Subgroups — #33051645

Order 18688.

750 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	#22618897	C₂	1	1	yes	maximal, characteristic
2	C₈×#11924028	C₂	1	1	yes	maximal, characteristic
3	#22618573	C₂	1	1	yes	maximal, characteristic
4	#22608230	C₂	1	2	yes	maximal, complemented
5	#22608225	C₂	1	2	yes	maximal, complemented
6	#22608223	C₂	1	1	yes	maximal, characteristic, complemented
7	#22608220	C₂	1	1	yes	maximal, characteristic, complemented
8	#22608213	C₂	1	2	yes	maximal, complemented
9	C₈×#11924012	C₂	1	1	yes	maximal, characteristic, complemented
10	C₈×#11924011	C₂	1	2	yes	maximal, complemented
11	C₂×C₈²×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₄×#11924028	C₂²	1	1	yes	characteristic
13	#17988717	C₄	1	2	yes	—
14	C₈×Dic₁₄₆	C₄	1	2	yes	—
15	#17988666	C₄	1	2	yes	—
16	#17988658	C₄	1	4	yes	—
17	C₈×D₂₉₂	C₄	1	2	yes	—
18	C₈×C₂²⋊D₇₃	C₄	1	4	yes	—
19	#17988646	C₄	1	4	yes	—
20	#17988640	C₂²	1	1	yes	characteristic
21	#17988640	C₂²	1	1	yes	characteristic
22	#17988638	C₂²	1	2	yes	—
23	#17988638	C₂²	1	2	yes	—
24	#17988638	C₂²	1	2	yes	—
25	#17988638	C₂²	1	2	yes	—
26	#17988613	C₄	1	2	yes	—
27	#17988553	C₄	1	2	yes	—
28	C₄×#11924012	C₂²	1	1	yes	characteristic
29	C₄×#11924011	C₂²	1	2	yes	—
30	C₄×C₈×D₇₃	C₂²	1	2	yes	—
31	#17988309	C₂²	1	2	yes	—
32	#17988309	C₂²	1	2	yes	—
33	#17988309	C₂²	1	2	yes	—
34	#17988309	C₂²	1	2	yes	—
35	#17988308	C₂²	1	2	yes	—
36	C₈×#10596831	C₂²	1	2	yes	—
37	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
38	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
39	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
40	C₈²×C₇₃	C₂²	1	1	yes	characteristic
41	C₈²×C₇₃	C₂²	1	1	yes	characteristic
42	C₈²×C₇₃	C₂²	1	1	yes	characteristic
43	C₈²×C₇₃	C₂²	1	1	yes	characteristic
44	C₂×#11924028	C₂×C₄	1	1	yes	characteristic
45	C₂×#11924012	C₂×C₄	1	2	yes	—
46	C₂×#11924011	C₂×C₄	1	4	yes	—
47	C₂×C₈×D₇₃	C₂×C₄	1	4	yes	—
48	C₄×Dic₁₄₆	C₂×C₄	1	1	yes	characteristic
49	C₄×Dic₁₄₆	C₂×C₄	1	1	yes	characteristic
50	C₄×D₂₉₂	C₂×C₄	1	1	yes	characteristic
51	C₄×D₂₉₂	C₂×C₄	1	1	yes	characteristic
52	C₄×C₂²⋊D₇₃	C₂×C₄	1	2	yes	—
53	C₄×C₂²⋊D₇₃	C₂×C₄	1	2	yes	—
54	#14556550	C₂×C₄	1	1	yes	characteristic
55	#14556548	C₂×C₄	1	2	yes	—
56	C₄²×D₇₃	C₂³	1	2	yes	—
57	C₄⋊#10596831	C₂×C₄	1	2	yes	—
58	C₄⋊#10596831	C₂×C₄	1	2	yes	—
59	#14556520	C₂×C₄	1	4	yes	—
60	#14556520	C₂×C₄	1	4	yes	—
61	#14556520	C₂×C₄	1	4	yes	—
62	#14556520	C₂×C₄	1	4	yes	—
63	C₂²⋊#10596831	C₂×C₄	1	2	yes	—
64	C₂²⋊#10596831	C₂×C₄	1	2	yes	—
65	#14556503	C₂×C₄	1	4	yes	—
66	#14556503	C₂×C₄	1	4	yes	—
67	#14556503	C₂×C₄	1	4	yes	—
68	#14556503	C₂×C₄	1	4	yes	—
69	#14556502	C₂×C₄	1	2	yes	—
70	#14556501	C₂×C₄	1	4	yes	—
71	C₄×#10596831	C₂³	1	2	yes	—
72	C₄×#10596831	C₂³	1	2	yes	—
73	C₄×#10596831	C₂³	1	2	yes	—
74	C₈×Dic₇₃	C₂×C₄	1	4	yes	—
75	C₂²×C₈×C₇₃	C₂×C₄	1	2	yes	—
76	C₂²×C₈×C₇₃	C₂×C₄	1	2	yes	—
77	C₂²×C₈×C₇₃	C₂×C₄	1	2	yes	—
78	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
79	C₄×C₈×C₇₃	C₂×C₄	1	2	yes	—
80	C₄×C₈×C₇₃	C₂×C₄	1	2	yes	—
81	C₄×C₈×C₇₃	C₂×C₄	1	2	yes	—
82	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
83	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
84	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
85	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
86	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
87	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
88	#11924028	—	2	4	no	—
89	C₂×Dic₁₄₆	C₄²	1	1	yes	characteristic
90	C₂×D₂₉₂	C₄²	1	1	yes	characteristic
91	C₂×C₂²⋊D₇₃	C₄²	1	2	yes	—
92	C₂×C₄×D₇₃	C₂²×C₄	1	2	yes	—
93	C₂×C₄×D₇₃	C₂²×C₄	1	2	yes	—
94	C₂×C₄×D₇₃	C₂²×C₄	1	2	yes	—
95	#11924012	—	2	4	no	—
96	#11924011	—	2	8	no	—
97	C₂×#10596831	C₂²×C₄	1	4	yes	—
98	C₂×#10596831	C₂²×C₄	1	4	yes	—
99	C₂×#10596831	C₂²×C₄	1	4	yes	—
100	C₈×D₇₃	—	2	8	no	—
101	C₄⋊Dic₇₃	C₄²	1	1	yes	characteristic
102	C₄⋊Dic₇₃	C₄²	1	1	yes	characteristic
103	#11924007	C₄²	1	2	yes	—
104	#11924007	C₄²	1	2	yes	—
105	C₂²⋊Dic₇₃	C₄²	1	2	yes	—
106	#11924005	C₄²	1	2	yes	—
107	#11924005	C₄²	1	2	yes	—
108	C₄×Dic₇₃	C₂²×C₄	1	2	yes	—
109	C₄×Dic₇₃	C₂²×C₄	1	2	yes	—
110	C₄×Dic₇₃	C₂²×C₄	1	2	yes	—
111	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
112	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
113	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
114	C₂×C₈×C₇₃	—	2	4	no	—
115	C₂×C₈×C₇₃	—	2	4	no	—
116	C₂×C₈×C₇₃	—	2	4	no	—
117	C₂×C₈×C₇₃	C₂²×C₄	1	2	yes	—
118	C₂×C₈×C₇₃	C₂²×C₄	1	2	yes	—
119	C₂×C₈×C₇₃	C₂²×C₄	1	2	yes	—
120	C₂×C₈×C₇₃	C₂²×C₄	1	2	yes	—
121	C₂×C₈×C₇₃	C₂²×C₄	1	2	yes	—
122	C₂×C₈×C₇₃	C₂²×C₄	1	2	yes	—
123	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
124	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
125	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
126	C₄²×C₇₃	C₂⁴	1	1	yes	characteristic
127	C₂³×C₇₃	C₂×C₄²	1	1	yes	characteristic
128	C₂²×D₇₃	C₂×C₄²	1	2	yes	—
129	C₂×C₄×C₇₃	—	2	2	no	—
130	C₂×C₄×C₇₃	—	2	2	no	—
131	C₂×C₄×C₇₃	C₂×C₄²	1	1	yes	characteristic
132	C₂×C₄×C₇₃	C₂×C₄²	1	1	yes	characteristic
133	C₂×C₄×C₇₃	C₂×C₄²	1	1	yes	characteristic
134	C₂×C₄×C₇₃	#131	1	4	yes	—
135	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
136	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
137	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
138	C₂²⋊D₇₃	—	2	4	no	—
139	C₂×Dic₇₃	C₂×C₄²	1	2	yes	—
140	C₂×Dic₇₃	C₂×C₄²	1	2	yes	—
141	C₂×Dic₇₃	C₂×C₄²	1	2	yes	—
142	D₂₉₂	—	2	2	no	—
143	C₄×D₇₃	—	2	4	no	—
144	C₄×D₇₃	—	2	4	no	—
145	C₄×D₇₃	—	2	4	no	—
146	Dic₁₄₆	—	2	2	no	—
147	C₈×C₇₃	—	2	2	no	—
148	C₈×C₇₃	—	2	2	no	—
149	C₈×C₇₃	—	2	2	no	—
150	C₈×C₇₃	—	2	2	no	—
151	C₈×C₇₃	#131	1	4	yes	—
152	C₈×C₇₃	#131	1	4	yes	—
153	#10596831	—	2	4	no	—
154	#10596831	—	2	4	no	—
155	#10596831	—	2	4	no	—
156	C₂²×C₇₃	—	2	1	no	—
157	C₂²×C₇₃	—	2	1	no	—
158	C₂²×C₇₃	#405	1	2	yes	—
159	C₂²×C₇₃	C₂²×C₄²	1	1	yes	characteristic
160	C₂×D₇₃	—	2	2	no	—
161	C₂×D₇₃	—	2	2	no	—
162	C₂×D₇₃	—	2	2	no	—
163	C₄×C₇₃	—	2	1	no	—
164	C₄×C₇₃	—	2	1	no	—
165	C₄×C₇₃	#405	1	2	yes	—
166	C₄×C₇₃	C₂×#131	1	2	yes	—
167	C₄×C₇₃	C₂×#131	1	2	yes	—
168	C₄×C₇₃	C₂×#131	1	2	yes	—
169	Dic₇₃	—	2	2	no	—
170	Dic₇₃	—	2	2	no	—
171	Dic₇₃	—	2	2	no	—
172	#17328	—	73	1	no	maximal, Sylow
173	C₂×C₇₃	—	2	2	no	—
174	C₂×C₇₃	C₂×#405	1	1	yes	characteristic
175	C₂×C₇₃	C₄×#131	1	1	yes	characteristic
176	C₂×C₇₃	C₄×#131	1	1	yes	characteristic
177	D₇₃	—	4	2	no	—
178	#2965	—	73	1	no	—
179	C₈×#41	—	73	1	no	—
180	#2922	—	73	1	no	—
181	#1454	—	73	2	no	—
182	#1453	—	73	2	no	—
183	#1452	—	73	1	no	—
184	#1451	—	73	1	no	—
185	#1450	—	73	2	no	—
186	C₈×M₁₆	—	73	1	no	—
187	C₈×M₁₆	—	73	2	no	—
188	C₂×C₈²	D₇₃	1	1	yes	characteristic, complemented
189	C₇₃	#17328	1	1	yes	characteristic, complemented, Sylow
190	C₄×#41	—	73	1	no	—
191	#446	—	73	2	no	—
192	C₈×Q₈	—	73	2	no	—
193	#436	—	73	2	no	—
194	#435	—	73	4	no	—
195	C₈×D₄	—	73	4	no	—
196	C₈×D₄	—	73	2	no	—
197	#433	—	73	4	no	—
198	#431	—	73	1	no	—
199	#431	—	73	1	no	—
200	#431	—	73	2	no	—
201	#431	—	73	2	no	—
202	#431	—	73	2	no	—
203	#431	—	73	2	no	—
204	#424	—	73	2	no	—
205	#408	—	73	2	no	—
206	C₄×M₁₆	—	73	1	no	—
207	C₄×M₁₆	—	73	2	no	—
208	C₂×C₄×C₈	—	73	2	no	—
209	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
210	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
211	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
212	#322	—	73	2	no	—
213	#322	—	73	2	no	—
214	#322	—	73	2	no	—
215	#322	—	73	2	no	—
216	#322	—	73	2	no	—
217	C₈²	—	73	2	no	—
218	C₈²	C₂×D₇₃	1	1	yes	characteristic
219	C₈²	C₂×D₇₃	1	1	yes	characteristic
220	C₈²	C₂×D₇₃	1	1	yes	characteristic
221	C₈²	C₂×D₇₃	1	1	yes	characteristic
222	C₂×#41	—	73	1	no	—
223	C₂×M₁₆	—	73	2	no	—
224	C₂×M₁₆	—	73	4	no	—
225	C₂²×C₈	—	73	4	no	—
226	C₂²×C₈	C₄×D₇₃	1	2	yes	—
227	C₂²×C₈	C₄×D₇₃	1	2	yes	—
228	C₂²×C₈	C₄×D₇₃	1	2	yes	—
229	C₄×Q₈	—	73	1	no	—
230	C₄×Q₈	—	73	1	no	—
231	C₄×D₄	—	73	2	no	—
232	C₄×D₄	—	73	2	no	—
233	C₄×D₄	—	73	1	no	—
234	C₄×D₄	—	73	1	no	—
235	#117	—	73	1	no	—
236	#117	—	73	2	no	—
237	C₂×C₄²	—	73	2	no	—
238	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
239	C₄⋊C₈	—	73	4	no	—
240	C₄⋊C₈	—	73	4	no	—
241	C₄⋊C₈	—	73	4	no	—
242	C₄⋊C₈	—	73	4	no	—
243	C₄⋊C₈	—	73	2	no	—
244	C₄⋊C₈	—	73	2	no	—
245	C₂²⋊C₈	—	73	2	no	—
246	C₂²⋊C₈	—	73	2	no	—
247	C₂²⋊C₈	—	73	4	no	—
248	C₂²⋊C₈	—	73	4	no	—
249	C₂²⋊C₈	—	73	4	no	—
250	C₂²⋊C₈	—	73	4	no	—
251	#97	—	73	4	no	—
252	#97	—	73	2	no	—
253	C₄×C₈	—	73	4	no	—
254	C₄×C₈	—	73	2	no	—
255	C₄×C₈	—	73	2	no	—
256	C₄×C₈	—	73	2	no	—
257	C₄×C₈	C₄×D₇₃	1	2	yes	—
258	C₄×C₈	C₄×D₇₃	1	2	yes	—
259	C₄×C₈	C₄×D₇₃	1	2	yes	—
260	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
261	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
262	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
263	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
264	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
265	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
266	#41	—	146	4	no	—
267	C₂×Q₈	—	73	1	no	—
268	C₂×D₄	—	73	2	no	—
269	C₂×D₄	—	73	1	no	—
270	C₂²×C₄	—	73	2	no	—
271	C₂²×C₄	—	73	2	no	—
272	C₂²×C₄	—	73	2	no	—
273	C₂²×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
274	C₂²×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
275	C₂²×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
276	M₁₆	—	146	4	no	—
277	M₁₆	—	146	8	no	—
278	C₂×C₈	—	2	4	no	—
279	C₂×C₈	—	2	4	no	—
280	C₂×C₈	—	2	4	no	—
281	C₂×C₈	—	73	4	no	—
282	C₂×C₈	—	73	4	no	—
283	C₂×C₈	—	73	4	no	—
284	C₂×C₈	—	146	8	no	—
285	C₂×C₈	C₂×C₄×D₇₃	1	2	yes	—
286	C₂×C₈	C₂×C₄×D₇₃	1	2	yes	—
287	C₂×C₈	C₂×C₄×D₇₃	1	2	yes	—
288	C₂×C₈	C₂×C₄×D₇₃	1	2	yes	—
289	C₂×C₈	C₂×C₄×D₇₃	1	2	yes	—
290	C₂×C₈	C₂×C₄×D₇₃	1	2	yes	—
291	C₄⋊C₄	—	73	2	no	—
292	C₄⋊C₄	—	73	2	no	—
293	C₄⋊C₄	—	73	1	no	—
294	C₄⋊C₄	—	73	1	no	—
295	C₂²⋊C₄	—	73	2	no	—
296	C₂²⋊C₄	—	73	2	no	—
297	C₂²⋊C₄	—	73	2	no	—
298	C₄²	—	73	2	no	—
299	C₄²	—	73	2	no	—
300	C₄²	—	73	2	no	—
301	C₄²	C₂×C₄×D₇₃	1	1	yes	characteristic
302	C₄²	C₂×C₄×D₇₃	1	1	yes	characteristic
303	C₄²	C₂×C₄×D₇₃	1	1	yes	characteristic
304	C₄²	C₂³×D₇₃	1	1	yes	characteristic
305	C₂³	—	73	2	no	—
306	C₂³	C₄²×D₇₃	1	1	yes	characteristic
307	Q₈	—	146	2	no	—
308	D₄	—	146	4	no	—
309	D₄	—	146	2	no	—
310	C₂×C₄	—	2	2	no	—
311	C₂×C₄	—	2	2	no	—
312	C₂×C₄	—	73	2	no	—
313	C₂×C₄	—	73	2	no	—
314	C₂×C₄	—	73	2	no	—
315	C₂×C₄	—	146	4	no	—
316	C₂×C₄	—	146	4	no	—
317	C₂×C₄	—	146	4	no	—
318	C₂×C₄	C₄²×D₇₃	1	1	yes	characteristic
319	C₂×C₄	C₄²×D₇₃	1	1	yes	characteristic
320	C₂×C₄	C₄²×D₇₃	1	1	yes	characteristic
321	C₂×C₄	#14556597	1	4	yes	—
322	C₂×C₄	C₂²×C₄×D₇₃	1	1	yes	characteristic
323	C₂×C₄	C₂²×C₄×D₇₃	1	1	yes	characteristic
324	C₂×C₄	C₂²×C₄×D₇₃	1	1	yes	characteristic
325	C₈	—	2	2	no	—
326	C₈	—	2	2	no	—
327	C₈	—	2	2	no	—
328	C₈	—	2	2	no	—
329	C₈	—	146	4	no	—
330	C₈	—	146	4	no	—
331	C₈	—	146	4	no	—
332	C₈	#14556597	1	4	yes	—
333	C₈	#14556597	1	4	yes	—
334	C₂²	—	2	1	no	—
335	C₂²	—	2	1	no	—
336	C₂²	—	146	2	no	—
337	C₂²	—	146	2	no	—
338	C₂²	—	146	2	no	—
339	C₂²	#17988541	1	2	yes	—
340	C₂²	C₂×C₄²×D₇₃	1	1	yes	characteristic
341	C₄	—	2	1	no	—
342	C₄	—	2	1	no	—
343	C₄	—	146	2	no	—
344	C₄	—	146	2	no	—
345	C₄	—	146	2	no	—
346	C₄	#17988541	1	2	yes	—
347	C₄	C₂×#14556597	1	2	yes	—
348	C₄	C₂×#14556597	1	2	yes	—
349	C₄	C₂×#14556597	1	2	yes	—
350	C₂	—	2	2	no	—
351	C₂	—	292	2	no	—
352	C₂	C₂×#17988541	1	1	yes	characteristic
353	C₂	C₄×#14556597	1	1	yes	characteristic
354	C₂	C₄×#14556597	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₈	C₂²×D₇₃
commutator, socle	C₂²×C₇₃	C₂²×C₄²
Frattini	C₄²	C₂³×D₇₃
Fitting	C₂×C₈²×C₇₃	C₂
prime radical	C₂×#11924028	C₂×C₄
cosocle	C₄²×C₇₃	C₂⁴
hypercenter	C₂×C₈²	D₇₃
hypocenter	C₇₃	#17328
LCS	G ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₄×C₈ ◁ C₂×C₈²

Complements

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

N	H ≅ G/N	type
4. #22608230	C₂	semidirect
5. #22608225	C₂	semidirect
6. #22608223	C₂	semidirect
7. #22608220	C₂	semidirect
8. #22608213	C₂	semidirect
9. C₈×#11924012	C₂	semidirect
10. C₈×#11924011	C₂	semidirect
11. C₂×C₈²×C₇₃	C₂	semidirect
188. C₂×C₈²	D₇₃	semidirect
189. C₇₃	#17328	semidirect

Maximal subgroups

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

subgroup	index
#22618897	2
C₈×#11924028	2
#22618573	2
#22608230	2
#22608225	2
#22608223	2
#22608220	2
#22608213	2
C₈×#11924012	2
C₈×#11924011	2
C₂×C₈²×C₇₃	2
#17328	73