Subgroups — #33241544

Order 18688.

558 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22620515	C₂	1	1	yes	maximal, characteristic, complemented
2	#22620508	C₂	1	1	yes	maximal, characteristic
3	#22612213	C₂	1	1	yes	maximal, characteristic, complemented
4	#22612158	C₂	1	1	yes	maximal, characteristic, complemented
5	#22612154	C₂	1	1	yes	maximal, characteristic, complemented
6	#22612024	C₂	1	4	yes	maximal, complemented
7	SD₃₂×Dic₇₃	C₂	1	4	yes	maximal, complemented
8	C₄×C₁₆×D₇₃	C₂	1	1	yes	maximal, characteristic, complemented
9	C₇₃×#2248	C₂	1	1	yes	maximal, characteristic, complemented
10	#17989142	—	2	1	no	—
11	#17989135	—	2	1	no	—
12	C₂×C₁₆×D₇₃	C₂²	1	2	yes	—
13	#17988992	C₂²	1	1	yes	characteristic
14	#17988992	C₂²	1	1	yes	characteristic
15	#17988937	—	2	1	no	—
16	#17988933	—	2	1	no	—
17	#17988825	—	2	1	no	—
18	#17988825	C₂²	1	2	yes	—
19	#17988820	—	2	1	no	—
20	#17988820	C₂²	1	2	yes	—
21	Q₁₆×Dic₇₃	—	2	1	no	—
22	Q₁₆×Dic₇₃	C₂²	1	2	yes	—
23	D₈×Dic₇₃	—	2	1	no	—
24	D₈×Dic₇₃	C₂²	1	2	yes	—
25	C₄×C₈×D₇₃	C₂²	1	1	yes	characteristic
26	#17988415	C₂²	1	4	yes	—
27	#17988391	C₂²	1	4	yes	—
28	#17988388	C₂²	1	4	yes	—
29	C₄×#11924003	C₂²	1	1	yes	characteristic
30	C₁₆×Dic₇₃	C₂²	1	2	yes	—
31	C₂×SD₃₂×C₇₃	C₂²	1	4	yes	—
32	#494×C₇₃	C₂²	1	1	yes	characteristic
33	#493×C₇₃	C₂²	1	1	yes	characteristic
34	C₄×C₁₆×C₇₃	C₂²	1	1	yes	characteristic
35	C₂×#11924027	—	4	1	no	—
36	C₂×C₈×D₇₃	D₄	1	1	yes	characteristic
37	C₂×C₈×D₇₃	C₂³	1	1	yes	characteristic
38	#14556591	—	2	1	no	—
39	#14556591	—	2	1	no	—
40	#14556582	—	4	1	no	—
41	#14556578	—	4	1	no	—
42	#14556571	—	4	1	no	—
43	#14556567	—	2	2	no	—
44	#14556562	—	2	2	no	—
45	#14556562	—	4	1	no	—
46	Q₈×Dic₇₃	—	2	2	no	—
47	#14556556	—	4	1	no	—
48	D₄×Dic₇₃	—	2	2	no	—
49	D₄×Dic₇₃	—	4	1	no	—
50	#14556549	—	4	1	no	—
51	C₄²×D₇₃	D₄	1	1	yes	characteristic
52	C₂×#11924003	C₂³	1	2	yes	—
53	C₁₆×D₇₃	—	2	4	no	—
54	#14556525	—	2	1	no	—
55	#14556525	—	2	1	no	—
56	#14556525	C₂³	1	2	yes	—
57	#14556525	C₂³	1	2	yes	—
58	#14556515	—	2	2	no	—
59	#14556515	—	2	2	no	—
60	#14556512	—	2	2	no	—
61	#14556512	—	2	2	no	—
62	C₄×#10596831	D₄	1	1	yes	characteristic
63	C₈×Dic₇₃	D₄	1	1	yes	characteristic
64	C₈×Dic₇₃	C₂³	1	1	yes	characteristic
65	C₂×Q₁₆×C₇₃	—	2	1	no	—
66	C₂×Q₁₆×C₇₃	C₂³	1	2	yes	—
67	C₂×D₈×C₇₃	—	2	1	no	—
68	C₂×D₈×C₇₃	C₂³	1	2	yes	—
69	C₄⋊Q₈×C₇₃	—	2	1	no	—
70	#127×C₇₃	—	2	1	no	—
71	SD₃₂×C₇₃	—	2	8	no	—
72	C₂×C₁₆×C₇₃	C₂³	1	2	yes	—
73	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
74	#11924027	—	8	4	no	—
75	C₂×Dic₁₄₆	—	4	1	no	—
76	C₂×Dic₁₄₆	—	4	1	no	—
77	C₂×C₂²⋊D₇₃	—	4	2	no	—
78	C₂²×Dic₇₃	—	4	2	no	—
79	C₂×C₄×D₇₃	D₈	1	1	yes	characteristic
80	C₂×C₄×D₇₃	D₈	1	1	yes	characteristic
81	C₂×C₄×D₇₃	C₂×D₄	1	1	yes	characteristic
82	C₂×#10596831	C₂×D₄	1	1	yes	characteristic
83	C₂×#10596831	C₂×D₄	1	1	yes	characteristic
84	C₈×D₇₃	—	2	2	no	—
85	C₈×D₇₃	—	2	2	no	—
86	C₄⋊Dic₇₃	—	2	2	no	—
87	C₄⋊Dic₇₃	—	2	2	no	—
88	C₄⋊Dic₇₃	—	4	1	no	—
89	C₄⋊Dic₇₃	—	4	1	no	—
90	C₄⋊Dic₇₃	—	4	2	no	—
91	#11924007	—	4	2	no	—
92	C₂²⋊Dic₇₃	—	4	2	no	—
93	C₂²⋊Dic₇₃	—	4	2	no	—
94	#11924005	—	4	2	no	—
95	C₄×Dic₇₃	—	4	2	no	—
96	C₄×Dic₇₃	D₈	1	1	yes	characteristic
97	C₄×Dic₇₃	D₈	1	1	yes	characteristic
98	C₄×Dic₇₃	C₂×D₄	1	1	yes	characteristic
99	#11924003	—	2	2	no	—
100	C₂×Q₈×C₇₃	—	2	2	no	—
101	C₂×D₄×C₇₃	—	2	2	no	—
102	C₂×D₄×C₇₃	—	4	1	no	—
103	C₂×D₄×C₇₃	—	4	1	no	—
104	Q₁₆×C₇₃	—	2	4	no	—
105	Q₁₆×C₇₃	—	4	2	no	—
106	D₈×C₇₃	—	2	4	no	—
107	D₈×C₇₃	—	4	2	no	—
108	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
109	C₂×C₈×C₇₃	C₂⁴	1	1	yes	characteristic
110	C₄⋊C₄×C₇₃	—	4	1	no	—
111	C₄⋊C₄×C₇₃	—	4	1	no	—
112	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
113	C₁₆×C₇₃	#41	1	4	yes	complemented
114	C₂³×C₇₃	—	4	2	no	—
115	C₂²×D₇₃	C₂×D₈	1	1	yes	characteristic
116	Q₈×C₇₃	—	4	2	no	—
117	Q₈×C₇₃	—	4	2	no	—
118	D₄×C₇₃	—	4	2	no	—
119	D₄×C₇₃	—	4	2	no	—
120	D₄×C₇₃	—	8	2	no	—
121	D₄×C₇₃	—	8	2	no	—
122	C₂×C₄×C₇₃	—	4	2	no	—
123	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
124	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
125	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
126	C₂²⋊D₇₃	—	8	4	no	—
127	C₂×Dic₇₃	—	4	2	no	—
128	C₂×Dic₇₃	—	4	2	no	—
129	C₂×Dic₇₃	—	8	4	no	—
130	C₂×Dic₇₃	C₂×D₈	1	1	yes	characteristic
131	C₂×Dic₇₃	C₂×D₈	1	1	yes	characteristic
132	C₂×Dic₇₃	C₂×D₈	1	1	yes	characteristic
133	C₄×D₇₃	—	2	1	no	—
134	C₄×D₇₃	—	2	1	no	—
135	C₄×D₇₃	—	2	2	no	—
136	C₄×D₇₃	SD₃₂	1	4	yes	complemented
137	Dic₁₄₆	—	8	2	no	—
138	Dic₁₄₆	—	8	2	no	—
139	C₈×C₇₃	#135	1	2	yes	—
140	C₈×C₇₃	C₂×#41	1	2	yes	—
141	#10596831	—	2	1	no	—
142	#10596831	—	2	1	no	—
143	C₂²×C₇₃	—	8	2	no	—
144	C₂²×C₇₃	—	8	2	no	—
145	C₂²×C₇₃	—	8	2	no	—
146	C₂²×C₇₃	C₂²×D₈	1	1	yes	characteristic
147	C₂×D₇₃	—	2	1	no	—
148	C₂×D₇₃	—	2	1	no	—
149	C₂×D₇₃	C₂×SD₃₂	1	2	yes	—
150	C₄×C₇₃	—	8	2	no	—
151	C₄×C₇₃	C₂×SD₃₂	1	2	yes	—
152	C₄×C₇₃	#508	1	2	yes	—
153	C₄×C₇₃	#532	1	1	yes	characteristic
154	C₄×C₇₃	C₂×#135	1	1	yes	characteristic
155	Dic₇₃	—	2	1	no	—
156	Dic₇₃	—	2	1	no	—
157	Dic₇₃	—	8	2	no	—
158	Dic₇₃	—	8	2	no	—
159	Dic₇₃	C₂×SD₃₂	1	2	yes	—
160	#33736	—	73	1	no	maximal, Sylow
161	C₂×C₇₃	—	16	2	no	—
162	C₂×C₇₃	#3169	1	1	yes	characteristic
163	C₂×C₇₃	C₂²×SD₃₂	1	1	yes	characteristic
164	C₂×C₇₃	C₂×#508	1	1	yes	characteristic
165	D₇₃	—	2	2	no	—
166	#3170	—	73	1	no	—
167	#3169	—	73	1	no	—
168	#2254	—	73	1	no	—
169	#2248	D₇₃	1	1	yes	characteristic, complemented
170	#2241	—	73	1	no	—
171	#2240	—	73	1	no	—
172	#2217	—	73	4	no	—
173	C₄×SD₃₂	—	73	4	no	—
174	C₂×C₄×C₁₆	—	73	1	no	—
175	C₇₃	#33736	1	1	yes	characteristic, complemented, Sylow
176	#533	—	146	1	no	—
177	#532	—	146	1	no	—
178	C₂×SD₃₂	C₂×D₇₃	1	4	yes	—
179	C₂²×C₁₆	—	73	2	no	—
180	#500	—	73	1	no	—
181	#500	—	73	1	no	—
182	#494	C₂×D₇₃	1	1	yes	characteristic
183	#493	C₂×D₇₃	1	1	yes	characteristic
184	#487	—	146	1	no	—
185	#486	—	146	1	no	—
186	#467	—	73	2	no	—
187	#467	—	146	1	no	—
188	#466	—	73	2	no	—
189	#466	—	146	1	no	—
190	C₄×Q₁₆	—	73	2	no	—
191	C₄×Q₁₆	—	146	1	no	—
192	C₄×D₈	—	73	2	no	—
193	C₄×D₈	—	146	1	no	—
194	C₂×C₄×C₈	—	73	1	no	—
195	#367	—	73	4	no	—
196	#358	—	73	4	no	—
197	#357	—	73	4	no	—
198	C₄×C₁₆	—	73	2	no	—
199	C₄×C₁₆	—	73	1	no	—
200	C₄×C₁₆	C₂×D₇₃	1	1	yes	characteristic
201	C₂×#41	—	292	1	no	—
202	C₂×Q₁₆	—	2	1	no	—
203	C₂×Q₁₆	C₂²×D₇₃	1	2	yes	—
204	C₂×D₈	—	2	1	no	—
205	C₂×D₈	C₂²×D₇₃	1	2	yes	—
206	C₂²×C₈	—	73	1	no	—
207	C₂²×C₈	—	73	1	no	—
208	C₄⋊Q₈	—	2	1	no	—
209	C₄⋊Q₈	—	146	1	no	—
210	C₄⋊Q₈	—	146	1	no	—
211	#127	—	2	1	no	—
212	#125	—	292	1	no	—
213	#124	—	292	1	no	—
214	C₂²⋊Q₈	—	292	1	no	—
215	C₂²⋊Q₈	—	146	2	no	—
216	#121	—	292	1	no	—
217	#121	—	146	2	no	—
218	C₄×Q₈	—	292	1	no	—
219	C₄×Q₈	—	146	2	no	—
220	C₄×D₄	—	292	1	no	—
221	C₄×D₄	—	146	2	no	—
222	#117	—	292	1	no	—
223	C₂×C₄²	—	73	1	no	—
224	SD₃₂	—	2	8	no	—
225	C₂×C₁₆	—	73	2	no	—
226	C₂×C₁₆	—	146	4	no	—
227	C₂×C₁₆	C₂²×D₇₃	1	2	yes	—
228	#107	—	73	2	no	—
229	#107	—	73	2	no	—
230	#107	—	146	1	no	—
231	#107	—	146	1	no	—
232	#103	—	146	2	no	—
233	#103	—	146	2	no	—
234	#102	—	146	2	no	—
235	#102	—	146	2	no	—
236	C₄×C₈	—	73	1	no	—
237	C₄×C₈	—	73	1	no	—
238	C₄×C₈	—	73	1	no	—
239	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
240	#41	—	584	4	no	—
241	C₂×Q₈	—	2	2	no	—
242	C₂×Q₈	—	292	1	no	—
243	C₂×Q₈	—	292	1	no	—
244	C₂×D₄	—	2	2	no	—
245	C₂×D₄	—	4	1	no	—
246	C₂×D₄	—	4	1	no	—
247	C₂×D₄	—	292	2	no	—
248	C₂²×C₄	—	292	2	no	—
249	C₂²×C₄	—	73	1	no	—
250	C₂²×C₄	—	73	1	no	—
251	C₂²×C₄	—	73	1	no	—
252	Q₁₆	—	2	4	no	—
253	Q₁₆	—	4	2	no	—
254	D₈	—	2	4	no	—
255	D₈	—	4	2	no	—
256	C₂×C₈	—	73	1	no	—
257	C₂×C₈	—	73	1	no	—
258	C₂×C₈	—	146	2	no	—
259	C₂×C₈	—	146	2	no	—
260	C₂×C₈	D₄×D₇₃	1	1	yes	characteristic
261	C₂×C₈	C₂³×D₇₃	1	1	yes	characteristic
262	C₄⋊C₄	—	4	1	no	—
263	C₄⋊C₄	—	4	1	no	—
264	C₄⋊C₄	—	292	2	no	—
265	C₄⋊C₄	—	292	2	no	—
266	C₄⋊C₄	—	292	1	no	—
267	C₄⋊C₄	—	292	1	no	—
268	C₄⋊C₄	—	146	2	no	—
269	C₄⋊C₄	—	146	2	no	—
270	C₂²⋊C₄	—	292	2	no	—
271	C₂²⋊C₄	—	292	2	no	—
272	C₂²⋊C₄	—	292	2	no	—
273	C₄²	—	292	2	no	—
274	C₄²	—	73	1	no	—
275	C₄²	—	73	1	no	—
276	C₄²	—	73	1	no	—
277	C₄²	D₄×D₇₃	1	1	yes	characteristic
278	C₁₆	—	146	2	no	—
279	C₁₆	#11924027	1	4	yes	complemented
280	C₂³	—	4	2	no	—
281	C₂³	—	73	1	no	—
282	Q₈	—	4	2	no	—
283	Q₈	—	4	2	no	—
284	Q₈	—	584	2	no	—
285	Q₈	—	584	2	no	—
286	D₄	—	4	2	no	—
287	D₄	—	4	2	no	—
288	D₄	—	8	2	no	—
289	D₄	—	8	2	no	—
290	D₄	—	584	4	no	—
291	C₂×C₄	—	4	2	no	—
292	C₂×C₄	—	292	2	no	—
293	C₂×C₄	—	292	2	no	—
294	C₂×C₄	—	584	4	no	—
295	C₂×C₄	—	73	1	no	—
296	C₂×C₄	—	73	4	no	—
297	C₂×C₄	—	73	1	no	—
298	C₂×C₄	—	73	1	no	—
299	C₂×C₄	—	146	1	no	—
300	C₂×C₄	—	146	2	no	—
301	C₂×C₄	—	146	1	no	—
302	C₂×C₄	D₈×D₇₃	1	1	yes	characteristic
303	C₂×C₄	D₈×D₇₃	1	1	yes	characteristic
304	C₂×C₄	C₂×D₄×D₇₃	1	1	yes	characteristic
305	C₈	—	146	1	no	—
306	C₈	—	146	1	no	—
307	C₈	#14556610	1	2	yes	—
308	C₈	C₂×#11924027	1	2	yes	—
309	C₂²	—	8	2	no	—
310	C₂²	—	8	2	no	—
311	C₂²	—	8	2	no	—
312	C₂²	—	73	2	no	—
313	C₂²	—	146	1	no	—
314	C₂²	—	146	1	no	—
315	C₂²	C₂×D₈×D₇₃	1	1	yes	characteristic
316	C₄	—	8	2	no	—
317	C₄	—	584	2	no	—
318	C₄	—	584	2	no	—
319	C₄	—	73	2	no	—
320	C₄	—	146	1	no	—
321	C₄	—	146	1	no	—
322	C₄	SD₃₂×D₇₃	1	2	yes	—
323	C₄	#17989018	1	2	yes	—
324	C₄	#17989135	1	1	yes	characteristic
325	C₄	C₂×#14556610	1	1	yes	characteristic
326	C₂	—	16	2	no	—
327	C₂	—	146	2	no	—
328	C₂	#22620508	1	1	yes	characteristic
329	C₂	C₂×SD₃₂×D₇₃	1	1	yes	characteristic
330	C₂	C₂×#17989018	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×#2248	C₂
socle	C₂²×C₇₃	C₂²×D₈
prime radical	#22620508	C₂
hypercenter	#2248	D₇₃
hypocenter	C₇₃	#33736
LCS	G ▷ C₂×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2248

Complements

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

N	H ≅ G/N	type
1. #22620515	C₂	semidirect
3. #22612213	C₂	semidirect
4. #22612158	C₂	semidirect
5. #22612154	C₂	semidirect
6. #22612024	C₂	semidirect
7. SD₃₂×Dic₇₃	C₂	semidirect
8. C₄×C₁₆×D₇₃	C₂	semidirect
9. C₇₃×#2248	C₂	semidirect
113. C₁₆×C₇₃	#41	semidirect
136. C₄×D₇₃	SD₃₂	semidirect
169. #2248	D₇₃	semidirect
175. C₇₃	#33736	semidirect
279. C₁₆	#11924027	semidirect

Maximal subgroups

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

subgroup	index
#22620515	2
#22620508	2
#22612213	2
#22612158	2
#22612154	2
#22612024	2
SD₃₂×Dic₇₃	2
C₄×C₁₆×D₇₃	2
C₇₃×#2248	2
#33736	73