Subgroups — #29838072

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	#21458832	C₂	1	1	yes	maximal, characteristic
2	#21448088	C₂	1	2	yes	maximal, complemented
3	#21448076	C₂	1	1	yes	maximal, characteristic
4	#21447881	C₂	1	2	yes	maximal, complemented
5	#21447866	C₂	1	1	yes	maximal, characteristic, complemented
6	#21447861	C₂	1	2	yes	maximal, complemented
7	#21447859	C₂	1	1	yes	maximal, characteristic
8	#21447728	C₂	1	1	yes	maximal, characteristic, complemented
9	#21447681	C₂	1	2	yes	maximal, complemented
10	C₄×#14519723	C₂	1	1	yes	maximal, characteristic, complemented
11	C₆₁×#3395	C₂	1	1	yes	maximal, characteristic, complemented
12	#16840497	—	2	1	no	—
13	C₂×#14519723	C₂²	1	1	yes	characteristic
14	C₂×#14519723	C₂²	1	2	yes	—
15	C₂×#14519720	C₂²	1	2	yes	—
16	C₂×#14519720	C₂²	1	2	yes	—
17	C₂×#14519720	C₂²	1	2	yes	—
18	#16840146	C₂²	1	2	yes	—
19	#16840139	C₂²	1	1	yes	characteristic
20	#16839978	—	2	2	no	—
21	#16839963	—	2	1	no	—
22	C₄×Dic₂₄₄	C₂²	1	1	yes	characteristic
23	C₄×D₄₈₈	C₂²	1	2	yes	—
24	#16839587	C₂²	1	1	yes	characteristic
25	C₄⋊#11758185	C₂²	1	1	yes	characteristic
26	C₄⋊#11758185	C₂²	1	2	yes	—
27	#16839566	C₂²	1	2	yes	—
28	#16839566	C₂²	1	1	yes	characteristic
29	#16839565	C₂²	1	1	yes	characteristic
30	#16839562	C₂²	1	2	yes	—
31	#16839538	C₂²	1	2	yes	—
32	C₄×#11758185	C₂²	1	1	yes	characteristic
33	C₆₁×#557	—	2	1	no	—
34	C₆₁×#494	C₂²	1	1	yes	characteristic
35	C₆₁×#494	C₂²	1	2	yes	—
36	C₆₁×#462	—	2	1	no	—
37	C₆₁×#462	—	2	2	no	—
38	C₈×Q₈×C₆₁	C₂²	1	1	yes	characteristic
39	C₄×Q₁₆×C₆₁	C₂²	1	2	yes	—
40	C₄×Q₁₆×C₆₁	C₂²	1	1	yes	characteristic
41	C₂×Dic₂₄₄	C₂³	1	1	yes	characteristic
42	C₂×#11758190	—	2	2	no	—
43	C₂×D₄₈₈	—	2	1	no	—
44	C₂×D₄₈₈	C₂³	1	2	yes	—
45	#14519774	—	2	1	no	—
46	#14519766	—	4	1	no	—
47	#14519765	—	2	2	no	—
48	#14519762	—	4	2	no	—
49	C₄×Dic₁₂₂	—	2	1	no	—
50	C₄×D₂₄₄	—	2	2	no	—
51	#14519739	—	4	1	no	—
52	#14519723	—	2	2	no	—
53	#14519723	—	4	2	no	—
54	#14519720	—	2	4	no	—
55	#14519720	—	4	2	no	—
56	#14519720	—	4	2	no	—
57	C₂×#11758185	C₂³	1	1	yes	characteristic
58	C₂×#11758185	C₂³	1	2	yes	—
59	C₂×#11758185	C₂³	1	1	yes	characteristic
60	#14519711	C₂³	1	1	yes	characteristic
61	#14519702	—	2	1	no	—
62	#14519699	—	2	2	no	—
63	C₂×Q₁₆×C₆₁	—	2	2	no	—
64	C₂×Q₁₆×C₆₁	—	2	1	no	—
65	C₂×Q₁₆×C₆₁	C₂³	1	2	yes	—
66	C₂×Q₁₆×C₆₁	C₂³	1	1	yes	characteristic
67	C₄⋊Q₈×C₆₁	—	2	2	no	—
68	C₄⋊Q₈×C₆₁	—	2	1	no	—
69	#125×C₆₁	—	4	1	no	—
70	#125×C₆₁	—	4	2	no	—
71	C₄×Q₈×C₆₁	—	2	2	no	—
72	C₄×Q₈×C₆₁	—	2	1	no	—
73	C₄×Q₈×C₆₁	—	4	1	no	—
74	C₄×Q₈×C₆₁	D₄	1	1	yes	characteristic
75	#107×C₆₁	C₂³	1	1	yes	characteristic
76	C₄⋊C₈×C₆₁	D₄	1	1	yes	characteristic
77	C₄⋊C₈×C₆₁	D₄	1	2	yes	—
78	#103×C₆₁	—	2	2	no	—
79	#103×C₆₁	—	2	1	no	—
80	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
81	C₄×C₈×C₆₁	C₂³	1	2	yes	—
82	C₂×Q₈×C₆₁	—	2	2	no	—
83	C₂×Q₈×C₆₁	—	2	1	no	—
84	C₂×Q₈×C₆₁	D₈	1	1	yes	characteristic
85	C₂×D₂₄₄	—	2	2	no	—
86	C₂×D₂₄₄	—	4	1	no	—
87	C₂×D₂₄₄	—	4	1	no	—
88	C₂×C₄×D₆₁	—	4	2	no	—
89	C₂×Dic₁₂₂	—	2	1	no	—
90	Q₁₆×C₆₁	—	2	2	no	—
91	Q₁₆×C₆₁	—	2	1	no	—
92	Q₁₆×C₆₁	—	2	1	no	—
93	Q₁₆×C₆₁	—	2	2	no	—
94	Q₁₆×C₆₁	—	8	2	no	—
95	Q₁₆×C₆₁	—	8	1	no	—
96	C₂×C₈×C₆₁	C₂×D₄	1	2	yes	—
97	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
98	C₂×C₈×C₆₁	C₂⁴	1	1	yes	characteristic
99	C₄⋊C₄×C₆₁	—	2	1	no	—
100	C₄⋊C₄×C₆₁	—	4	1	no	—
101	C₄⋊C₄×C₆₁	—	4	2	no	—
102	C₄⋊C₄×C₆₁	—	4	2	no	—
103	C₄⋊C₄×C₆₁	—	4	1	no	—
104	C₄⋊C₄×C₆₁	—	4	1	no	—
105	C₄⋊C₄×C₆₁	—	4	2	no	—
106	C₄⋊C₄×C₆₁	D₈	1	1	yes	characteristic
107	C₄⋊C₄×C₆₁	D₈	1	2	yes	—
108	C₄²×C₆₁	—	4	2	no	—
109	C₄²×C₆₁	—	4	1	no	—
110	C₄²×C₆₁	C₂×D₄	1	2	yes	—
111	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
112	#11758198	—	4	2	no	—
113	#11758198	—	4	2	no	—
114	#11758198	—	4	2	no	—
115	C₄⋊Dic₆₁	—	2	1	no	—
116	#11758196	—	4	1	no	—
117	C₄×Dic₆₁	—	4	1	no	—
118	Dic₂₄₄	—	2	1	no	—
119	Dic₂₄₄	—	2	1	no	—
120	D₄₈₈	—	2	2	no	—
121	D₄₈₈	—	2	2	no	—
122	D₄₈₈	—	8	1	no	—
123	#11758190	—	8	2	no	—
124	#11758185	—	2	1	no	—
125	#11758185	—	2	2	no	—
126	#11758185	#41	1	2	yes	—
127	C₂²×D₆₁	—	4	2	no	—
128	Q₈×C₆₁	—	2	2	no	—
129	Q₈×C₆₁	—	4	2	no	—
130	Q₈×C₆₁	—	4	1	no	—
131	Q₈×C₆₁	—	4	1	no	—
132	Q₈×C₆₁	—	4	2	no	—
133	C₂×C₄×C₆₁	—	4	1	no	—
134	C₂×C₄×C₆₁	—	4	2	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	2	yes	—
138	C₂×C₄×C₆₁	C₂×D₈	1	2	yes	—
139	C₂×C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
140	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
141	C₂×Dic₆₁	—	4	1	no	—
142	C₂×Dic₆₁	—	4	1	no	—
143	D₂₄₄	—	4	2	no	—
144	D₂₄₄	—	4	2	no	—
145	D₂₄₄	—	8	2	no	—
146	D₂₄₄	—	16	1	no	—
147	C₄×D₆₁	—	16	2	no	—
148	Dic₁₂₂	—	4	1	no	—
149	Dic₁₂₂	—	4	1	no	—
150	C₈×C₆₁	—	2	2	no	—
151	C₈×C₆₁	—	2	1	no	—
152	C₈×C₆₁	C₂×#41	1	1	yes	characteristic
153	C₈×C₆₁	H(5,𝔽₂)	1	1	yes	characteristic
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	2	no	—
160	C₄×C₆₁	—	2	2	no	—
161	C₄×C₆₁	—	2	1	no	—
162	C₄×C₆₁	—	8	2	no	—
163	C₄×C₆₁	—	8	1	no	—
164	C₄×C₆₁	—	8	1	no	—
165	C₄×C₆₁	#510	1	2	yes	—
166	C₄×C₆₁	#550	1	1	yes	characteristic
167	C₄×C₆₁	#576	1	1	yes	characteristic
168	Dic₆₁	—	8	1	no	—
169	Dic₆₁	—	8	1	no	—
170	#3396	—	61	1	no	—
171	#3395	D₆₁	1	1	yes	characteristic, complemented
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	—	61	1	no	—
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	—	2	1	no	—
187	C₂×Q₃₂	—	61	1	no	—
188	C₂×Q₃₂	—	61	2	no	—
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	C₂×D₆₁	1	2	yes	—
194	#494	C₂×D₆₁	1	1	yes	characteristic
195	#493	—	61	1	no	—
196	#463	—	122	2	no	—
197	#462	—	2	1	no	—
198	#462	—	2	2	no	—
199	#460	—	122	1	no	—
200	C₈×Q₈	C₂×D₆₁	1	1	yes	characteristic
201	C₄×Q₁₆	—	61	1	no	—
202	C₄×Q₁₆	C₂×D₆₁	1	1	yes	characteristic
203	C₄×Q₁₆	C₂×D₆₁	1	2	yes	—
204	C₄×D₈	—	61	2	no	—
205	#366	—	61	1	no	—
206	C₄⋊C₁₆	—	61	1	no	—
207	C₄⋊C₁₆	—	61	2	no	—
208	#358	—	61	1	no	—
209	#358	—	61	2	no	—
210	#358	—	61	1	no	—
211	#357	—	61	2	no	—
212	#346	—	61	2	no	—
213	C₄×C₁₆	—	61	1	no	—
214	C₆₁	#33967	1	1	yes	characteristic, complemented, Sylow
215	C₂×Q₁₆	—	2	2	no	—
216	C₂×Q₁₆	—	2	1	no	—
217	C₂×Q₁₆	—	61	1	no	—
218	C₂×Q₁₆	C₂²×D₆₁	1	2	yes	—
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	—	2	2	no	—
224	C₄⋊Q₈	—	2	1	no	—
225	#127	—	122	1	no	—
226	#125	—	4	1	no	—
227	#125	—	4	2	no	—
228	#124	—	122	2	no	—
229	#124	—	244	1	no	—
230	#123	—	244	2	no	—
231	C₄×Q₈	—	2	2	no	—
232	C₄×Q₈	—	2	1	no	—
233	C₄×Q₈	—	4	1	no	—
234	C₄×Q₈	—	122	1	no	—
235	C₄×Q₈	C₂²⋊D₆₁	1	1	yes	characteristic
236	C₄×D₄	—	122	2	no	—
237	C₄×D₄	—	244	1	no	—
238	Q₃₂	—	122	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	1	no	—
244	C₂×C₁₆	—	61	1	no	—
245	C₂×C₁₆	—	61	2	no	—
246	#107	—	61	1	no	—
247	#107	C₂²×D₆₁	1	1	yes	characteristic
248	C₄⋊C₈	C₂²⋊D₆₁	1	1	yes	characteristic
249	C₄⋊C₈	C₂²⋊D₆₁	1	2	yes	—
250	#103	—	2	2	no	—
251	#103	—	2	1	no	—
252	#103	—	122	1	no	—
253	#102	—	122	2	no	—
254	C₄×C₈	C₂²×D₆₁	1	1	yes	characteristic
255	C₄×C₈	C₂²×D₆₁	1	2	yes	—
256	C₂×Q₈	—	2	2	no	—
257	C₂×Q₈	—	2	1	no	—
258	C₂×Q₈	—	122	1	no	—
259	C₂×Q₈	#11758199	1	1	yes	characteristic
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	2	no	—
265	Q₁₆	—	2	1	no	—
266	Q₁₆	—	2	1	no	—
267	Q₁₆	—	2	2	no	—
268	Q₁₆	—	8	1	no	—
269	Q₁₆	—	8	2	no	—
270	Q₁₆	—	122	1	no	—
271	Q₁₆	—	122	1	no	—
272	SD₁₆	—	488	2	no	—
273	D₈	—	122	2	no	—
274	D₈	—	122	2	no	—
275	D₈	—	488	1	no	—
276	C₂×C₈	C₂×C₂²⋊D₆₁	1	2	yes	—
277	C₂×C₈	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
278	C₂×C₈	C₂³×D₆₁	1	1	yes	characteristic
279	C₄⋊C₄	—	2	1	no	—
280	C₄⋊C₄	—	4	2	no	—
281	C₄⋊C₄	—	4	1	no	—
282	C₄⋊C₄	—	4	2	no	—
283	C₄⋊C₄	—	4	1	no	—
284	C₄⋊C₄	—	4	2	no	—
285	C₄⋊C₄	—	4	1	no	—
286	C₄⋊C₄	—	122	1	no	—
287	C₄⋊C₄	—	244	1	no	—
288	C₄⋊C₄	#11758199	1	1	yes	characteristic
289	C₄⋊C₄	#11758199	1	2	yes	—
290	C₂²⋊C₄	—	244	2	no	—
291	C₂²⋊C₄	—	244	2	no	—
292	C₂²⋊C₄	—	244	2	no	—
293	C₄²	—	4	2	no	—
294	C₄²	—	4	1	no	—
295	C₄²	—	244	1	no	—
296	C₄²	C₂×C₂²⋊D₆₁	1	2	yes	—
297	C₄²	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
298	C₁₆	—	61	2	no	—
299	C₁₆	—	122	1	no	—
300	C₁₆	—	122	2	no	—
301	C₂³	—	244	2	no	—
302	Q₈	—	2	2	no	—
303	Q₈	—	4	2	no	—
304	Q₈	—	4	1	no	—
305	Q₈	—	4	2	no	—
306	Q₈	—	4	1	no	—
307	Q₈	—	244	1	no	—
308	Q₈	—	244	1	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	2	no	—
315	C₂×C₄	—	4	1	no	—
316	C₂×C₄	—	976	2	no	—
317	C₂×C₄	—	244	1	no	—
318	C₂×C₄	—	244	1	no	—
319	C₂×C₄	C₂×#11758199	1	2	yes	—
320	C₂×C₄	C₂×#11758199	1	2	yes	—
321	C₂×C₄	C₂×#11758199	1	1	yes	characteristic
322	C₂×C₄	C₂×#11758199	1	1	yes	characteristic
323	C₂×C₄	C₂²×C₂²⋊D₆₁	1	1	yes	characteristic
324	C₈	—	2	1	no	—
325	C₈	—	2	2	no	—
326	C₈	#41×D₆₁	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	C₂²×#11758199	1	1	yes	characteristic
332	C₄	—	2	2	no	—
333	C₄	—	2	2	no	—
334	C₄	—	2	1	no	—
335	C₄	—	8	1	no	—
336	C₄	—	8	2	no	—
337	C₄	—	8	1	no	—
338	C₄	—	488	1	no	—
339	C₄	—	488	1	no	—
340	C₄	#16840203	1	2	yes	—
341	C₄	#16840431	1	1	yes	characteristic
342	C₄	#16840565	1	1	yes	characteristic
343	C₂	—	976	2	no	—
344	C₂	#21458791	1	1	yes	characteristic
345	C₂	C₂×#16840203	1	1	yes	characteristic
346	C₂	#21458963	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×#11758199
commutator, cosocle	C₂×C₈×C₆₁	C₂⁴
Frattini	C₂×C₈	C₂³×D₆₁
Fitting	C₆₁×#3395	C₂
socle	C₂²×C₆₁	C₂²×D₈
prime radical	#16840139	C₂²
hypercenter	#3395	D₆₁
hypocenter	C₆₁	#33967
LCS	G ▷ C₂×C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ C₈×Q₈ ◁ #3395

Complements

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

N	H ≅ G/N	type
2. #21448088	C₂	semidirect
4. #21447881	C₂	semidirect
5. #21447866	C₂	semidirect
6. #21447861	C₂	semidirect
8. #21447728	C₂	semidirect
9. #21447681	C₂	semidirect
10. C₄×#14519723	C₂	semidirect
11. C₆₁×#3395	C₂	semidirect
171. #3395	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
#21458832	2
#21448088	2
#21448076	2
#21447881	2
#21447866	2
#21447861	2
#21447859	2
#21447728	2
#21447681	2
C₄×#14519723	2
C₆₁×#3395	2
#33967	61