Subgroups — #29573156

Order 15616.

450 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16839839	C₂	1	1	yes	maximal, characteristic
2	#21445990	C₂	1	1	yes	maximal, characteristic, complemented
3	#21444565	C₂	1	1	yes	maximal, characteristic, complemented
4	#21444245	C₂	1	1	yes	maximal, characteristic
5	#21444243	C₂	1	1	yes	maximal, characteristic
6	#21443716	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₁×#1301	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×#11758189	—	2	1	no	—
9	C₂×C₄×D₂₄₄	C₂²	1	1	yes	characteristic
10	#16839839	—	2	1	no	—
11	#16839839	—	2	1	no	—
12	#16839839	C₄	1	2	yes	—
13	#16839839	C₄	1	2	yes	—
14	#16839815	C₂²	1	1	yes	characteristic
15	C₂×#14519712	—	2	1	no	—
16	C₂×#14519712	—	2	1	no	—
17	#16839786	C₂²	1	1	yes	characteristic
18	C₂×#14519689	—	2	1	no	—
19	C₂×C₄×#92418	C₂²	1	1	yes	characteristic
20	#16839514	—	2	1	no	—
21	#16839512	—	2	1	no	—
22	#16839491	—	2	2	no	—
23	#16839489	—	2	2	no	—
24	C₆₁×#431	C₂²	1	1	yes	characteristic
25	C₆₁×#423	C₂²	1	1	yes	characteristic
26	C₂×C₄⋊C₈×C₆₁	C₂²	1	1	yes	characteristic
27	C₂²×D₂₄₄	Q₈	1	1	yes	characteristic
28	C₂²×C₄×D₆₁	—	2	1	no	—
29	C₂×#11758194	—	2	1	no	—
30	C₂×#11758189	—	2	2	no	—
31	C₂×#11758189	—	2	2	no	—
32	C₂×#11758189	—	4	2	no	—
33	C₂×#11758189	—	4	2	no	—
34	C₂²×#92418	D₄	1	1	yes	characteristic
35	C₂²×#92418	D₄	1	1	yes	characteristic
36	C₄×D₂₄₄	—	2	1	no	—
37	C₄×D₂₄₄	—	2	1	no	—
38	C₄×D₂₄₄	C₂×C₄	1	2	yes	—
39	C₄×D₂₄₄	D₄	1	2	yes	—
40	C₂×C₄⋊Dic₆₁	Q₈	1	1	yes	characteristic
41	C₂×#11758198	—	2	1	no	—
42	#14519712	—	4	2	no	—
43	#14519712	—	4	2	no	—
44	C₄⋊#92418	—	2	1	no	—
45	C₄⋊#92418	—	2	1	no	—
46	C₄⋊#92418	—	2	1	no	—
47	C₂²⋊#92418	—	2	1	no	—
48	#14519689	—	2	2	no	—
49	#14519689	—	2	2	no	—
50	C₄×#92418	—	2	1	no	—
51	C₄×#92418	C₂×C₄	1	1	yes	characteristic
52	C₄×#92418	C₂×C₄	1	1	yes	characteristic
53	C₄×#92418	D₄	1	1	yes	characteristic
54	C₄×#92418	D₄	1	1	yes	characteristic
55	C₂×M₁₆×C₆₁	—	2	1	no	—
56	C₂²×C₈×C₆₁	—	2	1	no	—
57	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
58	C₄⋊C₈×C₆₁	—	2	1	no	—
59	C₄⋊C₈×C₆₁	—	2	1	no	—
60	C₄⋊C₈×C₆₁	—	2	1	no	—
61	C₄⋊C₈×C₆₁	—	2	1	no	—
62	C₄⋊C₈×C₆₁	C₂×C₄	1	2	yes	—
63	C₂²⋊C₈×C₆₁	—	2	1	no	—
64	C₄×C₈×C₆₁	—	2	1	no	—
65	C₂³×D₆₁	—	2	1	no	—
66	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₆₁	C₂×Q₈	1	1	yes	characteristic
68	C₂²×C₄×C₆₁	#41	1	1	yes	characteristic
69	C₂²×Dic₆₁	—	2	1	no	—
70	C₂×D₂₄₄	—	2	1	no	—
71	C₂×D₂₄₄	—	4	2	no	—
72	C₂×D₂₄₄	C₄⋊C₄	1	2	yes	—
73	C₂×C₄×D₆₁	—	2	2	no	—
74	C₂×C₄×D₆₁	—	2	2	no	—
75	C₂×C₄×D₆₁	—	4	1	no	—
76	C₂×C₄×D₆₁	—	4	1	no	—
77	C₂×C₄×D₆₁	—	4	1	no	—
78	C₂×C₄×D₆₁	—	4	1	no	—
79	M₁₆×C₆₁	—	4	1	no	—
80	M₁₆×C₆₁	—	4	1	no	—
81	C₂×C₈×C₆₁	—	2	2	no	—
82	C₂×C₈×C₆₁	—	4	1	no	—
83	C₂×C₈×C₆₁	—	4	1	no	—
84	C₂×C₈×C₆₁	—	4	1	no	—
85	C₂×C₈×C₆₁	—	4	1	no	—
86	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
87	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
88	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
89	C₄²×C₆₁	#41	1	1	yes	characteristic
90	#11758198	—	2	2	no	—
91	#11758198	—	2	2	no	—
92	C₄⋊Dic₆₁	—	2	1	no	—
93	C₄⋊Dic₆₁	C₄⋊C₄	1	2	yes	—
94	#11758194	—	4	1	no	—
95	#11758194	—	4	1	no	—
96	C₂×#92418	—	2	1	no	—
97	C₂×#92418	—	2	1	no	—
98	C₂×#92418	—	4	1	no	—
99	C₂×#92418	—	4	1	no	—
100	C₂×#92418	C₂²⋊C₄	1	1	yes	characteristic
101	C₂×#92418	C₂²⋊C₄	1	1	yes	characteristic
102	C₂×#92418	C₂²⋊C₄	1	1	yes	characteristic
103	C₂×#92418	C₂²⋊C₄	1	1	yes	characteristic
104	C₂×#92418	D₈	1	2	yes	—
105	C₂×#92418	SD₁₆	1	2	yes	—
106	#11758189	—	4	4	no	—
107	#11758189	—	4	4	no	—
108	C₂³×C₆₁	C₂²⋊Q₈	1	1	yes	characteristic
109	C₂²×D₆₁	—	2	2	no	—
110	C₂²×D₆₁	—	4	2	no	—
111	C₂²×D₆₁	—	4	2	no	—
112	C₂²×D₆₁	—	4	1	no	—
113	C₂²×D₆₁	—	4	1	no	—
114	C₂×C₄×C₆₁	—	2	2	no	—
115	C₂×C₄×C₆₁	—	4	1	no	—
116	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
117	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
118	C₂×C₄×C₆₁	#117	1	1	yes	characteristic
119	C₂×C₄×C₆₁	#121	1	1	yes	characteristic
120	C₂×C₄×C₆₁	#121	1	1	yes	characteristic
121	C₂×C₄×C₆₁	C₂²⋊Q₈	1	1	yes	characteristic
122	C₂×C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
123	C₂×C₄×C₆₁	C₂×SD₁₆	1	1	yes	characteristic
124	C₂×C₄×C₆₁	AGL(1,ℤ/8)	1	1	yes	characteristic
125	C₂×C₄×C₆₁	#137	1	1	yes	characteristic
126	C₂×Dic₆₁	—	2	2	no	—
127	C₂×Dic₆₁	—	4	1	no	—
128	C₂×Dic₆₁	—	4	1	no	—
129	D₂₄₄	—	2	4	no	—
130	D₂₄₄	—	4	2	no	—
131	C₄×D₆₁	—	4	2	no	—
132	C₄×D₆₁	—	4	2	no	—
133	C₄×D₆₁	—	4	2	no	—
134	C₄×D₆₁	—	4	2	no	—
135	C₈×C₆₁	—	4	2	no	—
136	C₈×C₆₁	—	8	1	no	—
137	C₈×C₆₁	—	8	1	no	—
138	#92418	—	2	1	no	—
139	#92418	—	2	1	no	—
140	#92418	—	8	1	no	—
141	#92418	—	8	1	no	—
142	#92418	#102	1	2	yes	—
143	#92418	#102	1	2	yes	—
144	#8662	—	61	1	no	maximal, Sylow
145	C₂²×C₆₁	—	2	1	no	—
146	C₂²×C₆₁	—	2	1	no	—
147	C₂²×C₆₁	#380	1	1	yes	characteristic
148	C₂²×C₆₁	#474	1	1	yes	characteristic
149	C₂²×C₆₁	#476	1	1	yes	characteristic
150	C₂×D₆₁	—	4	2	no	—
151	C₂×D₆₁	—	4	2	no	—
152	C₂×D₆₁	—	4	2	no	—
153	C₂×D₆₁	—	8	2	no	—
154	C₂×D₆₁	—	8	2	no	—
155	C₄×C₆₁	—	2	1	no	—
156	C₄×C₆₁	—	2	1	no	—
157	C₄×C₆₁	—	2	1	no	—
158	C₄×C₆₁	C₂×#102	1	1	yes	characteristic
159	C₄×C₆₁	#417	1	1	yes	characteristic
160	C₄×C₆₁	#430	1	2	yes	—
161	C₄×C₆₁	#459	1	1	yes	characteristic
162	C₄×C₆₁	#461	1	1	yes	characteristic
163	Dic₆₁	—	4	2	no	—
164	C₂×#436	—	61	1	no	—
165	#1848	—	61	1	no	—
166	#1564	—	61	1	no	—
167	#1486	—	61	1	no	—
168	#1486	—	61	1	no	—
169	#1301	—	61	1	no	—
170	#1301	D₆₁	1	1	yes	characteristic, complemented
171	C₂×C₆₁	—	2	1	no	—
172	C₂×C₆₁	—	2	1	no	—
173	C₂×C₆₁	#1802	1	1	yes	characteristic
174	C₂×C₆₁	#1811	1	1	yes	characteristic
175	C₂×C₆₁	#1814	1	1	yes	characteristic
176	D₆₁	—	8	2	no	—
177	C₂²×M₁₆	—	122	1	no	—
178	C₂×C₄×D₄	—	61	1	no	—
179	#436	—	61	2	no	—
180	#436	—	61	2	no	—
181	#436	—	122	1	no	—
182	#436	—	122	1	no	—
183	#431	—	61	1	no	—
184	#431	C₂×D₆₁	1	1	yes	characteristic
185	C₂×#108	—	122	1	no	—
186	C₂×#108	—	122	1	no	—
187	#423	—	61	1	no	—
188	#423	C₂×D₆₁	1	1	yes	characteristic
189	C₂×C₄⋊C₈	C₂×D₆₁	1	1	yes	characteristic
190	C₂×C₂²⋊C₈	—	122	1	no	—
191	C₂×C₄×C₈	—	61	1	no	—
192	#335	—	122	1	no	—
193	#334	—	122	1	no	—
194	D₄⋊C₈	—	122	2	no	—
195	D₄⋊C₈	—	122	2	no	—
196	C₆₁	#8662	1	1	yes	characteristic, complemented, Sylow
197	C₂²×D₄	—	61	1	no	—
198	C₂³×C₄	—	122	1	no	—
199	C₂×M₁₆	—	2	1	no	—
200	C₂×M₁₆	—	122	2	no	—
201	C₂×M₁₆	—	122	2	no	—
202	C₂×M₁₆	—	122	1	no	—
203	C₂×M₁₆	—	244	2	no	—
204	C₂×M₁₆	—	244	2	no	—
205	C₂²×C₈	—	2	1	no	—
206	C₂²×C₈	—	61	1	no	—
207	C₂²×C₈	—	61	1	no	—
208	C₄×D₄	—	61	2	no	—
209	C₄×D₄	—	61	2	no	—
210	C₄×D₄	—	122	1	no	—
211	C₄×D₄	—	122	1	no	—
212	C₂×C₄⋊C₄	—	61	1	no	—
213	C₂×C₂²⋊C₄	—	122	1	no	—
214	C₂×C₄²	C₂²×D₆₁	1	1	yes	characteristic
215	#108	—	244	2	no	—
216	#108	—	244	2	no	—
217	C₄⋊C₈	—	2	1	no	—
218	C₄⋊C₈	—	2	1	no	—
219	C₄⋊C₈	—	2	1	no	—
220	C₄⋊C₈	—	2	1	no	—
221	C₄⋊C₈	—	122	1	no	—
222	C₄⋊C₈	—	122	1	no	—
223	C₄⋊C₈	—	122	1	no	—
224	C₄⋊C₈	C₄×D₆₁	1	2	yes	—
225	C₂²⋊C₈	—	2	1	no	—
226	C₂²⋊C₈	—	122	2	no	—
227	C₂²⋊C₈	—	122	2	no	—
228	C₂²⋊C₈	—	122	1	no	—
229	C₄×C₈	—	2	1	no	—
230	C₄×C₈	—	61	1	no	—
231	C₄×C₈	—	61	1	no	—
232	C₄×C₈	—	61	1	no	—
233	C₄×C₈	—	61	1	no	—
234	C₄×C₈	—	122	1	no	—
235	C₂⁴	—	122	1	no	—
236	C₂×D₄	—	61	2	no	—
237	C₂×D₄	—	122	1	no	—
238	C₂×D₄	—	244	2	no	—
239	C₂²×C₄	—	122	1	no	—
240	C₂²×C₄	—	122	2	no	—
241	C₂²×C₄	—	122	2	no	—
242	C₂²×C₄	—	244	1	no	—
243	C₂²×C₄	—	244	1	no	—
244	C₂²×C₄	—	244	1	no	—
245	C₂²×C₄	—	244	1	no	—
246	C₂²×C₄	#11758222	1	1	yes	characteristic
247	C₂²×C₄	D₄×D₆₁	1	1	yes	characteristic
248	C₂²×C₄	Q₈×D₆₁	1	1	yes	characteristic
249	M₁₆	—	4	1	no	—
250	M₁₆	—	4	1	no	—
251	M₁₆	—	244	4	no	—
252	M₁₆	—	244	4	no	—
253	M₁₆	—	244	1	no	—
254	M₁₆	—	244	1	no	—
255	C₂×C₈	—	2	2	no	—
256	C₂×C₈	—	4	1	no	—
257	C₂×C₈	—	4	1	no	—
258	C₂×C₈	—	4	1	no	—
259	C₂×C₈	—	4	1	no	—
260	C₂×C₈	—	61	1	no	—
261	C₂×C₈	—	61	1	no	—
262	C₂×C₈	—	61	2	no	—
263	C₂×C₈	—	61	2	no	—
264	C₂×C₈	—	61	1	no	—
265	C₂×C₈	—	61	1	no	—
266	C₂×C₈	—	122	1	no	—
267	C₂×C₈	—	122	1	no	—
268	C₂×C₈	—	244	1	no	—
269	C₂×C₈	—	244	1	no	—
270	C₄⋊C₄	—	61	2	no	—
271	C₄⋊C₄	—	122	1	no	—
272	C₂²⋊C₄	—	122	2	no	—
273	C₂²⋊C₄	—	122	2	no	—
274	C₄²	C₂×C₄×D₆₁	1	1	yes	characteristic
275	C₄²	#11758222	1	1	yes	characteristic
276	C₄²	D₄×D₆₁	1	1	yes	characteristic
277	C₄²	D₄×D₆₁	1	1	yes	characteristic
278	C₂³	—	122	2	no	—
279	C₂³	—	244	2	no	—
280	C₂³	—	244	2	no	—
281	C₂³	—	244	1	no	—
282	C₂³	—	244	1	no	—
283	C₂³	#14519755	1	1	yes	characteristic
284	D₄	—	122	4	no	—
285	D₄	—	244	2	no	—
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	4	1	no	—
288	C₂×C₄	—	122	2	no	—
289	C₂×C₄	—	244	1	no	—
290	C₂×C₄	—	244	1	no	—
291	C₂×C₄	—	244	2	no	—
292	C₂×C₄	—	244	2	no	—
293	C₂×C₄	—	244	2	no	—
294	C₂×C₄	—	244	2	no	—
295	C₂×C₄	C₂²⋊C₄×D₆₁	1	1	yes	characteristic
296	C₂×C₄	C₄⋊C₄×D₆₁	1	1	yes	characteristic
297	C₂×C₄	#14519734	1	1	yes	characteristic
298	C₂×C₄	#14519750	1	1	yes	characteristic
299	C₂×C₄	#14519750	1	1	yes	characteristic
300	C₂×C₄	#14519755	1	1	yes	characteristic
301	C₂×C₄	D₈×D₆₁	1	1	yes	characteristic
302	C₂×C₄	SD₁₆×D₆₁	1	1	yes	characteristic
303	C₂×C₄	#14519804	1	1	yes	characteristic
304	C₂×C₄	#14519809	1	1	yes	characteristic
305	C₈	—	4	2	no	—
306	C₈	—	8	1	no	—
307	C₈	—	8	1	no	—
308	C₈	—	61	2	no	—
309	C₈	—	61	2	no	—
310	C₈	—	122	1	no	—
311	C₈	—	122	1	no	—
312	C₈	—	488	1	no	—
313	C₈	—	488	1	no	—
314	C₂²	—	2	1	no	—
315	C₂²	—	2	1	no	—
316	C₂²	—	488	2	no	—
317	C₂²	—	488	2	no	—
318	C₂²	—	244	2	no	—
319	C₂²	—	244	2	no	—
320	C₂²	—	244	2	no	—
321	C₂²	#16839621	1	1	yes	characteristic
322	C₂²	#16840041	1	1	yes	characteristic
323	C₂²	#16840051	1	1	yes	characteristic
324	C₄	—	2	1	no	—
325	C₄	—	2	1	no	—
326	C₄	—	2	1	no	—
327	C₄	—	244	2	no	—
328	C₄	#102×D₆₁	1	1	yes	characteristic
329	C₄	#16839762	1	1	yes	characteristic
330	C₄	#16839809	1	2	yes	—
331	C₄	#16839956	1	1	yes	characteristic
332	C₄	#16839966	1	1	yes	characteristic
333	C₂	—	2	1	no	—
334	C₂	—	2	1	no	—
335	C₂	—	488	2	no	—
336	C₂	#21445818	1	1	yes	characteristic
337	C₂	#21445860	1	1	yes	characteristic
338	C₂	#21445872	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#14519750
commutator	C₄²×C₆₁	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₁
Fitting	C₆₁×#1301	C₂
socle	C₂²×C₆₁	#380
prime radical	C₂²×D₂₄₄	Q₈
cosocle	C₂×C₄²×C₆₁	C₂³
hypercenter	#1301	D₆₁
hypocenter	C₆₁	#8662
LCS	G ▷ C₄²×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1301

Complements

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

N	H ≅ G/N	type
2. #21445990	C₂	semidirect
3. #21444565	C₂	semidirect
6. #21443716	C₂	semidirect
7. C₆₁×#1301	C₂	semidirect
170. #1301	D₆₁	semidirect
196. C₆₁	#8662	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16839839	2
#21445990	2
#21444565	2
#21444245	2
#21444243	2
#21443716	2
C₆₁×#1301	2
#8662	61