Subgroups — #33241352

Order 18688.

622 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#500×D₇₃	C₂	1	1	yes	maximal, characteristic, complemented
2	#22620358	C₂	1	1	yes	maximal, characteristic
3	#22612166	C₂	1	4	yes	maximal, complemented
4	#22612160	C₂	1	1	yes	maximal, characteristic, complemented
5	#22612159	C₂	1	1	yes	maximal, characteristic, complemented
6	#22612154	C₂	1	1	yes	maximal, characteristic, complemented
7	#22611640	C₂	1	4	yes	maximal, complemented
8	C₄×#14556530	C₂	1	1	yes	maximal, characteristic, complemented
9	C₇₃×#2240	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#14556610	—	2	1	no	—
11	C₂×#14556610	C₂²	1	2	yes	—
12	#17989135	—	2	1	no	—
13	C₄⋊Q₈×D₇₃	—	2	1	no	—
14	C₂×#14556530	C₂²	1	2	yes	—
15	#17988992	C₂²	1	1	yes	characteristic
16	#17988991	—	2	1	no	—
17	#17988991	C₂²	1	2	yes	—
18	#17988990	—	2	1	no	—
19	#17988970	—	2	1	no	—
20	#17988970	C₂²	1	2	yes	—
21	#17988967	C₂²	1	1	yes	characteristic
22	#17988959	—	2	1	no	—
23	#107×D₇₃	—	2	1	no	—
24	#107×D₇₃	C₂²	1	2	yes	—
25	C₄×C₈×D₇₃	C₂²	1	1	yes	characteristic
26	#17988392	C₂²	1	4	yes	—
27	#17988390	C₂²	1	4	yes	—
28	#17988388	C₂²	1	4	yes	—
29	#17988362	C₂²	1	2	yes	—
30	C₄×#11924003	C₂²	1	1	yes	characteristic
31	#500×C₇₃	C₂²	1	1	yes	characteristic
32	#493×C₇₃	C₂²	1	1	yes	characteristic
33	#357×C₇₃	C₂²	1	4	yes	—
34	C₄×C₁₆×C₇₃	C₂²	1	1	yes	characteristic
35	C₂×#11924027	—	2	2	no	—
36	C₂×Q₈×D₇₃	—	4	1	no	—
37	#14556610	—	2	8	no	—
38	#14556610	—	4	4	no	—
39	C₂×Dic₂₉₂	—	2	1	no	—
40	C₂×Dic₂₉₂	C₂³	1	2	yes	—
41	C₂×#11924016	—	2	2	no	—
42	C₂×#11924016	—	2	2	no	—
43	C₂×C₈×D₇₃	D₄	1	1	yes	characteristic
44	C₂×C₈×D₇₃	C₂³	1	1	yes	characteristic
45	#14556591	—	2	1	no	—
46	#14556591	—	2	1	no	—
47	#14556590	—	2	2	no	—
48	#14556590	—	4	1	no	—
49	#14556589	—	4	1	no	—
50	#14556578	—	2	2	no	—
51	#14556562	—	4	1	no	—
52	#14556562	—	4	1	no	—
53	D₄×Dic₇₃	—	4	1	no	—
54	D₄×Dic₇₃	—	4	1	no	—
55	C₄⋊C₄×D₇₃	—	2	2	no	—
56	C₄⋊C₄×D₇₃	—	4	1	no	—
57	C₄²×D₇₃	D₄	1	1	yes	characteristic
58	#14556530	—	2	4	no	—
59	C₂×#11924003	C₂³	1	2	yes	—
60	#14556525	—	2	1	no	—
61	#14556525	C₂³	1	2	yes	—
62	#14556524	—	2	2	no	—
63	#14556524	—	2	2	no	—
64	C₄×#10596831	D₄	1	1	yes	characteristic
65	C₈×Dic₇₃	D₄	1	1	yes	characteristic
66	C₈×Dic₇₃	C₂³	1	1	yes	characteristic
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	C₂×C₁₆×C₇₃	C₂³	1	2	yes	—
72	#107×C₇₃	—	2	1	no	—
73	#107×C₇₃	C₂³	1	2	yes	—
74	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
75	#11924027	—	4	4	no	—
76	#11924027	—	4	4	no	—
77	C₂×Dic₁₄₆	—	2	2	no	—
78	C₂×Dic₁₄₆	—	4	1	no	—
79	C₂×Dic₁₄₆	—	4	2	no	—
80	Q₈×D₇₃	—	8	4	no	—
81	C₂×C₂²⋊D₇₃	—	4	2	no	—
82	C₂²×Dic₇₃	—	4	2	no	—
83	C₂×C₄×D₇₃	—	4	2	no	—
84	C₂×C₄×D₇₃	D₈	1	1	yes	characteristic
85	C₂×C₄×D₇₃	D₈	1	1	yes	characteristic
86	C₂×C₄×D₇₃	C₂×D₄	1	1	yes	characteristic
87	Dic₂₉₂	—	2	4	no	—
88	Dic₂₉₂	—	4	2	no	—
89	#11924016	—	4	4	no	—
90	#11924016	—	4	4	no	—
91	C₂×#10596831	C₂×D₄	1	1	yes	characteristic
92	C₂×#10596831	C₂×D₄	1	1	yes	characteristic
93	C₈×D₇₃	SD₁₆	1	4	yes	—
94	C₈×D₇₃	#41	1	4	yes	—
95	C₄⋊Dic₇₃	—	2	2	no	—
96	C₄⋊Dic₇₃	—	4	1	no	—
97	C₄⋊Dic₇₃	—	4	1	no	—
98	C₄⋊Dic₇₃	—	4	1	no	—
99	#11924007	—	4	2	no	—
100	#11924007	—	4	2	no	—
101	C₂²⋊Dic₇₃	—	4	2	no	—
102	C₂²⋊Dic₇₃	—	4	2	no	—
103	C₄×Dic₇₃	D₈	1	1	yes	characteristic
104	C₄×Dic₇₃	D₈	1	1	yes	characteristic
105	C₄×Dic₇₃	C₂×D₄	1	1	yes	characteristic
106	#11924003	—	2	2	no	—
107	C₂×Q₈×C₇₃	—	4	1	no	—
108	C₂×D₄×C₇₃	—	2	2	no	—
109	C₂×D₄×C₇₃	—	4	1	no	—
110	C₂×D₄×C₇₃	—	4	1	no	—
111	D₈×C₇₃	—	2	4	no	—
112	D₈×C₇₃	—	4	2	no	—
113	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
114	C₂×C₈×C₇₃	C₂⁴	1	1	yes	characteristic
115	C₄⋊C₄×C₇₃	—	2	2	no	—
116	C₄⋊C₄×C₇₃	—	4	1	no	—
117	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
118	C₁₆×C₇₃	—	2	2	no	—
119	C₂³×C₇₃	—	4	2	no	—
120	C₂²×D₇₃	C₂×D₈	1	1	yes	characteristic
121	Q₈×C₇₃	—	8	2	no	—
122	D₄×C₇₃	—	4	2	no	—
123	D₄×C₇₃	—	4	2	no	—
124	D₄×C₇₃	—	8	2	no	—
125	D₄×C₇₃	—	8	2	no	—
126	C₂×C₄×C₇₃	—	4	2	no	—
127	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
128	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
129	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
130	C₂²⋊D₇₃	—	8	4	no	—
131	C₂×Dic₇₃	—	4	2	no	—
132	C₂×Dic₇₃	—	4	2	no	—
133	C₂×Dic₇₃	—	8	4	no	—
134	C₂×Dic₇₃	C₂×D₈	1	1	yes	characteristic
135	C₂×Dic₇₃	C₂×D₈	1	1	yes	characteristic
136	C₂×Dic₇₃	C₂×D₈	1	1	yes	characteristic
137	C₄×D₇₃	—	2	2	no	—
138	C₄×D₇₃	—	2	2	no	—
139	C₄×D₇₃	—	8	4	no	—
140	C₄×D₇₃	#124	1	2	yes	—
141	C₄×D₇₃	C₂×SD₁₆	1	2	yes	—
142	Dic₁₄₆	—	4	2	no	—
143	Dic₁₄₆	—	4	2	no	—
144	Dic₁₄₆	—	8	4	no	—
145	Dic₁₄₆	—	8	2	no	—
146	C₈×C₇₃	C₂×SD₁₆	1	2	yes	—
147	C₈×C₇₃	C₂×#41	1	2	yes	—
148	#10596831	#124	1	2	yes	—
149	#10596831	C₂×SD₁₆	1	2	yes	—
150	C₂²×C₇₃	—	8	2	no	—
151	C₂²×C₇₃	—	8	2	no	—
152	C₂²×C₇₃	—	8	2	no	—
153	C₂²×C₇₃	C₂²×D₈	1	1	yes	characteristic
154	C₂×D₇₃	—	2	1	no	—
155	C₂×D₇₃	—	2	1	no	—
156	C₂×D₇₃	#486	1	2	yes	—
157	C₄×C₇₃	—	8	2	no	—
158	C₄×C₇₃	#509	1	2	yes	—
159	C₄×C₇₃	#510	1	2	yes	—
160	C₄×C₇₃	C₂×#124	1	1	yes	characteristic
161	C₄×C₇₃	C₂²×SD₁₆	1	1	yes	characteristic
162	Dic₇₃	—	2	1	no	—
163	Dic₇₃	—	2	1	no	—
164	Dic₇₃	—	8	2	no	—
165	Dic₇₃	—	8	2	no	—
166	Dic₇₃	#486	1	2	yes	—
167	#33707	—	73	1	no	maximal, Sylow
168	C₂×C₇₃	—	16	2	no	—
169	C₂×C₇₃	C₂×#486	1	1	yes	characteristic
170	C₂×C₇₃	C₂×#509	1	1	yes	characteristic
171	C₂×C₇₃	C₂×#510	1	1	yes	characteristic
172	D₇₃	—	2	2	no	—
173	C₂×#500	—	73	1	no	—
174	#3147	—	73	1	no	—
175	#2243	—	73	4	no	—
176	#2241	—	73	1	no	—
177	#2241	—	73	1	no	—
178	#2240	—	73	1	no	—
179	#2240	D₇₃	1	1	yes	characteristic, complemented
180	#2139	—	73	4	no	—
181	C₄×M₃₂	—	73	1	no	—
182	C₇₃	#33707	1	1	yes	characteristic, complemented, Sylow
183	C₂×#135	—	73	2	no	—
184	C₂×#135	—	146	1	no	—
185	#532	—	146	1	no	—
186	C₂×C₄⋊Q₈	—	146	1	no	—
187	C₂×M₃₂	—	73	2	no	—
188	#500	—	73	2	no	—
189	#500	—	73	1	no	—
190	#500	—	146	1	no	—
191	#500	—	146	1	no	—
192	#500	C₂×D₇₃	1	1	yes	characteristic
193	#495	—	73	2	no	—
194	#495	—	146	1	no	—
195	#494	—	73	1	no	—
196	#493	C₂×D₇₃	1	1	yes	characteristic
197	#492	—	146	1	no	—
198	C₂×#107	—	73	2	no	—
199	C₂×#107	—	146	1	no	—
200	C₂×C₄×C₈	—	73	1	no	—
201	#358	—	73	4	no	—
202	#358	—	73	4	no	—
203	#357	—	73	4	no	—
204	#357	C₂×D₇₃	1	4	yes	—
205	#346	—	73	2	no	—
206	C₄×C₁₆	—	73	1	no	—
207	C₄×C₁₆	C₂×D₇₃	1	1	yes	characteristic
208	C₂×#41	—	146	2	no	—
209	C₂²×Q₈	—	292	1	no	—
210	#135	—	292	4	no	—
211	#135	—	146	8	no	—
212	C₂×Q₁₆	—	73	2	no	—
213	C₂×Q₁₆	—	146	1	no	—
214	C₂×SD₁₆	—	146	2	no	—
215	C₂×SD₁₆	—	146	2	no	—
216	C₂×D₈	—	2	1	no	—
217	C₂×D₈	C₂²×D₇₃	1	2	yes	—
218	C₂²×C₈	—	73	1	no	—
219	C₂²×C₈	—	73	1	no	—
220	C₄⋊Q₈	—	2	1	no	—
221	C₄⋊Q₈	—	292	1	no	—
222	C₄⋊Q₈	—	292	1	no	—
223	C₄⋊Q₈	—	146	2	no	—
224	C₄⋊Q₈	—	146	1	no	—
225	C₄⋊Q₈	—	146	1	no	—
226	#127	—	2	1	no	—
227	#124	—	146	2	no	—
228	#121	—	292	1	no	—
229	#121	—	292	1	no	—
230	C₄×D₄	—	292	1	no	—
231	C₄×D₄	—	292	1	no	—
232	C₂×C₄⋊C₄	—	292	1	no	—
233	C₂×C₄⋊C₄	—	146	2	no	—
234	C₂×C₄²	—	73	1	no	—
235	M₃₂	—	146	4	no	—
236	C₂×C₁₆	—	73	2	no	—
237	C₂×C₁₆	C₂²×D₇₃	1	2	yes	—
238	#107	—	2	1	no	—
239	#107	—	73	2	no	—
240	#107	—	146	2	no	—
241	#107	—	146	2	no	—
242	#107	—	146	1	no	—
243	#107	C₂²×D₇₃	1	2	yes	—
244	C₄×C₈	—	73	1	no	—
245	C₄×C₈	—	73	1	no	—
246	C₄×C₈	—	73	1	no	—
247	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
248	#41	—	292	4	no	—
249	#41	—	292	4	no	—
250	C₂×Q₈	—	4	1	no	—
251	C₂×Q₈	—	292	2	no	—
252	C₂×Q₈	—	292	1	no	—
253	C₂×Q₈	—	584	4	no	—
254	C₂×Q₈	—	146	2	no	—
255	C₂×D₄	—	2	2	no	—
256	C₂×D₄	—	4	1	no	—
257	C₂×D₄	—	4	1	no	—
258	C₂×D₄	—	292	2	no	—
259	C₂²×C₄	—	292	2	no	—
260	C₂²×C₄	—	292	2	no	—
261	C₂²×C₄	—	73	1	no	—
262	C₂²×C₄	—	73	1	no	—
263	C₂²×C₄	—	73	1	no	—
264	Q₁₆	—	292	2	no	—
265	Q₁₆	—	146	4	no	—
266	SD₁₆	—	292	4	no	—
267	SD₁₆	—	292	4	no	—
268	D₈	—	2	4	no	—
269	D₈	—	4	2	no	—
270	C₂×C₈	—	73	1	no	—
271	C₂×C₈	—	73	1	no	—
272	C₂×C₈	—	73	4	no	—
273	C₂×C₈	—	73	4	no	—
274	C₂×C₈	D₄×D₇₃	1	1	yes	characteristic
275	C₂×C₈	C₂³×D₇₃	1	1	yes	characteristic
276	C₄⋊C₄	—	2	2	no	—
277	C₄⋊C₄	—	4	1	no	—
278	C₄⋊C₄	—	292	2	no	—
279	C₄⋊C₄	—	292	2	no	—
280	C₄⋊C₄	—	292	1	no	—
281	C₄⋊C₄	—	292	1	no	—
282	C₄⋊C₄	—	292	1	no	—
283	C₄⋊C₄	—	146	2	no	—
284	C₂²⋊C₄	—	292	2	no	—
285	C₂²⋊C₄	—	292	2	no	—
286	C₄²	—	73	1	no	—
287	C₄²	—	73	1	no	—
288	C₄²	—	73	1	no	—
289	C₄²	D₄×D₇₃	1	1	yes	characteristic
290	C₁₆	—	2	2	no	—
291	C₁₆	—	146	2	no	—
292	C₂³	—	4	2	no	—
293	C₂³	—	73	1	no	—
294	Q₈	—	8	2	no	—
295	Q₈	—	292	2	no	—
296	Q₈	—	292	2	no	—
297	Q₈	—	584	4	no	—
298	Q₈	—	584	2	no	—
299	D₄	—	4	2	no	—
300	D₄	—	4	2	no	—
301	D₄	—	8	2	no	—
302	D₄	—	8	2	no	—
303	D₄	—	584	4	no	—
304	C₂×C₄	—	4	2	no	—
305	C₂×C₄	—	292	2	no	—
306	C₂×C₄	—	292	2	no	—
307	C₂×C₄	—	584	4	no	—
308	C₂×C₄	—	584	4	no	—
309	C₂×C₄	—	73	1	no	—
310	C₂×C₄	—	73	2	no	—
311	C₂×C₄	—	73	2	no	—
312	C₂×C₄	—	73	1	no	—
313	C₂×C₄	—	73	1	no	—
314	C₂×C₄	—	146	2	no	—
315	C₂×C₄	—	146	2	no	—
316	C₂×C₄	D₈×D₇₃	1	1	yes	characteristic
317	C₂×C₄	D₈×D₇₃	1	1	yes	characteristic
318	C₂×C₄	C₂×D₄×D₇₃	1	1	yes	characteristic
319	C₈	—	73	2	no	—
320	C₈	—	73	2	no	—
321	C₈	SD₁₆×D₇₃	1	2	yes	—
322	C₈	#41×D₇₃	1	2	yes	—
323	C₂²	—	8	2	no	—
324	C₂²	—	8	2	no	—
325	C₂²	—	8	2	no	—
326	C₂²	—	73	2	no	—
327	C₂²	—	146	1	no	—
328	C₂²	—	146	1	no	—
329	C₂²	C₂×D₈×D₇₃	1	1	yes	characteristic
330	C₄	—	8	2	no	—
331	C₄	—	584	2	no	—
332	C₄	—	584	2	no	—
333	C₄	—	73	2	no	—
334	C₄	—	146	1	no	—
335	C₄	—	146	1	no	—
336	C₄	#17989021	1	2	yes	—
337	C₄	#17989026	1	2	yes	—
338	C₄	#124×D₇₃	1	1	yes	characteristic
339	C₄	C₂×SD₁₆×D₇₃	1	1	yes	characteristic
340	C₂	—	16	2	no	—
341	C₂	—	146	2	no	—
342	C₂	#486×D₇₃	1	1	yes	characteristic
343	C₂	C₂×#17989021	1	1	yes	characteristic
344	C₂	C₂×#17989026	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×#2240	C₂
socle	C₂²×C₇₃	C₂²×D₈
prime radical	#22620358	C₂
hypercenter	#2240	D₇₃
hypocenter	C₇₃	#33707
LCS	G ▷ C₂×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2240

Complements

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

N	H ≅ G/N	type
1. #500×D₇₃	C₂	semidirect
3. #22612166	C₂	semidirect
4. #22612160	C₂	semidirect
5. #22612159	C₂	semidirect
6. #22612154	C₂	semidirect
7. #22611640	C₂	semidirect
8. C₄×#14556530	C₂	semidirect
9. C₇₃×#2240	C₂	semidirect
179. #2240	D₇₃	semidirect
182. C₇₃	#33707	semidirect

Maximal subgroups

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

subgroup	index
#500×D₇₃	2
#22620358	2
#22612166	2
#22612160	2
#22612159	2
#22612154	2
#22611640	2
C₄×#14556530	2
C₇₃×#2240	2
#33707	73