Subgroups — #33068351

Order 18688.

606 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22620442	C₂	1	1	yes	maximal, characteristic
2	#22619969	C₂	1	2	yes	maximal
3	C₈×Q₈×D₇₃	C₂	1	1	yes	maximal, characteristic
4	#22609336	C₂	1	1	yes	maximal, characteristic, complemented
5	#22609335	C₂	1	1	yes	maximal, characteristic, complemented
6	#22609322	C₂	1	1	yes	maximal, characteristic, complemented
7	#22609312	C₂	1	2	yes	maximal, complemented
8	#22609311	C₂	1	2	yes	maximal, complemented
9	#22608678	C₂	1	1	yes	maximal, characteristic
10	#22608587	C₂	1	2	yes	maximal
11	C₇₃×#1674	C₂	1	1	yes	maximal, characteristic, complemented
12	#17989142	C₂²	1	2	yes	—
13	C₄×Q₈×D₇₃	C₂²	1	1	yes	characteristic
14	#17988992	C₂²	1	1	yes	characteristic
15	#17988988	—	2	1	no	—
16	#17988988	—	2	1	no	—
17	#17988983	C₂²	1	2	yes	—
18	#17988921	—	2	2	no	—
19	#17988883	C₂²	1	2	yes	—
20	#17988882	C₂²	1	2	yes	—
21	#17988794	C₂²	1	2	yes	—
22	#17988741	—	2	2	no	—
23	Q₈×#10596831	C₂²	1	1	yes	characteristic
24	C₈×Dic₁₄₆	—	2	1	no	—
25	C₈×Dic₁₄₆	C₂²	1	1	yes	characteristic
26	#17988711	—	2	1	no	—
27	#17988711	C₂²	1	1	yes	characteristic
28	Q₁₆×Dic₇₃	—	2	2	no	—
29	#17988679	—	2	2	no	—
30	#17988639	C₂²	1	2	yes	—
31	#17988624	—	2	1	no	—
32	#17988624	—	2	1	no	—
33	C₄⋊C₈×D₇₃	—	2	1	no	—
34	C₄⋊C₈×D₇₃	C₂²	1	1	yes	characteristic
35	C₄×C₈×D₇₃	—	2	1	no	—
36	C₄×C₈×D₇₃	C₂²	1	1	yes	characteristic
37	#17988340	C₂²	1	1	yes	characteristic
38	#17988337	C₂²	1	2	yes	—
39	#17988335	C₂²	1	2	yes	—
40	#17988335	C₂²	1	2	yes	—
41	#17988333	C₂²	1	2	yes	—
42	#500×C₇₃	C₂²	1	1	yes	characteristic
43	#462×C₇₃	C₂²	1	2	yes	—
44	C₈×Q₈×C₇₃	C₂²	1	1	yes	characteristic
45	#335×C₇₃	C₂²	1	1	yes	characteristic
46	#332×C₇₃	C₂²	1	2	yes	—
47	C₂×Q₈×D₇₃	D₄	1	1	yes	characteristic
48	C₂×#11924018	—	2	2	no	—
49	C₂×C₈×D₇₃	—	2	1	no	—
50	C₂×C₈×D₇₃	D₄	1	1	yes	characteristic
51	C₂×C₈×D₇₃	D₄	1	1	yes	characteristic
52	#14556591	C₂³	1	2	yes	—
53	#14556582	—	2	2	no	—
54	#14556571	—	2	2	no	—
55	#14556567	—	2	2	no	—
56	Q₈×Dic₇₃	—	2	2	no	—
57	Q₈×Dic₇₃	D₄	1	1	yes	characteristic
58	C₄×Dic₁₄₆	—	2	1	no	—
59	C₄×Dic₁₄₆	C₂³	1	1	yes	characteristic
60	#14556556	—	2	1	no	—
61	#14556556	—	2	2	no	—
62	#14556556	D₄	1	1	yes	characteristic
63	#14556549	—	2	2	no	—
64	C₄⋊C₄×D₇₃	—	2	1	no	—
65	C₄⋊C₄×D₇₃	D₄	1	1	yes	characteristic
66	C₄²×D₇₃	—	2	1	no	—
67	C₄²×D₇₃	C₂³	1	1	yes	characteristic
68	#14556525	—	2	2	no	—
69	#14556525	—	2	1	no	—
70	#14556525	—	2	1	no	—
71	#14556524	—	2	2	no	—
72	#14556522	—	2	2	no	—
73	#14556522	—	2	2	no	—
74	C₄⋊#10596831	—	2	1	no	—
75	C₄⋊#10596831	C₂³	1	2	yes	—
76	C₄⋊#10596831	C₂³	1	1	yes	characteristic
77	#14556520	—	2	1	no	—
78	#14556520	—	2	1	no	—
79	#14556520	—	2	1	no	—
80	#14556516	—	2	2	no	—
81	#14556515	—	2	2	no	—
82	#14556515	—	2	2	no	—
83	#14556503	—	2	2	no	—
84	C₄×#10596831	—	2	1	no	—
85	C₄×#10596831	C₂³	1	1	yes	characteristic
86	C₈×Dic₇₃	—	2	1	no	—
87	C₈×Dic₇₃	—	2	2	no	—
88	C₈×Dic₇₃	—	2	1	no	—
89	C₈×Dic₇₃	D₄	1	1	yes	characteristic
90	C₈×Dic₇₃	D₄	1	1	yes	characteristic
91	C₂×Q₁₆×C₇₃	—	2	2	no	—
92	C₄⋊Q₈×C₇₃	C₂³	1	2	yes	—
93	C₄×Q₈×C₇₃	C₂³	1	1	yes	characteristic
94	#107×C₇₃	—	2	1	no	—
95	#107×C₇₃	—	2	1	no	—
96	C₄⋊C₈×C₇₃	—	2	1	no	—
97	C₄⋊C₈×C₇₃	C₂³	1	2	yes	—
98	C₄⋊C₈×C₇₃	C₂³	1	1	yes	characteristic
99	#103×C₇₃	—	2	2	no	—
100	C₄×C₈×C₇₃	—	2	1	no	—
101	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
102	C₂×Dic₁₄₆	—	2	1	no	—
103	C₂×Dic₁₄₆	—	2	2	no	—
104	C₂×Dic₁₄₆	C₂×D₄	1	1	yes	characteristic
105	Q₈×D₇₃	—	2	2	no	—
106	Q₈×D₇₃	—	2	2	no	—
107	C₂×C₄×D₇₃	—	2	1	no	—
108	C₂×C₄×D₇₃	—	2	1	no	—
109	C₂×C₄×D₇₃	C₂×D₄	1	1	yes	characteristic
110	C₂×C₄×D₇₃	C₂×D₄	1	1	yes	characteristic
111	C₂×C₄×D₇₃	#41	1	1	yes	characteristic
112	#11924018	—	4	4	no	—
113	C₂×#10596831	—	2	1	no	—
114	C₂×#10596831	—	2	2	no	—
115	C₂×#10596831	C₂×D₄	1	1	yes	characteristic
116	C₂×#10596831	C₂×D₄	1	1	yes	characteristic
117	C₈×D₇₃	—	2	2	no	—
118	C₈×D₇₃	—	4	2	no	—
119	C₈×D₇₃	Q₁₆	1	4	yes	complemented
120	C₄⋊Dic₇₃	—	2	1	no	—
121	C₄⋊Dic₇₃	—	2	2	no	—
122	C₄⋊Dic₇₃	—	2	2	no	—
123	C₄⋊Dic₇₃	—	4	2	no	—
124	C₄⋊Dic₇₃	C₂×D₄	1	1	yes	characteristic
125	#11924007	—	2	1	no	—
126	#11924007	—	2	1	no	—
127	#11924007	—	2	1	no	—
128	#11924007	—	4	2	no	—
129	#11924005	—	4	2	no	—
130	C₄×Dic₇₃	—	2	1	no	—
131	C₄×Dic₇₃	—	2	1	no	—
132	C₄×Dic₇₃	—	2	1	no	—
133	C₄×Dic₇₃	—	4	2	no	—
134	C₄×Dic₇₃	C₂×D₄	1	1	yes	characteristic
135	C₄×Dic₇₃	C₂×D₄	1	1	yes	characteristic
136	C₄×Dic₇₃	#41	1	1	yes	characteristic
137	C₂×Q₈×C₇₃	—	2	2	no	—
138	C₂×Q₈×C₇₃	C₂×D₄	1	1	yes	characteristic
139	Q₁₆×C₇₃	—	4	4	no	—
140	C₂×C₈×C₇₃	—	2	1	no	—
141	C₂×C₈×C₇₃	—	2	2	no	—
142	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
143	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
144	C₄⋊C₄×C₇₃	—	2	1	no	—
145	C₄⋊C₄×C₇₃	—	2	2	no	—
146	C₄⋊C₄×C₇₃	—	2	2	no	—
147	C₄⋊C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
148	C₄²×C₇₃	—	2	1	no	—
149	C₄²×C₇₃	C₂⁴	1	1	yes	characteristic
150	C₂²×D₇₃	#121	1	1	yes	characteristic
151	Q₈×C₇₃	—	2	1	no	—
152	Q₈×C₇₃	—	4	2	no	—
153	Q₈×C₇₃	—	4	2	no	—
154	Q₈×C₇₃	#135	1	2	yes	—
155	C₂×C₄×C₇₃	—	2	1	no	—
156	C₂×C₄×C₇₃	—	2	1	no	—
157	C₂×C₄×C₇₃	—	4	2	no	—
158	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
159	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
160	C₂×C₄×C₇₃	C₂×#41	1	1	yes	characteristic
161	C₂×Dic₇₃	—	2	1	no	—
162	C₂×Dic₇₃	—	2	1	no	—
163	C₂×Dic₇₃	—	4	2	no	—
164	C₂×Dic₇₃	#121	1	1	yes	characteristic
165	C₂×Dic₇₃	#121	1	1	yes	characteristic
166	C₂×Dic₇₃	#121	1	1	yes	characteristic
167	C₄×D₇₃	—	2	4	no	—
168	C₄×D₇₃	—	2	1	no	—
169	C₄×D₇₃	—	2	1	no	—
170	C₄×D₇₃	—	4	1	no	—
171	C₄×D₇₃	—	4	2	no	—
172	C₄×D₇₃	C₂×Q₁₆	1	2	yes	—
173	C₄×D₇₃	#137	1	2	yes	—
174	Dic₁₄₆	—	2	4	no	—
175	Dic₁₄₆	—	2	1	no	—
176	Dic₁₄₆	—	8	2	no	—
177	Dic₁₄₆	#135	1	2	yes	—
178	C₈×C₇₃	—	2	1	no	—
179	C₈×C₇₃	—	4	2	no	—
180	C₈×C₇₃	—	4	1	no	—
181	C₈×C₇₃	C₂×Q₁₆	1	2	yes	—
182	#10596831	—	2	1	no	—
183	#10596831	—	4	2	no	—
184	#10596831	—	4	1	no	—
185	#10596831	C₂×Q₁₆	1	2	yes	—
186	C₂²×C₇₃	C₂×#121	1	1	yes	characteristic
187	C₂×D₇₃	—	2	1	no	—
188	C₂×D₇₃	—	2	1	no	—
189	C₂×D₇₃	#462	1	2	yes	—
190	C₄×C₇₃	—	2	2	no	—
191	C₄×C₇₃	—	2	1	no	—
192	C₄×C₇₃	—	4	1	no	—
193	C₄×C₇₃	—	8	2	no	—
194	C₄×C₇₃	C₂²×Q₁₆	1	1	yes	characteristic
195	C₄×C₇₃	C₂×#135	1	1	yes	characteristic
196	C₄×C₇₃	C₂×#135	1	1	yes	characteristic
197	C₄×C₇₃	C₂×#137	1	1	yes	characteristic
198	Dic₇₃	—	2	1	no	—
199	Dic₇₃	—	2	2	no	—
200	Dic₇₃	—	2	1	no	—
201	Dic₇₃	—	4	1	no	—
202	Dic₇₃	—	8	2	no	—
203	Dic₇₃	#462	1	2	yes	—
204	#19118	—	73	1	no	maximal, Sylow
205	C₂×C₇₃	C₂×#462	1	1	yes	characteristic
206	C₂×C₇₃	#3050	1	1	yes	characteristic
207	C₂×C₇₃	#3067	1	1	yes	characteristic
208	D₇₃	—	2	2	no	—
209	#3160	—	73	1	no	—
210	#3104	—	73	2	no	—
211	C₂×C₈×Q₈	—	73	1	no	—
212	#1676	—	73	1	no	—
213	#1676	—	73	1	no	—
214	#1674	—	73	1	no	—
215	#1674	D₇₃	1	1	yes	characteristic, complemented
216	#1672	—	73	2	no	—
217	#1672	—	73	2	no	—
218	#1564	—	73	1	no	—
219	#1542	—	73	2	no	—
220	C₇₃	#19118	1	1	yes	characteristic, complemented, Sylow
221	#533	—	73	2	no	—
222	C₂×C₄×Q₈	—	73	1	no	—
223	#500	—	73	1	no	—
224	#500	C₂×D₇₃	1	1	yes	characteristic
225	#499	—	146	1	no	—
226	#499	—	146	1	no	—
227	#498	—	73	2	no	—
228	#484	—	146	2	no	—
229	#477	—	73	2	no	—
230	#477	—	73	2	no	—
231	#462	—	73	2	no	—
232	#462	C₂×D₇₃	1	2	yes	—
233	#451	—	146	2	no	—
234	C₈×Q₈	—	73	1	no	—
235	C₈×Q₈	—	73	1	no	—
236	C₈×Q₈	—	73	1	no	—
237	C₈×Q₈	—	146	1	no	—
238	C₈×Q₈	—	146	1	no	—
239	C₈×Q₈	C₂×D₇₃	1	1	yes	characteristic
240	C₄×Q₁₆	—	146	2	no	—
241	C₄×Q₁₆	—	146	2	no	—
242	#431	—	73	2	no	—
243	#427	—	146	1	no	—
244	#427	—	146	1	no	—
245	C₂×C₄⋊C₈	—	73	1	no	—
246	C₂×C₄⋊C₈	—	146	1	no	—
247	C₂×C₄×C₈	—	73	1	no	—
248	C₂×C₄×C₈	—	146	1	no	—
249	#335	—	73	1	no	—
250	#335	C₂×D₇₃	1	1	yes	characteristic
251	#334	—	73	2	no	—
252	#333	—	73	2	no	—
253	#333	—	73	2	no	—
254	#332	—	73	2	no	—
255	#332	C₂×D₇₃	1	2	yes	—
256	C₂²×Q₈	—	73	1	no	—
257	C₂×Q₁₆	—	2	2	no	—
258	C₂×Q₁₆	—	146	2	no	—
259	C₂²×C₈	—	73	1	no	—
260	C₂²×C₈	—	73	1	no	—
261	C₂²×C₈	—	146	1	no	—
262	C₄⋊Q₈	—	73	2	no	—
263	C₄⋊Q₈	C₂²×D₇₃	1	2	yes	—
264	#125	—	146	2	no	—
265	C₂²⋊Q₈	—	146	2	no	—
266	C₂²⋊Q₈	—	146	2	no	—
267	C₄×Q₈	—	73	1	no	—
268	C₄×Q₈	—	73	1	no	—
269	C₄×Q₈	—	73	1	no	—
270	C₄×Q₈	—	146	1	no	—
271	C₄×Q₈	—	146	2	no	—
272	C₄×Q₈	—	146	2	no	—
273	C₄×Q₈	—	146	1	no	—
274	C₄×Q₈	C₂²×D₇₃	1	1	yes	characteristic
275	#117	—	146	2	no	—
276	C₂×C₄⋊C₄	—	73	1	no	—
277	C₂×C₄⋊C₄	—	146	1	no	—
278	C₂×C₄²	—	73	1	no	—
279	C₂×C₄²	—	146	1	no	—
280	#107	—	2	1	no	—
281	#107	—	2	1	no	—
282	#107	—	146	2	no	—
283	#107	—	146	1	no	—
284	#107	—	146	1	no	—
285	#107	—	146	2	no	—
286	#106	—	146	2	no	—
287	#106	—	146	2	no	—
288	C₄⋊C₈	—	2	1	no	—
289	C₄⋊C₈	—	73	1	no	—
290	C₄⋊C₈	—	73	2	no	—
291	C₄⋊C₈	—	146	1	no	—
292	C₄⋊C₈	—	146	1	no	—
293	C₄⋊C₈	—	146	1	no	—
294	C₄⋊C₈	—	146	1	no	—
295	C₄⋊C₈	C₂²×D₇₃	1	1	yes	characteristic
296	C₄⋊C₈	C₂²×D₇₃	1	2	yes	—
297	#103	—	2	2	no	—
298	#103	—	146	2	no	—
299	#103	—	146	2	no	—
300	#103	—	146	2	no	—
301	C₂²⋊C₈	—	146	2	no	—
302	C₄×C₈	—	2	1	no	—
303	C₄×C₈	—	73	1	no	—
304	C₄×C₈	—	73	1	no	—
305	C₄×C₈	—	73	1	no	—
306	C₄×C₈	—	146	1	no	—
307	C₄×C₈	—	146	2	no	—
308	C₄×C₈	—	146	1	no	—
309	C₄×C₈	—	146	1	no	—
310	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
311	C₂×Q₈	—	2	2	no	—
312	C₂×Q₈	—	73	1	no	—
313	C₂×Q₈	—	146	1	no	—
314	C₂×Q₈	—	146	2	no	—
315	C₂×Q₈	—	146	2	no	—
316	C₂×Q₈	—	146	2	no	—
317	C₂×Q₈	D₄×D₇₃	1	1	yes	characteristic
318	C₂²×C₄	—	73	1	no	—
319	C₂²×C₄	—	73	1	no	—
320	C₂²×C₄	—	73	1	no	—
321	C₂²×C₄	—	146	1	no	—
322	C₂²×C₄	—	146	1	no	—
323	Q₁₆	—	4	4	no	—
324	Q₁₆	—	292	4	no	—
325	C₂×C₈	—	2	1	no	—
326	C₂×C₈	—	2	2	no	—
327	C₂×C₈	—	292	2	no	—
328	C₂×C₈	—	73	1	no	—
329	C₂×C₈	—	73	4	no	—
330	C₂×C₈	—	73	1	no	—
331	C₂×C₈	—	146	2	no	—
332	C₂×C₈	—	146	1	no	—
333	C₂×C₈	—	146	2	no	—
334	C₂×C₈	D₄×D₇₃	1	1	yes	characteristic
335	C₂×C₈	D₄×D₇₃	1	1	yes	characteristic
336	C₄⋊C₄	—	2	1	no	—
337	C₄⋊C₄	—	2	2	no	—
338	C₄⋊C₄	—	2	2	no	—
339	C₄⋊C₄	—	292	2	no	—
340	C₄⋊C₄	—	292	2	no	—
341	C₄⋊C₄	—	73	1	no	—
342	C₄⋊C₄	—	146	1	no	—
343	C₄⋊C₄	—	146	1	no	—
344	C₄⋊C₄	—	146	1	no	—
345	C₄⋊C₄	—	146	2	no	—
346	C₄⋊C₄	—	146	2	no	—
347	C₄⋊C₄	—	146	1	no	—
348	C₄⋊C₄	D₄×D₇₃	1	1	yes	characteristic
349	C₂²⋊C₄	—	292	2	no	—
350	C₄²	—	2	1	no	—
351	C₄²	—	292	2	no	—
352	C₄²	—	73	1	no	—
353	C₄²	—	73	1	no	—
354	C₄²	—	73	1	no	—
355	C₄²	—	146	1	no	—
356	C₄²	—	146	1	no	—
357	C₄²	—	146	1	no	—
358	C₄²	C₂³×D₇₃	1	1	yes	characteristic
359	C₂³	—	73	1	no	—
360	Q₈	—	2	1	no	—
361	Q₈	—	4	2	no	—
362	Q₈	—	4	2	no	—
363	Q₈	—	584	2	no	—
364	Q₈	—	73	2	no	—
365	Q₈	—	146	4	no	—
366	Q₈	—	146	1	no	—
367	Q₈	#14556610	1	2	yes	—
368	C₂×C₄	—	2	1	no	—
369	C₂×C₄	—	2	1	no	—
370	C₂×C₄	—	4	2	no	—
371	C₂×C₄	—	292	1	no	—
372	C₂×C₄	—	292	2	no	—
373	C₂×C₄	—	292	2	no	—
374	C₂×C₄	—	73	1	no	—
375	C₂×C₄	—	73	2	no	—
376	C₂×C₄	—	73	2	no	—
377	C₂×C₄	—	73	1	no	—
378	C₂×C₄	—	73	1	no	—
379	C₂×C₄	—	146	4	no	—
380	C₂×C₄	—	146	1	no	—
381	C₂×C₄	—	146	1	no	—
382	C₂×C₄	—	146	1	no	—
383	C₂×C₄	—	146	1	no	—
384	C₂×C₄	C₂×D₄×D₇₃	1	1	yes	characteristic
385	C₂×C₄	C₂×D₄×D₇₃	1	1	yes	characteristic
386	C₂×C₄	#41×D₇₃	1	1	yes	characteristic
387	C₈	—	2	1	no	—
388	C₈	—	4	1	no	—
389	C₈	—	4	2	no	—
390	C₈	—	292	1	no	—
391	C₈	—	292	2	no	—
392	C₈	—	73	2	no	—
393	C₈	—	146	1	no	—
394	C₈	Q₁₆×D₇₃	1	2	yes	—
395	C₂²	—	73	2	no	—
396	C₂²	—	146	1	no	—
397	C₂²	—	146	1	no	—
398	C₂²	#121×D₇₃	1	1	yes	characteristic
399	C₄	—	2	1	no	—
400	C₄	—	2	2	no	—
401	C₄	—	4	1	no	—
402	C₄	—	8	2	no	—
403	C₄	—	292	1	no	—
404	C₄	—	584	2	no	—
405	C₄	—	73	2	no	—
406	C₄	—	146	1	no	—
407	C₄	—	146	1	no	—
408	C₄	—	146	2	no	—
409	C₄	C₂×Q₁₆×D₇₃	1	1	yes	characteristic
410	C₄	#135×D₇₃	1	1	yes	characteristic
411	C₄	C₂×#14556610	1	1	yes	characteristic
412	C₄	#137×D₇₃	1	1	yes	characteristic
413	C₂	—	146	2	no	—
414	C₂	#462×D₇₃	1	1	yes	characteristic
415	C₂	#22619510	1	1	yes	characteristic
416	C₂	#22619653	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#121×D₇₃
commutator, cosocle	C₄²×C₇₃	C₂⁴
Frattini	C₄²	C₂³×D₇₃
Fitting	C₇₃×#1674	C₂
socle	C₂²×C₇₃	C₂×#121
prime radical	C₂²×D₇₃	#121
hypercenter	#1674	D₇₃
hypocenter	C₇₃	#19118
LCS	G ▷ C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #1674

Complements

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

N	H ≅ G/N	type
4. #22609336	C₂	semidirect
5. #22609335	C₂	semidirect
6. #22609322	C₂	semidirect
7. #22609312	C₂	semidirect
8. #22609311	C₂	semidirect
11. C₇₃×#1674	C₂	semidirect
119. C₈×D₇₃	Q₁₆	semidirect
215. #1674	D₇₃	semidirect
220. C₇₃	#19118	semidirect

Maximal subgroups

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

subgroup	index
#22620442	2
#22619969	2
C₈×Q₈×D₇₃	2
#22609336	2
#22609335	2
#22609322	2
#22609312	2
#22609311	2
#22608678	2
#22608587	2
C₇₃×#1674	2
#19118	73