Subgroups — #33241582

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	#22620573	C₂	1	2	yes	maximal, complemented
2	#22612192	C₂	1	1	yes	maximal, characteristic, complemented
3	#22612170	C₂	1	2	yes	maximal, complemented
4	#22612045	C₂	1	2	yes	maximal, complemented
5	#22612023	C₂	1	1	yes	maximal, characteristic, complemented
6	#22612018	C₂	1	1	yes	maximal, characteristic
7	#22611794	C₂	1	2	yes	maximal, complemented
8	#22611789	C₂	1	1	yes	maximal, characteristic, complemented
9	#22611777	C₂	1	1	yes	maximal, characteristic, complemented
10	#22611526	C₂	1	1	yes	maximal, characteristic, complemented
11	C₇₃×#2252	C₂	1	1	yes	maximal, characteristic, complemented
12	#17989211	—	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	C₂×#14556530	C₂²	1	1	yes	characteristic
17	C₂×C₁₆×D₇₃	C₂²	1	1	yes	characteristic
18	#17988971	C₂²	1	2	yes	—
19	#17988949	—	2	2	no	—
20	#17988834	—	2	2	no	—
21	#17988829	C₂²	1	2	yes	—
22	#17988824	C₂²	1	2	yes	—
23	#17988686	—	2	2	no	—
24	#17988681	C₂²	1	2	yes	—
25	#17988669	C₂²	1	2	yes	—
26	#17988541	C₂²	1	1	yes	characteristic
27	#17988390	C₂²	1	2	yes	—
28	#17988390	C₂²	1	2	yes	—
29	#17988387	C₂²	1	2	yes	—
30	#17988387	C₂²	1	2	yes	—
31	#17988363	C₂²	1	1	yes	characteristic
32	#17988362	C₂²	1	1	yes	characteristic
33	C₁₆×Dic₇₃	C₂²	1	1	yes	characteristic
34	#500×C₇₃	C₂²	1	2	yes	—
35	#366×C₇₃	C₂²	1	1	yes	characteristic
36	#366×C₇₃	C₂²	1	1	yes	characteristic
37	#366×C₇₃	C₂²	1	2	yes	—
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	C₂×#11924011	D₄	1	1	yes	characteristic
43	C₂×C₈×D₇₃	C₂³	1	1	yes	characteristic
44	#14556586	—	4	2	no	—
45	#14556580	—	4	2	no	—
46	#14556579	—	2	2	no	—
47	#14556576	—	4	2	no	—
48	#14556571	—	2	2	no	—
49	#14556567	—	4	2	no	—
50	#14556566	—	2	2	no	—
51	#14556556	—	2	2	no	—
52	#14556553	—	2	2	no	—
53	#14556548	D₄	1	1	yes	characteristic
54	Dic₅₈₄	—	2	2	no	—
55	#14556536	—	2	4	no	—
56	D₁₁₆₈	—	2	2	no	—
57	#14556530	—	2	2	no	—
58	C₂×#11924003	C₂³	1	1	yes	characteristic
59	C₂×#11924003	C₂³	1	1	yes	characteristic
60	C₁₆×D₇₃	—	2	2	no	—
61	#14556514	—	2	2	no	—
62	#14556514	—	2	2	no	—
63	#14556511	—	2	2	no	—
64	#14556511	—	2	2	no	—
65	#14556502	D₄	1	1	yes	characteristic
66	#14556501	D₄	1	1	yes	characteristic
67	C₈×Dic₇₃	C₂³	1	1	yes	characteristic
68	C₄⋊Q₈×C₇₃	—	2	2	no	—
69	C₂×C₁₆×C₇₃	C₂³	1	1	yes	characteristic
70	C₂×C₁₆×C₇₃	C₂³	1	1	yes	characteristic
71	#107×C₇₃	—	2	2	no	—
72	#107×C₇₃	C₂³	1	2	yes	—
73	#107×C₇₃	C₂³	1	2	yes	—
74	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
75	C₂×Dic₁₄₆	—	2	2	no	—
76	C₂×D₂₉₂	—	2	2	no	—
77	C₂×D₂₉₂	—	4	2	no	—
78	C₂×C₄×D₇₃	—	4	2	no	—
79	C₂×C₄×D₇₃	C₂×D₄	1	1	yes	characteristic
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	#11924011	—	2	2	no	—
86	C₂×#10596831	C₂×D₄	1	1	yes	characteristic
87	C₂×#10596831	C₂×D₄	1	1	yes	characteristic
88	C₈×D₇₃	—	2	1	no	—
89	C₈×D₇₃	—	2	1	no	—
90	C₄⋊Dic₇₃	—	4	2	no	—
91	#11924007	—	4	2	no	—
92	#11924007	D₈	1	1	yes	characteristic
93	#11924007	D₈	1	1	yes	characteristic
94	#11924005	—	4	2	no	—
95	#11924005	—	4	2	no	—
96	#11924005	—	4	2	no	—
97	#11924005	—	4	2	no	—
98	#11924005	—	4	2	no	—
99	#11924005	D₈	1	1	yes	characteristic
100	#11924005	D₈	1	1	yes	characteristic
101	C₄×Dic₇₃	—	4	2	no	—
102	C₄×Dic₇₃	C₂×D₄	1	1	yes	characteristic
103	#11924003	—	2	1	no	—
104	#11924003	—	2	1	no	—
105	C₂×Q₈×C₇₃	—	4	2	no	—
106	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
107	C₂×C₈×C₇₃	C₂⁴	1	1	yes	characteristic
108	C₄⋊C₄×C₇₃	—	2	2	no	—
109	C₄⋊C₄×C₇₃	—	2	2	no	—
110	C₄⋊C₄×C₇₃	—	4	2	no	—
111	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
112	C₁₆×C₇₃	#41	1	2	yes	—
113	C₁₆×C₇₃	#41	1	2	yes	—
114	C₂²×D₇₃	—	4	2	no	—
115	C₂²×D₇₃	C₂×D₈	1	1	yes	characteristic
116	Q₈×C₇₃	—	16	2	no	—
117	C₂×C₄×C₇₃	—	4	2	no	—
118	C₂×C₄×C₇₃	—	4	2	no	—
119	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
120	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
121	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
122	C₂×Dic₇₃	—	4	2	no	—
123	C₂×Dic₇₃	C₂×D₈	1	1	yes	characteristic
124	C₂×Dic₇₃	C₂×D₈	1	1	yes	characteristic
125	C₂×Dic₇₃	C₂×D₈	1	1	yes	characteristic
126	D₂₉₂	—	4	2	no	—
127	D₂₉₂	—	4	2	no	—
128	D₂₉₂	—	16	2	no	—
129	C₄×D₇₃	—	2	1	no	—
130	C₄×D₇₃	—	2	1	no	—
131	C₄×D₇₃	—	16	2	no	—
132	Dic₁₄₆	—	4	2	no	—
133	Dic₁₄₆	—	4	2	no	—
134	C₈×C₇₃	#135	1	2	yes	—
135	C₈×C₇₃	C₂×#41	1	1	yes	characteristic
136	C₈×C₇₃	C₂×#41	1	1	yes	characteristic
137	#10596831	—	2	1	no	—
138	#10596831	—	2	1	no	—
139	C₂²×C₇₃	C₂²×D₈	1	1	yes	characteristic
140	C₂×D₇₃	—	2	1	no	—
141	C₂×D₇₃	—	2	1	no	—
142	C₂×D₇₃	—	2	1	no	—
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	—	8	2	no	—
149	C₄×C₇₃	—	8	2	no	—
150	C₄×C₇₃	#532	1	1	yes	characteristic
151	C₄×C₇₃	C₂×#135	1	1	yes	characteristic
152	Dic₇₃	—	2	1	no	—
153	Dic₇₃	—	2	1	no	—
154	Dic₇₃	—	2	1	no	—
155	Dic₇₃	—	8	2	no	—
156	#33739	—	73	1	no	maximal, Sylow
157	C₂×C₇₃	#3169	1	1	yes	characteristic
158	C₂×C₇₃	#3415	1	1	yes	characteristic
159	C₂×C₇₃	#3417	1	1	yes	characteristic
160	D₇₃	—	4	1	no	—
161	D₇₃	—	16	2	no	—
162	#3175	—	73	2	no	—
163	#2252	D₇₃	1	1	yes	characteristic, complemented
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	—	73	1	no	—
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	—	73	1	no	—
179	C₂²×C₁₆	—	73	1	no	—
180	#500	C₂×D₇₃	1	2	yes	—
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	—	73	1	no	—
190	#366	C₂×D₇₃	1	1	yes	characteristic
191	#366	C₂×D₇₃	1	1	yes	characteristic
192	#366	C₂×D₇₃	1	2	yes	—
193	#358	—	73	2	no	—
194	#358	—	73	2	no	—
195	#357	—	73	2	no	—
196	#357	—	73	2	no	—
197	#346	—	73	1	no	—
198	#346	—	73	1	no	—
199	C₄×C₁₆	—	73	1	no	—
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	—	73	1	no	—
205	C₂²×C₈	—	73	1	no	—
206	C₄⋊Q₈	—	2	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	—	73	1	no	—
217	Q₃₂	—	146	2	no	—
218	SD₃₂	—	146	4	no	—
219	D₁₆	—	146	2	no	—
220	M₃₂	—	146	2	no	—
221	C₂×C₁₆	—	73	1	no	—
222	C₂×C₁₆	—	73	1	no	—
223	C₂×C₁₆	—	146	2	no	—
224	C₂×C₁₆	C₂²×D₇₃	1	1	yes	characteristic
225	C₂×C₁₆	C₂²×D₇₃	1	1	yes	characteristic
226	#107	—	2	2	no	—
227	#107	C₂²×D₇₃	1	2	yes	—
228	#107	C₂²×D₇₃	1	2	yes	—
229	#103	—	146	2	no	—
230	#103	—	146	2	no	—
231	#102	—	146	2	no	—
232	#102	—	146	2	no	—
233	#97	—	73	1	no	—
234	#97	—	73	1	no	—
235	C₄×C₈	—	73	1	no	—
236	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
237	C₂×Q₈	—	4	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	—	73	1	no	—
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	—	146	2	no	—
249	C₂×C₈	—	73	1	no	—
250	C₂×C₈	—	73	1	no	—
251	C₂×C₈	—	146	1	no	—
252	C₂×C₈	—	146	1	no	—
253	C₂×C₈	D₄×D₇₃	1	1	yes	characteristic
254	C₂×C₈	C₂³×D₇₃	1	1	yes	characteristic
255	C₄⋊C₄	—	2	2	no	—
256	C₄⋊C₄	—	2	2	no	—
257	C₄⋊C₄	—	4	2	no	—
258	C₄⋊C₄	—	292	2	no	—
259	C₄⋊C₄	—	292	2	no	—
260	C₄⋊C₄	—	73	1	no	—
261	C₄⋊C₄	—	73	1	no	—
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	—	73	1	no	—
268	C₂²⋊C₄	—	73	1	no	—
269	C₄²	—	292	2	no	—
270	C₄²	—	73	1	no	—
271	C₄²	D₄×D₇₃	1	1	yes	characteristic
272	C₁₆	—	146	1	no	—
273	C₁₆	—	146	1	no	—
274	C₁₆	#11924029	1	2	yes	—
275	C₁₆	#11924029	1	2	yes	—
276	C₂³	—	292	2	no	—
277	C₂³	—	73	1	no	—
278	Q₈	—	16	2	no	—
279	Q₈	—	292	2	no	—
280	Q₈	—	292	2	no	—
281	D₄	—	292	2	no	—
282	D₄	—	292	2	no	—
283	D₄	—	1168	2	no	—
284	C₂×C₄	—	4	2	no	—
285	C₂×C₄	—	4	2	no	—
286	C₂×C₄	—	292	2	no	—
287	C₂×C₄	—	73	1	no	—
288	C₂×C₄	—	73	1	no	—
289	C₂×C₄	—	73	1	no	—
290	C₂×C₄	—	1168	2	no	—
291	C₂×C₄	—	146	1	no	—
292	C₂×C₄	—	146	1	no	—
293	C₂×C₄	D₈×D₇₃	1	1	yes	characteristic
294	C₂×C₄	D₈×D₇₃	1	1	yes	characteristic
295	C₂×C₄	C₂×D₄×D₇₃	1	1	yes	characteristic
296	C₈	—	146	1	no	—
297	C₈	—	146	1	no	—
298	C₈	#14556614	1	2	yes	—
299	C₈	C₂×#11924029	1	1	yes	characteristic
300	C₈	C₂×#11924029	1	1	yes	characteristic
301	C₂²	—	584	2	no	—
302	C₂²	—	584	2	no	—
303	C₂²	—	584	2	no	—
304	C₂²	—	146	1	no	—
305	C₂²	—	146	1	no	—
306	C₂²	—	146	1	no	—
307	C₂²	C₂×D₈×D₇₃	1	1	yes	characteristic
308	C₄	—	2	1	no	—
309	C₄	—	2	1	no	—
310	C₄	—	8	2	no	—
311	C₄	—	8	2	no	—
312	C₄	—	584	2	no	—
313	C₄	—	146	1	no	—
314	C₄	—	146	1	no	—
315	C₄	—	146	1	no	—
316	C₄	#17989138	1	1	yes	characteristic
317	C₄	C₂×#14556614	1	1	yes	characteristic
318	C₂	—	292	1	no	—
319	C₂	—	1168	2	no	—
320	C₂	#22620511	1	1	yes	characteristic
321	C₂	#22623068	1	1	yes	characteristic
322	C₂	#22623080	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₈×D₇₃
commutator, cosocle	C₂×C₈×C₇₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₇₃
Fitting	C₇₃×#2252	C₂
socle	C₂²×C₇₃	C₂²×D₈
prime radical	#22612018	C₂
hypercenter	#2252	D₇₃
hypocenter	C₇₃	#33739
LCS	G ▷ C₂×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2252

Complements

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

N	H ≅ G/N	type
1. #22620573	C₂	semidirect
2. #22612192	C₂	semidirect
3. #22612170	C₂	semidirect
4. #22612045	C₂	semidirect
5. #22612023	C₂	semidirect
7. #22611794	C₂	semidirect
8. #22611789	C₂	semidirect
9. #22611777	C₂	semidirect
10. #22611526	C₂	semidirect
11. C₇₃×#2252	C₂	semidirect
163. #2252	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
#22620573	2
#22612192	2
#22612170	2
#22612045	2
#22612023	2
#22612018	2
#22611794	2
#22611789	2
#22611777	2
#22611526	2
C₇₃×#2252	2
#33739	73