Subgroups — #29596505

Order 15616.

376 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21458921	C₂	1	1	yes	maximal, characteristic
2	#21447584	C₂	1	1	yes	maximal, characteristic, complemented
3	#21447448	C₂	1	1	yes	maximal, characteristic
4	#21445656	C₂	1	1	yes	maximal, characteristic
5	#21443904	C₂	1	1	yes	maximal, characteristic, complemented
6	#21443901	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₁×#2150	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#14519783	C₂²	1	1	yes	characteristic
9	#16840180	—	2	2	no	—
10	#16840178	—	2	2	no	—
11	C₂×#14519717	C₂²	1	1	yes	characteristic
12	C₂×#14519717	C₂²	1	1	yes	characteristic
13	M₃₂×D₆₁	—	2	2	no	—
14	C₂×#14519705	C₄	1	1	yes	characteristic
15	#16839759	C₄	1	1	yes	characteristic
16	#16839717	C₂²	1	1	yes	characteristic
17	#16839582	—	2	2	no	—
18	#16839547	—	2	2	no	—
19	#16839527	C₄	1	1	yes	characteristic
20	#16839522	C₄	1	1	yes	characteristic
21	C₂×M₃₂×C₆₁	C₂²	1	1	yes	characteristic
22	C₂×M₃₂×C₆₁	C₂²	1	1	yes	characteristic
23	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
24	C₆₁×C₄⋊C₁₆	—	2	2	no	—
25	C₂×#11758222	C₂×C₄	1	1	yes	characteristic
26	#14519783	—	2	1	no	—
27	#14519783	—	2	1	no	—
28	#14519783	D₄	1	2	yes	—
29	#14519783	Q₈	1	2	yes	—
30	C₂×#11758194	C₂×C₄	1	1	yes	characteristic
31	C₂×#11758194	C₂×C₄	1	1	yes	characteristic
32	C₂×#11758189	—	2	1	no	—
33	C₂×C₈×D₆₁	—	2	1	no	—
34	#14519736	C₂×C₄	1	1	yes	characteristic
35	#14519717	—	2	1	no	—
36	#14519717	—	2	1	no	—
37	#14519717	—	2	1	no	—
38	#14519717	D₄	1	2	yes	—
39	#14519716	—	2	4	no	—
40	C₂×#11758185	—	2	1	no	—
41	C₂×#11758185	—	2	1	no	—
42	C₁₆×D₆₁	—	2	4	no	—
43	#14519705	C₈	1	2	yes	—
44	#14519705	C₈	1	2	yes	—
45	#14519702	—	2	1	no	—
46	#14519699	—	2	1	no	—
47	#14519687	—	2	1	no	—
48	C₈×Dic₆₁	—	2	1	no	—
49	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
50	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
51	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
52	M₃₂×C₆₁	—	2	1	no	—
53	M₃₂×C₆₁	—	2	1	no	—
54	M₃₂×C₆₁	—	2	1	no	—
55	M₃₂×C₆₁	—	2	1	no	—
56	C₂×C₁₆×C₆₁	—	2	1	no	—
57	C₂×C₁₆×C₆₁	D₄	1	2	yes	—
58	C₄×C₈×C₆₁	—	2	1	no	—
59	C₄×C₈×C₆₁	—	2	1	no	—
60	C₂²×C₄×C₆₁	—	2	1	no	—
61	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
62	C₂×C₂²⋊D₆₁	—	2	1	no	—
63	#11758222	—	2	1	no	—
64	#11758222	—	2	1	no	—
65	#11758222	C₄⋊C₄	1	2	yes	—
66	#11758222	C₂×C₈	1	2	yes	—
67	C₂×D₂₄₄	M₁₆	1	1	yes	characteristic
68	C₂×C₄×D₆₁	—	2	1	no	—
69	C₂×Dic₁₂₂	M₁₆	1	1	yes	characteristic
70	C₂×C₈×C₆₁	—	2	1	no	—
71	C₂×C₈×C₆₁	—	2	1	no	—
72	C₂×C₈×C₆₁	—	2	1	no	—
73	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
74	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
75	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
76	C₂×C₈×C₆₁	C₂×Q₈	1	1	yes	characteristic
77	C₂×C₈×C₆₁	#41	1	1	yes	characteristic
78	C₂×C₈×C₆₁	#41	1	1	yes	characteristic
79	C₄²×C₆₁	—	2	1	no	—
80	C₄²×C₆₁	C₂×C₈	1	1	yes	characteristic
81	C₄²×C₆₁	C₂×C₈	1	1	yes	characteristic
82	C₂²⋊Dic₆₁	—	2	1	no	—
83	C₄⋊Dic₆₁	—	2	1	no	—
84	C₄×Dic₆₁	—	2	1	no	—
85	#11758194	—	2	1	no	—
86	#11758194	—	2	1	no	—
87	#11758194	C₄⋊C₄	1	2	yes	—
88	#11758194	C₂×C₈	1	2	yes	—
89	C₂×#92418	—	2	1	no	—
90	C₂×#92418	—	2	1	no	—
91	#11758189	—	2	2	no	—
92	#11758189	—	2	2	no	—
93	C₈×D₆₁	—	2	2	no	—
94	C₈×D₆₁	—	2	2	no	—
95	C₁₆×C₆₁	—	2	2	no	—
96	C₁₆×C₆₁	—	4	1	no	—
97	#11758185	—	2	2	no	—
98	#11758185	—	4	1	no	—
99	C₂³×C₆₁	#131	1	1	yes	characteristic
100	C₂²×D₆₁	—	2	1	no	—
101	C₂×C₄×C₆₁	—	2	1	no	—
102	C₂×C₄×C₆₁	—	2	1	no	—
103	C₂×C₄×C₆₁	—	4	2	no	—
104	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
105	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
106	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
107	C₂×C₄×C₆₁	C₂²×C₈	1	1	yes	characteristic
108	C₂×C₄×C₆₁	C₂×M₁₆	1	1	yes	characteristic
109	C₂×C₄×C₆₁	#131	1	1	yes	characteristic
110	C₂²⋊D₆₁	—	2	2	no	—
111	C₂²⋊D₆₁	—	2	2	no	—
112	C₂×Dic₆₁	—	2	1	no	—
113	C₂×Dic₆₁	—	4	1	no	—
114	D₂₄₄	—	2	1	no	—
115	D₂₄₄	C₄⋊C₈	1	2	yes	—
116	C₄×D₆₁	—	2	2	no	—
117	C₄×D₆₁	—	2	2	no	—
118	Dic₁₂₂	—	2	1	no	—
119	Dic₁₂₂	C₄⋊C₈	1	2	yes	—
120	C₈×C₆₁	—	4	1	no	—
121	C₈×C₆₁	#120	1	1	yes	characteristic
122	C₈×C₆₁	C₂²⋊Q₈	1	1	yes	characteristic
123	C₈×C₆₁	C₂²⋊Q₈	1	1	yes	characteristic
124	C₈×C₆₁	#123	1	1	yes	characteristic
125	#92418	—	2	2	no	—
126	#92418	—	2	2	no	—
127	#12918	—	61	1	no	maximal, Sylow
128	C₂²×C₆₁	—	4	1	no	—
129	C₂²×C₆₁	#424	1	1	yes	characteristic
130	C₂²×C₆₁	C₈×D₄	1	1	yes	characteristic
131	C₂²×C₆₁	#435	1	1	yes	characteristic
132	C₂×D₆₁	—	2	2	no	—
133	C₂×D₆₁	—	8	1	no	—
134	C₄×C₆₁	—	2	2	no	—
135	C₄×C₆₁	—	4	1	no	—
136	C₄×C₆₁	#385	1	1	yes	characteristic
137	C₄×C₆₁	C₂×C₄⋊C₈	1	1	yes	characteristic
138	C₄×C₆₁	C₈×D₄	1	1	yes	characteristic
139	C₄×C₆₁	#435	1	1	yes	characteristic
140	Dic₆₁	—	2	2	no	—
141	Dic₆₁	—	8	1	no	—
142	#3407	—	61	1	no	—
143	#2153	—	61	1	no	—
144	#2150	D₆₁	1	1	yes	characteristic, complemented
145	#2117	—	61	1	no	—
146	#1764	—	61	1	no	—
147	#1378	—	61	1	no	—
148	#1376	—	61	1	no	—
149	C₂×C₆₁	—	2	1	no	—
150	C₂×C₆₁	—	2	1	no	—
151	C₂×C₆₁	—	2	1	no	—
152	C₂×C₆₁	#1850	1	1	yes	characteristic
153	D₆₁	—	4	2	no	—
154	C₂×#131	—	61	1	no	—
155	#504	—	122	2	no	—
156	#504	—	122	2	no	—
157	C₂×M₃₂	—	61	1	no	—
158	C₂×M₃₂	—	61	1	no	—
159	C₂×M₃₂	—	122	2	no	—
160	C₂×M₃₂	C₂×D₆₁	1	1	yes	characteristic
161	C₂×M₃₂	C₂×D₆₁	1	1	yes	characteristic
162	C₂×#104	—	61	1	no	—
163	#416	—	61	1	no	—
164	#405	—	61	1	no	—
165	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
166	#364	—	122	2	no	—
167	C₄⋊C₁₆	—	2	2	no	—
168	#350	—	122	2	no	—
169	#341	—	61	1	no	—
170	#339	—	61	1	no	—
171	C₆₁	#12918	1	1	yes	characteristic, complemented, Sylow
172	C₂×#41	—	61	1	no	—
173	#131	—	61	2	no	—
174	#131	—	61	2	no	—
175	#131	—	122	1	no	—
176	#131	—	122	1	no	—
177	C₂×M₁₆	—	61	1	no	—
178	C₂×M₁₆	—	61	1	no	—
179	C₂×M₁₆	—	122	1	no	—
180	C₂²×C₈	—	122	1	no	—
181	C₂²×C₈	C₄×D₆₁	1	1	yes	characteristic
182	C₂²×C₈	C₂²×D₆₁	1	1	yes	characteristic
183	#117	—	61	1	no	—
184	C₂×C₄²	C₄×D₆₁	1	1	yes	characteristic
185	M₃₂	—	2	1	no	—
186	M₃₂	—	2	1	no	—
187	M₃₂	—	2	1	no	—
188	M₃₂	—	2	1	no	—
189	M₃₂	—	61	2	no	—
190	M₃₂	—	122	4	no	—
191	M₃₂	—	122	1	no	—
192	M₃₂	—	122	1	no	—
193	M₃₂	—	122	1	no	—
194	C₂×C₁₆	—	2	1	no	—
195	C₂×C₁₆	—	122	4	no	—
196	C₂×C₁₆	—	122	1	no	—
197	C₂×C₁₆	—	122	1	no	—
198	C₂×C₁₆	C₂²⋊D₆₁	1	2	yes	—
199	#104	—	61	2	no	—
200	#104	—	61	2	no	—
201	#103	—	122	1	no	—
202	#102	—	122	1	no	—
203	#97	—	122	1	no	—
204	C₄×C₈	—	2	1	no	—
205	C₄×C₈	—	2	1	no	—
206	C₄×C₈	—	122	1	no	—
207	#41	—	61	2	no	—
208	#41	—	61	2	no	—
209	#41	—	122	1	no	—
210	#41	—	122	1	no	—
211	C₂×Q₈	—	61	1	no	—
212	C₂×D₄	—	61	1	no	—
213	C₂×D₄	—	122	1	no	—
214	C₂²×C₄	—	2	1	no	—
215	C₂²×C₄	—	122	1	no	—
216	C₂²×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
217	M₁₆	—	61	2	no	—
218	M₁₆	—	61	2	no	—
219	M₁₆	—	122	1	no	—
220	M₁₆	—	122	1	no	—
221	M₁₆	—	122	2	no	—
222	M₁₆	—	122	2	no	—
223	C₂×C₈	—	2	1	no	—
224	C₂×C₈	—	2	1	no	—
225	C₂×C₈	—	2	1	no	—
226	C₂×C₈	—	122	1	no	—
227	C₂×C₈	—	122	1	no	—
228	C₂×C₈	—	122	2	no	—
229	C₂×C₈	—	122	2	no	—
230	C₂×C₈	#11758222	1	1	yes	characteristic
231	C₂×C₈	D₄×D₆₁	1	1	yes	characteristic
232	C₂×C₈	D₄×D₆₁	1	1	yes	characteristic
233	C₂×C₈	Q₈×D₆₁	1	1	yes	characteristic
234	C₂×C₈	#11758226	1	1	yes	characteristic
235	C₂×C₈	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
236	C₄⋊C₄	—	122	1	no	—
237	C₂²⋊C₄	—	122	1	no	—
238	C₄²	—	2	1	no	—
239	C₄²	—	122	1	no	—
240	C₄²	C₈×D₆₁	1	1	yes	characteristic
241	C₄²	C₈×D₆₁	1	1	yes	characteristic
242	C₁₆	—	2	2	no	—
243	C₁₆	—	4	1	no	—
244	C₁₆	—	122	2	no	—
245	C₁₆	—	244	1	no	—
246	C₂³	—	122	1	no	—
247	C₂³	#14519783	1	1	yes	characteristic
248	Q₈	—	61	2	no	—
249	Q₈	—	122	1	no	—
250	D₄	—	61	2	no	—
251	D₄	—	122	2	no	—
252	D₄	—	122	2	no	—
253	D₄	—	122	1	no	—
254	C₂×C₄	—	2	1	no	—
255	C₂×C₄	—	2	1	no	—
256	C₂×C₄	—	4	2	no	—
257	C₂×C₄	—	122	1	no	—
258	C₂×C₄	—	122	2	no	—
259	C₂×C₄	—	122	2	no	—
260	C₂×C₄	—	244	1	no	—
261	C₂×C₄	C₄⋊C₄×D₆₁	1	1	yes	characteristic
262	C₂×C₄	C₄×C₂²⋊D₆₁	1	1	yes	characteristic
263	C₂×C₄	#14519739	1	1	yes	characteristic
264	C₂×C₄	C₂×C₈×D₆₁	1	1	yes	characteristic
265	C₂×C₄	M₁₆×D₆₁	1	1	yes	characteristic
266	C₂×C₄	#14519785	1	1	yes	characteristic
267	C₈	—	4	1	no	—
268	C₈	—	122	2	no	—
269	C₈	—	122	2	no	—
270	C₈	#14519745	1	1	yes	characteristic
271	C₈	#14519753	1	1	yes	characteristic
272	C₈	#14519755	1	1	yes	characteristic
273	C₈	#14519762	1	1	yes	characteristic
274	C₂²	—	4	1	no	—
275	C₂²	—	122	2	no	—
276	C₂²	—	488	1	no	—
277	C₂²	#16839787	1	1	yes	characteristic
278	C₂²	C₈×C₂²⋊D₆₁	1	1	yes	characteristic
279	C₂²	#16839832	1	1	yes	characteristic
280	C₄	—	2	2	no	—
281	C₄	—	4	1	no	—
282	C₄	—	122	2	no	—
283	C₄	—	488	1	no	—
284	C₄	#16839633	1	1	yes	characteristic
285	C₄	C₄⋊C₈×D₆₁	1	1	yes	characteristic
286	C₄	#16839827	1	1	yes	characteristic
287	C₄	#16839831	1	1	yes	characteristic
288	C₂	—	2	1	no	—
289	C₂	—	2	1	no	—
290	C₂	—	2	1	no	—
291	C₂	—	244	2	no	—
292	C₂	#21445995	1	1	yes	characteristic

	subgroup	G/H
center	C₈	#14519745
commutator	C₂×C₄×C₆₁	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₆₁
Fitting	C₆₁×#2150	C₂
socle	C₂×C₆₁	#1850
prime radical	C₂×#11758222	C₂×C₄
cosocle	C₂²×C₈×C₆₁	C₂³
hypercenter	#2150	D₆₁
hypocenter	C₆₁	#12918
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ #2150

Complements

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

N	H ≅ G/N	type
2. #21447584	C₂	semidirect
5. #21443904	C₂	semidirect
6. #21443901	C₂	semidirect
7. C₆₁×#2150	C₂	semidirect
144. #2150	D₆₁	semidirect
171. C₆₁	#12918	semidirect

Maximal subgroups

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

subgroup	index
#21458921	2
#21447584	2
#21447448	2
#21445656	2
#21443904	2
#21443901	2
C₆₁×#2150	2
#12918	61