Subgroups — #33240483

Order 18688.

650 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₃₂×D₇₃	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#17989003	C₂	1	1	yes	maximal, characteristic
3	#22619581	C₂	1	2	yes	maximal
4	#22612038	C₂	1	2	yes	maximal, complemented
5	#22612037	C₂	1	2	yes	maximal, complemented
6	#22612023	C₂	1	1	yes	maximal, characteristic, complemented
7	#22612022	C₂	1	1	yes	maximal, characteristic, complemented
8	#22612019	C₂	1	1	yes	maximal, characteristic, complemented
9	#22611712	C₂	1	1	yes	maximal, characteristic, complemented
10	#22611649	C₂	1	2	yes	maximal, complemented
11	C₇₃×#2216	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×Q₁₆×D₇₃	C₂²	1	2	yes	—
13	C₂×#14556597	C₂²	1	1	yes	characteristic
14	#17989160	—	2	2	no	—
15	C₂×Dic₅₈₄	C₂²	1	1	yes	characteristic
16	C₂×#14556537	C₂²	1	2	yes	—
17	Q₃₂×D₇₃	—	2	4	no	—
18	#17989003	—	2	4	no	—
19	C₂×#14556531	C₂²	1	1	yes	characteristic
20	C₂×#14556530	C₂²	1	1	yes	characteristic
21	C₂×#14556530	C₂²	1	1	yes	characteristic
22	C₂×C₁₆×D₇₃	C₂²	1	1	yes	characteristic
23	C₂×C₁₆×D₇₃	C₂²	1	1	yes	characteristic
24	#17988844	—	2	2	no	—
25	#17988843	—	2	2	no	—
26	#17988829	C₂²	1	2	yes	—
27	#17988828	C₂²	1	2	yes	—
28	#17988825	C₂²	1	2	yes	—
29	#17988624	C₂²	1	2	yes	—
30	#17988594	—	2	2	no	—
31	#17988414	C₂²	1	2	yes	—
32	#17988413	C₂²	1	1	yes	characteristic
33	#17988392	C₂²	1	2	yes	—
34	#17988391	C₂²	1	2	yes	—
35	#17988390	C₂²	1	2	yes	—
36	C₂×Q₃₂×C₇₃	C₂²	1	1	yes	characteristic
37	C₂²×C₁₆×C₇₃	C₂²	1	1	yes	characteristic
38	#467×C₇₃	C₂²	1	2	yes	—
39	#366×C₇₃	C₂²	1	1	yes	characteristic
40	#358×C₇₃	C₂²	1	2	yes	—
41	C₂×#11924028	D₄	1	1	yes	characteristic
42	C₂×Q₈×D₇₃	—	2	2	no	—
43	C₂×Dic₂₉₂	C₂³	1	2	yes	—
44	C₂×#11924018	—	2	2	no	—
45	Q₁₆×D₇₃	—	2	4	no	—
46	Q₁₆×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	#14556590	—	4	2	no	—
55	#14556577	—	4	2	no	—
56	#14556576	—	4	2	no	—
57	#14556573	—	4	2	no	—
58	#14556571	—	2	2	no	—
59	#14556570	—	2	2	no	—
60	#14556567	—	2	2	no	—
61	#14556549	—	2	2	no	—
62	Dic₅₈₄	—	2	2	no	—
63	#14556537	—	2	4	no	—
64	#14556531	—	2	2	no	—
65	#14556530	—	2	2	no	—
66	#14556530	—	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	—
71	#14556525	C₂³	1	2	yes	—
72	#14556522	—	2	2	no	—
73	#14556516	—	2	2	no	—
74	#14556515	—	2	2	no	—
75	#14556514	—	2	2	no	—
76	C₂×Q₁₆×C₇₃	C₂³	1	2	yes	—
77	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
78	C₂²⋊Q₈×C₇₃	—	2	2	no	—
79	Q₃₂×C₇₃	—	2	2	no	—
80	C₂×C₁₆×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	#103×C₇₃	—	2	2	no	—
85	#11924028	—	2	2	no	—
86	#11924028	—	2	2	no	—
87	C₂×Dic₁₄₆	—	2	2	no	—
88	C₂×Dic₁₄₆	—	4	2	no	—
89	C₂×Dic₁₄₆	D₈	1	1	yes	characteristic
90	Q₈×D₇₃	—	4	4	no	—
91	Q₈×D₇₃	—	4	4	no	—
92	C₂×D₂₉₂	D₈	1	1	yes	characteristic
93	C₂×C₂²⋊D₇₃	D₈	1	1	yes	characteristic
94	C₂×C₂²⋊D₇₃	D₈	1	1	yes	characteristic
95	C₂×C₄×D₇₃	—	4	2	no	—
96	C₂×C₄×D₇₃	C₂×D₄	1	1	yes	characteristic
97	C₂×C₄×D₇₃	C₂×D₄	1	1	yes	characteristic
98	Dic₂₉₂	—	2	2	no	—
99	Dic₂₉₂	—	2	2	no	—
100	#11924018	—	4	4	no	—
101	#11924012	—	2	2	no	—
102	#11924011	—	2	2	no	—
103	#11924011	—	2	2	no	—
104	C₂×#10596831	C₂×D₄	1	1	yes	characteristic
105	C₂×#10596831	C₂×D₄	1	1	yes	characteristic
106	C₈×D₇₃	—	2	1	no	—
107	C₈×D₇₃	—	2	1	no	—
108	C₈×D₇₃	C₂×D₄	1	2	yes	—
109	C₈×D₇₃	#41	1	2	yes	—
110	C₄⋊Dic₇₃	—	2	2	no	—
111	C₄⋊Dic₇₃	—	4	2	no	—
112	#11924007	—	4	2	no	—
113	#11924007	—	4	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×Q₈×C₇₃	—	2	2	no	—
122	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
123	Q₁₆×C₇₃	—	2	2	no	—
124	Q₁₆×C₇₃	—	2	2	no	—
125	C₂×C₈×C₇₃	—	2	1	no	—
126	C₂×C₈×C₇₃	—	2	1	no	—
127	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
128	C₂×C₈×C₇₃	C₂⁴	1	1	yes	characteristic
129	C₄⋊C₄×C₇₃	—	2	2	no	—
130	C₄⋊C₄×C₇₃	—	4	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	C₂×D₈	1	1	yes	characteristic
135	C₂²×D₇₃	C₂×D₈	1	1	yes	characteristic
136	C₂²×D₇₃	C₂×D₈	1	1	yes	characteristic
137	Q₈×C₇₃	—	4	2	no	—
138	Q₈×C₇₃	—	4	2	no	—
139	C₂×C₄×C₇₃	—	2	1	no	—
140	C₂×C₄×C₇₃	—	2	1	no	—
141	C₂×C₄×C₇₃	—	4	2	no	—
142	C₂×C₄×C₇₃	—	4	2	no	—
143	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
144	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
145	C₂²⋊D₇₃	—	2	2	no	—
146	C₂²⋊D₇₃	—	2	2	no	—
147	C₂×Dic₇₃	—	4	2	no	—
148	C₂×Dic₇₃	—	4	2	no	—
149	C₂×Dic₇₃	C₂×D₈	1	1	yes	characteristic
150	C₂×Dic₇₃	C₂×D₈	1	1	yes	characteristic
151	D₂₉₂	—	2	2	no	—
152	C₄×D₇₃	—	2	1	no	—
153	C₄×D₇₃	—	2	1	no	—
154	C₄×D₇₃	—	8	4	no	—
155	C₄×D₇₃	#121	1	2	yes	—
156	C₄×D₇₃	#135	1	2	yes	—
157	Dic₁₄₆	—	2	2	no	—
158	Dic₁₄₆	—	4	2	no	—
159	Dic₁₄₆	—	4	2	no	—
160	Dic₁₄₆	—	8	4	no	—
161	C₈×C₇₃	—	2	1	no	—
162	C₈×C₇₃	C₂²×D₄	1	1	yes	characteristic
163	C₈×C₇₃	C₂×#41	1	1	yes	characteristic
164	#10596831	—	2	1	no	—
165	#10596831	#121	1	2	yes	—
166	C₂²×C₇₃	—	2	1	no	—
167	C₂²×C₇₃	—	2	1	no	—
168	C₂²×C₇₃	—	2	1	no	—
169	C₂²×C₇₃	C₂²×D₈	1	1	yes	characteristic
170	C₂×D₇₃	—	2	1	no	—
171	C₂×D₇₃	—	2	1	no	—
172	C₂×D₇₃	—	2	1	no	—
173	C₂×D₇₃	—	2	1	no	—
174	C₂×D₇₃	—	2	1	no	—
175	C₂×D₇₃	#466	1	2	yes	—
176	C₄×C₇₃	—	2	1	no	—
177	C₄×C₇₃	—	8	2	no	—
178	C₄×C₇₃	—	8	2	no	—
179	C₄×C₇₃	C₂×#121	1	1	yes	characteristic
180	C₄×C₇₃	C₂×#135	1	1	yes	characteristic
181	Dic₇₃	—	2	1	no	—
182	Dic₇₃	—	8	2	no	—
183	Dic₇₃	—	8	2	no	—
184	Dic₇₃	#466	1	2	yes	—
185	#33608	—	73	1	no	maximal, Sylow
186	C₂×C₇₃	—	4	1	no	—
187	C₂×C₇₃	C₂×#466	1	1	yes	characteristic
188	C₂×C₇₃	#3417	1	1	yes	characteristic
189	C₂×C₇₃	#3417	1	1	yes	characteristic
190	D₇₃	—	2	2	no	—
191	D₇₃	—	4	1	no	—
192	C₂²×Q₃₂	—	73	1	no	—
193	C₂×#504	—	73	1	no	—
194	#3059	—	73	2	no	—
195	#2219	—	73	2	no	—
196	#2219	—	73	2	no	—
197	#2216	—	73	1	no	—
198	#2216	—	73	1	no	—
199	#2216	—	73	1	no	—
200	#2216	D₇₃	1	1	yes	characteristic, complemented
201	#2158	—	73	1	no	—
202	#2141	—	73	2	no	—
203	C₇₃	#33608	1	1	yes	characteristic, complemented, Sylow
204	C₂²×Q₁₆	—	73	2	no	—
205	C₂×#131	—	73	1	no	—
206	#536	—	146	2	no	—
207	C₂×Q₃₂	—	73	2	no	—
208	C₂×Q₃₂	—	73	1	no	—
209	C₂×Q₃₂	—	146	4	no	—
210	C₂×Q₃₂	C₂×D₇₃	1	1	yes	characteristic
211	#504	—	146	4	no	—
212	C₂×M₃₂	—	73	1	no	—
213	C₂×M₃₂	—	73	1	no	—
214	C₂×M₃₂	—	73	1	no	—
215	C₂²×C₁₆	—	73	1	no	—
216	C₂²×C₁₆	—	73	1	no	—
217	C₂²×C₁₆	C₂×D₇₃	1	1	yes	characteristic
218	#470	—	146	2	no	—
219	#470	—	146	2	no	—
220	#467	—	73	2	no	—
221	#467	—	73	2	no	—
222	#467	—	73	2	no	—
223	#467	C₂×D₇₃	1	2	yes	—
224	#427	—	73	2	no	—
225	#419	—	146	2	no	—
226	#367	—	73	2	no	—
227	#366	—	73	1	no	—
228	#366	C₂×D₇₃	1	1	yes	characteristic
229	#358	—	73	2	no	—
230	#358	—	73	2	no	—
231	#358	—	73	2	no	—
232	#358	C₂×D₇₃	1	2	yes	—
233	C₂×#41	—	73	1	no	—
234	C₂²×Q₈	—	146	2	no	—
235	C₂×Q₁₆	—	73	2	no	—
236	C₂×Q₁₆	—	146	4	no	—
237	C₂×Q₁₆	—	146	4	no	—
238	C₂×Q₁₆	—	146	2	no	—
239	C₂×Q₁₆	C₂²×D₇₃	1	2	yes	—
240	#131	—	146	2	no	—
241	#131	—	146	2	no	—
242	C₂×M₁₆	—	73	1	no	—
243	C₂×M₁₆	—	73	1	no	—
244	C₂×M₁₆	—	73	1	no	—
245	C₂²×C₈	—	73	1	no	—
246	C₂²×C₈	—	73	1	no	—
247	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
248	C₄⋊Q₈	—	292	2	no	—
249	#124	—	292	2	no	—
250	#123	—	292	2	no	—
251	#123	—	292	2	no	—
252	C₂²⋊Q₈	—	2	2	no	—
253	C₂²⋊Q₈	—	146	2	no	—
254	C₂²⋊Q₈	—	146	2	no	—
255	C₂²⋊Q₈	—	146	2	no	—
256	#117	—	146	2	no	—
257	Q₃₂	—	2	2	no	—
258	Q₃₂	—	146	4	no	—
259	Q₃₂	—	146	2	no	—
260	M₃₂	—	146	2	no	—
261	M₃₂	—	146	2	no	—
262	M₃₂	—	146	2	no	—
263	C₂×C₁₆	—	2	2	no	—
264	C₂×C₁₆	—	73	1	no	—
265	C₂×C₁₆	—	73	4	no	—
266	C₂×C₁₆	—	73	1	no	—
267	C₂×C₁₆	—	146	2	no	—
268	C₂×C₁₆	C₂²×D₇₃	1	1	yes	characteristic
269	C₂×C₁₆	C₂²×D₇₃	1	1	yes	characteristic
270	#107	—	73	2	no	—
271	#107	C₂²×D₇₃	1	2	yes	—
272	#106	—	146	2	no	—
273	#103	—	2	2	no	—
274	#103	—	146	2	no	—
275	#103	—	146	2	no	—
276	#103	—	146	2	no	—
277	#41	—	146	2	no	—
278	#41	—	146	2	no	—
279	C₂×Q₈	—	2	2	no	—
280	C₂×Q₈	—	292	2	no	—
281	C₂×Q₈	—	292	4	no	—
282	C₂×Q₈	—	292	4	no	—
283	C₂×Q₈	—	73	1	no	—
284	C₂×Q₈	—	146	2	no	—
285	C₂×D₄	—	73	1	no	—
286	C₂×D₄	—	73	1	no	—
287	C₂×D₄	—	73	1	no	—
288	C₂²×C₄	—	292	2	no	—
289	C₂²×C₄	—	73	1	no	—
290	C₂²×C₄	—	73	1	no	—
291	C₂²×C₄	D₄×D₇₃	1	1	yes	characteristic
292	Q₁₆	—	2	2	no	—
293	Q₁₆	—	2	2	no	—
294	Q₁₆	—	292	4	no	—
295	Q₁₆	—	146	2	no	—
296	Q₁₆	—	146	2	no	—
297	M₁₆	—	146	2	no	—
298	M₁₆	—	146	2	no	—
299	M₁₆	—	146	2	no	—
300	C₂×C₈	—	2	1	no	—
301	C₂×C₈	—	2	1	no	—
302	C₂×C₈	—	73	1	no	—
303	C₂×C₈	—	73	2	no	—
304	C₂×C₈	—	73	2	no	—
305	C₂×C₈	—	73	1	no	—
306	C₂×C₈	—	146	1	no	—
307	C₂×C₈	—	146	1	no	—
308	C₂×C₈	D₄×D₇₃	1	1	yes	characteristic
309	C₂×C₈	C₂³×D₇₃	1	1	yes	characteristic
310	C₄⋊C₄	—	2	2	no	—
311	C₄⋊C₄	—	4	2	no	—
312	C₄⋊C₄	—	292	2	no	—
313	C₄⋊C₄	—	292	2	no	—
314	C₄⋊C₄	—	292	2	no	—
315	C₄⋊C₄	—	146	2	no	—
316	C₂²⋊C₄	—	4	2	no	—
317	C₂²⋊C₄	—	292	2	no	—
318	C₂²⋊C₄	—	292	2	no	—
319	C₂²⋊C₄	—	292	2	no	—
320	C₂²⋊C₄	—	292	2	no	—
321	C₄²	—	292	2	no	—
322	C₁₆	—	2	1	no	—
323	C₁₆	—	73	2	no	—
324	C₁₆	—	146	1	no	—
325	C₁₆	D₄×D₇₃	1	2	yes	—
326	C₂³	—	73	1	no	—
327	C₂³	—	73	1	no	—
328	C₂³	D₈×D₇₃	1	1	yes	characteristic
329	Q₈	—	4	2	no	—
330	Q₈	—	4	2	no	—
331	Q₈	—	292	2	no	—
332	Q₈	—	292	2	no	—
333	Q₈	—	584	4	no	—
334	Q₈	—	146	2	no	—
335	D₄	—	146	2	no	—
336	D₄	—	146	2	no	—
337	D₄	—	146	2	no	—
338	C₂×C₄	—	2	1	no	—
339	C₂×C₄	—	2	1	no	—
340	C₂×C₄	—	4	2	no	—
341	C₂×C₄	—	4	2	no	—
342	C₂×C₄	—	292	2	no	—
343	C₂×C₄	—	292	2	no	—
344	C₂×C₄	—	584	4	no	—
345	C₂×C₄	—	73	1	no	—
346	C₂×C₄	—	73	2	no	—
347	C₂×C₄	—	73	2	no	—
348	C₂×C₄	—	73	1	no	—
349	C₂×C₄	—	146	1	no	—
350	C₂×C₄	—	146	1	no	—
351	C₂×C₄	D₈×D₇₃	1	1	yes	characteristic
352	C₂×C₄	C₂×D₄×D₇₃	1	1	yes	characteristic
353	C₈	—	2	1	no	—
354	C₈	—	73	2	no	—
355	C₈	—	146	1	no	—
356	C₈	C₂×D₄×D₇₃	1	1	yes	characteristic
357	C₈	#41×D₇₃	1	1	yes	characteristic
358	C₂²	—	2	1	no	—
359	C₂²	—	2	1	no	—
360	C₂²	—	2	1	no	—
361	C₂²	—	73	2	no	—
362	C₂²	—	146	1	no	—
363	C₂²	—	146	1	no	—
364	C₂²	—	146	1	no	—
365	C₂²	—	146	1	no	—
366	C₂²	—	146	1	no	—
367	C₂²	C₂×D₈×D₇₃	1	1	yes	characteristic
368	C₄	—	2	1	no	—
369	C₄	—	8	2	no	—
370	C₄	—	8	2	no	—
371	C₄	—	584	2	no	—
372	C₄	—	584	2	no	—
373	C₄	—	73	2	no	—
374	C₄	—	146	1	no	—
375	C₄	#121×D₇₃	1	1	yes	characteristic
376	C₄	#135×D₇₃	1	1	yes	characteristic
377	C₂	—	4	1	no	—
378	C₂	—	292	1	no	—
379	C₂	—	146	2	no	—
380	C₂	#466×D₇₃	1	1	yes	characteristic
381	C₂	#22623080	1	1	yes	characteristic
382	C₂	#22623081	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×#2216	C₂
socle	C₂²×C₇₃	C₂²×D₈
prime radical	C₂×#11924028	D₄
hypercenter	#2216	D₇₃
hypocenter	C₇₃	#33608
LCS	G ▷ C₂×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #2216

Complements

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

N	H ≅ G/N	type
1. C₂×Q₃₂×D₇₃	C₂	semidirect
4. #22612038	C₂	semidirect
5. #22612037	C₂	semidirect
6. #22612023	C₂	semidirect
7. #22612022	C₂	semidirect
8. #22612019	C₂	semidirect
9. #22611712	C₂	semidirect
10. #22611649	C₂	semidirect
11. C₇₃×#2216	C₂	semidirect
200. #2216	D₇₃	semidirect
203. C₇₃	#33608	semidirect

Maximal subgroups

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

subgroup	index
C₂×Q₃₂×D₇₃	2
C₂×#17989003	2
#22619581	2
#22612038	2
#22612037	2
#22612023	2
#22612022	2
#22612019	2
#22611712	2
#22611649	2
C₇₃×#2216	2
#33608	73