Subgroups — #33069932

Order 18688.

608 conjugacy classes of subgroups.

The subgroup lattice has 430 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, complemented
2	#22620096	C₂	1	2	yes	maximal
3	C₈×#11924029	C₂	1	1	yes	maximal, characteristic
4	#22609327	C₂	1	1	yes	maximal, characteristic, complemented
5	#22609325	C₂	1	1	yes	maximal, characteristic, complemented
6	#22609321	C₂	1	1	yes	maximal, characteristic, complemented
7	#22609315	C₂	1	2	yes	maximal, complemented
8	#22609309	C₂	1	2	yes	maximal, complemented
9	#22608693	C₂	1	1	yes	maximal, characteristic, complemented
10	#22608640	C₂	1	2	yes	maximal, complemented
11	C₇₃×#1676	C₂	1	1	yes	maximal, characteristic, complemented
12	#17989142	C₂²	1	2	yes	—
13	C₄×#11924029	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	#17988930	—	2	2	no	—
19	#17988874	C₂²	1	2	yes	—
20	#17988797	C₂²	1	2	yes	—
21	#17988793	C₂²	1	2	yes	—
22	#17988752	—	2	2	no	—
23	Q₈×#10596831	C₂²	1	1	yes	characteristic
24	#17988691	—	2	2	no	—
25	#17988686	—	2	2	no	—
26	C₈×D₂₉₂	—	2	1	no	—
27	C₈×D₂₉₂	C₂²	1	1	yes	characteristic
28	#17988652	—	2	1	no	—
29	#17988652	C₂²	1	1	yes	characteristic
30	#17988643	C₂²	1	2	yes	—
31	#17988639	—	2	1	no	—
32	#17988639	C₂²	1	1	yes	characteristic
33	#17988624	—	2	1	no	—
34	#17988624	—	2	1	no	—
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	#17988336	C₂²	1	2	yes	—
40	#17988334	C₂²	1	2	yes	—
41	#17988332	C₂²	1	2	yes	—
42	#500×C₇₃	C₂²	1	1	yes	characteristic
43	#477×C₇₃	C₂²	1	2	yes	—
44	C₈×Q₈×C₇₃	C₂²	1	1	yes	characteristic
45	#335×C₇₃	C₂²	1	1	yes	characteristic
46	#333×C₇₃	C₂²	1	2	yes	—
47	C₂×#11924029	D₄	1	1	yes	characteristic
48	C₂×#11924018	—	2	2	no	—
49	C₂×#11924017	—	2	2	no	—
50	C₂×C₈×D₇₃	—	2	1	no	—
51	C₂×C₈×D₇₃	—	2	1	no	—
52	C₂×C₈×D₇₃	D₄	1	1	yes	characteristic
53	C₂×C₈×D₇₃	D₄	1	1	yes	characteristic
54	#14556591	C₂³	1	2	yes	—
55	#14556582	—	2	2	no	—
56	#14556571	—	2	2	no	—
57	#14556567	—	2	2	no	—
58	Q₈×Dic₇₃	—	2	2	no	—
59	Q₈×Dic₇₃	D₄	1	1	yes	characteristic
60	#14556556	—	2	2	no	—
61	C₄×D₂₉₂	—	2	1	no	—
62	C₄×D₂₉₂	C₂³	1	1	yes	characteristic
63	#14556553	—	2	1	no	—
64	#14556553	D₄	1	1	yes	characteristic
65	#14556549	—	2	1	no	—
66	#14556549	—	2	2	no	—
67	#14556549	D₄	1	1	yes	characteristic
68	C₄²×D₇₃	—	2	1	no	—
69	C₄²×D₇₃	C₂³	1	1	yes	characteristic
70	#14556525	—	2	1	no	—
71	#14556525	—	2	1	no	—
72	#14556522	—	2	2	no	—
73	#14556522	—	2	2	no	—
74	#14556522	—	2	2	no	—
75	C₄⋊#10596831	—	2	1	no	—
76	C₄⋊#10596831	C₂³	1	2	yes	—
77	C₄⋊#10596831	C₂³	1	1	yes	characteristic
78	#14556516	—	2	2	no	—
79	#14556514	—	2	2	no	—
80	#14556511	—	2	2	no	—
81	#14556503	—	2	1	no	—
82	#14556503	—	2	2	no	—
83	#14556503	—	2	1	no	—
84	#14556503	—	2	1	no	—
85	#14556501	—	2	2	no	—
86	C₄×#10596831	—	2	1	no	—
87	C₄×#10596831	C₂³	1	1	yes	characteristic
88	C₈×Dic₇₃	—	2	1	no	—
89	C₈×Dic₇₃	D₄	1	1	yes	characteristic
90	C₈×Dic₇₃	D₄	1	1	yes	characteristic
91	C₄⋊Q₈×C₇₃	C₂³	1	2	yes	—
92	C₄×Q₈×C₇₃	C₂³	1	1	yes	characteristic
93	#107×C₇₃	—	2	1	no	—
94	#107×C₇₃	—	2	1	no	—
95	#107×C₇₃	—	2	2	no	—
96	C₄⋊C₈×C₇₃	—	2	1	no	—
97	C₄⋊C₈×C₇₃	C₂³	1	1	yes	characteristic
98	C₄⋊C₈×C₇₃	C₂³	1	2	yes	—
99	#103×C₇₃	—	2	2	no	—
100	C₄×C₈×C₇₃	—	2	1	no	—
101	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
102	#11924029	—	2	2	no	—
103	#11924029	—	2	2	no	—
104	C₂×Dic₁₄₆	—	2	2	no	—
105	C₂×D₂₉₂	—	2	1	no	—
106	C₂×D₂₉₂	C₂×D₄	1	1	yes	characteristic
107	C₂×C₄×D₇₃	—	2	1	no	—
108	C₂×C₄×D₇₃	—	2	1	no	—
109	C₂×C₄×D₇₃	—	2	1	no	—
110	C₂×C₄×D₇₃	C₂×D₄	1	1	yes	characteristic
111	C₂×C₄×D₇₃	C₂×D₄	1	1	yes	characteristic
112	C₂×C₄×D₇₃	#41	1	1	yes	characteristic
113	#11924018	—	4	4	no	—
114	#11924017	—	4	4	no	—
115	C₂×#10596831	—	2	1	no	—
116	C₂×#10596831	—	2	2	no	—
117	C₂×#10596831	C₂×D₄	1	1	yes	characteristic
118	C₂×#10596831	C₂×D₄	1	1	yes	characteristic
119	C₈×D₇₃	—	2	2	no	—
120	C₈×D₇₃	—	2	2	no	—
121	C₈×D₇₃	—	4	2	no	—
122	C₈×D₇₃	—	4	2	no	—
123	C₄⋊Dic₇₃	—	2	1	no	—
124	C₄⋊Dic₇₃	—	2	2	no	—
125	C₄⋊Dic₇₃	—	2	2	no	—
126	C₄⋊Dic₇₃	—	4	2	no	—
127	C₄⋊Dic₇₃	C₂×D₄	1	1	yes	characteristic
128	#11924007	—	4	2	no	—
129	#11924005	—	2	1	no	—
130	#11924005	—	2	1	no	—
131	#11924005	—	2	1	no	—
132	#11924005	—	4	2	no	—
133	C₄×Dic₇₃	—	2	1	no	—
134	C₄×Dic₇₃	—	2	1	no	—
135	C₄×Dic₇₃	—	4	2	no	—
136	C₄×Dic₇₃	C₂×D₄	1	1	yes	characteristic
137	C₄×Dic₇₃	C₂×D₄	1	1	yes	characteristic
138	C₄×Dic₇₃	#41	1	1	yes	characteristic
139	C₂×Q₈×C₇₃	—	2	2	no	—
140	C₂×Q₈×C₇₃	C₂×D₄	1	1	yes	characteristic
141	C₂×C₈×C₇₃	—	2	1	no	—
142	C₂×C₈×C₇₃	—	2	2	no	—
143	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
144	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
145	C₄⋊C₄×C₇₃	—	2	1	no	—
146	C₄⋊C₄×C₇₃	—	2	2	no	—
147	C₄⋊C₄×C₇₃	—	2	2	no	—
148	C₄⋊C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
149	C₄²×C₇₃	—	2	1	no	—
150	C₄²×C₇₃	C₂⁴	1	1	yes	characteristic
151	C₂²×D₇₃	—	2	1	no	—
152	C₂²×D₇₃	#121	1	1	yes	characteristic
153	Q₈×C₇₃	—	2	1	no	—
154	Q₈×C₇₃	—	8	2	no	—
155	Q₈×C₇₃	#135	1	2	yes	—
156	C₂×C₄×C₇₃	—	2	1	no	—
157	C₂×C₄×C₇₃	—	2	1	no	—
158	C₂×C₄×C₇₃	—	4	2	no	—
159	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
160	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
161	C₂×C₄×C₇₃	C₂×#41	1	1	yes	characteristic
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	D₂₉₂	—	2	4	no	—
168	D₂₉₂	—	2	1	no	—
169	D₂₉₂	#135	1	2	yes	—
170	C₄×D₇₃	—	2	4	no	—
171	C₄×D₇₃	—	2	1	no	—
172	C₄×D₇₃	—	2	1	no	—
173	C₄×D₇₃	—	2	1	no	—
174	C₄×D₇₃	—	2	1	no	—
175	C₄×D₇₃	—	4	1	no	—
176	C₄×D₇₃	—	4	2	no	—
177	C₄×D₇₃	—	4	1	no	—
178	C₄×D₇₃	—	4	1	no	—
179	Dic₁₄₆	—	4	2	no	—
180	Dic₁₄₆	—	4	2	no	—
181	C₈×C₇₃	—	2	1	no	—
182	C₈×C₇₃	—	4	2	no	—
183	C₈×C₇₃	—	4	1	no	—
184	C₈×C₇₃	#137	1	2	yes	—
185	#10596831	—	2	1	no	—
186	#10596831	—	4	2	no	—
187	#10596831	—	4	1	no	—
188	#10596831	#137	1	2	yes	—
189	C₂²×C₇₃	C₂×#121	1	1	yes	characteristic
190	C₂×D₇₃	—	2	1	no	—
191	C₂×D₇₃	—	2	1	no	—
192	C₂×D₇₃	—	2	2	no	—
193	C₂×D₇₃	—	2	1	no	—
194	C₂×D₇₃	—	4	1	no	—
195	C₂×D₇₃	—	4	1	no	—
196	C₄×C₇₃	—	2	2	no	—
197	C₄×C₇₃	—	2	1	no	—
198	C₄×C₇₃	—	4	1	no	—
199	C₄×C₇₃	—	8	2	no	—
200	C₄×C₇₃	C₂×#135	1	1	yes	characteristic
201	C₄×C₇₃	C₂×#135	1	1	yes	characteristic
202	C₄×C₇₃	C₂×#137	1	1	yes	characteristic
203	C₄×C₇₃	C₂×#137	1	1	yes	characteristic
204	Dic₇₃	—	2	1	no	—
205	Dic₇₃	—	2	1	no	—
206	Dic₇₃	—	2	1	no	—
207	Dic₇₃	—	4	1	no	—
208	Dic₇₃	—	8	2	no	—
209	#19273	—	73	1	no	maximal, Sylow
210	C₂×C₇₃	#3039	1	1	yes	characteristic
211	C₂×C₇₃	#3050	1	1	yes	characteristic
212	C₂×C₇₃	#3067	1	1	yes	characteristic
213	D₇₃	—	4	1	no	—
214	D₇₃	—	4	2	no	—
215	#3160	—	73	1	no	—
216	#3116	—	73	2	no	—
217	C₈×#41	—	73	1	no	—
218	#1676	D₇₃	1	1	yes	characteristic, complemented
219	#1675	—	73	1	no	—
220	#1674	—	73	1	no	—
221	#1673	—	73	1	no	—
222	#1672	—	73	2	no	—
223	#1671	—	73	2	no	—
224	#1568	—	73	1	no	—
225	#1554	—	73	2	no	—
226	C₇₃	#19273	1	1	yes	characteristic, complemented, Sylow
227	#533	—	73	2	no	—
228	C₄×#41	—	73	1	no	—
229	#500	—	73	1	no	—
230	#500	C₂×D₇₃	1	1	yes	characteristic
231	#499	—	146	1	no	—
232	#499	—	146	1	no	—
233	#498	—	73	2	no	—
234	#485	—	146	2	no	—
235	#477	C₂×D₇₃	1	2	yes	—
236	#476	—	73	2	no	—
237	#462	—	73	2	no	—
238	#461	—	73	2	no	—
239	#452	—	146	2	no	—
240	C₈×Q₈	—	73	1	no	—
241	C₈×Q₈	C₂×D₇₃	1	1	yes	characteristic
242	#441	—	146	2	no	—
243	#440	—	146	2	no	—
244	C₈×D₄	—	73	1	no	—
245	C₈×D₄	—	73	1	no	—
246	C₈×D₄	—	146	1	no	—
247	C₈×D₄	—	146	1	no	—
248	#432	—	73	2	no	—
249	#431	—	73	1	no	—
250	#431	—	146	1	no	—
251	#427	—	146	1	no	—
252	#427	—	146	1	no	—
253	C₂×C₄×C₈	—	73	1	no	—
254	C₂×C₄×C₈	—	146	1	no	—
255	#335	—	73	1	no	—
256	#335	C₂×D₇₃	1	1	yes	characteristic
257	#334	—	73	2	no	—
258	#333	—	73	2	no	—
259	#333	C₂×D₇₃	1	2	yes	—
260	#332	—	73	2	no	—
261	#332	—	73	2	no	—
262	C₂×#41	—	73	1	no	—
263	C₂×Q₁₆	—	146	2	no	—
264	C₂×SD₁₆	—	146	2	no	—
265	C₂²×C₈	—	73	1	no	—
266	C₂²×C₈	—	73	1	no	—
267	C₂²×C₈	—	146	1	no	—
268	C₂²×C₈	—	146	1	no	—
269	C₄⋊Q₈	—	73	2	no	—
270	C₄⋊Q₈	C₂²×D₇₃	1	2	yes	—
271	#125	—	146	2	no	—
272	C₂²⋊Q₈	—	146	2	no	—
273	C₂²⋊Q₈	—	146	2	no	—
274	C₄×Q₈	—	73	1	no	—
275	C₄×Q₈	—	146	2	no	—
276	C₄×Q₈	—	146	2	no	—
277	C₄×Q₈	C₂²×D₇₃	1	1	yes	characteristic
278	C₄×D₄	—	73	1	no	—
279	C₄×D₄	—	73	1	no	—
280	C₄×D₄	—	146	1	no	—
281	C₄×D₄	—	146	1	no	—
282	#117	—	73	1	no	—
283	#117	—	146	2	no	—
284	#117	—	146	1	no	—
285	C₂×C₄²	—	73	1	no	—
286	C₂×C₄²	—	146	1	no	—
287	#107	—	2	1	no	—
288	#107	—	2	1	no	—
289	#107	—	2	2	no	—
290	#107	—	146	1	no	—
291	#107	—	146	1	no	—
292	#106	—	146	2	no	—
293	#106	—	146	2	no	—
294	#106	—	146	2	no	—
295	C₄⋊C₈	—	2	1	no	—
296	C₄⋊C₈	—	73	1	no	—
297	C₄⋊C₈	—	73	2	no	—
298	C₄⋊C₈	—	146	1	no	—
299	C₄⋊C₈	C₂²×D₇₃	1	1	yes	characteristic
300	C₄⋊C₈	C₂²×D₇₃	1	2	yes	—
301	#103	—	2	2	no	—
302	#103	—	146	2	no	—
303	#103	—	146	2	no	—
304	#102	—	146	2	no	—
305	C₂²⋊C₈	—	146	1	no	—
306	C₂²⋊C₈	—	146	1	no	—
307	C₂²⋊C₈	—	146	2	no	—
308	C₂²⋊C₈	—	146	1	no	—
309	#97	—	146	2	no	—
310	C₄×C₈	—	2	1	no	—
311	C₄×C₈	—	73	1	no	—
312	C₄×C₈	—	73	1	no	—
313	C₄×C₈	—	73	1	no	—
314	C₄×C₈	—	146	1	no	—
315	C₄×C₈	—	146	1	no	—
316	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
317	#41	—	146	2	no	—
318	#41	—	146	2	no	—
319	C₂×Q₈	—	2	2	no	—
320	C₂×Q₈	—	146	2	no	—
321	C₂×Q₈	D₄×D₇₃	1	1	yes	characteristic
322	C₂×D₄	—	73	1	no	—
323	C₂×D₄	—	146	1	no	—
324	C₂²×C₄	—	73	1	no	—
325	C₂²×C₄	—	73	1	no	—
326	C₂²×C₄	—	73	1	no	—
327	C₂²×C₄	—	146	1	no	—
328	C₂²×C₄	—	146	1	no	—
329	C₂²×C₄	—	146	1	no	—
330	Q₁₆	—	292	4	no	—
331	SD₁₆	—	292	4	no	—
332	C₂×C₈	—	2	1	no	—
333	C₂×C₈	—	2	2	no	—
334	C₂×C₈	—	292	2	no	—
335	C₂×C₈	—	292	2	no	—
336	C₂×C₈	—	73	1	no	—
337	C₂×C₈	—	73	1	no	—
338	C₂×C₈	—	146	2	no	—
339	C₂×C₈	—	146	1	no	—
340	C₂×C₈	—	146	2	no	—
341	C₂×C₈	—	146	2	no	—
342	C₂×C₈	D₄×D₇₃	1	1	yes	characteristic
343	C₂×C₈	D₄×D₇₃	1	1	yes	characteristic
344	C₄⋊C₄	—	2	1	no	—
345	C₄⋊C₄	—	2	2	no	—
346	C₄⋊C₄	—	2	2	no	—
347	C₄⋊C₄	—	292	2	no	—
348	C₄⋊C₄	—	292	2	no	—
349	C₄⋊C₄	—	73	1	no	—
350	C₄⋊C₄	—	146	1	no	—
351	C₄⋊C₄	—	146	2	no	—
352	C₄⋊C₄	—	146	2	no	—
353	C₄⋊C₄	D₄×D₇₃	1	1	yes	characteristic
354	C₂²⋊C₄	—	292	2	no	—
355	C₂²⋊C₄	—	146	1	no	—
356	C₂²⋊C₄	—	146	1	no	—
357	C₂²⋊C₄	—	146	1	no	—
358	C₄²	—	2	1	no	—
359	C₄²	—	292	2	no	—
360	C₄²	—	73	1	no	—
361	C₄²	—	73	1	no	—
362	C₄²	—	73	1	no	—
363	C₄²	—	146	1	no	—
364	C₄²	—	146	1	no	—
365	C₄²	C₂³×D₇₃	1	1	yes	characteristic
366	C₂³	—	73	1	no	—
367	C₂³	—	146	1	no	—
368	Q₈	—	2	1	no	—
369	Q₈	—	8	2	no	—
370	Q₈	—	292	2	no	—
371	Q₈	—	292	2	no	—
372	Q₈	#14556614	1	2	yes	—
373	D₄	—	73	2	no	—
374	D₄	—	146	4	no	—
375	D₄	—	146	1	no	—
376	C₂×C₄	—	2	1	no	—
377	C₂×C₄	—	2	1	no	—
378	C₂×C₄	—	4	2	no	—
379	C₂×C₄	—	292	1	no	—
380	C₂×C₄	—	292	2	no	—
381	C₂×C₄	—	292	1	no	—
382	C₂×C₄	—	292	1	no	—
383	C₂×C₄	—	292	2	no	—
384	C₂×C₄	—	73	1	no	—
385	C₂×C₄	—	73	1	no	—
386	C₂×C₄	—	73	1	no	—
387	C₂×C₄	—	146	4	no	—
388	C₂×C₄	—	146	1	no	—
389	C₂×C₄	—	146	1	no	—
390	C₂×C₄	—	146	1	no	—
391	C₂×C₄	—	146	1	no	—
392	C₂×C₄	—	146	1	no	—
393	C₂×C₄	C₂×D₄×D₇₃	1	1	yes	characteristic
394	C₂×C₄	C₂×D₄×D₇₃	1	1	yes	characteristic
395	C₂×C₄	#41×D₇₃	1	1	yes	characteristic
396	C₈	—	2	1	no	—
397	C₈	—	4	1	no	—
398	C₈	—	4	2	no	—
399	C₈	—	292	1	no	—
400	C₈	—	292	2	no	—
401	C₈	—	73	2	no	—
402	C₈	—	146	1	no	—
403	C₈	#14556623	1	2	yes	—
404	C₂²	—	292	1	no	—
405	C₂²	—	292	1	no	—
406	C₂²	—	146	1	no	—
407	C₂²	—	146	1	no	—
408	C₂²	—	146	2	no	—
409	C₂²	—	146	1	no	—
410	C₂²	#121×D₇₃	1	1	yes	characteristic
411	C₄	—	2	1	no	—
412	C₄	—	2	2	no	—
413	C₄	—	4	1	no	—
414	C₄	—	8	2	no	—
415	C₄	—	292	1	no	—
416	C₄	—	584	2	no	—
417	C₄	—	146	1	no	—
418	C₄	—	146	1	no	—
419	C₄	—	146	1	no	—
420	C₄	#135×D₇₃	1	1	yes	characteristic
421	C₄	C₂×#14556614	1	1	yes	characteristic
422	C₄	#137×D₇₃	1	1	yes	characteristic
423	C₄	C₂×#14556623	1	1	yes	characteristic
424	C₂	—	292	2	no	—
425	C₂	—	292	1	no	—
426	C₂	#22619415	1	1	yes	characteristic
427	C₂	#22619514	1	1	yes	characteristic
428	C₂	#22619649	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#121×D₇₃
commutator, cosocle	C₄²×C₇₃	C₂⁴
Frattini	C₄²	C₂³×D₇₃
Fitting	C₇₃×#1676	C₂
socle	C₂²×C₇₃	C₂×#121
prime radical	C₂×#11924029	D₄
hypercenter	#1676	D₇₃
hypocenter	C₇₃	#19273
LCS	G ▷ C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #1676

Complements

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

N	H ≅ G/N	type
1. #22620442	C₂	semidirect
4. #22609327	C₂	semidirect
5. #22609325	C₂	semidirect
6. #22609321	C₂	semidirect
7. #22609315	C₂	semidirect
8. #22609309	C₂	semidirect
9. #22608693	C₂	semidirect
10. #22608640	C₂	semidirect
11. C₇₃×#1676	C₂	semidirect
218. #1676	D₇₃	semidirect
226. C₇₃	#19273	semidirect

Maximal subgroups

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

subgroup	index
#22620442	2
#22620096	2
C₈×#11924029	2
#22609327	2
#22609325	2
#22609321	2
#22609315	2
#22609309	2
#22608693	2
#22608640	2
C₇₃×#1676	2
#19273	73