Subgroups — #33243652

Order 18688.

498 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22622941	C₂	1	1	yes	maximal, characteristic, complemented
2	#22622796	C₂	1	1	yes	maximal, characteristic, complemented
3	#22612164	C₂	1	2	yes	maximal, complemented
4	#22612155	C₂	1	1	yes	maximal, characteristic, complemented
5	#22612087	C₂	1	2	yes	maximal, complemented
6	#22612083	C₂	1	1	yes	maximal, characteristic, complemented
7	#22611988	C₂	1	2	yes	maximal, complemented
8	#22611966	C₂	1	1	yes	maximal, characteristic
9	C₁₆×Dic₁₄₆	C₂	1	1	yes	maximal, characteristic, complemented
10	#22611775	C₂	1	2	yes	maximal, complemented
11	C₄×SD₃₂×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
12	#17989322	—	2	1	no	—
13	#17989294	—	2	1	no	—
14	C₂×#14556532	C₂²	1	2	yes	—
15	#17988992	C₂²	1	1	yes	characteristic
16	#17988988	C₂²	1	2	yes	—
17	#17988972	C₂²	1	2	yes	—
18	#17988964	C₂²	1	1	yes	characteristic
19	#17988885	—	2	2	no	—
20	#17988877	—	2	1	no	—
21	#17988805	—	2	2	no	—
22	#17988787	—	2	1	no	—
23	C₈×Dic₁₄₆	C₂²	1	1	yes	characteristic
24	D₈×Dic₇₃	C₂²	1	2	yes	—
25	#17988669	C₂²	1	2	yes	—
26	#17988412	C₂²	1	2	yes	—
27	C₄⋊#11924003	C₂²	1	1	yes	characteristic
28	#17988405	C₂²	1	2	yes	—
29	#17988391	C₂²	1	2	yes	—
30	#17988389	C₂²	1	2	yes	—
31	#17988389	C₂²	1	2	yes	—
32	#17988387	C₂²	1	2	yes	—
33	C₁₆×Dic₇₃	C₂²	1	2	yes	—
34	C₂×SD₃₂×C₇₃	C₂²	1	1	yes	characteristic
35	C₄×Q₁₆×C₇₃	C₂²	1	1	yes	characteristic
36	C₄×D₈×C₇₃	C₂²	1	1	yes	characteristic
37	#367×C₇₃	C₂²	1	1	yes	characteristic
38	#358×C₇₃	C₂²	1	1	yes	characteristic
39	#357×C₇₃	C₂²	1	1	yes	characteristic
40	C₄×C₁₆×C₇₃	C₂²	1	1	yes	characteristic
41	C₂×#11924015	—	2	2	no	—
42	C₂×D₅₈₄	—	2	1	no	—
43	C₂×D₅₈₄	C₂³	1	2	yes	—
44	#14556591	—	2	1	no	—
45	#14556588	—	2	1	no	—
46	#14556582	—	2	2	no	—
47	#14556580	—	2	2	no	—
48	#14556579	—	4	1	no	—
49	#14556576	—	4	2	no	—
50	#14556570	—	4	1	no	—
51	#14556568	—	4	2	no	—
52	C₄×Dic₁₄₆	D₄	1	1	yes	characteristic
53	C₄×D₂₉₂	—	4	1	no	—
54	D₄×Dic₇₃	—	2	2	no	—
55	#14556553	—	2	2	no	—
56	C₂×C₄⋊Dic₇₃	—	4	1	no	—
57	#14556532	—	2	4	no	—
58	C₂×#11924003	C₂³	1	2	yes	—
59	#14556525	—	2	1	no	—
60	#14556525	C₂³	1	2	yes	—
61	#14556522	—	2	2	no	—
62	C₄⋊#10596831	D₄	1	1	yes	characteristic
63	#14556520	D₄	1	2	yes	—
64	#14556512	—	2	2	no	—
65	#14556511	—	2	2	no	—
66	C₈×Dic₇₃	C₂³	1	2	yes	—
67	C₂×Q₁₆×C₇₃	C₂³	1	1	yes	characteristic
68	C₂×D₈×C₇₃	C₂³	1	1	yes	characteristic
69	C₄×Q₈×C₇₃	—	2	1	no	—
70	C₄×D₄×C₇₃	—	2	1	no	—
71	SD₃₂×C₇₃	—	2	2	no	—
72	C₂×C₁₆×C₇₃	C₂³	1	1	yes	characteristic
73	C₂×C₁₆×C₇₃	C₂³	1	1	yes	characteristic
74	#107×C₇₃	C₂³	1	1	yes	characteristic
75	#107×C₇₃	C₂³	1	1	yes	characteristic
76	#103×C₇₃	—	2	1	no	—
77	#102×C₇₃	—	2	1	no	—
78	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
79	C₂×Dic₁₄₆	—	4	1	no	—
80	C₂×Dic₁₄₆	D₈	1	1	yes	characteristic
81	C₂×D₂₉₂	—	2	2	no	—
82	C₂×D₂₉₂	—	4	1	no	—
83	C₂×D₂₉₂	—	4	1	no	—
84	C₂²×Dic₇₃	—	4	2	no	—
85	C₂×C₄×D₇₃	—	4	2	no	—
86	#11924015	—	8	2	no	—
87	D₅₈₄	—	2	2	no	—
88	D₅₈₄	—	2	2	no	—
89	D₅₈₄	—	8	1	no	—
90	C₂×#10596831	C₂×D₄	1	2	yes	—
91	C₄⋊Dic₇₃	—	2	2	no	—
92	C₄⋊Dic₇₃	—	4	1	no	—
93	C₄⋊Dic₇₃	—	4	2	no	—
94	C₄⋊Dic₇₃	D₈	1	1	yes	characteristic
95	#11924007	—	4	2	no	—
96	#11924007	D₈	1	2	yes	—
97	C₂²⋊Dic₇₃	—	4	2	no	—
98	#11924005	—	4	2	no	—
99	#11924005	—	4	2	no	—
100	#11924005	—	4	2	no	—
101	C₄×Dic₇₃	C₂×D₄	1	2	yes	—
102	#11924003	—	2	2	no	—
103	C₂×Q₈×C₇₃	—	2	1	no	—
104	C₂×D₄×C₇₃	—	2	1	no	—
105	C₂²×C₄×C₇₃	—	4	1	no	—
106	Q₁₆×C₇₃	—	2	1	no	—
107	Q₁₆×C₇₃	—	2	1	no	—
108	D₈×C₇₃	—	2	1	no	—
109	D₈×C₇₃	#41	1	2	yes	—
110	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
111	C₂×C₈×C₇₃	C₂⁴	1	1	yes	characteristic
112	C₄⋊C₄×C₇₃	—	2	1	no	—
113	C₄⋊C₄×C₇₃	—	2	1	no	—
114	C₄⋊C₄×C₇₃	—	4	1	no	—
115	C₂²⋊C₄×C₇₃	—	4	1	no	—
116	C₄²×C₇₃	—	4	1	no	—
117	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
118	C₁₆×C₇₃	—	2	1	no	—
119	C₁₆×C₇₃	—	2	1	no	—
120	C₂³×C₇₃	—	4	1	no	—
121	C₂²×D₇₃	—	4	2	no	—
122	Q₈×C₇₃	—	4	1	no	—
123	Q₈×C₇₃	—	4	1	no	—
124	D₄×C₇₃	—	2	2	no	—
125	D₄×C₇₃	—	4	1	no	—
126	C₂×C₄×C₇₃	—	4	1	no	—
127	C₂×C₄×C₇₃	—	4	1	no	—
128	C₂×C₄×C₇₃	—	4	1	no	—
129	C₂×C₄×C₇₃	—	8	2	no	—
130	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
131	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
132	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
133	C₂×Dic₇₃	—	4	2	no	—
134	C₂×Dic₇₃	—	16	2	no	—
135	C₂×Dic₇₃	C₂×D₈	1	2	yes	—
136	C₂×Dic₇₃	C₂×D₈	1	2	yes	—
137	D₂₉₂	—	4	2	no	—
138	D₂₉₂	—	4	2	no	—
139	D₂₉₂	—	8	2	no	—
140	D₂₉₂	—	16	1	no	—
141	C₄×D₇₃	—	16	2	no	—
142	Dic₁₄₆	—	2	2	no	—
143	Dic₁₄₆	—	16	1	no	—
144	C₈×C₇₃	—	2	1	no	—
145	C₈×C₇₃	C₂×#41	1	1	yes	characteristic
146	C₈×C₇₃	2₋¹⁺⁴	1	1	yes	characteristic
147	#10596831	—	2	2	no	—
148	C₂²×C₇₃	—	4	2	no	—
149	C₂²×C₇₃	—	8	1	no	—
150	C₂²×C₇₃	—	8	1	no	—
151	C₂²×C₇₃	C₂²×D₈	1	1	yes	characteristic
152	C₂×D₇₃	—	8	2	no	—
153	C₂×D₇₃	—	8	2	no	—
154	C₂×D₇₃	—	8	2	no	—
155	C₄×C₇₃	—	2	1	no	—
156	C₄×C₇₃	—	8	1	no	—
157	C₄×C₇₃	—	8	1	no	—
158	C₄×C₇₃	—	8	1	no	—
159	C₄×C₇₃	#508	1	2	yes	—
160	C₄×C₇₃	#548	1	1	yes	characteristic
161	C₄×C₇₃	#577	1	1	yes	characteristic
162	Dic₇₃	—	2	2	no	—
163	Dic₇₃	—	2	2	no	—
164	Dic₇₃	—	8	2	no	—
165	#33928	—	73	1	no	maximal, Sylow
166	C₂×C₇₃	—	8	2	no	—
167	C₂×C₇₃	#3358	1	1	yes	characteristic
168	C₂×C₇₃	C₂×#508	1	1	yes	characteristic
169	C₂×C₇₃	#3415	1	1	yes	characteristic
170	D₇₃	—	16	2	no	—
171	#3396	—	73	1	no	—
172	#3380	—	73	1	no	—
173	#2242	—	73	2	no	—
174	#2240	—	73	1	no	—
175	#2228	—	73	2	no	—
176	#2227	—	73	1	no	—
177	#2210	—	73	2	no	—
178	#2206	—	73	1	no	—
179	Q₈×C₁₆	—	73	1	no	—
180	C₄×SD₃₂	D₇₃	1	1	yes	characteristic, complemented
181	C₄×D₁₆	—	73	2	no	—
182	C₇₃	#33928	1	1	yes	characteristic, complemented, Sylow
183	#559	—	146	1	no	—
184	#554	—	146	1	no	—
185	C₂×SD₃₂	C₂×D₇₃	1	1	yes	characteristic
186	C₂×D₁₆	—	73	2	no	—
187	#500	—	73	1	no	—
188	#499	—	73	2	no	—
189	#495	—	73	2	no	—
190	#493	—	73	1	no	—
191	#478	—	146	2	no	—
192	#476	—	146	1	no	—
193	#463	—	146	2	no	—
194	#460	—	146	1	no	—
195	C₈×Q₈	—	73	1	no	—
196	C₄×Q₁₆	C₂×D₇₃	1	1	yes	characteristic
197	C₄×D₈	—	73	2	no	—
198	C₄×D₈	—	73	2	no	—
199	C₄×D₈	C₂×D₇₃	1	1	yes	characteristic
200	#367	C₂×D₇₃	1	1	yes	characteristic
201	#366	—	73	2	no	—
202	C₄⋊C₁₆	—	73	2	no	—
203	C₄⋊C₁₆	—	73	1	no	—
204	#358	—	73	2	no	—
205	#358	C₂×D₇₃	1	1	yes	characteristic
206	#357	—	73	2	no	—
207	#357	—	73	2	no	—
208	#357	—	73	2	no	—
209	#357	C₂×D₇₃	1	1	yes	characteristic
210	C₄×C₁₆	—	73	2	no	—
211	C₄×C₁₆	C₂×D₇₃	1	1	yes	characteristic
212	C₂×Q₁₆	C₂²×D₇₃	1	1	yes	characteristic
213	C₂×SD₁₆	—	146	2	no	—
214	C₂×D₈	—	73	2	no	—
215	C₂×D₈	—	146	1	no	—
216	C₂×D₈	C₂²×D₇₃	1	1	yes	characteristic
217	C₄⋊Q₈	—	146	1	no	—
218	#127	—	146	1	no	—
219	#125	—	146	2	no	—
220	#124	—	292	1	no	—
221	#124	—	146	2	no	—
222	#123	—	292	2	no	—
223	C₂²⋊Q₈	—	292	2	no	—
224	C₂²⋊Q₈	—	292	1	no	—
225	C₄×Q₈	—	2	1	no	—
226	C₄×Q₈	—	73	1	no	—
227	C₄×D₄	—	2	1	no	—
228	C₄×D₄	—	292	1	no	—
229	C₄×D₄	—	146	2	no	—
230	C₄×D₄	—	146	2	no	—
231	C₂×C₄⋊C₄	—	292	1	no	—
232	SD₃₂	—	2	2	no	—
233	D₁₆	—	146	4	no	—
234	C₂×C₁₆	—	73	2	no	—
235	C₂×C₁₆	C₂²×D₇₃	1	1	yes	characteristic
236	C₂×C₁₆	C₂²×D₇₃	1	1	yes	characteristic
237	#107	—	73	2	no	—
238	#107	—	146	1	no	—
239	#107	C₂²×D₇₃	1	1	yes	characteristic
240	#107	C₂²×D₇₃	1	1	yes	characteristic
241	#106	—	146	2	no	—
242	C₄⋊C₈	—	73	2	no	—
243	C₄⋊C₈	—	73	1	no	—
244	#103	—	2	1	no	—
245	#102	—	2	1	no	—
246	#102	—	146	2	no	—
247	#102	—	146	2	no	—
248	C₄×C₈	—	73	2	no	—
249	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
250	C₂×Q₈	—	2	1	no	—
251	C₂×Q₈	—	292	1	no	—
252	C₂×Q₈	—	73	1	no	—
253	C₂×D₄	—	2	1	no	—
254	C₂×D₄	—	292	1	no	—
255	C₂×D₄	—	292	1	no	—
256	C₂×D₄	—	146	2	no	—
257	C₂²×C₄	—	4	1	no	—
258	C₂²×C₄	—	292	2	no	—
259	C₂²×C₄	—	292	2	no	—
260	Q₁₆	—	2	1	no	—
261	Q₁₆	—	2	1	no	—
262	SD₁₆	—	584	2	no	—
263	D₈	—	2	1	no	—
264	D₈	—	584	1	no	—
265	D₈	—	146	2	no	—
266	D₈	—	146	2	no	—
267	D₈	#11924028	1	2	yes	—
268	C₂×C₈	—	73	2	no	—
269	C₂×C₈	D₄×D₇₃	1	1	yes	characteristic
270	C₂×C₈	C₂³×D₇₃	1	1	yes	characteristic
271	C₄⋊C₄	—	2	1	no	—
272	C₄⋊C₄	—	2	1	no	—
273	C₄⋊C₄	—	4	1	no	—
274	C₄⋊C₄	—	292	2	no	—
275	C₄⋊C₄	—	292	2	no	—
276	C₄⋊C₄	—	292	1	no	—
277	C₄⋊C₄	—	73	2	no	—
278	C₄⋊C₄	—	73	1	no	—
279	C₄⋊C₄	—	146	2	no	—
280	C₂²⋊C₄	—	4	1	no	—
281	C₂²⋊C₄	—	292	2	no	—
282	C₂²⋊C₄	—	292	2	no	—
283	C₂²⋊C₄	—	292	2	no	—
284	C₂²⋊C₄	—	292	2	no	—
285	C₄²	—	4	1	no	—
286	C₄²	—	73	2	no	—
287	C₄²	D₄×D₇₃	1	1	yes	characteristic
288	C₁₆	—	2	1	no	—
289	C₁₆	—	2	1	no	—
290	C₁₆	—	146	2	no	—
291	C₂³	—	4	1	no	—
292	C₂³	—	292	2	no	—
293	Q₈	—	4	1	no	—
294	Q₈	—	4	1	no	—
295	Q₈	—	1168	1	no	—
296	Q₈	—	146	2	no	—
297	D₄	—	2	2	no	—
298	D₄	—	4	1	no	—
299	D₄	—	292	2	no	—
300	D₄	—	292	2	no	—
301	D₄	—	584	2	no	—
302	D₄	—	1168	1	no	—
303	C₂×C₄	—	4	1	no	—
304	C₂×C₄	—	4	1	no	—
305	C₂×C₄	—	4	1	no	—
306	C₂×C₄	—	8	2	no	—
307	C₂×C₄	—	292	2	no	—
308	C₂×C₄	—	73	2	no	—
309	C₂×C₄	—	73	2	no	—
310	C₂×C₄	—	1168	2	no	—
311	C₂×C₄	—	1168	2	no	—
312	C₂×C₄	D₈×D₇₃	1	1	yes	characteristic
313	C₂×C₄	D₈×D₇₃	1	1	yes	characteristic
314	C₂×C₄	C₂×D₄×D₇₃	1	1	yes	characteristic
315	C₈	—	2	1	no	—
316	C₈	—	146	2	no	—
317	C₈	C₂×#11924028	1	1	yes	characteristic
318	C₈	#14556638	1	1	yes	characteristic
319	C₂²	—	4	2	no	—
320	C₂²	—	8	1	no	—
321	C₂²	—	8	1	no	—
322	C₂²	—	584	2	no	—
323	C₂²	—	584	2	no	—
324	C₂²	—	584	2	no	—
325	C₂²	C₂×D₈×D₇₃	1	1	yes	characteristic
326	C₄	—	2	1	no	—
327	C₄	—	8	1	no	—
328	C₄	—	8	1	no	—
329	C₄	—	8	1	no	—
330	C₄	—	584	2	no	—
331	C₄	—	146	2	no	—
332	C₄	—	146	2	no	—
333	C₄	#17989018	1	2	yes	—
334	C₄	#17989246	1	1	yes	characteristic
335	C₄	#17989394	1	1	yes	characteristic
336	C₂	—	8	2	no	—
337	C₂	—	1168	2	no	—
338	C₂	#22622564	1	1	yes	characteristic
339	C₂	C₂×#17989018	1	1	yes	characteristic
340	C₂	#22623069	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₈×D₇₃
commutator, cosocle	C₂×C₈×C₇₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₇₃
Fitting	C₄×SD₃₂×C₇₃	C₂
socle	C₂²×C₇₃	C₂²×D₈
prime radical	#22611966	C₂
hypercenter	C₄×SD₃₂	D₇₃
hypocenter	C₇₃	#33928
LCS	G ▷ C₂×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×SD₃₂

Complements

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

N	H ≅ G/N	type
1. #22622941	C₂	semidirect
2. #22622796	C₂	semidirect
3. #22612164	C₂	semidirect
4. #22612155	C₂	semidirect
5. #22612087	C₂	semidirect
6. #22612083	C₂	semidirect
7. #22611988	C₂	semidirect
9. C₁₆×Dic₁₄₆	C₂	semidirect
10. #22611775	C₂	semidirect
11. C₄×SD₃₂×C₇₃	C₂	semidirect
180. C₄×SD₃₂	D₇₃	semidirect
182. C₇₃	#33928	semidirect

Maximal subgroups

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

subgroup	index
#22622941	2
#22622796	2
#22612164	2
#22612155	2
#22612087	2
#22612083	2
#22611988	2
#22611966	2
C₁₆×Dic₁₄₆	2
#22611775	2
C₄×SD₃₂×C₇₃	2
#33928	73