Subgroups — #33090692

Order 18688.

646 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22612409	C₂	1	1	yes	maximal, characteristic
2	#22610872	C₂	1	1	yes	maximal, characteristic
3	#22610654	C₂	1	1	yes	maximal, characteristic
4	#22610652	C₂	1	1	yes	maximal, characteristic
5	#22610496	C₂	1	2	yes	maximal
6	#22610494	C₂	1	2	yes	maximal
7	#22609787	C₂	1	2	yes	maximal
8	#22609786	C₂	1	1	yes	maximal, characteristic
9	#22609681	C₂	1	2	yes	maximal
10	#97×Dic₇₃	C₂	1	1	yes	maximal, characteristic
11	C₇₃×#1985	C₂	1	1	yes	maximal, characteristic
12	C₂×C₄⋊#10596831	C₂²	1	1	yes	characteristic
13	C₂×#14556520	C₂²	1	2	yes	—
14	C₂×#14556520	C₂²	1	2	yes	—
15	C₂×#14556502	C₂²	1	1	yes	characteristic
16	C₂×#14556501	C₂²	1	2	yes	—
17	C₂×#14556501	C₂²	1	1	yes	characteristic
18	C₂×#14556501	C₂²	1	1	yes	characteristic
19	C₂×C₈×Dic₇₃	C₂²	1	1	yes	characteristic
20	C₂×C₈×Dic₇₃	C₂²	1	1	yes	characteristic
21	#17988494	C₄	1	2	yes	—
22	#17988486	C₄	1	2	yes	—
23	#17988473	C₄	1	4	yes	—
24	C₄×C₄⋊Dic₇₃	C₂²	1	2	yes	—
25	C₄×C₄⋊Dic₇₃	C₂²	1	1	yes	characteristic
26	#17988433	C₂²	1	2	yes	—
27	C₄²×Dic₇₃	C₂²	1	1	yes	characteristic
28	#17988342	C₂²	1	2	yes	—
29	#17988342	C₂²	1	2	yes	—
30	#17988341	C₂²	1	2	yes	—
31	#17988341	C₂²	1	2	yes	—
32	#17988341	C₂²	1	2	yes	—
33	C₂×C₄×Q₈×C₇₃	C₂²	1	1	yes	characteristic
34	C₂×C₄⋊C₈×C₇₃	C₂²	1	1	yes	characteristic
35	C₂×#97×C₇₃	C₂²	1	1	yes	characteristic
36	#336×C₇₃	C₂²	1	2	yes	—
37	#336×C₇₃	C₂²	1	2	yes	—
38	C₂²×#10596831	C₂³	1	2	yes	—
39	C₂²×#10596831	C₂³	1	1	yes	characteristic
40	C₂²×#10596831	C₂³	1	1	yes	characteristic
41	C₂×C₄⋊Dic₇₃	C₂×C₄	1	4	yes	—
42	C₂×C₄⋊Dic₇₃	C₂×C₄	1	2	yes	—
43	C₂×C₄×Dic₇₃	C₂×C₄	1	2	yes	—
44	C₂×C₄×Dic₇₃	C₂×C₄	1	2	yes	—
45	C₂×C₄×Dic₇₃	C₂³	1	1	yes	characteristic
46	C₂×C₄×Dic₇₃	C₂³	1	2	yes	—
47	C₂×C₄×Dic₇₃	C₂³	1	1	yes	characteristic
48	C₄⋊#10596831	—	2	2	no	—
49	#14556520	—	2	4	no	—
50	#14556520	—	2	4	no	—
51	#14556502	—	2	2	no	—
52	#14556501	—	2	2	no	—
53	#14556501	—	2	4	no	—
54	#14556501	—	2	2	no	—
55	C₈×Dic₇₃	—	2	2	no	—
56	C₈×Dic₇₃	Q₈	1	4	yes	complemented
57	#14556498	C₂×C₄	1	4	yes	—
58	#14556498	C₂×C₄	1	4	yes	—
59	#14556498	C₂×C₄	1	2	yes	—
60	#14556498	C₂×C₄	1	4	yes	—
61	C₂²×Q₈×C₇₃	C₂×C₄	1	1	yes	characteristic
62	C₂²×C₈×C₇₃	C₂³	1	2	yes	—
63	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
64	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
65	C₄×Q₈×C₇₃	—	2	2	no	—
66	C₄×Q₈×C₇₃	—	2	2	no	—
67	C₂×C₄⋊C₄×C₇₃	C₂×C₄	1	2	yes	—
68	C₂×C₄⋊C₄×C₇₃	C₂×C₄	1	1	yes	characteristic
69	C₂×C₄²×C₇₃	C₂³	1	2	yes	—
70	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
71	C₄⋊C₈×C₇₃	—	2	2	no	—
72	#97×C₇₃	—	2	2	no	—
73	C₂²×Dic₇₃	C₂²×C₄	1	4	yes	—
74	C₂²×Dic₇₃	C₂²×C₄	1	2	yes	—
75	C₂²×Dic₇₃	C₂²×C₄	1	2	yes	—
76	C₂×#10596831	—	2	1	no	—
77	C₂×#10596831	—	2	1	no	—
78	C₂×#10596831	—	2	1	no	—
79	C₂×#10596831	—	2	1	no	—
80	C₂×#10596831	—	2	2	no	—
81	C₂×#10596831	—	2	2	no	—
82	C₂×#10596831	—	2	2	no	—
83	C₂×#10596831	C₂×Q₈	1	2	yes	—
84	C₂×#10596831	#41	1	2	yes	—
85	C₄⋊Dic₇₃	—	4	4	no	—
86	C₄⋊Dic₇₃	—	4	2	no	—
87	C₄×Dic₇₃	—	2	1	no	—
88	C₄×Dic₇₃	—	2	1	no	—
89	C₄×Dic₇₃	—	2	2	no	—
90	C₄×Dic₇₃	—	2	2	no	—
91	C₄×Dic₇₃	—	4	2	no	—
92	C₄×Dic₇₃	—	4	2	no	—
93	C₄×Dic₇₃	C₂×Q₈	1	2	yes	—
94	C₄×Dic₇₃	#41	1	2	yes	—
95	C₂×Q₈×C₇₃	—	2	2	no	—
96	C₂×Q₈×C₇₃	—	4	2	no	—
97	C₂²×C₄×C₇₃	C₂²×C₄	1	2	yes	—
98	C₂²×C₄×C₇₃	C₂²×C₄	1	2	yes	—
99	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
100	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
101	C₂²×C₄×C₇₃	C₂⁴	1	1	yes	characteristic
102	C₂×C₈×C₇₃	—	2	1	no	—
103	C₂×C₈×C₇₃	—	2	1	no	—
104	C₂×C₈×C₇₃	—	2	1	no	—
105	C₂×C₈×C₇₃	—	2	1	no	—
106	C₂×C₈×C₇₃	—	2	2	no	—
107	C₂×C₈×C₇₃	—	2	2	no	—
108	C₂×C₈×C₇₃	—	2	2	no	—
109	C₂×C₈×C₇₃	C₂×Q₈	1	2	yes	—
110	C₂×C₈×C₇₃	#41	1	2	yes	—
111	C₄⋊C₄×C₇₃	—	2	2	no	—
112	C₄⋊C₄×C₇₃	—	2	4	no	—
113	C₄²×C₇₃	—	2	2	no	—
114	C₄²×C₇₃	—	2	2	no	—
115	C₄²×C₇₃	#41	1	2	yes	—
116	C₄²×C₇₃	#41	1	2	yes	—
117	C₂³×C₇₃	C₂³×C₄	1	1	yes	characteristic
118	Q₈×C₇₃	—	4	4	no	—
119	C₂×C₄×C₇₃	—	2	2	no	—
120	C₂×C₄×C₇₃	—	2	2	no	—
121	C₂×C₄×C₇₃	—	2	2	no	—
122	C₂×C₄×C₇₃	—	2	2	no	—
123	C₂×C₄×C₇₃	—	2	4	no	—
124	C₂×C₄×C₇₃	—	2	4	no	—
125	C₂×C₄×C₇₃	#117	1	2	yes	—
126	C₂×C₄×C₇₃	#117	1	2	yes	—
127	C₂×C₄×C₇₃	C₂²×Q₈	1	1	yes	characteristic
128	C₂×C₄×C₇₃	C₂×#41	1	1	yes	characteristic
129	C₂×C₄×C₇₃	C₂×#41	1	1	yes	characteristic
130	C₂×C₄×C₇₃	C₂×#41	1	1	yes	characteristic
131	C₂×C₄×C₇₃	C₂×#41	1	1	yes	characteristic
132	C₂×C₄×C₇₃	C₂×#41	1	1	yes	characteristic
133	C₂×Dic₇₃	—	2	2	no	—
134	C₂×Dic₇₃	—	2	2	no	—
135	C₂×Dic₇₃	—	2	2	no	—
136	C₂×Dic₇₃	—	2	4	no	—
137	C₂×Dic₇₃	—	2	2	no	—
138	C₂×Dic₇₃	—	2	2	no	—
139	C₂×Dic₇₃	—	2	4	no	—
140	C₂×Dic₇₃	—	2	4	no	—
141	C₂×Dic₇₃	C₄×Q₈	1	4	yes	—
142	C₈×C₇₃	—	2	2	no	—
143	C₈×C₇₃	—	2	2	no	—
144	C₈×C₇₃	—	4	2	no	—
145	#10596831	—	2	2	no	—
146	#10596831	—	2	2	no	—
147	#10596831	—	4	2	no	—
148	C₂²×C₇₃	C₂×#117	1	1	yes	characteristic
149	C₂²×C₇₃	C₂×C₄×Q₈	1	1	yes	characteristic
150	C₂²×C₇₃	C₄×#41	1	1	yes	characteristic
151	C₂²×C₇₃	#568	1	2	yes	—
152	C₂²×C₇₃	#568	1	2	yes	—
153	C₄×C₇₃	—	2	2	no	—
154	C₄×C₇₃	—	2	2	no	—
155	C₄×C₇₃	—	4	2	no	—
156	C₄×C₇₃	—	4	2	no	—
157	C₄×C₇₃	#529	1	1	yes	characteristic
158	C₄×C₇₃	#529	1	1	yes	characteristic
159	C₄×C₇₃	#533	1	1	yes	characteristic
160	C₄×C₇₃	#533	1	1	yes	characteristic
161	Dic₇₃	—	2	4	no	—
162	Dic₇₃	—	4	2	no	—
163	Dic₇₃	—	4	4	no	—
164	#21342	—	73	1	no	maximal, Sylow
165	C₂×C₇₃	#2295	1	1	yes	characteristic
166	C₂×C₇₃	#2921	1	1	yes	characteristic
167	C₂×C₇₃	#2923	1	1	yes	characteristic
168	C₂×C₇₃	#2961	1	2	yes	—
169	C₂×C₇₃	#2967	1	2	yes	—
170	#2295	—	73	1	no	—
171	#1985	—	73	1	no	—
172	#1985	D₇₃	1	1	yes	characteristic
173	#1940	—	73	1	no	—
174	#1940	—	73	1	no	—
175	#1917	—	73	2	no	—
176	#1917	—	73	2	no	—
177	#1768	—	73	2	no	—
178	#1767	—	73	1	no	—
179	#1746	—	73	2	no	—
180	C₄×#97	—	73	1	no	—
181	C₇₃	#21342	1	1	yes	characteristic, complemented, Sylow
182	C₂×C₄×Q₈	C₂×D₇₃	1	1	yes	characteristic
183	C₂×C₄⋊C₈	—	73	2	no	—
184	C₂×C₄⋊C₈	—	73	2	no	—
185	C₂×C₄⋊C₈	—	73	1	no	—
186	C₂×C₄⋊C₈	C₂×D₇₃	1	1	yes	characteristic
187	C₂×#97	—	73	1	no	—
188	C₂×#97	—	73	1	no	—
189	C₂×#97	—	73	2	no	—
190	C₂×#97	—	73	1	no	—
191	C₂×#97	C₂×D₇₃	1	1	yes	characteristic
192	C₂×C₄×C₈	—	73	1	no	—
193	C₂×C₄×C₈	—	73	1	no	—
194	#391	—	73	2	no	—
195	#389	—	73	2	no	—
196	#387	—	73	4	no	—
197	C₄×C₄⋊C₄	—	73	2	no	—
198	C₄×C₄⋊C₄	—	73	1	no	—
199	#376	—	73	2	no	—
200	C₄³	—	73	1	no	—
201	#336	—	73	2	no	—
202	#336	—	73	2	no	—
203	#336	—	73	2	no	—
204	#336	—	73	2	no	—
205	#336	—	73	2	no	—
206	#336	C₂×D₇₃	1	2	yes	—
207	#336	C₂×D₇₃	1	2	yes	—
208	C₂²×Q₈	C₄×D₇₃	1	1	yes	characteristic
209	C₂²×C₈	—	73	1	no	—
210	C₂²×C₈	—	73	1	no	—
211	C₂²×C₈	—	73	2	no	—
212	C₂²×C₈	C₂²×D₇₃	1	2	yes	—
213	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
214	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
215	C₄×Q₈	—	2	2	no	—
216	C₄×Q₈	—	2	2	no	—
217	C₂×C₄⋊C₄	—	73	4	no	—
218	C₂×C₄⋊C₄	—	73	2	no	—
219	C₂×C₄⋊C₄	C₄×D₇₃	1	2	yes	—
220	C₂×C₄⋊C₄	C₄×D₇₃	1	1	yes	characteristic
221	C₂×C₄²	—	73	2	no	—
222	C₂×C₄²	—	73	2	no	—
223	C₂×C₄²	—	73	1	no	—
224	C₂×C₄²	—	73	1	no	—
225	C₂×C₄²	—	73	2	no	—
226	C₂×C₄²	C₂²×D₇₃	1	2	yes	—
227	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
228	C₄⋊C₈	—	2	2	no	—
229	C₄⋊C₈	—	146	4	no	—
230	C₄⋊C₈	—	146	4	no	—
231	C₄⋊C₈	—	146	2	no	—
232	#97	—	2	2	no	—
233	#97	—	146	4	no	—
234	#97	—	146	2	no	—
235	#97	—	146	2	no	—
236	#97	—	146	2	no	—
237	C₄×C₈	—	73	4	no	—
238	C₄×C₈	—	146	2	no	—
239	#95	—	73	4	no	—
240	#95	—	73	4	no	—
241	#95	—	73	2	no	—
242	#95	—	73	4	no	—
243	C₂×Q₈	—	2	2	no	—
244	C₂×Q₈	—	4	2	no	—
245	C₂²×C₄	—	73	2	no	—
246	C₂²×C₄	—	73	4	no	—
247	C₂²×C₄	—	73	2	no	—
248	C₂²×C₄	C₂×C₄×D₇₃	1	2	yes	—
249	C₂²×C₄	C₂×C₄×D₇₃	1	2	yes	—
250	C₂²×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
251	C₂²×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
252	C₂²×C₄	C₂³×D₇₃	1	1	yes	characteristic
253	C₂×C₈	—	2	1	no	—
254	C₂×C₈	—	2	1	no	—
255	C₂×C₈	—	2	1	no	—
256	C₂×C₈	—	2	1	no	—
257	C₂×C₈	—	2	2	no	—
258	C₂×C₈	—	2	2	no	—
259	C₂×C₈	—	2	2	no	—
260	C₂×C₈	—	73	2	no	—
261	C₂×C₈	—	73	2	no	—
262	C₂×C₈	—	146	2	no	—
263	C₂×C₈	—	146	2	no	—
264	C₂×C₈	—	146	2	no	—
265	C₂×C₈	—	146	1	no	—
266	C₂×C₈	—	146	1	no	—
267	C₂×C₈	—	146	1	no	—
268	C₂×C₈	—	146	1	no	—
269	C₂×C₈	Q₈×D₇₃	1	2	yes	—
270	C₂×C₈	#11924029	1	2	yes	—
271	C₄⋊C₄	—	2	2	no	—
272	C₄⋊C₄	—	2	4	no	—
273	C₄⋊C₄	—	292	2	no	—
274	C₄⋊C₄	—	292	4	no	—
275	C₄²	—	2	2	no	—
276	C₄²	—	2	2	no	—
277	C₄²	—	292	2	no	—
278	C₄²	—	292	2	no	—
279	C₄²	—	73	2	no	—
280	C₄²	—	73	2	no	—
281	C₄²	—	146	1	no	—
282	C₄²	—	146	2	no	—
283	C₄²	—	146	1	no	—
284	C₄²	—	146	2	no	—
285	C₄²	#11924027	1	2	yes	—
286	C₄²	#11924027	1	2	yes	—
287	C₂³	C₂²×C₄×D₇₃	1	1	yes	characteristic
288	Q₈	—	4	4	no	—
289	C₂×C₄	—	2	2	no	—
290	C₂×C₄	—	2	2	no	—
291	C₂×C₄	—	2	2	no	—
292	C₂×C₄	—	2	2	no	—
293	C₂×C₄	—	2	4	no	—
294	C₂×C₄	—	2	4	no	—
295	C₂×C₄	—	73	4	no	—
296	C₂×C₄	—	146	2	no	—
297	C₂×C₄	—	146	4	no	—
298	C₂×C₄	—	146	2	no	—
299	C₂×C₄	—	146	4	no	—
300	C₂×C₄	—	146	2	no	—
301	C₂×C₄	—	146	4	no	—
302	C₂×C₄	—	146	2	no	—
303	C₂×C₄	—	146	2	no	—
304	C₂×C₄	#14556547	1	2	yes	—
305	C₂×C₄	#14556547	1	2	yes	—
306	C₂×C₄	C₂×Q₈×D₇₃	1	1	yes	characteristic
307	C₂×C₄	#41×D₇₃	1	1	yes	characteristic
308	C₂×C₄	#41×D₇₃	1	1	yes	characteristic
309	C₂×C₄	C₂×#11924027	1	1	yes	characteristic
310	C₂×C₄	C₂×#11924027	1	1	yes	characteristic
311	C₂×C₄	C₂×#11924029	1	1	yes	characteristic
312	C₈	—	2	2	no	—
313	C₈	—	2	2	no	—
314	C₈	—	4	2	no	—
315	C₈	—	292	2	no	—
316	C₈	—	146	2	no	—
317	C₈	—	146	2	no	—
318	C₂²	C₂×#14556547	1	1	yes	characteristic
319	C₂²	C₄×Q₈×D₇₃	1	1	yes	characteristic
320	C₂²	C₄×#11924029	1	1	yes	characteristic
321	C₂²	#17989352	1	2	yes	—
322	C₂²	#17989353	1	2	yes	—
323	C₄	—	2	2	no	—
324	C₄	—	2	2	no	—
325	C₄	—	4	2	no	—
326	C₄	—	4	2	no	—
327	C₄	—	292	2	no	—
328	C₄	—	292	4	no	—
329	C₄	—	146	4	no	—
330	C₄	#17989116	1	1	yes	characteristic
331	C₄	#17989123	1	1	yes	characteristic
332	C₄	#17989139	1	1	yes	characteristic
333	C₄	#17989142	1	1	yes	characteristic
334	C₂	#22612406	1	1	yes	characteristic
335	C₂	#22618566	1	1	yes	characteristic
336	C₂	#22618578	1	1	yes	characteristic
337	C₂	#22618867	1	2	yes	—
338	C₂	#22618915	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₇₃
commutator, socle, prime radical	C₂³×C₇₃	C₂³×C₄
Fitting	C₇₃×#1985	C₂
cosocle	C₂²×C₄×C₇₃	C₂⁴
hypercenter	#1985	D₇₃
hypocenter	C₇₃	#21342
LCS	G ▷ C₂³×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂²×C₄ ◁ #1985

Complements

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

N	H ≅ G/N	type
56. C₈×Dic₇₃	Q₈	semidirect
181. C₇₃	#21342	semidirect

Maximal subgroups

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

subgroup	index
#22612409	2
#22610872	2
#22610654	2
#22610652	2
#22610496	2
#22610494	2
#22609787	2
#22609786	2
#22609681	2
#97×Dic₇₃	2
C₇₃×#1985	2
#21342	73