Subgroups — #29570595

Order 15616.

494 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16839841	C₂	1	1	yes	maximal, characteristic
2	C₄×#14519693	C₂	1	1	yes	maximal, characteristic
3	#21444139	C₂	1	1	yes	maximal, characteristic
4	C₈×#11758194	C₂	1	1	yes	maximal, characteristic, complemented
5	#21443670	C₂	1	1	yes	maximal, characteristic, complemented
6	#21443668	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₁×#1451	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×#11758189	C₄	1	2	yes	—
9	C₂×C₄×D₂₄₄	C₂²	1	1	yes	characteristic
10	#16839841	—	2	1	no	—
11	#16839841	—	2	1	no	—
12	#16839841	—	2	1	no	—
13	#16839841	—	2	1	no	—
14	#16839815	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊#92418	C₂²	1	1	yes	characteristic
16	C₂×#14519693	C₄	1	2	yes	—
17	C₂×#14519689	C₄	1	2	yes	—
18	C₄×#11758194	C₂²	1	1	yes	characteristic
19	#16839530	C₄	1	2	yes	—
20	#16839485	C₄	1	4	yes	—
21	#16839484	—	2	1	no	—
22	C₈×#92418	—	2	1	no	—
23	C₆₁×#431	C₂²	1	1	yes	characteristic
24	C₄×M₁₆×C₆₁	C₂²	1	1	yes	characteristic
25	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
26	C₆₁×#322	—	2	1	no	—
27	C₆₁×#322	—	2	1	no	—
28	C₂²×D₂₄₄	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₄×D₆₁	C₂×C₄	1	2	yes	—
30	C₂×#11758194	C₂×C₄	1	2	yes	—
31	C₂×#11758189	—	2	2	no	—
32	C₂×#11758189	—	2	2	no	—
33	C₂×#11758189	—	4	4	no	—
34	C₂²×#92418	C₂×C₄	1	2	yes	—
35	C₄×D₂₄₄	—	2	1	no	—
36	C₄×D₂₄₄	—	2	1	no	—
37	C₄×D₂₄₄	—	2	1	no	—
38	C₄×D₂₄₄	—	2	1	no	—
39	C₂×C₄⋊Dic₆₁	C₂×C₄	1	1	yes	characteristic
40	C₂×#11758198	C₂×C₄	1	2	yes	—
41	C₄⋊#92418	—	2	1	no	—
42	C₄⋊#92418	—	2	1	no	—
43	C₄⋊#92418	—	2	1	no	—
44	#14519693	—	2	4	no	—
45	C₂²⋊#92418	C₂×C₄	1	2	yes	—
46	#14519689	—	2	2	no	—
47	#14519689	—	2	2	no	—
48	#14519688	—	2	1	no	—
49	C₄×#92418	—	2	1	no	—
50	C₄×#92418	—	2	1	no	—
51	C₂×M₁₆×C₆₁	C₂×C₄	1	2	yes	—
52	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
53	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
54	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
55	C₄⋊C₈×C₆₁	—	2	1	no	—
56	C₂²⋊C₈×C₆₁	C₂×C₄	1	2	yes	—
57	#97×C₆₁	—	2	1	no	—
58	C₄×C₈×C₆₁	—	2	1	no	—
59	C₄×C₈×C₆₁	—	2	1	no	—
60	C₄×C₈×C₆₁	D₄	1	1	yes	characteristic
61	C₄×C₈×C₆₁	D₄	1	1	yes	characteristic
62	C₄×C₈×C₆₁	D₄	1	1	yes	characteristic
63	C₄×C₈×C₆₁	D₄	1	1	yes	characteristic
64	C₂³×D₆₁	C₄²	1	2	yes	—
65	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
66	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
67	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
68	C₂²×Dic₆₁	C₄²	1	2	yes	—
69	C₂×D₂₄₄	—	2	1	no	—
70	C₂×D₂₄₄	—	2	1	no	—
71	C₂×D₂₄₄	—	2	2	no	—
72	C₂×D₂₄₄	—	2	2	no	—
73	C₂×C₄×D₆₁	—	2	2	no	—
74	C₂×C₄×D₆₁	—	2	2	no	—
75	C₂×C₄×D₆₁	—	4	2	no	—
76	C₂×C₄×D₆₁	—	4	2	no	—
77	M₁₆×C₆₁	—	2	2	no	—
78	M₁₆×C₆₁	—	2	2	no	—
79	C₂×C₈×C₆₁	—	2	2	no	—
80	C₂×C₈×C₆₁	—	2	2	no	—
81	C₂×C₈×C₆₁	—	2	2	no	—
82	C₂×C₈×C₆₁	—	4	1	no	—
83	C₂×C₈×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
84	C₂×C₈×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
85	C₂×C₈×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
86	C₂×C₈×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	C₄²×C₆₁	#41	1	1	yes	characteristic
91	#11758198	—	2	2	no	—
92	#11758198	—	2	2	no	—
93	C₄⋊Dic₆₁	—	2	1	no	—
94	C₄⋊Dic₆₁	—	2	1	no	—
95	#11758194	—	2	2	no	—
96	#11758194	M₁₆	1	4	yes	complemented
97	C₂×#92418	—	2	2	no	—
98	C₂×#92418	—	2	2	no	—
99	C₂×#92418	—	2	2	no	—
100	C₂×#92418	—	4	2	no	—
101	#11758189	—	4	4	no	—
102	#11758189	—	4	4	no	—
103	C₂³×C₆₁	C₂×C₄²	1	1	yes	characteristic
104	C₂²×D₆₁	—	2	2	no	—
105	C₂²×D₆₁	—	2	2	no	—
106	C₂²×D₆₁	—	2	2	no	—
107	C₂²×D₆₁	—	4	2	no	—
108	C₂²×D₆₁	—	4	2	no	—
109	C₂×C₄×C₆₁	—	2	1	no	—
110	C₂×C₄×C₆₁	—	2	1	no	—
111	C₂×C₄×C₆₁	—	4	1	no	—
112	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
113	C₂×C₄×C₆₁	#117	1	1	yes	characteristic
114	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
115	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
116	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
117	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
118	C₂×C₄×C₆₁	C₂×M₁₆	1	2	yes	—
119	C₂×C₄×C₆₁	#131	1	2	yes	—
120	C₂×Dic₆₁	—	2	2	no	—
121	C₂×Dic₆₁	—	4	2	no	—
122	D₂₄₄	—	4	2	no	—
123	D₂₄₄	—	4	2	no	—
124	C₄×D₆₁	—	4	4	no	—
125	C₄×D₆₁	—	4	2	no	—
126	C₄×D₆₁	—	4	2	no	—
127	C₈×C₆₁	—	2	1	no	—
128	C₈×C₆₁	—	2	1	no	—
129	C₈×C₆₁	—	2	1	no	—
130	C₈×C₆₁	—	4	2	no	—
131	C₈×C₆₁	—	4	2	no	—
132	C₈×C₆₁	#100	1	2	yes	complemented
133	#92418	—	2	4	no	—
134	#92418	—	4	2	no	—
135	#92418	—	4	2	no	—
136	#8166	—	61	1	no	maximal, Sylow
137	C₂²×C₆₁	—	2	1	no	—
138	C₂²×C₆₁	—	2	1	no	—
139	C₂²×C₆₁	C₄×C₂²⋊C₄	1	1	yes	characteristic
140	C₂²×C₆₁	C₄×M₁₆	1	1	yes	characteristic
141	C₂²×C₆₁	#405	1	1	yes	characteristic
142	C₂×D₆₁	—	4	2	no	—
143	C₂×D₆₁	—	4	2	no	—
144	C₂×D₆₁	—	4	2	no	—
145	C₂×D₆₁	—	8	2	no	—
146	C₂×D₆₁	—	8	2	no	—
147	C₄×C₆₁	—	2	1	no	—
148	C₄×C₆₁	—	2	1	no	—
149	C₄×C₆₁	—	2	1	no	—
150	C₄×C₆₁	C₂×#100	1	1	yes	characteristic
151	C₄×C₆₁	#413	1	1	yes	characteristic
152	C₄×C₆₁	#435	1	2	yes	—
153	C₄×C₆₁	#436	1	2	yes	—
154	Dic₆₁	—	4	2	no	—
155	C₂×#436	—	61	1	no	—
156	C₄×#100	—	61	1	no	—
157	#1460	—	61	1	no	—
158	#1451	D₆₁	1	1	yes	characteristic, complemented
159	C₈×M₁₆	—	61	1	no	—
160	#1283	—	61	1	no	—
161	#1283	—	61	1	no	—
162	C₂×C₆₁	—	2	1	no	—
163	C₂×C₆₁	—	2	1	no	—
164	C₂×C₆₁	#1753	1	1	yes	characteristic
165	C₂×C₆₁	C₄×#100	1	1	yes	characteristic
166	C₂×C₆₁	#1757	1	1	yes	characteristic
167	D₆₁	—	8	2	no	—
168	C₂²×M₁₆	—	61	2	no	—
169	C₂×C₄×D₄	—	61	1	no	—
170	#436	—	122	1	no	—
171	#436	—	122	1	no	—
172	#436	—	122	1	no	—
173	#436	—	122	1	no	—
174	#431	—	61	1	no	—
175	#431	C₂×D₆₁	1	1	yes	characteristic
176	C₂×C₄⋊C₈	—	61	1	no	—
177	C₂×#100	—	61	2	no	—
178	C₂×C₂²⋊C₈	—	61	2	no	—
179	C₄×M₁₆	—	61	1	no	—
180	C₄×M₁₆	C₂×D₆₁	1	1	yes	characteristic
181	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
182	#343	—	61	2	no	—
183	#323	—	61	4	no	—
184	#322	—	2	1	no	—
185	#322	—	2	1	no	—
186	#322	—	122	1	no	—
187	C₈²	—	122	1	no	—
188	C₆₁	#8166	1	1	yes	characteristic, complemented, Sylow
189	C₂²×D₄	—	61	1	no	—
190	C₂³×C₄	—	61	2	no	—
191	C₂×M₁₆	—	61	2	no	—
192	C₂×M₁₆	—	122	2	no	—
193	C₂×M₁₆	—	122	2	no	—
194	C₂×M₁₆	—	244	4	no	—
195	C₂×M₁₆	C₄×D₆₁	1	2	yes	—
196	C₂²×C₈	—	61	2	no	—
197	C₂²×C₈	C₄×D₆₁	1	1	yes	characteristic
198	C₂²×C₈	C₄×D₆₁	1	1	yes	characteristic
199	C₄×D₄	—	122	1	no	—
200	C₄×D₄	—	122	1	no	—
201	C₄×D₄	—	122	1	no	—
202	C₄×D₄	—	122	1	no	—
203	C₂×C₄⋊C₄	—	61	1	no	—
204	C₂×C₂²⋊C₄	—	61	2	no	—
205	C₂×C₄²	C₂²×D₆₁	1	1	yes	characteristic
206	C₄⋊C₈	—	2	1	no	—
207	C₄⋊C₈	—	122	1	no	—
208	C₄⋊C₈	—	122	1	no	—
209	C₄⋊C₈	—	122	1	no	—
210	#100	—	122	4	no	—
211	C₂²⋊C₈	—	61	2	no	—
212	C₂²⋊C₈	—	122	2	no	—
213	C₂²⋊C₈	—	122	2	no	—
214	C₂²⋊C₈	C₄×D₆₁	1	2	yes	—
215	#97	—	2	1	no	—
216	#97	—	122	1	no	—
217	C₄×C₈	—	2	1	no	—
218	C₄×C₈	—	2	1	no	—
219	C₄×C₈	—	122	1	no	—
220	C₄×C₈	—	122	1	no	—
221	C₄×C₈	D₂₄₄	1	1	yes	characteristic
222	C₄×C₈	D₂₄₄	1	1	yes	characteristic
223	C₄×C₈	C₂²⋊D₆₁	1	1	yes	characteristic
224	C₄×C₈	C₂²⋊D₆₁	1	1	yes	characteristic
225	C₂⁴	—	61	2	no	—
226	C₂×D₄	—	122	1	no	—
227	C₂×D₄	—	122	1	no	—
228	C₂×D₄	—	122	2	no	—
229	C₂×D₄	—	122	2	no	—
230	C₂²×C₄	—	61	2	no	—
231	C₂²×C₄	—	122	2	no	—
232	C₂²×C₄	—	122	2	no	—
233	C₂²×C₄	—	244	2	no	—
234	C₂²×C₄	—	244	2	no	—
235	C₂²×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
236	C₂²×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
237	C₂²×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
238	M₁₆	—	2	2	no	—
239	M₁₆	—	2	2	no	—
240	M₁₆	—	61	4	no	—
241	M₁₆	—	122	2	no	—
242	M₁₆	—	244	4	no	—
243	M₁₆	—	244	4	no	—
244	C₂×C₈	—	2	2	no	—
245	C₂×C₈	—	2	2	no	—
246	C₂×C₈	—	2	2	no	—
247	C₂×C₈	—	4	1	no	—
248	C₂×C₈	—	122	2	no	—
249	C₂×C₈	—	122	2	no	—
250	C₂×C₈	—	122	2	no	—
251	C₂×C₈	—	244	2	no	—
252	C₂×C₈	#11758198	1	1	yes	characteristic
253	C₂×C₈	#11758198	1	1	yes	characteristic
254	C₂×C₈	#11758198	1	1	yes	characteristic
255	C₂×C₈	#11758198	1	1	yes	characteristic
256	C₄⋊C₄	—	122	1	no	—
257	C₄⋊C₄	—	122	1	no	—
258	C₂²⋊C₄	—	122	2	no	—
259	C₂²⋊C₄	—	122	2	no	—
260	C₄²	C₂×D₂₄₄	1	1	yes	characteristic
261	C₄²	#11758222	1	1	yes	characteristic
262	C₄²	#11758222	1	1	yes	characteristic
263	C₄²	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
264	C₂³	—	122	2	no	—
265	C₂³	—	122	2	no	—
266	C₂³	—	122	2	no	—
267	C₂³	—	244	2	no	—
268	C₂³	—	244	2	no	—
269	C₂³	C₄²×D₆₁	1	1	yes	characteristic
270	D₄	—	244	2	no	—
271	D₄	—	244	2	no	—
272	C₂×C₄	—	2	1	no	—
273	C₂×C₄	—	2	1	no	—
274	C₂×C₄	—	4	1	no	—
275	C₂×C₄	—	122	2	no	—
276	C₂×C₄	—	244	4	no	—
277	C₂×C₄	—	244	2	no	—
278	C₂×C₄	—	244	2	no	—
279	C₂×C₄	—	244	2	no	—
280	C₂×C₄	C₂×#11758198	1	1	yes	characteristic
281	C₂×C₄	#14519734	1	1	yes	characteristic
282	C₂×C₄	C₄×C₂²⋊D₆₁	1	1	yes	characteristic
283	C₂×C₄	C₄×C₂²⋊D₆₁	1	1	yes	characteristic
284	C₂×C₄	C₄×D₂₄₄	1	1	yes	characteristic
285	C₂×C₄	C₄×D₂₄₄	1	1	yes	characteristic
286	C₂×C₄	M₁₆×D₆₁	1	2	yes	—
287	C₂×C₄	#14519785	1	2	yes	—
288	C₈	—	2	1	no	—
289	C₈	—	2	1	no	—
290	C₈	—	2	1	no	—
291	C₈	—	4	2	no	—
292	C₈	—	4	2	no	—
293	C₈	—	122	4	no	—
294	C₈	—	244	2	no	—
295	C₈	—	244	2	no	—
296	C₈	#14519693	1	2	yes	complemented
297	C₂²	—	2	1	no	—
298	C₂²	—	2	1	no	—
299	C₂²	—	488	2	no	—
300	C₂²	—	488	2	no	—
301	C₂²	—	244	2	no	—
302	C₂²	—	244	2	no	—
303	C₂²	—	244	2	no	—
304	C₂²	C₄×#11758198	1	1	yes	characteristic
305	C₂²	#16839714	1	1	yes	characteristic
306	C₂²	#16839719	1	1	yes	characteristic
307	C₄	—	2	1	no	—
308	C₄	—	2	1	no	—
309	C₄	—	2	1	no	—
310	C₄	—	244	2	no	—
311	C₄	C₂×#14519693	1	1	yes	characteristic
312	C₄	#16839744	1	1	yes	characteristic
313	C₄	#16839835	1	2	yes	—
314	C₄	#16839838	1	2	yes	—
315	C₂	—	2	1	no	—
316	C₂	—	2	1	no	—
317	C₂	—	488	2	no	—
318	C₂	#21445604	1	1	yes	characteristic
319	C₂	C₄×#14519693	1	1	yes	characteristic
320	C₂	#21445620	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×#11758198
commutator	C₂³×C₆₁	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₆₁
Fitting	C₆₁×#1451	C₂
socle	C₂²×C₆₁	C₄×C₂²⋊C₄
prime radical	C₂²×D₂₄₄	C₂×C₄
cosocle	C₂×C₄²×C₆₁	C₂³
hypercenter	#1451	D₆₁
hypocenter	C₆₁	#8166
LCS	G ▷ C₂³×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ #1451

Complements

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

N	H ≅ G/N	type
4. C₈×#11758194	C₂	semidirect
5. #21443670	C₂	semidirect
6. #21443668	C₂	semidirect
7. C₆₁×#1451	C₂	semidirect
96. #11758194	M₁₆	semidirect
132. C₈×C₆₁	#100	semidirect
158. #1451	D₆₁	semidirect
188. C₆₁	#8166	semidirect
296. C₈	#14519693	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16839841	2
C₄×#14519693	2
#21444139	2
C₈×#11758194	2
#21443670	2
#21443668	2
C₆₁×#1451	2
#8166	61