₂²	1	1	yes	characteristic, complemented
38	#466×C₇₃	C₂²	1	2	yes	complemented
39	#367×C₇₃	C₂²	1	1	yes	characteristic, complemented
40	#357×C₇₃	C₂²	1	2	yes	complemented
41	C₂×#11924028	D₄	1	1	yes	characteristic
42	C₂×D₄×D₇₃	—	2	2	no	—
43	C₂×D₅₈₄	C₂³	1	2	yes	—
44	C₂×#11924013	—	2	2	no	—
45	D₈×D₇₃	—	2	4	no	—
46	D₈×D₇₃	—	2	4	no	—
47	#14556597	—	2	2	no	—
48	#14556597	—	2	2	no	—
49	C₂×#11924012	D₄	1	1	yes	characteristic
50	C₂×#11924011	D₄	1	1	yes	characteristic
51	C₂×#11924011	D₄	1	1	yes	characteristic
52	C₂×C₈×D₇₃	C₂³	1	1	yes	characteristic
53	C₂×C₈×D₇₃	C₂³	1	1	yes	characteristic
54	#14556587	—	4	2	no	—
55	#14556577	—	4	2	no	—
56	#14556566	—	2	2	no	—
57	#14556565	—	2	2	no	—
58	#14556562	—	2	2	no	—
59	#14556561	—	4	2	no	—
60	#14556559	—	4	2	no	—
61	#14556549	—	2	2	no	—
62	D₁₁₆₈	—	2	2	no	—
63	#14556532	—	2	4	no	—
64	#14556530	—	2	2	no	—
65	#14556530	—	2	2	no	—
66	C₂×#11924003	—	2	2	no	—
67	C₂×#11924003	C₂³	1	1	yes	characteristic
68	C₂×#11924003	C₂³	1	1	yes	characteristic
69	C₁₆×D₇₃	—	2	2	no	—
70	C₁₆×D₇₃	D₄	1	4	yes	complemented
71	#14556525	C₂³	1	2	yes	—
72	#14556522	—	2	2	no	—
73	#14556513	—	2	2	no	—
74	#14556512	—	2	2	no	—
75	#14556511	—	2	2	no	—
76	C₂×D₈×C₇₃	C₂³	1	2	yes	—
77	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
78	#121×C₇₃	—	2	2	no	—
79	D₁₆×C₇₃	—	2	2	no	—
80	M₃₂×C₇₃	—	2	2	no	—
81	C₂×C₁₆×C₇₃	C₂³	1	1	yes	characteristic
82	C₂×C₁₆×C₇₃	C₂³	1	1	yes	characteristic
83	#107×C₇₃	C₂³	1	2	yes	—
84	#102×C₇₃	—	2	2	no	—
85	C₂³×D₇₃	—	4	2	no	—
86	#11924028	—	2	2	no	—
87	#11924028	—	2	2	no	—
88	C₂×Dic₁₄₆	D₈	1	1	yes	characteristic
89	C₂×D₂₉₂	—	2	2	no	—
90	C₂×D₂₉₂	—	4	2	no	—
91	C₂×D₂₉₂	D₈	1	1	yes	characteristic
92	C₂×C₂²⋊D₇₃	—	4	2	no	—
93	C₂×C₂²⋊D₇₃	—	4	2	no	—
94	C₂×C₂²⋊D₇₃	—	4	2	no	—
95	C₂×C₂²⋊D₇₃	D₈	1	1	yes	characteristic
96	C₂×C₂²⋊D₇₃	D₈	1	1	yes	characteristic
97	D₄×D₇₃	—	4	4	no	—
98	D₄×D₇₃	—	4	4	no	—
99	C₂×C₄×D₇₃	C₂×D₄	1	1	yes	characteristic
100	C₂×C₄×D₇₃	C₂×D₄	1	1	yes	characteristic
101	D₅₈₄	—	2	2	no	—
102	D₅₈₄	—	2	2	no	—
103	#11924013	—	4	4	no	—
104	#11924012	—	2	2	no	—
105	#11924011	—	2	2	no	—
106	#11924011	—	2	2	no	—
107	C₂×#10596831	C₂×D₄	1	1	yes	characteristic
108	C₂×#10596831	C₂×D₄	1	1	yes	characteristic
109	C₈×D₇₃	—	2	1	no	—
110	C₈×D₇₃	—	2	1	no	—
111	C₈×D₇₃	C₂×D₄	1	2	yes	—
112	C₈×D₇₃	#41	1	2	yes	—
113	C₄⋊Dic₇₃	—	2	2	no	—
114	C₂²⋊Dic₇₃	—	4	2	no	—
115	#11924005	—	4	2	no	—
116	#11924005	—	4	2	no	—
117	#11924005	—	4	2	no	—
118	C₄×Dic₇₃	—	4	2	no	—
119	#11924003	—	2	1	no	—
120	#11924003	C₂×D₄	1	2	yes	—
121	C₂×D₄×C₇₃	—	2	2	no	—
122	C₂×D₄×C₇₃	—	4	2	no	—
123	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
124	D₈×C₇₃	—	2	2	no	—
125	D₈×C₇₃	—	2	2	no	—
126	C₂×C₈×C₇₃	—	2	1	no	—
127	C₂×C₈×C₇₃	—	2	1	no	—
128	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
129	C₂×C₈×C₇₃	C₂⁴	1	1	yes	characteristic
130	C₄⋊C₄×C₇₃	—	2	2	no	—
131	C₂²⋊C₄×C₇₃	—	4	2	no	—
132	C₁₆×C₇₃	—	2	1	no	—
133	C₁₆×C₇₃	C₂×D₄	1	2	yes	—
134	C₂³×C₇₃	—	4	2	no	—
135	C₂³×C₇₃	C₂×D₈	1	1	yes	characteristic
136	C₂²×D₇₃	—	4	2	no	—
137	C₂²×D₇₃	—	8	4	no	—
138	C₂²×D₇₃	—	16	2	no	—
139	C₂²×D₇₃	—	16	2	no	—
140	C₂²×D₇₃	C₂×D₈	1	1	yes	characteristic
141	C₂²×D₇₃	C₂×D₈	1	1	yes	characteristic
142	D₄×C₇₃	—	4	2	no	—
143	D₄×C₇₃	—	4	2	no	—
144	D₄×C₇₃	—	16	2	no	—
145	C₂×C₄×C₇₃	—	2	1	no	—
146	C₂×C₄×C₇₃	—	2	1	no	—
147	C₂×C₄×C₇₃	—	4	2	no	—
148	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
149	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
150	C₂²⋊D₇₃	—	2	2	no	—
151	C₂²⋊D₇₃	—	2	2	no	—
152	C₂²⋊D₇₃	—	8	4	no	—
153	C₂²⋊D₇₃	—	16	2	no	—
154	C₂²⋊D₇₃	—	16	2	no	—
155	C₂×Dic₇₃	—	4	2	no	—
156	C₂×Dic₇₃	C₂×D₈	1	1	yes	characteristic
157	C₂×Dic₇₃	C₂×D₈	1	1	yes	characteristic
158	D₂₉₂	—	2	2	no	—
159	D₂₉₂	—	4	2	no	—
160	D₂₉₂	—	4	2	no	—
161	D₂₉₂	—	16	2	no	—
162	C₄×D₇₃	—	2	1	no	—
163	C₄×D₇₃	—	2	1	no	—
164	C₄×D₇₃	#121	1	2	yes	—
165	C₄×D₇₃	#135	1	2	yes	—
166	Dic₁₄₆	—	2	2	no	—
167	C₈×C₇₃	—	2	1	no	—
168	C₈×C₇₃	C₂²×D₄	1	1	yes	characteristic
169	C₈×C₇₃	C₂×#41	1	1	yes	characteristic
170	#10596831	—	2	1	no	—
171	#10596831	#121	1	2	yes	—
172	C₂²×C₇₃	—	2	2	no	—
173	C₂²×C₇₃	—	2	1	no	—
174	C₂²×C₇₃	—	8	2	no	—
175	C₂²×C₇₃	—	8	2	no	—
176	C₂²×C₇₃	—	8	2	no	—
177	C₂²×C₇₃	C₂²×D₈	1	1	yes	characteristic
178	C₂×D₇₃	—	2	2	no	—
179	C₂×D₇₃	—	2	2	no	—
180	C₂×D₇₃	—	2	1	no	—
181	C₂×D₇₃	—	8	2	no	—
182	C₂×D₇₃	—	8	2	no	—
183	C₂×D₇₃	—	8	2	no	—
184	C₂×D₇₃	—	16	8	no	—
185	C₂×D₇₃	#466	1	2	yes	—
186	C₄×C₇₃	—	2	1	no	—
187	C₄×C₇₃	—	8	2	no	—
188	C₄×C₇₃	C₂×#121	1	1	yes	characteristic
189	C₄×C₇₃	C₂×#135	1	1	yes	characteristic
190	Dic₇₃	—	2	1	no	—
191	Dic₇₃	—	8	2	no	—
192	Dic₇₃	#466	1	2	yes	—
193	#33607	—	73	1	no	maximal, Sylow
194	C₂×C₇₃	—	4	1	no	—
195	C₂×C₇₃	—	16	2	no	—
196	C₂×C₇₃	C₂×#466	1	1	yes	characteristic
197	C₂×C₇₃	#3415	1	2	yes	—
198	D₇₃	—	2	2	no	—
199	D₇₃	—	4	1	no	—
200	D₇₃	—	16	2	no	—
201	C₂²×D₁₆	—	73	1	no	—
202	C₂×#504	—	73	1	no	—
203	#3058	—	73	2	no	—
204	#2218	—	73	1	no	—
205	#2218	—	73	1	no	—
206	#2218	—	73	1	no	—
207	#2218	D₇₃	1	1	yes	characteristic, complemented
208	#2215	—	73	2	no	—
209	#2215	—	73	2	no	—
210	#2158	—	73	1	no	—
211	#2140	—	73	2	no	—
212	C₇₃	#33607	1	1	yes	characteristic, complemented, Sylow
213	C₂²×D₈	—	73	2	no	—
214	C₂×#131	—	73	1	no	—
215	#535	—	146	2	no	—
216	C₂×D₁₆	—	73	2	no	—
217	C₂×D₁₆	—	73	1	no	—
218	C₂×D₁₆	—	146	4	no	—
219	C₂×D₁₆	C₂×D₇₃	1	1	yes	characteristic
220	#504	—	146	4	no	—
221	C₂×M₃₂	—	73	1	no	—
222	C₂×M₃₂	—	73	1	no	—
223	C₂×M₃₂	C₂×D₇₃	1	1	yes	characteristic, complemented
224	C₂²×C₁₆	—	73	1	no	—
225	C₂²×C₁₆	—	73	1	no	—
226	C₂²×C₁₆	—	73	1	no	—
227	#469	—	146	2	no	—
228	#469	—	146	2	no	—
229	#466	—	73	2	no	—
230	#466	—	73	2	no	—
231	#466	—	73	2	no	—
232	#466	C₂×D₇₃	1	2	yes	complemented
233	#427	—	73	2	no	—
234	#418	—	146	2	no	—
235	#367	—	73	1	no	—
236	#367	C₂×D₇₃	1	1	yes	characteristic, complemented
237	#366	—	73	2	no	—
238	#357	—	73	2	no	—
239	#357	—	73	2	no	—
240	#357	—	73	2	no	—
241	#357	C₂×D₇₃	1	2	yes	complemented
242	C₂×#41	—	73	1	no	—
243	C₂²×D₄	—	146	2	no	—
244	C₂×D₈	—	73	2	no	—
245	C₂×D₈	—	146	4	no	—
246	C₂×D₈	—	146	4	no	—
247	C₂×D₈	—	146	2	no	—
248	C₂×D₈	C₂²×D₇₃	1	2	yes	—
249	#131	—	146	2	no	—
250	#131	—	146	2	no	—
251	C₂×M₁₆	—	73	1	no	—
252	C₂×M₁₆	—	73	1	no	—
253	C₂×M₁₆	—	73	1	no	—
254	C₂²×C₈	—	73	1	no	—
255	C₂²×C₈	—	73	1	no	—
256	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
257	#127	—	292	2	no	—
258	#124	—	292	2	no	—
259	#121	—	2	2	no	—
260	#121	—	146	2	no	—
261	#121	—	146	2	no	—
262	#121	—	146	2	no	—
263	#120	—	292	2	no	—
264	#120	—	292	2	no	—
265	#117	—	146	2	no	—
266	D₁₆	—	2	2	no	—
267	D₁₆	—	146	4	no	—
268	D₁₆	—	146	2	no	—
269	M₃₂	—	2	2	no	—
270	M₃₂	—	146	2	no	—
271	M₃₂	—	146	2	no	—
272	C₂×C₁₆	—	73	1	no	—
273	C₂×C₁₆	—	73	4	no	—
274	C₂×C₁₆	—	73	1	no	—
275	C₂×C₁₆	—	146	2	no	—
276	C₂×C₁₆	—	146	2	no	—
277	C₂×C₁₆	C₂²×D₇₃	1	1	yes	characteristic
278	C₂×C₁₆	C₂²×D₇₃	1	1	yes	characteristic
279	#107	—	73	2	no	—
280	#107	C₂²×D₇₃	1	2	yes	—
281	#106	—	146	2	no	—
282	#102	—	2	2	no	—
283	#102	—	146	2	no	—
284	#102	—	146	2	no	—
285	#102	—	146	2	no	—
286	C₂⁴	—	292	2	no	—
287	#41	—	146	2	no	—
288	#41	—	146	2	no	—
289	C₂×Q₈	—	73	1	no	—
290	C₂×D₄	—	2	2	no	—
291	C₂×D₄	—	4	2	no	—
292	C₂×D₄	—	292	2	no	—
293	C₂×D₄	—	292	2	no	—
294	C₂×D₄	—	292	2	no	—
295	C₂×D₄	—	292	4	no	—
296	C₂×D₄	—	292	4	no	—
297	C₂×D₄	—	292	2	no	—
298	C₂×D₄	—	73	1	no	—
299	C₂×D₄	—	73	1	no	—
300	C₂×D₄	—	73	1	no	—
301	C₂×D₄	—	146	2	no	—
302	C₂²×C₄	—	73	1	no	—
303	C₂²×C₄	—	73	1	no	—
304	C₂²×C₄	D₄×D₇₃	1	1	yes	characteristic
305	D₈	—	2	2	no	—
306	D₈	—	2	2	no	—
307	D₈	—	292	4	no	—
308	D₈	—	146	2	no	—
309	D₈	—	146	2	no	—
310	M₁₆	—	146	2	no	—
311	M₁₆	—	146	2	no	—
312	M₁₆	—	146	2	no	—
313	C₂×C₈	—	2	1	no	—
314	C₂×C₈	—	2	1	no	—
315	C₂×C₈	—	73	1	no	—
316	C₂×C₈	—	73	2	no	—
317	C₂×C₈	—	73	2	no	—
318	C₂×C₈	—	73	1	no	—
319	C₂×C₈	—	146	1	no	—
320	C₂×C₈	—	146	1	no	—
321	C₂×C₈	D₄×D₇₃	1	1	yes	characteristic
322	C₂×C₈	C₂³×D₇₃	1	1	yes	characteristic
323	C₄⋊C₄	—	2	2	no	—
324	C₄⋊C₄	—	146	2	no	—
325	C₂²⋊C₄	—	4	2	no	—
326	C₂²⋊C₄	—	292	2	no	—
327	C₂²⋊C₄	—	292	2	no	—
328	C₂²⋊C₄	—	292	2	no	—
329	C₂²⋊C₄	—	292	2	no	—
330	C₄²	—	292	2	no	—
331	C₁₆	—	2	1	no	—
332	C₁₆	—	73	2	no	—
333	C₁₆	—	146	1	no	—
334	C₁₆	D₄×D₇₃	1	2	yes	—
335	C₂³	—	4	2	no	—
336	C₂³	—	292	2	no	—
337	C₂³	—	584	4	no	—
338	C₂³	—	73	1	no	—
339	C₂³	—	73	1	no	—
340	C₂³	—	1168	2	no	—
341	C₂³	—	1168	2	no	—
342	C₂³	D₈×D₇₃	1	1	yes	characteristic
343	Q₈	—	146	2	no	—
344	D₄	—	4	2	no	—
345	D₄	—	4	2	no	—
346	D₄	—	16	2	no	—
347	D₄	—	292	2	no	—
348	D₄	—	292	2	no	—
349	D₄	—	584	4	no	—
350	D₄	—	1168	2	no	—
351	D₄	—	1168	2	no	—
352	D₄	—	1168	2	no	—
353	D₄	—	146	2	no	—
354	D₄	—	146	2	no	—
355	D₄	—	146	2	no	—
356	C₂×C₄	—	2	1	no	—
357	C₂×C₄	—	2	1	no	—
358	C₂×C₄	—	4	2	no	—
359	C₂×C₄	—	292	2	no	—
360	C₂×C₄	—	73	1	no	—
361	C₂×C₄	—	73	2	no	—
362	C₂×C₄	—	73	2	no	—
363	C₂×C₄	—	73	1	no	—
364	C₂×C₄	—	146	1	no	—
365	C₂×C₄	—	146	1	no	—
366	C₂×C₄	D₈×D₇₃	1	1	yes	characteristic
367	C₂×C₄	C₂×D₄×D₇₃	1	1	yes	characteristic
368	C₈	—	2	1	no	—
369	C₈	—	73	2	no	—
370	C₈	—	146	1	no	—
371	C₈	C₂×D₄×D₇₃	1	1	yes	characteristic
372	C₈	#41×D₇₃	1	1	yes	characteristic
373	C₂²	—	2	2	no	—
374	C₂²	—	2	1	no	—
375	C₂²	—	8	2	no	—
376	C₂²	—	8	2	no	—
377	C₂²	—	8	2	no	—
378	C₂²	—	584	2	no	—
379	C₂²	—	584	2	no	—
380	C₂²	—	584	2	no	—
381	C₂²	—	73	2	no	—
382	C₂²	—	1168	8	no	—
383	C₂²	—	146	2	no	—
384	C₂²	—	146	1	no	—
385	C₂²	—	146	2	no	—
386	C₂²	C₂×D₈×D₇₃	1	1	yes	characteristic
387	C₄	—	2	1	no	—
388	C₄	—	8	2	no	—
389	C₄	—	584	2	no	—
390	C₄	—	73	2	no	—
391	C₄	—	146	1	no	—
392	C₄	#121×D₇₃	1	1	yes	characteristic
393	C₄	#135×D₇₃	1	1	yes	characteristic
394	C₂	—	4	1	no	—
395	C₂	—	16	2	no	—
396	C₂	—	292	1	no	—
397	C₂	—	1168	2	no	—
398	C₂	—	146	2	no	—
399	C₂	#466×D₇₃	1	1	yes	characteristic
400	C₂	#22623071	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₈×D₇₃
commutator, cosocle	C₂×C₈×C₇₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₇₃
Fitting	C₇₃×#2218	C₂
socle	C₂²×C₇₃	C₂²×D₈
hypercenter	#2218	D₇₃
hypocenter	C₇₃	#33607
LCS	G ▷ C₂×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #2218

Complements

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

N	H ≅ G/N	type
1. C₂×D₁₆×D₇₃	C₂	semidirect
2. C₂×#17989005	C₂	semidirect
3. #22619574	C₂	semidirect
4. #22612035	C₂	semidirect
5. #22612034	C₂	semidirect
6. #22612031	C₂	semidirect
7. #22612016	C₂	semidirect
8. #22612015	C₂	semidirect
9. #22611714	C₂	semidirect
10. #22611645	C₂	semidirect
11. C₇₃×#2218	C₂	semidirect
20. C₂×#14556530	C₂²	semidirect
21. C₂²×#11924003	C₂²	semidirect
23. C₂×C₁₆×D₇₃	C₂²	semidirect
26. #17988824	C₂²	semidirect
27. #17988823	C₂²	semidirect
28. #17988820	C₂²	semidirect
31. #17988415	C₂²	semidirect
32. #17988412	C₂²	semidirect
33. #17988389	C₂²	semidirect
34. #17988388	C₂²	semidirect
35. #17988387	C₂²	semidirect
37. C₂×M₃₂×C₇₃	C₂²	semidirect
38. #466×C₇₃	C₂²	semidirect
39. #367×C₇₃	C₂²	semidirect
40. #357×C₇₃	C₂²	semidirect
70. C₁₆×D₇₃	D₄	semidirect
207. #2218	D₇₃	semidirect
212. C₇₃	#33607	semidirect
223. C₂×M₃₂	C₂×D₇₃	semidirect
232. #466	C₂×D₇₃	semidirect
236. #367	C₂×D₇₃	semidirect
241. #357	C₂×D₇₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₁₆×D₇₃	2
C₂×#17989005	2
#22619574	2
#22612035	2
#22612034	2
#22612031	2
#22612016	2
#22612015	2
#22611714	2
#22611645	2
C₇₃×#2218	2
#33607	73