Subgroups — #33241579

Order 18688.

508 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22620570	C₂	1	2	yes	maximal, complemented
2	#22612206	C₂	1	1	yes	maximal, characteristic, complemented
3	#22612192	C₂	1	1	yes	maximal, characteristic, complemented
4	#22612168	C₂	1	2	yes	maximal, complemented
5	#22612044	C₂	1	2	yes	maximal, complemented
6	#22612022	C₂	1	1	yes	maximal, characteristic, complemented
7	#22612017	C₂	1	1	yes	maximal, characteristic
8	#22611797	C₂	1	2	yes	maximal, complemented
9	C₄×Dic₅₈₄	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×D₁₁₆₈	C₂	1	1	yes	maximal, characteristic, complemented
11	C₇₃×#2108	C₂	1	1	yes	maximal, characteristic, complemented
12	#17989208	—	2	2	no	—
13	C₂×Dic₅₈₄	C₂²	1	1	yes	characteristic
14	C₂×#14556536	C₂²	1	2	yes	—
15	C₂×D₁₁₆₈	C₂²	1	1	yes	characteristic
16	#17988992	C₂²	1	2	yes	—
17	#17988971	C₂²	1	2	yes	—
18	#17988947	—	2	2	no	—
19	#17988833	—	2	2	no	—
20	#17988828	C₂²	1	2	yes	—
21	#17988823	C₂²	1	2	yes	—
22	#17988689	—	2	2	no	—
23	C₄×Dic₂₉₂	C₂²	1	2	yes	—
24	C₄×D₅₈₄	C₂²	1	2	yes	—
25	#17988413	C₂²	1	1	yes	characteristic
26	#17988413	C₂²	1	1	yes	characteristic
27	#17988413	C₂²	1	2	yes	—
28	#17988392	C₂²	1	2	yes	—
29	#17988392	C₂²	1	2	yes	—
30	#17988389	C₂²	1	2	yes	—
31	#17988389	C₂²	1	2	yes	—
32	C₂×M₃₂×C₇₃	C₂²	1	1	yes	characteristic
33	C₂²×C₁₆×C₇₃	C₂²	1	1	yes	characteristic
34	#405×C₇₃	C₂²	1	1	yes	characteristic
35	#346×C₇₃	C₂²	1	1	yes	characteristic
36	#346×C₇₃	C₂²	1	1	yes	characteristic
37	C₄×C₁₆×C₇₃	C₂²	1	1	yes	characteristic
38	C₄×C₁₆×C₇₃	C₂²	1	1	yes	characteristic
39	C₂×Dic₂₉₂	C₂³	1	2	yes	—
40	C₂×#11924017	—	2	2	no	—
41	C₂×D₅₈₄	C₂³	1	2	yes	—
42	#14556591	—	2	2	no	—
43	#14556586	—	4	2	no	—
44	#14556579	—	2	2	no	—
45	#14556577	—	4	2	no	—
46	#14556573	—	4	2	no	—
47	#14556570	—	2	2	no	—
48	#14556569	—	4	2	no	—
49	#14556565	—	2	2	no	—
50	C₄×Dic₁₄₆	—	2	2	no	—
51	C₄×D₂₉₂	—	2	2	no	—
52	Dic₅₈₄	—	2	2	no	—
53	#14556536	—	2	4	no	—
54	D₁₁₆₈	—	2	2	no	—
55	#14556525	—	2	2	no	—
56	#14556525	C₂³	1	2	yes	—
57	#14556525	C₂³	1	2	yes	—
58	#14556516	—	2	2	no	—
59	#14556516	—	2	2	no	—
60	#14556513	—	2	2	no	—
61	#14556513	—	2	2	no	—
62	C₂×M₁₆×C₇₃	D₄	1	1	yes	characteristic
63	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
64	#117×C₇₃	D₄	1	1	yes	characteristic
65	M₃₂×C₇₃	—	2	2	no	—
66	C₂×C₁₆×C₇₃	—	2	2	no	—
67	C₂×C₁₆×C₇₃	C₂³	1	1	yes	characteristic
68	C₂×C₁₆×C₇₃	C₂³	1	1	yes	characteristic
69	C₂×C₁₆×C₇₃	C₂³	1	1	yes	characteristic
70	C₂×C₁₆×C₇₃	C₂³	1	1	yes	characteristic
71	#97×C₇₃	D₄	1	1	yes	characteristic
72	#97×C₇₃	D₄	1	1	yes	characteristic
73	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
74	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
75	C₂×Dic₁₄₆	—	2	2	no	—
76	C₂×Dic₁₄₆	—	4	2	no	—
77	C₂×D₂₉₂	—	2	2	no	—
78	C₂×C₂²⋊D₇₃	—	4	2	no	—
79	C₂×C₄×D₇₃	—	4	2	no	—
80	Dic₂₉₂	—	2	2	no	—
81	Dic₂₉₂	—	2	2	no	—
82	#11924017	—	4	4	no	—
83	D₅₈₄	—	2	2	no	—
84	D₅₈₄	—	2	2	no	—
85	C₄⋊Dic₇₃	—	2	2	no	—
86	C₄⋊Dic₇₃	—	2	2	no	—
87	C₄⋊Dic₇₃	—	4	2	no	—
88	#11924007	—	4	2	no	—
89	#11924007	—	4	2	no	—
90	C₂²⋊Dic₇₃	—	4	2	no	—
91	#11924005	—	4	2	no	—
92	#11924005	—	4	2	no	—
93	#11924005	—	4	2	no	—
94	#11924005	—	4	2	no	—
95	C₄×Dic₇₃	—	4	2	no	—
96	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
97	M₁₆×C₇₃	—	2	2	no	—
98	C₂×C₈×C₇₃	—	2	1	no	—
99	C₂×C₈×C₇₃	—	2	1	no	—
100	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
101	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
102	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
103	C₂×C₈×C₇₃	C₂⁴	1	1	yes	characteristic
104	C₄⋊C₄×C₇₃	D₈	1	1	yes	characteristic
105	C₄⋊C₄×C₇₃	D₈	1	1	yes	characteristic
106	C₂²⋊C₄×C₇₃	D₈	1	1	yes	characteristic
107	C₂²⋊C₄×C₇₃	D₈	1	1	yes	characteristic
108	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
109	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
110	C₁₆×C₇₃	—	2	1	no	—
111	C₁₆×C₇₃	—	2	1	no	—
112	C₁₆×C₇₃	#41	1	2	yes	—
113	C₁₆×C₇₃	#41	1	2	yes	—
114	C₂³×C₇₃	C₂×D₈	1	1	yes	characteristic
115	C₂²×D₇₃	—	4	2	no	—
116	C₂×C₄×C₇₃	—	2	1	no	—
117	C₂×C₄×C₇₃	—	2	1	no	—
118	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
119	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
120	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
121	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
122	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
123	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
124	C₂²⋊D₇₃	—	16	2	no	—
125	C₂×Dic₇₃	—	4	2	no	—
126	C₂×Dic₇₃	—	4	2	no	—
127	C₂×Dic₇₃	—	4	2	no	—
128	D₂₉₂	—	4	2	no	—
129	D₂₉₂	—	4	2	no	—
130	C₄×D₇₃	—	16	2	no	—
131	Dic₁₄₆	—	4	2	no	—
132	Dic₁₄₆	—	4	2	no	—
133	Dic₁₄₆	—	16	2	no	—
134	C₈×C₇₃	—	2	1	no	—
135	C₈×C₇₃	—	2	1	no	—
136	C₈×C₇₃	#135	1	2	yes	—
137	C₈×C₇₃	C₂×#41	1	1	yes	characteristic
138	C₈×C₇₃	C₂×#41	1	1	yes	characteristic
139	C₂²×C₇₃	—	2	1	no	—
140	C₂²×C₇₃	—	2	1	no	—
141	C₂²×C₇₃	—	2	1	no	—
142	C₂²×C₇₃	C₂²×D₈	1	1	yes	characteristic
143	C₂×D₇₃	—	8	2	no	—
144	C₂×D₇₃	—	8	2	no	—
145	C₂×D₇₃	—	8	2	no	—
146	C₄×C₇₃	—	2	1	no	—
147	C₄×C₇₃	—	2	1	no	—
148	C₄×C₇₃	—	2	1	no	—
149	C₄×C₇₃	—	2	1	no	—
150	C₄×C₇₃	—	2	1	no	—
151	C₄×C₇₃	#532	1	1	yes	characteristic
152	C₄×C₇₃	C₂×#135	1	1	yes	characteristic
153	Dic₇₃	—	8	2	no	—
154	Dic₇₃	—	8	2	no	—
155	Dic₇₃	—	8	2	no	—
156	#33739	—	73	1	no	maximal, Sylow
157	C₂×C₇₃	—	4	1	no	—
158	C₂×C₇₃	#3169	1	1	yes	characteristic
159	C₂×C₇₃	#3415	1	1	yes	characteristic
160	C₂×C₇₃	#3417	1	1	yes	characteristic
161	D₇₃	—	16	2	no	—
162	#3175	—	73	2	no	—
163	#2252	—	73	1	no	—
164	#2249	—	73	1	no	—
165	#2243	—	73	2	no	—
166	#2220	—	73	2	no	—
167	#2216	—	73	1	no	—
168	#2215	—	73	1	no	—
169	#2175	—	73	2	no	—
170	C₄×Q₃₂	—	73	1	no	—
171	C₄×D₁₆	—	73	1	no	—
172	#2108	D₇₃	1	1	yes	characteristic, complemented
173	C₇₃	#33739	1	1	yes	characteristic, complemented, Sylow
174	#542	—	146	2	no	—
175	C₂×Q₃₂	—	73	1	no	—
176	C₂×SD₃₂	—	73	2	no	—
177	C₂×D₁₆	—	73	1	no	—
178	C₂×M₃₂	C₂×D₇₃	1	1	yes	characteristic
179	C₂²×C₁₆	C₂×D₇₃	1	1	yes	characteristic
180	#500	—	73	2	no	—
181	#495	—	73	2	no	—
182	#489	—	146	2	no	—
183	#468	—	146	2	no	—
184	#467	—	73	2	no	—
185	#466	—	73	2	no	—
186	#440	—	146	2	no	—
187	C₄×Q₁₆	—	73	2	no	—
188	C₄×D₈	—	73	2	no	—
189	#405	C₂×D₇₃	1	1	yes	characteristic
190	#366	—	73	1	no	—
191	#366	—	73	1	no	—
192	#366	—	73	2	no	—
193	#358	—	73	2	no	—
194	#358	—	73	2	no	—
195	#357	—	73	2	no	—
196	#357	—	73	2	no	—
197	#346	C₂×D₇₃	1	1	yes	characteristic
198	#346	C₂×D₇₃	1	1	yes	characteristic
199	C₄×C₁₆	C₂×D₇₃	1	1	yes	characteristic
200	C₄×C₁₆	C₂×D₇₃	1	1	yes	characteristic
201	C₂×Q₁₆	—	73	2	no	—
202	C₂×SD₁₆	—	146	2	no	—
203	C₂×D₈	—	73	2	no	—
204	C₂×M₁₆	D₂₉₂	1	1	yes	characteristic
205	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
206	C₄⋊Q₈	—	146	2	no	—
207	#126	—	292	2	no	—
208	#124	—	292	2	no	—
209	#124	—	146	2	no	—
210	#123	—	292	2	no	—
211	C₂²⋊Q₈	—	292	2	no	—
212	C₂²⋊Q₈	—	146	2	no	—
213	#121	—	146	2	no	—
214	C₄×Q₈	—	146	2	no	—
215	C₄×D₄	—	146	2	no	—
216	#117	D₂₉₂	1	1	yes	characteristic
217	Q₃₂	—	146	2	no	—
218	SD₃₂	—	146	4	no	—
219	D₁₆	—	146	2	no	—
220	M₃₂	—	2	2	no	—
221	C₂×C₁₆	—	2	2	no	—
222	C₂×C₁₆	C₂²×D₇₃	1	1	yes	characteristic
223	C₂×C₁₆	C₂²×D₇₃	1	1	yes	characteristic
224	C₂×C₁₆	C₂²×D₇₃	1	1	yes	characteristic
225	C₂×C₁₆	C₂²×D₇₃	1	1	yes	characteristic
226	#107	—	73	2	no	—
227	#107	—	73	2	no	—
228	#107	—	146	2	no	—
229	#103	—	146	2	no	—
230	#103	—	146	2	no	—
231	#102	—	146	2	no	—
232	#102	—	146	2	no	—
233	#97	D₂₉₂	1	1	yes	characteristic
234	#97	D₂₉₂	1	1	yes	characteristic
235	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
236	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
237	C₂×Q₈	—	292	2	no	—
238	C₂×Q₈	—	146	2	no	—
239	C₂×D₄	—	292	2	no	—
240	C₂×D₄	—	146	2	no	—
241	C₂²×C₄	—	292	2	no	—
242	C₂²×C₄	C₂×D₂₉₂	1	1	yes	characteristic
243	Q₁₆	—	146	2	no	—
244	Q₁₆	—	146	2	no	—
245	SD₁₆	—	292	4	no	—
246	D₈	—	146	2	no	—
247	D₈	—	146	2	no	—
248	M₁₆	—	2	2	no	—
249	C₂×C₈	—	2	1	no	—
250	C₂×C₈	—	2	1	no	—
251	C₂×C₈	C₂×D₂₉₂	1	1	yes	characteristic
252	C₂×C₈	C₂×D₂₉₂	1	1	yes	characteristic
253	C₂×C₈	C₂×D₂₉₂	1	1	yes	characteristic
254	C₂×C₈	C₂³×D₇₃	1	1	yes	characteristic
255	C₄⋊C₄	—	292	2	no	—
256	C₄⋊C₄	—	292	2	no	—
257	C₄⋊C₄	—	292	2	no	—
258	C₄⋊C₄	—	146	2	no	—
259	C₄⋊C₄	—	146	2	no	—
260	C₄⋊C₄	D₅₈₄	1	1	yes	characteristic
261	C₄⋊C₄	D₅₈₄	1	1	yes	characteristic
262	C₂²⋊C₄	—	292	2	no	—
263	C₂²⋊C₄	—	292	2	no	—
264	C₂²⋊C₄	—	292	2	no	—
265	C₂²⋊C₄	—	292	2	no	—
266	C₂²⋊C₄	—	292	2	no	—
267	C₂²⋊C₄	D₅₈₄	1	1	yes	characteristic
268	C₂²⋊C₄	D₅₈₄	1	1	yes	characteristic
269	C₄²	—	292	2	no	—
270	C₄²	C₂×D₂₉₂	1	1	yes	characteristic
271	C₄²	C₂×D₂₉₂	1	1	yes	characteristic
272	C₁₆	—	2	1	no	—
273	C₁₆	—	2	1	no	—
274	C₁₆	#11924028	1	2	yes	—
275	C₁₆	#11924028	1	2	yes	—
276	C₂³	—	292	2	no	—
277	C₂³	C₂×D₅₈₄	1	1	yes	characteristic
278	Q₈	—	292	2	no	—
279	Q₈	—	292	2	no	—
280	Q₈	—	1168	2	no	—
281	D₄	—	292	2	no	—
282	D₄	—	292	2	no	—
283	D₄	—	1168	2	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	2	1	no	—
286	C₂×C₄	—	292	2	no	—
287	C₂×C₄	—	292	2	no	—
288	C₂×C₄	—	292	2	no	—
289	C₂×C₄	—	1168	2	no	—
290	C₂×C₄	C₂×D₅₈₄	1	1	yes	characteristic
291	C₂×C₄	C₂×D₅₈₄	1	1	yes	characteristic
292	C₂×C₄	C₂×D₅₈₄	1	1	yes	characteristic
293	C₂×C₄	C₂×D₅₈₄	1	1	yes	characteristic
294	C₂×C₄	C₂×D₅₈₄	1	1	yes	characteristic
295	C₂×C₄	C₂²×D₂₉₂	1	1	yes	characteristic
296	C₈	—	2	1	no	—
297	C₈	—	2	1	no	—
298	C₈	#14556613	1	2	yes	—
299	C₈	C₂×#11924028	1	1	yes	characteristic
300	C₈	C₂×#11924028	1	1	yes	characteristic
301	C₂²	—	2	1	no	—
302	C₂²	—	2	1	no	—
303	C₂²	—	2	1	no	—
304	C₂²	—	584	2	no	—
305	C₂²	—	584	2	no	—
306	C₂²	—	584	2	no	—
307	C₂²	C₂²×D₅₈₄	1	1	yes	characteristic
308	C₄	—	2	1	no	—
309	C₄	—	2	1	no	—
310	C₄	—	2	1	no	—
311	C₄	—	2	1	no	—
312	C₄	—	2	1	no	—
313	C₄	—	584	2	no	—
314	C₄	—	584	2	no	—
315	C₄	—	584	2	no	—
316	C₄	#17989137	1	1	yes	characteristic
317	C₄	C₂×#14556613	1	1	yes	characteristic
318	C₂	—	4	1	no	—
319	C₂	—	1168	2	no	—
320	C₂	#22620510	1	1	yes	characteristic
321	C₂	#22623072	1	1	yes	characteristic
322	C₂	#22623084	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×D₅₈₄
commutator, cosocle	C₂×C₈×C₇₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₇₃
Fitting	C₇₃×#2108	C₂
socle	C₂²×C₇₃	C₂²×D₈
prime radical	#22612017	C₂
hypercenter	#2108	D₇₃
hypocenter	C₇₃	#33739
LCS	G ▷ C₂×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ #117 ◁ #405 ◁ #2108

Complements

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

N	H ≅ G/N	type
1. #22620570	C₂	semidirect
2. #22612206	C₂	semidirect
3. #22612192	C₂	semidirect
4. #22612168	C₂	semidirect
5. #22612044	C₂	semidirect
6. #22612022	C₂	semidirect
8. #22611797	C₂	semidirect
9. C₄×Dic₅₈₄	C₂	semidirect
10. C₄×D₁₁₆₈	C₂	semidirect
11. C₇₃×#2108	C₂	semidirect
172. #2108	D₇₃	semidirect
173. C₇₃	#33739	semidirect

Maximal subgroups

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

subgroup	index
#22620570	2
#22612206	2
#22612192	2
#22612168	2
#22612044	2
#22612022	2
#22612017	2
#22611797	2
C₄×Dic₅₈₄	2
C₄×D₁₁₆₈	2
C₇₃×#2108	2
#33739	73