Subgroups — #29570718

Order 15616.

554 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	#21454775	C₂	1	1	yes	maximal, characteristic
2	C₄×#14519705	C₂	1	1	yes	maximal, characteristic
3	C₄²×#92418	C₂	1	1	yes	maximal, characteristic, complemented
4	#21444199	C₂	1	1	yes	maximal, characteristic
5	#21444100	C₂	1	1	yes	maximal, characteristic, complemented
6	#21443655	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₁×#1276	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×#11758222	C₂²	1	1	yes	characteristic
9	#16839886	C₄	1	2	yes	complemented
10	#16839827	C₄	1	2	yes	complemented
11	#16839825	—	2	2	no	—
12	#16839813	—	2	1	no	—
13	C₂×#14519705	C₄	1	2	yes	—
14	C₄×#11758194	C₂²	1	1	yes	characteristic
15	C₂×C₄×#92418	—	2	2	no	—
16	C₂×C₄×#92418	C₂²	1	1	yes	characteristic
17	C₂×C₄×#92418	C₂²	1	1	yes	characteristic
18	C₄×C₈×D₆₁	—	2	1	no	—
19	#16839522	C₄	1	2	yes	—
20	#16839492	C₄	1	2	yes	complemented
21	#16839489	C₄	1	2	yes	complemented
22	#16839484	—	2	1	no	—
23	C₈×#92418	—	2	1	no	—
24	C₆₁×#431	C₂²	1	1	yes	characteristic
25	C₆₁×#431	C₂²	1	1	yes	characteristic
26	C₄³×C₆₁	C₂²	1	1	yes	characteristic
27	C₂×#11758222	C₂×C₄	1	1	yes	characteristic
28	C₂×#11758194	C₂×C₄	1	2	yes	—
29	C₂²×#92418	—	2	1	no	—
30	C₂²×#92418	D₄	1	2	yes	—
31	C₂×C₈×D₆₁	—	2	2	no	—
32	C₄×Dic₁₂₂	C₂×C₄	1	1	yes	characteristic
33	C₄×Dic₁₂₂	C₂×C₄	1	1	yes	characteristic
34	C₄×D₂₄₄	C₂×C₄	1	1	yes	characteristic
35	C₄×D₂₄₄	C₂×C₄	1	1	yes	characteristic
36	C₄×C₂²⋊D₆₁	—	2	1	no	—
37	C₄×C₂²⋊D₆₁	—	2	1	no	—
38	#14519736	C₂×C₄	1	1	yes	characteristic
39	#14519734	—	2	1	no	—
40	C₄²×D₆₁	—	2	1	no	—
41	#14519706	—	2	2	no	—
42	#14519705	C₈	1	8	yes	complemented
43	#14519689	—	2	2	no	—
44	#14519688	—	2	1	no	—
45	C₄×#92418	—	2	8	no	—
46	C₄×#92418	—	2	1	no	—
47	C₄×#92418	—	2	1	no	—
48	C₄×#92418	C₂×C₄	1	2	yes	—
49	C₄×#92418	C₂×C₄	1	2	yes	—
50	C₄×#92418	D₄	1	1	yes	characteristic
51	C₄×#92418	D₄	1	1	yes	characteristic
52	C₈×Dic₆₁	—	2	2	no	—
53	C₂×C₄²×C₆₁	—	2	1	no	—
54	C₂×C₄²×C₆₁	—	2	1	no	—
55	C₂×C₄²×C₆₁	C₂×C₄	1	2	yes	—
56	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
57	C₄⋊C₈×C₆₁	C₂×C₄	1	2	yes	—
58	C₄⋊C₈×C₆₁	C₂×C₄	1	2	yes	—
59	C₂²⋊C₈×C₆₁	—	2	1	no	—
60	C₂²⋊C₈×C₆₁	—	2	1	no	—
61	C₄×C₈×C₆₁	—	2	1	no	—
62	C₄×C₈×C₆₁	—	2	1	no	—
63	C₂²×C₄×C₆₁	—	2	2	no	—
64	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
65	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₆₁	#41	1	1	yes	characteristic
67	C₂×C₂²⋊D₆₁	—	2	1	no	—
68	#11758222	—	2	2	no	—
69	#11758222	C₂×C₈	1	4	yes	—
70	C₂×D₂₄₄	C₄²	1	1	yes	characteristic
71	C₂×C₄×D₆₁	—	2	1	no	—
72	C₂×C₄×D₆₁	—	2	1	no	—
73	C₂×C₄×D₆₁	—	2	1	no	—
74	C₂×Dic₁₂₂	C₄²	1	1	yes	characteristic
75	C₂×C₈×C₆₁	—	2	2	no	—
76	C₂×C₈×C₆₁	—	2	2	no	—
77	C₄²×C₆₁	—	2	4	no	—
78	C₄²×C₆₁	—	2	4	no	—
79	C₄²×C₆₁	—	2	2	no	—
80	C₄²×C₆₁	C₂×C₈	1	4	yes	complemented
81	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
82	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
83	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
84	C₄²×C₆₁	#41	1	1	yes	characteristic
85	C₂²⋊Dic₆₁	—	2	1	no	—
86	#11758198	—	2	1	no	—
87	#11758198	—	2	1	no	—
88	C₄⋊Dic₆₁	C₄²	1	1	yes	characteristic
89	C₄⋊Dic₆₁	C₄²	1	1	yes	characteristic
90	#11758196	—	2	1	no	—
91	#11758196	—	2	1	no	—
92	C₄×Dic₆₁	—	2	1	no	—
93	C₄×Dic₆₁	—	2	1	no	—
94	C₄×Dic₆₁	—	2	1	no	—
95	#11758194	—	2	2	no	—
96	#11758194	C₂×C₈	1	4	yes	complemented
97	C₂×#92418	—	2	4	no	—
98	C₂×#92418	—	2	2	no	—
99	C₂×#92418	—	2	2	no	—
100	C₂×#92418	—	2	4	no	—
101	C₂×#92418	C₂²⋊C₄	1	2	yes	—
102	C₂×#92418	C₂²⋊C₄	1	2	yes	—
103	C₈×D₆₁	—	4	4	no	—
104	C₂³×C₆₁	C₄×D₄	1	1	yes	characteristic
105	C₂²×D₆₁	—	2	1	no	—
106	C₂×C₄×C₆₁	—	2	4	no	—
107	C₂×C₄×C₆₁	—	2	4	no	—
108	C₂×C₄×C₆₁	—	2	4	no	—
109	C₂×C₄×C₆₁	—	2	2	no	—
110	C₂×C₄×C₆₁	—	2	2	no	—
111	C₂×C₄×C₆₁	C₂×C₄²	1	1	yes	characteristic
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	C₂²×C₈	1	2	yes	—
118	C₂×C₄×C₆₁	#131	1	2	yes	—
119	C₂²⋊D₆₁	—	2	4	no	—
120	C₂×Dic₆₁	—	2	1	no	—
121	C₂×Dic₆₁	—	2	1	no	—
122	C₂×Dic₆₁	—	2	1	no	—
123	D₂₄₄	—	2	1	no	—
124	D₂₄₄	C₄×C₈	1	2	yes	complemented
125	C₄×D₆₁	—	2	4	no	—
126	C₄×D₆₁	—	4	2	no	—
127	C₄×D₆₁	—	4	2	no	—
128	Dic₁₂₂	—	2	1	no	—
129	Dic₁₂₂	C₄×C₈	1	2	yes	complemented
130	C₈×C₆₁	—	4	2	no	—
131	C₈×C₆₁	—	4	2	no	—
132	#92418	—	2	4	no	—
133	#92418	—	2	2	no	—
134	#92418	—	2	4	no	—
135	#92418	#104	1	4	yes	complemented
136	C₈×#104	—	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	C₈×D₄	1	2	yes	—
141	C₂×D₆₁	—	2	2	no	—
142	C₂×D₆₁	—	4	1	no	—
143	C₂×D₆₁	—	4	1	no	—
144	C₄×C₆₁	—	2	8	no	—
145	C₄×C₆₁	—	2	1	no	—
146	C₄×C₆₁	—	2	1	no	—
147	C₄×C₆₁	C₂×C₄×C₈	1	1	yes	characteristic
148	C₄×C₆₁	#405	1	1	yes	characteristic
149	C₄×C₆₁	C₂×#104	1	2	yes	—
150	C₄×C₆₁	C₈×D₄	1	2	yes	—
151	C₄×C₆₁	#443	1	2	yes	—
152	Dic₆₁	—	2	2	no	—
153	Dic₆₁	—	4	1	no	—
154	Dic₆₁	—	4	1	no	—
155	C₈×#41	—	61	1	no	—
156	C₄×#104	—	61	1	no	—
157	C₄²×C₈	—	61	1	no	—
158	#1476	—	61	1	no	—
159	C₈×M₁₆	—	61	1	no	—
160	#1278	—	61	1	no	—
161	#1276	D₆₁	1	1	yes	characteristic, complemented
162	C₂×C₆₁	—	2	2	no	—
163	C₂×C₆₁	C₈×C₂²⋊C₄	1	1	yes	characteristic
164	C₂×C₆₁	C₄×#104	1	1	yes	characteristic
165	C₂×C₆₁	#1764	1	1	yes	characteristic
166	D₆₁	—	4	2	no	—
167	C₄×#41	—	61	1	no	—
168	C₈×Q₈	—	61	2	no	—
169	C₈×D₄	—	61	2	no	—
170	C₈×D₄	—	122	2	no	—
171	#431	—	122	1	no	—
172	#431	C₂×D₆₁	1	1	yes	characteristic
173	#431	C₂×D₆₁	1	1	yes	characteristic
174	C₂×#104	—	61	2	no	—
175	C₄×M₁₆	—	61	1	no	—
176	C₂×C₄×C₈	—	61	1	no	—
177	C₂×C₄×C₈	—	61	1	no	—
178	C₂×C₄×C₈	—	122	1	no	—
179	C₂×C₄×C₈	—	122	2	no	—
180	C₄³	C₂×D₆₁	1	1	yes	characteristic
181	#339	—	61	2	no	—
182	Q₈⋊C₈	—	61	2	no	—
183	D₄⋊C₈	—	61	2	no	—
184	#322	—	122	1	no	—
185	C₈²	—	122	1	no	—
186	C₆₁	C₈×#104	1	1	yes	characteristic, complemented, Sylow
187	C₂×#41	—	61	1	no	—
188	C₂×M₁₆	—	61	2	no	—
189	C₂²×C₈	—	61	2	no	—
190	C₂²×C₈	—	122	2	no	—
191	C₂²×C₈	—	122	1	no	—
192	C₄×Q₈	—	61	1	no	—
193	C₄×Q₈	—	61	1	no	—
194	C₄×D₄	—	61	1	no	—
195	C₄×D₄	—	61	1	no	—
196	C₄×D₄	—	122	1	no	—
197	C₄×D₄	—	122	1	no	—
198	#117	—	61	1	no	—
199	#117	—	122	1	no	—
200	C₂×C₄²	—	2	1	no	—
201	C₂×C₄²	—	2	1	no	—
202	C₂×C₄²	—	122	1	no	—
203	C₂×C₄²	C₄×D₆₁	1	2	yes	—
204	C₂×C₄²	C₂²×D₆₁	1	1	yes	characteristic
205	C₄⋊C₈	—	122	2	no	—
206	C₄⋊C₈	C₄×D₆₁	1	2	yes	—
207	C₄⋊C₈	C₄×D₆₁	1	2	yes	—
208	#104	—	61	8	no	—
209	C₂²⋊C₈	—	2	1	no	—
210	C₂²⋊C₈	—	2	1	no	—
211	C₂²⋊C₈	—	122	2	no	—
212	#97	—	122	1	no	—
213	C₄×C₈	—	2	1	no	—
214	C₄×C₈	—	2	1	no	—
215	C₄×C₈	—	61	2	no	—
216	C₄×C₈	—	61	2	no	—
217	C₄×C₈	—	61	1	no	—
218	C₄×C₈	—	61	1	no	—
219	C₄×C₈	—	122	8	no	—
220	C₄×C₈	—	122	2	no	—
221	C₄×C₈	—	122	1	no	—
222	C₄×C₈	—	122	1	no	—
223	#41	—	61	4	no	—
224	#41	—	122	2	no	—
225	C₂×Q₈	—	61	1	no	—
226	C₂×D₄	—	61	1	no	—
227	C₂×D₄	—	122	1	no	—
228	C₂²×C₄	—	2	2	no	—
229	C₂²×C₄	—	122	1	no	—
230	C₂²×C₄	—	122	1	no	—
231	C₂²×C₄	—	122	1	no	—
232	C₂²×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
233	C₂²×C₄	D₄×D₆₁	1	1	yes	characteristic
234	C₂²×C₄	#11758224	1	1	yes	characteristic
235	M₁₆	—	61	4	no	—
236	M₁₆	—	122	2	no	—
237	C₂×C₈	—	2	2	no	—
238	C₂×C₈	—	2	2	no	—
239	C₂×C₈	—	61	2	no	—
240	C₂×C₈	—	61	2	no	—
241	C₂×C₈	—	122	4	no	—
242	C₂×C₈	—	122	4	no	—
243	C₂×C₈	—	122	2	no	—
244	C₂×C₈	—	122	2	no	—
245	C₂×C₈	—	244	4	no	—
246	C₄⋊C₄	—	61	1	no	—
247	C₄⋊C₄	—	61	1	no	—
248	C₄⋊C₄	—	122	1	no	—
249	C₄⋊C₄	—	122	1	no	—
250	C₂²⋊C₄	—	122	1	no	—
251	C₂²⋊C₄	—	122	1	no	—
252	C₂²⋊C₄	—	122	1	no	—
253	C₄²	—	2	4	no	—
254	C₄²	—	2	4	no	—
255	C₄²	—	2	2	no	—
256	C₄²	—	122	1	no	—
257	C₄²	—	122	1	no	—
258	C₄²	—	122	1	no	—
259	C₄²	C₈×D₆₁	1	4	yes	complemented
260	C₄²	C₂×C₄×D₆₁	1	1	yes	characteristic
261	C₄²	C₂×C₄×D₆₁	1	1	yes	characteristic
262	C₄²	D₄×D₆₁	1	1	yes	characteristic
263	C₄²	#11758226	1	1	yes	characteristic
264	C₂³	—	122	1	no	—
265	C₂³	#14519737	1	1	yes	characteristic
266	Q₈	—	61	2	no	—
267	Q₈	—	122	1	no	—
268	D₄	—	61	2	no	—
269	D₄	—	122	4	no	—
270	D₄	—	122	1	no	—
271	C₂×C₄	—	2	4	no	—
272	C₂×C₄	—	2	4	no	—
273	C₂×C₄	—	2	4	no	—
274	C₂×C₄	—	2	2	no	—
275	C₂×C₄	—	2	2	no	—
276	C₂×C₄	—	122	1	no	—
277	C₂×C₄	—	122	4	no	—
278	C₂×C₄	—	122	1	no	—
279	C₂×C₄	—	122	1	no	—
280	C₂×C₄	—	244	2	no	—
281	C₂×C₄	—	244	2	no	—
282	C₂×C₄	C₄²×D₆₁	1	1	yes	characteristic
283	C₂×C₄	C₂²⋊C₄×D₆₁	1	1	yes	characteristic
284	C₂×C₄	#14519735	1	1	yes	characteristic
285	C₂×C₄	#14519737	1	1	yes	characteristic
286	C₂×C₄	#14519739	1	1	yes	characteristic
287	C₂×C₄	#14519739	1	1	yes	characteristic
288	C₂×C₄	C₂×C₈×D₆₁	1	2	yes	—
289	C₂×C₄	#14519785	1	2	yes	—
290	C₈	—	4	2	no	—
291	C₈	—	4	2	no	—
292	C₈	—	61	4	no	—
293	C₈	—	122	4	no	—
294	C₈	—	122	2	no	—
295	C₈	—	122	4	no	—
296	C₂²	—	2	1	no	—
297	C₂²	—	2	1	no	—
298	C₂²	—	122	2	no	—
299	C₂²	—	244	1	no	—
300	C₂²	—	244	1	no	—
301	C₂²	#16839609	1	1	yes	characteristic
302	C₂²	#16839825	1	2	yes	—
303	C₄	—	2	8	no	—
304	C₄	—	2	1	no	—
305	C₄	—	2	1	no	—
306	C₄	—	122	2	no	—
307	C₄	—	244	1	no	—
308	C₄	—	244	1	no	—
309	C₄	C₄×C₈×D₆₁	1	1	yes	characteristic
310	C₄	#16839719	1	1	yes	characteristic
311	C₄	#104×D₆₁	1	2	yes	—
312	C₄	#16839827	1	2	yes	—
313	C₄	#16839876	1	2	yes	—
314	C₂	—	2	2	no	—
315	C₂	—	244	2	no	—
316	C₂	#21445589	1	1	yes	characteristic
317	C₂	#21445623	1	1	yes	characteristic
318	C₂	#21445653	1	1	yes	characteristic

	subgroup	G/H
center	C₄²	D₄×D₆₁
commutator	C₄×C₆₁	C₂×C₄×C₈
Frattini	C₂×C₄²	C₂²×D₆₁
Fitting	C₆₁×#1276	C₂
socle	C₂²×C₆₁	C₄×C₂²⋊C₄
prime radical	C₂×#11758222	C₂×C₄
cosocle	C₂×C₄²×C₆₁	C₂³
hypercenter	#1276	D₆₁
hypocenter	C₆₁	C₈×#104
LCS	G ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₄² ◁ C₂×C₄² ◁ #1276

Complements

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

N	H ≅ G/N	type
3. C₄²×#92418	C₂	semidirect
5. #21444100	C₂	semidirect
6. #21443655	C₂	semidirect
7. C₆₁×#1276	C₂	semidirect
9. #16839886	C₄	semidirect
10. #16839827	C₄	semidirect
20. #16839492	C₄	semidirect
21. #16839489	C₄	semidirect
42. #14519705	C₈	semidirect
80. C₄²×C₆₁	C₂×C₈	semidirect
96. #11758194	C₂×C₈	semidirect
124. D₂₄₄	C₄×C₈	semidirect
129. Dic₁₂₂	C₄×C₈	semidirect
135. #92418	#104	semidirect
161. #1276	D₆₁	semidirect
186. C₆₁	C₈×#104	semidirect
259. C₄²	C₈×D₆₁	semidirect

Maximal subgroups

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

subgroup	index
#21454775	2
C₄×#14519705	2
C₄²×#92418	2
#21444199	2
#21444100	2
#21443655	2
C₆₁×#1276	2
C₈×#104	61