Subgroups — #33241387

Order 18688.

834 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17989020	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#17989018	C₂	1	2	yes	maximal, complemented
3	C₂×#17989016	C₂	1	1	yes	maximal, characteristic, complemented
4	#22620356	C₂	1	2	yes	maximal, complemented
5	#22612192	C₂	1	1	yes	maximal, characteristic, complemented
6	#22612190	C₂	1	2	yes	maximal, complemented
7	#22612185	C₂	1	2	yes	maximal, complemented
8	#22612182	C₂	1	1	yes	maximal, characteristic, complemented
9	#22612179	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×C₁₆×D₇₃	C₂	1	1	yes	maximal, characteristic, complemented
11	C₇₃×#2249	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#14556614	C₂²	1	2	yes	—
13	C₂×#14556612	—	2	2	no	—
14	C₂×#14556610	C₂²	1	2	yes	—
15	#17989136	—	2	2	no	—
16	#17989020	—	2	4	no	—
17	#17989018	—	2	8	no	—
18	#17989016	—	2	4	no	—
19	C₂×Dic₅₈₄	C₂²	1	1	yes	characteristic
20	C₂×#14556537	C₂²	1	2	yes	—
21	C₂×#14556536	C₂²	1	2	yes	—
22	C₂×#14556535	C₂²	1	2	yes	—
23	C₂×#14556534	C₂²	1	2	yes	—
24	C₂×D₁₁₆₈	C₂²	1	1	yes	characteristic
25	C₂×#14556532	C₂²	1	2	yes	—
26	C₂×C₁₆×D₇₃	C₂²	1	1	yes	characteristic
27	C₂×C₁₆×D₇₃	C₂²	1	1	yes	characteristic
28	#17988971	C₂²	1	2	yes	—
29	#17988969	—	2	2	no	—
30	#17988965	C₂²	1	2	yes	—
31	#17988962	C₂²	1	2	yes	—
32	#17988958	—	2	2	no	—
33	C₄×C₈×D₇₃	C₂²	1	1	yes	characteristic
34	C₄×#11924003	C₂²	1	1	yes	characteristic
35	C₁₆×Dic₇₃	C₂²	1	1	yes	characteristic
36	C₁₆×Dic₇₃	C₂²	1	1	yes	characteristic
37	C₂×Q₃₂×C₇₃	C₂²	1	1	yes	characteristic
38	C₂×SD₃₂×C₇₃	C₂²	1	2	yes	—
39	C₂×D₁₆×C₇₃	C₂²	1	1	yes	characteristic
40	#495×C₇₃	C₂²	1	2	yes	—
41	C₄×C₁₆×C₇₃	C₂²	1	1	yes	characteristic
42	C₂×#11924029	—	2	2	no	—
43	C₂×#11924027	—	2	2	no	—
44	#14556614	—	2	4	no	—
45	#14556614	—	2	4	no	—
46	#14556612	—	4	8	no	—
47	#14556610	—	2	4	no	—
48	#14556610	—	2	4	no	—
49	C₂×Dic₂₉₂	C₂³	1	2	yes	—
50	C₂×#11924018	—	2	2	no	—
51	C₂×#11924017	—	2	2	no	—
52	C₂×#11924016	—	2	2	no	—
53	C₂×#11924015	—	2	2	no	—
54	C₂×D₅₈₄	C₂³	1	2	yes	—
55	C₂×#11924013	—	2	2	no	—
56	C₂×C₈×D₇₃	D₄	1	1	yes	characteristic
57	C₂×C₈×D₇₃	C₂³	1	1	yes	characteristic
58	#14556589	—	2	2	no	—
59	#14556587	—	2	2	no	—
60	#14556579	—	2	2	no	—
61	#14556564	—	4	4	no	—
62	#14556551	—	4	4	no	—
63	C₄²×D₇₃	D₄	1	1	yes	characteristic
64	Dic₅₈₄	—	2	2	no	—
65	#14556537	—	2	4	no	—
66	#14556536	—	2	4	no	—
67	#14556535	—	2	4	no	—
68	#14556534	—	2	4	no	—
69	D₁₁₆₈	—	2	2	no	—
70	#14556532	—	2	4	no	—
71	C₂×#11924003	C₂³	1	1	yes	characteristic
72	C₂×#11924003	C₂³	1	1	yes	characteristic
73	C₁₆×D₇₃	D₄	1	4	yes	complemented
74	C₁₆×D₇₃	D₄	1	4	yes	complemented
75	C₄×#10596831	D₄	1	1	yes	characteristic
76	C₈×Dic₇₃	D₄	1	1	yes	characteristic
77	C₈×Dic₇₃	C₂³	1	1	yes	characteristic
78	C₂×Q₁₆×C₇₃	C₂³	1	2	yes	—
79	C₂×SD₁₆×C₇₃	—	2	2	no	—
80	C₂×D₈×C₇₃	C₂³	1	2	yes	—
81	#124×C₇₃	—	2	2	no	—
82	Q₃₂×C₇₃	—	2	2	no	—
83	SD₃₂×C₇₃	—	2	4	no	—
84	D₁₆×C₇₃	—	2	2	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	#11924027	—	4	4	no	—
91	#11924027	—	4	4	no	—
92	C₂×Dic₁₄₆	—	2	2	no	—
93	C₂×D₂₉₂	—	2	2	no	—
94	C₂×D₂₉₂	—	4	2	no	—
95	C₂×C₂²⋊D₇₃	—	4	2	no	—
96	C₂×C₂²⋊D₇₃	—	4	4	no	—
97	C₂²×Dic₇₃	—	4	2	no	—
98	C₂×C₄×D₇₃	—	4	2	no	—
99	C₂×C₄×D₇₃	D₈	1	2	yes	—
100	C₂×C₄×D₇₃	C₂×D₄	1	1	yes	characteristic
101	Dic₂₉₂	—	2	2	no	—
102	Dic₂₉₂	—	2	2	no	—
103	#11924018	—	4	4	no	—
104	#11924017	—	4	4	no	—
105	#11924016	—	4	4	no	—
106	#11924015	—	4	4	no	—
107	D₅₈₄	—	2	2	no	—
108	D₅₈₄	—	2	2	no	—
109	#11924013	—	4	4	no	—
110	C₂×#10596831	C₂×D₄	1	1	yes	characteristic
111	C₂×#10596831	C₂×D₄	1	1	yes	characteristic
112	C₈×D₇₃	D₈	1	4	yes	—
113	C₈×D₇₃	C₂×D₄	1	2	yes	—
114	C₈×D₇₃	C₂×D₄	1	2	yes	—
115	#11924007	—	4	4	no	—
116	#11924005	—	4	4	no	—
117	C₄×Dic₇₃	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×D₄×C₇₃	—	2	2	no	—
123	Q₁₆×C₇₃	—	2	2	no	—
124	Q₁₆×C₇₃	—	2	2	no	—
125	SD₁₆×C₇₃	—	4	4	no	—
126	D₈×C₇₃	—	2	2	no	—
127	D₈×C₇₃	—	2	2	no	—
128	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
129	C₂×C₈×C₇₃	C₂⁴	1	1	yes	characteristic
130	C₂²⋊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	C₂³×C₇₃	—	4	2	no	—
135	C₂²×D₇₃	—	4	2	no	—
136	C₂²×D₇₃	C₂×D₈	1	1	yes	characteristic
137	Q₈×C₇₃	—	4	2	no	—
138	Q₈×C₇₃	—	4	2	no	—
139	D₄×C₇₃	—	4	2	no	—
140	D₄×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	C₂²⋊D₇₃	—	8	4	no	—
145	C₂²⋊D₇₃	—	8	8	no	—
146	C₂×Dic₇₃	—	4	2	no	—
147	C₂×Dic₇₃	—	8	4	no	—
148	C₂×Dic₇₃	C₂×D₈	1	1	yes	characteristic
149	C₂×Dic₇₃	C₂×D₈	1	2	yes	—
150	D₂₉₂	—	4	2	no	—
151	D₂₉₂	—	4	2	no	—
152	D₂₉₂	—	8	4	no	—
153	C₄×D₇₃	—	2	4	no	—
154	C₄×D₇₃	—	8	4	no	—
155	C₄×D₇₃	#127	1	2	yes	—
156	C₄×D₇₃	C₂×D₈	1	2	yes	—
157	Dic₁₄₆	—	4	2	no	—
158	Dic₁₄₆	—	4	2	no	—
159	C₈×C₇₃	C₂×D₈	1	2	yes	—
160	C₈×C₇₃	C₂²×D₄	1	1	yes	characteristic
161	C₈×C₇₃	C₂²×D₄	1	1	yes	characteristic
162	#10596831	#127	1	2	yes	—
163	#10596831	C₂×D₈	1	2	yes	—
164	C₂²×C₇₃	—	8	2	no	—
165	C₂²×C₇₃	—	8	4	no	—
166	C₂²×C₇₃	C₂²×D₈	1	1	yes	characteristic
167	C₂×D₇₃	—	2	2	no	—
168	C₂×D₇₃	—	8	2	no	—
169	C₂×D₇₃	—	8	4	no	—
170	C₂×D₇₃	#493	1	2	yes	—
171	C₄×C₇₃	—	2	2	no	—
172	C₄×C₇₃	—	8	2	no	—
173	C₄×C₇₃	C₂×#127	1	1	yes	characteristic
174	C₄×C₇₃	C₂²×D₈	1	1	yes	characteristic
175	Dic₇₃	—	8	2	no	—
176	Dic₇₃	#493	1	2	yes	—
177	Dic₇₃	#508	1	4	yes	complemented
178	#33713	—	73	1	no	maximal, Sylow
179	C₂×C₇₃	—	16	2	no	—
180	C₂×C₇₃	C₂×#493	1	1	yes	characteristic
181	C₂×C₇₃	C₂×#508	1	2	yes	—
182	D₇₃	—	2	2	no	—
183	D₇₃	—	16	2	no	—
184	C₂×#508	—	73	1	no	—
185	C₂×#508	—	73	2	no	—
186	C₂×#508	—	73	1	no	—
187	#3147	—	73	2	no	—
188	#2249	—	73	2	no	—
189	#2249	—	73	1	no	—
190	#2249	D₇₃	1	1	yes	characteristic, complemented
191	#2248	—	73	2	no	—
192	#2247	—	73	1	no	—
193	#2246	—	73	1	no	—
194	C₂×C₄×C₁₆	—	73	1	no	—
195	C₇₃	#33713	1	1	yes	characteristic, complemented, Sylow
196	C₂×#135	—	73	2	no	—
197	C₂×#135	—	73	2	no	—
198	C₂×#135	—	146	2	no	—
199	#532	—	146	2	no	—
200	#508	—	146	4	no	—
201	#508	—	146	8	no	—
202	#508	—	146	4	no	—
203	C₂×Q₃₂	—	73	2	no	—
204	C₂×Q₃₂	—	73	1	no	—
205	C₂×Q₃₂	C₂×D₇₃	1	1	yes	characteristic
206	C₂×SD₃₂	—	73	2	no	—
207	C₂×SD₃₂	—	73	2	no	—
208	C₂×SD₃₂	—	73	2	no	—
209	C₂×SD₃₂	C₂×D₇₃	1	2	yes	—
210	C₂×D₁₆	—	73	2	no	—
211	C₂×D₁₆	—	73	1	no	—
212	C₂×D₁₆	C₂×D₇₃	1	1	yes	characteristic
213	C₂²×C₁₆	—	73	1	no	—
214	C₂²×C₁₆	—	73	1	no	—
215	#495	—	73	2	no	—
216	#495	—	146	2	no	—
217	#495	C₂×D₇₃	1	2	yes	—
218	#494	—	73	2	no	—
219	#493	—	73	2	no	—
220	#492	—	146	2	no	—
221	C₂×C₄×C₈	—	73	1	no	—
222	C₄×C₁₆	—	73	1	no	—
223	C₄×C₁₆	—	73	1	no	—
224	C₄×C₁₆	—	73	1	no	—
225	C₄×C₁₆	C₂×D₇₃	1	1	yes	characteristic
226	C₂×#41	—	146	2	no	—
227	C₂×#41	—	146	2	no	—
228	#135	—	292	8	no	—
229	#135	—	146	4	no	—
230	#135	—	146	4	no	—
231	#135	—	146	4	no	—
232	#135	—	146	4	no	—
233	C₂×Q₁₆	—	73	2	no	—
234	C₂×Q₁₆	—	146	2	no	—
235	C₂×Q₁₆	C₂²×D₇₃	1	2	yes	—
236	C₂×SD₁₆	—	2	2	no	—
237	C₂×SD₁₆	—	146	2	no	—
238	C₂×SD₁₆	—	146	2	no	—
239	C₂×SD₁₆	—	146	2	no	—
240	C₂×D₈	—	73	2	no	—
241	C₂×D₈	—	146	2	no	—
242	C₂×D₈	C₂²×D₇₃	1	2	yes	—
243	C₂²×C₈	—	73	1	no	—
244	C₂²×C₈	—	73	1	no	—
245	C₄⋊Q₈	—	146	2	no	—
246	#127	—	146	2	no	—
247	#124	—	2	2	no	—
248	#124	—	146	2	no	—
249	#121	—	292	4	no	—
250	C₄×D₄	—	292	4	no	—
251	C₂×C₄²	—	73	1	no	—
252	Q₃₂	—	2	2	no	—
253	Q₃₂	—	146	4	no	—
254	Q₃₂	—	146	2	no	—
255	SD₃₂	—	2	4	no	—
256	SD₃₂	—	146	4	no	—
257	SD₃₂	—	146	4	no	—
258	SD₃₂	—	146	4	no	—
259	D₁₆	—	2	2	no	—
260	D₁₆	—	146	4	no	—
261	D₁₆	—	146	2	no	—
262	C₂×C₁₆	—	73	1	no	—
263	C₂×C₁₆	—	73	1	no	—
264	C₂×C₁₆	—	73	4	no	—
265	C₂×C₁₆	—	73	4	no	—
266	C₂×C₁₆	C₂²×D₇₃	1	1	yes	characteristic
267	C₂×C₁₆	C₂²×D₇₃	1	1	yes	characteristic
268	C₄×C₈	—	73	1	no	—
269	C₄×C₈	—	73	1	no	—
270	C₄×C₈	—	73	1	no	—
271	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
272	#41	—	292	4	no	—
273	#41	—	292	4	no	—
274	#41	—	292	4	no	—
275	#41	—	292	4	no	—
276	C₂×Q₈	—	2	2	no	—
277	C₂×Q₈	—	146	2	no	—
278	C₂×D₄	—	2	2	no	—
279	C₂×D₄	—	292	2	no	—
280	C₂×D₄	—	292	2	no	—
281	C₂×D₄	—	292	4	no	—
282	C₂×D₄	—	146	2	no	—
283	C₂²×C₄	—	292	2	no	—
284	C₂²×C₄	—	292	2	no	—
285	C₂²×C₄	—	73	1	no	—
286	C₂²×C₄	—	73	2	no	—
287	Q₁₆	—	2	2	no	—
288	Q₁₆	—	2	2	no	—
289	Q₁₆	—	292	4	no	—
290	Q₁₆	—	146	2	no	—
291	Q₁₆	—	146	2	no	—
292	SD₁₆	—	4	4	no	—
293	SD₁₆	—	292	4	no	—
294	SD₁₆	—	292	4	no	—
295	SD₁₆	—	292	4	no	—
296	D₈	—	2	2	no	—
297	D₈	—	2	2	no	—
298	D₈	—	292	4	no	—
299	D₈	—	146	2	no	—
300	D₈	—	146	2	no	—
301	C₂×C₈	—	73	1	no	—
302	C₂×C₈	—	73	1	no	—
303	C₂×C₈	—	73	4	no	—
304	C₂×C₈	—	73	2	no	—
305	C₂×C₈	—	73	2	no	—
306	C₂×C₈	D₄×D₇₃	1	1	yes	characteristic
307	C₂×C₈	C₂³×D₇₃	1	1	yes	characteristic
308	C₄⋊C₄	—	292	4	no	—
309	C₂²⋊C₄	—	4	4	no	—
310	C₂²⋊C₄	—	292	4	no	—
311	C₄²	—	73	2	no	—
312	C₄²	—	73	1	no	—
313	C₄²	D₄×D₇₃	1	1	yes	characteristic
314	C₁₆	—	73	2	no	—
315	C₁₆	—	73	2	no	—
316	C₁₆	D₄×D₇₃	1	2	yes	—
317	C₁₆	D₄×D₇₃	1	2	yes	—
318	C₂³	—	4	2	no	—
319	C₂³	—	292	2	no	—
320	C₂³	—	73	1	no	—
321	Q₈	—	4	2	no	—
322	Q₈	—	4	2	no	—
323	Q₈	—	292	2	no	—
324	Q₈	—	292	2	no	—
325	D₄	—	4	2	no	—
326	D₄	—	4	2	no	—
327	D₄	—	292	2	no	—
328	D₄	—	292	2	no	—
329	D₄	—	584	4	no	—
330	D₄	—	584	4	no	—
331	D₄	—	584	8	no	—
332	C₂×C₄	—	4	2	no	—
333	C₂×C₄	—	292	2	no	—
334	C₂×C₄	—	584	4	no	—
335	C₂×C₄	—	584	4	no	—
336	C₂×C₄	—	73	2	no	—
337	C₂×C₄	—	73	1	no	—
338	C₂×C₄	—	73	2	no	—
339	C₂×C₄	—	73	2	no	—
340	C₂×C₄	—	146	4	no	—
341	C₂×C₄	D₈×D₇₃	1	2	yes	—
342	C₂×C₄	C₂×D₄×D₇₃	1	1	yes	characteristic
343	C₈	—	73	2	no	—
344	C₈	—	73	2	no	—
345	C₈	D₈×D₇₃	1	2	yes	—
346	C₈	C₂×D₄×D₇₃	1	1	yes	characteristic
347	C₈	C₂×D₄×D₇₃	1	1	yes	characteristic
348	C₂²	—	8	4	no	—
349	C₂²	—	8	2	no	—
350	C₂²	—	584	4	no	—
351	C₂²	—	584	2	no	—
352	C₂²	—	73	2	no	—
353	C₂²	—	146	2	no	—
354	C₂²	C₂×D₈×D₇₃	1	1	yes	characteristic
355	C₄	—	2	2	no	—
356	C₄	—	8	2	no	—
357	C₄	—	584	2	no	—
358	C₄	—	73	4	no	—
359	C₄	—	73	2	no	—
360	C₄	#127×D₇₃	1	1	yes	characteristic
361	C₄	C₂×D₈×D₇₃	1	1	yes	characteristic
362	C₂	—	16	2	no	—
363	C₂	—	1168	2	no	—
364	C₂	—	146	2	no	—
365	C₂	#493×D₇₃	1	1	yes	characteristic
366	C₂	#508×D₇₃	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₈×D₇₃
commutator, cosocle	C₂×C₈×C₇₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₇₃
Fitting	C₇₃×#2249	C₂
socle	C₂²×C₇₃	C₂²×D₈
hypercenter	#2249	D₇₃
hypocenter	C₇₃	#33713
LCS	G ▷ C₂×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2249

Complements

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

N	H ≅ G/N	type
1. C₂×#17989020	C₂	semidirect
2. C₂×#17989018	C₂	semidirect
3. C₂×#17989016	C₂	semidirect
4. #22620356	C₂	semidirect
5. #22612192	C₂	semidirect
6. #22612190	C₂	semidirect
7. #22612185	C₂	semidirect
8. #22612182	C₂	semidirect
9. #22612179	C₂	semidirect
10. C₄×C₁₆×D₇₃	C₂	semidirect
11. C₇₃×#2249	C₂	semidirect
73. C₁₆×D₇₃	D₄	semidirect
74. C₁₆×D₇₃	D₄	semidirect
177. Dic₇₃	#508	semidirect
190. #2249	D₇₃	semidirect
195. C₇₃	#33713	semidirect

Maximal subgroups

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

subgroup	index
C₂×#17989020	2
C₂×#17989018	2
C₂×#17989016	2
#22620356	2
#22612192	2
#22612190	2
#22612185	2
#22612182	2
#22612179	2
C₄×C₁₆×D₇₃	2
C₇₃×#2249	2
#33713	73