Subgroups — #29838074

Order 15616.

510 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21458829	C₂	1	1	yes	maximal, characteristic
2	#21458824	C₂	1	1	yes	maximal, characteristic, complemented
3	#21448087	C₂	1	2	yes	maximal, complemented
4	#21448075	C₂	1	1	yes	maximal, characteristic
5	#21447884	C₂	1	2	yes	maximal, complemented
6	#21447869	C₂	1	1	yes	maximal, characteristic, complemented
7	#21447863	C₂	1	2	yes	maximal, complemented
8	#21447859	C₂	1	1	yes	maximal, characteristic
9	#21447727	C₂	1	1	yes	maximal, characteristic, complemented
10	#21447682	C₂	1	2	yes	maximal, complemented
11	C₄×Q₃₂×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
12	#16840497	—	2	1	no	—
13	#16840490	—	2	1	no	—
14	C₂×#14519723	C₂²	1	2	yes	—
15	C₂×#14519722	C₂²	1	2	yes	—
16	C₂×#14519722	C₂²	1	2	yes	—
17	C₂×#14519720	C₂²	1	2	yes	—
18	#16840147	C₂²	1	2	yes	—
19	#16840142	C₂²	1	1	yes	characteristic
20	#16840140	C₂²	1	2	yes	—
21	#16840139	C₂²	1	1	yes	characteristic
22	#16839980	—	2	2	no	—
23	#16839972	—	2	1	no	—
24	#16839971	—	2	2	no	—
25	#16839962	—	2	1	no	—
26	C₈×Dic₁₂₂	C₂²	1	1	yes	characteristic
27	#16839856	C₂²	1	2	yes	—
28	#16839844	C₂²	1	2	yes	—
29	C₄⋊#11758185	C₂²	1	1	yes	characteristic
30	#16839580	C₂²	1	2	yes	—
31	#16839565	C₂²	1	2	yes	—
32	#16839562	C₂²	1	2	yes	—
33	#16839537	C₂²	1	2	yes	—
34	C₂×Q₃₂×C₆₁	C₂²	1	1	yes	characteristic
35	C₄×Q₁₆×C₆₁	C₂²	1	1	yes	characteristic
36	C₄×Q₁₆×C₆₁	C₂²	1	1	yes	characteristic
37	C₆₁×#366	C₂²	1	1	yes	characteristic
38	C₆₁×#358	C₂²	1	1	yes	characteristic
39	C₆₁×#358	C₂²	1	1	yes	characteristic
40	C₄×C₁₆×C₆₁	C₂²	1	1	yes	characteristic
41	C₂×Dic₂₄₄	—	2	1	no	—
42	C₂×Dic₂₄₄	C₂³	1	2	yes	—
43	C₂×#11758202	—	2	2	no	—
44	C₂×#11758201	—	2	2	no	—
45	C₂×D₄₈₈	—	2	1	no	—
46	C₂×D₄₈₈	C₂³	1	2	yes	—
47	#14519777	—	2	1	no	—
48	#14519775	—	2	2	no	—
49	#14519774	—	2	1	no	—
50	#14519769	—	4	1	no	—
51	#14519767	—	4	2	no	—
52	#14519766	—	2	2	no	—
53	#14519765	—	4	1	no	—
54	#14519762	—	4	2	no	—
55	C₄×Dic₁₂₂	—	4	1	no	—
56	C₄×Dic₁₂₂	D₄	1	1	yes	characteristic
57	#14519742	—	2	2	no	—
58	C₄×D₂₄₄	—	4	1	no	—
59	#14519739	—	2	2	no	—
60	#14519723	—	4	2	no	—
61	#14519722	—	2	4	no	—
62	#14519722	—	4	2	no	—
63	#14519720	—	4	2	no	—
64	C₂×#11758185	C₂³	1	2	yes	—
65	C₄⋊#92418	D₄	1	1	yes	characteristic
66	#14519706	D₄	1	2	yes	—
67	#14519700	—	2	2	no	—
68	#14519697	—	2	2	no	—
69	C₈×Dic₆₁	C₂³	1	2	yes	—
70	C₂×Q₁₆×C₆₁	C₂³	1	1	yes	characteristic
71	C₂×Q₁₆×C₆₁	C₂³	1	1	yes	characteristic
72	C₄×Q₈×C₆₁	—	2	1	no	—
73	C₄×Q₈×C₆₁	—	2	1	no	—
74	Q₃₂×C₆₁	—	2	2	no	—
75	C₂×C₁₆×C₆₁	C₂³	1	1	yes	characteristic
76	C₂×C₁₆×C₆₁	C₂³	1	1	yes	characteristic
77	#107×C₆₁	C₂³	1	1	yes	characteristic
78	#107×C₆₁	C₂³	1	1	yes	characteristic
79	#103×C₆₁	—	2	1	no	—
80	#103×C₆₁	—	2	1	no	—
81	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
82	C₂×Q₈×C₆₁	—	2	1	no	—
83	C₂×Q₈×C₆₁	—	2	1	no	—
84	C₂×D₂₄₄	—	2	2	no	—
85	C₂×D₂₄₄	—	4	1	no	—
86	C₂×D₂₄₄	—	4	1	no	—
87	C₂×C₄×D₆₁	—	4	2	no	—
88	C₂×Dic₁₂₂	—	2	2	no	—
89	C₂×Dic₁₂₂	D₈	1	1	yes	characteristic
90	Q₁₆×C₆₁	—	2	1	no	—
91	Q₁₆×C₆₁	—	2	1	no	—
92	Q₁₆×C₆₁	—	2	1	no	—
93	Q₁₆×C₆₁	—	2	1	no	—
94	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
95	C₂×C₈×C₆₁	C₂⁴	1	1	yes	characteristic
96	C₄⋊C₄×C₆₁	—	2	1	no	—
97	C₄⋊C₄×C₆₁	—	2	1	no	—
98	C₄⋊C₄×C₆₁	—	4	1	no	—
99	C₄⋊C₄×C₆₁	—	4	1	no	—
100	C₄²×C₆₁	—	4	1	no	—
101	C₄²×C₆₁	—	4	1	no	—
102	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
103	#11758202	—	8	2	no	—
104	#11758201	—	8	2	no	—
105	#11758198	—	4	2	no	—
106	#11758198	—	4	2	no	—
107	#11758198	—	4	2	no	—
108	C₄⋊Dic₆₁	—	4	1	no	—
109	C₄⋊Dic₆₁	—	4	1	no	—
110	C₄⋊Dic₆₁	D₈	1	1	yes	characteristic
111	#11758196	—	4	2	no	—
112	#11758196	—	4	2	no	—
113	#11758196	—	4	2	no	—
114	#11758196	D₈	1	2	yes	—
115	C₄×Dic₆₁	—	4	2	no	—
116	C₄×Dic₆₁	C₂×D₄	1	2	yes	—
117	C₂×#92418	C₂×D₄	1	2	yes	—
118	Dic₂₄₄	—	2	2	no	—
119	Dic₂₄₄	—	2	2	no	—
120	Dic₂₄₄	—	8	1	no	—
121	D₄₈₈	—	2	2	no	—
122	D₄₈₈	—	2	2	no	—
123	D₄₈₈	—	8	1	no	—
124	C₁₆×C₆₁	—	2	1	no	—
125	C₁₆×C₆₁	#41	1	2	yes	—
126	#11758185	—	2	2	no	—
127	C₂²×D₆₁	—	4	2	no	—
128	Q₈×C₆₁	—	4	1	no	—
129	Q₈×C₆₁	—	4	1	no	—
130	Q₈×C₆₁	—	4	1	no	—
131	Q₈×C₆₁	—	4	1	no	—
132	C₂×C₄×C₆₁	—	4	1	no	—
133	C₂×C₄×C₆₁	—	4	1	no	—
134	C₂×C₄×C₆₁	—	4	1	no	—
135	C₂×C₄×C₆₁	—	4	1	no	—
136	C₂×C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
137	C₂×C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
138	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
139	C₂×Dic₆₁	—	4	2	no	—
140	C₂×Dic₆₁	C₂×D₈	1	2	yes	—
141	C₂×Dic₆₁	C₂×D₈	1	2	yes	—
142	D₂₄₄	—	4	2	no	—
143	D₂₄₄	—	4	2	no	—
144	D₂₄₄	—	8	2	no	—
145	D₂₄₄	—	16	1	no	—
146	C₄×D₆₁	—	16	2	no	—
147	Dic₁₂₂	—	2	2	no	—
148	Dic₁₂₂	—	4	2	no	—
149	Dic₁₂₂	—	4	2	no	—
150	C₈×C₆₁	—	2	1	no	—
151	C₈×C₆₁	C₂×#41	1	1	yes	characteristic
152	C₈×C₆₁	H(5,𝔽₂)	1	1	yes	characteristic
153	#92418	—	2	2	no	—
154	#33967	—	61	1	no	maximal, Sylow
155	C₂²×C₆₁	C₂²×D₈	1	1	yes	characteristic
156	C₂×D₆₁	—	8	2	no	—
157	C₂×D₆₁	—	8	2	no	—
158	C₂×D₆₁	—	8	2	no	—
159	C₄×C₆₁	—	2	1	no	—
160	C₄×C₆₁	—	8	1	no	—
161	C₄×C₆₁	—	8	1	no	—
162	C₄×C₆₁	—	8	1	no	—
163	C₄×C₆₁	—	8	1	no	—
164	C₄×C₆₁	#510	1	2	yes	—
165	C₄×C₆₁	#550	1	1	yes	characteristic
166	C₄×C₆₁	#576	1	1	yes	characteristic
167	Dic₆₁	—	2	2	no	—
168	Dic₆₁	—	2	2	no	—
169	Dic₆₁	—	8	2	no	—
170	#3396	—	61	1	no	—
171	#3395	—	61	1	no	—
172	#2251	—	61	2	no	—
173	#2248	—	61	1	no	—
174	#2211	—	61	2	no	—
175	#2209	—	61	1	no	—
176	#2208	—	61	2	no	—
177	#2207	—	61	1	no	—
178	#2183	—	61	1	no	—
179	#2175	—	61	2	no	—
180	C₄×Q₃₂	D₆₁	1	1	yes	characteristic, complemented
181	C₂×C₆₁	#3391	1	1	yes	characteristic
182	C₂×C₆₁	C₂×#510	1	1	yes	characteristic
183	C₂×C₆₁	#3416	1	1	yes	characteristic
184	D₆₁	—	16	2	no	—
185	#559	—	122	1	no	—
186	#557	—	122	1	no	—
187	C₂×Q₃₂	—	61	2	no	—
188	C₂×Q₃₂	C₂×D₆₁	1	1	yes	characteristic
189	C₂×SD₃₂	—	61	2	no	—
190	C₂×SD₃₂	—	61	2	no	—
191	C₂×SD₃₂	—	61	2	no	—
192	#495	—	61	2	no	—
193	#494	—	61	2	no	—
194	#494	—	61	1	no	—
195	#493	—	61	1	no	—
196	#463	—	122	2	no	—
197	#462	—	122	2	no	—
198	#462	—	122	1	no	—
199	#460	—	122	1	no	—
200	C₈×Q₈	—	61	1	no	—
201	C₄×Q₁₆	—	61	2	no	—
202	C₄×Q₁₆	C₂×D₆₁	1	1	yes	characteristic
203	C₄×Q₁₆	C₂×D₆₁	1	1	yes	characteristic
204	C₄×D₈	—	61	2	no	—
205	#366	C₂×D₆₁	1	1	yes	characteristic
206	C₄⋊C₁₆	—	61	2	no	—
207	C₄⋊C₁₆	—	61	1	no	—
208	#358	—	61	2	no	—
209	#358	C₂×D₆₁	1	1	yes	characteristic
210	#358	C₂×D₆₁	1	1	yes	characteristic
211	#357	—	61	2	no	—
212	#346	—	61	2	no	—
213	C₄×C₁₆	C₂×D₆₁	1	1	yes	characteristic
214	C₆₁	#33967	1	1	yes	characteristic, complemented, Sylow
215	C₂×Q₁₆	—	61	2	no	—
216	C₂×Q₁₆	—	122	2	no	—
217	C₂×Q₁₆	—	122	1	no	—
218	C₂×Q₁₆	C₂²×D₆₁	1	1	yes	characteristic
219	C₂×Q₁₆	C₂²×D₆₁	1	1	yes	characteristic
220	C₂×SD₁₆	—	122	2	no	—
221	C₂×D₈	—	61	2	no	—
222	C₂×D₈	—	122	1	no	—
223	C₄⋊Q₈	—	122	2	no	—
224	C₄⋊Q₈	—	122	1	no	—
225	#127	—	122	1	no	—
226	#125	—	244	1	no	—
227	#125	—	244	2	no	—
228	#124	—	122	2	no	—
229	#124	—	244	1	no	—
230	#123	—	244	2	no	—
231	C₄×Q₈	—	2	1	no	—
232	C₄×Q₈	—	2	1	no	—
233	C₄×Q₈	—	61	1	no	—
234	C₄×Q₈	—	122	2	no	—
235	C₄×Q₈	—	244	1	no	—
236	C₄×D₄	—	122	2	no	—
237	C₄×D₄	—	244	1	no	—
238	Q₃₂	—	2	2	no	—
239	Q₃₂	—	244	2	no	—
240	SD₃₂	—	122	4	no	—
241	SD₃₂	—	244	2	no	—
242	SD₃₂	—	244	2	no	—
243	C₂×C₁₆	—	61	2	no	—
244	C₂×C₁₆	C₂²×D₆₁	1	1	yes	characteristic
245	C₂×C₁₆	C₂²×D₆₁	1	1	yes	characteristic
246	#107	C₂²×D₆₁	1	1	yes	characteristic
247	#107	C₂²×D₆₁	1	1	yes	characteristic
248	C₄⋊C₈	—	61	2	no	—
249	C₄⋊C₈	—	61	1	no	—
250	#103	—	2	1	no	—
251	#103	—	2	1	no	—
252	#103	—	122	2	no	—
253	#102	—	122	2	no	—
254	C₄×C₈	—	61	2	no	—
255	C₄×C₈	C₂²×D₆₁	1	1	yes	characteristic
256	C₂×Q₈	—	2	1	no	—
257	C₂×Q₈	—	2	1	no	—
258	C₂×Q₈	—	61	1	no	—
259	C₂×Q₈	—	122	2	no	—
260	C₂×D₄	—	122	2	no	—
261	C₂×D₄	—	244	1	no	—
262	C₂×D₄	—	244	1	no	—
263	C₂²×C₄	—	244	2	no	—
264	Q₁₆	—	2	1	no	—
265	Q₁₆	—	2	1	no	—
266	Q₁₆	—	2	1	no	—
267	Q₁₆	—	2	1	no	—
268	Q₁₆	—	122	2	no	—
269	Q₁₆	—	122	2	no	—
270	Q₁₆	—	488	1	no	—
271	Q₁₆	—	488	2	no	—
272	SD₁₆	—	488	2	no	—
273	D₈	—	122	2	no	—
274	D₈	—	122	2	no	—
275	D₈	—	488	1	no	—
276	C₂×C₈	—	61	2	no	—
277	C₂×C₈	D₄×D₆₁	1	1	yes	characteristic
278	C₂×C₈	C₂³×D₆₁	1	1	yes	characteristic
279	C₄⋊C₄	—	2	1	no	—
280	C₄⋊C₄	—	2	1	no	—
281	C₄⋊C₄	—	4	1	no	—
282	C₄⋊C₄	—	4	1	no	—
283	C₄⋊C₄	—	61	2	no	—
284	C₄⋊C₄	—	61	1	no	—
285	C₄⋊C₄	—	244	2	no	—
286	C₄⋊C₄	—	244	2	no	—
287	C₄⋊C₄	—	244	2	no	—
288	C₄⋊C₄	—	244	1	no	—
289	C₄⋊C₄	—	244	1	no	—
290	C₂²⋊C₄	—	244	2	no	—
291	C₂²⋊C₄	—	244	2	no	—
292	C₂²⋊C₄	—	244	2	no	—
293	C₄²	—	4	1	no	—
294	C₄²	—	4	1	no	—
295	C₄²	—	61	2	no	—
296	C₄²	—	244	2	no	—
297	C₄²	D₄×D₆₁	1	1	yes	characteristic
298	C₁₆	—	2	1	no	—
299	C₁₆	—	122	2	no	—
300	C₁₆	#11758222	1	2	yes	—
301	C₂³	—	244	2	no	—
302	Q₈	—	4	1	no	—
303	Q₈	—	4	1	no	—
304	Q₈	—	4	1	no	—
305	Q₈	—	4	1	no	—
306	Q₈	—	122	2	no	—
307	Q₈	—	244	2	no	—
308	Q₈	—	244	2	no	—
309	D₄	—	976	1	no	—
310	D₄	—	488	2	no	—
311	D₄	—	244	2	no	—
312	D₄	—	244	2	no	—
313	C₂×C₄	—	4	1	no	—
314	C₂×C₄	—	4	1	no	—
315	C₂×C₄	—	4	1	no	—
316	C₂×C₄	—	4	1	no	—
317	C₂×C₄	—	61	2	no	—
318	C₂×C₄	—	61	2	no	—
319	C₂×C₄	—	976	2	no	—
320	C₂×C₄	—	244	2	no	—
321	C₂×C₄	D₈×D₆₁	1	1	yes	characteristic
322	C₂×C₄	D₈×D₆₁	1	1	yes	characteristic
323	C₂×C₄	C₂×D₄×D₆₁	1	1	yes	characteristic
324	C₈	—	2	1	no	—
325	C₈	—	122	2	no	—
326	C₈	C₂×#11758222	1	1	yes	characteristic
327	C₈	#14519823	1	1	yes	characteristic
328	C₂²	—	488	2	no	—
329	C₂²	—	488	2	no	—
330	C₂²	—	488	2	no	—
331	C₂²	C₂×D₈×D₆₁	1	1	yes	characteristic
332	C₄	—	2	1	no	—
333	C₄	—	8	1	no	—
334	C₄	—	8	1	no	—
335	C₄	—	8	1	no	—
336	C₄	—	8	1	no	—
337	C₄	—	122	2	no	—
338	C₄	—	122	2	no	—
339	C₄	—	488	2	no	—
340	C₄	#16840204	1	2	yes	—
341	C₄	#16840433	1	1	yes	characteristic
342	C₄	#16840564	1	1	yes	characteristic
343	C₂	—	976	2	no	—
344	C₂	#21458793	1	1	yes	characteristic
345	C₂	C₂×#16840204	1	1	yes	characteristic
346	C₂	#21458962	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₈×D₆₁
commutator, cosocle	C₂×C₈×C₆₁	C₂⁴
Frattini	C₂×C₈	C₂³×D₆₁
Fitting	C₄×Q₃₂×C₆₁	C₂
socle	C₂²×C₆₁	C₂²×D₈
prime radical	#16840139	C₂²
hypercenter	C₄×Q₃₂	D₆₁
hypocenter	C₆₁	#33967
LCS	G ▷ C₂×C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×Q₃₂

Complements

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

N	H ≅ G/N	type
2. #21458824	C₂	semidirect
3. #21448087	C₂	semidirect
5. #21447884	C₂	semidirect
6. #21447869	C₂	semidirect
7. #21447863	C₂	semidirect
9. #21447727	C₂	semidirect
10. #21447682	C₂	semidirect
11. C₄×Q₃₂×C₆₁	C₂	semidirect
180. C₄×Q₃₂	D₆₁	semidirect
214. C₆₁	#33967	semidirect

Maximal subgroups

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

subgroup	index
#21458829	2
#21458824	2
#21448087	2
#21448075	2
#21447884	2
#21447869	2
#21447863	2
#21447859	2
#21447727	2
#21447682	2
C₄×Q₃₂×C₆₁	2
#33967	61