Subgroups — #33241588

Order 18688.

532 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22620577	C₂	1	2	yes	maximal, complemented
2	#22612209	C₂	1	1	yes	maximal, characteristic, complemented
3	#22612179	C₂	1	1	yes	maximal, characteristic, complemented
4	#22612171	C₂	1	2	yes	maximal, complemented
5	#22612035	C₂	1	1	yes	maximal, characteristic, complemented
6	#22612031	C₂	1	1	yes	maximal, characteristic, complemented
7	#22612016	C₂	1	2	yes	maximal, complemented
8	#22611808	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄×#14556532	C₂	1	2	yes	maximal, complemented
10	#22611529	C₂	1	1	yes	maximal, characteristic, complemented
11	C₇₃×#2177	C₂	1	1	yes	maximal, characteristic, complemented
12	#17989188	—	2	2	no	—
13	C₂×D₁₁₆₈	C₂²	1	1	yes	characteristic
14	C₂×#14556532	C₂²	1	2	yes	—
15	C₂×#14556530	C₂²	1	1	yes	characteristic
16	C₂×C₁₆×D₇₃	C₂²	1	1	yes	characteristic
17	#17988988	C₂²	1	2	yes	—
18	#17988962	C₂²	1	2	yes	—
19	#17988950	—	2	2	no	—
20	#17988839	—	2	2	no	—
21	#17988824	C₂²	1	2	yes	—
22	#17988820	C₂²	1	2	yes	—
23	#17988697	—	2	2	no	—
24	C₄×D₅₈₄	C₂²	1	2	yes	—
25	#17988541	C₂²	1	1	yes	characteristic
26	#17988415	C₂²	1	1	yes	characteristic
27	#17988412	C₂²	1	2	yes	—
28	#17988389	C₂²	1	2	yes	—
29	#17988388	C₂²	1	2	yes	—
30	#17988387	C₂²	1	2	yes	—
31	#17988362	C₂²	1	1	yes	characteristic
32	C₄×#11924003	C₂²	1	1	yes	characteristic
33	C₁₆×Dic₇₃	C₂²	1	1	yes	characteristic
34	C₂×D₁₆×C₇₃	C₂²	1	1	yes	characteristic
35	C₄×D₈×C₇₃	C₂²	1	2	yes	—
36	#367×C₇₃	C₂²	1	1	yes	characteristic
37	#357×C₇₃	C₂²	1	2	yes	—
38	#346×C₇₃	C₂²	1	1	yes	characteristic
39	C₂×D₅₈₄	C₂³	1	2	yes	—
40	C₂×#11924013	—	2	2	no	—
41	C₂×#11924011	D₄	1	1	yes	characteristic
42	C₂×C₈×D₇₃	C₂³	1	1	yes	characteristic
43	#14556587	—	2	2	no	—
44	#14556582	—	2	2	no	—
45	#14556574	—	4	2	no	—
46	#14556573	—	4	2	no	—
47	#14556566	—	2	2	no	—
48	#14556565	—	4	2	no	—
49	#14556564	—	4	2	no	—
50	#14556562	—	2	2	no	—
51	C₄×D₂₉₂	—	2	2	no	—
52	#14556548	D₄	1	1	yes	characteristic
53	D₁₁₆₈	—	2	2	no	—
54	#14556532	—	2	4	no	—
55	#14556530	—	2	2	no	—
56	C₂×#11924003	C₂³	1	1	yes	characteristic
57	C₂×#11924003	C₂³	1	1	yes	characteristic
58	C₁₆×D₇₃	—	2	2	no	—
59	#14556525	C₂³	1	2	yes	—
60	#14556522	—	2	2	no	—
61	#14556513	—	2	2	no	—
62	#14556512	—	2	2	no	—
63	#14556511	—	2	2	no	—
64	#14556502	D₄	1	1	yes	characteristic
65	#14556501	D₄	1	1	yes	characteristic
66	C₈×Dic₇₃	C₂³	1	1	yes	characteristic
67	C₂×D₈×C₇₃	C₂³	1	2	yes	—
68	C₄×D₄×C₇₃	—	2	2	no	—
69	D₁₆×C₇₃	—	2	2	no	—
70	C₂×C₁₆×C₇₃	C₂³	1	1	yes	characteristic
71	C₂×C₁₆×C₇₃	C₂³	1	1	yes	characteristic
72	#107×C₇₃	C₂³	1	2	yes	—
73	#102×C₇₃	—	2	2	no	—
74	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
75	C₂×D₂₉₂	—	2	2	no	—
76	C₂×D₂₉₂	—	4	2	no	—
77	C₂×C₂²⋊D₇₃	—	4	2	no	—
78	C₂×C₂²⋊D₇₃	—	4	2	no	—
79	C₂×C₂²⋊D₇₃	—	4	2	no	—
80	C₂×C₄×D₇₃	—	4	2	no	—
81	C₂×C₄×D₇₃	C₂×D₄	1	1	yes	characteristic
82	D₅₈₄	—	2	2	no	—
83	D₅₈₄	—	2	2	no	—
84	#11924013	—	4	4	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	—	2	2	no	—
91	C₄⋊Dic₇₃	—	4	2	no	—
92	#11924007	—	4	2	no	—
93	#11924007	D₈	1	2	yes	—
94	C₂²⋊Dic₇₃	—	4	2	no	—
95	#11924005	—	4	2	no	—
96	#11924005	—	4	2	no	—
97	#11924005	D₈	1	2	yes	—
98	C₄×Dic₇₃	C₂×D₄	1	1	yes	characteristic
99	#11924003	—	2	1	no	—
100	#11924003	#41	1	2	yes	—
101	C₂×D₄×C₇₃	—	2	2	no	—
102	C₂²×C₄×C₇₃	—	4	2	no	—
103	D₈×C₇₃	—	2	2	no	—
104	D₈×C₇₃	—	2	2	no	—
105	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
106	C₂×C₈×C₇₃	C₂⁴	1	1	yes	characteristic
107	C₄⋊C₄×C₇₃	—	2	2	no	—
108	C₂²⋊C₄×C₇₃	—	4	2	no	—
109	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
110	C₁₆×C₇₃	—	2	1	no	—
111	C₁₆×C₇₃	#41	1	2	yes	—
112	C₂³×C₇₃	—	4	2	no	—
113	C₂²×D₇₃	—	4	2	no	—
114	C₂²×D₇₃	C₂×D₈	1	1	yes	characteristic
115	D₄×C₇₃	—	4	2	no	—
116	D₄×C₇₃	—	4	2	no	—
117	C₂×C₄×C₇₃	—	4	2	no	—
118	C₂×C₄×C₇₃	—	16	2	no	—
119	C₂×C₄×C₇₃	C₂×D₈	1	2	yes	—
120	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
121	C₂²⋊D₇₃	—	16	2	no	—
122	C₂²⋊D₇₃	—	16	2	no	—
123	C₂²⋊D₇₃	—	16	2	no	—
124	C₂×Dic₇₃	—	4	2	no	—
125	C₂×Dic₇₃	C₂×D₈	1	2	yes	—
126	C₂×Dic₇₃	C₂×D₈	1	1	yes	characteristic
127	D₂₉₂	—	4	2	no	—
128	D₂₉₂	—	4	2	no	—
129	D₂₉₂	—	16	2	no	—
130	C₄×D₇₃	—	2	1	no	—
131	C₄×D₇₃	—	2	1	no	—
132	C₄×D₇₃	—	16	2	no	—
133	C₈×C₇₃	—	2	1	no	—
134	C₈×C₇₃	C₂×#41	1	1	yes	characteristic
135	C₈×C₇₃	C₂×#41	1	1	yes	characteristic
136	#10596831	—	2	1	no	—
137	#10596831	#135	1	2	yes	—
138	C₂²×C₇₃	—	8	2	no	—
139	C₂²×C₇₃	—	8	2	no	—
140	C₂²×C₇₃	—	8	2	no	—
141	C₂²×C₇₃	C₂²×D₈	1	1	yes	characteristic
142	C₂×D₇₃	—	2	2	no	—
143	C₂×D₇₃	—	2	1	no	—
144	C₂×D₇₃	—	8	2	no	—
145	C₂×D₇₃	—	8	2	no	—
146	C₂×D₇₃	—	8	2	no	—
147	C₄×C₇₃	—	2	2	no	—
148	C₄×C₇₃	—	8	2	no	—
149	C₄×C₇₃	#532	1	1	yes	characteristic
150	C₄×C₇₃	C₂×#135	1	1	yes	characteristic
151	Dic₇₃	—	2	2	no	—
152	Dic₇₃	—	2	1	no	—
153	Dic₇₃	—	8	2	no	—
154	#33740	—	73	1	no	maximal, Sylow
155	C₂×C₇₃	—	16	2	no	—
156	C₂×C₇₃	#3169	1	1	yes	characteristic
157	C₂×C₇₃	#3415	1	2	yes	—
158	D₇₃	—	4	1	no	—
159	D₇₃	—	16	2	no	—
160	#3176	—	73	2	no	—
161	#2253	—	73	1	no	—
162	#2246	—	73	1	no	—
163	#2244	—	73	2	no	—
164	#2218	—	73	1	no	—
165	#2218	—	73	1	no	—
166	#2215	—	73	2	no	—
167	#2177	—	73	1	no	—
168	#2177	D₇₃	1	1	yes	characteristic, complemented
169	C₄×D₁₆	—	73	2	no	—
170	#2108	—	73	1	no	—
171	C₇₃	#33740	1	1	yes	characteristic, complemented, Sylow
172	#540	—	146	2	no	—
173	C₂×D₁₆	—	73	2	no	—
174	C₂×D₁₆	—	73	1	no	—
175	C₂×D₁₆	C₂×D₇₃	1	1	yes	characteristic
176	C₂×M₃₂	—	73	1	no	—
177	C₂²×C₁₆	—	73	1	no	—
178	#499	—	73	2	no	—
179	#493	—	73	2	no	—
180	#490	—	146	2	no	—
181	#469	—	146	2	no	—
182	#466	—	73	2	no	—
183	#466	—	73	2	no	—
184	#442	—	146	2	no	—
185	C₄×D₈	—	73	2	no	—
186	C₄×D₈	C₂×D₇₃	1	2	yes	—
187	#405	—	73	1	no	—
188	#367	—	73	1	no	—
189	#367	C₂×D₇₃	1	1	yes	characteristic
190	#366	—	73	2	no	—
191	#357	—	73	2	no	—
192	#357	—	73	2	no	—
193	#357	—	73	2	no	—
194	#357	C₂×D₇₃	1	2	yes	—
195	#346	—	73	1	no	—
196	#346	C₂×D₇₃	1	1	yes	characteristic
197	C₄×C₁₆	—	73	1	no	—
198	C₄×C₁₆	—	73	1	no	—
199	C₂×D₈	—	73	2	no	—
200	C₂×D₈	—	146	2	no	—
201	C₂×D₈	C₂²×D₇₃	1	2	yes	—
202	C₂×M₁₆	—	73	1	no	—
203	C₂²×C₈	—	73	1	no	—
204	#127	—	146	2	no	—
205	#125	—	146	2	no	—
206	#123	—	292	2	no	—
207	#123	—	292	2	no	—
208	#121	—	292	2	no	—
209	#121	—	292	2	no	—
210	#121	—	146	2	no	—
211	#121	—	146	2	no	—
212	C₄×D₄	—	2	2	no	—
213	C₄×D₄	—	146	2	no	—
214	#117	—	73	1	no	—
215	D₁₆	—	2	2	no	—
216	D₁₆	—	146	4	no	—
217	D₁₆	—	146	2	no	—
218	M₃₂	—	146	2	no	—
219	C₂×C₁₆	—	73	1	no	—
220	C₂×C₁₆	—	73	1	no	—
221	C₂×C₁₆	—	146	2	no	—
222	C₂×C₁₆	C₂²×D₇₃	1	1	yes	characteristic
223	C₂×C₁₆	C₂²×D₇₃	1	1	yes	characteristic
224	#107	—	73	2	no	—
225	#107	C₂²×D₇₃	1	2	yes	—
226	#106	—	146	2	no	—
227	#102	—	2	2	no	—
228	#102	—	146	2	no	—
229	#102	—	146	2	no	—
230	#102	—	146	2	no	—
231	#97	—	73	1	no	—
232	#97	—	73	1	no	—
233	C₄×C₈	—	73	1	no	—
234	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
235	C₂×D₄	—	2	2	no	—
236	C₂×D₄	—	292	2	no	—
237	C₂×D₄	—	292	2	no	—
238	C₂×D₄	—	292	2	no	—
239	C₂×D₄	—	292	2	no	—
240	C₂×D₄	—	146	2	no	—
241	C₂²×C₄	—	4	2	no	—
242	C₂²×C₄	—	292	2	no	—
243	C₂²×C₄	—	73	1	no	—
244	D₈	—	2	2	no	—
245	D₈	—	2	2	no	—
246	D₈	—	292	4	no	—
247	D₈	—	146	2	no	—
248	D₈	—	146	2	no	—
249	M₁₆	—	146	2	no	—
250	C₂×C₈	—	73	1	no	—
251	C₂×C₈	—	73	1	no	—
252	C₂×C₈	—	146	1	no	—
253	C₂×C₈	—	146	1	no	—
254	C₂×C₈	D₄×D₇₃	1	1	yes	characteristic
255	C₂×C₈	C₂³×D₇₃	1	1	yes	characteristic
256	C₄⋊C₄	—	2	2	no	—
257	C₄⋊C₄	—	292	2	no	—
258	C₄⋊C₄	—	292	2	no	—
259	C₄⋊C₄	—	73	2	no	—
260	C₄⋊C₄	—	146	2	no	—
261	C₂²⋊C₄	—	4	2	no	—
262	C₂²⋊C₄	—	292	2	no	—
263	C₂²⋊C₄	—	292	2	no	—
264	C₂²⋊C₄	—	292	2	no	—
265	C₂²⋊C₄	—	73	2	no	—
266	C₄²	—	73	1	no	—
267	C₄²	D₄×D₇₃	1	1	yes	characteristic
268	C₁₆	—	2	1	no	—
269	C₁₆	—	73	2	no	—
270	C₁₆	—	146	1	no	—
271	C₁₆	#11924028	1	2	yes	—
272	C₂³	—	4	2	no	—
273	C₂³	—	292	2	no	—
274	C₂³	—	73	1	no	—
275	D₄	—	4	2	no	—
276	D₄	—	4	2	no	—
277	D₄	—	292	2	no	—
278	D₄	—	292	2	no	—
279	D₄	—	1168	2	no	—
280	D₄	—	1168	2	no	—
281	D₄	—	1168	2	no	—
282	D₄	—	1168	2	no	—
283	C₂×C₄	—	4	2	no	—
284	C₂×C₄	—	16	2	no	—
285	C₂×C₄	—	292	2	no	—
286	C₂×C₄	—	73	2	no	—
287	C₂×C₄	—	73	1	no	—
288	C₂×C₄	—	1168	2	no	—
289	C₂×C₄	—	146	1	no	—
290	C₂×C₄	—	146	1	no	—
291	C₂×C₄	D₈×D₇₃	1	2	yes	—
292	C₂×C₄	C₂×D₄×D₇₃	1	1	yes	characteristic
293	C₈	—	2	1	no	—
294	C₈	—	73	2	no	—
295	C₈	—	146	1	no	—
296	C₈	#41×D₇₃	1	1	yes	characteristic
297	C₈	C₂×#11924028	1	1	yes	characteristic
298	C₂²	—	8	2	no	—
299	C₂²	—	8	2	no	—
300	C₂²	—	8	2	no	—
301	C₂²	—	584	2	no	—
302	C₂²	—	584	2	no	—
303	C₂²	—	584	2	no	—
304	C₂²	—	146	1	no	—
305	C₂²	—	146	2	no	—
306	C₂²	C₂×D₈×D₇₃	1	1	yes	characteristic
307	C₄	—	2	2	no	—
308	C₄	—	8	2	no	—
309	C₄	—	584	2	no	—
310	C₄	—	146	1	no	—
311	C₄	—	146	2	no	—
312	C₄	#17989134	1	1	yes	characteristic
313	C₄	#135×D₇₃	1	1	yes	characteristic
314	C₂	—	16	2	no	—
315	C₂	—	292	1	no	—
316	C₂	—	1168	2	no	—
317	C₂	#22620507	1	1	yes	characteristic
318	C₂	#22623071	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₈×D₇₃
commutator, cosocle	C₂×C₈×C₇₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₇₃
Fitting	C₇₃×#2177	C₂
socle	C₂²×C₇₃	C₂²×D₈
hypercenter	#2177	D₇₃
hypocenter	C₇₃	#33740
LCS	G ▷ C₂×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2177

Complements

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

N	H ≅ G/N	type
1. #22620577	C₂	semidirect
2. #22612209	C₂	semidirect
3. #22612179	C₂	semidirect
4. #22612171	C₂	semidirect
5. #22612035	C₂	semidirect
6. #22612031	C₂	semidirect
7. #22612016	C₂	semidirect
8. #22611808	C₂	semidirect
9. C₄×#14556532	C₂	semidirect
10. #22611529	C₂	semidirect
11. C₇₃×#2177	C₂	semidirect
168. #2177	D₇₃	semidirect
171. C₇₃	#33740	semidirect

Maximal subgroups

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

subgroup	index
#22620577	2
#22612209	2
#22612179	2
#22612171	2
#22612035	2
#22612031	2
#22612016	2
#22611808	2
C₄×#14556532	2
#22611529	2
C₇₃×#2177	2
#33740	73