Subgroups — #33055028

Order 18688.

666 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22619044	C₂	1	2	yes	maximal
2	C₄×#14556597	C₂	1	1	yes	maximal, characteristic
3	#22608697	C₂	1	4	yes	maximal, complemented
4	#22608691	C₂	1	2	yes	maximal, complemented
5	#22608685	C₂	1	2	yes	maximal, complemented
6	#22608684	C₂	1	2	yes	maximal, complemented
7	C₂×#17988338	C₂	1	1	yes	maximal, characteristic, complemented
8	C₇₃×#1564	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#14556597	C₄	1	1	yes	characteristic
10	C₂×#14556597	C₄	1	1	yes	characteristic
11	C₄×#11924028	C₂²	1	1	yes	characteristic
12	#17988717	—	2	2	no	—
13	#17988666	—	2	2	no	—
14	#17988655	—	2	4	no	—
15	#17988643	C₂²	1	4	yes	—
16	#17988609	C₂²	1	4	yes	—
17	C₂×C₄⋊#10596831	C₂²	1	2	yes	—
18	#17988542	C₄	1	1	yes	characteristic
19	#17988542	C₄	1	1	yes	characteristic
20	#17988541	C₄	1	2	yes	—
21	#17988541	C₄	1	2	yes	—
22	C₄×#11924012	C₂²	1	1	yes	characteristic
23	C₄×#11924011	C₂²	1	2	yes	—
24	C₄×C₈×D₇₃	C₂²	1	2	yes	—
25	#17988339	C₂²	1	4	yes	—
26	#17988338	C₂²	1	2	yes	—
27	#17988338	C₂²	1	2	yes	—
28	#17988337	C₂²	1	4	yes	—
29	#431×C₇₃	C₂²	1	2	yes	—
30	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
31	#335×C₇₃	C₂²	1	2	yes	—
32	#334×C₇₃	C₂²	1	1	yes	characteristic
33	#334×C₇₃	C₂²	1	1	yes	characteristic
34	C₂×#11924028	C₂×C₄	1	1	yes	characteristic
35	#14556597	—	2	4	no	—
36	#14556597	—	2	4	no	—
37	C₂×#11924012	C₂×C₄	1	1	yes	characteristic
38	C₂×#11924012	C₂×C₄	1	1	yes	characteristic
39	C₂×#11924011	—	2	4	no	—
40	C₂×#11924011	C₂×C₄	1	2	yes	—
41	C₂×#11924011	C₂×C₄	1	2	yes	—
42	C₂²×#10596831	—	2	2	no	—
43	C₂×C₈×D₇₃	C₂×C₄	1	2	yes	—
44	C₂×C₈×D₇₃	C₂×C₄	1	2	yes	—
45	C₄×Dic₁₄₆	D₄	1	1	yes	characteristic
46	C₄×Dic₁₄₆	Q₈	1	1	yes	characteristic
47	C₄×D₂₉₂	D₄	1	1	yes	characteristic
48	C₄×D₂₉₂	Q₈	1	1	yes	characteristic
49	C₄×C₂²⋊D₇₃	D₄	1	2	yes	—
50	C₄×C₂²⋊D₇₃	Q₈	1	2	yes	—
51	#14556550	C₂×C₄	1	1	yes	characteristic
52	#14556548	C₂×C₄	1	2	yes	—
53	C₄²×D₇₃	C₂³	1	2	yes	—
54	C₄⋊#10596831	—	2	2	no	—
55	C₄⋊#10596831	C₂³	1	4	yes	—
56	#14556520	—	2	4	no	—
57	#14556503	—	2	4	no	—
58	#14556502	C₂×C₄	1	2	yes	—
59	#14556501	—	2	4	no	—
60	#14556501	C₂×C₄	1	2	yes	—
61	#14556501	C₂×C₄	1	2	yes	—
62	C₄×#10596831	C₂³	1	2	yes	—
63	C₈×Dic₇₃	C₂×C₄	1	2	yes	—
64	C₈×Dic₇₃	C₂×C₄	1	2	yes	—
65	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
66	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
67	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
68	C₄⋊C₈×C₇₃	C₂³	1	2	yes	—
69	C₄⋊C₈×C₇₃	C₂³	1	2	yes	—
70	C₂²⋊C₈×C₇₃	—	2	2	no	—
71	C₄×C₈×C₇₃	—	2	2	no	—
72	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
73	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
74	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
75	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
76	#11924028	—	2	2	no	—
77	#11924028	—	2	2	no	—
78	C₂×Dic₁₄₆	C₄⋊C₄	1	1	yes	characteristic
79	C₂×D₂₉₂	C₄⋊C₄	1	1	yes	characteristic
80	C₂×C₂²⋊D₇₃	C₄⋊C₄	1	2	yes	—
81	C₂×C₄×D₇₃	C₂²×C₄	1	2	yes	—
82	C₂×C₄×D₇₃	C₂×D₄	1	2	yes	—
83	C₂×C₄×D₇₃	C₂×Q₈	1	2	yes	—
84	#11924012	—	2	2	no	—
85	#11924012	M₁₆	1	4	yes	complemented
86	#11924011	—	2	4	no	—
87	#11924011	—	2	4	no	—
88	#11924011	—	8	4	no	—
89	C₂×#10596831	—	2	4	no	—
90	C₂×#10596831	—	8	2	no	—
91	C₂×#10596831	C₂²×C₄	1	2	yes	—
92	C₂×#10596831	C₂²×C₄	1	2	yes	—
93	C₈×D₇₃	—	2	4	no	—
94	C₈×D₇₃	—	2	4	no	—
95	C₄⋊Dic₇₃	C₄⋊C₄	1	1	yes	characteristic
96	C₄⋊Dic₇₃	C₄⋊C₄	1	1	yes	characteristic
97	#11924007	C₄⋊C₄	1	2	yes	—
98	#11924007	C₄⋊C₄	1	2	yes	—
99	C₂²⋊Dic₇₃	C₄⋊C₄	1	2	yes	—
100	#11924005	C₄⋊C₄	1	2	yes	—
101	#11924005	C₄⋊C₄	1	2	yes	—
102	C₄×Dic₇₃	C₂²×C₄	1	2	yes	—
103	C₄×Dic₇₃	C₂×D₄	1	2	yes	—
104	C₄×Dic₇₃	C₂×Q₈	1	2	yes	—
105	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
106	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
107	C₂²×C₄×C₇₃	C₂×Q₈	1	1	yes	characteristic
108	C₂×C₈×C₇₃	—	2	2	no	—
109	C₂×C₈×C₇₃	—	2	2	no	—
110	C₂×C₈×C₇₃	—	2	2	no	—
111	C₂×C₈×C₇₃	M₁₆	1	4	yes	complemented
112	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
113	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
114	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
115	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
116	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
117	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
118	C₄²×C₇₃	C₂×Q₈	1	1	yes	characteristic
119	C₄²×C₇₃	C₂⁴	1	1	yes	characteristic
120	C₂³×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
121	C₂²×D₇₃	C₂×C₄⋊C₄	1	2	yes	—
122	C₂×C₄×C₇₃	—	2	2	no	—
123	C₂×C₄×C₇₃	—	4	1	no	—
124	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
125	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
126	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
127	C₂×C₄×C₇₃	C₂×M₁₆	1	2	yes	—
128	C₂×C₄×C₇₃	C₂×M₁₆	1	2	yes	—
129	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
130	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
131	C₂×C₄×C₇₃	C₂²×Q₈	1	1	yes	characteristic
132	C₂²⋊D₇₃	—	2	4	no	—
133	C₂×Dic₇₃	C₂×C₄⋊C₄	1	2	yes	—
134	C₂×Dic₇₃	C₂×C₄⋊C₄	1	2	yes	—
135	C₂×Dic₇₃	C₂×C₄⋊C₄	1	2	yes	—
136	D₂₉₂	—	2	2	no	—
137	C₄×D₇₃	—	2	2	no	—
138	C₄×D₇₃	—	2	2	no	—
139	C₄×D₇₃	—	2	4	no	—
140	C₄×D₇₃	—	4	2	no	—
141	Dic₁₄₆	—	2	2	no	—
142	C₈×C₇₃	—	2	1	no	—
143	C₈×C₇₃	—	2	1	no	—
144	C₈×C₇₃	—	4	2	no	—
145	C₈×C₇₃	—	4	2	no	—
146	C₈×C₇₃	C₂×M₁₆	1	2	yes	—
147	C₈×C₇₃	C₂×M₁₆	1	2	yes	—
148	#10596831	—	2	2	no	—
149	#10596831	—	4	4	no	—
150	#10596831	C₂×M₁₆	1	4	yes	—
151	C₂²×C₇₃	—	2	1	no	—
152	C₂²×C₇₃	—	2	1	no	—
153	C₂²×C₇₃	#423	1	2	yes	—
154	C₂²×C₇₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
155	C₂×D₇₃	—	2	2	no	—
156	C₂×D₇₃	—	2	2	no	—
157	C₂×D₇₃	—	2	2	no	—
158	C₄×C₇₃	—	2	1	no	—
159	C₄×C₇₃	—	2	1	no	—
160	C₄×C₇₃	—	2	1	no	—
161	C₄×C₇₃	#423	1	2	yes	—
162	C₄×C₇₃	C₂²×M₁₆	1	1	yes	characteristic
163	C₄×C₇₃	C₂²×M₁₆	1	1	yes	characteristic
164	C₄×C₇₃	#577	1	2	yes	—
165	Dic₇₃	—	2	2	no	—
166	Dic₇₃	—	2	2	no	—
167	Dic₇₃	—	2	2	no	—
168	#17798	—	73	1	no	maximal, Sylow
169	C₂×C₇₃	—	2	2	no	—
170	C₂×C₇₃	C₂×#423	1	1	yes	characteristic
171	C₂×C₇₃	#2912	1	1	yes	characteristic
172	C₂×C₇₃	#2916	1	1	yes	characteristic
173	D₇₃	—	4	2	no	—
174	#2989	—	73	2	no	—
175	C₄×#131	—	73	1	no	—
176	#1569	—	73	4	no	—
177	#1568	—	73	2	no	—
178	#1566	—	73	2	no	—
179	#1565	—	73	2	no	—
180	#1564	D₇₃	1	1	yes	characteristic, complemented
181	C₂×#334	—	73	1	no	—
182	C₇₃	#17798	1	1	yes	characteristic, complemented, Sylow
183	C₂×#131	—	73	1	no	—
184	C₂×#131	—	73	1	no	—
185	C₄×#41	—	73	1	no	—
186	#446	—	146	2	no	—
187	#436	—	146	2	no	—
188	#435	—	146	4	no	—
189	#432	—	73	4	no	—
190	#431	C₂×D₇₃	1	2	yes	—
191	#423	—	73	4	no	—
192	C₂×C₄⋊C₈	—	73	2	no	—
193	#405	—	73	1	no	—
194	#405	—	73	1	no	—
195	#405	—	73	2	no	—
196	#405	—	73	2	no	—
197	C₄×M₁₆	—	73	1	no	—
198	C₄×M₁₆	—	73	2	no	—
199	C₂×C₄×C₈	—	73	2	no	—
200	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
201	#335	—	73	4	no	—
202	#335	C₂×D₇₃	1	2	yes	—
203	#334	—	73	4	no	—
204	#334	—	73	2	no	—
205	#334	—	73	2	no	—
206	#334	C₂×D₇₃	1	1	yes	characteristic
207	#334	C₂×D₇₃	1	1	yes	characteristic
208	C₂×#41	—	73	1	no	—
209	#131	—	146	4	no	—
210	#131	—	146	4	no	—
211	C₂×M₁₆	—	73	1	no	—
212	C₂×M₁₆	—	73	1	no	—
213	C₂×M₁₆	—	73	2	no	—
214	C₂×M₁₆	—	73	2	no	—
215	C₂×M₁₆	—	146	4	no	—
216	C₂²×C₈	—	73	2	no	—
217	C₂²×C₈	—	73	2	no	—
218	C₂²×C₈	—	146	2	no	—
219	C₂²×C₈	C₄×D₇₃	1	1	yes	characteristic
220	C₂²×C₈	C₄×D₇₃	1	1	yes	characteristic
221	C₄×Q₈	—	73	1	no	—
222	C₄×Q₈	—	73	1	no	—
223	C₄×D₄	—	73	2	no	—
224	C₄×D₄	—	73	2	no	—
225	C₄×D₄	—	73	1	no	—
226	C₄×D₄	—	73	1	no	—
227	#117	—	73	1	no	—
228	#117	—	73	2	no	—
229	C₂×C₄²	—	73	2	no	—
230	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
231	C₄⋊C₈	—	73	4	no	—
232	C₄⋊C₈	—	146	4	no	—
233	C₄⋊C₈	—	146	2	no	—
234	C₄⋊C₈	C₂²×D₇₃	1	2	yes	—
235	C₄⋊C₈	C₂²×D₇₃	1	2	yes	—
236	C₂²⋊C₈	—	2	2	no	—
237	C₂²⋊C₈	—	146	4	no	—
238	#97	—	73	2	no	—
239	#97	—	73	2	no	—
240	#97	—	73	2	no	—
241	#97	—	146	4	no	—
242	C₄×C₈	—	2	2	no	—
243	C₄×C₈	—	73	2	no	—
244	C₄×C₈	—	73	2	no	—
245	C₄×C₈	—	73	2	no	—
246	C₄×C₈	C₄×D₇₃	1	1	yes	characteristic
247	C₄×C₈	C₄×D₇₃	1	1	yes	characteristic
248	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
249	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
250	#41	—	146	2	no	—
251	#41	—	146	2	no	—
252	C₂×Q₈	—	73	1	no	—
253	C₂×D₄	—	73	2	no	—
254	C₂×D₄	—	73	1	no	—
255	C₂²×C₄	—	73	2	no	—
256	C₂²×C₄	—	73	2	no	—
257	C₂²×C₄	—	73	2	no	—
258	C₂²×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
259	C₂²×C₄	D₄×D₇₃	1	1	yes	characteristic
260	C₂²×C₄	Q₈×D₇₃	1	1	yes	characteristic
261	M₁₆	—	584	4	no	—
262	M₁₆	—	73	4	no	—
263	M₁₆	—	146	4	no	—
264	M₁₆	—	146	2	no	—
265	M₁₆	—	146	4	no	—
266	C₂×C₈	—	2	2	no	—
267	C₂×C₈	—	2	2	no	—
268	C₂×C₈	—	2	2	no	—
269	C₂×C₈	—	584	2	no	—
270	C₂×C₈	—	73	2	no	—
271	C₂×C₈	—	73	2	no	—
272	C₂×C₈	—	146	4	no	—
273	C₂×C₈	—	146	4	no	—
274	C₂×C₈	—	146	4	no	—
275	C₂×C₈	#11924011	1	4	yes	complemented
276	C₂×C₈	C₂×C₄×D₇₃	1	1	yes	characteristic
277	C₂×C₈	C₂×C₄×D₇₃	1	1	yes	characteristic
278	C₂×C₈	C₂×C₄×D₇₃	1	1	yes	characteristic
279	C₂×C₈	C₂×C₄×D₇₃	1	1	yes	characteristic
280	C₄⋊C₄	—	73	2	no	—
281	C₄⋊C₄	—	73	2	no	—
282	C₄⋊C₄	—	73	1	no	—
283	C₄⋊C₄	—	73	1	no	—
284	C₂²⋊C₄	—	73	2	no	—
285	C₂²⋊C₄	—	73	2	no	—
286	C₂²⋊C₄	—	73	2	no	—
287	C₄²	—	73	2	no	—
288	C₄²	—	73	2	no	—
289	C₄²	—	73	2	no	—
290	C₄²	C₂×C₄×D₇₃	1	1	yes	characteristic
291	C₄²	D₄×D₇₃	1	1	yes	characteristic
292	C₄²	Q₈×D₇₃	1	1	yes	characteristic
293	C₄²	C₂³×D₇₃	1	1	yes	characteristic
294	C₂³	—	73	2	no	—
295	C₂³	C₄⋊C₄×D₇₃	1	1	yes	characteristic
296	Q₈	—	146	2	no	—
297	D₄	—	146	4	no	—
298	D₄	—	146	2	no	—
299	C₂×C₄	—	2	2	no	—
300	C₂×C₄	—	4	1	no	—
301	C₂×C₄	—	292	2	no	—
302	C₂×C₄	—	73	2	no	—
303	C₂×C₄	—	73	2	no	—
304	C₂×C₄	—	73	2	no	—
305	C₂×C₄	—	146	4	no	—
306	C₂×C₄	—	146	2	no	—
307	C₂×C₄	—	146	2	no	—
308	C₂×C₄	C₄⋊C₄×D₇₃	1	1	yes	characteristic
309	C₂×C₄	C₄⋊C₄×D₇₃	1	1	yes	characteristic
310	C₂×C₄	C₄⋊C₄×D₇₃	1	1	yes	characteristic
311	C₂×C₄	M₁₆×D₇₃	1	2	yes	—
312	C₂×C₄	C₂×#11924011	1	2	yes	—
313	C₂×C₄	C₂²×C₄×D₇₃	1	1	yes	characteristic
314	C₂×C₄	C₂×D₄×D₇₃	1	1	yes	characteristic
315	C₂×C₄	C₂×Q₈×D₇₃	1	1	yes	characteristic
316	C₈	—	2	1	no	—
317	C₈	—	2	1	no	—
318	C₈	—	4	2	no	—
319	C₈	—	4	2	no	—
320	C₈	—	292	4	no	—
321	C₈	—	73	4	no	—
322	C₈	—	146	2	no	—
323	C₈	C₂×#11924011	1	2	yes	—
324	C₈	C₂×#11924011	1	2	yes	—
325	C₂²	—	2	1	no	—
326	C₂²	—	2	1	no	—
327	C₂²	—	146	2	no	—
328	C₂²	—	146	2	no	—
329	C₂²	—	146	2	no	—
330	C₂²	#17988609	1	2	yes	—
331	C₂²	C₂×C₄⋊C₄×D₇₃	1	1	yes	characteristic
332	C₄	—	2	1	no	—
333	C₄	—	2	1	no	—
334	C₄	—	2	1	no	—
335	C₄	—	146	2	no	—
336	C₄	—	146	2	no	—
337	C₄	—	146	2	no	—
338	C₄	#17988609	1	2	yes	—
339	C₄	C₂×M₁₆×D₇₃	1	1	yes	characteristic
340	C₄	C₂²×#11924011	1	1	yes	characteristic
341	C₄	#17989394	1	2	yes	—
342	C₂	—	2	2	no	—
343	C₂	—	292	2	no	—
344	C₂	C₂×#17988609	1	1	yes	characteristic
345	C₂	#22618523	1	1	yes	characteristic
346	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	C₂×C₄
cosocle	C₄²×C₇₃	C₂⁴
hypercenter	#1564	D₇₃
hypocenter	C₇₃	#17798
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. #22608697	C₂	semidirect
4. #22608691	C₂	semidirect
5. #22608685	C₂	semidirect
6. #22608684	C₂	semidirect
7. C₂×#17988338	C₂	semidirect
8. C₇₃×#1564	C₂	semidirect
85. #11924012	M₁₆	semidirect
111. C₂×C₈×C₇₃	M₁₆	semidirect
180. #1564	D₇₃	semidirect
182. C₇₃	#17798	semidirect
275. C₂×C₈	#11924011	semidirect

Maximal subgroups

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

subgroup	index
#22619044	2
C₄×#14556597	2
#22608697	2
#22608691	2
#22608685	2
#22608684	2
C₂×#17988338	2
C₇₃×#1564	2
#17798	73