Subgroups — #33056838

Order 18688.

584 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	#22620490	C₂	1	1	yes	maximal, characteristic
2	#22619025	C₂	1	1	yes	maximal, characteristic, complemented
3	#22619006	C₂	1	1	yes	maximal, characteristic
4	#22608811	C₂	1	2	yes	maximal, complemented
5	#22608790	C₂	1	2	yes	maximal, complemented
6	#22608776	C₂	1	2	yes	maximal, complemented
7	#22608769	C₂	1	2	yes	maximal, complemented
8	#22608694	C₂	1	1	yes	maximal, characteristic, complemented
9	#22608405	C₂	1	1	yes	maximal, characteristic, complemented
10	#22608401	C₂	1	1	yes	maximal, characteristic
11	C₇₃×#1452	C₂	1	1	yes	maximal, characteristic, complemented
12	#17989144	C₂²	1	1	yes	characteristic
13	#17989137	C₂²	1	1	yes	characteristic
14	#17988985	C₄	1	1	yes	characteristic
15	#17988960	C₄	1	1	yes	characteristic
16	#17988938	C₄	1	1	yes	characteristic
17	#17988934	C₄	1	1	yes	characteristic
18	#17988816	C₄	1	2	yes	—
19	#17988816	C₄	1	2	yes	—
20	#17988717	C₂²	1	2	yes	—
21	#17988716	—	2	2	no	—
22	C₈×Dic₁₄₆	C₂²	1	2	yes	—
23	C₄×#11924017	C₂²	1	2	yes	—
24	C₄×#11924017	C₂²	1	2	yes	—
25	#17988666	C₂²	1	2	yes	—
26	#17988657	—	2	2	no	—
27	C₈×D₂₉₂	C₂²	1	2	yes	—
28	#17988645	—	2	2	no	—
29	#17988644	C₂²	1	1	yes	characteristic
30	#17988644	C₂²	1	1	yes	characteristic
31	#17988552	—	2	2	no	—
32	#17988340	C₂²	1	2	yes	—
33	#17988338	C₂²	1	2	yes	—
34	#17988319	C₂²	1	2	yes	—
35	#17988319	C₂²	1	2	yes	—
36	#17988316	C₂²	1	2	yes	—
37	#17988316	C₂²	1	2	yes	—
38	#431×C₇₃	C₂²	1	1	yes	characteristic
39	#431×C₇₃	C₂²	1	1	yes	characteristic
40	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
41	#322×C₇₃	C₂²	1	1	yes	characteristic
42	#322×C₇₃	C₂²	1	1	yes	characteristic
43	#322×C₇₃	C₂²	1	1	yes	characteristic
44	#322×C₇₃	C₂²	1	1	yes	characteristic
45	C₂×#11924028	—	2	1	no	—
46	C₂×#11924017	C₂×C₄	1	2	yes	—
47	C₂×#11924017	C₂×C₄	1	2	yes	—
48	C₂×#11924011	—	2	2	no	—
49	C₂×C₈×D₇₃	—	2	2	no	—
50	#14556591	C₂×C₄	1	1	yes	characteristic
51	#14556591	C₂×C₄	1	1	yes	characteristic
52	#14556591	C₂×C₄	1	1	yes	characteristic
53	#14556588	C₂×C₄	1	1	yes	characteristic
54	#14556583	—	2	1	no	—
55	#14556579	—	2	1	no	—
56	#14556570	—	2	1	no	—
57	#14556570	C₂×C₄	1	2	yes	—
58	#14556565	—	2	1	no	—
59	#14556565	C₂×C₄	1	2	yes	—
60	C₄×Dic₁₄₆	—	2	1	no	—
61	C₄×Dic₁₄₆	C₂³	1	2	yes	—
62	C₄×D₂₉₂	—	2	1	no	—
63	C₄×D₂₉₂	C₂³	1	2	yes	—
64	#14556550	—	2	1	no	—
65	#14556523	C₂×C₄	1	2	yes	—
66	#14556523	C₂×C₄	1	2	yes	—
67	C₄⋊#10596831	C₂³	1	2	yes	—
68	C₄⋊#10596831	C₂³	1	2	yes	—
69	#14556520	—	2	2	no	—
70	#14556520	—	2	2	no	—
71	#14556516	C₂×C₄	1	2	yes	—
72	#14556516	C₂×C₄	1	2	yes	—
73	#14556513	C₂×C₄	1	2	yes	—
74	#14556513	C₂×C₄	1	2	yes	—
75	C₂²⋊#10596831	—	2	1	no	—
76	C₂²⋊#10596831	—	2	1	no	—
77	#14556503	—	2	2	no	—
78	#14556503	—	2	2	no	—
79	#14556502	—	2	1	no	—
80	#14556502	—	2	1	no	—
81	#14556501	—	2	2	no	—
82	C₈×Dic₇₃	—	2	2	no	—
83	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
84	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
85	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
86	C₄⋊C₈×C₇₃	—	2	1	no	—
87	C₄⋊C₈×C₇₃	D₄	1	2	yes	—
88	C₂²⋊C₈×C₇₃	—	2	1	no	—
89	C₂²⋊C₈×C₇₃	D₄	1	2	yes	—
90	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
91	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
92	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
93	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
94	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
95	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
96	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
97	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
98	#11924028	—	8	2	no	—
99	C₂×Dic₁₄₆	—	2	1	no	—
100	C₂×Dic₁₄₆	—	2	1	no	—
101	C₂×Dic₁₄₆	C₂²×C₄	1	2	yes	—
102	C₂×D₂₉₂	—	2	1	no	—
103	C₂×D₂₉₂	—	2	1	no	—
104	C₂×D₂₉₂	C₂²×C₄	1	2	yes	—
105	C₂×C₂²⋊D₇₃	—	2	2	no	—
106	C₂×C₄×D₇₃	—	2	2	no	—
107	#11924017	—	4	2	no	—
108	#11924017	—	4	2	no	—
109	#11924011	—	8	2	no	—
110	C₂×#10596831	—	2	2	no	—
111	C₂×#10596831	—	2	2	no	—
112	C₈×D₇₃	—	8	2	no	—
113	C₄⋊Dic₇₃	—	2	1	no	—
114	C₄⋊Dic₇₃	—	2	1	no	—
115	C₄⋊Dic₇₃	—	2	1	no	—
116	C₄⋊Dic₇₃	—	2	1	no	—
117	C₄⋊Dic₇₃	C₂²×C₄	1	2	yes	—
118	C₄⋊Dic₇₃	C₂²×C₄	1	2	yes	—
119	#11924007	—	2	2	no	—
120	#11924007	—	2	2	no	—
121	C₂²⋊Dic₇₃	—	2	2	no	—
122	#11924005	—	2	2	no	—
123	#11924005	—	2	2	no	—
124	C₄×Dic₇₃	—	2	2	no	—
125	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
126	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
127	C₂²×C₄×C₇₃	#41	1	1	yes	characteristic
128	C₂×C₈×C₇₃	—	2	1	no	—
129	C₂×C₈×C₇₃	—	2	1	no	—
130	C₂×C₈×C₇₃	—	4	1	no	—
131	C₂×C₈×C₇₃	—	4	1	no	—
132	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
133	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
134	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
135	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
136	C₂×C₈×C₇₃	C₂×D₄	1	2	yes	—
137	C₂×C₈×C₇₃	C₂×D₄	1	2	yes	—
138	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
139	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
140	C₄²×C₇₃	#41	1	1	yes	characteristic
141	C₄²×C₇₃	C₂⁴	1	1	yes	characteristic
142	C₂³×C₇₃	C₄×D₄	1	1	yes	characteristic
143	C₂²×D₇₃	—	2	2	no	—
144	C₂×C₄×C₇₃	—	2	1	no	—
145	C₂×C₄×C₇₃	—	2	1	no	—
146	C₂×C₄×C₇₃	—	2	2	no	—
147	C₂×C₄×C₇₃	—	4	1	no	—
148	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
149	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
150	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
151	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
152	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
153	C₂×C₄×C₇₃	C₂×#41	1	1	yes	characteristic
154	C₂²⋊D₇₃	—	8	2	no	—
155	C₂×Dic₇₃	—	2	2	no	—
156	C₂×Dic₇₃	—	2	2	no	—
157	C₂×Dic₇₃	—	2	2	no	—
158	D₂₉₂	—	2	2	no	—
159	D₂₉₂	—	2	2	no	—
160	D₂₉₂	—	8	1	no	—
161	D₂₉₂	—	8	1	no	—
162	C₄×D₇₃	—	4	2	no	—
163	C₄×D₇₃	—	4	2	no	—
164	Dic₁₄₆	—	2	2	no	—
165	Dic₁₄₆	—	2	2	no	—
166	Dic₁₄₆	—	8	1	no	—
167	Dic₁₄₆	—	8	1	no	—
168	C₈×C₇₃	—	2	1	no	—
169	C₈×C₇₃	—	2	1	no	—
170	C₈×C₇₃	—	2	1	no	—
171	C₈×C₇₃	—	2	1	no	—
172	C₈×C₇₃	—	2	2	no	—
173	C₈×C₇₃	—	2	2	no	—
174	C₈×C₇₃	—	4	1	no	—
175	C₈×C₇₃	—	4	1	no	—
176	#10596831	—	4	2	no	—
177	#10596831	—	4	2	no	—
178	C₂²×C₇₃	—	2	1	no	—
179	C₂²×C₇₃	—	2	1	no	—
180	C₂²×C₇₃	—	2	1	no	—
181	C₂²×C₇₃	C₂×C₄×D₄	1	1	yes	characteristic
182	C₂×D₇₃	—	4	2	no	—
183	C₂×D₇₃	—	4	2	no	—
184	C₂×D₇₃	—	4	2	no	—
185	C₄×C₇₃	—	2	1	no	—
186	C₄×C₇₃	—	2	1	no	—
187	C₄×C₇₃	—	2	1	no	—
188	C₄×C₇₃	—	2	1	no	—
189	C₄×C₇₃	#568	1	1	yes	characteristic
190	C₄×C₇₃	#568	1	1	yes	characteristic
191	C₄×C₇₃	#577	1	2	yes	—
192	Dic₇₃	—	4	2	no	—
193	Dic₇₃	—	4	2	no	—
194	Dic₇₃	—	4	2	no	—
195	#17990	—	73	1	no	maximal, Sylow
196	C₂×C₇₃	—	4	1	no	—
197	C₂×C₇₃	#2932	1	1	yes	characteristic
198	C₂×C₇₃	#2949	1	1	yes	characteristic
199	C₂×C₇₃	#2957	1	1	yes	characteristic
200	D₇₃	—	8	2	no	—
201	#3167	—	73	1	no	—
202	#2985	—	73	1	no	—
203	#2981	—	73	1	no	—
204	#1588	—	73	2	no	—
205	#1585	—	73	2	no	—
206	#1583	—	73	2	no	—
207	#1582	—	73	2	no	—
208	#1568	—	73	1	no	—
209	#1497	—	73	1	no	—
210	#1496	—	73	1	no	—
211	#1452	D₇₃	1	1	yes	characteristic, complemented
212	C₇₃	#17990	1	1	yes	characteristic, complemented, Sylow
213	#533	—	73	1	no	—
214	#532	—	73	1	no	—
215	#498	—	73	1	no	—
216	#492	—	73	1	no	—
217	#487	—	73	1	no	—
218	#486	—	73	1	no	—
219	#465	—	73	2	no	—
220	#465	—	73	2	no	—
221	#446	—	73	2	no	—
222	#446	—	146	2	no	—
223	C₈×Q₈	—	73	2	no	—
224	C₄×SD₁₆	—	73	2	no	—
225	C₄×SD₁₆	—	73	2	no	—
226	#436	—	73	2	no	—
227	#435	—	146	2	no	—
228	C₈×D₄	—	73	2	no	—
229	#433	—	146	2	no	—
230	#432	—	73	1	no	—
231	#432	—	73	1	no	—
232	#431	C₂×D₇₃	1	1	yes	characteristic
233	#431	C₂×D₇₃	1	1	yes	characteristic
234	#408	—	146	2	no	—
235	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
236	#335	—	73	2	no	—
237	#334	—	73	2	no	—
238	Q₈⋊C₈	—	73	2	no	—
239	Q₈⋊C₈	—	73	2	no	—
240	D₄⋊C₈	—	73	2	no	—
241	D₄⋊C₈	—	73	2	no	—
242	#322	C₂×D₇₃	1	1	yes	characteristic
243	#322	C₂×D₇₃	1	1	yes	characteristic
244	#322	C₂×D₇₃	1	1	yes	characteristic
245	#322	C₂×D₇₃	1	1	yes	characteristic
246	C₂×#41	—	146	1	no	—
247	C₂×SD₁₆	—	73	2	no	—
248	C₂×SD₁₆	—	73	2	no	—
249	C₂×M₁₆	—	146	2	no	—
250	C₂²×C₈	—	146	2	no	—
251	C₂²×C₈	C₄×D₇₃	1	1	yes	characteristic
252	C₂²×C₈	C₄×D₇₃	1	1	yes	characteristic
253	C₄⋊Q₈	—	73	1	no	—
254	C₄⋊Q₈	—	73	1	no	—
255	C₄⋊Q₈	—	73	1	no	—
256	#127	—	73	1	no	—
257	#125	—	146	1	no	—
258	#124	—	146	1	no	—
259	C₂²⋊Q₈	—	73	2	no	—
260	C₂²⋊Q₈	—	146	1	no	—
261	#121	—	73	2	no	—
262	#121	—	146	1	no	—
263	C₄×Q₈	—	73	2	no	—
264	C₄×Q₈	—	146	1	no	—
265	C₄×D₄	—	73	2	no	—
266	C₄×D₄	—	146	1	no	—
267	#117	—	146	1	no	—
268	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
269	#106	—	73	2	no	—
270	#106	—	73	2	no	—
271	C₄⋊C₈	—	2	1	no	—
272	C₄⋊C₈	—	73	2	no	—
273	C₄⋊C₈	—	73	2	no	—
274	C₄⋊C₈	—	146	2	no	—
275	C₄⋊C₈	—	146	2	no	—
276	C₄⋊C₈	D₂₉₂	1	2	yes	—
277	#103	—	73	2	no	—
278	#103	—	73	2	no	—
279	#102	—	73	2	no	—
280	#102	—	73	2	no	—
281	C₂²⋊C₈	—	2	1	no	—
282	C₂²⋊C₈	—	146	1	no	—
283	C₂²⋊C₈	—	146	1	no	—
284	C₂²⋊C₈	—	146	2	no	—
285	C₂²⋊C₈	—	146	2	no	—
286	C₂²⋊C₈	D₂₉₂	1	2	yes	—
287	#97	—	146	2	no	—
288	#97	—	146	1	no	—
289	#97	—	146	1	no	—
290	C₄×C₈	—	146	2	no	—
291	C₄×C₈	C₄×D₇₃	1	1	yes	characteristic
292	C₄×C₈	C₄×D₇₃	1	1	yes	characteristic
293	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
294	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
295	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
296	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
297	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
298	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
299	#41	—	584	2	no	—
300	C₂×Q₈	—	73	2	no	—
301	C₂×Q₈	—	146	1	no	—
302	C₂×Q₈	—	146	1	no	—
303	C₂×D₄	—	73	2	no	—
304	C₂×D₄	—	146	2	no	—
305	C₂×D₄	—	146	1	no	—
306	C₂×D₄	—	146	1	no	—
307	C₂²×C₄	—	146	2	no	—
308	C₂²×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
309	C₂²×C₄	C₂×D₂₉₂	1	1	yes	characteristic
310	C₂²×C₄	#11924028	1	1	yes	characteristic
311	SD₁₆	—	292	2	no	—
312	SD₁₆	—	292	2	no	—
313	M₁₆	—	584	2	no	—
314	C₂×C₈	—	2	1	no	—
315	C₂×C₈	—	2	1	no	—
316	C₂×C₈	—	4	1	no	—
317	C₂×C₈	—	4	1	no	—
318	C₂×C₈	—	584	2	no	—
319	C₂×C₈	—	146	2	no	—
320	C₂×C₈	—	146	2	no	—
321	C₂×C₈	C₂×C₄×D₇₃	1	1	yes	characteristic
322	C₂×C₈	C₂×C₄×D₇₃	1	1	yes	characteristic
323	C₂×C₈	C₂×C₄×D₇₃	1	1	yes	characteristic
324	C₂×C₈	C₂×C₄×D₇₃	1	1	yes	characteristic
325	C₂×C₈	C₂×D₂₉₂	1	2	yes	—
326	C₂×C₈	C₂×D₂₉₂	1	2	yes	—
327	C₄⋊C₄	—	73	2	no	—
328	C₄⋊C₄	—	73	2	no	—
329	C₄⋊C₄	—	146	2	no	—
330	C₄⋊C₄	—	146	1	no	—
331	C₄⋊C₄	—	146	1	no	—
332	C₄⋊C₄	—	146	1	no	—
333	C₄⋊C₄	—	146	1	no	—
334	C₄⋊C₄	—	146	2	no	—
335	C₂²⋊C₄	—	146	2	no	—
336	C₂²⋊C₄	—	146	2	no	—
337	C₂²⋊C₄	—	146	2	no	—
338	C₄²	—	146	2	no	—
339	C₄²	C₂×C₄×D₇₃	1	1	yes	characteristic
340	C₄²	C₂×D₂₉₂	1	1	yes	characteristic
341	C₄²	#11924028	1	1	yes	characteristic
342	C₄²	C₂³×D₇₃	1	1	yes	characteristic
343	C₂³	—	146	2	no	—
344	C₂³	C₄×D₂₉₂	1	1	yes	characteristic
345	Q₈	—	584	1	no	—
346	Q₈	—	584	1	no	—
347	Q₈	—	146	2	no	—
348	Q₈	—	146	2	no	—
349	D₄	—	584	1	no	—
350	D₄	—	584	1	no	—
351	D₄	—	584	2	no	—
352	D₄	—	146	2	no	—
353	D₄	—	146	2	no	—
354	C₂×C₄	—	2	2	no	—
355	C₂×C₄	—	2	1	no	—
356	C₂×C₄	—	2	1	no	—
357	C₂×C₄	—	4	1	no	—
358	C₂×C₄	—	292	2	no	—
359	C₂×C₄	—	292	2	no	—
360	C₂×C₄	—	146	2	no	—
361	C₂×C₄	—	146	2	no	—
362	C₂×C₄	—	146	2	no	—
363	C₂×C₄	C₄×D₂₉₂	1	1	yes	characteristic
364	C₂×C₄	C₄×D₂₉₂	1	1	yes	characteristic
365	C₂×C₄	C₄×D₂₉₂	1	1	yes	characteristic
366	C₂×C₄	C₂²×C₄×D₇₃	1	1	yes	characteristic
367	C₂×C₄	C₂²×D₂₉₂	1	1	yes	characteristic
368	C₂×C₄	C₂×#11924028	1	1	yes	characteristic
369	C₈	—	2	2	no	—
370	C₈	—	2	2	no	—
371	C₈	—	2	1	no	—
372	C₈	—	2	1	no	—
373	C₈	—	2	1	no	—
374	C₈	—	2	1	no	—
375	C₈	—	4	1	no	—
376	C₈	—	4	1	no	—
377	C₈	—	292	2	no	—
378	C₈	—	292	2	no	—
379	C₂²	—	2	1	no	—
380	C₂²	—	2	1	no	—
381	C₂²	—	2	1	no	—
382	C₂²	—	292	2	no	—
383	C₂²	—	292	2	no	—
384	C₂²	—	292	2	no	—
385	C₂²	C₂×C₄×D₂₉₂	1	1	yes	characteristic
386	C₄	—	2	1	no	—
387	C₄	—	2	1	no	—
388	C₄	—	2	1	no	—
389	C₄	—	2	1	no	—
390	C₄	—	292	2	no	—
391	C₄	—	292	2	no	—
392	C₄	—	292	2	no	—
393	C₄	#17989353	1	1	yes	characteristic
394	C₄	#17989353	1	1	yes	characteristic
395	C₄	#17989399	1	2	yes	—
396	C₂	—	4	1	no	—
397	C₂	—	584	2	no	—
398	C₂	#22618650	1	1	yes	characteristic
399	C₂	#22618792	1	1	yes	characteristic
400	C₂	#22618849	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₂₉₂
commutator	C₂×C₄×C₇₃	C₂³×C₄
Frattini	C₄²	C₂³×D₇₃
Fitting	C₇₃×#1452	C₂
socle	C₂²×C₇₃	C₂×C₄×D₄
prime radical	#17989137	C₂²
cosocle	C₄²×C₇₃	C₂⁴
hypercenter	#1452	D₇₃
hypocenter	C₇₃	#17990
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ #431 ◁ #1452

Complements

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

N	H ≅ G/N	type
2. #22619025	C₂	semidirect
4. #22608811	C₂	semidirect
5. #22608790	C₂	semidirect
6. #22608776	C₂	semidirect
7. #22608769	C₂	semidirect
8. #22608694	C₂	semidirect
9. #22608405	C₂	semidirect
11. C₇₃×#1452	C₂	semidirect
211. #1452	D₇₃	semidirect
212. C₇₃	#17990	semidirect

Maximal subgroups

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

subgroup	index
#22620490	2
#22619025	2
#22619006	2
#22608811	2
#22608790	2
#22608776	2
#22608769	2
#22608694	2
#22608405	2
#22608401	2
C₇₃×#1452	2
#17990	73