Subgroups — #29680376

Order 15616.

616 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#131×Dic₆₁	C₂	1	1	yes	maximal, characteristic
2	#21445966	C₂	1	2	yes	maximal, complemented
3	#21445942	C₂	1	2	yes	maximal, complemented
4	#21445920	C₂	1	2	yes	maximal
5	#21445869	C₂	1	1	yes	maximal, characteristic
6	#21445700	C₂	1	2	yes	maximal, complemented
7	#21445621	C₂	1	2	yes	maximal
8	#21445573	C₂	1	2	yes	maximal, complemented
9	C₆₁×#1812	C₂	1	1	yes	maximal, characteristic
10	C₂×#14519784	C₂²	1	1	yes	characteristic
11	#41×Dic₆₁	C₂²	1	1	yes	characteristic
12	C₂×#14519713	C₂²	1	2	yes	—
13	#16839746	C₂²	1	2	yes	—
14	#16839720	C₄	1	2	yes	complemented
15	#16839720	C₂²	1	1	yes	characteristic
16	#16839718	C₂²	1	4	yes	—
17	#16839717	C₄	1	2	yes	complemented
18	#16839717	C₂²	1	1	yes	characteristic
19	M₁₆×Dic₆₁	C₄	1	2	yes	complemented
20	M₁₆×Dic₆₁	C₂²	1	1	yes	characteristic
21	C₂×C₈×Dic₆₁	C₄	1	2	yes	complemented
22	C₂×C₈×Dic₆₁	C₂²	1	1	yes	characteristic
23	#16839532	C₂²	1	4	yes	—
24	#16839521	C₂²	1	4	yes	—
25	#16839520	C₂²	1	2	yes	—
26	#16839518	C₂²	1	4	yes	—
27	C₂×#131×C₆₁	C₂²	1	1	yes	characteristic
28	C₆₁×#427	C₂²	1	2	yes	—
29	C₆₁×#410	C₂²	1	2	yes	—
30	C₆₁×H(3,ℤ/4)	C₂²	1	2	yes	—
31	#14519784	—	2	2	no	—
32	#14519784	—	2	2	no	—
33	C₂×#11758194	C₂×C₄	1	2	yes	—
34	C₂×#11758194	C₂³	1	4	yes	—
35	C₂×#11758194	C₂³	1	1	yes	characteristic
36	C₂²×#92418	C₂×C₄	1	2	yes	—
37	C₂²×#92418	C₂³	1	1	yes	characteristic
38	Q₈×Dic₆₁	—	2	1	no	—
39	D₄×Dic₆₁	—	2	2	no	—
40	D₄×Dic₆₁	—	2	1	no	—
41	#14519736	C₂×C₄	1	2	yes	—
42	#14519736	C₂³	1	1	yes	characteristic
43	C₂×C₄×Dic₆₁	C₂×C₄	1	2	yes	—
44	C₂×C₄×Dic₆₁	C₂³	1	1	yes	characteristic
45	#14519713	—	2	2	no	—
46	#14519713	—	2	2	no	—
47	#14519696	—	2	2	no	—
48	#14519694	—	2	2	no	—
49	#14519688	—	2	4	no	—
50	#14519687	C₂×C₄	1	2	yes	—
51	#14519687	C₂×C₄	1	2	yes	—
52	#14519687	C₂×C₄	1	2	yes	—
53	C₄×#92418	—	2	4	no	—
54	C₈×Dic₆₁	C₂×C₄	1	1	yes	characteristic
55	C₈×Dic₆₁	C₂×C₄	1	2	yes	—
56	C₈×Dic₆₁	C₂×C₄	1	2	yes	—
57	C₈×Dic₆₁	C₂×C₄	1	2	yes	—
58	C₈×Dic₆₁	C₂×C₄	1	1	yes	characteristic
59	C₈×Dic₆₁	C₂×C₄	1	1	yes	characteristic
60	C₈×Dic₆₁	C₂×C₄	1	1	yes	characteristic
61	C₂×#41×C₆₁	C₂³	1	1	yes	characteristic
62	#131×C₆₁	—	2	2	no	—
63	#131×C₆₁	—	2	2	no	—
64	C₂×M₁₆×C₆₁	C₂×C₄	1	2	yes	—
65	C₂×M₁₆×C₆₁	C₂³	1	1	yes	characteristic
66	C₂²×C₈×C₆₁	C₂×C₄	1	2	yes	—
67	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
68	#117×C₆₁	C₂³	1	4	yes	—
69	#107×C₆₁	—	2	2	no	—
70	#106×C₆₁	—	2	2	no	—
71	#99×C₆₁	—	2	4	no	—
72	#41×C₆₁	—	2	2	no	—
73	#41×C₆₁	—	2	2	no	—
74	C₂×Q₈×C₆₁	#41	1	1	yes	characteristic
75	C₂×D₄×C₆₁	#41	1	2	yes	—
76	C₂×D₄×C₆₁	#41	1	1	yes	characteristic
77	C₂²×C₄×C₆₁	C₂²×C₄	1	2	yes	—
78	C₂²×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
79	C₂²×Dic₆₁	—	2	2	no	—
80	C₂²×Dic₆₁	—	2	1	no	—
81	M₁₆×C₆₁	—	2	2	no	—
82	M₁₆×C₆₁	—	2	4	no	—
83	C₂×C₈×C₆₁	—	2	2	no	—
84	C₂×C₈×C₆₁	—	2	2	no	—
85	C₂×C₈×C₆₁	C₂²×C₄	1	2	yes	—
86	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
87	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
88	C₂×C₈×C₆₁	#41	1	2	yes	—
89	C₂×C₈×C₆₁	#41	1	2	yes	—
90	C₄⋊C₄×C₆₁	—	2	4	no	—
91	C₂²⋊C₄×C₆₁	—	2	4	no	—
92	C₄²×C₆₁	—	2	4	no	—
93	C₂²⋊Dic₆₁	—	2	2	no	—
94	C₂²⋊Dic₆₁	—	2	1	no	—
95	C₄⋊Dic₆₁	—	2	2	no	—
96	C₄⋊Dic₆₁	—	2	1	no	—
97	C₄×Dic₆₁	—	2	2	no	—
98	C₄×Dic₆₁	—	2	1	no	—
99	C₄×Dic₆₁	C₂²×C₄	1	1	yes	characteristic
100	C₄×Dic₆₁	C₂²×C₄	1	2	yes	—
101	C₄×Dic₆₁	C₂²×C₄	1	1	yes	characteristic
102	#11758194	—	2	2	no	—
103	#11758194	—	2	4	no	—
104	#11758194	—	2	4	no	—
105	#11758194	—	2	4	no	—
106	C₂×#92418	—	2	2	no	—
107	C₂×#92418	—	2	2	no	—
108	C₂×#92418	—	2	4	no	—
109	C₂×#92418	C₂²×C₄	1	2	yes	—
110	C₂×#92418	C₂²×C₄	1	1	yes	characteristic
111	C₂×#92418	C₂²×C₄	1	1	yes	characteristic
112	C₂×#92418	#41	1	2	yes	—
113	C₂×#92418	#41	1	2	yes	—
114	C₂³×C₆₁	#117	1	2	yes	—
115	C₂³×C₆₁	C₂×#41	1	1	yes	characteristic
116	Q₈×C₆₁	—	2	2	no	—
117	D₄×C₆₁	—	2	4	no	—
118	D₄×C₆₁	—	2	2	no	—
119	C₂×C₄×C₆₁	—	2	2	no	—
120	C₂×C₄×C₆₁	—	2	2	no	—
121	C₂×C₄×C₆₁	—	4	4	no	—
122	C₂×C₄×C₆₁	#117	1	2	yes	—
123	C₂×C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
124	C₂×C₄×C₆₁	C₂×#41	1	2	yes	—
125	C₂×C₄×C₆₁	C₂×#41	1	2	yes	—
126	C₂×C₄×C₆₁	C₂×#41	1	1	yes	characteristic
127	C₂×Dic₆₁	—	2	1	no	—
128	C₂×Dic₆₁	—	2	2	no	—
129	C₂×Dic₆₁	—	2	1	no	—
130	C₂×Dic₆₁	—	4	2	no	—
131	C₂×Dic₆₁	—	4	4	no	—
132	C₈×C₆₁	—	2	2	no	—
133	C₈×C₆₁	#117	1	1	yes	characteristic
134	C₈×C₆₁	#117	1	1	yes	characteristic
135	C₈×C₆₁	#117	1	2	yes	—
136	#92418	—	2	2	no	—
137	#92418	—	4	4	no	—
138	#92418	#117	1	1	yes	characteristic
139	#92418	#117	1	1	yes	characteristic
140	#92418	#117	1	2	yes	—
141	#20787	—	61	1	no	maximal, Sylow
142	C₂²×C₆₁	—	2	2	no	—
143	C₂²×C₆₁	—	4	1	no	—
144	C₂²×C₆₁	—	4	2	no	—
145	C₂²×C₆₁	C₂×#117	1	1	yes	characteristic
146	C₂²×C₆₁	#529	1	2	yes	—
147	C₄×C₆₁	—	2	2	no	—
148	C₄×C₆₁	—	8	4	no	—
149	C₄×C₆₁	C₂×#117	1	1	yes	characteristic
150	C₄×C₆₁	C₂×#117	1	1	yes	characteristic
151	C₄×C₆₁	#529	1	2	yes	—
152	Dic₆₁	—	2	2	no	—
153	Dic₆₁	—	4	2	no	—
154	Dic₆₁	—	4	1	no	—
155	C₄×#131	—	61	1	no	—
156	#1841	—	61	2	no	—
157	#1833	—	61	2	no	—
158	#1827	—	61	2	no	—
159	#1813	—	61	1	no	—
160	#1812	D₆₁	1	1	yes	characteristic
161	#1772	—	61	2	no	—
162	#1757	—	61	2	no	—
163	#1747	—	61	2	no	—
164	C₂×C₆₁	—	2	1	no	—
165	C₂×C₆₁	—	2	2	no	—
166	C₂×C₆₁	—	4	2	no	—
167	C₂×C₆₁	#2283	1	1	yes	characteristic
168	C₂×#131	—	61	1	no	—
169	C₂×#131	C₂×D₆₁	1	1	yes	characteristic
170	C₄×#41	—	61	1	no	—
171	C₂×#108	—	61	2	no	—
172	#427	C₂×D₆₁	1	2	yes	—
173	#413	—	61	2	no	—
174	#410	C₂×D₆₁	1	2	yes	—
175	#405	—	61	2	no	—
176	#405	—	61	1	no	—
177	#405	—	61	2	no	—
178	#405	—	61	1	no	—
179	#405	—	61	4	no	—
180	C₄×M₁₆	—	61	2	no	—
181	C₄×M₁₆	—	61	1	no	—
182	C₂×C₄×C₈	—	61	2	no	—
183	C₂×C₄×C₈	—	61	1	no	—
184	#344	—	61	4	no	—
185	#339	—	61	4	no	—
186	#338	—	61	2	no	—
187	H(3,ℤ/4)	—	61	4	no	—
188	H(3,ℤ/4)	C₂×D₆₁	1	2	yes	—
189	C₆₁	#20787	1	1	yes	characteristic, complemented, Sylow
190	C₂×#41	C₂²×D₆₁	1	1	yes	characteristic
191	#131	—	2	2	no	—
192	#131	—	2	2	no	—
193	#131	—	122	2	no	—
194	#131	—	122	2	no	—
195	C₂×M₁₆	—	61	2	no	—
196	C₂×M₁₆	—	61	1	no	—
197	C₂×M₁₆	—	61	4	no	—
198	C₂×M₁₆	C₄×D₆₁	1	2	yes	—
199	C₂×M₁₆	C₂²×D₆₁	1	1	yes	characteristic
200	C₂²×C₈	—	61	1	no	—
201	C₂²×C₈	—	61	2	no	—
202	C₂²×C₈	C₄×D₆₁	1	2	yes	—
203	C₂²×C₈	C₂²×D₆₁	1	1	yes	characteristic
204	C₄×Q₈	—	122	1	no	—
205	C₄×D₄	—	122	2	no	—
206	C₄×D₄	—	122	1	no	—
207	#117	—	61	2	no	—
208	#117	—	61	1	no	—
209	#117	C₂²×D₆₁	1	4	yes	—
210	C₂×C₄²	—	61	2	no	—
211	C₂×C₄²	—	61	1	no	—
212	#108	—	122	2	no	—
213	#108	—	122	2	no	—
214	#107	—	2	2	no	—
215	#106	—	2	2	no	—
216	#101	—	122	2	no	—
217	#100	—	122	2	no	—
218	#99	—	2	4	no	—
219	#97	—	61	2	no	—
220	#97	—	61	2	no	—
221	#97	—	61	2	no	—
222	#97	—	122	4	no	—
223	C₄×C₈	—	61	1	no	—
224	C₄×C₈	—	61	1	no	—
225	C₄×C₈	—	61	2	no	—
226	C₄×C₈	—	61	2	no	—
227	C₄×C₈	—	61	1	no	—
228	C₄×C₈	—	61	1	no	—
229	C₄×C₈	—	61	2	no	—
230	C₄×C₈	—	122	4	no	—
231	#41	—	2	2	no	—
232	#41	—	2	2	no	—
233	C₂×Q₈	#11758226	1	1	yes	characteristic
234	C₂×D₄	#11758224	1	2	yes	—
235	C₂×D₄	#11758226	1	1	yes	characteristic
236	C₂²×C₄	—	122	1	no	—
237	C₂²×C₄	—	122	2	no	—
238	C₂²×C₄	C₂×C₄×D₆₁	1	2	yes	—
239	C₂²×C₄	C₂³×D₆₁	1	1	yes	characteristic
240	M₁₆	—	2	2	no	—
241	M₁₆	—	2	4	no	—
242	M₁₆	—	122	4	no	—
243	M₁₆	—	122	2	no	—
244	M₁₆	—	122	4	no	—
245	M₁₆	—	122	4	no	—
246	C₂×C₈	—	2	2	no	—
247	C₂×C₈	—	2	2	no	—
248	C₂×C₈	—	61	1	no	—
249	C₂×C₈	—	61	1	no	—
250	C₂×C₈	—	61	2	no	—
251	C₂×C₈	—	61	2	no	—
252	C₂×C₈	—	61	2	no	—
253	C₂×C₈	—	122	2	no	—
254	C₂×C₈	—	122	2	no	—
255	C₂×C₈	—	122	4	no	—
256	C₂×C₈	C₂×C₄×D₆₁	1	2	yes	—
257	C₂×C₈	C₂×C₄×D₆₁	1	1	yes	characteristic
258	C₂×C₈	C₂×C₄×D₆₁	1	1	yes	characteristic
259	C₂×C₈	#11758224	1	2	yes	—
260	C₂×C₈	#11758224	1	2	yes	—
261	C₄⋊C₄	—	2	4	no	—
262	C₄⋊C₄	—	122	1	no	—
263	C₄⋊C₄	—	122	2	no	—
264	C₂²⋊C₄	—	2	4	no	—
265	C₂²⋊C₄	—	122	1	no	—
266	C₂²⋊C₄	—	122	2	no	—
267	C₄²	—	2	4	no	—
268	C₄²	—	61	1	no	—
269	C₄²	—	61	2	no	—
270	C₄²	—	61	1	no	—
271	C₄²	—	122	1	no	—
272	C₄²	—	122	2	no	—
273	C₂³	#14519735	1	2	yes	—
274	C₂³	C₂×#11758224	1	1	yes	characteristic
275	Q₈	—	2	2	no	—
276	D₄	—	2	4	no	—
277	D₄	—	2	2	no	—
278	C₂×C₄	—	2	2	no	—
279	C₂×C₄	—	2	2	no	—
280	C₂×C₄	—	4	4	no	—
281	C₂×C₄	—	122	1	no	—
282	C₂×C₄	—	122	1	no	—
283	C₂×C₄	—	122	2	no	—
284	C₂×C₄	—	244	2	no	—
285	C₂×C₄	—	244	4	no	—
286	C₂×C₄	#14519735	1	2	yes	—
287	C₂×C₄	C₂²×C₄×D₆₁	1	1	yes	characteristic
288	C₂×C₄	#41×D₆₁	1	2	yes	—
289	C₂×C₄	C₂×#11758224	1	2	yes	—
290	C₂×C₄	C₂×#11758226	1	1	yes	characteristic
291	C₈	—	2	2	no	—
292	C₈	—	61	1	no	—
293	C₈	—	61	1	no	—
294	C₈	—	61	2	no	—
295	C₈	—	122	2	no	—
296	C₈	—	244	4	no	—
297	C₈	#14519733	1	1	yes	characteristic
298	C₈	#14519733	1	1	yes	characteristic
299	C₈	#14519733	1	2	yes	—
300	C₂²	—	2	2	no	—
301	C₂²	—	4	1	no	—
302	C₂²	—	4	2	no	—
303	C₂²	C₂×#14519735	1	1	yes	characteristic
304	C₂²	#16840293	1	2	yes	—
305	C₄	—	2	2	no	—
306	C₄	—	8	4	no	—
307	C₄	—	122	2	no	—
308	C₄	—	244	1	no	—
309	C₄	—	244	2	no	—
310	C₄	#117×D₆₁	1	1	yes	characteristic
311	C₄	C₂×#14519733	1	1	yes	characteristic
312	C₄	#16840298	1	2	yes	—
313	C₂	—	2	2	no	—
314	C₂	—	2	1	no	—
315	C₂	—	4	2	no	—
316	C₂	#21448214	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#117×D₆₁
commutator	C₂×C₄×C₆₁	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₆₁
Fitting	C₆₁×#1812	C₂
socle	C₂×C₆₁	#2283
prime radical	C₂×#41×C₆₁	C₂³
cosocle	C₂²×C₄×C₆₁	C₂⁴
hypercenter	#1812	D₆₁
hypocenter	C₆₁	#20787
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₄ ◁ C₂×M₁₆ ◁ #1812

Complements

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

N	H ≅ G/N	type
2. #21445966	C₂	semidirect
3. #21445942	C₂	semidirect
6. #21445700	C₂	semidirect
8. #21445573	C₂	semidirect
14. #16839720	C₄	semidirect
17. #16839717	C₄	semidirect
19. M₁₆×Dic₆₁	C₄	semidirect
21. C₂×C₈×Dic₆₁	C₄	semidirect
189. C₆₁	#20787	semidirect

Maximal subgroups

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

subgroup	index
#131×Dic₆₁	2
#21445966	2
#21445942	2
#21445920	2
#21445869	2
#21445700	2
#21445621	2
#21445573	2
C₆₁×#1812	2
#20787	61