Subgroups — #33069278

Order 18688.

658 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22620876	C₂	1	1	yes	maximal, characteristic, complemented
2	#22620869	C₂	1	1	yes	maximal, characteristic
3	C₂×#17988934	C₂	1	1	yes	maximal, characteristic
4	#22609499	C₂	1	4	yes	maximal, complemented
5	#22609375	C₂	1	4	yes	maximal, complemented
6	#22608679	C₂	1	1	yes	maximal, characteristic, complemented
7	#22608617	C₂	1	1	yes	maximal, characteristic, complemented
8	#22608600	C₂	1	1	yes	maximal, characteristic
9	C₇₃×#2960	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#14556591	C₂²	1	1	yes	characteristic
11	C₂×#14556588	C₂²	1	1	yes	characteristic
12	#17988966	—	2	1	no	—
13	#17988961	—	2	1	no	—
14	#17988939	—	2	1	no	—
15	#17988935	—	2	1	no	—
16	#17988934	—	2	1	no	—
17	#17988934	—	2	1	no	—
18	#17988934	C₂²	1	2	yes	—
19	#17988934	C₂²	1	2	yes	—
20	#17988797	C₂²	1	4	yes	complemented
21	#17988787	C₂²	1	4	yes	—
22	#17988742	—	2	2	no	—
23	#17988737	—	2	2	no	—
24	#17988640	C₂²	1	1	yes	characteristic
25	#17988640	C₂²	1	1	yes	characteristic
26	C₂×#14556513	—	2	1	no	—
27	C₂×#14556513	—	2	1	no	—
28	#17988340	C₂²	1	2	yes	complemented
29	#17988338	C₂²	1	2	yes	complemented
30	#17988332	C₂²	1	4	yes	complemented
31	#17988330	C₂²	1	4	yes	—
32	#533×C₇₃	C₂²	1	1	yes	characteristic
33	#446×C₇₃	—	2	1	no	—
34	#446×C₇₃	C₂²	1	2	yes	complemented
35	C₈×Q₈×C₇₃	—	2	1	no	—
36	C₈×Q₈×C₇₃	C₂²	1	2	yes	complemented
37	#424×C₇₃	—	2	1	no	—
38	#423×C₇₃	C₂²	1	1	yes	characteristic
39	#405×C₇₃	—	2	1	no	—
40	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
41	C₂²×Dic₁₄₆	—	2	1	no	—
42	C₂²×D₂₉₂	—	2	1	no	—
43	C₂²×D₂₉₂	—	2	1	no	—
44	C₂²×D₂₉₂	—	2	1	no	—
45	C₂×#11924018	—	2	4	no	—
46	C₂×#11924015	—	2	4	no	—
47	#14556591	—	2	1	no	—
48	#14556591	—	2	1	no	—
49	#14556591	—	2	1	no	—
50	#14556591	C₂³	1	2	yes	—
51	#14556588	—	2	1	no	—
52	#14556588	—	2	1	no	—
53	#14556588	—	2	1	no	—
54	#14556588	C₂³	1	2	yes	—
55	C₂×C₄⋊Dic₇₃	—	2	1	no	—
56	C₂×C₄⋊Dic₇₃	—	2	1	no	—
57	C₄⋊#10596831	C₂³	1	2	yes	—
58	C₄⋊#10596831	C₂³	1	2	yes	—
59	#14556514	—	2	4	no	—
60	#14556513	—	2	2	no	—
61	#14556513	—	2	2	no	—
62	#14556513	—	2	2	no	—
63	#14556513	—	2	2	no	—
64	#14556511	—	2	4	no	—
65	C₂²⋊#10596831	—	2	1	no	—
66	C₂²⋊#10596831	—	2	1	no	—
67	C₄×#10596831	—	2	1	no	—
68	C₄×#10596831	—	2	1	no	—
69	C₂×M₁₆×C₇₃	—	2	1	no	—
70	C₂²×C₈×C₇₃	D₄	1	1	yes	characteristic
71	C₂²×C₈×C₇₃	D₄	1	1	yes	characteristic
72	C₄⋊Q₈×C₇₃	D₄	1	1	yes	characteristic
73	C₄⋊Q₈×C₇₃	D₄	1	1	yes	characteristic
74	#125×C₇₃	—	2	1	no	—
75	C₂²⋊Q₈×C₇₃	—	2	1	no	—
76	C₂²⋊Q₈×C₇₃	D₄	1	2	yes	—
77	C₄×Q₈×C₇₃	—	2	1	no	—
78	C₄×Q₈×C₇₃	C₂³	1	2	yes	—
79	#117×C₇₃	—	2	1	no	—
80	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
81	C₄⋊C₈×C₇₃	—	2	2	no	—
82	C₄⋊C₈×C₇₃	—	2	2	no	—
83	C₄⋊C₈×C₇₃	—	2	1	no	—
84	C₄⋊C₈×C₇₃	C₂³	1	2	yes	—
85	#97×C₇₃	—	2	2	no	—
86	C₄×C₈×C₇₃	—	2	2	no	—
87	C₄×C₈×C₇₃	D₄	1	1	yes	characteristic
88	C₄×C₈×C₇₃	D₄	1	1	yes	characteristic
89	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
90	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
91	C₂³×D₇₃	—	4	1	no	—
92	C₂×Dic₁₄₆	—	2	2	no	—
93	C₂×Dic₁₄₆	—	4	1	no	—
94	C₂×Dic₁₄₆	—	8	2	no	—
95	C₂×D₂₉₂	—	2	2	no	—
96	C₂×D₂₉₂	—	2	2	no	—
97	C₂×D₂₉₂	—	2	2	no	—
98	C₂×D₂₉₂	—	4	2	no	—
99	C₂×D₂₉₂	—	4	2	no	—
100	C₂×D₂₉₂	—	4	1	no	—
101	C₂×D₂₉₂	—	4	1	no	—
102	C₂×D₂₉₂	—	4	1	no	—
103	C₂×D₂₉₂	—	8	2	no	—
104	C₂×D₂₉₂	—	8	2	no	—
105	C₂²×Dic₇₃	—	4	1	no	—
106	#11924018	—	8	4	no	—
107	#11924015	—	8	4	no	—
108	C₂×#10596831	—	2	2	no	—
109	C₂×#10596831	—	2	2	no	—
110	C₄⋊Dic₇₃	—	2	2	no	—
111	C₄⋊Dic₇₃	—	2	2	no	—
112	C₄⋊Dic₇₃	—	4	1	no	—
113	C₄⋊Dic₇₃	—	4	1	no	—
114	C₂×Q₈×C₇₃	—	2	1	no	—
115	C₂×Q₈×C₇₃	C₂×D₄	1	2	yes	—
116	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
117	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
118	C₂²×C₄×C₇₃	#41	1	1	yes	characteristic
119	M₁₆×C₇₃	—	4	2	no	—
120	C₂×C₈×C₇₃	—	2	2	no	—
121	C₂×C₈×C₇₃	—	2	2	no	—
122	C₂×C₈×C₇₃	—	2	2	no	—
123	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
124	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
125	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
126	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
127	C₄⋊C₄×C₇₃	—	2	1	no	—
128	C₄⋊C₄×C₇₃	—	2	1	no	—
129	C₄⋊C₄×C₇₃	—	2	1	no	—
130	C₄⋊C₄×C₇₃	—	2	2	no	—
131	C₄⋊C₄×C₇₃	—	2	2	no	—
132	C₄⋊C₄×C₇₃	C₂×D₄	1	2	yes	—
133	C₂²⋊C₄×C₇₃	—	2	2	no	—
134	C₄²×C₇₃	—	2	2	no	—
135	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
136	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
137	C₄²×C₇₃	#41	1	1	yes	characteristic
138	C₄²×C₇₃	C₂⁴	1	1	yes	characteristic
139	C₂³×C₇₃	#121	1	1	yes	characteristic
140	C₂²×D₇₃	—	4	2	no	—
141	C₂²×D₇₃	—	8	2	no	—
142	C₂²×D₇₃	—	8	2	no	—
143	C₂²×D₇₃	—	8	2	no	—
144	Q₈×C₇₃	—	2	4	no	—
145	Q₈×C₇₃	—	4	2	no	—
146	C₂×C₄×C₇₃	—	2	2	no	—
147	C₂×C₄×C₇₃	—	2	2	no	—
148	C₂×C₄×C₇₃	—	2	2	no	—
149	C₂×C₄×C₇₃	—	2	2	no	—
150	C₂×C₄×C₇₃	#121	1	1	yes	characteristic
151	C₂×C₄×C₇₃	#121	1	1	yes	characteristic
152	C₂×C₄×C₇₃	#121	1	1	yes	characteristic
153	C₂×C₄×C₇₃	#135	1	2	yes	—
154	C₂×C₄×C₇₃	#137	1	2	yes	—
155	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
156	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
157	C₂×C₄×C₇₃	C₂×#41	1	1	yes	characteristic
158	C₂×Dic₇₃	—	4	2	no	—
159	C₂×Dic₇₃	—	8	2	no	—
160	D₂₉₂	—	4	2	no	—
161	D₂₉₂	—	4	2	no	—
162	D₂₉₂	—	4	4	no	—
163	D₂₉₂	—	8	2	no	—
164	D₂₉₂	—	8	2	no	—
165	D₂₉₂	—	8	2	no	—
166	D₂₉₂	—	8	2	no	—
167	Dic₁₄₆	—	4	4	no	—
168	Dic₁₄₆	—	8	2	no	—
169	C₈×C₇₃	—	2	1	no	—
170	C₈×C₇₃	—	2	1	no	—
171	C₈×C₇₃	—	2	1	no	—
172	C₈×C₇₃	—	2	1	no	—
173	C₈×C₇₃	—	4	2	no	—
174	#10596831	—	4	2	no	—
175	#10596831	—	4	2	no	—
176	C₂²×C₇₃	—	2	1	no	—
177	C₂²×C₇₃	—	2	1	no	—
178	C₂²×C₇₃	#464	1	2	yes	—
179	C₂²×C₇₃	C₂×#121	1	1	yes	characteristic
180	C₂×D₇₃	—	8	2	no	—
181	C₂×D₇₃	—	8	2	no	—
182	C₂×D₇₃	—	8	2	no	—
183	C₂×D₇₃	—	16	4	no	—
184	C₄×C₇₃	—	2	1	no	—
185	C₄×C₇₃	—	2	1	no	—
186	C₄×C₇₃	—	4	2	no	—
187	C₄×C₇₃	—	4	2	no	—
188	C₄×C₇₃	#464	1	2	yes	—
189	C₄×C₇₃	#471	1	2	yes	—
190	C₄×C₇₃	C₂×#135	1	1	yes	characteristic
191	C₄×C₇₃	C₂×#137	1	1	yes	characteristic
192	C₄×C₇₃	#576	1	2	yes	—
193	Dic₇₃	—	8	2	no	—
194	#19214	—	73	1	no	maximal, Sylow
195	C₂×C₇₃	—	2	2	no	—
196	C₂×C₇₃	C₂×#464	1	1	yes	characteristic
197	C₂×C₇₃	#3058	1	1	yes	characteristic
198	C₂×C₇₃	C₂×#471	1	1	yes	characteristic
199	D₇₃	—	16	2	no	—
200	#3209	—	73	1	no	—
201	#3208	—	73	1	no	—
202	C₂×#486	—	73	1	no	—
203	#2960	D₇₃	1	1	yes	characteristic, complemented
204	#1702	—	73	4	no	—
205	#1682	—	73	4	no	—
206	#1564	—	73	1	no	—
207	#1549	—	73	1	no	—
208	#1545	—	73	1	no	—
209	C₇₃	#19214	1	1	yes	characteristic, complemented, Sylow
210	#533	C₂×D₇₃	1	1	yes	characteristic
211	C₂×C₄⋊Q₈	—	73	1	no	—
212	C₂×#127	—	73	1	no	—
213	#494	—	146	1	no	—
214	#492	—	146	1	no	—
215	#487	—	146	1	no	—
216	#486	—	73	2	no	—
217	#486	—	73	2	no	—
218	#486	—	146	1	no	—
219	#486	—	146	1	no	—
220	#486	—	146	1	no	—
221	#462	—	73	4	no	—
222	#460	—	73	4	no	—
223	#451	—	146	2	no	—
224	#450	—	146	2	no	—
225	#446	—	2	1	no	—
226	#446	C₂×D₇₃	1	2	yes	complemented
227	C₈×Q₈	—	2	1	no	—
228	C₈×Q₈	C₂×D₇₃	1	2	yes	complemented
229	#431	—	73	1	no	—
230	#431	—	73	1	no	—
231	#424	—	2	1	no	—
232	#423	C₂×D₇₃	1	1	yes	characteristic
233	C₂×#102	—	146	1	no	—
234	C₂×#102	—	146	1	no	—
235	#405	—	2	1	no	—
236	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
237	#335	—	73	2	no	—
238	#334	—	73	2	no	—
239	#332	—	73	4	no	—
240	#331	—	73	4	no	—
241	C₂²×Q₈	—	146	1	no	—
242	C₂²×D₄	—	146	1	no	—
243	C₂²×D₄	—	146	1	no	—
244	C₂²×D₄	—	146	1	no	—
245	C₂×Q₁₆	—	146	4	no	—
246	C₂×SD₁₆	—	146	4	no	—
247	C₂×M₁₆	—	2	1	no	—
248	C₂²×C₈	C₂²⋊D₇₃	1	1	yes	characteristic
249	C₂²×C₈	C₂²⋊D₇₃	1	1	yes	characteristic
250	C₄⋊Q₈	—	73	2	no	—
251	C₄⋊Q₈	—	146	1	no	—
252	C₄⋊Q₈	—	146	1	no	—
253	C₄⋊Q₈	—	146	1	no	—
254	C₄⋊Q₈	D₂₉₂	1	1	yes	characteristic
255	C₄⋊Q₈	D₂₉₂	1	1	yes	characteristic
256	#127	—	73	2	no	—
257	#127	—	146	1	no	—
258	#127	—	146	1	no	—
259	#127	—	146	1	no	—
260	#125	—	2	1	no	—
261	C₂²⋊Q₈	—	2	1	no	—
262	C₂²⋊Q₈	D₂₉₂	1	2	yes	—
263	C₄×Q₈	—	2	1	no	—
264	C₄×Q₈	C₂²×D₇₃	1	2	yes	—
265	#117	—	2	1	no	—
266	C₂×C₄⋊C₄	—	146	1	no	—
267	C₂×C₄⋊C₄	—	146	1	no	—
268	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
269	C₄⋊C₈	—	2	2	no	—
270	C₄⋊C₈	—	2	2	no	—
271	C₄⋊C₈	—	2	1	no	—
272	C₄⋊C₈	—	73	2	no	—
273	C₄⋊C₈	—	73	2	no	—
274	C₄⋊C₈	C₂²×D₇₃	1	2	yes	—
275	#103	—	146	4	no	—
276	#102	—	146	2	no	—
277	#102	—	146	2	no	—
278	#102	—	146	2	no	—
279	#102	—	146	2	no	—
280	#102	—	146	4	no	—
281	C₂²⋊C₈	—	146	1	no	—
282	C₂²⋊C₈	—	146	1	no	—
283	#97	—	2	2	no	—
284	C₄×C₈	—	2	2	no	—
285	C₄×C₈	—	146	1	no	—
286	C₄×C₈	—	146	1	no	—
287	C₄×C₈	C₂²⋊D₇₃	1	1	yes	characteristic
288	C₄×C₈	C₂²⋊D₇₃	1	1	yes	characteristic
289	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
290	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
291	C₂⁴	—	292	1	no	—
292	C₂×Q₈	—	2	1	no	—
293	C₂×Q₈	—	292	1	no	—
294	C₂×Q₈	—	584	2	no	—
295	C₂×Q₈	—	146	2	no	—
296	C₂×Q₈	C₂×D₂₉₂	1	2	yes	—
297	C₂×D₄	—	292	1	no	—
298	C₂×D₄	—	292	1	no	—
299	C₂×D₄	—	292	1	no	—
300	C₂×D₄	—	292	2	no	—
301	C₂×D₄	—	292	2	no	—
302	C₂×D₄	—	584	2	no	—
303	C₂×D₄	—	584	2	no	—
304	C₂×D₄	—	146	2	no	—
305	C₂×D₄	—	146	2	no	—
306	C₂×D₄	—	146	2	no	—
307	C₂²×C₄	—	292	1	no	—
308	C₂²×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
309	C₂²×C₄	C₂×D₂₉₂	1	1	yes	characteristic
310	C₂²×C₄	#11924028	1	1	yes	characteristic
311	Q₁₆	—	584	4	no	—
312	SD₁₆	—	584	4	no	—
313	M₁₆	—	4	2	no	—
314	C₂×C₈	—	2	2	no	—
315	C₂×C₈	—	2	2	no	—
316	C₂×C₈	—	2	2	no	—
317	C₂×C₈	—	146	2	no	—
318	C₂×C₈	—	146	2	no	—
319	C₂×C₈	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
320	C₂×C₈	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
321	C₂×C₈	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
322	C₂×C₈	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
323	C₄⋊C₄	—	2	1	no	—
324	C₄⋊C₄	—	2	1	no	—
325	C₄⋊C₄	—	2	1	no	—
326	C₄⋊C₄	—	2	2	no	—
327	C₄⋊C₄	—	2	2	no	—
328	C₄⋊C₄	—	292	1	no	—
329	C₄⋊C₄	—	292	1	no	—
330	C₄⋊C₄	—	146	2	no	—
331	C₄⋊C₄	—	146	2	no	—
332	C₄⋊C₄	C₂×D₂₉₂	1	2	yes	—
333	C₂²⋊C₄	—	2	2	no	—
334	C₄²	—	2	2	no	—
335	C₄²	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
336	C₄²	C₂×D₂₉₂	1	1	yes	characteristic
337	C₄²	#11924028	1	1	yes	characteristic
338	C₄²	C₂³×D₇₃	1	1	yes	characteristic
339	C₂³	—	292	2	no	—
340	C₂³	—	584	2	no	—
341	C₂³	—	584	2	no	—
342	C₂³	—	584	2	no	—
343	C₂³	#14556565	1	1	yes	characteristic
344	Q₈	—	2	4	no	—
345	Q₈	—	4	2	no	—
346	Q₈	—	292	4	no	—
347	Q₈	—	584	2	no	—
348	D₄	—	292	2	no	—
349	D₄	—	292	4	no	—
350	D₄	—	292	2	no	—
351	D₄	—	584	2	no	—
352	D₄	—	584	2	no	—
353	D₄	—	584	2	no	—
354	D₄	—	584	2	no	—
355	C₂×C₄	—	2	2	no	—
356	C₂×C₄	—	2	2	no	—
357	C₂×C₄	—	2	2	no	—
358	C₂×C₄	—	2	2	no	—
359	C₂×C₄	—	292	2	no	—
360	C₂×C₄	—	584	2	no	—
361	C₂×C₄	#14556565	1	1	yes	characteristic
362	C₂×C₄	#14556565	1	1	yes	characteristic
363	C₂×C₄	#14556565	1	1	yes	characteristic
364	C₂×C₄	#14556611	1	2	yes	—
365	C₂×C₄	#14556622	1	2	yes	—
366	C₂×C₄	C₂²×C₂²⋊D₇₃	1	1	yes	characteristic
367	C₂×C₄	C₂²×D₂₉₂	1	1	yes	characteristic
368	C₂×C₄	C₂×#11924028	1	1	yes	characteristic
369	C₈	—	2	1	no	—
370	C₈	—	2	1	no	—
371	C₈	—	2	1	no	—
372	C₈	—	2	1	no	—
373	C₈	—	4	2	no	—
374	C₈	—	292	2	no	—
375	C₈	—	292	2	no	—
376	C₂²	—	2	1	no	—
377	C₂²	—	2	1	no	—
378	C₂²	—	584	2	no	—
379	C₂²	—	584	2	no	—
380	C₂²	—	584	2	no	—
381	C₂²	—	1168	4	no	—
382	C₂²	#17988809	1	2	yes	—
383	C₂²	C₂×#14556565	1	1	yes	characteristic
384	C₄	—	2	1	no	—
385	C₄	—	2	1	no	—
386	C₄	—	4	2	no	—
387	C₄	—	4	2	no	—
388	C₄	—	584	2	no	—
389	C₄	#17988809	1	2	yes	—
390	C₄	#17988850	1	2	yes	—
391	C₄	C₂×#14556611	1	1	yes	characteristic
392	C₄	C₂×#14556622	1	1	yes	characteristic
393	C₄	#17989392	1	2	yes	—
394	C₂	—	2	2	no	—
395	C₂	—	1168	2	no	—
396	C₂	C₂×#17988809	1	1	yes	characteristic
397	C₂	#22619579	1	1	yes	characteristic
398	C₂	C₂×#17988850	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#14556565
commutator, cosocle	C₄²×C₇₃	C₂⁴
Frattini	C₄²	C₂³×D₇₃
Fitting	C₇₃×#2960	C₂
socle	C₂²×C₇₃	C₂×#121
prime radical	C₂×#14556588	C₂²
hypercenter	#2960	D₇₃
hypocenter	C₇₃	#19214
LCS	G ▷ C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960

Complements

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

N	H ≅ G/N	type
1. #22620876	C₂	semidirect
4. #22609499	C₂	semidirect
5. #22609375	C₂	semidirect
6. #22608679	C₂	semidirect
7. #22608617	C₂	semidirect
9. C₇₃×#2960	C₂	semidirect
20. #17988797	C₂²	semidirect
28. #17988340	C₂²	semidirect
29. #17988338	C₂²	semidirect
30. #17988332	C₂²	semidirect
34. #446×C₇₃	C₂²	semidirect
36. C₈×Q₈×C₇₃	C₂²	semidirect
203. #2960	D₇₃	semidirect
209. C₇₃	#19214	semidirect
226. #446	C₂×D₇₃	semidirect
228. C₈×Q₈	C₂×D₇₃	semidirect

Maximal subgroups

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

subgroup	index
#22620876	2
#22620869	2
C₂×#17988934	2
#22609499	2
#22609375	2
#22608679	2
#22608617	2
#22608600	2
C₇₃×#2960	2
#19214	73