Subgroups — #33002877

Order 18688.

508 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₁₆×D₇₃	C₂	1	1	yes	maximal, characteristic, complemented
2	#22618627	C₂	1	1	yes	maximal, characteristic
3	C₂×#17988372	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#17988370	C₂	1	1	yes	maximal, characteristic, complemented
5	#22611541	C₂	1	1	yes	maximal, characteristic, complemented
6	#22608020	C₂	1	1	yes	maximal, characteristic, complemented
7	C₇₃×#1379	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#14556597	C₂²	1	1	yes	characteristic
9	C₂²×C₈×D₇₃	C₂²	1	1	yes	characteristic
10	#17989099	C₄	1	1	yes	characteristic
11	C₂×#14556531	C₂²	1	1	yes	characteristic
12	C₂²×#11924003	C₂²	1	1	yes	characteristic
13	C₂×C₁₆×D₇₃	—	2	2	no	—
14	C₂×C₁₆×D₇₃	—	2	2	no	—
15	C₂×C₁₆×D₇₃	—	2	2	no	—
16	#17988660	—	2	2	no	—
17	C₈×D₂₉₂	—	2	2	no	—
18	#17988614	C₄	1	1	yes	characteristic
19	#17988555	C₄	1	1	yes	characteristic
20	#17988550	C₄	1	1	yes	characteristic
21	#17988372	—	2	2	no	—
22	#17988370	—	2	2	no	—
23	#17988366	—	2	2	no	—
24	C₂×M₃₂×C₇₃	C₂²	1	1	yes	characteristic
25	C₂²×C₁₆×C₇₃	C₂²	1	1	yes	characteristic
26	#405×C₇₃	C₂²	1	1	yes	characteristic
27	C₂×#11924028	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₄×D₇₃	C₂×C₄	1	1	yes	characteristic
29	#14556597	—	2	1	no	—
30	#14556597	—	2	1	no	—
31	#14556597	—	2	1	no	—
32	#14556597	—	2	1	no	—
33	C₂×#11924012	C₂×C₄	1	1	yes	characteristic
34	C₂×#11924011	—	2	1	no	—
35	C₂²×#10596831	C₂×C₄	1	1	yes	characteristic
36	C₂×C₈×D₇₃	—	2	1	no	—
37	C₂×C₈×D₇₃	—	2	1	no	—
38	C₂×C₈×D₇₃	—	2	1	no	—
39	C₂×C₈×D₇₃	—	2	1	no	—
40	C₂×C₈×D₇₃	—	2	1	no	—
41	C₂×C₈×D₇₃	D₄	1	2	yes	—
42	C₂×C₈×D₇₃	D₄	1	2	yes	—
43	#14556575	—	2	1	no	—
44	#14556571	C₈	1	2	yes	—
45	#14556566	C₈	1	2	yes	—
46	#14556561	—	2	1	no	—
47	C₄×D₂₉₂	—	2	2	no	—
48	#14556531	—	2	1	no	—
49	#14556531	—	2	1	no	—
50	C₂×#11924003	—	2	1	no	—
51	C₂×#11924003	—	2	1	no	—
52	C₂×#11924003	—	2	1	no	—
53	C₂×#11924003	D₄	1	2	yes	complemented
54	C₁₆×D₇₃	—	2	8	no	—
55	C₄⋊#10596831	—	2	2	no	—
56	#14556503	—	2	2	no	—
57	#14556503	—	2	2	no	—
58	C₂×M₁₆×C₇₃	C₂×C₄	1	1	yes	characteristic
59	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
60	#117×C₇₃	C₂×C₄	1	1	yes	characteristic
61	M₃₂×C₇₃	—	2	1	no	—
62	M₃₂×C₇₃	—	2	1	no	—
63	C₂×C₁₆×C₇₃	—	2	1	no	—
64	C₂×C₁₆×C₇₃	—	2	1	no	—
65	C₂×C₁₆×C₇₃	—	2	1	no	—
66	C₂×C₁₆×C₇₃	D₄	1	2	yes	complemented
67	#97×C₇₃	—	2	1	no	—
68	C₄×C₈×C₇₃	—	2	1	no	—
69	C₂³×D₇₃	M₁₆	1	1	yes	characteristic
70	#11924028	—	2	1	no	—
71	#11924028	—	2	1	no	—
72	#11924028	—	2	1	no	—
73	#11924028	—	2	1	no	—
74	C₂×Dic₁₄₆	C₂×C₈	1	1	yes	characteristic
75	C₂×D₂₉₂	—	2	2	no	—
76	C₂×D₂₉₂	C₂×C₈	1	1	yes	characteristic
77	C₂×C₂²⋊D₇₃	—	2	1	no	—
78	C₂²×Dic₇₃	M₁₆	1	1	yes	characteristic
79	C₂×C₄×D₇₃	—	2	1	no	—
80	C₂×C₄×D₇₃	—	2	1	no	—
81	C₂×C₄×D₇₃	—	2	1	no	—
82	C₂×C₄×D₇₃	—	2	1	no	—
83	C₂×C₄×D₇₃	—	2	1	no	—
84	C₂×C₄×D₇₃	C₂²⋊C₄	1	2	yes	—
85	C₂×C₄×D₇₃	C₂×C₈	1	2	yes	—
86	#11924012	—	2	1	no	—
87	#11924012	—	2	1	no	—
88	#11924011	—	4	1	no	—
89	#11924011	—	4	1	no	—
90	C₂×#10596831	—	2	1	no	—
91	C₂×#10596831	—	2	1	no	—
92	C₂×#10596831	—	2	1	no	—
93	C₂×#10596831	C₂²⋊C₄	1	2	yes	—
94	C₈×D₇₃	—	2	2	no	—
95	C₈×D₇₃	—	2	2	no	—
96	C₈×D₇₃	—	2	2	no	—
97	C₈×D₇₃	—	2	2	no	—
98	C₈×D₇₃	—	4	1	no	—
99	C₈×D₇₃	—	4	1	no	—
100	C₄⋊Dic₇₃	—	2	2	no	—
101	#11924005	—	2	2	no	—
102	#11924005	—	2	2	no	—
103	#11924003	—	2	2	no	—
104	#11924003	—	4	1	no	—
105	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
106	M₁₆×C₇₃	—	2	1	no	—
107	M₁₆×C₇₃	—	2	1	no	—
108	C₂×C₈×C₇₃	—	2	1	no	—
109	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
110	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
111	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
112	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
113	C₂×C₈×C₇₃	#41	1	1	yes	characteristic
114	C₂×C₈×C₇₃	#41	1	1	yes	characteristic
115	C₄⋊C₄×C₇₃	C₂×C₈	1	2	yes	—
116	C₂²⋊C₄×C₇₃	—	2	1	no	—
117	C₄²×C₇₃	—	2	1	no	—
118	C₁₆×C₇₃	—	2	2	no	—
119	C₁₆×C₇₃	—	4	1	no	—
120	C₂³×C₇₃	C₂×M₁₆	1	1	yes	characteristic
121	C₂²×D₇₃	—	2	1	no	—
122	C₂²×D₇₃	—	2	1	no	—
123	C₂²×D₇₃	—	2	4	no	—
124	C₂²×D₇₃	—	2	1	no	—
125	C₂²×D₇₃	C₂²⋊C₈	1	2	yes	—
126	C₂×C₄×C₇₃	—	2	2	no	—
127	C₂×C₄×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
128	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
129	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
130	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
131	C₂×C₄×C₇₃	#131	1	1	yes	characteristic
132	C₂×C₄×C₇₃	#131	1	1	yes	characteristic
133	C₂²⋊D₇₃	—	4	1	no	—
134	C₂²⋊D₇₃	—	4	1	no	—
135	C₂×Dic₇₃	—	2	1	no	—
136	C₂×Dic₇₃	—	2	1	no	—
137	C₂×Dic₇₃	—	2	1	no	—
138	C₂×Dic₇₃	C₂²⋊C₈	1	2	yes	—
139	D₂₉₂	—	2	1	no	—
140	D₂₉₂	—	2	1	no	—
141	D₂₉₂	—	4	4	no	—
142	C₄×D₇₃	—	2	2	no	—
143	C₄×D₇₃	—	2	2	no	—
144	C₄×D₇₃	—	2	2	no	—
145	C₄×D₇₃	—	2	2	no	—
146	C₄×D₇₃	—	4	1	no	—
147	C₄×D₇₃	—	4	1	no	—
148	Dic₁₄₆	—	2	1	no	—
149	Dic₁₄₆	—	2	1	no	—
150	C₈×C₇₃	—	4	1	no	—
151	C₈×C₇₃	#120	1	1	yes	characteristic
152	C₈×C₇₃	#121	1	1	yes	characteristic
153	C₈×C₇₃	#121	1	1	yes	characteristic
154	C₈×C₇₃	#123	1	1	yes	characteristic
155	#10596831	—	2	2	no	—
156	#10596831	—	4	1	no	—
157	C₂²×C₇₃	—	2	2	no	—
158	C₂²×C₇₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
159	C₂²×C₇₃	#435	1	1	yes	characteristic
160	C₂²×C₇₃	#435	1	1	yes	characteristic
161	C₂×D₇₃	—	2	4	no	—
162	C₂×D₇₃	—	2	4	no	—
163	C₂×D₇₃	—	2	2	no	—
164	C₂×D₇₃	—	2	2	no	—
165	C₂×D₇₃	—	4	2	no	—
166	C₂×D₇₃	—	4	1	no	—
167	C₂×D₇₃	#350	1	4	yes	complemented
168	C₄×C₇₃	—	4	2	no	—
169	C₄×C₇₃	#386	1	1	yes	characteristic
170	C₄×C₇₃	#408	1	1	yes	characteristic
171	C₄×C₇₃	C₈×D₄	1	1	yes	characteristic
172	C₄×C₇₃	C₈×D₄	1	1	yes	characteristic
173	Dic₇₃	—	2	2	no	—
174	Dic₇₃	—	4	1	no	—
175	#12983	—	73	1	no	maximal, Sylow
176	C₂×C₇₃	—	2	1	no	—
177	C₂×C₇₃	—	2	1	no	—
178	C₂×C₇₃	#1869	1	1	yes	characteristic
179	C₂×C₇₃	C₂×#350	1	2	yes	—
180	D₇₃	—	2	4	no	—
181	D₇₃	—	8	1	no	—
182	C₂³×C₁₆	—	73	1	no	—
183	#2929	—	73	1	no	—
184	C₂×#350	—	73	1	no	—
185	C₂×#350	—	73	1	no	—
186	#2112	—	73	1	no	—
187	#1379	—	73	1	no	—
188	#1379	D₇₃	1	1	yes	characteristic, complemented
189	C₇₃	#12983	1	1	yes	characteristic, complemented, Sylow
190	C₂×#131	—	73	1	no	—
191	C₂³×C₈	—	73	1	no	—
192	#525	—	73	1	no	—
193	C₂×M₃₂	—	73	1	no	—
194	C₂×M₃₂	C₂×D₇₃	1	1	yes	characteristic
195	C₂²×C₁₆	—	73	1	no	—
196	C₂²×C₁₆	—	146	2	no	—
197	C₂²×C₁₆	—	146	2	no	—
198	C₂²×C₁₆	—	146	2	no	—
199	C₂²×C₁₆	C₂×D₇₃	1	1	yes	characteristic
200	#435	—	146	2	no	—
201	C₈×D₄	—	146	2	no	—
202	#424	—	73	1	no	—
203	#408	—	73	1	no	—
204	#407	—	73	1	no	—
205	#405	C₂×D₇₃	1	1	yes	characteristic
206	#350	—	146	2	no	—
207	#350	—	146	2	no	—
208	C₂²⋊C₁₆	—	146	2	no	—
209	C₂×#41	—	73	1	no	—
210	C₂³×C₄	—	73	1	no	—
211	#131	—	146	1	no	—
212	#131	—	146	1	no	—
213	#131	—	146	1	no	—
214	#131	—	146	1	no	—
215	C₂×M₁₆	—	73	1	no	—
216	C₂×M₁₆	—	146	1	no	—
217	C₂×M₁₆	C₄×D₇₃	1	1	yes	characteristic
218	C₂²×C₈	—	73	1	no	—
219	C₂²×C₈	—	73	2	no	—
220	C₂²×C₈	—	73	2	no	—
221	C₂²×C₈	—	146	1	no	—
222	C₂²×C₈	—	146	1	no	—
223	C₂²×C₈	—	146	1	no	—
224	C₂²×C₈	—	146	1	no	—
225	C₂²×C₈	—	146	1	no	—
226	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
227	#123	—	146	1	no	—
228	C₂²⋊Q₈	—	73	2	no	—
229	#121	—	73	2	no	—
230	#120	—	146	1	no	—
231	C₄×D₄	—	146	2	no	—
232	#117	C₄×D₇₃	1	1	yes	characteristic
233	M₃₂	—	2	1	no	—
234	M₃₂	—	2	1	no	—
235	M₃₂	—	146	1	no	—
236	M₃₂	—	146	1	no	—
237	C₂×C₁₆	—	2	1	no	—
238	C₂×C₁₆	—	2	1	no	—
239	C₂×C₁₆	—	2	1	no	—
240	C₂×C₁₆	—	73	2	no	—
241	C₂×C₁₆	—	146	8	no	—
242	C₂×C₁₆	—	146	1	no	—
243	C₂×C₁₆	—	146	1	no	—
244	C₂×C₁₆	—	146	1	no	—
245	C₂×C₁₆	D₂₉₂	1	2	yes	complemented
246	C₄⋊C₈	—	146	2	no	—
247	C₂²⋊C₈	—	146	2	no	—
248	C₂²⋊C₈	—	146	2	no	—
249	#97	—	2	1	no	—
250	C₄×C₈	—	2	1	no	—
251	C₂⁴	—	73	1	no	—
252	#41	—	146	1	no	—
253	#41	—	146	1	no	—
254	#41	—	146	1	no	—
255	#41	—	146	1	no	—
256	C₂×Q₈	—	73	1	no	—
257	C₂×D₄	—	73	1	no	—
258	C₂×D₄	—	146	2	no	—
259	C₂×D₄	—	146	1	no	—
260	C₂²×C₄	—	73	1	no	—
261	C₂²×C₄	—	73	2	no	—
262	C₂²×C₄	—	73	2	no	—
263	C₂²×C₄	—	146	1	no	—
264	C₂²×C₄	—	146	1	no	—
265	C₂²×C₄	—	146	1	no	—
266	C₂²×C₄	—	146	1	no	—
267	C₂²×C₄	—	146	1	no	—
268	C₂²×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
269	M₁₆	—	2	1	no	—
270	M₁₆	—	2	1	no	—
271	M₁₆	—	292	1	no	—
272	M₁₆	—	292	1	no	—
273	M₁₆	—	146	1	no	—
274	M₁₆	—	146	1	no	—
275	C₂×C₈	—	2	1	no	—
276	C₂×C₈	—	292	1	no	—
277	C₂×C₈	—	292	1	no	—
278	C₂×C₈	—	73	2	no	—
279	C₂×C₈	—	146	1	no	—
280	C₂×C₈	—	146	1	no	—
281	C₂×C₈	—	146	1	no	—
282	C₂×C₈	—	146	2	no	—
283	C₂×C₈	—	146	2	no	—
284	C₂×C₈	—	146	2	no	—
285	C₂×C₈	—	146	2	no	—
286	C₂×C₈	D₄×D₇₃	1	1	yes	characteristic
287	C₂×C₈	D₄×D₇₃	1	1	yes	characteristic
288	C₂×C₈	D₄×D₇₃	1	1	yes	characteristic
289	C₂×C₈	C₂×D₂₉₂	1	1	yes	characteristic
290	C₂×C₈	#11924028	1	1	yes	characteristic
291	C₂×C₈	#11924029	1	1	yes	characteristic
292	C₄⋊C₄	—	146	2	no	—
293	C₄⋊C₄	C₈×D₇₃	1	2	yes	—
294	C₂²⋊C₄	—	2	1	no	—
295	C₂²⋊C₄	—	146	2	no	—
296	C₂²⋊C₄	—	146	2	no	—
297	C₄²	—	2	1	no	—
298	C₁₆	—	2	2	no	—
299	C₁₆	—	4	1	no	—
300	C₁₆	—	292	1	no	—
301	C₁₆	—	146	2	no	—
302	C₂³	—	73	2	no	—
303	C₂³	—	146	4	no	—
304	C₂³	—	146	1	no	—
305	C₂³	—	146	1	no	—
306	C₂³	—	146	1	no	—
307	C₂³	M₁₆×D₇₃	1	1	yes	characteristic
308	Q₈	—	146	1	no	—
309	Q₈	—	146	1	no	—
310	D₄	—	292	4	no	—
311	D₄	—	292	1	no	—
312	D₄	—	292	1	no	—
313	D₄	—	146	1	no	—
314	D₄	—	146	1	no	—
315	C₂×C₄	—	2	2	no	—
316	C₂×C₄	—	292	1	no	—
317	C₂×C₄	—	292	1	no	—
318	C₂×C₄	—	73	2	no	—
319	C₂×C₄	—	146	1	no	—
320	C₂×C₄	—	146	1	no	—
321	C₂×C₄	—	146	1	no	—
322	C₂×C₄	—	146	2	no	—
323	C₂×C₄	—	146	2	no	—
324	C₂×C₄	—	146	2	no	—
325	C₂×C₄	—	146	2	no	—
326	C₂×C₄	C₂²⋊C₄×D₇₃	1	1	yes	characteristic
327	C₂×C₄	#14556553	1	1	yes	characteristic
328	C₂×C₄	C₄×D₂₉₂	1	1	yes	characteristic
329	C₂×C₄	C₂×C₈×D₇₃	1	1	yes	characteristic
330	C₂×C₄	#14556597	1	1	yes	characteristic
331	C₂×C₄	#14556599	1	1	yes	characteristic
332	C₈	—	4	1	no	—
333	C₈	—	292	1	no	—
334	C₈	—	146	2	no	—
335	C₈	#14556561	1	1	yes	characteristic
336	C₈	#14556564	1	1	yes	characteristic
337	C₈	#14556566	1	1	yes	characteristic
338	C₈	#14556576	1	1	yes	characteristic
339	C₂²	—	2	2	no	—
340	C₂²	—	292	1	no	—
341	C₂²	—	292	2	no	—
342	C₂²	—	73	4	no	—
343	C₂²	—	146	4	no	—
344	C₂²	—	146	4	no	—
345	C₂²	—	146	2	no	—
346	C₂²	—	146	2	no	—
347	C₂²	C₂²⋊C₈×D₇₃	1	1	yes	characteristic
348	C₂²	#17988657	1	1	yes	characteristic
349	C₂²	#17988660	1	1	yes	characteristic
350	C₄	—	4	2	no	—
351	C₄	—	292	1	no	—
352	C₄	—	146	2	no	—
353	C₄	#17988465	1	1	yes	characteristic
354	C₄	#17988552	1	1	yes	characteristic
355	C₄	#17988652	1	1	yes	characteristic
356	C₄	C₈×D₂₉₂	1	1	yes	characteristic
357	C₂	—	2	1	no	—
358	C₂	—	2	1	no	—
359	C₂	—	584	1	no	—
360	C₂	—	146	4	no	—
361	C₂	#22610222	1	1	yes	characteristic
362	C₂	#350×D₇₃	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	D₄×D₇₃
commutator	C₂×C₄×C₇₃	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₇₃
Fitting	C₇₃×#1379	C₂
socle	C₂²×C₇₃	C₂×C₂²⋊C₈
prime radical	#17989099	C₄
cosocle	C₂²×C₈×C₇₃	C₂³
hypercenter	#1379	D₇₃
hypocenter	C₇₃	#12983
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ #1379

Complements

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

N	H ≅ G/N	type
1. C₂²×C₁₆×D₇₃	C₂	semidirect
3. C₂×#17988372	C₂	semidirect
4. C₂×#17988370	C₂	semidirect
5. #22611541	C₂	semidirect
6. #22608020	C₂	semidirect
7. C₇₃×#1379	C₂	semidirect
53. C₂×#11924003	D₄	semidirect
66. C₂×C₁₆×C₇₃	D₄	semidirect
167. C₂×D₇₃	#350	semidirect
188. #1379	D₇₃	semidirect
189. C₇₃	#12983	semidirect
245. C₂×C₁₆	D₂₉₂	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₁₆×D₇₃	2
#22618627	2
C₂×#17988372	2
C₂×#17988370	2
#22611541	2
#22608020	2
C₇₃×#1379	2
#12983	73