Subgroups — #32979844

Order 18688.

482 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22618954	C₂	1	1	yes	maximal, characteristic
2	#22610724	C₂	1	1	yes	maximal, characteristic
3	C₄×#14556507	C₂	1	1	yes	maximal, characteristic
4	#22608681	C₂	1	1	yes	maximal, characteristic, complemented
5	#22607769	C₂	1	1	yes	maximal, characteristic, complemented
6	#22607767	C₂	1	1	yes	maximal, characteristic, complemented
7	C₇₃×#1565	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄×D₂₉₂	C₂²	1	1	yes	characteristic
9	C₈×D₂₉₂	—	2	1	no	—
10	C₈×D₂₉₂	—	2	1	no	—
11	#17988652	—	2	1	no	—
12	#17988652	—	2	1	no	—
13	#17988640	C₂²	1	1	yes	characteristic
14	C₂×#14556507	C₄	1	1	yes	characteristic
15	C₂×#14556507	C₄	1	1	yes	characteristic
16	C₂×#14556503	—	2	2	no	—
17	C₂²⋊C₈×D₇₃	—	2	2	no	—
18	C₄×#11924012	C₂²	1	1	yes	characteristic
19	C₂×C₄×#10596831	C₂²	1	1	yes	characteristic
20	#17988355	C₄	1	1	yes	characteristic
21	#17988355	C₄	1	1	yes	characteristic
22	#17988339	—	2	1	no	—
23	#17988337	—	2	1	no	—
24	#431×C₇₃	C₂²	1	1	yes	characteristic
25	C₂×C₄⋊C₈×C₇₃	C₂²	1	1	yes	characteristic
26	C₄×M₁₆×C₇₃	C₂²	1	1	yes	characteristic
27	#335×C₇₃	—	2	1	no	—
28	#334×C₇₃	—	2	1	no	—
29	C₂²×D₂₉₂	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₄×D₇₃	D₄	1	2	yes	—
31	C₂×#11924012	C₂×C₄	1	1	yes	characteristic
32	C₂×#11924012	C₂×C₄	1	1	yes	characteristic
33	C₂²×#10596831	—	2	1	no	—
34	C₂×C₈×D₇₃	—	2	2	no	—
35	C₂×C₈×D₇₃	—	2	2	no	—
36	C₄×D₂₉₂	—	2	1	no	—
37	C₄×D₂₉₂	—	2	1	no	—
38	C₄×D₂₉₂	—	2	1	no	—
39	C₄×D₂₉₂	—	2	1	no	—
40	C₂×C₄⋊Dic₇₃	C₂×C₄	1	1	yes	characteristic
41	C₂×#11924005	Q₈	1	2	yes	—
42	C₄⋊#10596831	D₄	1	1	yes	characteristic
43	C₄⋊#10596831	D₄	1	1	yes	characteristic
44	#14556507	—	2	2	no	—
45	#14556507	C₈	1	4	yes	complemented
46	C₂²⋊#10596831	—	2	1	no	—
47	#14556503	—	2	2	no	—
48	#14556503	—	2	2	no	—
49	#14556503	—	4	4	no	—
50	#14556502	—	2	1	no	—
51	C₄×#10596831	—	2	1	no	—
52	C₄×#10596831	—	2	1	no	—
53	C₄×#10596831	—	2	1	no	—
54	C₄×#10596831	—	2	1	no	—
55	C₂×M₁₆×C₇₃	C₂×C₄	1	1	yes	characteristic
56	C₂×M₁₆×C₇₃	C₂×C₄	1	1	yes	characteristic
57	C₂²×C₈×C₇₃	—	2	1	no	—
58	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
59	C₄⋊C₈×C₇₃	—	2	1	no	—
60	C₄⋊C₈×C₇₃	—	2	1	no	—
61	C₄⋊C₈×C₇₃	D₄	1	1	yes	characteristic
62	C₄⋊C₈×C₇₃	D₄	1	1	yes	characteristic
63	C₂²⋊C₈×C₇₃	—	2	1	no	—
64	#97×C₇₃	—	2	1	no	—
65	C₄×C₈×C₇₃	—	2	1	no	—
66	C₄×C₈×C₇₃	—	2	1	no	—
67	C₂³×D₇₃	C₄⋊C₄	1	2	yes	—
68	C₂×D₂₉₂	—	2	1	no	—
69	C₂×D₂₉₂	—	2	1	no	—
70	C₂×D₂₉₂	—	2	1	no	—
71	C₂×D₂₉₂	—	2	1	no	—
72	C₂×D₂₉₂	C₂×C₈	1	2	yes	—
73	C₂×D₂₉₂	M₁₆	1	2	yes	—
74	C₂²×Dic₇₃	C₄⋊C₄	1	2	yes	—
75	C₂×C₄×D₇₃	—	2	2	no	—
76	C₂×C₄×D₇₃	—	2	2	no	—
77	C₂×C₄×D₇₃	—	4	2	no	—
78	C₂×C₄×D₇₃	—	4	2	no	—
79	#11924012	—	2	1	no	—
80	#11924012	—	2	1	no	—
81	#11924012	—	2	1	no	—
82	#11924012	C₂×C₈	1	2	yes	complemented
83	C₂×#10596831	—	2	1	no	—
84	C₂×#10596831	—	2	2	no	—
85	C₂×#10596831	—	2	1	no	—
86	C₂×#10596831	—	2	1	no	—
87	C₂×#10596831	—	2	1	no	—
88	C₂×#10596831	—	4	1	no	—
89	C₂×#10596831	—	4	1	no	—
90	C₈×D₇₃	—	8	2	no	—
91	C₈×D₇₃	—	8	2	no	—
92	C₄⋊Dic₇₃	—	2	1	no	—
93	C₄⋊Dic₇₃	—	2	1	no	—
94	#11924005	—	2	2	no	—
95	#11924005	—	2	2	no	—
96	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
97	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
98	C₂²×C₄×C₇₃	C₂×Q₈	1	1	yes	characteristic
99	M₁₆×C₇₃	—	2	1	no	—
100	M₁₆×C₇₃	—	2	1	no	—
101	M₁₆×C₇₃	—	2	1	no	—
102	M₁₆×C₇₃	C₂×C₈	1	2	yes	complemented
103	C₂×C₈×C₇₃	—	2	1	no	—
104	C₂×C₈×C₇₃	—	2	1	no	—
105	C₂×C₈×C₇₃	—	2	1	no	—
106	C₂×C₈×C₇₃	—	2	1	no	—
107	C₂×C₈×C₇₃	—	4	2	no	—
108	C₂×C₈×C₇₃	—	4	1	no	—
109	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
110	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
111	C₄²×C₇₃	#41	1	1	yes	characteristic
112	C₄²×C₇₃	#41	1	1	yes	characteristic
113	C₂³×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
114	C₂²×D₇₃	—	2	2	no	—
115	C₂²×D₇₃	—	2	2	no	—
116	C₂²×D₇₃	—	4	2	no	—
117	C₂²×D₇₃	—	4	2	no	—
118	C₂²×D₇₃	C₄⋊C₈	1	4	yes	—
119	C₂×C₄×C₇₃	—	2	1	no	—
120	C₂×C₄×C₇₃	—	2	1	no	—
121	C₂×C₄×C₇₃	—	4	1	no	—
122	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
123	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
124	C₂×C₄×C₇₃	#120	1	1	yes	characteristic
125	C₂×C₄×C₇₃	C₂²⋊Q₈	1	1	yes	characteristic
126	C₂×C₄×C₇₃	C₂²⋊Q₈	1	1	yes	characteristic
127	C₂×C₄×C₇₃	#123	1	1	yes	characteristic
128	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
129	C₂×C₄×C₇₃	C₂×M₁₆	1	1	yes	characteristic
130	C₂×C₄×C₇₃	#131	1	1	yes	characteristic
131	C₂×C₄×C₇₃	#131	1	1	yes	characteristic
132	C₂×Dic₇₃	—	2	2	no	—
133	C₂×Dic₇₃	—	4	2	no	—
134	D₂₉₂	—	2	2	no	—
135	D₂₉₂	—	2	2	no	—
136	D₂₉₂	—	2	2	no	—
137	D₂₉₂	—	2	2	no	—
138	C₄×D₇₃	—	4	2	no	—
139	C₄×D₇₃	—	4	2	no	—
140	C₄×D₇₃	—	8	2	no	—
141	C₈×C₇₃	—	2	2	no	—
142	C₈×C₇₃	—	4	1	no	—
143	C₈×C₇₃	—	4	1	no	—
144	C₈×C₇₃	—	4	1	no	—
145	C₈×C₇₃	—	8	1	no	—
146	#10596831	—	2	2	no	—
147	#10596831	—	4	2	no	—
148	#10596831	—	4	1	no	—
149	#10596831	—	4	1	no	—
150	#10596831	—	4	1	no	—
151	C₂²×C₇₃	—	2	1	no	—
152	C₂²×C₇₃	—	2	1	no	—
153	C₂²×C₇₃	#385	1	1	yes	characteristic
154	C₂²×C₇₃	C₂×C₄⋊C₈	1	1	yes	characteristic
155	C₂²×C₇₃	#424	1	1	yes	characteristic
156	C₂×D₇₃	—	2	4	no	—
157	C₂×D₇₃	—	4	2	no	—
158	C₂×D₇₃	—	4	2	no	—
159	C₂×D₇₃	—	4	4	no	—
160	C₂×D₇₃	—	8	2	no	—
161	C₄×C₇₃	—	2	1	no	—
162	C₄×C₇₃	—	2	1	no	—
163	C₄×C₇₃	—	2	1	no	—
164	C₄×C₇₃	C₈×D₄	1	1	yes	characteristic
165	C₄×C₇₃	C₈×D₄	1	1	yes	characteristic
166	C₄×C₇₃	#435	1	1	yes	characteristic
167	C₄×C₇₃	#435	1	1	yes	characteristic
168	C₄×C₇₃	#453	1	1	yes	characteristic
169	C₄×C₇₃	#455	1	1	yes	characteristic
170	Dic₇₃	—	4	2	no	—
171	#8835	—	73	1	no	maximal, Sylow
172	C₂×C₇₃	—	2	1	no	—
173	C₂×C₇₃	—	2	1	no	—
174	C₂×C₇₃	#1850	1	1	yes	characteristic
175	C₂×C₇₃	#1856	1	1	yes	characteristic
176	C₂×C₇₃	#1858	1	1	yes	characteristic
177	D₇₃	—	4	4	no	—
178	#2972	—	73	1	no	—
179	#1954	—	73	1	no	—
180	C₄×#100	—	73	1	no	—
181	#1565	—	73	1	no	—
182	#1565	D₇₃	1	1	yes	characteristic, complemented
183	#1278	—	73	1	no	—
184	#1278	—	73	1	no	—
185	C₇₃	#8835	1	1	yes	characteristic, complemented, Sylow
186	C₂×C₄×D₄	—	73	1	no	—
187	C₈×D₄	—	146	1	no	—
188	C₈×D₄	—	146	1	no	—
189	C₈×D₄	—	146	1	no	—
190	C₈×D₄	—	146	1	no	—
191	#431	—	73	1	no	—
192	#431	C₂×D₇₃	1	1	yes	characteristic
193	C₂×C₄⋊C₈	C₂×D₇₃	1	1	yes	characteristic
194	C₂×#100	—	73	1	no	—
195	C₂×#100	—	73	1	no	—
196	C₂×C₂²⋊C₈	—	146	2	no	—
197	C₂×C₂²⋊C₈	—	146	2	no	—
198	C₄×M₁₆	—	73	1	no	—
199	C₄×M₁₆	C₂×D₇₃	1	1	yes	characteristic
200	C₂×C₄×C₈	—	73	1	no	—
201	#343	—	73	1	no	—
202	#343	—	73	1	no	—
203	#335	—	2	1	no	—
204	#335	—	146	1	no	—
205	#334	—	2	1	no	—
206	#334	—	146	1	no	—
207	C₂²×D₄	—	73	1	no	—
208	C₂³×C₄	—	73	2	no	—
209	C₂×M₁₆	—	73	1	no	—
210	C₂×M₁₆	—	73	1	no	—
211	C₂×M₁₆	C₄×D₇₃	1	1	yes	characteristic
212	C₂×M₁₆	C₄×D₇₃	1	1	yes	characteristic
213	C₂²×C₈	—	2	1	no	—
214	C₂²×C₈	—	146	2	no	—
215	C₂²×C₈	—	146	2	no	—
216	C₂²×C₈	—	146	1	no	—
217	C₄×D₄	—	146	1	no	—
218	C₄×D₄	—	146	1	no	—
219	C₄×D₄	—	146	1	no	—
220	C₄×D₄	—	146	1	no	—
221	C₂×C₄⋊C₄	—	73	1	no	—
222	C₂×C₂²⋊C₄	—	73	2	no	—
223	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
224	C₄⋊C₈	—	2	1	no	—
225	C₄⋊C₈	—	2	1	no	—
226	C₄⋊C₈	—	73	1	no	—
227	C₄⋊C₈	—	73	1	no	—
228	C₄⋊C₈	C₂²⋊D₇₃	1	1	yes	characteristic
229	C₄⋊C₈	C₂²⋊D₇₃	1	1	yes	characteristic
230	#100	—	73	4	no	—
231	#100	—	146	2	no	—
232	C₂²⋊C₈	—	2	1	no	—
233	C₂²⋊C₈	—	292	4	no	—
234	C₂²⋊C₈	—	146	2	no	—
235	C₂²⋊C₈	—	146	2	no	—
236	C₂²⋊C₈	—	146	1	no	—
237	#97	—	2	1	no	—
238	#97	—	146	1	no	—
239	C₄×C₈	—	2	1	no	—
240	C₄×C₈	—	2	1	no	—
241	C₄×C₈	—	146	1	no	—
242	C₄×C₈	—	146	1	no	—
243	C₄×C₈	—	146	1	no	—
244	C₄×C₈	—	146	1	no	—
245	C₂⁴	—	73	2	no	—
246	C₂×D₄	—	73	2	no	—
247	C₂×D₄	—	73	2	no	—
248	C₂×D₄	—	146	1	no	—
249	C₂×D₄	—	146	1	no	—
250	C₂×D₄	—	146	1	no	—
251	C₂×D₄	—	146	1	no	—
252	C₂²×C₄	—	292	2	no	—
253	C₂²×C₄	—	292	2	no	—
254	C₂²×C₄	—	73	2	no	—
255	C₂²×C₄	—	146	2	no	—
256	C₂²×C₄	—	146	2	no	—
257	C₂²×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
258	C₂²×C₄	D₄×D₇₃	1	1	yes	characteristic
259	C₂²×C₄	Q₈×D₇₃	1	1	yes	characteristic
260	M₁₆	—	2	1	no	—
261	M₁₆	—	2	1	no	—
262	M₁₆	—	2	1	no	—
263	M₁₆	—	73	2	no	—
264	M₁₆	—	146	1	no	—
265	M₁₆	—	146	1	no	—
266	M₁₆	—	146	1	no	—
267	M₁₆	C₈×D₇₃	1	2	yes	complemented
268	C₂×C₈	—	2	1	no	—
269	C₂×C₈	—	2	1	no	—
270	C₂×C₈	—	2	1	no	—
271	C₂×C₈	—	2	1	no	—
272	C₂×C₈	—	4	2	no	—
273	C₂×C₈	—	4	1	no	—
274	C₂×C₈	—	292	1	no	—
275	C₂×C₈	—	292	1	no	—
276	C₂×C₈	—	584	2	no	—
277	C₂×C₈	—	584	2	no	—
278	C₂×C₈	—	146	2	no	—
279	C₂×C₈	—	146	1	no	—
280	C₂×C₈	—	146	1	no	—
281	C₂×C₈	—	146	1	no	—
282	C₂×C₈	—	146	1	no	—
283	C₄⋊C₄	—	146	1	no	—
284	C₄⋊C₄	—	146	1	no	—
285	C₂²⋊C₄	—	146	2	no	—
286	C₂²⋊C₄	—	146	2	no	—
287	C₄²	D₄×D₇₃	1	1	yes	characteristic
288	C₄²	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
289	C₄²	#11924028	1	1	yes	characteristic
290	C₄²	#11924029	1	1	yes	characteristic
291	C₂³	—	292	2	no	—
292	C₂³	—	292	2	no	—
293	C₂³	—	73	4	no	—
294	C₂³	—	146	2	no	—
295	C₂³	—	146	2	no	—
296	C₂³	C₄⋊C₄×D₇₃	1	1	yes	characteristic
297	D₄	—	146	2	no	—
298	D₄	—	146	2	no	—
299	D₄	—	146	2	no	—
300	D₄	—	146	2	no	—
301	C₂×C₄	—	2	1	no	—
302	C₂×C₄	—	2	1	no	—
303	C₂×C₄	—	4	1	no	—
304	C₂×C₄	—	292	2	no	—
305	C₂×C₄	—	292	2	no	—
306	C₂×C₄	—	292	2	no	—
307	C₂×C₄	—	584	2	no	—
308	C₂×C₄	—	146	2	no	—
309	C₂×C₄	C₄×C₂²⋊D₇₃	1	1	yes	characteristic
310	C₂×C₄	#14556553	1	1	yes	characteristic
311	C₂×C₄	#14556559	1	1	yes	characteristic
312	C₂×C₄	#14556567	1	1	yes	characteristic
313	C₂×C₄	#14556569	1	1	yes	characteristic
314	C₂×C₄	#14556576	1	1	yes	characteristic
315	C₂×C₄	C₂×C₈×D₇₃	1	1	yes	characteristic
316	C₂×C₄	M₁₆×D₇₃	1	1	yes	characteristic
317	C₂×C₄	#14556597	1	1	yes	characteristic
318	C₂×C₄	#14556599	1	1	yes	characteristic
319	C₈	—	2	2	no	—
320	C₈	—	4	1	no	—
321	C₈	—	4	1	no	—
322	C₈	—	4	1	no	—
323	C₈	—	8	1	no	—
324	C₈	—	292	2	no	—
325	C₈	—	292	1	no	—
326	C₈	—	292	1	no	—
327	C₈	—	292	1	no	—
328	C₈	—	146	2	no	—
329	C₂²	—	2	1	no	—
330	C₂²	—	2	1	no	—
331	C₂²	—	292	4	no	—
332	C₂²	—	292	2	no	—
333	C₂²	—	292	2	no	—
334	C₂²	—	584	2	no	—
335	C₂²	—	146	4	no	—
336	C₂²	#17988458	1	1	yes	characteristic
337	C₂²	C₄⋊C₈×D₇₃	1	1	yes	characteristic
338	C₂²	#17988612	1	1	yes	characteristic
339	C₄	—	2	1	no	—
340	C₄	—	2	1	no	—
341	C₄	—	2	1	no	—
342	C₄	—	292	2	no	—
343	C₄	C₈×C₂²⋊D₇₃	1	1	yes	characteristic
344	C₄	#17988652	1	1	yes	characteristic
345	C₄	#17988656	1	1	yes	characteristic
346	C₄	#17988657	1	1	yes	characteristic
347	C₄	#17988753	1	1	yes	characteristic
348	C₄	#17988763	1	1	yes	characteristic
349	C₂	—	2	1	no	—
350	C₂	—	2	1	no	—
351	C₂	—	292	4	no	—
352	C₂	#22610107	1	1	yes	characteristic
353	C₂	#22610139	1	1	yes	characteristic
354	C₂	#22610149	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#14556559
commutator	C₂×C₄×C₇₃	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₇₃
Fitting	C₇₃×#1565	C₂
socle	C₂²×C₇₃	#385
prime radical	C₂²×D₂₉₂	C₂×C₄
cosocle	C₂×C₄²×C₇₃	C₂³
hypercenter	#1565	D₇₃
hypocenter	C₇₃	#8835
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1565

Complements

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

N	H ≅ G/N	type
4. #22608681	C₂	semidirect
5. #22607769	C₂	semidirect
6. #22607767	C₂	semidirect
7. C₇₃×#1565	C₂	semidirect
45. #14556507	C₈	semidirect
82. #11924012	C₂×C₈	semidirect
102. M₁₆×C₇₃	C₂×C₈	semidirect
182. #1565	D₇₃	semidirect
185. C₇₃	#8835	semidirect
267. M₁₆	C₈×D₇₃	semidirect

Maximal subgroups

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

subgroup	index
#22618954	2
#22610724	2
C₄×#14556507	2
#22608681	2
#22607769	2
#22607767	2
C₇₃×#1565	2
#8835	73