Subgroups — #33069196

Order 18688.

646 conjugacy classes of subgroups.

The subgroup lattice has 424 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22621074	C₂	1	2	yes	maximal
2	#494×D₇₃	C₂	1	1	yes	maximal, characteristic
3	#22618862	C₂	1	1	yes	maximal, characteristic
4	#22609426	C₂	1	1	yes	maximal, characteristic, complemented
5	#22609425	C₂	1	1	yes	maximal, characteristic, complemented
6	#22609422	C₂	1	1	yes	maximal, characteristic, complemented
7	#22609300	C₂	1	2	yes	maximal, complemented
8	#22609299	C₂	1	2	yes	maximal, complemented
9	#22608676	C₂	1	1	yes	maximal, characteristic
10	#22608601	C₂	1	2	yes	maximal
11	C₇₃×#1690	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×Q₁₆×D₇₃	—	2	1	no	—
13	C₂×Q₁₆×D₇₃	—	2	1	no	—
14	#17989141	C₂²	1	1	yes	characteristic
15	C₄⋊Q₈×D₇₃	C₂²	1	2	yes	—
16	#17988984	—	2	2	no	—
17	#17988967	C₂²	1	1	yes	characteristic
18	#17988966	C₂²	1	2	yes	—
19	#17988965	—	2	1	no	—
20	#17988965	—	2	1	no	—
21	#17988936	—	2	2	no	—
22	#17988919	—	2	2	no	—
23	#17988916	—	2	2	no	—
24	#17988873	C₂²	1	2	yes	—
25	#17988872	C₂²	1	2	yes	—
26	#17988869	C₂²	1	2	yes	—
27	#17988717	C₂²	1	1	yes	characteristic
28	#17988715	—	2	1	no	—
29	Q₈×#10596831	C₂²	1	1	yes	characteristic
30	#17988711	—	2	1	no	—
31	#17988711	C₂²	1	1	yes	characteristic
32	#17988639	C₂²	1	2	yes	—
33	#17988612	—	2	1	no	—
34	#17988609	C₂²	1	1	yes	characteristic
35	#17988544	—	2	1	no	—
36	C₄×C₈×D₇₃	C₂²	1	1	yes	characteristic
37	#17988339	C₂²	1	2	yes	—
38	#17988338	C₂²	1	1	yes	characteristic
39	#17988334	C₂²	1	2	yes	—
40	#17988333	C₂²	1	2	yes	—
41	#17988332	C₂²	1	2	yes	—
42	#494×C₇₃	C₂²	1	1	yes	characteristic
43	#475×C₇₃	C₂²	1	2	yes	—
44	#446×C₇₃	C₂²	1	1	yes	characteristic
45	#334×C₇₃	C₂²	1	1	yes	characteristic
46	#332×C₇₃	C₂²	1	2	yes	—
47	C₂×Q₈×D₇₃	—	2	2	no	—
48	C₂×Dic₂₉₂	—	2	1	no	—
49	C₂×Dic₂₉₂	—	2	1	no	—
50	C₂×#11924018	—	2	2	no	—
51	C₂×#11924018	—	2	2	no	—
52	Q₁₆×D₇₃	—	4	4	no	—
53	Q₁₆×D₇₃	—	4	4	no	—
54	C₂×#11924011	—	2	1	no	—
55	C₂×C₈×D₇₃	D₄	1	1	yes	characteristic
56	C₂×C₈×D₇₃	D₄	1	1	yes	characteristic
57	#14556591	C₂³	1	2	yes	—
58	#14556590	—	2	2	no	—
59	#14556590	—	2	2	no	—
60	#14556590	D₄	1	1	yes	characteristic
61	#14556589	—	2	2	no	—
62	#14556589	D₄	1	1	yes	characteristic
63	#14556581	—	2	1	no	—
64	#14556571	D₄	1	1	yes	characteristic
65	#14556569	—	2	1	no	—
66	#14556567	D₄	1	1	yes	characteristic
67	C₄×Dic₁₄₆	—	2	1	no	—
68	C₄×Dic₁₄₆	C₂³	1	1	yes	characteristic
69	#14556548	—	2	1	no	—
70	C₄⋊C₄×D₇₃	—	2	2	no	—
71	C₄⋊C₄×D₇₃	—	2	2	no	—
72	C₄²×D₇₃	C₂³	1	1	yes	characteristic
73	#14556523	—	2	2	no	—
74	#14556522	—	2	2	no	—
75	#14556522	—	2	2	no	—
76	C₄⋊#10596831	—	2	1	no	—
77	C₄⋊#10596831	C₂³	1	1	yes	characteristic
78	C₄⋊#10596831	C₂³	1	2	yes	—
79	#14556520	—	2	1	no	—
80	#14556520	—	2	1	no	—
81	#14556520	—	2	1	no	—
82	#14556516	—	2	2	no	—
83	#14556515	—	2	2	no	—
84	#14556514	—	2	2	no	—
85	#14556503	—	2	2	no	—
86	#14556501	—	2	1	no	—
87	C₄×#10596831	—	2	1	no	—
88	C₄×#10596831	C₂³	1	1	yes	characteristic
89	C₈×Dic₇₃	—	2	1	no	—
90	C₈×Dic₇₃	—	2	2	no	—
91	C₈×Dic₇₃	D₄	1	1	yes	characteristic
92	C₈×Dic₇₃	D₄	1	1	yes	characteristic
93	C₂×Q₁₆×C₇₃	—	2	1	no	—
94	C₂×Q₁₆×C₇₃	—	2	1	no	—
95	C₄⋊Q₈×C₇₃	C₂³	1	2	yes	—
96	C₄×Q₈×C₇₃	C₂³	1	1	yes	characteristic
97	#106×C₇₃	—	2	2	no	—
98	C₄⋊C₈×C₇₃	—	2	1	no	—
99	C₄⋊C₈×C₇₃	C₂³	1	1	yes	characteristic
100	C₄⋊C₈×C₇₃	C₂³	1	2	yes	—
101	#103×C₇₃	—	2	2	no	—
102	#97×C₇₃	—	2	1	no	—
103	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
104	C₂×Dic₁₄₆	—	2	1	no	—
105	C₂×Dic₁₄₆	—	2	2	no	—
106	C₂×Dic₁₄₆	—	4	2	no	—
107	C₂×Dic₁₄₆	C₂×D₄	1	1	yes	characteristic
108	Q₈×D₇₃	—	4	4	no	—
109	Q₈×D₇₃	—	4	4	no	—
110	C₂×C₄×D₇₃	—	4	2	no	—
111	C₂×C₄×D₇₃	C₂×D₄	1	1	yes	characteristic
112	C₂×C₄×D₇₃	C₂×D₄	1	1	yes	characteristic
113	C₂×C₄×D₇₃	#41	1	1	yes	characteristic
114	Dic₂₉₂	—	4	2	no	—
115	Dic₂₉₂	—	4	2	no	—
116	#11924018	—	4	4	no	—
117	#11924018	—	4	4	no	—
118	#11924011	—	4	2	no	—
119	C₂×#10596831	—	2	2	no	—
120	C₂×#10596831	—	2	1	no	—
121	C₂×#10596831	C₂×D₄	1	1	yes	characteristic
122	C₂×#10596831	C₂×D₄	1	1	yes	characteristic
123	C₈×D₇₃	—	2	2	no	—
124	C₈×D₇₃	SD₁₆	1	4	yes	—
125	C₄⋊Dic₇₃	—	2	1	no	—
126	C₄⋊Dic₇₃	—	2	2	no	—
127	C₄⋊Dic₇₃	—	2	2	no	—
128	C₄⋊Dic₇₃	C₂×D₄	1	1	yes	characteristic
129	#11924007	—	2	1	no	—
130	#11924007	—	2	1	no	—
131	#11924007	—	2	1	no	—
132	#11924007	—	2	1	no	—
133	#11924007	—	2	1	no	—
134	#11924007	—	4	2	no	—
135	#11924007	—	4	2	no	—
136	#11924005	—	2	1	no	—
137	#11924005	—	2	1	no	—
138	C₄×Dic₇₃	—	2	1	no	—
139	C₄×Dic₇₃	C₂×D₄	1	1	yes	characteristic
140	C₄×Dic₇₃	C₂×D₄	1	1	yes	characteristic
141	C₄×Dic₇₃	#41	1	1	yes	characteristic
142	C₂×Q₈×C₇₃	—	2	2	no	—
143	C₂×Q₈×C₇₃	C₂×D₄	1	1	yes	characteristic
144	Q₁₆×C₇₃	—	4	2	no	—
145	Q₁₆×C₇₃	—	4	2	no	—
146	C₂×C₈×C₇₃	—	2	1	no	—
147	C₂×C₈×C₇₃	—	2	2	no	—
148	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
149	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
150	C₄⋊C₄×C₇₃	—	2	1	no	—
151	C₄⋊C₄×C₇₃	—	2	2	no	—
152	C₄⋊C₄×C₇₃	—	2	2	no	—
153	C₄⋊C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
154	C₄²×C₇₃	—	2	1	no	—
155	C₄²×C₇₃	C₂⁴	1	1	yes	characteristic
156	C₂²×D₇₃	#121	1	1	yes	characteristic
157	Q₈×C₇₃	—	2	2	no	—
158	Q₈×C₇₃	—	4	2	no	—
159	Q₈×C₇₃	—	4	2	no	—
160	C₂×C₄×C₇₃	—	2	1	no	—
161	C₂×C₄×C₇₃	—	2	1	no	—
162	C₂×C₄×C₇₃	—	4	2	no	—
163	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
164	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
165	C₂×C₄×C₇₃	C₂×#41	1	1	yes	characteristic
166	C₂×Dic₇₃	—	2	1	no	—
167	C₂×Dic₇₃	—	2	1	no	—
168	C₂×Dic₇₃	—	4	2	no	—
169	C₂×Dic₇₃	#121	1	1	yes	characteristic
170	C₂×Dic₇₃	#121	1	1	yes	characteristic
171	C₂×Dic₇₃	#121	1	1	yes	characteristic
172	C₄×D₇₃	—	2	2	no	—
173	C₄×D₇₃	—	2	2	no	—
174	C₄×D₇₃	—	8	4	no	—
175	C₄×D₇₃	C₂×SD₁₆	1	2	yes	—
176	C₄×D₇₃	AGL(1,ℤ/8)	1	2	yes	—
177	Dic₁₄₆	—	2	2	no	—
178	Dic₁₄₆	—	4	2	no	—
179	Dic₁₄₆	—	4	2	no	—
180	Dic₁₄₆	—	4	2	no	—
181	Dic₁₄₆	—	8	4	no	—
182	C₈×C₇₃	—	2	1	no	—
183	C₈×C₇₃	—	4	1	no	—
184	C₈×C₇₃	—	4	2	no	—
185	C₈×C₇₃	C₂×SD₁₆	1	2	yes	—
186	#10596831	—	2	1	no	—
187	#10596831	—	4	1	no	—
188	#10596831	—	4	2	no	—
189	#10596831	C₂×SD₁₆	1	2	yes	—
190	C₂²×C₇₃	C₂×#121	1	1	yes	characteristic
191	C₂×D₇₃	—	2	1	no	—
192	C₂×D₇₃	—	2	1	no	—
193	C₂×D₇₃	#460	1	2	yes	—
194	C₄×C₇₃	—	2	1	no	—
195	C₄×C₇₃	—	4	1	no	—
196	C₄×C₇₃	—	4	1	no	—
197	C₄×C₇₃	—	8	2	no	—
198	C₄×C₇₃	C₂²×SD₁₆	1	1	yes	characteristic
199	C₄×C₇₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
200	C₄×C₇₃	#578	1	2	yes	—
201	Dic₇₃	—	2	1	no	—
202	Dic₇₃	—	4	1	no	—
203	Dic₇₃	—	4	1	no	—
204	Dic₇₃	—	8	2	no	—
205	Dic₇₃	#460	1	2	yes	—
206	Dic₇₃	#473	1	2	yes	—
207	#19204	—	73	1	no	maximal, Sylow
208	C₂×C₇₃	C₂×#460	1	1	yes	characteristic
209	C₂×C₇₃	#3059	1	1	yes	characteristic
210	C₂×C₇₃	C₂×#473	1	1	yes	characteristic
211	D₇₃	—	2	2	no	—
212	#3229	—	73	2	no	—
213	C₂×#494	—	73	1	no	—
214	#2960	—	73	1	no	—
215	#1690	—	73	1	no	—
216	#1690	—	73	1	no	—
217	#1690	—	73	1	no	—
218	#1690	D₇₃	1	1	yes	characteristic, complemented
219	#1670	—	73	2	no	—
220	#1670	—	73	2	no	—
221	#1564	—	73	1	no	—
222	#1546	—	73	2	no	—
223	C₇₃	#19204	1	1	yes	characteristic, complemented, Sylow
224	C₂²×Q₁₆	—	146	1	no	—
225	C₂²×Q₁₆	—	146	1	no	—
226	#533	—	73	1	no	—
227	C₂×C₄⋊Q₈	—	73	2	no	—
228	#498	—	146	2	no	—
229	#494	—	73	2	no	—
230	#494	—	73	1	no	—
231	#494	—	146	1	no	—
232	#494	—	146	1	no	—
233	#494	C₂×D₇₃	1	1	yes	characteristic
234	#487	—	146	2	no	—
235	#483	—	146	2	no	—
236	#483	—	146	2	no	—
237	#475	—	73	2	no	—
238	#475	—	73	2	no	—
239	#475	—	73	2	no	—
240	#475	C₂×D₇₃	1	2	yes	—
241	#446	—	73	1	no	—
242	#446	—	146	1	no	—
243	#446	C₂×D₇₃	1	1	yes	characteristic
244	C₈×Q₈	—	73	1	no	—
245	C₈×Q₈	—	73	1	no	—
246	C₈×Q₈	—	146	1	no	—
247	#431	—	73	2	no	—
248	#424	—	146	1	no	—
249	#423	—	73	1	no	—
250	#405	—	146	1	no	—
251	C₂×C₄×C₈	—	73	1	no	—
252	#335	—	73	2	no	—
253	#334	—	73	1	no	—
254	#334	C₂×D₇₃	1	1	yes	characteristic
255	#332	—	73	2	no	—
256	#332	—	73	2	no	—
257	#332	—	73	2	no	—
258	#332	C₂×D₇₃	1	2	yes	—
259	C₂²×Q₈	—	146	2	no	—
260	C₂×Q₁₆	—	2	1	no	—
261	C₂×Q₁₆	—	2	1	no	—
262	C₂×Q₁₆	—	292	4	no	—
263	C₂×Q₁₆	—	292	4	no	—
264	C₂×Q₁₆	—	146	2	no	—
265	C₂×Q₁₆	—	146	2	no	—
266	C₂×Q₁₆	—	146	1	no	—
267	C₂×Q₁₆	—	146	1	no	—
268	C₂×M₁₆	—	146	1	no	—
269	C₂²×C₈	—	73	1	no	—
270	C₂²×C₈	—	73	1	no	—
271	C₄⋊Q₈	—	73	1	no	—
272	C₄⋊Q₈	—	73	1	no	—
273	C₄⋊Q₈	—	73	2	no	—
274	C₄⋊Q₈	—	146	2	no	—
275	C₄⋊Q₈	—	146	2	no	—
276	C₄⋊Q₈	—	146	2	no	—
277	C₄⋊Q₈	C₂²×D₇₃	1	2	yes	—
278	#125	—	146	1	no	—
279	C₂²⋊Q₈	—	73	1	no	—
280	C₂²⋊Q₈	—	73	1	no	—
281	C₂²⋊Q₈	—	146	1	no	—
282	C₄×Q₈	—	73	1	no	—
283	C₄×Q₈	—	146	1	no	—
284	C₄×Q₈	C₂²×D₇₃	1	1	yes	characteristic
285	#117	—	146	1	no	—
286	C₂×C₄⋊C₄	—	146	2	no	—
287	C₂×C₄⋊C₄	—	146	2	no	—
288	C₂×C₄²	—	73	1	no	—
289	#106	—	2	2	no	—
290	#106	—	146	2	no	—
291	#106	—	146	2	no	—
292	#106	—	146	2	no	—
293	C₄⋊C₈	—	2	1	no	—
294	C₄⋊C₈	—	73	1	no	—
295	C₄⋊C₈	—	73	2	no	—
296	C₄⋊C₈	—	146	1	no	—
297	C₄⋊C₈	—	146	1	no	—
298	C₄⋊C₈	—	146	1	no	—
299	C₄⋊C₈	—	146	1	no	—
300	C₄⋊C₈	C₂²×D₇₃	1	1	yes	characteristic
301	C₄⋊C₈	C₂²×D₇₃	1	2	yes	—
302	#103	—	2	2	no	—
303	#103	—	146	2	no	—
304	#103	—	146	2	no	—
305	#103	—	146	2	no	—
306	C₂²⋊C₈	—	146	2	no	—
307	#97	—	2	1	no	—
308	#97	—	146	1	no	—
309	C₄×C₈	—	73	1	no	—
310	C₄×C₈	—	73	1	no	—
311	C₄×C₈	—	73	1	no	—
312	C₄×C₈	—	146	1	no	—
313	C₄×C₈	—	146	2	no	—
314	C₄×C₈	—	146	1	no	—
315	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
316	C₂×Q₈	—	2	2	no	—
317	C₂×Q₈	—	292	2	no	—
318	C₂×Q₈	—	292	4	no	—
319	C₂×Q₈	—	292	4	no	—
320	C₂×Q₈	—	73	1	no	—
321	C₂×Q₈	—	146	1	no	—
322	C₂×Q₈	—	146	2	no	—
323	C₂×Q₈	D₄×D₇₃	1	1	yes	characteristic
324	C₂²×C₄	—	292	2	no	—
325	C₂²×C₄	—	73	1	no	—
326	C₂²×C₄	—	73	1	no	—
327	C₂²×C₄	—	73	1	no	—
328	Q₁₆	—	4	2	no	—
329	Q₁₆	—	4	2	no	—
330	Q₁₆	—	292	2	no	—
331	Q₁₆	—	292	4	no	—
332	Q₁₆	—	292	4	no	—
333	Q₁₆	—	292	2	no	—
334	M₁₆	—	292	2	no	—
335	C₂×C₈	—	2	1	no	—
336	C₂×C₈	—	2	2	no	—
337	C₂×C₈	—	73	1	no	—
338	C₂×C₈	—	73	4	no	—
339	C₂×C₈	—	73	1	no	—
340	C₂×C₈	—	146	2	no	—
341	C₂×C₈	—	146	1	no	—
342	C₂×C₈	—	146	2	no	—
343	C₂×C₈	D₄×D₇₃	1	1	yes	characteristic
344	C₂×C₈	D₄×D₇₃	1	1	yes	characteristic
345	C₄⋊C₄	—	2	1	no	—
346	C₄⋊C₄	—	2	2	no	—
347	C₄⋊C₄	—	2	2	no	—
348	C₄⋊C₄	—	292	2	no	—
349	C₄⋊C₄	—	292	2	no	—
350	C₄⋊C₄	—	73	1	no	—
351	C₄⋊C₄	—	146	1	no	—
352	C₄⋊C₄	—	146	1	no	—
353	C₄⋊C₄	—	146	1	no	—
354	C₄⋊C₄	—	146	1	no	—
355	C₄⋊C₄	—	146	1	no	—
356	C₄⋊C₄	—	146	2	no	—
357	C₄⋊C₄	—	146	2	no	—
358	C₄⋊C₄	—	146	1	no	—
359	C₄⋊C₄	D₄×D₇₃	1	1	yes	characteristic
360	C₂²⋊C₄	—	146	1	no	—
361	C₂²⋊C₄	—	146	1	no	—
362	C₄²	—	2	1	no	—
363	C₄²	—	73	1	no	—
364	C₄²	—	73	1	no	—
365	C₄²	—	73	1	no	—
366	C₄²	—	146	1	no	—
367	C₄²	C₂³×D₇₃	1	1	yes	characteristic
368	C₂³	—	73	1	no	—
369	Q₈	—	2	2	no	—
370	Q₈	—	4	2	no	—
371	Q₈	—	4	2	no	—
372	Q₈	—	292	2	no	—
373	Q₈	—	292	2	no	—
374	Q₈	—	292	2	no	—
375	Q₈	—	584	4	no	—
376	Q₈	—	146	2	no	—
377	C₂×C₄	—	2	1	no	—
378	C₂×C₄	—	2	1	no	—
379	C₂×C₄	—	4	2	no	—
380	C₂×C₄	—	292	2	no	—
381	C₂×C₄	—	584	4	no	—
382	C₂×C₄	—	73	1	no	—
383	C₂×C₄	—	73	1	no	—
384	C₂×C₄	—	73	2	no	—
385	C₂×C₄	—	73	2	no	—
386	C₂×C₄	—	73	1	no	—
387	C₂×C₄	—	146	2	no	—
388	C₂×C₄	—	146	1	no	—
389	C₂×C₄	—	146	1	no	—
390	C₂×C₄	—	146	2	no	—
391	C₂×C₄	C₂×D₄×D₇₃	1	1	yes	characteristic
392	C₂×C₄	C₂×D₄×D₇₃	1	1	yes	characteristic
393	C₂×C₄	#41×D₇₃	1	1	yes	characteristic
394	C₈	—	2	1	no	—
395	C₈	—	4	2	no	—
396	C₈	—	4	1	no	—
397	C₈	—	292	2	no	—
398	C₈	—	292	1	no	—
399	C₈	—	73	2	no	—
400	C₈	—	146	1	no	—
401	C₈	SD₁₆×D₇₃	1	2	yes	—
402	C₂²	—	73	2	no	—
403	C₂²	—	146	1	no	—
404	C₂²	—	146	1	no	—
405	C₂²	#121×D₇₃	1	1	yes	characteristic
406	C₄	—	2	1	no	—
407	C₄	—	4	1	no	—
408	C₄	—	4	1	no	—
409	C₄	—	8	2	no	—
410	C₄	—	292	1	no	—
411	C₄	—	292	1	no	—
412	C₄	—	584	2	no	—
413	C₄	—	73	2	no	—
414	C₄	—	73	2	no	—
415	C₄	—	146	1	no	—
416	C₄	C₂×SD₁₆×D₇₃	1	1	yes	characteristic
417	C₄	AGL(1,ℤ/8)×D₇₃	1	1	yes	characteristic
418	C₄	#17989403	1	2	yes	—
419	C₂	—	146	2	no	—
420	C₂	#460×D₇₃	1	1	yes	characteristic
421	C₂	#22619584	1	1	yes	characteristic
422	C₂	#473×D₇₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#121×D₇₃
commutator, cosocle	C₄²×C₇₃	C₂⁴
Frattini	C₄²	C₂³×D₇₃
Fitting	C₇₃×#1690	C₂
socle	C₂²×C₇₃	C₂×#121
prime radical	C₂²×D₇₃	#121
hypercenter	#1690	D₇₃
hypocenter	C₇₃	#19204
LCS	G ▷ C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #1690

Complements

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

N	H ≅ G/N	type
4. #22609426	C₂	semidirect
5. #22609425	C₂	semidirect
6. #22609422	C₂	semidirect
7. #22609300	C₂	semidirect
8. #22609299	C₂	semidirect
11. C₇₃×#1690	C₂	semidirect
218. #1690	D₇₃	semidirect
223. C₇₃	#19204	semidirect

Maximal subgroups

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

subgroup	index
#22621074	2
#494×D₇₃	2
#22618862	2
#22609426	2
#22609425	2
#22609422	2
#22609300	2
#22609299	2
#22608676	2
#22608601	2
C₇₃×#1690	2
#19204	73