Subgroups — #33239392

Order 18688.

652 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22623064	C₂	1	1	yes	maximal, characteristic
2	#443×D₇₃	C₂	1	2	yes	maximal, complemented
3	#22611817	C₂	1	1	yes	maximal, characteristic, complemented
4	#22611816	C₂	1	1	yes	maximal, characteristic, complemented
5	#22611814	C₂	1	2	yes	maximal, complemented
6	#22611813	C₂	1	1	yes	maximal, characteristic, complemented
7	#22611810	C₂	1	2	yes	maximal, complemented
8	#22611724	C₂	1	1	yes	maximal, characteristic, complemented
9	#22611657	C₂	1	2	yes	maximal, complemented
10	C₄×#14556530	C₂	1	1	yes	maximal, characteristic, complemented
11	C₇₃×#2178	C₂	1	1	yes	maximal, characteristic, complemented
12	#135×D₇₃	C₂²	1	2	yes	—
13	#131×D₇₃	—	2	2	no	—
14	#17989029	C₄	1	2	yes	complemented
15	#17989026	C₄	1	2	yes	complemented
16	#17989024	C₄	1	2	yes	complemented
17	#17989021	C₄	1	2	yes	complemented
18	#17989019	C₂²	1	1	yes	characteristic
19	#17989017	C₂²	1	2	yes	—
20	C₂×#14556530	C₂²	1	1	yes	characteristic
21	C₂×#14556530	C₂²	1	1	yes	characteristic
22	#17988705	C₂²	1	2	yes	complemented
23	#17988704	C₂²	1	2	yes	complemented
24	#17988703	C₂²	1	2	yes	complemented
25	#17988702	—	2	2	no	—
26	#17988701	—	2	2	no	—
27	#108×D₇₃	C₂²	1	2	yes	—
28	#104×D₇₃	—	2	2	no	—
29	C₄×C₈×D₇₃	C₂²	1	1	yes	characteristic
30	#17988417	C₂²	1	1	yes	characteristic, complemented
31	#17988416	C₂²	1	2	yes	complemented
32	#17988395	C₂²	1	2	yes	complemented
33	#17988394	C₂²	1	2	yes	complemented
34	#17988393	C₂²	1	2	yes	complemented
35	#17988362	C₂²	1	1	yes	characteristic, complemented
36	#17988362	C₂²	1	1	yes	characteristic, complemented
37	C₄×#11924003	C₂²	1	1	yes	characteristic, complemented
38	#508×C₇₃	C₂²	1	1	yes	characteristic
39	#443×C₇₃	C₂²	1	2	yes	complemented
40	#368×C₇₃	C₂²	1	1	yes	characteristic, complemented
41	#359×C₇₃	C₂²	1	2	yes	complemented
42	C₄×C₁₆×C₇₃	C₂²	1	1	yes	characteristic, complemented
43	#41×D₇₃	—	2	2	no	—
44	#14556614	C₂×C₄	1	2	yes	—
45	#14556613	C₂³	1	2	yes	—
46	#14556612	—	2	2	no	—
47	#14556611	—	2	2	no	—
48	#14556610	C₂×C₄	1	2	yes	—
49	Q₁₆×D₇₃	C₂×C₄	1	2	yes	—
50	SD₁₆×D₇₃	—	2	2	no	—
51	D₈×D₇₃	C₂×C₄	1	2	yes	—
52	#14556599	—	2	2	no	—
53	#14556599	—	4	2	no	—
54	#14556598	—	2	2	no	—
55	#14556597	—	2	2	no	—
56	#14556597	—	4	2	no	—
57	M₁₆×D₇₃	—	2	2	no	—
58	M₁₆×D₇₃	—	4	2	no	—
59	C₂×C₈×D₇₃	—	4	2	no	—
60	C₂×C₈×D₇₃	D₄	1	1	yes	characteristic
61	C₂×C₈×D₇₃	C₂³	1	1	yes	characteristic
62	C₄²×D₇₃	D₄	1	1	yes	characteristic
63	Dic₅₈₄	C₂×C₄	1	1	yes	characteristic, complemented
64	#14556537	C₂×C₄	1	2	yes	complemented
65	#14556536	C₂×C₄	1	2	yes	complemented
66	#14556535	C₂×C₄	1	2	yes	complemented
67	#14556534	C₂×C₄	1	2	yes	complemented
68	D₁₁₆₈	C₂×C₄	1	1	yes	characteristic, complemented
69	#14556532	C₂×C₄	1	2	yes	complemented
70	#14556530	—	2	2	no	—
71	#14556530	C₂×C₄	1	2	yes	—
72	#14556530	C₂×C₄	1	2	yes	—
73	C₂×#11924003	C₂³	1	1	yes	characteristic
74	C₂×#11924003	C₂³	1	1	yes	characteristic
75	#14556527	C₂³	1	2	yes	—
76	#14556526	—	2	2	no	—
77	#14556519	—	2	2	no	—
78	#14556518	—	2	2	no	—
79	#14556517	—	2	2	no	—
80	C₄×#10596831	D₄	1	1	yes	characteristic
81	C₈×Dic₇₃	D₄	1	1	yes	characteristic
82	C₈×Dic₇₃	C₂³	1	1	yes	characteristic
83	#135×C₇₃	C₂³	1	2	yes	—
84	#131×C₇₃	—	2	2	no	—
85	Q₃₂×C₇₃	C₂×C₄	1	1	yes	characteristic, complemented
86	SD₃₂×C₇₃	C₂×C₄	1	2	yes	complemented
87	D₁₆×C₇₃	C₂×C₄	1	1	yes	characteristic, complemented
88	C₂×C₁₆×C₇₃	C₂³	1	1	yes	characteristic
89	C₂×C₁₆×C₇₃	C₂³	1	1	yes	characteristic
90	#108×C₇₃	C₂³	1	2	yes	—
91	#104×C₇₃	—	2	2	no	—
92	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
93	#11924029	—	2	2	no	—
94	#11924028	—	2	2	no	—
95	#11924028	—	4	2	no	—
96	#11924027	—	2	2	no	—
97	#11924027	—	4	2	no	—
98	Q₈×D₇₃	—	2	2	no	—
99	D₄×D₇₃	—	2	2	no	—
100	D₄×D₇₃	—	4	2	no	—
101	C₂×C₄×D₇₃	—	2	1	no	—
102	C₂×C₄×D₇₃	—	4	2	no	—
103	C₂×C₄×D₇₃	C₂×D₄	1	1	yes	characteristic
104	Dic₂₉₂	C₂²×C₄	1	2	yes	—
105	#11924018	—	2	2	no	—
106	#11924017	—	2	2	no	—
107	#11924016	—	2	2	no	—
108	#11924015	—	2	2	no	—
109	D₅₈₄	C₂²×C₄	1	2	yes	—
110	#11924013	—	2	2	no	—
111	#11924012	—	2	2	no	—
112	#11924012	—	4	2	no	—
113	#11924011	—	4	2	no	—
114	#11924011	—	4	2	no	—
115	#11924011	—	4	2	no	—
116	C₂×#10596831	—	4	2	no	—
117	C₂×#10596831	C₂×D₄	1	1	yes	characteristic
118	C₂×#10596831	C₂×D₄	1	1	yes	characteristic
119	C₈×D₇₃	—	4	2	no	—
120	C₈×D₇₃	—	4	2	no	—
121	C₈×D₇₃	—	4	2	no	—
122	C₈×D₇₃	D₈	1	2	yes	—
123	C₈×D₇₃	D₈	1	2	yes	—
124	C₈×D₇₃	C₂²×C₄	1	2	yes	—
125	C₈×D₇₃	#41	1	2	yes	—
126	C₄×Dic₇₃	—	2	1	no	—
127	C₄×Dic₇₃	C₂×D₄	1	1	yes	characteristic
128	#11924003	—	2	1	no	—
129	#11924003	C₂²×C₄	1	2	yes	—
130	#41×C₇₃	—	2	2	no	—
131	Q₁₆×C₇₃	C₂²×C₄	1	2	yes	—
132	SD₁₆×C₇₃	—	2	2	no	—
133	D₈×C₇₃	C₂²×C₄	1	2	yes	—
134	M₁₆×C₇₃	—	2	2	no	—
135	M₁₆×C₇₃	—	4	2	no	—
136	C₂×C₈×C₇₃	—	4	2	no	—
137	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
138	C₂×C₈×C₇₃	C₂⁴	1	1	yes	characteristic
139	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
140	C₁₆×C₇₃	—	2	1	no	—
141	C₁₆×C₇₃	C₂²×C₄	1	1	yes	characteristic
142	C₁₆×C₇₃	C₂²×C₄	1	1	yes	characteristic
143	C₂²×D₇₃	—	4	2	no	—
144	C₂²×D₇₃	#135	1	1	yes	characteristic
145	Q₈×C₇₃	—	2	2	no	—
146	D₄×C₇₃	—	2	2	no	—
147	D₄×C₇₃	—	4	2	no	—
148	C₂×C₄×C₇₃	—	2	1	no	—
149	C₂×C₄×C₇₃	—	4	2	no	—
150	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
151	C₂²⋊D₇₃	—	4	2	no	—
152	C₂²⋊D₇₃	—	4	2	no	—
153	C₂²⋊D₇₃	—	4	2	no	—
154	C₂×Dic₇₃	—	2	1	no	—
155	C₂×Dic₇₃	—	4	2	no	—
156	C₂×Dic₇₃	#135	1	1	yes	characteristic
157	D₂₉₂	—	2	2	no	—
158	D₂₉₂	—	4	2	no	—
159	C₄×D₇₃	—	4	2	no	—
160	C₄×D₇₃	—	4	2	no	—
161	C₄×D₇₃	—	4	2	no	—
162	C₄×D₇₃	—	4	2	no	—
163	C₄×D₇₃	C₄×D₄	1	2	yes	—
164	C₄×D₇₃	C₂×D₈	1	2	yes	—
165	Dic₁₄₆	—	2	2	no	—
166	Dic₁₄₆	—	4	2	no	—
167	C₈×C₇₃	—	4	2	no	—
168	C₈×C₇₃	C₂×D₈	1	1	yes	characteristic
169	C₈×C₇₃	C₂×D₈	1	1	yes	characteristic
170	C₈×C₇₃	C₂³×C₄	1	1	yes	characteristic
171	C₈×C₇₃	C₂×#41	1	1	yes	characteristic
172	#10596831	—	4	2	no	—
173	#10596831	C₄×D₄	1	2	yes	—
174	#10596831	C₂×D₈	1	2	yes	—
175	C₂²×C₇₃	—	4	2	no	—
176	C₂²×C₇₃	C₂×#135	1	1	yes	characteristic
177	C₂×D₇₃	—	4	1	no	—
178	C₂×D₇₃	—	4	2	no	—
179	C₂×D₇₃	—	8	4	no	—
180	C₂×D₇₃	C₄×D₈	1	2	yes	—
181	C₄×C₇₃	—	2	2	no	—
182	C₄×C₇₃	—	4	2	no	—
183	C₄×C₇₃	C₂×C₄×D₄	1	1	yes	characteristic
184	C₄×C₇₃	C₂²×D₈	1	1	yes	characteristic
185	Dic₇₃	—	4	1	no	—
186	Dic₇₃	—	4	2	no	—
187	Dic₇₃	C₄×D₈	1	2	yes	—
188	#33463	—	73	1	no	maximal, Sylow
189	C₂×C₇₃	—	2	1	no	—
190	C₂×C₇₃	—	8	2	no	—
191	C₂×C₇₃	C₂×C₄×D₈	1	1	yes	characteristic
192	D₇₃	—	2	2	no	—
193	D₇₃	—	8	2	no	—
194	#3414	—	73	1	no	—
195	C₂×#443	—	73	2	no	—
196	#2179	—	73	1	no	—
197	#2179	—	73	1	no	—
198	#2179	—	73	2	no	—
199	#2178	—	73	2	no	—
200	#2178	—	73	1	no	—
201	#2178	D₇₃	1	1	yes	characteristic, complemented
202	#2161	—	73	1	no	—
203	#2143	—	73	2	no	—
204	C₄×M₃₂	—	73	1	no	—
205	C₇₃	#33463	1	1	yes	characteristic, complemented, Sylow
206	C₂×#135	—	73	2	no	—
207	C₂×#131	—	146	2	no	—
208	#510	—	73	2	no	—
209	#510	—	73	2	no	—
210	#509	—	73	2	no	—
211	#509	—	73	2	no	—
212	#508	—	73	2	no	—
213	#508	—	73	1	no	—
214	#508	C₂×D₇₃	1	1	yes	characteristic
215	C₂×M₃₂	—	73	1	no	—
216	C₂×M₃₂	—	73	1	no	—
217	#443	—	73	2	no	—
218	#443	—	73	2	no	—
219	#443	—	73	2	no	—
220	#443	—	146	2	no	—
221	#443	—	146	2	no	—
222	#443	C₂×D₇₃	1	2	yes	complemented
223	C₂×#108	—	73	2	no	—
224	C₂×#104	—	146	2	no	—
225	C₂×C₄×C₈	—	73	1	no	—
226	#368	—	73	1	no	—
227	#368	—	73	2	no	—
228	#368	C₂×D₇₃	1	1	yes	characteristic, complemented
229	#359	—	73	2	no	—
230	#359	—	73	2	no	—
231	#359	—	73	2	no	—
232	#359	C₂×D₇₃	1	2	yes	complemented
233	#346	—	73	1	no	—
234	#346	—	73	1	no	—
235	C₄×C₁₆	—	73	1	no	—
236	C₄×C₁₆	C₂×D₇₃	1	1	yes	characteristic, complemented
237	C₂×#41	—	146	2	no	—
238	#135	—	73	2	no	—
239	#135	—	73	2	no	—
240	#135	—	73	2	no	—
241	#135	—	146	2	no	—
242	#135	—	146	2	no	—
243	#135	C₂²×D₇₃	1	2	yes	—
244	C₂×Q₁₆	—	73	2	no	—
245	C₂×SD₁₆	—	146	2	no	—
246	C₂×D₈	—	73	2	no	—
247	#131	—	2	2	no	—
248	#131	—	292	2	no	—
249	#131	—	292	2	no	—
250	#131	—	146	2	no	—
251	#131	—	146	2	no	—
252	#131	—	146	2	no	—
253	C₂×M₁₆	—	292	2	no	—
254	C₂×M₁₆	—	146	2	no	—
255	C₂²×C₈	—	292	2	no	—
256	C₂²×C₈	—	73	1	no	—
257	C₂²×C₈	—	73	1	no	—
258	C₂×C₄²	—	73	1	no	—
259	Q₃₂	—	73	2	no	—
260	Q₃₂	—	73	1	no	—
261	Q₃₂	C₄×D₇₃	1	1	yes	characteristic, complemented
262	SD₃₂	—	73	2	no	—
263	SD₃₂	—	73	2	no	—
264	SD₃₂	—	73	2	no	—
265	SD₃₂	C₄×D₇₃	1	2	yes	complemented
266	D₁₆	—	73	2	no	—
267	D₁₆	—	73	1	no	—
268	D₁₆	C₄×D₇₃	1	1	yes	characteristic, complemented
269	M₃₂	—	73	2	no	—
270	M₃₂	—	73	2	no	—
271	M₃₂	—	146	2	no	—
272	C₂×C₁₆	—	73	1	no	—
273	C₂×C₁₆	—	73	1	no	—
274	C₂×C₁₆	C₂²×D₇₃	1	1	yes	characteristic
275	C₂×C₁₆	C₂²×D₇₃	1	1	yes	characteristic
276	#108	—	73	2	no	—
277	#108	—	146	2	no	—
278	#108	C₂²×D₇₃	1	2	yes	—
279	#104	—	2	2	no	—
280	#104	—	146	2	no	—
281	#104	—	146	2	no	—
282	#104	—	146	2	no	—
283	C₄×C₈	—	73	1	no	—
284	C₄×C₈	—	73	1	no	—
285	C₄×C₈	—	73	1	no	—
286	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
287	#41	—	2	2	no	—
288	#41	—	292	2	no	—
289	#41	—	292	2	no	—
290	#41	—	146	2	no	—
291	#41	—	146	2	no	—
292	#41	—	146	2	no	—
293	C₂×Q₈	—	146	2	no	—
294	C₂×D₄	—	292	2	no	—
295	C₂×D₄	—	146	2	no	—
296	C₂²×C₄	—	292	2	no	—
297	C₂²×C₄	—	73	1	no	—
298	C₂²×C₄	—	146	1	no	—
299	Q₁₆	—	73	2	no	—
300	Q₁₆	—	146	2	no	—
301	Q₁₆	C₂×C₄×D₇₃	1	2	yes	—
302	SD₁₆	—	2	2	no	—
303	SD₁₆	—	146	2	no	—
304	SD₁₆	—	146	2	no	—
305	SD₁₆	—	146	2	no	—
306	D₈	—	73	2	no	—
307	D₈	—	146	2	no	—
308	D₈	C₂×C₄×D₇₃	1	2	yes	—
309	M₁₆	—	2	2	no	—
310	M₁₆	—	4	2	no	—
311	M₁₆	—	292	2	no	—
312	M₁₆	—	292	2	no	—
313	M₁₆	—	292	2	no	—
314	M₁₆	—	292	2	no	—
315	M₁₆	—	146	2	no	—
316	C₂×C₈	—	4	2	no	—
317	C₂×C₈	—	292	2	no	—
318	C₂×C₈	—	292	2	no	—
319	C₂×C₈	—	292	2	no	—
320	C₂×C₈	—	292	2	no	—
321	C₂×C₈	—	73	1	no	—
322	C₂×C₈	—	73	1	no	—
323	C₂×C₈	—	73	2	no	—
324	C₂×C₈	—	73	2	no	—
325	C₂×C₈	—	73	2	no	—
326	C₂×C₈	—	73	2	no	—
327	C₂×C₈	D₄×D₇₃	1	1	yes	characteristic
328	C₂×C₈	C₂³×D₇₃	1	1	yes	characteristic
329	C₄²	—	73	1	no	—
330	C₄²	—	146	1	no	—
331	C₄²	D₄×D₇₃	1	1	yes	characteristic
332	C₁₆	—	2	1	no	—
333	C₁₆	—	73	2	no	—
334	C₁₆	—	146	1	no	—
335	C₁₆	C₂×C₄×D₇₃	1	1	yes	characteristic
336	C₁₆	C₂×C₄×D₇₃	1	1	yes	characteristic
337	C₂³	—	292	2	no	—
338	C₂³	—	73	1	no	—
339	Q₈	—	2	2	no	—
340	Q₈	—	292	2	no	—
341	Q₈	—	146	2	no	—
342	D₄	—	2	2	no	—
343	D₄	—	4	2	no	—
344	D₄	—	292	2	no	—
345	D₄	—	292	2	no	—
346	D₄	—	292	2	no	—
347	D₄	—	292	2	no	—
348	D₄	—	146	2	no	—
349	C₂×C₄	—	2	1	no	—
350	C₂×C₄	—	4	2	no	—
351	C₂×C₄	—	292	2	no	—
352	C₂×C₄	—	292	2	no	—
353	C₂×C₄	—	292	2	no	—
354	C₂×C₄	—	292	2	no	—
355	C₂×C₄	—	292	2	no	—
356	C₂×C₄	—	73	1	no	—
357	C₂×C₄	—	73	2	no	—
358	C₂×C₄	—	73	2	no	—
359	C₂×C₄	—	146	1	no	—
360	C₂×C₄	C₂×D₄×D₇₃	1	1	yes	characteristic
361	C₈	—	4	2	no	—
362	C₈	—	292	2	no	—
363	C₈	—	73	2	no	—
364	C₈	—	73	2	no	—
365	C₈	D₈×D₇₃	1	1	yes	characteristic
366	C₈	D₈×D₇₃	1	1	yes	characteristic
367	C₈	C₂²×C₄×D₇₃	1	1	yes	characteristic
368	C₈	#41×D₇₃	1	1	yes	characteristic
369	C₂²	—	4	2	no	—
370	C₂²	—	292	1	no	—
371	C₂²	—	292	2	no	—
372	C₂²	—	584	4	no	—
373	C₂²	—	73	2	no	—
374	C₂²	#135×D₇₃	1	1	yes	characteristic
375	C₄	—	2	2	no	—
376	C₄	—	4	2	no	—
377	C₄	—	292	1	no	—
378	C₄	—	292	2	no	—
379	C₄	—	73	2	no	—
380	C₄	C₄×D₄×D₇₃	1	1	yes	characteristic
381	C₄	C₂×D₈×D₇₃	1	1	yes	characteristic
382	C₂	—	2	1	no	—
383	C₂	—	8	2	no	—
384	C₂	—	584	2	no	—
385	C₂	—	146	2	no	—
386	C₂	C₄×D₈×D₇₃	1	1	yes	characteristic

	subgroup	G/H
center	C₈	D₈×D₇₃
commutator	C₈×C₇₃	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₇₃
Fitting	C₇₃×#2178	C₂
socle	C₂×C₇₃	C₂×C₄×D₈
prime radical	#22623064	C₂
cosocle	C₂×C₈×C₇₃	C₂⁴
hypercenter	#2178	D₇₃
hypocenter	C₇₃	#33463
LCS	G ▷ C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₈ ◁ C₂×C₈ ◁ C₄×C₈ ◁ #2178

Complements

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

N	H ≅ G/N	type
2. #443×D₇₃	C₂	semidirect
3. #22611817	C₂	semidirect
4. #22611816	C₂	semidirect
5. #22611814	C₂	semidirect
6. #22611813	C₂	semidirect
7. #22611810	C₂	semidirect
8. #22611724	C₂	semidirect
9. #22611657	C₂	semidirect
10. C₄×#14556530	C₂	semidirect
11. C₇₃×#2178	C₂	semidirect
14. #17989029	C₄	semidirect
15. #17989026	C₄	semidirect
16. #17989024	C₄	semidirect
17. #17989021	C₄	semidirect
22. #17988705	C₂²	semidirect
23. #17988704	C₂²	semidirect
24. #17988703	C₂²	semidirect
30. #17988417	C₂²	semidirect
31. #17988416	C₂²	semidirect
32. #17988395	C₂²	semidirect
33. #17988394	C₂²	semidirect
34. #17988393	C₂²	semidirect
35. #17988362	C₂²	semidirect
36. #17988362	C₂²	semidirect
37. C₄×#11924003	C₂²	semidirect
39. #443×C₇₃	C₂²	semidirect
40. #368×C₇₃	C₂²	semidirect
41. #359×C₇₃	C₂²	semidirect
42. C₄×C₁₆×C₇₃	C₂²	semidirect
63. Dic₅₈₄	C₂×C₄	semidirect
64. #14556537	C₂×C₄	semidirect
65. #14556536	C₂×C₄	semidirect
66. #14556535	C₂×C₄	semidirect
67. #14556534	C₂×C₄	semidirect
68. D₁₁₆₈	C₂×C₄	semidirect
69. #14556532	C₂×C₄	semidirect
85. Q₃₂×C₇₃	C₂×C₄	semidirect
86. SD₃₂×C₇₃	C₂×C₄	semidirect
87. D₁₆×C₇₃	C₂×C₄	semidirect
201. #2178	D₇₃	semidirect
205. C₇₃	#33463	semidirect
222. #443	C₂×D₇₃	semidirect
228. #368	C₂×D₇₃	semidirect
232. #359	C₂×D₇₃	semidirect
236. C₄×C₁₆	C₂×D₇₃	semidirect
261. Q₃₂	C₄×D₇₃	semidirect
265. SD₃₂	C₄×D₇₃	semidirect
268. D₁₆	C₄×D₇₃	semidirect

Maximal subgroups

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

subgroup	index
#22623064	2
#443×D₇₃	2
#22611817	2
#22611816	2
#22611814	2
#22611813	2
#22611810	2
#22611724	2
#22611657	2
C₄×#14556530	2
C₇₃×#2178	2
#33463	73