Subgroups — #33241146

Order 18688.

580 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22618784	C₂	1	1	yes	maximal, characteristic, complemented
2	#22612119	C₂	1	2	yes	maximal, complemented
3	#22612085	C₂	1	2	yes	maximal, complemented
4	#22611963	C₂	1	2	yes	maximal, complemented
5	#22611842	C₂	1	2	yes	maximal, complemented
6	#22611805	C₂	1	2	yes	maximal, complemented
7	C₄×#14556532	C₂	1	2	yes	maximal, complemented
8	#22611738	C₂	1	1	yes	maximal, characteristic, complemented
9	C₇₃×#3176	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×D₅₈₄	C₂²	1	1	yes	characteristic
11	#17989061	—	2	1	no	—
12	C₂×#14556532	C₂²	1	2	yes	—
13	C₂×#14556532	C₂²	1	2	yes	—
14	#17988700	—	2	1	no	—
15	#17988700	—	2	1	no	—
16	C₄×D₅₈₄	C₂²	1	2	yes	—
17	C₄×D₅₈₄	C₂²	1	2	yes	—
18	#17988625	C₂²	1	1	yes	characteristic
19	#17988593	—	2	1	no	—
20	#17988414	C₂²	1	2	yes	complemented
21	#17988412	C₂²	1	2	yes	complemented
22	C₄⋊#11924003	C₂²	1	2	yes	complemented
23	#17988388	C₂²	1	2	yes	complemented
24	#17988388	C₂²	1	2	yes	complemented
25	#17988387	C₂²	1	2	yes	—
26	#17988387	C₂²	1	2	yes	—
27	C₂²⋊#11924003	C₂²	1	2	yes	—
28	#17988363	C₂²	1	1	yes	characteristic, complemented
29	C₄×#11924003	C₂²	1	1	yes	characteristic, complemented
30	#540×C₇₃	—	2	1	no	—
31	#499×C₇₃	C₂²	1	1	yes	characteristic, complemented
32	#493×C₇₃	C₂²	1	1	yes	characteristic, complemented
33	#490×C₇₃	—	2	1	no	—
34	#469×C₇₃	—	2	1	no	—
35	#466×C₇₃	C₂²	1	2	yes	—
36	#442×C₇₃	—	2	1	no	—
37	C₄×D₈×C₇₃	C₂²	1	2	yes	complemented
38	#405×C₇₃	C₂²	1	1	yes	characteristic
39	C₂²×D₂₉₂	—	2	1	no	—
40	C₂×D₅₈₄	—	2	4	no	—
41	C₂×D₅₈₄	—	2	1	no	—
42	C₂×D₅₈₄	C₂³	1	2	yes	—
43	C₄×D₂₉₂	—	2	2	no	—
44	C₄×D₂₉₂	—	2	2	no	—
45	#14556553	—	4	2	no	—
46	#14556550	—	2	1	no	—
47	C₂²⋊C₄×D₇₃	—	4	2	no	—
48	#14556532	—	2	4	no	—
49	#14556532	—	2	4	no	—
50	C₂×#11924003	C₂³	1	2	yes	—
51	C₂×#11924003	C₂³	1	2	yes	—
52	#14556525	C₂³	1	2	yes	—
53	#14556523	—	2	1	no	—
54	#14556513	—	2	2	no	—
55	#14556513	—	2	2	no	—
56	C₂×D₈×C₇₃	—	2	1	no	—
57	C₂×D₈×C₇₃	C₂³	1	2	yes	—
58	C₂×M₁₆×C₇₃	D₄	1	1	yes	characteristic
59	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
60	#127×C₇₃	—	2	1	no	—
61	#125×C₇₃	—	2	1	no	—
62	#123×C₇₃	—	4	2	no	—
63	#121×C₇₃	—	2	2	no	—
64	#121×C₇₃	—	4	2	no	—
65	C₄×D₄×C₇₃	—	2	2	no	—
66	#117×C₇₃	D₄	1	1	yes	characteristic
67	#107×C₇₃	C₂³	1	2	yes	—
68	#106×C₇₃	—	2	1	no	—
69	#102×C₇₃	—	2	2	no	—
70	#102×C₇₃	—	2	2	no	—
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	C₂³×D₇₃	—	4	1	no	—
76	C₂×D₂₉₂	—	2	4	no	—
77	C₂×D₂₉₂	—	2	2	no	—
78	C₂×D₂₉₂	—	4	2	no	—
79	C₂×D₂₉₂	—	4	1	no	—
80	C₂×C₄×D₇₃	—	4	2	no	—
81	C₂×C₄×D₇₃	—	4	2	no	—
82	D₅₈₄	—	2	4	no	—
83	D₅₈₄	—	2	2	no	—
84	D₅₈₄	#41	1	4	yes	—
85	C₄⋊Dic₇₃	—	2	2	no	—
86	C₂²⋊Dic₇₃	—	4	1	no	—
87	#11924005	—	4	2	no	—
88	#11924005	—	4	2	no	—
89	C₄×Dic₇₃	—	4	1	no	—
90	#11924003	—	2	2	no	—
91	#11924003	—	2	2	no	—
92	C₂×D₄×C₇₃	—	2	2	no	—
93	C₂×D₄×C₇₃	—	4	2	no	—
94	C₂×D₄×C₇₃	—	4	2	no	—
95	C₂²×C₄×C₇₃	—	4	2	no	—
96	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
97	D₈×C₇₃	—	2	2	no	—
98	D₈×C₇₃	—	2	2	no	—
99	D₈×C₇₃	—	4	2	no	—
100	M₁₆×C₇₃	—	4	1	no	—
101	C₂×C₈×C₇₃	—	2	2	no	—
102	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
103	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
104	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
105	C₂×C₈×C₇₃	C₂⁴	1	1	yes	characteristic
106	C₄⋊C₄×C₇₃	—	2	2	no	—
107	C₄⋊C₄×C₇₃	—	4	2	no	—
108	C₄⋊C₄×C₇₃	D₈	1	2	yes	—
109	C₂²⋊C₄×C₇₃	—	4	2	no	—
110	C₂²⋊C₄×C₇₃	—	4	2	no	—
111	C₂²⋊C₄×C₇₃	D₈	1	2	yes	—
112	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
113	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
114	C₂³×C₇₃	—	4	2	no	—
115	C₂³×C₇₃	C₂×D₈	1	1	yes	characteristic
116	C₂²×D₇₃	—	4	2	no	—
117	C₂²×D₇₃	—	4	4	no	—
118	C₂²×D₇₃	—	16	2	no	—
119	D₄×C₇₃	—	4	2	no	—
120	D₄×C₇₃	—	4	2	no	—
121	D₄×C₇₃	—	16	2	no	—
122	D₄×C₇₃	—	16	2	no	—
123	C₂×C₄×C₇₃	—	2	1	no	—
124	C₂×C₄×C₇₃	—	2	1	no	—
125	C₂×C₄×C₇₃	—	4	2	no	—
126	C₂×C₄×C₇₃	—	16	2	no	—
127	C₂×C₄×C₇₃	C₂×D₈	1	2	yes	—
128	C₂×C₄×C₇₃	C₂×D₈	1	2	yes	—
129	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
130	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
131	C₂×Dic₇₃	—	4	2	no	—
132	D₂₉₂	—	2	4	no	—
133	D₂₉₂	—	4	4	no	—
134	D₂₉₂	—	4	2	no	—
135	C₄×D₇₃	—	16	2	no	—
136	C₄×D₇₃	—	16	2	no	—
137	C₈×C₇₃	—	2	1	no	—
138	C₈×C₇₃	—	2	1	no	—
139	C₈×C₇₃	—	2	1	no	—
140	C₈×C₇₃	C₂×#41	1	2	yes	—
141	C₂²×C₇₃	—	2	1	no	—
142	C₂²×C₇₃	—	2	2	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	—	4	4	no	—
148	C₂×D₇₃	—	8	2	no	—
149	C₂×D₇₃	—	8	2	no	—
150	C₂×D₇₃	—	16	4	no	—
151	C₄×C₇₃	—	2	1	no	—
152	C₄×C₇₃	—	2	2	no	—
153	C₄×C₇₃	—	2	2	no	—
154	C₄×C₇₃	—	8	2	no	—
155	C₄×C₇₃	#525	1	1	yes	characteristic
156	C₄×C₇₃	#575	1	1	yes	characteristic
157	Dic₇₃	—	8	2	no	—
158	#33682	—	73	1	no	maximal, Sylow
159	C₂×C₇₃	—	4	1	no	—
160	C₂×C₇₃	—	16	2	no	—
161	C₂×C₇₃	#3100	1	1	yes	characteristic
162	C₂×C₇₃	#3415	1	2	yes	—
163	D₇₃	—	8	4	no	—
164	#3176	D₇₃	1	1	yes	characteristic, complemented
165	#2948	—	73	1	no	—
166	#2234	—	73	2	no	—
167	#2228	—	73	2	no	—
168	#2206	—	73	2	no	—
169	#2184	—	73	2	no	—
170	#2177	—	73	2	no	—
171	C₄×D₁₆	—	73	2	no	—
172	#2164	—	73	1	no	—
173	C₇₃	#33682	1	1	yes	characteristic, complemented, Sylow
174	C₂²×D₈	—	73	1	no	—
175	#540	—	2	1	no	—
176	#518	—	146	1	no	—
177	C₂×D₁₆	—	73	2	no	—
178	C₂×D₁₆	—	73	2	no	—
179	#499	C₂×D₇₃	1	1	yes	characteristic, complemented
180	#493	C₂×D₇₃	1	1	yes	characteristic, complemented
181	#490	—	2	1	no	—
182	#469	—	2	1	no	—
183	#466	C₂×D₇₃	1	2	yes	—
184	#442	—	2	1	no	—
185	#442	—	146	1	no	—
186	#442	—	146	1	no	—
187	C₄×D₈	—	73	2	no	—
188	C₄×D₈	—	73	2	no	—
189	C₄×D₈	C₂×D₇₃	1	2	yes	complemented
190	#427	—	73	1	no	—
191	#418	—	146	1	no	—
192	#405	C₂×D₇₃	1	1	yes	characteristic
193	#367	—	73	2	no	—
194	#366	—	73	2	no	—
195	C₄⋊C₁₆	—	73	2	no	—
196	#357	—	73	2	no	—
197	#357	—	73	2	no	—
198	#357	—	73	2	no	—
199	#357	—	73	2	no	—
200	C₂²⋊C₁₆	—	73	2	no	—
201	#346	—	73	1	no	—
202	C₄×C₁₆	—	73	1	no	—
203	C₂²×D₄	—	146	1	no	—
204	C₂×D₈	—	2	1	no	—
205	C₂×D₈	—	73	2	no	—
206	C₂×D₈	—	146	4	no	—
207	C₂×D₈	—	146	1	no	—
208	C₂×D₈	C₂²×D₇₃	1	2	yes	—
209	C₂×M₁₆	C₂²⋊D₇₃	1	1	yes	characteristic
210	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
211	#127	—	2	1	no	—
212	#125	—	2	1	no	—
213	#123	—	4	2	no	—
214	#121	—	2	2	no	—
215	#121	—	4	2	no	—
216	C₄×D₄	—	2	2	no	—
217	C₄×D₄	—	292	2	no	—
218	C₄×D₄	—	146	2	no	—
219	C₄×D₄	—	146	2	no	—
220	#117	—	146	1	no	—
221	#117	C₂²⋊D₇₃	1	1	yes	characteristic
222	C₂×C₂²⋊C₄	—	292	2	no	—
223	D₁₆	—	146	4	no	—
224	D₁₆	—	146	4	no	—
225	C₂×C₁₆	—	73	2	no	—
226	C₂×C₁₆	—	73	2	no	—
227	#107	—	73	2	no	—
228	#107	C₂²×D₇₃	1	2	yes	—
229	#106	—	2	1	no	—
230	#106	—	146	1	no	—
231	#102	—	2	2	no	—
232	#102	—	2	2	no	—
233	#102	—	146	2	no	—
234	#102	—	146	2	no	—
235	#97	C₂²⋊D₇₃	1	1	yes	characteristic
236	#97	C₂²⋊D₇₃	1	1	yes	characteristic
237	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
238	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
239	C₂⁴	—	292	1	no	—
240	C₂×D₄	—	2	2	no	—
241	C₂×D₄	—	4	2	no	—
242	C₂×D₄	—	4	2	no	—
243	C₂×D₄	—	292	2	no	—
244	C₂×D₄	—	292	1	no	—
245	C₂×D₄	—	146	4	no	—
246	C₂×D₄	—	146	2	no	—
247	C₂²×C₄	—	4	2	no	—
248	C₂²×C₄	—	292	2	no	—
249	C₂²×C₄	—	292	2	no	—
250	C₂²×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
251	D₈	—	2	2	no	—
252	D₈	—	2	2	no	—
253	D₈	—	4	2	no	—
254	D₈	—	73	4	no	—
255	D₈	—	146	4	no	—
256	D₈	—	146	2	no	—
257	M₁₆	—	4	1	no	—
258	C₂×C₈	—	2	2	no	—
259	C₂×C₈	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
260	C₂×C₈	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
261	C₂×C₈	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
262	C₂×C₈	C₂³×D₇₃	1	1	yes	characteristic
263	C₄⋊C₄	—	2	2	no	—
264	C₄⋊C₄	—	4	2	no	—
265	C₄⋊C₄	—	146	2	no	—
266	C₄⋊C₄	#11924013	1	2	yes	—
267	C₂²⋊C₄	—	4	2	no	—
268	C₂²⋊C₄	—	4	2	no	—
269	C₂²⋊C₄	—	292	1	no	—
270	C₂²⋊C₄	—	292	2	no	—
271	C₂²⋊C₄	—	292	2	no	—
272	C₂²⋊C₄	#11924013	1	2	yes	—
273	C₄²	—	292	1	no	—
274	C₄²	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
275	C₄²	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
276	C₁₆	—	146	2	no	—
277	C₁₆	—	146	2	no	—
278	C₂³	—	4	2	no	—
279	C₂³	—	292	4	no	—
280	C₂³	—	292	2	no	—
281	C₂³	—	1168	2	no	—
282	C₂³	C₂×#11924013	1	1	yes	characteristic
283	D₄	—	4	2	no	—
284	D₄	—	4	2	no	—
285	D₄	—	16	2	no	—
286	D₄	—	16	2	no	—
287	D₄	—	292	4	no	—
288	D₄	—	292	2	no	—
289	D₄	—	146	4	no	—
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	4	2	no	—
293	C₂×C₄	—	16	2	no	—
294	C₂×C₄	—	292	2	no	—
295	C₂×C₄	—	1168	2	no	—
296	C₂×C₄	—	1168	2	no	—
297	C₂×C₄	C₂×#11924013	1	1	yes	characteristic
298	C₂×C₄	C₂×#11924013	1	2	yes	—
299	C₂×C₄	C₂×#11924013	1	2	yes	—
300	C₂×C₄	C₂²×C₂²⋊D₇₃	1	1	yes	characteristic
301	C₈	—	2	1	no	—
302	C₈	—	2	1	no	—
303	C₈	—	2	1	no	—
304	C₈	#41×D₇₃	1	2	yes	—
305	C₂²	—	2	1	no	—
306	C₂²	—	2	2	no	—
307	C₂²	—	8	2	no	—
308	C₂²	—	8	2	no	—
309	C₂²	—	8	2	no	—
310	C₂²	—	292	4	no	—
311	C₂²	—	584	2	no	—
312	C₂²	—	584	2	no	—
313	C₂²	—	1168	4	no	—
314	C₂²	C₂²×#11924013	1	1	yes	characteristic
315	C₄	—	2	1	no	—
316	C₄	—	2	2	no	—
317	C₄	—	2	2	no	—
318	C₄	—	8	2	no	—
319	C₄	—	584	2	no	—
320	C₄	#17989094	1	1	yes	characteristic
321	C₄	#17989385	1	1	yes	characteristic
322	C₂	—	4	1	no	—
323	C₂	—	16	2	no	—
324	C₂	—	584	4	no	—
325	C₂	#22619927	1	1	yes	characteristic
326	C₂	#22623070	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂²×#11924013
commutator, cosocle	C₂×C₈×C₇₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₇₃
Fitting	C₇₃×#3176	C₂
socle	C₂²×C₇₃	C₂²×D₈
hypercenter	#3176	D₇₃
hypocenter	C₇₃	#33682
LCS	G ▷ C₂×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ #117 ◁ #405 ◁ #3176

Complements

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

N	H ≅ G/N	type
1. #22618784	C₂	semidirect
2. #22612119	C₂	semidirect
3. #22612085	C₂	semidirect
4. #22611963	C₂	semidirect
5. #22611842	C₂	semidirect
6. #22611805	C₂	semidirect
7. C₄×#14556532	C₂	semidirect
8. #22611738	C₂	semidirect
9. C₇₃×#3176	C₂	semidirect
20. #17988414	C₂²	semidirect
21. #17988412	C₂²	semidirect
22. C₄⋊#11924003	C₂²	semidirect
23. #17988388	C₂²	semidirect
24. #17988388	C₂²	semidirect
28. #17988363	C₂²	semidirect
29. C₄×#11924003	C₂²	semidirect
31. #499×C₇₃	C₂²	semidirect
32. #493×C₇₃	C₂²	semidirect
37. C₄×D₈×C₇₃	C₂²	semidirect
164. #3176	D₇₃	semidirect
173. C₇₃	#33682	semidirect
179. #499	C₂×D₇₃	semidirect
180. #493	C₂×D₇₃	semidirect
189. C₄×D₈	C₂×D₇₃	semidirect

Maximal subgroups

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

subgroup	index
#22618784	2
#22612119	2
#22612085	2
#22611963	2
#22611842	2
#22611805	2
C₄×#14556532	2
#22611738	2
C₇₃×#3176	2
#33682	73