Subgroups — #33241439

Order 18688.

794 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17989029	C₂	1	2	yes	maximal, complemented
2	C₂×#17989020	C₂	1	1	yes	maximal, characteristic
3	C₂×Q₃₂×D₇₃	C₂	1	1	yes	maximal, characteristic, complemented
4	#22620419	C₂	1	2	yes	maximal, complemented
5	#22612200	C₂	1	2	yes	maximal, complemented
6	#22612199	C₂	1	2	yes	maximal, complemented
7	#22612193	C₂	1	1	yes	maximal, characteristic, complemented
8	#22612192	C₂	1	1	yes	maximal, characteristic, complemented
9	#22612182	C₂	1	1	yes	maximal, characteristic, complemented
10	#22611526	C₂	1	1	yes	maximal, characteristic, complemented
11	C₇₃×#2247	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#14556623	—	2	2	no	—
13	C₂×#14556614	C₂²	1	2	yes	—
14	C₂×Q₁₆×D₇₃	C₂²	1	2	yes	—
15	#17989162	—	2	2	no	—
16	#17989029	—	2	8	no	—
17	#17989020	—	2	4	no	—
18	C₂×Dic₅₈₄	C₂²	1	1	yes	characteristic
19	C₂×#14556537	C₂²	1	2	yes	—
20	C₂×#14556537	C₂²	1	2	yes	—
21	Q₃₂×D₇₃	—	2	4	no	—
22	C₂×#14556536	C₂²	1	2	yes	—
23	C₂×#14556534	C₂²	1	2	yes	—
24	C₂×#14556534	C₂²	1	2	yes	—
25	C₂×D₁₁₆₈	C₂²	1	1	yes	characteristic
26	C₂×#14556530	C₂²	1	1	yes	characteristic
27	C₂×C₁₆×D₇₃	C₂²	1	1	yes	characteristic
28	#17988980	—	2	2	no	—
29	#17988979	—	2	2	no	—
30	#17988972	C₂²	1	2	yes	—
31	#17988971	C₂²	1	2	yes	—
32	#17988965	C₂²	1	2	yes	—
33	#17988541	C₂²	1	1	yes	characteristic
34	#17988363	C₂²	1	1	yes	characteristic
35	#17988362	C₂²	1	1	yes	characteristic
36	C₁₆×Dic₇₃	C₂²	1	1	yes	characteristic
37	C₂×Q₃₂×C₇₃	C₂²	1	1	yes	characteristic
38	C₂×Q₃₂×C₇₃	C₂²	1	1	yes	characteristic
39	C₂×Q₃₂×C₇₃	C₂²	1	2	yes	—
40	#494×C₇₃	C₂²	1	2	yes	—
41	C₄×C₁₆×C₇₃	C₂²	1	1	yes	characteristic
42	C₂×#11924029	—	2	2	no	—
43	C₂×Q₈×D₇₃	—	2	2	no	—
44	#14556623	—	4	8	no	—
45	#14556614	—	2	4	no	—
46	#14556614	—	2	4	no	—
47	C₂×Dic₂₉₂	C₂³	1	2	yes	—
48	C₂×#11924018	—	2	2	no	—
49	C₂×#11924018	—	2	2	no	—
50	Q₁₆×D₇₃	—	2	4	no	—
51	Q₁₆×D₇₃	—	2	4	no	—
52	C₂×#11924017	—	2	2	no	—
53	C₂×#11924015	—	2	2	no	—
54	C₂×#11924015	—	2	2	no	—
55	C₂×D₅₈₄	C₂³	1	2	yes	—
56	C₂×#11924011	D₄	1	1	yes	characteristic
57	C₂×C₈×D₇₃	C₂³	1	1	yes	characteristic
58	#14556589	—	2	2	no	—
59	#14556580	—	2	2	no	—
60	#14556579	—	2	2	no	—
61	#14556576	—	4	4	no	—
62	#14556569	—	4	4	no	—
63	#14556548	D₄	1	1	yes	characteristic
64	Dic₅₈₄	—	2	2	no	—
65	#14556537	—	2	4	no	—
66	#14556537	—	2	4	no	—
67	#14556536	—	2	4	no	—
68	#14556534	—	2	4	no	—
69	#14556534	—	2	4	no	—
70	D₁₁₆₈	—	2	2	no	—
71	#14556530	D₄	1	4	yes	—
72	C₂×#11924003	C₂³	1	1	yes	characteristic
73	C₂×#11924003	C₂³	1	1	yes	characteristic
74	C₁₆×D₇₃	D₄	1	4	yes	—
75	#14556502	D₄	1	1	yes	characteristic
76	#14556501	D₄	1	1	yes	characteristic
77	C₈×Dic₇₃	C₂³	1	1	yes	characteristic
78	C₂×Q₁₆×C₇₃	—	2	2	no	—
79	C₂×Q₁₆×C₇₃	C₂³	1	2	yes	—
80	C₂×Q₁₆×C₇₃	C₂³	1	2	yes	—
81	C₄⋊Q₈×C₇₃	—	2	2	no	—
82	Q₃₂×C₇₃	—	2	2	no	—
83	Q₃₂×C₇₃	—	2	2	no	—
84	Q₃₂×C₇₃	—	2	4	no	—
85	C₂×C₁₆×C₇₃	C₂³	1	1	yes	characteristic
86	C₂×C₁₆×C₇₃	C₂³	1	1	yes	characteristic
87	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
88	#11924029	—	4	4	no	—
89	#11924029	—	4	4	no	—
90	C₂×Dic₁₄₆	—	2	2	no	—
91	C₂×Dic₁₄₆	—	4	2	no	—
92	Q₈×D₇₃	—	4	4	no	—
93	Q₈×D₇₃	—	4	4	no	—
94	C₂×D₂₉₂	—	2	2	no	—
95	C₂×D₂₉₂	—	4	2	no	—
96	C₂×C₄×D₇₃	—	4	2	no	—
97	C₂×C₄×D₇₃	—	4	2	no	—
98	C₂×C₄×D₇₃	C₂×D₄	1	1	yes	characteristic
99	Dic₂₉₂	—	2	2	no	—
100	Dic₂₉₂	—	2	2	no	—
101	#11924018	—	4	4	no	—
102	#11924018	—	4	4	no	—
103	#11924017	—	4	4	no	—
104	#11924015	—	4	4	no	—
105	#11924015	—	4	4	no	—
106	D₅₈₄	—	2	2	no	—
107	D₅₈₄	—	2	2	no	—
108	#11924011	D₈	1	4	yes	—
109	C₂×#10596831	C₂×D₄	1	1	yes	characteristic
110	C₂×#10596831	C₂×D₄	1	1	yes	characteristic
111	C₈×D₇₃	C₂×D₄	1	2	yes	—
112	C₈×D₇₃	C₂×D₄	1	2	yes	—
113	#11924007	—	4	4	no	—
114	#11924007	D₈	1	2	yes	—
115	#11924005	—	4	4	no	—
116	#11924005	—	4	4	no	—
117	#11924005	D₈	1	2	yes	—
118	C₄×Dic₇₃	C₂×D₄	1	1	yes	characteristic
119	#11924003	C₂×D₄	1	2	yes	—
120	#11924003	C₂×D₄	1	2	yes	—
121	C₂×Q₈×C₇₃	—	2	2	no	—
122	C₂×Q₈×C₇₃	—	2	2	no	—
123	Q₁₆×C₇₃	—	2	2	no	—
124	Q₁₆×C₇₃	—	2	2	no	—
125	Q₁₆×C₇₃	—	2	2	no	—
126	Q₁₆×C₇₃	—	2	2	no	—
127	Q₁₆×C₇₃	—	4	4	no	—
128	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
129	C₂×C₈×C₇₃	C₂⁴	1	1	yes	characteristic
130	C₄⋊C₄×C₇₃	—	4	4	no	—
131	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
132	C₁₆×C₇₃	C₂×D₄	1	2	yes	—
133	C₁₆×C₇₃	C₂×D₄	1	2	yes	—
134	C₂²×D₇₃	—	4	2	no	—
135	C₂²×D₇₃	C₂×D₈	1	1	yes	characteristic
136	Q₈×C₇₃	—	4	2	no	—
137	Q₈×C₇₃	—	4	2	no	—
138	Q₈×C₇₃	—	4	2	no	—
139	Q₈×C₇₃	—	4	2	no	—
140	C₂×C₄×C₇₃	—	4	2	no	—
141	C₂×C₄×C₇₃	—	4	2	no	—
142	C₂×C₄×C₇₃	C₂×D₈	1	2	yes	—
143	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
144	C₂×Dic₇₃	—	4	2	no	—
145	C₂×Dic₇₃	C₂×D₈	1	2	yes	—
146	C₂×Dic₇₃	C₂×D₈	1	1	yes	characteristic
147	D₂₉₂	—	4	2	no	—
148	D₂₉₂	—	4	2	no	—
149	D₂₉₂	—	8	4	no	—
150	C₄×D₇₃	—	8	4	no	—
151	C₄×D₇₃	—	8	4	no	—
152	C₄×D₇₃	#127	1	2	yes	—
153	C₄×D₇₃	C₂×D₈	1	2	yes	—
154	Dic₁₄₆	—	4	2	no	—
155	Dic₁₄₆	—	4	2	no	—
156	Dic₁₄₆	—	8	4	no	—
157	C₈×C₇₃	C₂×D₈	1	2	yes	—
158	C₈×C₇₃	C₂²×D₄	1	1	yes	characteristic
159	C₈×C₇₃	C₂²×D₄	1	1	yes	characteristic
160	#10596831	#127	1	2	yes	—
161	#10596831	C₂×D₈	1	2	yes	—
162	C₂²×C₇₃	C₂²×D₈	1	1	yes	characteristic
163	C₂×D₇₃	—	2	2	no	—
164	C₂×D₇₃	—	8	2	no	—
165	C₂×D₇₃	—	8	4	no	—
166	C₂×D₇₃	#493	1	2	yes	—
167	C₄×C₇₃	—	2	2	no	—
168	C₄×C₇₃	—	8	2	no	—
169	C₄×C₇₃	—	8	2	no	—
170	C₄×C₇₃	C₂×#127	1	1	yes	characteristic
171	C₄×C₇₃	C₂²×D₈	1	1	yes	characteristic
172	Dic₇₃	—	2	2	no	—
173	Dic₇₃	—	8	2	no	—
174	Dic₇₃	#493	1	2	yes	—
175	#33721	—	73	1	no	maximal, Sylow
176	C₂×C₇₃	C₂×#493	1	1	yes	characteristic
177	C₂×C₇₃	#3417	1	2	yes	—
178	D₇₃	—	2	2	no	—
179	D₇₃	—	16	2	no	—
180	C₂×#510	—	73	2	no	—
181	C₂×#508	—	73	1	no	—
182	C₂²×Q₃₂	—	73	1	no	—
183	#3156	—	73	2	no	—
184	#2251	—	73	2	no	—
185	#2251	—	73	2	no	—
186	#2249	—	73	1	no	—
187	#2249	—	73	1	no	—
188	#2247	—	73	1	no	—
189	#2247	D₇₃	1	1	yes	characteristic, complemented
190	#2108	—	73	1	no	—
191	C₇₃	#33721	1	1	yes	characteristic, complemented, Sylow
192	C₂×#137	—	146	2	no	—
193	C₂×#135	—	73	2	no	—
194	C₂²×Q₁₆	—	73	2	no	—
195	#536	—	146	2	no	—
196	#510	—	146	8	no	—
197	#508	—	146	4	no	—
198	C₂×Q₃₂	—	73	2	no	—
199	C₂×Q₃₂	—	73	2	no	—
200	C₂×Q₃₂	—	73	1	no	—
201	C₂×Q₃₂	—	146	4	no	—
202	C₂×Q₃₂	C₂×D₇₃	1	1	yes	characteristic
203	C₂×Q₃₂	C₂×D₇₃	1	1	yes	characteristic
204	C₂×Q₃₂	C₂×D₇₃	1	2	yes	—
205	C₂×SD₃₂	—	73	2	no	—
206	C₂×SD₃₂	—	73	2	no	—
207	C₂×SD₃₂	—	73	2	no	—
208	C₂×D₁₆	—	73	1	no	—
209	C₂×M₃₂	—	73	1	no	—
210	C₂²×C₁₆	—	73	1	no	—
211	#497	—	146	2	no	—
212	#497	—	146	2	no	—
213	#495	—	73	2	no	—
214	#495	—	73	2	no	—
215	#494	—	73	2	no	—
216	#494	C₂×D₇₃	1	2	yes	—
217	#405	—	73	1	no	—
218	#346	—	73	1	no	—
219	#346	—	73	1	no	—
220	C₄×C₁₆	—	73	1	no	—
221	C₄×C₁₆	C₂×D₇₃	1	1	yes	characteristic
222	C₂×#41	—	146	2	no	—
223	C₂²×Q₈	—	146	2	no	—
224	#137	—	292	8	no	—
225	#135	—	146	4	no	—
226	#135	—	146	4	no	—
227	C₂×Q₁₆	—	2	2	no	—
228	C₂×Q₁₆	—	73	2	no	—
229	C₂×Q₁₆	—	146	4	no	—
230	C₂×Q₁₆	—	146	4	no	—
231	C₂×Q₁₆	—	146	2	no	—
232	C₂×Q₁₆	—	146	2	no	—
233	C₂×Q₁₆	C₂²×D₇₃	1	2	yes	—
234	C₂×Q₁₆	C₂²×D₇₃	1	2	yes	—
235	C₂×SD₁₆	—	146	2	no	—
236	C₂×SD₁₆	—	146	2	no	—
237	C₂×SD₁₆	—	146	2	no	—
238	C₂×D₈	—	73	2	no	—
239	C₂×M₁₆	—	73	1	no	—
240	C₂²×C₈	—	73	1	no	—
241	C₄⋊Q₈	—	2	2	no	—
242	C₄⋊Q₈	—	146	2	no	—
243	#124	—	146	2	no	—
244	#124	—	146	2	no	—
245	#123	—	292	4	no	—
246	C₂²⋊Q₈	—	292	4	no	—
247	#117	—	73	1	no	—
248	Q₃₂	—	2	2	no	—
249	Q₃₂	—	2	2	no	—
250	Q₃₂	—	2	4	no	—
251	Q₃₂	—	146	4	no	—
252	Q₃₂	—	146	4	no	—
253	Q₃₂	—	146	2	no	—
254	SD₃₂	—	146	4	no	—
255	SD₃₂	—	146	4	no	—
256	SD₃₂	—	146	4	no	—
257	D₁₆	—	146	2	no	—
258	M₃₂	—	73	4	no	—
259	C₂×C₁₆	—	73	1	no	—
260	C₂×C₁₆	—	73	1	no	—
261	C₂×C₁₆	—	73	4	no	—
262	C₂×C₁₆	C₂²×D₇₃	1	1	yes	characteristic
263	C₂×C₁₆	C₂²×D₇₃	1	1	yes	characteristic
264	#97	—	73	1	no	—
265	#97	—	73	1	no	—
266	C₄×C₈	—	73	1	no	—
267	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
268	#41	—	292	4	no	—
269	#41	—	292	4	no	—
270	C₂×Q₈	—	2	2	no	—
271	C₂×Q₈	—	2	2	no	—
272	C₂×Q₈	—	292	2	no	—
273	C₂×Q₈	—	292	4	no	—
274	C₂×Q₈	—	292	4	no	—
275	C₂×Q₈	—	146	2	no	—
276	C₂×D₄	—	292	2	no	—
277	C₂×D₄	—	146	2	no	—
278	C₂²×C₄	—	292	2	no	—
279	C₂²×C₄	—	292	2	no	—
280	C₂²×C₄	—	73	1	no	—
281	Q₁₆	—	2	2	no	—
282	Q₁₆	—	2	2	no	—
283	Q₁₆	—	2	2	no	—
284	Q₁₆	—	2	2	no	—
285	Q₁₆	—	4	4	no	—
286	Q₁₆	—	292	4	no	—
287	Q₁₆	—	292	4	no	—
288	Q₁₆	—	146	2	no	—
289	Q₁₆	—	146	2	no	—
290	SD₁₆	—	292	4	no	—
291	SD₁₆	—	292	4	no	—
292	SD₁₆	—	292	4	no	—
293	D₈	—	146	2	no	—
294	D₈	—	146	2	no	—
295	M₁₆	—	73	4	no	—
296	C₂×C₈	—	73	1	no	—
297	C₂×C₈	—	73	1	no	—
298	C₂×C₈	—	73	2	no	—
299	C₂×C₈	—	73	2	no	—
300	C₂×C₈	D₄×D₇₃	1	1	yes	characteristic
301	C₂×C₈	C₂³×D₇₃	1	1	yes	characteristic
302	C₄⋊C₄	—	4	4	no	—
303	C₄⋊C₄	—	292	4	no	—
304	C₄⋊C₄	—	73	2	no	—
305	C₂²⋊C₄	—	292	4	no	—
306	C₂²⋊C₄	—	292	4	no	—
307	C₂²⋊C₄	—	73	2	no	—
308	C₄²	—	73	1	no	—
309	C₄²	D₄×D₇₃	1	1	yes	characteristic
310	C₁₆	—	73	2	no	—
311	C₁₆	—	73	2	no	—
312	C₁₆	D₄×D₇₃	1	2	yes	—
313	C₁₆	D₄×D₇₃	1	2	yes	—
314	C₂³	—	292	2	no	—
315	C₂³	—	73	1	no	—
316	Q₈	—	4	2	no	—
317	Q₈	—	4	2	no	—
318	Q₈	—	4	2	no	—
319	Q₈	—	4	2	no	—
320	Q₈	—	292	2	no	—
321	Q₈	—	292	2	no	—
322	Q₈	—	584	4	no	—
323	D₄	—	292	2	no	—
324	D₄	—	292	2	no	—
325	D₄	—	584	4	no	—
326	C₂×C₄	—	4	2	no	—
327	C₂×C₄	—	4	2	no	—
328	C₂×C₄	—	292	2	no	—
329	C₂×C₄	—	584	4	no	—
330	C₂×C₄	—	584	4	no	—
331	C₂×C₄	—	73	1	no	—
332	C₂×C₄	—	73	2	no	—
333	C₂×C₄	—	73	2	no	—
334	C₂×C₄	—	73	2	no	—
335	C₂×C₄	D₈×D₇₃	1	2	yes	—
336	C₂×C₄	C₂×D₄×D₇₃	1	1	yes	characteristic
337	C₈	—	73	2	no	—
338	C₈	—	73	2	no	—
339	C₈	D₈×D₇₃	1	2	yes	—
340	C₈	C₂×D₄×D₇₃	1	1	yes	characteristic
341	C₈	C₂×D₄×D₇₃	1	1	yes	characteristic
342	C₂²	—	584	4	no	—
343	C₂²	—	584	2	no	—
344	C₂²	—	73	2	no	—
345	C₂²	—	146	2	no	—
346	C₂²	C₂×D₈×D₇₃	1	1	yes	characteristic
347	C₄	—	2	2	no	—
348	C₄	—	8	2	no	—
349	C₄	—	8	2	no	—
350	C₄	—	584	2	no	—
351	C₄	—	73	2	no	—
352	C₄	—	146	2	no	—
353	C₄	#127×D₇₃	1	1	yes	characteristic
354	C₄	C₂×D₈×D₇₃	1	1	yes	characteristic
355	C₂	—	1168	2	no	—
356	C₂	—	146	2	no	—
357	C₂	#493×D₇₃	1	1	yes	characteristic
358	C₂	#22623080	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₈×D₇₃
commutator, cosocle	C₂×C₈×C₇₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₇₃
Fitting	C₇₃×#2247	C₂
socle	C₂²×C₇₃	C₂²×D₈
prime radical	C₂×#17989020	C₂
hypercenter	#2247	D₇₃
hypocenter	C₇₃	#33721
LCS	G ▷ C₂×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2247

Complements

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

N	H ≅ G/N	type
1. C₂×#17989029	C₂	semidirect
3. C₂×Q₃₂×D₇₃	C₂	semidirect
4. #22620419	C₂	semidirect
5. #22612200	C₂	semidirect
6. #22612199	C₂	semidirect
7. #22612193	C₂	semidirect
8. #22612192	C₂	semidirect
9. #22612182	C₂	semidirect
10. #22611526	C₂	semidirect
11. C₇₃×#2247	C₂	semidirect
189. #2247	D₇₃	semidirect
191. C₇₃	#33721	semidirect

Maximal subgroups

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

subgroup	index
C₂×#17989029	2
C₂×#17989020	2
C₂×Q₃₂×D₇₃	2
#22620419	2
#22612200	2
#22612199	2
#22612193	2
#22612192	2
#22612182	2
#22611526	2
C₇₃×#2247	2
#33721	73