Subgroups — #33057744

Order 18688.

722 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	#22620510	C₂	1	1	yes	maximal, characteristic
2	#22618677	C₂	1	2	yes	maximal, complemented
3	#22608817	C₂	1	2	yes	maximal, complemented
4	#22608762	C₂	1	4	yes	maximal, complemented
5	C₈×D₅₈₄	C₂	1	2	yes	maximal, complemented
6	C₂×#17988340	C₂	1	1	yes	maximal, characteristic, complemented
7	#22608331	C₂	1	2	yes	maximal, complemented
8	C₈×M₁₆×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#14556599	—	2	2	no	—
10	#17989137	C₂²	1	2	yes	—
11	#17988992	C₄	1	1	yes	characteristic
12	#17988964	C₄	1	1	yes	characteristic
13	#17988934	C₄	1	2	yes	—
14	#17988823	C₄	1	2	yes	—
15	#17988823	C₄	1	2	yes	—
16	C₄×D₅₈₄	C₂²	1	2	yes	—
17	C₄×D₅₈₄	C₂²	1	2	yes	—
18	#17988666	C₂²	1	4	yes	—
19	#17988660	—	2	4	no	—
20	C₈×D₂₉₂	C₂²	1	4	yes	—
21	C₂×C₄⋊#10596831	C₂²	1	2	yes	—
22	#17988555	—	2	2	no	—
23	#17988340	C₂²	1	1	yes	characteristic
24	#17988340	C₂²	1	2	yes	—
25	#17988340	C₂²	1	1	yes	characteristic
26	#17988316	C₂²	1	4	yes	—
27	#17988316	C₂²	1	4	yes	—
28	#431×C₇₃	C₂²	1	1	yes	characteristic
29	C₄×M₁₆×C₇₃	C₂²	1	1	yes	characteristic
30	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
31	#322×C₇₃	C₂²	1	2	yes	—
32	C₈²×C₇₃	C₂²	1	2	yes	—
33	C₂×#11924028	—	2	2	no	—
34	C₂×D₅₈₄	C₂×C₄	1	2	yes	—
35	C₂×D₅₈₄	C₂×C₄	1	2	yes	—
36	#14556599	—	4	8	no	—
37	C₂×#11924011	—	2	4	no	—
38	C₂²×#10596831	—	2	2	no	—
39	C₂×C₈×D₇₃	—	2	4	no	—
40	#14556591	C₂×C₄	1	2	yes	—
41	#14556588	C₂×C₄	1	2	yes	—
42	#14556579	—	2	2	no	—
43	#14556565	—	2	2	no	—
44	#14556565	C₂×C₄	1	4	yes	—
45	C₄×D₂₉₂	—	2	2	no	—
46	C₄×D₂₉₂	C₂³	1	4	yes	—
47	#14556525	C₂×C₄	1	2	yes	—
48	#14556525	C₂×C₄	1	2	yes	—
49	C₄⋊#10596831	—	2	2	no	—
50	C₄⋊#10596831	C₂³	1	2	yes	—
51	C₄⋊#10596831	C₂³	1	2	yes	—
52	#14556513	C₂×C₄	1	4	yes	—
53	#14556513	C₂×C₄	1	4	yes	—
54	#14556503	—	2	4	no	—
55	#14556503	—	2	4	no	—
56	C₂×M₁₆×C₇₃	D₄	1	1	yes	characteristic
57	C₂×M₁₆×C₇₃	D₄	1	1	yes	characteristic
58	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
59	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
60	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
61	C₄⋊C₈×C₇₃	—	2	1	no	—
62	C₂²⋊C₈×C₇₃	—	2	1	no	—
63	#97×C₇₃	D₄	1	2	yes	—
64	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
65	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
66	C₄×C₈×C₇₃	C₂³	1	2	yes	—
67	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
68	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
69	C₄×C₈×C₇₃	C₂³	1	2	yes	—
70	#11924028	—	4	4	no	—
71	#11924028	—	4	4	no	—
72	C₂×Dic₁₄₆	—	2	2	no	—
73	C₂×D₂₉₂	—	2	2	no	—
74	C₂×D₂₉₂	—	2	2	no	—
75	C₂×D₂₉₂	C₂²×C₄	1	4	yes	—
76	C₂×C₂²⋊D₇₃	—	2	4	no	—
77	C₂×C₄×D₇₃	—	2	4	no	—
78	D₅₈₄	—	2	4	no	—
79	D₅₈₄	—	4	2	no	—
80	#11924011	—	4	8	no	—
81	C₂×#10596831	—	2	2	no	—
82	C₂×#10596831	—	2	2	no	—
83	C₂×#10596831	—	4	4	no	—
84	C₈×D₇₃	—	4	8	no	—
85	C₄⋊Dic₇₃	—	2	2	no	—
86	C₄⋊Dic₇₃	C₂²×C₄	1	4	yes	—
87	#11924005	—	2	4	no	—
88	#11924005	—	2	4	no	—
89	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
90	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
91	C₂²×C₄×C₇₃	#41	1	1	yes	characteristic
92	M₁₆×C₇₃	—	2	2	no	—
93	M₁₆×C₇₃	D₈	1	4	yes	complemented
94	C₂×C₈×C₇₃	—	2	2	no	—
95	C₂×C₈×C₇₃	—	2	2	no	—
96	C₂×C₈×C₇₃	—	4	1	no	—
97	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
98	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
99	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
100	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
101	C₂×C₈×C₇₃	C₂×D₄	1	2	yes	—
102	C₂×C₈×C₇₃	C₂×D₄	1	2	yes	—
103	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
104	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
105	C₄²×C₇₃	#41	1	1	yes	characteristic
106	C₄²×C₇₃	C₂⁴	1	1	yes	characteristic
107	C₂³×C₇₃	C₄×D₄	1	1	yes	characteristic
108	C₂²×D₇₃	—	2	4	no	—
109	C₂×C₄×C₇₃	—	2	1	no	—
110	C₂×C₄×C₇₃	—	2	1	no	—
111	C₂×C₄×C₇₃	—	4	1	no	—
112	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
113	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
114	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
115	C₂×C₄×C₇₃	C₂×D₈	1	2	yes	—
116	C₂×C₄×C₇₃	#135	1	2	yes	—
117	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
118	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
119	C₂×C₄×C₇₃	C₂×#41	1	1	yes	characteristic
120	C₂²⋊D₇₃	—	4	8	no	—
121	C₂×Dic₇₃	—	2	4	no	—
122	D₂₉₂	—	2	4	no	—
123	D₂₉₂	—	2	4	no	—
124	D₂₉₂	—	4	4	no	—
125	D₂₉₂	—	8	2	no	—
126	C₄×D₇₃	—	4	4	no	—
127	C₄×D₇₃	—	4	4	no	—
128	Dic₁₄₆	—	4	4	no	—
129	C₈×C₇₃	—	2	1	no	—
130	C₈×C₇₃	—	2	1	no	—
131	C₈×C₇₃	—	2	1	no	—
132	C₈×C₇₃	—	2	2	no	—
133	C₈×C₇₃	—	4	2	no	—
134	C₈×C₇₃	#131	1	2	yes	complemented
135	C₈×C₇₃	C₂×D₈	1	4	yes	—
136	#10596831	—	4	2	no	—
137	#10596831	—	4	2	no	—
138	C₂²×C₇₃	—	2	1	no	—
139	C₂²×C₇₃	—	2	1	no	—
140	C₂²×C₇₃	C₄×D₈	1	2	yes	—
141	C₂²×C₇₃	C₂×C₄×D₄	1	1	yes	characteristic
142	C₂×D₇₃	—	4	4	no	—
143	C₂×D₇₃	—	4	4	no	—
144	C₂×D₇₃	—	4	4	no	—
145	C₄×C₇₃	—	2	1	no	—
146	C₄×C₇₃	—	2	1	no	—
147	C₄×C₇₃	—	2	1	no	—
148	C₄×C₇₃	C₄×D₈	1	2	yes	—
149	C₄×C₇₃	C₂×#131	1	1	yes	characteristic
150	C₄×C₇₃	#568	1	1	yes	characteristic
151	C₄×C₇₃	C₂²×D₈	1	1	yes	characteristic
152	C₄×C₇₃	C₂×#135	1	1	yes	characteristic
153	Dic₇₃	—	4	4	no	—
154	#18066	—	73	1	no	maximal, Sylow
155	C₂×C₇₃	—	2	2	no	—
156	C₂×C₇₃	#2935	1	1	yes	characteristic
157	C₂×C₇₃	C₂×C₄×D₈	1	1	yes	characteristic
158	C₂×C₇₃	#2957	1	1	yes	characteristic
159	D₇₃	—	8	4	no	—
160	#3169	—	73	1	no	—
161	#2935	—	73	2	no	—
162	#1589	—	73	2	no	—
163	#1581	—	73	4	no	—
164	C₈×D₈	—	73	2	no	—
165	C₂×#335	—	73	1	no	—
166	#1480	—	73	2	no	—
167	C₈×M₁₆	D₇₃	1	1	yes	characteristic, complemented
168	C₇₃	#18066	1	1	yes	characteristic, complemented, Sylow
169	C₂×#131	—	146	2	no	—
170	#532	—	73	2	no	—
171	#500	—	73	1	no	—
172	#493	—	73	1	no	—
173	#486	—	73	2	no	—
174	#466	—	73	2	no	—
175	#466	—	73	2	no	—
176	C₄×D₈	—	73	2	no	—
177	C₄×D₈	—	73	2	no	—
178	#436	—	73	4	no	—
179	#435	—	146	4	no	—
180	C₈×D₄	—	73	4	no	—
181	#431	C₂×D₇₃	1	1	yes	characteristic
182	C₂×C₄⋊C₈	—	73	2	no	—
183	#408	—	146	2	no	—
184	C₄×M₁₆	C₂×D₇₃	1	1	yes	characteristic
185	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
186	#335	—	73	1	no	—
187	#335	—	73	2	no	—
188	#335	—	73	1	no	—
189	D₄⋊C₈	—	73	4	no	—
190	D₄⋊C₈	—	73	4	no	—
191	#322	C₂×D₇₃	1	2	yes	—
192	C₈²	C₂×D₇₃	1	2	yes	—
193	C₂×#41	—	146	2	no	—
194	C₂×D₈	—	73	2	no	—
195	C₂×D₈	—	73	2	no	—
196	#131	—	292	8	no	—
197	C₂×M₁₆	—	146	4	no	—
198	C₂×M₁₆	D₂₉₂	1	1	yes	characteristic
199	C₂×M₁₆	D₂₉₂	1	1	yes	characteristic
200	C₂²×C₈	—	146	2	no	—
201	C₂²×C₈	—	146	4	no	—
202	C₂²×C₈	C₄×D₇₃	1	1	yes	characteristic
203	C₂²×C₈	C₄×D₇₃	1	1	yes	characteristic
204	C₄⋊Q₈	—	73	2	no	—
205	#127	—	73	2	no	—
206	#124	—	146	2	no	—
207	#121	—	73	4	no	—
208	#121	—	146	2	no	—
209	C₄×D₄	—	73	4	no	—
210	C₄×D₄	—	146	2	no	—
211	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
212	#107	—	73	2	no	—
213	#107	—	73	2	no	—
214	C₄⋊C₈	—	2	1	no	—
215	C₄⋊C₈	—	73	2	no	—
216	C₄⋊C₈	—	73	2	no	—
217	C₄⋊C₈	—	146	2	no	—
218	#102	—	73	4	no	—
219	#102	—	73	4	no	—
220	C₂²⋊C₈	—	2	1	no	—
221	C₂²⋊C₈	—	146	4	no	—
222	C₂²⋊C₈	—	146	4	no	—
223	#97	D₂₉₂	1	2	yes	—
224	C₄×C₈	C₄×D₇₃	1	1	yes	characteristic
225	C₄×C₈	C₄×D₇₃	1	1	yes	characteristic
226	C₄×C₈	C₂²×D₇₃	1	2	yes	—
227	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
228	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
229	C₄×C₈	C₂²×D₇₃	1	2	yes	—
230	#41	—	292	4	no	—
231	#41	—	292	4	no	—
232	C₂×Q₈	—	146	2	no	—
233	C₂×D₄	—	73	4	no	—
234	C₂×D₄	—	146	4	no	—
235	C₂×D₄	—	146	2	no	—
236	C₂×D₄	—	146	2	no	—
237	C₂²×C₄	—	146	4	no	—
238	C₂²×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
239	C₂²×C₄	C₂×D₂₉₂	1	1	yes	characteristic
240	C₂²×C₄	#11924028	1	1	yes	characteristic
241	D₈	—	292	2	no	—
242	D₈	—	146	4	no	—
243	M₁₆	—	2	2	no	—
244	M₁₆	—	292	8	no	—
245	M₁₆	D₅₈₄	1	4	yes	complemented
246	C₂×C₈	—	2	2	no	—
247	C₂×C₈	—	2	2	no	—
248	C₂×C₈	—	4	1	no	—
249	C₂×C₈	—	292	4	no	—
250	C₂×C₈	—	292	8	no	—
251	C₂×C₈	—	146	2	no	—
252	C₂×C₈	—	146	2	no	—
253	C₂×C₈	C₂×C₄×D₇₃	1	1	yes	characteristic
254	C₂×C₈	C₂×C₄×D₇₃	1	1	yes	characteristic
255	C₂×C₈	C₂×C₄×D₇₃	1	1	yes	characteristic
256	C₂×C₈	C₂×C₄×D₇₃	1	1	yes	characteristic
257	C₂×C₈	C₂×D₂₉₂	1	2	yes	—
258	C₂×C₈	C₂×D₂₉₂	1	2	yes	—
259	C₄⋊C₄	—	73	4	no	—
260	C₄⋊C₄	—	146	2	no	—
261	C₂²⋊C₄	—	146	4	no	—
262	C₂²⋊C₄	—	146	4	no	—
263	C₄²	C₂×C₄×D₇₃	1	1	yes	characteristic
264	C₄²	C₂×D₂₉₂	1	1	yes	characteristic
265	C₄²	#11924028	1	1	yes	characteristic
266	C₄²	C₂³×D₇₃	1	1	yes	characteristic
267	C₂³	—	146	4	no	—
268	C₂³	C₄×D₂₉₂	1	1	yes	characteristic
269	Q₈	—	292	4	no	—
270	D₄	—	292	8	no	—
271	D₄	—	292	4	no	—
272	D₄	—	584	2	no	—
273	D₄	—	146	4	no	—
274	D₄	—	146	4	no	—
275	C₂×C₄	—	2	1	no	—
276	C₂×C₄	—	2	1	no	—
277	C₂×C₄	—	4	1	no	—
278	C₂×C₄	—	292	4	no	—
279	C₂×C₄	—	292	4	no	—
280	C₂×C₄	—	146	4	no	—
281	C₂×C₄	C₄×D₂₉₂	1	1	yes	characteristic
282	C₂×C₄	C₄×D₂₉₂	1	1	yes	characteristic
283	C₂×C₄	C₄×D₂₉₂	1	1	yes	characteristic
284	C₂×C₄	C₂×D₅₈₄	1	2	yes	—
285	C₂×C₄	#14556613	1	2	yes	—
286	C₂×C₄	C₂²×C₄×D₇₃	1	1	yes	characteristic
287	C₂×C₄	C₂²×D₂₉₂	1	1	yes	characteristic
288	C₂×C₄	C₂×#11924028	1	1	yes	characteristic
289	C₈	—	2	1	no	—
290	C₈	—	2	1	no	—
291	C₈	—	2	1	no	—
292	C₈	—	2	2	no	—
293	C₈	—	4	2	no	—
294	C₈	—	292	2	no	—
295	C₈	—	292	2	no	—
296	C₈	#14556599	1	2	yes	complemented
297	C₈	C₂×D₅₈₄	1	4	yes	—
298	C₂²	—	2	1	no	—
299	C₂²	—	2	1	no	—
300	C₂²	—	292	4	no	—
301	C₂²	—	292	4	no	—
302	C₂²	—	292	4	no	—
303	C₂²	C₄×D₅₈₄	1	2	yes	—
304	C₂²	C₂×C₄×D₂₉₂	1	1	yes	characteristic
305	C₄	—	2	1	no	—
306	C₄	—	2	1	no	—
307	C₄	—	2	1	no	—
308	C₄	—	292	4	no	—
309	C₄	C₄×D₅₈₄	1	2	yes	—
310	C₄	C₂×#14556599	1	1	yes	characteristic
311	C₄	#17989353	1	1	yes	characteristic
312	C₄	C₂²×D₅₈₄	1	1	yes	characteristic
313	C₄	C₂×#14556613	1	1	yes	characteristic
314	C₂	—	2	2	no	—
315	C₂	—	584	4	no	—
316	C₂	#22618677	1	1	yes	characteristic
317	C₂	C₂×C₄×D₅₈₄	1	1	yes	characteristic
318	C₂	#22618849	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₂₉₂
commutator	C₂×C₄×C₇₃	C₂³×C₄
Frattini	C₄²	C₂³×D₇₃
Fitting	C₈×M₁₆×C₇₃	C₂
socle	C₂²×C₇₃	C₂×C₄×D₄
prime radical	#22620510	C₂
cosocle	C₄²×C₇₃	C₂⁴
hypercenter	C₈×M₁₆	D₇₃
hypocenter	C₇₃	#18066
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄×M₁₆ ◁ C₈×M₁₆

Complements

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

N	H ≅ G/N	type
2. #22618677	C₂	semidirect
3. #22608817	C₂	semidirect
4. #22608762	C₂	semidirect
5. C₈×D₅₈₄	C₂	semidirect
6. C₂×#17988340	C₂	semidirect
7. #22608331	C₂	semidirect
8. C₈×M₁₆×C₇₃	C₂	semidirect
93. M₁₆×C₇₃	D₈	semidirect
134. C₈×C₇₃	#131	semidirect
167. C₈×M₁₆	D₇₃	semidirect
168. C₇₃	#18066	semidirect
245. M₁₆	D₅₈₄	semidirect
296. C₈	#14556599	semidirect

Maximal subgroups

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

subgroup	index
#22620510	2
#22618677	2
#22608817	2
#22608762	2
C₈×D₅₈₄	2
C₂×#17988340	2
#22608331	2
C₈×M₁₆×C₇₃	2
#18066	73