Subgroups — #33055243

Order 18688.

588 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	#22618897	C₂	1	1	yes	maximal, characteristic
2	#22618592	C₂	1	2	yes	maximal
3	#22608721	C₂	1	2	yes	maximal, complemented
4	#22608715	C₂	1	2	yes	maximal, complemented
5	#22608709	C₂	1	2	yes	maximal, complemented
6	#22608695	C₂	1	2	yes	maximal, complemented
7	#22608694	C₂	1	1	yes	maximal, characteristic, complemented
8	#22608680	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	2	yes	—
30	#17988339	C₂²	1	2	yes	—
31	#17988339	C₂²	1	2	yes	—
32	#17988338	C₂²	1	1	yes	characteristic
33	#17988338	C₂²	1	1	yes	characteristic
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	2	yes	—
39	#334×C₇₃	C₂²	1	1	yes	characteristic
40	#334×C₇₃	C₂²	1	1	yes	characteristic
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	2	no	—
87	C₂×Dic₁₄₆	C₄⋊C₄	1	1	yes	characteristic
88	C₂×D₂₉₂	C₄⋊C₄	1	1	yes	characteristic
89	C₂×C₂²⋊D₇₃	C₄⋊C₄	1	2	yes	—
90	C₂×C₄×D₇₃	C₂²×C₄	1	2	yes	—
91	C₂×C₄×D₇₃	C₂×D₄	1	2	yes	—
92	C₂×C₄×D₇₃	C₂×Q₈	1	2	yes	—
93	#11924011	—	2	4	no	—
94	#11924011	—	4	2	no	—
95	#11924011	—	4	4	no	—
96	C₂×#10596831	—	2	2	no	—
97	C₂×#10596831	—	2	2	no	—
98	C₂×#10596831	C₂²×C₄	1	2	yes	—
99	C₂×#10596831	C₂²×C₄	1	2	yes	—
100	C₈×D₇₃	—	4	4	no	—
101	C₄⋊Dic₇₃	C₄⋊C₄	1	1	yes	characteristic
102	C₄⋊Dic₇₃	C₄⋊C₄	1	1	yes	characteristic
103	#11924007	C₄⋊C₄	1	2	yes	—
104	#11924007	C₄⋊C₄	1	2	yes	—
105	C₂²⋊Dic₇₃	C₄⋊C₄	1	2	yes	—
106	#11924005	C₄⋊C₄	1	2	yes	—
107	#11924005	C₄⋊C₄	1	2	yes	—
108	C₄×Dic₇₃	C₂²×C₄	1	2	yes	—
109	C₄×Dic₇₃	C₂×D₄	1	2	yes	—
110	C₄×Dic₇₃	C₂×Q₈	1	2	yes	—
111	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
112	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
113	C₂²×C₄×C₇₃	C₂×Q₈	1	1	yes	characteristic
114	C₂×C₈×C₇₃	—	2	2	no	—
115	C₂×C₈×C₇₃	—	2	2	no	—
116	C₂×C₈×C₇₃	—	2	2	no	—
117	C₂×C₈×C₇₃	—	4	1	no	—
118	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
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	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
123	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
124	C₄²×C₇₃	C₂×Q₈	1	1	yes	characteristic
125	C₄²×C₇₃	C₂⁴	1	1	yes	characteristic
126	C₂³×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
127	C₂²×D₇₃	C₂×C₄⋊C₄	1	2	yes	—
128	C₂×C₄×C₇₃	—	2	1	no	—
129	C₂×C₄×C₇₃	—	2	1	no	—
130	C₂×C₄×C₇₃	—	2	2	no	—
131	C₂×C₄×C₇₃	—	4	1	no	—
132	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
133	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
134	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
135	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
136	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
137	C₂×C₄×C₇₃	C₂²×Q₈	1	1	yes	characteristic
138	C₂²⋊D₇₃	—	4	2	no	—
139	C₂×Dic₇₃	C₂×C₄⋊C₄	1	2	yes	—
140	C₂×Dic₇₃	C₂×C₄⋊C₄	1	2	yes	—
141	C₂×Dic₇₃	C₂×C₄⋊C₄	1	2	yes	—
142	D₂₉₂	—	4	1	no	—
143	C₄×D₇₃	—	2	2	no	—
144	C₄×D₇₃	—	2	2	no	—
145	C₄×D₇₃	—	2	2	no	—
146	C₄×D₇₃	—	2	2	no	—
147	C₄×D₇₃	—	4	2	no	—
148	Dic₁₄₆	—	4	1	no	—
149	C₈×C₇₃	—	2	1	no	—
150	C₈×C₇₃	—	2	1	no	—
151	C₈×C₇₃	—	2	1	no	—
152	C₈×C₇₃	—	4	2	no	—
153	C₈×C₇₃	—	4	2	no	—
154	C₈×C₇₃	#131	1	2	yes	—
155	#10596831	—	2	2	no	—
156	#10596831	—	2	2	no	—
157	#10596831	—	4	2	no	—
158	#10596831	—	4	2	no	—
159	C₂²×C₇₃	—	2	1	no	—
160	C₂²×C₇₃	—	2	1	no	—
161	C₂²×C₇₃	—	2	1	no	—
162	C₂²×C₇₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
163	C₂×D₇₃	—	2	2	no	—
164	C₂×D₇₃	—	2	2	no	—
165	C₂×D₇₃	—	2	2	no	—
166	C₄×C₇₃	—	2	1	no	—
167	C₄×C₇₃	—	2	1	no	—
168	C₄×C₇₃	—	2	1	no	—
169	C₄×C₇₃	—	2	1	no	—
170	C₄×C₇₃	C₂×#131	1	1	yes	characteristic
171	C₄×C₇₃	#568	1	1	yes	characteristic
172	C₄×C₇₃	#575	1	2	yes	—
173	Dic₇₃	—	2	2	no	—
174	Dic₇₃	—	2	2	no	—
175	Dic₇₃	—	2	2	no	—
176	#17831	—	73	1	no	maximal, Sylow
177	C₂×C₇₃	—	4	1	no	—
178	C₂×C₇₃	#2906	1	1	yes	characteristic
179	C₂×C₇₃	#2911	1	1	yes	characteristic
180	C₂×C₇₃	#2916	1	1	yes	characteristic
181	D₇₃	—	4	2	no	—
182	#2965	—	73	1	no	—
183	#2925	—	73	2	no	—
184	#1574	—	73	2	no	—
185	#1573	—	73	2	no	—
186	#1572	—	73	2	no	—
187	#1569	—	73	2	no	—
188	#1568	—	73	1	no	—
189	#1565	—	73	2	no	—
190	#1564	D₇₃	1	1	yes	characteristic, complemented
191	C₇₃	#17831	1	1	yes	characteristic, complemented, Sylow
192	C₄×#41	—	73	1	no	—
193	#446	—	73	1	no	—
194	#446	—	73	1	no	—
195	#436	—	73	1	no	—
196	#436	—	73	1	no	—
197	#435	—	73	2	no	—
198	#435	—	73	2	no	—
199	#433	—	146	2	no	—
200	#433	—	146	2	no	—
201	#432	—	73	2	no	—
202	#432	—	73	2	no	—
203	#431	—	73	1	no	—
204	#431	—	73	2	no	—
205	#431	—	73	2	no	—
206	#431	C₂×D₇₃	1	2	yes	—
207	#423	—	73	2	no	—
208	C₂×C₄⋊C₈	—	73	2	no	—
209	#408	—	146	2	no	—
210	#405	—	146	2	no	—
211	C₄×M₁₆	—	73	2	no	—
212	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
213	#335	—	73	2	no	—
214	#335	—	73	2	no	—
215	#335	—	73	2	no	—
216	#335	C₂×D₇₃	1	2	yes	—
217	#334	—	73	2	no	—
218	#334	—	73	2	no	—
219	#334	—	73	1	no	—
220	#334	—	73	1	no	—
221	#334	C₂×D₇₃	1	1	yes	characteristic
222	#334	C₂×D₇₃	1	1	yes	characteristic
223	C₂×#41	—	73	1	no	—
224	C₂×M₁₆	—	73	2	no	—
225	C₂×M₁₆	—	73	2	no	—
226	C₂×M₁₆	—	146	2	no	—
227	C₂²×C₈	—	146	2	no	—
228	C₂²×C₈	C₄×D₇₃	1	1	yes	characteristic
229	C₂²×C₈	C₄×D₇₃	1	1	yes	characteristic
230	C₄×Q₈	—	73	1	no	—
231	C₄×Q₈	—	73	1	no	—
232	C₄×D₄	—	73	2	no	—
233	C₄×D₄	—	73	2	no	—
234	C₄×D₄	—	73	1	no	—
235	C₄×D₄	—	73	1	no	—
236	#117	—	73	1	no	—
237	#117	—	73	2	no	—
238	C₂×C₄²	—	73	2	no	—
239	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
240	C₄⋊C₈	—	73	2	no	—
241	C₄⋊C₈	—	73	2	no	—
242	C₄⋊C₈	—	73	1	no	—
243	C₄⋊C₈	—	73	1	no	—
244	C₄⋊C₈	—	73	2	no	—
245	C₄⋊C₈	—	73	2	no	—
246	C₄⋊C₈	—	146	2	no	—
247	C₄⋊C₈	—	146	2	no	—
248	C₄⋊C₈	C₂²×D₇₃	1	2	yes	—
249	C₄⋊C₈	C₂²×D₇₃	1	2	yes	—
250	C₂²⋊C₈	—	2	2	no	—
251	C₂²⋊C₈	—	73	2	no	—
252	C₂²⋊C₈	—	73	2	no	—
253	C₂²⋊C₈	—	73	2	no	—
254	C₂²⋊C₈	—	146	2	no	—
255	C₂²⋊C₈	—	146	2	no	—
256	C₂²⋊C₈	—	146	2	no	—
257	#97	—	73	2	no	—
258	#97	—	73	2	no	—
259	#97	—	146	2	no	—
260	#97	—	146	2	no	—
261	C₄×C₈	—	2	2	no	—
262	C₄×C₈	—	73	2	no	—
263	C₄×C₈	—	146	2	no	—
264	C₄×C₈	C₄×D₇₃	1	1	yes	characteristic
265	C₄×C₈	C₄×D₇₃	1	1	yes	characteristic
266	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
267	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
268	#41	—	292	2	no	—
269	C₂×Q₈	—	73	1	no	—
270	C₂×D₄	—	73	2	no	—
271	C₂×D₄	—	73	1	no	—
272	C₂²×C₄	—	73	2	no	—
273	C₂²×C₄	—	73	2	no	—
274	C₂²×C₄	—	73	2	no	—
275	C₂²×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
276	C₂²×C₄	D₄×D₇₃	1	1	yes	characteristic
277	C₂²×C₄	Q₈×D₇₃	1	1	yes	characteristic
278	M₁₆	—	292	2	no	—
279	M₁₆	—	292	4	no	—
280	M₁₆	—	146	4	no	—
281	C₂×C₈	—	2	2	no	—
282	C₂×C₈	—	2	2	no	—
283	C₂×C₈	—	2	2	no	—
284	C₂×C₈	—	4	1	no	—
285	C₂×C₈	—	292	4	no	—
286	C₂×C₈	—	73	2	no	—
287	C₂×C₈	—	73	2	no	—
288	C₂×C₈	—	146	2	no	—
289	C₂×C₈	—	146	2	no	—
290	C₂×C₈	C₂×C₄×D₇₃	1	1	yes	characteristic
291	C₂×C₈	C₂×C₄×D₇₃	1	1	yes	characteristic
292	C₂×C₈	C₂×C₄×D₇₃	1	1	yes	characteristic
293	C₂×C₈	C₂×C₄×D₇₃	1	1	yes	characteristic
294	C₄⋊C₄	—	73	2	no	—
295	C₄⋊C₄	—	73	2	no	—
296	C₄⋊C₄	—	73	1	no	—
297	C₄⋊C₄	—	73	1	no	—
298	C₂²⋊C₄	—	73	2	no	—
299	C₂²⋊C₄	—	73	2	no	—
300	C₂²⋊C₄	—	73	2	no	—
301	C₄²	—	73	2	no	—
302	C₄²	—	73	2	no	—
303	C₄²	—	73	2	no	—
304	C₄²	C₂×C₄×D₇₃	1	1	yes	characteristic
305	C₄²	D₄×D₇₃	1	1	yes	characteristic
306	C₄²	Q₈×D₇₃	1	1	yes	characteristic
307	C₄²	C₂³×D₇₃	1	1	yes	characteristic
308	C₂³	—	73	2	no	—
309	C₂³	C₄⋊C₄×D₇₃	1	1	yes	characteristic
310	Q₈	—	292	1	no	—
311	D₄	—	292	1	no	—
312	D₄	—	292	2	no	—
313	C₂×C₄	—	2	2	no	—
314	C₂×C₄	—	2	1	no	—
315	C₂×C₄	—	2	1	no	—
316	C₂×C₄	—	4	1	no	—
317	C₂×C₄	—	292	2	no	—
318	C₂×C₄	—	73	2	no	—
319	C₂×C₄	—	73	2	no	—
320	C₂×C₄	—	73	2	no	—
321	C₂×C₄	—	146	2	no	—
322	C₂×C₄	—	146	2	no	—
323	C₂×C₄	—	146	2	no	—
324	C₂×C₄	—	146	2	no	—
325	C₂×C₄	C₄⋊C₄×D₇₃	1	1	yes	characteristic
326	C₂×C₄	C₄⋊C₄×D₇₃	1	1	yes	characteristic
327	C₂×C₄	C₄⋊C₄×D₇₃	1	1	yes	characteristic
328	C₂×C₄	C₂²×C₄×D₇₃	1	1	yes	characteristic
329	C₂×C₄	C₂×D₄×D₇₃	1	1	yes	characteristic
330	C₂×C₄	C₂×Q₈×D₇₃	1	1	yes	characteristic
331	C₈	—	2	1	no	—
332	C₈	—	2	1	no	—
333	C₈	—	2	1	no	—
334	C₈	—	4	2	no	—
335	C₈	—	4	2	no	—
336	C₈	—	292	2	no	—
337	C₈	—	292	2	no	—
338	C₈	—	146	2	no	—
339	C₈	—	146	2	no	—
340	C₈	#14556597	1	2	yes	—
341	C₂²	—	2	1	no	—
342	C₂²	—	2	1	no	—
343	C₂²	—	2	1	no	—
344	C₂²	—	146	2	no	—
345	C₂²	—	146	2	no	—
346	C₂²	—	146	2	no	—
347	C₂²	C₂×C₄⋊C₄×D₇₃	1	1	yes	characteristic
348	C₄	—	2	1	no	—
349	C₄	—	2	1	no	—
350	C₄	—	2	1	no	—
351	C₄	—	2	1	no	—
352	C₄	—	146	2	no	—
353	C₄	—	146	2	no	—
354	C₄	—	146	2	no	—
355	C₄	C₂×#14556597	1	1	yes	characteristic
356	C₄	#17989353	1	1	yes	characteristic
357	C₄	#17989384	1	2	yes	—
358	C₂	—	4	1	no	—
359	C₂	—	292	2	no	—
360	C₂	#22618496	1	1	yes	characteristic
361	C₂	#22618519	1	1	yes	characteristic
362	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	#17831
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. #22608721	C₂	semidirect
4. #22608715	C₂	semidirect
5. #22608709	C₂	semidirect
6. #22608695	C₂	semidirect
7. #22608694	C₂	semidirect
8. #22608680	C₂	semidirect
9. C₇₃×#1564	C₂	semidirect
190. #1564	D₇₃	semidirect
191. C₇₃	#17831	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
#22608721	2
#22608715	2
#22608709	2
#22608695	2
#22608694	2
#22608680	2
C₇₃×#1564	2
#17831	73