Subgroups — #33241518

Order 18688.

602 conjugacy classes of subgroups.

The subgroup lattice has 344 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22620507	C₂	1	2	yes	maximal, complemented
2	#22612205	C₂	1	1	yes	maximal, characteristic, complemented
3	#22612179	C₂	1	1	yes	maximal, characteristic, complemented
4	#22612153	C₂	1	2	yes	maximal, complemented
5	#22612018	C₂	1	1	yes	maximal, characteristic, complemented
6	#22612016	C₂	1	2	yes	maximal, complemented
7	#22612014	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×D₁₁₆₈	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄×#14556532	C₂	1	2	yes	maximal, complemented
10	C₄×C₁₆×D₇₃	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×D₁₆×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
12	#17989134	—	2	2	no	—
13	C₂×D₁₁₆₈	C₂²	1	1	yes	characteristic
14	C₂×#14556532	C₂²	1	2	yes	—
15	C₂×C₁₆×D₇₃	C₂²	1	1	yes	characteristic
16	C₂×C₁₆×D₇₃	C₂²	1	1	yes	characteristic
17	#17988991	C₂²	1	2	yes	—
18	#17988962	C₂²	1	2	yes	—
19	#17988932	—	2	2	no	—
20	#17988824	C₂²	1	2	yes	—
21	#17988822	—	2	2	no	—
22	#17988820	C₂²	1	2	yes	—
23	C₄×D₅₈₄	C₂²	1	2	yes	—
24	C₄×#11924013	—	2	2	no	—
25	C₄×C₈×D₇₃	C₂²	1	1	yes	characteristic
26	#17988413	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	C₄×#11924003	C₂²	1	1	yes	characteristic
32	C₁₆×Dic₇₃	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×#11924028	—	4	2	no	—
40	C₂×D₅₈₄	C₂³	1	2	yes	—
41	C₂×#11924013	—	2	2	no	—
42	C₂×C₈×D₇₃	D₄	1	1	yes	characteristic
43	C₂×C₈×D₇₃	C₂³	1	1	yes	characteristic
44	#14556590	—	2	2	no	—
45	#14556587	—	2	2	no	—
46	#14556577	—	4	2	no	—
47	#14556566	—	2	2	no	—
48	#14556564	—	4	2	no	—
49	#14556562	—	2	2	no	—
50	C₄×D₂₉₂	—	2	2	no	—
51	C₄×C₂²⋊D₇₃	—	4	2	no	—
52	C₄²×D₇₃	D₄	1	1	yes	characteristic
53	D₁₁₆₈	—	2	2	no	—
54	#14556532	—	2	4	no	—
55	C₂×#11924003	C₂³	1	1	yes	characteristic
56	C₂×#11924003	C₂³	1	1	yes	characteristic
57	C₁₆×D₇₃	—	2	2	no	—
58	C₁₆×D₇₃	—	2	2	no	—
59	#14556525	C₂³	1	2	yes	—
60	#14556524	—	2	2	no	—
61	#14556513	—	2	2	no	—
62	#14556512	—	2	2	no	—
63	#14556511	—	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	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	#11924028	—	8	8	no	—
76	C₂×Dic₁₄₆	—	4	2	no	—
77	C₂×D₂₉₂	—	2	2	no	—
78	C₂×D₂₉₂	—	4	2	no	—
79	C₂×C₂²⋊D₇₃	—	4	2	no	—
80	C₂×C₂²⋊D₇₃	—	4	2	no	—
81	C₂×C₂²⋊D₇₃	—	4	2	no	—
82	C₂×C₄×D₇₃	—	4	2	no	—
83	C₂×C₄×D₇₃	D₈	1	1	yes	characteristic
84	C₂×C₄×D₇₃	D₈	1	1	yes	characteristic
85	C₂×C₄×D₇₃	C₂×D₄	1	1	yes	characteristic
86	D₅₈₄	—	2	2	no	—
87	D₅₈₄	—	2	2	no	—
88	#11924013	—	4	4	no	—
89	C₂×#10596831	C₂×D₄	1	1	yes	characteristic
90	C₂×#10596831	C₂×D₄	1	1	yes	characteristic
91	C₈×D₇₃	—	2	1	no	—
92	C₈×D₇₃	—	2	1	no	—
93	C₈×D₇₃	—	2	2	no	—
94	C₄⋊Dic₇₃	—	2	2	no	—
95	#11924007	—	4	2	no	—
96	C₂²⋊Dic₇₃	—	4	2	no	—
97	#11924005	—	4	2	no	—
98	#11924005	—	4	2	no	—
99	C₄×Dic₇₃	D₈	1	1	yes	characteristic
100	C₄×Dic₇₃	D₈	1	1	yes	characteristic
101	C₄×Dic₇₃	C₂×D₄	1	1	yes	characteristic
102	#11924003	—	2	1	no	—
103	#11924003	#41	1	2	yes	complemented
104	C₂×D₄×C₇₃	—	2	2	no	—
105	C₂²×C₄×C₇₃	—	4	2	no	—
106	D₈×C₇₃	—	2	2	no	—
107	D₈×C₇₃	—	2	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	—	2	2	no	—
111	C₂²⋊C₄×C₇₃	—	4	2	no	—
112	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
113	C₁₆×C₇₃	—	2	1	no	—
114	C₁₆×C₇₃	#41	1	2	yes	complemented
115	C₂³×C₇₃	—	4	2	no	—
116	C₂²×D₇₃	—	4	2	no	—
117	C₂²×D₇₃	C₂×D₈	1	1	yes	characteristic
118	D₄×C₇₃	—	4	2	no	—
119	D₄×C₇₃	—	4	2	no	—
120	C₂×C₄×C₇₃	—	4	2	no	—
121	C₂×C₄×C₇₃	—	8	4	no	—
122	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
123	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
124	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
125	C₂²⋊D₇₃	—	8	4	no	—
126	C₂²⋊D₇₃	—	8	4	no	—
127	C₂²⋊D₇₃	—	8	4	no	—
128	C₂×Dic₇₃	—	4	2	no	—
129	C₂×Dic₇₃	C₂×D₈	1	1	yes	characteristic
130	C₂×Dic₇₃	C₂×D₈	1	1	yes	characteristic
131	C₂×Dic₇₃	C₂×D₈	1	1	yes	characteristic
132	D₂₉₂	—	4	2	no	—
133	D₂₉₂	—	4	2	no	—
134	D₂₉₂	—	8	4	no	—
135	C₄×D₇₃	—	2	1	no	—
136	C₄×D₇₃	—	2	1	no	—
137	C₄×D₇₃	—	2	2	no	—
138	C₄×D₇₃	—	8	4	no	—
139	C₄×D₇₃	D₁₆	1	4	yes	complemented
140	Dic₁₄₆	—	8	4	no	—
141	C₈×C₇₃	—	2	1	no	—
142	C₈×C₇₃	C₂×#41	1	1	yes	characteristic
143	C₈×C₇₃	C₂×#41	1	1	yes	characteristic
144	#10596831	—	2	1	no	—
145	#10596831	#135	1	2	yes	—
146	C₂²×C₇₃	—	8	2	no	—
147	C₂²×C₇₃	—	8	2	no	—
148	C₂²×C₇₃	—	8	2	no	—
149	C₂²×C₇₃	C₂²×D₈	1	1	yes	characteristic
150	C₂×D₇₃	—	2	1	no	—
151	C₂×D₇₃	—	2	1	no	—
152	C₂×D₇₃	—	8	2	no	—
153	C₂×D₇₃	—	8	2	no	—
154	C₂×D₇₃	—	8	2	no	—
155	C₂×D₇₃	C₂×D₁₆	1	2	yes	—
156	C₄×C₇₃	—	2	1	no	—
157	C₄×C₇₃	—	8	2	no	—
158	C₄×C₇₃	C₂×D₁₆	1	2	yes	—
159	C₄×C₇₃	#532	1	1	yes	characteristic
160	C₄×C₇₃	C₂×#135	1	1	yes	characteristic
161	Dic₇₃	—	2	1	no	—
162	Dic₇₃	—	8	2	no	—
163	Dic₇₃	C₂×D₁₆	1	2	yes	—
164	Dic₇₃	#508	1	2	yes	—
165	#33733	—	73	1	no	maximal, Sylow
166	C₂×C₇₃	—	16	2	no	—
167	C₂×C₇₃	#3169	1	1	yes	characteristic
168	C₂×C₇₃	C₂²×D₁₆	1	1	yes	characteristic
169	C₂×C₇₃	C₂×#508	1	1	yes	characteristic
170	D₇₃	—	2	2	no	—
171	D₇₃	—	16	2	no	—
172	#3169	—	73	2	no	—
173	#2252	—	73	1	no	—
174	#2246	—	73	1	no	—
175	#2240	—	73	2	no	—
176	#2215	—	73	1	no	—
177	#2215	—	73	1	no	—
178	#2215	—	73	2	no	—
179	C₄×D₁₆	—	73	2	no	—
180	C₄×D₁₆	—	73	1	no	—
181	C₄×D₁₆	D₇₃	1	1	yes	characteristic, complemented
182	C₂×C₄×C₁₆	—	73	1	no	—
183	C₇₃	#33733	1	1	yes	characteristic, complemented, Sylow
184	#532	—	146	2	no	—
185	C₂×D₁₆	—	73	2	no	—
186	C₂×D₁₆	—	73	1	no	—
187	C₂×D₁₆	C₂×D₇₃	1	1	yes	characteristic
188	C₂²×C₁₆	—	73	1	no	—
189	C₂²×C₁₆	—	73	1	no	—
190	#500	—	73	2	no	—
191	#493	—	73	2	no	—
192	#486	—	146	2	no	—
193	#466	—	73	2	no	—
194	#466	—	73	2	no	—
195	#466	—	146	2	no	—
196	C₄×D₈	—	73	2	no	—
197	C₄×D₈	—	146	2	no	—
198	C₄×D₈	C₂×D₇₃	1	2	yes	—
199	C₂×C₄×C₈	—	73	1	no	—
200	#366	—	73	1	no	—
201	#366	—	73	2	no	—
202	#366	C₂×D₇₃	1	1	yes	characteristic
203	#357	—	73	2	no	—
204	#357	—	73	2	no	—
205	#357	—	73	2	no	—
206	#357	C₂×D₇₃	1	2	yes	—
207	C₄×C₁₆	—	73	1	no	—
208	C₄×C₁₆	—	73	1	no	—
209	C₄×C₁₆	—	73	1	no	—
210	C₄×C₁₆	C₂×D₇₃	1	1	yes	characteristic
211	C₂×#41	—	292	2	no	—
212	C₂×D₈	—	73	2	no	—
213	C₂×D₈	—	146	2	no	—
214	C₂×D₈	C₂²×D₇₃	1	2	yes	—
215	C₂²×C₈	—	73	1	no	—
216	C₂²×C₈	—	73	1	no	—
217	C₄⋊Q₈	—	146	2	no	—
218	#127	—	146	2	no	—
219	#124	—	292	2	no	—
220	#121	—	292	2	no	—
221	#121	—	146	2	no	—
222	#121	—	146	2	no	—
223	C₄×D₄	—	2	2	no	—
224	C₄×D₄	—	292	2	no	—
225	C₄×D₄	—	146	2	no	—
226	C₂×C₄²	—	73	1	no	—
227	D₁₆	—	2	2	no	—
228	D₁₆	—	146	4	no	—
229	D₁₆	—	146	2	no	—
230	C₂×C₁₆	—	73	1	no	—
231	C₂×C₁₆	—	73	1	no	—
232	C₂×C₁₆	—	146	2	no	—
233	C₂×C₁₆	—	146	2	no	—
234	C₂×C₁₆	C₂²×D₇₃	1	1	yes	characteristic
235	C₂×C₁₆	C₂²×D₇₃	1	1	yes	characteristic
236	#107	—	73	2	no	—
237	#107	—	146	2	no	—
238	#107	C₂²×D₇₃	1	2	yes	—
239	#102	—	2	2	no	—
240	#102	—	146	2	no	—
241	#102	—	146	2	no	—
242	#102	—	146	2	no	—
243	C₄×C₈	—	73	1	no	—
244	C₄×C₈	—	73	1	no	—
245	C₄×C₈	—	73	1	no	—
246	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
247	#41	—	584	8	no	—
248	C₂×Q₈	—	292	2	no	—
249	C₂×D₄	—	2	2	no	—
250	C₂×D₄	—	292	2	no	—
251	C₂×D₄	—	292	2	no	—
252	C₂×D₄	—	292	2	no	—
253	C₂×D₄	—	292	2	no	—
254	C₂×D₄	—	146	2	no	—
255	C₂²×C₄	—	4	2	no	—
256	C₂²×C₄	—	292	2	no	—
257	C₂²×C₄	—	73	1	no	—
258	C₂²×C₄	—	73	1	no	—
259	C₂²×C₄	—	73	1	no	—
260	D₈	—	2	2	no	—
261	D₈	—	2	2	no	—
262	D₈	—	292	4	no	—
263	D₈	—	146	2	no	—
264	D₈	—	146	2	no	—
265	C₂×C₈	—	73	1	no	—
266	C₂×C₈	—	73	1	no	—
267	C₂×C₈	—	146	2	no	—
268	C₂×C₈	—	146	1	no	—
269	C₂×C₈	—	146	1	no	—
270	C₂×C₈	D₄×D₇₃	1	1	yes	characteristic
271	C₂×C₈	C₂³×D₇₃	1	1	yes	characteristic
272	C₄⋊C₄	—	2	2	no	—
273	C₄⋊C₄	—	292	2	no	—
274	C₄⋊C₄	—	146	2	no	—
275	C₂²⋊C₄	—	4	2	no	—
276	C₂²⋊C₄	—	292	2	no	—
277	C₂²⋊C₄	—	292	2	no	—
278	C₂²⋊C₄	—	292	2	no	—
279	C₄²	—	73	1	no	—
280	C₄²	—	73	1	no	—
281	C₄²	—	73	1	no	—
282	C₄²	D₄×D₇₃	1	1	yes	characteristic
283	C₁₆	—	2	1	no	—
284	C₁₆	—	73	2	no	—
285	C₁₆	—	146	1	no	—
286	C₁₆	#11924028	1	2	yes	complemented
287	C₂³	—	4	2	no	—
288	C₂³	—	292	2	no	—
289	C₂³	—	73	1	no	—
290	Q₈	—	584	4	no	—
291	D₄	—	4	2	no	—
292	D₄	—	4	2	no	—
293	D₄	—	292	2	no	—
294	D₄	—	292	2	no	—
295	D₄	—	584	4	no	—
296	D₄	—	584	4	no	—
297	D₄	—	584	4	no	—
298	D₄	—	584	4	no	—
299	C₂×C₄	—	4	2	no	—
300	C₂×C₄	—	8	4	no	—
301	C₂×C₄	—	292	2	no	—
302	C₂×C₄	—	584	4	no	—
303	C₂×C₄	—	73	1	no	—
304	C₂×C₄	—	73	1	no	—
305	C₂×C₄	—	73	4	no	—
306	C₂×C₄	—	73	1	no	—
307	C₂×C₄	—	146	2	no	—
308	C₂×C₄	—	146	1	no	—
309	C₂×C₄	—	146	1	no	—
310	C₂×C₄	D₈×D₇₃	1	1	yes	characteristic
311	C₂×C₄	D₈×D₇₃	1	1	yes	characteristic
312	C₂×C₄	C₂×D₄×D₇₃	1	1	yes	characteristic
313	C₈	—	2	1	no	—
314	C₈	—	73	2	no	—
315	C₈	—	146	1	no	—
316	C₈	#41×D₇₃	1	1	yes	characteristic
317	C₈	C₂×#11924028	1	1	yes	characteristic
318	C₂²	—	8	2	no	—
319	C₂²	—	8	2	no	—
320	C₂²	—	8	2	no	—
321	C₂²	—	584	2	no	—
322	C₂²	—	584	2	no	—
323	C₂²	—	584	2	no	—
324	C₂²	—	73	2	no	—
325	C₂²	—	146	1	no	—
326	C₂²	—	146	1	no	—
327	C₂²	C₂×D₈×D₇₃	1	1	yes	characteristic
328	C₄	—	2	1	no	—
329	C₄	—	8	2	no	—
330	C₄	—	584	2	no	—
331	C₄	—	73	2	no	—
332	C₄	—	73	2	no	—
333	C₄	—	146	1	no	—
334	C₄	D₁₆×D₇₃	1	2	yes	—
335	C₄	#17989134	1	1	yes	characteristic
336	C₄	#135×D₇₃	1	1	yes	characteristic
337	C₂	—	16	2	no	—
338	C₂	—	1168	2	no	—
339	C₂	—	146	2	no	—
340	C₂	#22620507	1	1	yes	characteristic
341	C₂	C₂×D₁₆×D₇₃	1	1	yes	characteristic
342	C₂	#508×D₇₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	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	#33733
LCS	G ▷ C₂×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈ ◁ C₄×D₁₆

Complements

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

N	H ≅ G/N	type
1. #22620507	C₂	semidirect
2. #22612205	C₂	semidirect
3. #22612179	C₂	semidirect
4. #22612153	C₂	semidirect
5. #22612018	C₂	semidirect
6. #22612016	C₂	semidirect
7. #22612014	C₂	semidirect
8. C₄×D₁₁₆₈	C₂	semidirect
9. C₄×#14556532	C₂	semidirect
10. C₄×C₁₆×D₇₃	C₂	semidirect
11. C₄×D₁₆×C₇₃	C₂	semidirect
103. #11924003	#41	semidirect
114. C₁₆×C₇₃	#41	semidirect
139. C₄×D₇₃	D₁₆	semidirect
181. C₄×D₁₆	D₇₃	semidirect
183. C₇₃	#33733	semidirect
286. C₁₆	#11924028	semidirect

Maximal subgroups

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

subgroup	index
#22620507	2
#22612205	2
#22612179	2
#22612153	2
#22612018	2
#22612016	2
#22612014	2
C₄×D₁₁₆₈	2
C₄×#14556532	2
C₄×C₁₆×D₇₃	2
C₄×D₁₆×C₇₃	2
#33733	73