Subgroups — #33241586

Order 18688.

532 conjugacy classes of subgroups.

The subgroup lattice has 336 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	#22612208	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	#22612033	C₂	1	2	yes	maximal, complemented
6	#22612018	C₂	1	1	yes	maximal, characteristic, complemented
7	#22612014	C₂	1	1	yes	maximal, characteristic, complemented
8	#22611805	C₂	1	2	yes	maximal, complemented
9	C₄×D₁₁₆₈	C₂	1	1	yes	maximal, characteristic, complemented
10	#22611526	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×D₁₆×C₇₃	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	#17988414	C₂²	1	2	yes	—
27	#17988413	C₂²	1	1	yes	characteristic
28	#17988389	C₂²	1	2	yes	—
29	#17988388	C₂²	1	2	yes	—
30	#17988387	C₂²	1	2	yes	—
31	#17988363	C₂²	1	1	yes	characteristic
32	#17988362	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	#366×C₇₃	C₂²	1	1	yes	characteristic
37	#357×C₇₃	C₂²	1	2	yes	—
38	C₄×C₁₆×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	1	yes	characteristic
94	#11924007	D₈	1	1	yes	characteristic
95	C₂²⋊Dic₇₃	—	4	2	no	—
96	#11924005	—	4	2	no	—
97	#11924005	—	4	2	no	—
98	#11924005	D₈	1	1	yes	characteristic
99	#11924005	D₈	1	1	yes	characteristic
100	C₄×Dic₇₃	C₂×D₄	1	1	yes	characteristic
101	#11924003	—	2	1	no	—
102	#11924003	#41	1	2	yes	—
103	C₂×D₄×C₇₃	—	2	2	no	—
104	C₂²×C₄×C₇₃	—	4	2	no	—
105	D₈×C₇₃	—	2	2	no	—
106	D₈×C₇₃	—	2	2	no	—
107	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
108	C₂×C₈×C₇₃	C₂⁴	1	1	yes	characteristic
109	C₄⋊C₄×C₇₃	—	2	2	no	—
110	C₂²⋊C₄×C₇₃	—	4	2	no	—
111	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
112	C₁₆×C₇₃	—	2	1	no	—
113	C₁₆×C₇₃	#41	1	2	yes	—
114	C₂³×C₇₃	—	4	2	no	—
115	C₂²×D₇₃	—	4	2	no	—
116	C₂²×D₇₃	C₂×D₈	1	1	yes	characteristic
117	D₄×C₇₃	—	4	2	no	—
118	D₄×C₇₃	—	4	2	no	—
119	C₂×C₄×C₇₃	—	4	2	no	—
120	C₂×C₄×C₇₃	—	16	2	no	—
121	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
122	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
123	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
124	C₂²⋊D₇₃	—	16	2	no	—
125	C₂²⋊D₇₃	—	16	2	no	—
126	C₂²⋊D₇₃	—	16	2	no	—
127	C₂×Dic₇₃	—	4	2	no	—
128	C₂×Dic₇₃	C₂×D₈	1	1	yes	characteristic
129	C₂×Dic₇₃	C₂×D₈	1	1	yes	characteristic
130	C₂×Dic₇₃	C₂×D₈	1	1	yes	characteristic
131	D₂₉₂	—	4	2	no	—
132	D₂₉₂	—	4	2	no	—
133	D₂₉₂	—	16	2	no	—
134	C₄×D₇₃	—	2	1	no	—
135	C₄×D₇₃	—	2	1	no	—
136	C₄×D₇₃	—	16	2	no	—
137	C₈×C₇₃	—	2	1	no	—
138	C₈×C₇₃	C₂×#41	1	1	yes	characteristic
139	C₈×C₇₃	C₂×#41	1	1	yes	characteristic
140	#10596831	—	2	1	no	—
141	#10596831	#135	1	2	yes	—
142	C₂²×C₇₃	—	8	2	no	—
143	C₂²×C₇₃	—	8	2	no	—
144	C₂²×C₇₃	—	8	2	no	—
145	C₂²×C₇₃	C₂²×D₈	1	1	yes	characteristic
146	C₂×D₇₃	—	2	1	no	—
147	C₂×D₇₃	—	2	1	no	—
148	C₂×D₇₃	—	2	1	no	—
149	C₂×D₇₃	—	8	2	no	—
150	C₂×D₇₃	—	8	2	no	—
151	C₂×D₇₃	—	8	2	no	—
152	C₄×C₇₃	—	2	1	no	—
153	C₄×C₇₃	—	2	1	no	—
154	C₄×C₇₃	—	8	2	no	—
155	C₄×C₇₃	#532	1	1	yes	characteristic
156	C₄×C₇₃	C₂×#135	1	1	yes	characteristic
157	Dic₇₃	—	2	1	no	—
158	Dic₇₃	—	2	1	no	—
159	Dic₇₃	—	2	1	no	—
160	Dic₇₃	—	8	2	no	—
161	#33740	—	73	1	no	maximal, Sylow
162	C₂×C₇₃	—	16	2	no	—
163	C₂×C₇₃	#3169	1	1	yes	characteristic
164	C₂×C₇₃	#3415	1	1	yes	characteristic
165	C₂×C₇₃	#3415	1	1	yes	characteristic
166	D₇₃	—	4	1	no	—
167	D₇₃	—	16	2	no	—
168	#3176	—	73	2	no	—
169	#2253	—	73	1	no	—
170	#2246	—	73	1	no	—
171	#2244	—	73	2	no	—
172	#2218	—	73	2	no	—
173	#2215	—	73	1	no	—
174	#2215	—	73	1	no	—
175	#2177	—	73	2	no	—
176	C₄×D₁₆	—	73	1	no	—
177	C₄×D₁₆	D₇₃	1	1	yes	characteristic, complemented
178	#2108	—	73	1	no	—
179	C₇₃	#33740	1	1	yes	characteristic, complemented, Sylow
180	#540	—	146	2	no	—
181	C₂×D₁₆	—	73	2	no	—
182	C₂×D₁₆	—	73	1	no	—
183	C₂×D₁₆	C₂×D₇₃	1	1	yes	characteristic
184	C₂×M₃₂	—	73	1	no	—
185	C₂²×C₁₆	—	73	1	no	—
186	#499	—	73	2	no	—
187	#493	—	73	2	no	—
188	#490	—	146	2	no	—
189	#469	—	146	2	no	—
190	#466	—	73	2	no	—
191	#466	—	73	2	no	—
192	#442	—	146	2	no	—
193	C₄×D₈	—	73	2	no	—
194	C₄×D₈	C₂×D₇₃	1	2	yes	—
195	#405	—	73	1	no	—
196	#367	—	73	2	no	—
197	#366	—	73	1	no	—
198	#366	C₂×D₇₃	1	1	yes	characteristic
199	#357	—	73	2	no	—
200	#357	—	73	2	no	—
201	#357	—	73	2	no	—
202	#357	C₂×D₇₃	1	2	yes	—
203	#346	—	73	1	no	—
204	#346	—	73	1	no	—
205	C₄×C₁₆	—	73	1	no	—
206	C₄×C₁₆	C₂×D₇₃	1	1	yes	characteristic
207	C₂×D₈	—	73	2	no	—
208	C₂×D₈	—	146	2	no	—
209	C₂×D₈	C₂²×D₇₃	1	2	yes	—
210	C₂×M₁₆	—	73	1	no	—
211	C₂²×C₈	—	73	1	no	—
212	#127	—	146	2	no	—
213	#125	—	146	2	no	—
214	#123	—	292	2	no	—
215	#123	—	292	2	no	—
216	#121	—	292	2	no	—
217	#121	—	292	2	no	—
218	#121	—	146	2	no	—
219	#121	—	146	2	no	—
220	C₄×D₄	—	2	2	no	—
221	C₄×D₄	—	146	2	no	—
222	#117	—	73	1	no	—
223	D₁₆	—	2	2	no	—
224	D₁₆	—	146	4	no	—
225	D₁₆	—	146	2	no	—
226	M₃₂	—	146	2	no	—
227	C₂×C₁₆	—	73	1	no	—
228	C₂×C₁₆	—	73	1	no	—
229	C₂×C₁₆	—	146	2	no	—
230	C₂×C₁₆	C₂²×D₇₃	1	1	yes	characteristic
231	C₂×C₁₆	C₂²×D₇₃	1	1	yes	characteristic
232	#107	—	73	2	no	—
233	#107	C₂²×D₇₃	1	2	yes	—
234	#106	—	146	2	no	—
235	#102	—	2	2	no	—
236	#102	—	146	2	no	—
237	#102	—	146	2	no	—
238	#102	—	146	2	no	—
239	#97	—	73	1	no	—
240	#97	—	73	1	no	—
241	C₄×C₈	—	73	1	no	—
242	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
243	C₂×D₄	—	2	2	no	—
244	C₂×D₄	—	292	2	no	—
245	C₂×D₄	—	292	2	no	—
246	C₂×D₄	—	292	2	no	—
247	C₂×D₄	—	292	2	no	—
248	C₂×D₄	—	146	2	no	—
249	C₂²×C₄	—	4	2	no	—
250	C₂²×C₄	—	292	2	no	—
251	C₂²×C₄	—	73	1	no	—
252	D₈	—	2	2	no	—
253	D₈	—	2	2	no	—
254	D₈	—	292	4	no	—
255	D₈	—	146	2	no	—
256	D₈	—	146	2	no	—
257	M₁₆	—	146	2	no	—
258	C₂×C₈	—	73	1	no	—
259	C₂×C₈	—	73	1	no	—
260	C₂×C₈	—	146	1	no	—
261	C₂×C₈	—	146	1	no	—
262	C₂×C₈	D₄×D₇₃	1	1	yes	characteristic
263	C₂×C₈	C₂³×D₇₃	1	1	yes	characteristic
264	C₄⋊C₄	—	2	2	no	—
265	C₄⋊C₄	—	292	2	no	—
266	C₄⋊C₄	—	292	2	no	—
267	C₄⋊C₄	—	73	1	no	—
268	C₄⋊C₄	—	73	1	no	—
269	C₄⋊C₄	—	146	2	no	—
270	C₂²⋊C₄	—	4	2	no	—
271	C₂²⋊C₄	—	292	2	no	—
272	C₂²⋊C₄	—	292	2	no	—
273	C₂²⋊C₄	—	292	2	no	—
274	C₂²⋊C₄	—	73	1	no	—
275	C₂²⋊C₄	—	73	1	no	—
276	C₄²	—	73	1	no	—
277	C₄²	D₄×D₇₃	1	1	yes	characteristic
278	C₁₆	—	2	1	no	—
279	C₁₆	—	73	2	no	—
280	C₁₆	—	146	1	no	—
281	C₁₆	#11924028	1	2	yes	—
282	C₂³	—	4	2	no	—
283	C₂³	—	292	2	no	—
284	C₂³	—	73	1	no	—
285	D₄	—	4	2	no	—
286	D₄	—	4	2	no	—
287	D₄	—	292	2	no	—
288	D₄	—	292	2	no	—
289	D₄	—	1168	2	no	—
290	D₄	—	1168	2	no	—
291	D₄	—	1168	2	no	—
292	D₄	—	1168	2	no	—
293	C₂×C₄	—	4	2	no	—
294	C₂×C₄	—	16	2	no	—
295	C₂×C₄	—	292	2	no	—
296	C₂×C₄	—	73	1	no	—
297	C₂×C₄	—	73	1	no	—
298	C₂×C₄	—	73	1	no	—
299	C₂×C₄	—	1168	2	no	—
300	C₂×C₄	—	146	1	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	—	2	1	no	—
306	C₈	—	73	2	no	—
307	C₈	—	146	1	no	—
308	C₈	#41×D₇₃	1	1	yes	characteristic
309	C₈	C₂×#11924028	1	1	yes	characteristic
310	C₂²	—	8	2	no	—
311	C₂²	—	8	2	no	—
312	C₂²	—	8	2	no	—
313	C₂²	—	584	2	no	—
314	C₂²	—	584	2	no	—
315	C₂²	—	584	2	no	—
316	C₂²	—	146	1	no	—
317	C₂²	—	146	1	no	—
318	C₂²	—	146	1	no	—
319	C₂²	C₂×D₈×D₇₃	1	1	yes	characteristic
320	C₄	—	2	1	no	—
321	C₄	—	2	1	no	—
322	C₄	—	8	2	no	—
323	C₄	—	584	2	no	—
324	C₄	—	146	1	no	—
325	C₄	—	146	1	no	—
326	C₄	—	146	1	no	—
327	C₄	#17989134	1	1	yes	characteristic
328	C₄	#135×D₇₃	1	1	yes	characteristic
329	C₂	—	16	2	no	—
330	C₂	—	292	1	no	—
331	C₂	—	1168	2	no	—
332	C₂	#22620507	1	1	yes	characteristic
333	C₂	#22623068	1	1	yes	characteristic
334	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×D₁₆×C₇₃	C₂
socle	C₂²×C₇₃	C₂²×D₈
hypercenter	C₄×D₁₆	D₇₃
hypocenter	C₇₃	#33740
LCS	G ▷ C₂×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×D₁₆

Complements

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

N	H ≅ G/N	type
1. #22620577	C₂	semidirect
2. #22612208	C₂	semidirect
3. #22612179	C₂	semidirect
4. #22612171	C₂	semidirect
5. #22612033	C₂	semidirect
6. #22612018	C₂	semidirect
7. #22612014	C₂	semidirect
8. #22611805	C₂	semidirect
9. C₄×D₁₁₆₈	C₂	semidirect
10. #22611526	C₂	semidirect
11. C₄×D₁₆×C₇₃	C₂	semidirect
177. C₄×D₁₆	D₇₃	semidirect
179. 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
#22612208	2
#22612179	2
#22612171	2
#22612033	2
#22612018	2
#22612014	2
#22611805	2
C₄×D₁₁₆₈	2
#22611526	2
C₄×D₁₆×C₇₃	2
#33740	73