Subgroups — #33240446

Order 18688.

690 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17989016	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×M₃₂×D₇₃	C₂	1	1	yes	maximal, characteristic, complemented
3	#22619510	C₂	1	2	yes	maximal, complemented
4	#22612043	C₂	1	2	yes	maximal, complemented
5	#22612042	C₂	1	2	yes	maximal, complemented
6	#22612040	C₂	1	1	yes	maximal, characteristic, complemented
7	#22612039	C₂	1	1	yes	maximal, characteristic, complemented
8	#22612031	C₂	1	1	yes	maximal, characteristic, complemented
9	#22611714	C₂	1	1	yes	maximal, characteristic, complemented
10	#22611635	C₂	1	2	yes	maximal, complemented
11	C₇₃×#2218	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#14556610	C₂²	1	2	yes	—
13	C₂²×C₈×D₇₃	C₂²	1	1	yes	characteristic
14	#17989097	—	2	2	no	—
15	#17989016	—	2	4	no	—
16	C₂×Dic₅₈₄	C₂²	1	1	yes	characteristic, complemented
17	C₂×#14556535	C₂²	1	2	yes	complemented
18	C₂×#14556531	C₂²	1	1	yes	characteristic
19	C₂×#14556530	C₂²	1	1	yes	characteristic
20	C₂×#14556530	C₂²	1	1	yes	characteristic
21	M₃₂×D₇₃	—	2	4	no	—
22	C₂×C₁₆×D₇₃	C₂²	1	1	yes	characteristic
23	C₂×C₁₆×D₇₃	C₂²	1	1	yes	characteristic
24	#17988829	C₂²	1	2	yes	—
25	#17988828	C₂²	1	2	yes	—
26	#17988820	C₂²	1	2	yes	—
27	#17988815	—	2	2	no	—
28	#17988814	—	2	2	no	—
29	#17988624	C₂²	1	2	yes	—
30	#17988580	—	2	2	no	—
31	#17988415	C₂²	1	1	yes	characteristic, complemented
32	#17988412	C₂²	1	2	yes	complemented
33	#17988392	C₂²	1	2	yes	complemented
34	#17988390	C₂²	1	2	yes	complemented
35	#17988388	C₂²	1	2	yes	complemented
36	C₂×D₁₆×C₇₃	C₂²	1	1	yes	characteristic, complemented
37	C₂×M₃₂×C₇₃	C₂²	1	1	yes	characteristic
38	#466×C₇₃	C₂²	1	2	yes	—
39	#367×C₇₃	C₂²	1	1	yes	characteristic, complemented
40	#357×C₇₃	C₂²	1	2	yes	complemented
41	C₂×#11924027	—	2	2	no	—
42	C₂²×C₄×D₇₃	D₄	1	1	yes	characteristic
43	#14556610	—	2	4	no	—
44	#14556610	—	2	4	no	—
45	C₂×Dic₂₉₂	C₂³	1	2	yes	—
46	C₂×#11924016	—	2	2	no	—
47	C₂²×#10596831	D₄	1	1	yes	characteristic
48	C₂×C₈×D₇₃	—	2	2	no	—
49	C₂×C₈×D₇₃	—	2	2	no	—
50	C₂×C₈×D₇₃	D₄	1	1	yes	characteristic
51	C₂×C₈×D₇₃	D₄	1	1	yes	characteristic
52	C₂×C₈×D₇₃	C₂³	1	1	yes	characteristic
53	C₂×C₈×D₇₃	C₂³	1	1	yes	characteristic
54	#14556573	—	4	2	no	—
55	#14556571	—	2	2	no	—
56	#14556570	—	2	2	no	—
57	#14556562	—	2	2	no	—
58	#14556559	—	4	2	no	—
59	D₄×Dic₇₃	—	4	2	no	—
60	#14556551	—	4	2	no	—
61	#14556549	—	2	2	no	—
62	Dic₅₈₄	—	2	2	no	—
63	#14556535	—	2	4	no	—
64	#14556531	—	2	2	no	—
65	#14556530	—	2	2	no	—
66	#14556530	—	2	2	no	—
67	C₂×#11924003	C₂³	1	1	yes	characteristic
68	C₂×#11924003	C₂³	1	1	yes	characteristic
69	C₁₆×D₇₃	—	2	2	no	—
70	C₁₆×D₇₃	D₄	1	4	yes	complemented
71	#14556525	C₂³	1	2	yes	—
72	#14556522	—	2	2	no	—
73	#14556516	—	2	2	no	—
74	#14556514	—	2	2	no	—
75	#14556512	—	2	2	no	—
76	C₂×D₈×C₇₃	C₂³	1	2	yes	—
77	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
78	#121×C₇₃	—	2	2	no	—
79	D₁₆×C₇₃	—	2	2	no	—
80	M₃₂×C₇₃	—	2	2	no	—
81	C₂×C₁₆×C₇₃	C₂³	1	1	yes	characteristic
82	C₂×C₁₆×C₇₃	C₂³	1	1	yes	characteristic
83	#107×C₇₃	C₂³	1	2	yes	—
84	#102×C₇₃	—	2	2	no	—
85	C₂³×D₇₃	D₈	1	1	yes	characteristic
86	#11924027	—	4	4	no	—
87	#11924027	—	4	4	no	—
88	C₂×Dic₁₄₆	—	2	2	no	—
89	C₂×C₂²⋊D₇₃	—	4	2	no	—
90	C₂×C₂²⋊D₇₃	—	4	2	no	—
91	C₂²×Dic₇₃	—	4	2	no	—
92	C₂²×Dic₇₃	D₈	1	1	yes	characteristic
93	C₂×C₄×D₇₃	—	2	2	no	—
94	C₂×C₄×D₇₃	—	2	2	no	—
95	C₂×C₄×D₇₃	D₈	1	1	yes	characteristic
96	C₂×C₄×D₇₃	D₈	1	1	yes	characteristic
97	C₂×C₄×D₇₃	C₂×D₄	1	1	yes	characteristic
98	C₂×C₄×D₇₃	C₂×D₄	1	1	yes	characteristic
99	Dic₂₉₂	—	2	2	no	—
100	Dic₂₉₂	—	2	2	no	—
101	#11924016	—	4	4	no	—
102	C₂×#10596831	—	2	2	no	—
103	C₂×#10596831	C₂×D₄	1	1	yes	characteristic
104	C₂×#10596831	C₂×D₄	1	1	yes	characteristic
105	C₈×D₇₃	—	2	2	no	—
106	C₈×D₇₃	—	2	1	no	—
107	C₈×D₇₃	—	2	1	no	—
108	C₈×D₇₃	—	2	2	no	—
109	C₈×D₇₃	C₂×D₄	1	2	yes	—
110	C₈×D₇₃	#41	1	2	yes	—
111	C₄⋊Dic₇₃	—	2	2	no	—
112	C₄⋊Dic₇₃	—	4	2	no	—
113	#11924007	—	4	2	no	—
114	C₂²⋊Dic₇₃	—	4	2	no	—
115	C₂²⋊Dic₇₃	—	4	2	no	—
116	#11924005	—	4	2	no	—
117	#11924005	—	4	2	no	—
118	C₄×Dic₇₃	—	4	2	no	—
119	#11924003	—	2	1	no	—
120	#11924003	C₂×D₄	1	2	yes	—
121	C₂×D₄×C₇₃	—	2	2	no	—
122	C₂×D₄×C₇₃	—	4	2	no	—
123	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
124	D₈×C₇₃	—	2	2	no	—
125	D₈×C₇₃	—	2	2	no	—
126	C₂×C₈×C₇₃	—	2	1	no	—
127	C₂×C₈×C₇₃	—	2	1	no	—
128	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
129	C₂×C₈×C₇₃	C₂⁴	1	1	yes	characteristic
130	C₄⋊C₄×C₇₃	—	2	2	no	—
131	C₂²⋊C₄×C₇₃	—	4	2	no	—
132	C₁₆×C₇₃	—	2	1	no	—
133	C₁₆×C₇₃	C₂×D₄	1	2	yes	—
134	C₂³×C₇₃	—	4	2	no	—
135	C₂³×C₇₃	C₂×D₈	1	1	yes	characteristic
136	C₂²×D₇₃	—	2	2	no	—
137	C₂²×D₇₃	—	4	2	no	—
138	C₂²×D₇₃	C₂×D₈	1	1	yes	characteristic
139	C₂²×D₇₃	C₂×D₈	1	1	yes	characteristic
140	D₄×C₇₃	—	4	2	no	—
141	D₄×C₇₃	—	4	2	no	—
142	D₄×C₇₃	—	8	4	no	—
143	C₂×C₄×C₇₃	—	2	1	no	—
144	C₂×C₄×C₇₃	—	2	1	no	—
145	C₂×C₄×C₇₃	—	4	2	no	—
146	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
147	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
148	C₂²⋊D₇₃	—	8	4	no	—
149	C₂²⋊D₇₃	—	8	4	no	—
150	C₂×Dic₇₃	—	2	2	no	—
151	C₂×Dic₇₃	—	4	2	no	—
152	C₂×Dic₇₃	—	4	2	no	—
153	C₂×Dic₇₃	—	8	4	no	—
154	C₂×Dic₇₃	C₂×D₈	1	1	yes	characteristic
155	C₂×Dic₇₃	C₂×D₈	1	1	yes	characteristic
156	C₄×D₇₃	—	2	2	no	—
157	C₄×D₇₃	—	2	1	no	—
158	C₄×D₇₃	—	2	1	no	—
159	C₄×D₇₃	—	2	2	no	—
160	C₄×D₇₃	#121	1	2	yes	—
161	C₄×D₇₃	#135	1	2	yes	—
162	Dic₁₄₆	—	4	2	no	—
163	Dic₁₄₆	—	4	2	no	—
164	C₈×C₇₃	—	2	1	no	—
165	C₈×C₇₃	C₂²×D₄	1	1	yes	characteristic
166	C₈×C₇₃	C₂×#41	1	1	yes	characteristic
167	#10596831	—	2	1	no	—
168	#10596831	#121	1	2	yes	—
169	C₂²×C₇₃	—	2	2	no	—
170	C₂²×C₇₃	—	2	1	no	—
171	C₂²×C₇₃	—	8	2	no	—
172	C₂²×C₇₃	—	8	2	no	—
173	C₂²×C₇₃	—	8	2	no	—
174	C₂²×C₇₃	C₂²×D₈	1	1	yes	characteristic
175	C₂×D₇₃	—	2	2	no	—
176	C₂×D₇₃	—	2	2	no	—
177	C₂×D₇₃	—	2	1	no	—
178	C₂×D₇₃	—	4	4	no	—
179	C₂×D₇₃	#466	1	2	yes	—
180	C₄×C₇₃	—	2	1	no	—
181	C₄×C₇₃	—	8	2	no	—
182	C₄×C₇₃	C₂×#121	1	1	yes	characteristic
183	C₄×C₇₃	C₂×#135	1	1	yes	characteristic
184	Dic₇₃	—	2	1	no	—
185	Dic₇₃	—	8	2	no	—
186	Dic₇₃	—	8	2	no	—
187	Dic₇₃	#466	1	2	yes	—
188	#33604	—	73	1	no	maximal, Sylow
189	C₂×C₇₃	—	4	1	no	—
190	C₂×C₇₃	—	16	2	no	—
191	C₂×C₇₃	C₂×#466	1	1	yes	characteristic
192	C₂×C₇₃	#3414	1	2	yes	—
193	D₇₃	—	2	2	no	—
194	D₇₃	—	4	1	no	—
195	C₂×#508	—	73	1	no	—
196	C₂²×M₃₂	—	73	1	no	—
197	#3050	—	73	2	no	—
198	#2220	—	73	2	no	—
199	#2220	—	73	2	no	—
200	#2219	—	73	1	no	—
201	#2219	—	73	1	no	—
202	#2218	—	73	1	no	—
203	#2218	D₇₃	1	1	yes	characteristic, complemented
204	#2158	—	73	1	no	—
205	#2138	—	73	2	no	—
206	C₇₃	#33604	1	1	yes	characteristic, complemented, Sylow
207	C₂×#135	—	73	2	no	—
208	C₂³×C₈	—	73	1	no	—
209	#525	—	146	2	no	—
210	#508	—	146	4	no	—
211	C₂×Q₃₂	—	73	1	no	—
212	C₂×SD₃₂	—	73	2	no	—
213	C₂×D₁₆	C₂×D₇₃	1	1	yes	characteristic, complemented
214	C₂×M₃₂	—	73	1	no	—
215	C₂×M₃₂	—	73	1	no	—
216	C₂×M₃₂	—	73	1	no	—
217	C₂×M₃₂	—	146	4	no	—
218	C₂×M₃₂	C₂×D₇₃	1	1	yes	characteristic
219	C₂²×C₁₆	—	73	1	no	—
220	C₂²×C₁₆	—	73	1	no	—
221	#467	—	73	2	no	—
222	#467	—	73	2	no	—
223	#466	—	73	2	no	—
224	#466	C₂×D₇₃	1	2	yes	—
225	#465	—	146	2	no	—
226	#465	—	146	2	no	—
227	#427	—	73	2	no	—
228	#416	—	146	2	no	—
229	#367	—	73	1	no	—
230	#367	C₂×D₇₃	1	1	yes	characteristic, complemented
231	#366	—	73	2	no	—
232	#358	—	73	2	no	—
233	#358	—	73	2	no	—
234	#357	—	73	2	no	—
235	#357	C₂×D₇₃	1	2	yes	complemented
236	C₂×#41	—	146	2	no	—
237	C₂³×C₄	—	73	1	no	—
238	#135	—	146	4	no	—
239	#135	—	146	4	no	—
240	C₂×Q₁₆	—	73	2	no	—
241	C₂×SD₁₆	—	146	2	no	—
242	C₂×D₈	C₂²×D₇₃	1	2	yes	—
243	C₂²×C₈	—	73	1	no	—
244	C₂²×C₈	—	73	1	no	—
245	C₂²×C₈	—	73	1	no	—
246	C₂²×C₈	—	73	1	no	—
247	C₂²×C₈	—	73	1	no	—
248	C₂²×C₈	—	146	2	no	—
249	C₂²×C₈	—	146	2	no	—
250	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
251	#123	—	292	2	no	—
252	C₂²⋊Q₈	—	146	2	no	—
253	C₂²⋊Q₈	—	146	2	no	—
254	#121	—	2	2	no	—
255	#121	—	146	2	no	—
256	#120	—	292	2	no	—
257	C₄×D₄	—	292	2	no	—
258	C₄×D₄	—	292	2	no	—
259	#117	—	146	2	no	—
260	Q₃₂	—	146	2	no	—
261	SD₃₂	—	146	4	no	—
262	D₁₆	—	2	2	no	—
263	M₃₂	—	2	2	no	—
264	M₃₂	—	146	2	no	—
265	M₃₂	—	146	2	no	—
266	M₃₂	—	146	2	no	—
267	C₂×C₁₆	—	73	1	no	—
268	C₂×C₁₆	—	73	4	no	—
269	C₂×C₁₆	—	73	1	no	—
270	C₂×C₁₆	—	146	2	no	—
271	C₂×C₁₆	C₂²×D₇₃	1	1	yes	characteristic
272	C₂×C₁₆	C₂²×D₇₃	1	1	yes	characteristic
273	#107	—	73	2	no	—
274	#107	C₂²×D₇₃	1	2	yes	—
275	#106	—	146	2	no	—
276	#103	—	146	2	no	—
277	#103	—	146	2	no	—
278	#102	—	2	2	no	—
279	#102	—	146	2	no	—
280	C₂⁴	—	73	1	no	—
281	#41	—	292	4	no	—
282	#41	—	292	4	no	—
283	C₂×Q₈	—	146	2	no	—
284	C₂×D₄	—	2	2	no	—
285	C₂×D₄	—	4	2	no	—
286	C₂×D₄	—	292	2	no	—
287	C₂×D₄	—	292	2	no	—
288	C₂²×C₄	—	292	2	no	—
289	C₂²×C₄	—	73	1	no	—
290	C₂²×C₄	—	73	1	no	—
291	C₂²×C₄	—	73	1	no	—
292	C₂²×C₄	—	73	1	no	—
293	C₂²×C₄	—	73	1	no	—
294	C₂²×C₄	—	146	2	no	—
295	C₂²×C₄	—	146	2	no	—
296	C₂²×C₄	D₄×D₇₃	1	1	yes	characteristic
297	Q₁₆	—	146	2	no	—
298	Q₁₆	—	146	2	no	—
299	SD₁₆	—	292	4	no	—
300	D₈	—	2	2	no	—
301	D₈	—	2	2	no	—
302	C₂×C₈	—	2	1	no	—
303	C₂×C₈	—	2	1	no	—
304	C₂×C₈	—	73	1	no	—
305	C₂×C₈	—	73	2	no	—
306	C₂×C₈	—	73	2	no	—
307	C₂×C₈	—	73	1	no	—
308	C₂×C₈	—	146	2	no	—
309	C₂×C₈	—	146	2	no	—
310	C₂×C₈	—	146	2	no	—
311	C₂×C₈	—	146	1	no	—
312	C₂×C₈	—	146	1	no	—
313	C₂×C₈	D₄×D₇₃	1	1	yes	characteristic
314	C₂×C₈	C₂³×D₇₃	1	1	yes	characteristic
315	C₄⋊C₄	—	2	2	no	—
316	C₄⋊C₄	—	292	2	no	—
317	C₄⋊C₄	—	292	2	no	—
318	C₄⋊C₄	—	146	2	no	—
319	C₂²⋊C₄	—	4	2	no	—
320	C₂²⋊C₄	—	292	2	no	—
321	C₂²⋊C₄	—	292	2	no	—
322	C₂²⋊C₄	—	292	2	no	—
323	C₂²⋊C₄	—	292	2	no	—
324	C₄²	—	292	2	no	—
325	C₁₆	—	2	1	no	—
326	C₁₆	—	73	2	no	—
327	C₁₆	—	146	1	no	—
328	C₁₆	D₄×D₇₃	1	2	yes	—
329	C₂³	—	4	2	no	—
330	C₂³	—	292	2	no	—
331	C₂³	—	73	1	no	—
332	C₂³	—	73	1	no	—
333	C₂³	—	146	2	no	—
334	C₂³	D₈×D₇₃	1	1	yes	characteristic
335	Q₈	—	292	2	no	—
336	Q₈	—	292	2	no	—
337	D₄	—	4	2	no	—
338	D₄	—	4	2	no	—
339	D₄	—	8	4	no	—
340	D₄	—	584	4	no	—
341	D₄	—	584	4	no	—
342	C₂×C₄	—	2	1	no	—
343	C₂×C₄	—	2	1	no	—
344	C₂×C₄	—	4	2	no	—
345	C₂×C₄	—	292	2	no	—
346	C₂×C₄	—	292	2	no	—
347	C₂×C₄	—	584	4	no	—
348	C₂×C₄	—	73	1	no	—
349	C₂×C₄	—	73	2	no	—
350	C₂×C₄	—	73	2	no	—
351	C₂×C₄	—	73	1	no	—
352	C₂×C₄	—	146	2	no	—
353	C₂×C₄	—	146	2	no	—
354	C₂×C₄	—	146	2	no	—
355	C₂×C₄	—	146	1	no	—
356	C₂×C₄	—	146	1	no	—
357	C₂×C₄	D₈×D₇₃	1	1	yes	characteristic
358	C₂×C₄	C₂×D₄×D₇₃	1	1	yes	characteristic
359	C₈	—	2	1	no	—
360	C₈	—	73	2	no	—
361	C₈	—	146	1	no	—
362	C₈	C₂×D₄×D₇₃	1	1	yes	characteristic
363	C₈	#41×D₇₃	1	1	yes	characteristic
364	C₂²	—	2	2	no	—
365	C₂²	—	2	1	no	—
366	C₂²	—	8	2	no	—
367	C₂²	—	8	2	no	—
368	C₂²	—	8	2	no	—
369	C₂²	—	292	4	no	—
370	C₂²	—	73	2	no	—
371	C₂²	—	146	2	no	—
372	C₂²	—	146	1	no	—
373	C₂²	—	146	2	no	—
374	C₂²	C₂×D₈×D₇₃	1	1	yes	characteristic
375	C₄	—	2	1	no	—
376	C₄	—	8	2	no	—
377	C₄	—	584	2	no	—
378	C₄	—	584	2	no	—
379	C₄	—	73	2	no	—
380	C₄	—	146	1	no	—
381	C₄	#121×D₇₃	1	1	yes	characteristic
382	C₄	#135×D₇₃	1	1	yes	characteristic
383	C₂	—	4	1	no	—
384	C₂	—	16	2	no	—
385	C₂	—	292	1	no	—
386	C₂	—	146	2	no	—
387	C₂	#466×D₇₃	1	1	yes	characteristic
388	C₂	#22623063	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₈×D₇₃
commutator, cosocle	C₂×C₈×C₇₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₇₃
Fitting	C₇₃×#2218	C₂
socle	C₂²×C₇₃	C₂²×D₈
hypercenter	#2218	D₇₃
hypocenter	C₇₃	#33604
LCS	G ▷ C₂×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #2218

Complements

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

N	H ≅ G/N	type
1. C₂×#17989016	C₂	semidirect
2. C₂×M₃₂×D₇₃	C₂	semidirect
3. #22619510	C₂	semidirect
4. #22612043	C₂	semidirect
5. #22612042	C₂	semidirect
6. #22612040	C₂	semidirect
7. #22612039	C₂	semidirect
8. #22612031	C₂	semidirect
9. #22611714	C₂	semidirect
10. #22611635	C₂	semidirect
11. C₇₃×#2218	C₂	semidirect
16. C₂×Dic₅₈₄	C₂²	semidirect
17. C₂×#14556535	C₂²	semidirect
31. #17988415	C₂²	semidirect
32. #17988412	C₂²	semidirect
33. #17988392	C₂²	semidirect
34. #17988390	C₂²	semidirect
35. #17988388	C₂²	semidirect
36. C₂×D₁₆×C₇₃	C₂²	semidirect
39. #367×C₇₃	C₂²	semidirect
40. #357×C₇₃	C₂²	semidirect
70. C₁₆×D₇₃	D₄	semidirect
203. #2218	D₇₃	semidirect
206. C₇₃	#33604	semidirect
213. C₂×D₁₆	C₂×D₇₃	semidirect
230. #367	C₂×D₇₃	semidirect
235. #357	C₂×D₇₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×#17989016	2
C₂×M₃₂×D₇₃	2
#22619510	2
#22612043	2
#22612042	2
#22612040	2
#22612039	2
#22612031	2
#22611714	2
#22611635	2
C₇₃×#2218	2
#33604	73