Subgroups — #33055263

Order 18688.

588 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22618897	C₂	1	1	yes	maximal, characteristic
2	#22618592	C₂	1	2	yes	maximal
3	#22608718	C₂	1	2	yes	maximal, complemented
4	#22608713	C₂	1	2	yes	maximal, complemented
5	#22608709	C₂	1	2	yes	maximal, complemented
6	#22608697	C₂	1	2	yes	maximal, complemented
7	#22608694	C₂	1	1	yes	maximal, characteristic, complemented
8	#22608682	C₂	1	2	yes	maximal, complemented
9	C₇₃×#1564	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×#11924028	C₂²	1	1	yes	characteristic
11	#17988717	C₄	1	1	yes	characteristic
12	#17988717	C₄	1	1	yes	characteristic
13	#17988666	C₄	1	1	yes	characteristic
14	#17988666	C₄	1	1	yes	characteristic
15	#17988658	C₄	1	2	yes	—
16	#17988658	C₄	1	2	yes	—
17	#17988646	—	2	2	no	—
18	#17988645	—	2	2	no	—
19	#17988644	C₂²	1	2	yes	—
20	#17988643	C₂²	1	2	yes	—
21	#17988640	C₂²	1	1	yes	characteristic
22	#17988639	C₂²	1	2	yes	—
23	#17988638	C₂²	1	2	yes	—
24	#17988609	C₂²	1	2	yes	—
25	C₄⋊C₈×D₇₃	C₂²	1	2	yes	—
26	#17988552	—	2	2	no	—
27	#17988541	—	2	2	no	—
28	C₄×#11924011	C₂²	1	2	yes	—
29	#17988340	C₂²	1	1	yes	characteristic
30	#17988340	C₂²	1	1	yes	characteristic
31	#17988339	C₂²	1	2	yes	—
32	#17988339	C₂²	1	2	yes	—
33	#17988338	C₂²	1	2	yes	—
34	#17988337	C₂²	1	2	yes	—
35	#17988337	C₂²	1	2	yes	—
36	#431×C₇₃	C₂²	1	2	yes	—
37	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
38	#335×C₇₃	C₂²	1	1	yes	characteristic
39	#335×C₇₃	C₂²	1	1	yes	characteristic
40	#334×C₇₃	C₂²	1	2	yes	—
41	C₂×#11924028	D₄	1	1	yes	characteristic
42	C₂×#11924011	—	2	2	no	—
43	C₂×#11924011	C₂×C₄	1	2	yes	—
44	C₂×#11924011	C₂×C₄	1	2	yes	—
45	C₂×C₈×D₇₃	—	2	2	no	—
46	C₄×Dic₁₄₆	C₂×C₄	1	1	yes	characteristic
47	C₄×Dic₁₄₆	Q₈	1	1	yes	characteristic
48	C₄×D₂₉₂	C₂×C₄	1	1	yes	characteristic
49	C₄×D₂₉₂	Q₈	1	1	yes	characteristic
50	C₄×C₂²⋊D₇₃	C₂×C₄	1	2	yes	—
51	C₄×C₂²⋊D₇₃	Q₈	1	2	yes	—
52	#14556550	D₄	1	1	yes	characteristic
53	#14556548	D₄	1	2	yes	—
54	C₄²×D₇₃	C₂³	1	2	yes	—
55	C₄⋊#10596831	C₂×C₄	1	1	yes	characteristic
56	C₄⋊#10596831	C₂×C₄	1	1	yes	characteristic
57	C₄⋊#10596831	C₂³	1	2	yes	—
58	C₄⋊#10596831	C₂³	1	2	yes	—
59	#14556520	—	2	2	no	—
60	#14556520	—	2	2	no	—
61	#14556520	C₂×C₄	1	2	yes	—
62	#14556520	C₂×C₄	1	2	yes	—
63	C₂²⋊#10596831	—	2	2	no	—
64	C₂²⋊#10596831	C₂×C₄	1	2	yes	—
65	#14556503	—	2	2	no	—
66	#14556503	—	2	2	no	—
67	#14556503	C₂×C₄	1	2	yes	—
68	#14556503	C₂×C₄	1	2	yes	—
69	#14556502	—	2	2	no	—
70	#14556501	—	2	2	no	—
71	#14556501	C₂×C₄	1	2	yes	—
72	#14556501	C₂×C₄	1	2	yes	—
73	C₄×#10596831	C₂³	1	2	yes	—
74	C₈×Dic₇₃	—	2	2	no	—
75	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
76	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
77	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
78	C₄⋊C₈×C₇₃	C₂³	1	2	yes	—
79	C₄⋊C₈×C₇₃	C₂³	1	2	yes	—
80	C₂²⋊C₈×C₇₃	—	2	2	no	—
81	C₄×C₈×C₇₃	—	2	2	no	—
82	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
83	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
84	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
85	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
86	#11924028	—	4	1	no	—
87	#11924028	—	4	1	no	—
88	C₂×Dic₁₄₆	C₄⋊C₄	1	1	yes	characteristic
89	C₂×D₂₉₂	C₄⋊C₄	1	1	yes	characteristic
90	C₂×C₂²⋊D₇₃	C₄⋊C₄	1	2	yes	—
91	C₂×C₄×D₇₃	C₂²×C₄	1	2	yes	—
92	C₂×C₄×D₇₃	C₂×D₄	1	2	yes	—
93	C₂×C₄×D₇₃	C₂×Q₈	1	2	yes	—
94	#11924011	—	2	4	no	—
95	#11924011	—	4	2	no	—
96	#11924011	—	4	4	no	—
97	C₂×#10596831	—	2	2	no	—
98	C₂×#10596831	—	2	2	no	—
99	C₂×#10596831	C₂²×C₄	1	2	yes	—
100	C₂×#10596831	C₂²×C₄	1	2	yes	—
101	C₈×D₇₃	—	4	4	no	—
102	C₄⋊Dic₇₃	C₄⋊C₄	1	1	yes	characteristic
103	C₄⋊Dic₇₃	C₄⋊C₄	1	1	yes	characteristic
104	#11924007	C₄⋊C₄	1	2	yes	—
105	#11924007	C₄⋊C₄	1	2	yes	—
106	C₂²⋊Dic₇₃	C₄⋊C₄	1	2	yes	—
107	#11924005	C₄⋊C₄	1	2	yes	—
108	#11924005	C₄⋊C₄	1	2	yes	—
109	C₄×Dic₇₃	C₂²×C₄	1	2	yes	—
110	C₄×Dic₇₃	C₂×D₄	1	2	yes	—
111	C₄×Dic₇₃	C₂×Q₈	1	2	yes	—
112	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
113	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
114	C₂²×C₄×C₇₃	C₂×Q₈	1	1	yes	characteristic
115	C₂×C₈×C₇₃	—	2	2	no	—
116	C₂×C₈×C₇₃	—	2	2	no	—
117	C₂×C₈×C₇₃	—	2	2	no	—
118	C₂×C₈×C₇₃	—	4	1	no	—
119	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
120	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
121	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
122	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
123	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
124	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
125	C₄²×C₇₃	C₂×Q₈	1	1	yes	characteristic
126	C₄²×C₇₃	C₂⁴	1	1	yes	characteristic
127	C₂³×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
128	C₂²×D₇₃	C₂×C₄⋊C₄	1	2	yes	—
129	C₂×C₄×C₇₃	—	2	1	no	—
130	C₂×C₄×C₇₃	—	2	1	no	—
131	C₂×C₄×C₇₃	—	2	1	no	—
132	C₂×C₄×C₇₃	—	2	1	no	—
133	C₂×C₄×C₇₃	—	4	1	no	—
134	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
135	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
136	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
137	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
138	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
139	C₂×C₄×C₇₃	C₂²×Q₈	1	1	yes	characteristic
140	C₂²⋊D₇₃	—	4	2	no	—
141	C₂×Dic₇₃	C₂×C₄⋊C₄	1	2	yes	—
142	C₂×Dic₇₃	C₂×C₄⋊C₄	1	2	yes	—
143	C₂×Dic₇₃	C₂×C₄⋊C₄	1	2	yes	—
144	D₂₉₂	—	4	1	no	—
145	C₄×D₇₃	—	2	2	no	—
146	C₄×D₇₃	—	2	2	no	—
147	C₄×D₇₃	—	2	2	no	—
148	C₄×D₇₃	—	2	2	no	—
149	C₄×D₇₃	—	4	2	no	—
150	Dic₁₄₆	—	4	1	no	—
151	C₈×C₇₃	—	2	1	no	—
152	C₈×C₇₃	—	2	1	no	—
153	C₈×C₇₃	—	2	1	no	—
154	C₈×C₇₃	—	4	2	no	—
155	C₈×C₇₃	—	4	2	no	—
156	C₈×C₇₃	#131	1	2	yes	—
157	#10596831	—	2	2	no	—
158	#10596831	—	2	2	no	—
159	#10596831	—	4	2	no	—
160	#10596831	—	4	2	no	—
161	C₂²×C₇₃	—	2	1	no	—
162	C₂²×C₇₃	—	2	1	no	—
163	C₂²×C₇₃	—	2	1	no	—
164	C₂²×C₇₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
165	C₂×D₇₃	—	2	2	no	—
166	C₂×D₇₃	—	2	2	no	—
167	C₂×D₇₃	—	2	2	no	—
168	C₄×C₇₃	—	2	1	no	—
169	C₄×C₇₃	—	2	1	no	—
170	C₄×C₇₃	—	2	1	no	—
171	C₄×C₇₃	—	2	1	no	—
172	C₄×C₇₃	C₂×#131	1	1	yes	characteristic
173	C₄×C₇₃	#568	1	1	yes	characteristic
174	C₄×C₇₃	#575	1	1	yes	characteristic
175	C₄×C₇₃	#575	1	1	yes	characteristic
176	Dic₇₃	—	2	2	no	—
177	Dic₇₃	—	2	2	no	—
178	Dic₇₃	—	2	2	no	—
179	#17833	—	73	1	no	maximal, Sylow
180	C₂×C₇₃	—	4	1	no	—
181	C₂×C₇₃	#2906	1	1	yes	characteristic
182	C₂×C₇₃	#2911	1	1	yes	characteristic
183	C₂×C₇₃	#2916	1	1	yes	characteristic
184	D₇₃	—	4	2	no	—
185	#2965	—	73	1	no	—
186	#2925	—	73	2	no	—
187	#1574	—	73	2	no	—
188	#1573	—	73	2	no	—
189	#1572	—	73	2	no	—
190	#1569	—	73	2	no	—
191	#1568	—	73	1	no	—
192	#1565	—	73	2	no	—
193	#1564	D₇₃	1	1	yes	characteristic, complemented
194	C₇₃	#17833	1	1	yes	characteristic, complemented, Sylow
195	C₄×#41	—	73	1	no	—
196	#446	—	73	1	no	—
197	#446	—	73	1	no	—
198	#436	—	73	1	no	—
199	#436	—	73	1	no	—
200	#435	—	73	2	no	—
201	#435	—	73	2	no	—
202	#433	—	146	2	no	—
203	#433	—	146	2	no	—
204	#432	—	73	2	no	—
205	#432	—	73	2	no	—
206	#431	—	73	1	no	—
207	#431	—	73	2	no	—
208	#431	—	73	2	no	—
209	#431	C₂×D₇₃	1	2	yes	—
210	#423	—	73	2	no	—
211	C₂×C₄⋊C₈	—	73	2	no	—
212	#408	—	146	2	no	—
213	#405	—	146	2	no	—
214	C₄×M₁₆	—	73	2	no	—
215	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
216	#335	—	73	2	no	—
217	#335	—	73	2	no	—
218	#335	—	73	1	no	—
219	#335	—	73	1	no	—
220	#335	C₂×D₇₃	1	1	yes	characteristic
221	#335	C₂×D₇₃	1	1	yes	characteristic
222	#334	—	73	2	no	—
223	#334	—	73	2	no	—
224	#334	—	73	2	no	—
225	#334	C₂×D₇₃	1	2	yes	—
226	C₂×#41	—	73	1	no	—
227	C₂×M₁₆	—	73	2	no	—
228	C₂×M₁₆	—	73	2	no	—
229	C₂×M₁₆	—	146	2	no	—
230	C₂²×C₈	—	146	2	no	—
231	C₂²×C₈	C₄×D₇₃	1	1	yes	characteristic
232	C₂²×C₈	C₄×D₇₃	1	1	yes	characteristic
233	C₄×Q₈	—	73	1	no	—
234	C₄×Q₈	—	73	1	no	—
235	C₄×D₄	—	73	2	no	—
236	C₄×D₄	—	73	2	no	—
237	C₄×D₄	—	73	1	no	—
238	C₄×D₄	—	73	1	no	—
239	#117	—	73	1	no	—
240	#117	—	73	2	no	—
241	C₂×C₄²	—	73	2	no	—
242	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
243	C₄⋊C₈	—	73	2	no	—
244	C₄⋊C₈	—	73	2	no	—
245	C₄⋊C₈	—	73	1	no	—
246	C₄⋊C₈	—	73	1	no	—
247	C₄⋊C₈	—	73	2	no	—
248	C₄⋊C₈	—	73	2	no	—
249	C₄⋊C₈	—	146	2	no	—
250	C₄⋊C₈	—	146	2	no	—
251	C₄⋊C₈	C₂²×D₇₃	1	2	yes	—
252	C₄⋊C₈	C₂²×D₇₃	1	2	yes	—
253	C₂²⋊C₈	—	2	2	no	—
254	C₂²⋊C₈	—	73	2	no	—
255	C₂²⋊C₈	—	73	2	no	—
256	C₂²⋊C₈	—	73	2	no	—
257	C₂²⋊C₈	—	146	2	no	—
258	C₂²⋊C₈	—	146	2	no	—
259	C₂²⋊C₈	—	146	2	no	—
260	#97	—	73	2	no	—
261	#97	—	73	2	no	—
262	#97	—	146	2	no	—
263	#97	—	146	2	no	—
264	C₄×C₈	—	2	2	no	—
265	C₄×C₈	—	73	2	no	—
266	C₄×C₈	—	146	2	no	—
267	C₄×C₈	C₄×D₇₃	1	1	yes	characteristic
268	C₄×C₈	C₄×D₇₃	1	1	yes	characteristic
269	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
270	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
271	#41	—	292	1	no	—
272	#41	—	292	1	no	—
273	C₂×Q₈	—	73	1	no	—
274	C₂×D₄	—	73	2	no	—
275	C₂×D₄	—	73	1	no	—
276	C₂²×C₄	—	73	2	no	—
277	C₂²×C₄	—	73	2	no	—
278	C₂²×C₄	—	73	2	no	—
279	C₂²×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
280	C₂²×C₄	D₄×D₇₃	1	1	yes	characteristic
281	C₂²×C₄	Q₈×D₇₃	1	1	yes	characteristic
282	M₁₆	—	292	2	no	—
283	M₁₆	—	292	4	no	—
284	M₁₆	—	146	4	no	—
285	C₂×C₈	—	2	2	no	—
286	C₂×C₈	—	2	2	no	—
287	C₂×C₈	—	2	2	no	—
288	C₂×C₈	—	4	1	no	—
289	C₂×C₈	—	292	4	no	—
290	C₂×C₈	—	73	2	no	—
291	C₂×C₈	—	73	2	no	—
292	C₂×C₈	—	146	2	no	—
293	C₂×C₈	—	146	2	no	—
294	C₂×C₈	C₂×C₄×D₇₃	1	1	yes	characteristic
295	C₂×C₈	C₂×C₄×D₇₃	1	1	yes	characteristic
296	C₂×C₈	C₂×C₄×D₇₃	1	1	yes	characteristic
297	C₂×C₈	C₂×C₄×D₇₃	1	1	yes	characteristic
298	C₄⋊C₄	—	73	2	no	—
299	C₄⋊C₄	—	73	2	no	—
300	C₄⋊C₄	—	73	1	no	—
301	C₄⋊C₄	—	73	1	no	—
302	C₂²⋊C₄	—	73	2	no	—
303	C₂²⋊C₄	—	73	2	no	—
304	C₂²⋊C₄	—	73	2	no	—
305	C₄²	—	73	2	no	—
306	C₄²	—	73	2	no	—
307	C₄²	—	73	2	no	—
308	C₄²	C₂×C₄×D₇₃	1	1	yes	characteristic
309	C₄²	D₄×D₇₃	1	1	yes	characteristic
310	C₄²	Q₈×D₇₃	1	1	yes	characteristic
311	C₄²	C₂³×D₇₃	1	1	yes	characteristic
312	C₂³	—	73	2	no	—
313	C₂³	C₄⋊C₄×D₇₃	1	1	yes	characteristic
314	Q₈	—	292	1	no	—
315	D₄	—	292	1	no	—
316	D₄	—	292	2	no	—
317	C₂×C₄	—	2	1	no	—
318	C₂×C₄	—	2	1	no	—
319	C₂×C₄	—	2	1	no	—
320	C₂×C₄	—	2	1	no	—
321	C₂×C₄	—	4	1	no	—
322	C₂×C₄	—	292	2	no	—
323	C₂×C₄	—	73	2	no	—
324	C₂×C₄	—	73	2	no	—
325	C₂×C₄	—	73	2	no	—
326	C₂×C₄	—	146	2	no	—
327	C₂×C₄	—	146	2	no	—
328	C₂×C₄	—	146	2	no	—
329	C₂×C₄	—	146	2	no	—
330	C₂×C₄	C₄⋊C₄×D₇₃	1	1	yes	characteristic
331	C₂×C₄	C₄⋊C₄×D₇₃	1	1	yes	characteristic
332	C₂×C₄	C₄⋊C₄×D₇₃	1	1	yes	characteristic
333	C₂×C₄	C₂²×C₄×D₇₃	1	1	yes	characteristic
334	C₂×C₄	C₂×D₄×D₇₃	1	1	yes	characteristic
335	C₂×C₄	C₂×Q₈×D₇₃	1	1	yes	characteristic
336	C₈	—	2	1	no	—
337	C₈	—	2	1	no	—
338	C₈	—	2	1	no	—
339	C₈	—	4	2	no	—
340	C₈	—	4	2	no	—
341	C₈	—	292	2	no	—
342	C₈	—	292	2	no	—
343	C₈	—	146	2	no	—
344	C₈	—	146	2	no	—
345	C₈	#14556597	1	2	yes	—
346	C₂²	—	2	1	no	—
347	C₂²	—	2	1	no	—
348	C₂²	—	2	1	no	—
349	C₂²	—	146	2	no	—
350	C₂²	—	146	2	no	—
351	C₂²	—	146	2	no	—
352	C₂²	C₂×C₄⋊C₄×D₇₃	1	1	yes	characteristic
353	C₄	—	2	1	no	—
354	C₄	—	2	1	no	—
355	C₄	—	2	1	no	—
356	C₄	—	2	1	no	—
357	C₄	—	146	2	no	—
358	C₄	—	146	2	no	—
359	C₄	—	146	2	no	—
360	C₄	C₂×#14556597	1	1	yes	characteristic
361	C₄	#17989353	1	1	yes	characteristic
362	C₄	#17989384	1	1	yes	characteristic
363	C₄	#17989384	1	1	yes	characteristic
364	C₂	—	4	1	no	—
365	C₂	—	292	2	no	—
366	C₂	#22618496	1	1	yes	characteristic
367	C₂	#22618519	1	1	yes	characteristic
368	C₂	#22618539	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₇₃
commutator	C₂×C₄×C₇₃	C₂³×C₄
Frattini	C₄²	C₂³×D₇₃
Fitting	C₇₃×#1564	C₂
socle	C₂²×C₇₃	C₂²×C₄⋊C₄
prime radical	C₂×#11924028	D₄
cosocle	C₄²×C₇₃	C₂⁴
hypercenter	#1564	D₇₃
hypocenter	C₇₃	#17833
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1564

Complements

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

N	H ≅ G/N	type
3. #22608718	C₂	semidirect
4. #22608713	C₂	semidirect
5. #22608709	C₂	semidirect
6. #22608697	C₂	semidirect
7. #22608694	C₂	semidirect
8. #22608682	C₂	semidirect
9. C₇₃×#1564	C₂	semidirect
193. #1564	D₇₃	semidirect
194. C₇₃	#17833	semidirect

Maximal subgroups

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

subgroup	index
#22618897	2
#22618592	2
#22608718	2
#22608713	2
#22608709	2
#22608697	2
#22608694	2
#22608682	2
C₇₃×#1564	2
#17833	73