Subgroups — #29680379

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	#21445968	C₂	1	2	yes	maximal, complemented
2	#21445943	C₂	1	2	yes	maximal, complemented
3	#21445923	C₂	1	2	yes	maximal
4	#21445869	C₂	1	1	yes	maximal, characteristic
5	#21445865	C₂	1	1	yes	maximal, characteristic
6	#21445698	C₂	1	2	yes	maximal, complemented
7	#21445619	C₂	1	2	yes	maximal
8	#21445574	C₂	1	2	yes	maximal, complemented
9	C₄×#131×C₆₁	C₂	1	1	yes	maximal, characteristic
10	C₂×#14519713	C₂²	1	2	yes	—
11	#16839800	C₂²	1	2	yes	—
12	#16839746	C₂²	1	2	yes	—
13	#16839736	C₂²	1	2	yes	—
14	#16839717	C₂²	1	4	yes	—
15	#16839533	C₂²	1	4	yes	—
16	#16839522	C₂²	1	4	yes	—
17	#16839520	C₂²	1	2	yes	—
18	#16839519	C₂²	1	2	yes	—
19	#16839518	C₂²	1	4	yes	—
20	C₂×#131×C₆₁	C₂²	1	1	yes	characteristic
21	C₂×#131×C₆₁	C₂²	1	1	yes	characteristic
22	C₄×#41×C₆₁	C₂²	1	1	yes	characteristic
23	C₆₁×#405	C₄	1	2	yes	complemented
24	C₆₁×#405	C₄	1	2	yes	complemented
25	C₆₁×#405	C₂²	1	1	yes	characteristic
26	C₆₁×#405	C₂²	1	1	yes	characteristic
27	C₄×M₁₆×C₆₁	C₄	1	2	yes	complemented
28	C₄×M₁₆×C₆₁	C₂²	1	1	yes	characteristic
29	C₂×C₄×C₈×C₆₁	C₄	1	2	yes	complemented
30	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
31	C₂×#11758194	C₂³	1	4	yes	—
32	#14519736	C₂³	1	4	yes	—
33	#14519713	—	2	2	no	—
34	#14519713	—	2	2	no	—
35	#14519711	—	2	2	no	—
36	#14519709	—	2	2	no	—
37	#14519696	—	2	2	no	—
38	#14519694	—	2	2	no	—
39	#14519692	—	2	4	no	—
40	#14519687	—	2	4	no	—
41	C₈×Dic₆₁	—	2	4	no	—
42	C₂×#41×C₆₁	C₂³	1	1	yes	characteristic
43	#131×C₆₁	—	2	2	no	—
44	#131×C₆₁	—	2	2	no	—
45	#131×C₆₁	—	2	2	no	—
46	#131×C₆₁	—	2	2	no	—
47	C₂×M₁₆×C₆₁	C₂×C₄	1	2	yes	—
48	C₂×M₁₆×C₆₁	C₂×C₄	1	2	yes	—
49	C₂×M₁₆×C₆₁	C₂³	1	1	yes	characteristic
50	C₂×M₁₆×C₆₁	C₂³	1	1	yes	characteristic
51	C₂²×C₈×C₆₁	C₂×C₄	1	2	yes	—
52	C₂²×C₈×C₆₁	C₂×C₄	1	2	yes	—
53	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
54	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
55	C₄×Q₈×C₆₁	—	2	1	no	—
56	C₄×D₄×C₆₁	—	2	2	no	—
57	C₄×D₄×C₆₁	—	2	1	no	—
58	#117×C₆₁	C₂×C₄	1	2	yes	—
59	#117×C₆₁	C₂³	1	1	yes	characteristic
60	C₂×C₄²×C₆₁	C₂×C₄	1	2	yes	—
61	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
62	#97×C₆₁	C₂×C₄	1	2	yes	—
63	#97×C₆₁	C₂×C₄	1	2	yes	—
64	#97×C₆₁	C₂×C₄	1	2	yes	—
65	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
66	C₄×C₈×C₆₁	C₂×C₄	1	2	yes	—
67	C₄×C₈×C₆₁	C₂×C₄	1	2	yes	—
68	C₄×C₈×C₆₁	C₂×C₄	1	2	yes	—
69	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
70	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
71	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
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	—	2	2	no	—
78	C₂²×C₄×C₆₁	—	2	1	no	—
79	C₂²×C₄×C₆₁	C₂²×C₄	1	2	yes	—
80	C₂²×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
81	M₁₆×C₆₁	—	2	2	no	—
82	M₁₆×C₆₁	—	2	2	no	—
83	M₁₆×C₆₁	—	2	4	no	—
84	M₁₆×C₆₁	—	2	4	no	—
85	C₂×C₈×C₆₁	—	2	2	no	—
86	C₂×C₈×C₆₁	—	2	2	no	—
87	C₂×C₈×C₆₁	—	2	2	no	—
88	C₂×C₈×C₆₁	—	2	2	no	—
89	C₂×C₈×C₆₁	C₂²×C₄	1	2	yes	—
90	C₂×C₈×C₆₁	C₂²×C₄	1	2	yes	—
91	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
92	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
93	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
94	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
95	C₂×C₈×C₆₁	#41	1	2	yes	—
96	C₂×C₈×C₆₁	#41	1	2	yes	—
97	C₂×C₈×C₆₁	#41	1	2	yes	—
98	C₂×C₈×C₆₁	#41	1	2	yes	—
99	C₄⋊C₄×C₆₁	—	2	2	no	—
100	C₄⋊C₄×C₆₁	—	2	1	no	—
101	C₂²⋊C₄×C₆₁	—	2	2	no	—
102	C₂²⋊C₄×C₆₁	—	2	1	no	—
103	C₄²×C₆₁	—	2	2	no	—
104	C₄²×C₆₁	—	2	1	no	—
105	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
106	C₄²×C₆₁	C₂²×C₄	1	2	yes	—
107	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
108	C₂²⋊Dic₆₁	—	2	4	no	—
109	C₄⋊Dic₆₁	—	2	4	no	—
110	C₄×Dic₆₁	—	2	4	no	—
111	#11758194	—	2	4	no	—
112	#11758194	—	2	4	no	—
113	C₂×#92418	—	2	4	no	—
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	1	no	—
120	C₂×C₄×C₆₁	—	2	2	no	—
121	C₂×C₄×C₆₁	—	2	2	no	—
122	C₂×C₄×C₆₁	—	2	2	no	—
123	C₂×C₄×C₆₁	—	2	1	no	—
124	C₂×C₄×C₆₁	—	4	2	no	—
125	C₂×C₄×C₆₁	—	4	4	no	—
126	C₂×C₄×C₆₁	#117	1	2	yes	—
127	C₂×C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
128	C₂×C₄×C₆₁	C₂×#41	1	2	yes	—
129	C₂×C₄×C₆₁	C₂×#41	1	2	yes	—
130	C₂×C₄×C₆₁	C₂×#41	1	1	yes	characteristic
131	C₂×Dic₆₁	—	4	4	no	—
132	C₈×C₆₁	—	2	2	no	—
133	C₈×C₆₁	—	2	2	no	—
134	C₈×C₆₁	#117	1	1	yes	characteristic
135	C₈×C₆₁	#117	1	1	yes	characteristic
136	C₈×C₆₁	#117	1	1	yes	characteristic
137	C₈×C₆₁	#117	1	2	yes	—
138	C₈×C₆₁	#117	1	2	yes	—
139	C₈×C₆₁	#117	1	1	yes	characteristic
140	#92418	—	4	4	no	—
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	—	2	2	no	—
149	C₄×C₆₁	—	4	2	no	—
150	C₄×C₆₁	—	4	1	no	—
151	C₄×C₆₁	C₂×#117	1	1	yes	characteristic
152	C₄×C₆₁	C₂×#117	1	1	yes	characteristic
153	C₄×C₆₁	#529	1	2	yes	—
154	Dic₆₁	—	8	4	no	—
155	C₄×#131	D₆₁	1	1	yes	characteristic
156	#1841	—	61	2	no	—
157	#1833	—	61	2	no	—
158	#1827	—	61	2	no	—
159	#1813	—	61	1	no	—
160	#1812	—	61	1	no	—
161	#1772	—	61	2	no	—
162	#1757	—	61	2	no	—
163	#1747	—	61	2	no	—
164	C₂×C₆₁	—	2	2	no	—
165	C₂×C₆₁	—	2	1	no	—
166	C₂×C₆₁	—	4	2	no	—
167	C₂×C₆₁	#2283	1	1	yes	characteristic
168	C₂×#131	C₂×D₆₁	1	1	yes	characteristic
169	C₂×#131	C₂×D₆₁	1	1	yes	characteristic
170	C₄×#41	C₂×D₆₁	1	1	yes	characteristic
171	C₂×#108	—	61	2	no	—
172	#427	—	61	2	no	—
173	#413	—	61	2	no	—
174	#410	—	61	2	no	—
175	#405	—	61	4	no	—
176	#405	Dic₆₁	1	2	yes	complemented
177	#405	Dic₆₁	1	2	yes	complemented
178	#405	C₂×D₆₁	1	1	yes	characteristic
179	#405	C₂×D₆₁	1	1	yes	characteristic
180	C₄×M₁₆	Dic₆₁	1	2	yes	complemented
181	C₄×M₁₆	C₂×D₆₁	1	1	yes	characteristic
182	C₂×C₄×C₈	Dic₆₁	1	2	yes	complemented
183	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
184	#344	—	61	4	no	—
185	#339	—	61	4	no	—
186	#338	—	61	2	no	—
187	H(3,ℤ/4)	—	61	2	no	—
188	H(3,ℤ/4)	—	61	4	no	—
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	—	2	2	no	—
194	#131	—	2	2	no	—
195	C₂×M₁₆	—	61	4	no	—
196	C₂×M₁₆	C₂×Dic₆₁	1	2	yes	—
197	C₂×M₁₆	C₂×Dic₆₁	1	2	yes	—
198	C₂×M₁₆	C₂²×D₆₁	1	1	yes	characteristic
199	C₂×M₁₆	C₂²×D₆₁	1	1	yes	characteristic
200	C₂²×C₈	C₂×Dic₆₁	1	2	yes	—
201	C₂²×C₈	C₂×Dic₆₁	1	2	yes	—
202	C₂²×C₈	C₂²×D₆₁	1	1	yes	characteristic
203	C₂²×C₈	C₂²×D₆₁	1	1	yes	characteristic
204	C₄×Q₈	—	2	1	no	—
205	C₄×D₄	—	2	1	no	—
206	C₄×D₄	—	2	2	no	—
207	#117	—	61	4	no	—
208	#117	C₂×Dic₆₁	1	2	yes	—
209	#117	C₂²×D₆₁	1	1	yes	characteristic
210	C₂×C₄²	C₂×Dic₆₁	1	2	yes	—
211	C₂×C₄²	C₂²×D₆₁	1	1	yes	characteristic
212	#108	—	122	2	no	—
213	#108	—	122	2	no	—
214	#107	—	122	2	no	—
215	#106	—	122	2	no	—
216	#101	—	122	2	no	—
217	#100	—	122	2	no	—
218	#99	—	122	4	no	—
219	#97	—	122	4	no	—
220	#97	C₂×Dic₆₁	1	2	yes	—
221	#97	C₂×Dic₆₁	1	2	yes	—
222	#97	C₂×Dic₆₁	1	2	yes	—
223	C₄×C₈	—	122	4	no	—
224	C₄×C₈	C₂×Dic₆₁	1	1	yes	characteristic
225	C₄×C₈	C₂×Dic₆₁	1	2	yes	—
226	C₄×C₈	C₂×Dic₆₁	1	2	yes	—
227	C₄×C₈	C₂×Dic₆₁	1	2	yes	—
228	C₄×C₈	C₂×Dic₆₁	1	1	yes	characteristic
229	C₄×C₈	C₂×Dic₆₁	1	1	yes	characteristic
230	C₄×C₈	C₂×Dic₆₁	1	1	yes	characteristic
231	#41	—	2	2	no	—
232	#41	—	2	2	no	—
233	C₂×Q₈	#11758222	1	1	yes	characteristic
234	C₂×D₄	#11758222	1	1	yes	characteristic
235	C₂×D₄	#11758222	1	2	yes	—
236	C₂²×C₄	—	2	1	no	—
237	C₂²×C₄	—	2	2	no	—
238	C₂²×C₄	C₂²×Dic₆₁	1	2	yes	—
239	C₂²×C₄	C₂³×D₆₁	1	1	yes	characteristic
240	M₁₆	—	2	2	no	—
241	M₁₆	—	2	2	no	—
242	M₁₆	—	2	4	no	—
243	M₁₆	—	2	4	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	—	2	2	no	—
249	C₂×C₈	—	2	2	no	—
250	C₂×C₈	—	122	4	no	—
251	C₂×C₈	#11758222	1	2	yes	—
252	C₂×C₈	#11758222	1	2	yes	—
253	C₂×C₈	#11758222	1	2	yes	—
254	C₂×C₈	#11758222	1	2	yes	—
255	C₂×C₈	C₂²×Dic₆₁	1	2	yes	—
256	C₂×C₈	C₂²×Dic₆₁	1	2	yes	—
257	C₂×C₈	C₂²×Dic₆₁	1	1	yes	characteristic
258	C₂×C₈	C₂²×Dic₆₁	1	1	yes	characteristic
259	C₂×C₈	C₂²×Dic₆₁	1	1	yes	characteristic
260	C₂×C₈	C₂²×Dic₆₁	1	1	yes	characteristic
261	C₄⋊C₄	—	2	1	no	—
262	C₄⋊C₄	—	2	2	no	—
263	C₄⋊C₄	—	122	4	no	—
264	C₂²⋊C₄	—	2	1	no	—
265	C₂²⋊C₄	—	2	2	no	—
266	C₂²⋊C₄	—	122	4	no	—
267	C₄²	—	2	1	no	—
268	C₄²	—	2	2	no	—
269	C₄²	—	122	4	no	—
270	C₄²	C₂²×Dic₆₁	1	1	yes	characteristic
271	C₄²	C₂²×Dic₆₁	1	2	yes	—
272	C₄²	C₂²×Dic₆₁	1	1	yes	characteristic
273	C₂³	#14519736	1	2	yes	—
274	C₂³	C₂×#11758222	1	1	yes	characteristic
275	Q₈	—	2	2	no	—
276	D₄	—	2	4	no	—
277	D₄	—	2	2	no	—
278	C₂×C₄	—	2	1	no	—
279	C₂×C₄	—	2	2	no	—
280	C₂×C₄	—	2	2	no	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	2	2	no	—
283	C₂×C₄	—	4	2	no	—
284	C₂×C₄	—	4	4	no	—
285	C₂×C₄	—	244	4	no	—
286	C₂×C₄	#14519736	1	2	yes	—
287	C₂×C₄	C₂³×Dic₆₁	1	1	yes	characteristic
288	C₂×C₄	C₂×#11758222	1	2	yes	—
289	C₂×C₄	C₂×#11758222	1	2	yes	—
290	C₂×C₄	C₂×#11758222	1	1	yes	characteristic
291	C₈	—	2	2	no	—
292	C₈	—	2	2	no	—
293	C₈	—	244	4	no	—
294	C₈	#14519736	1	1	yes	characteristic
295	C₈	#14519736	1	1	yes	characteristic
296	C₈	#14519736	1	1	yes	characteristic
297	C₈	#14519736	1	2	yes	—
298	C₈	#14519736	1	2	yes	—
299	C₈	#14519736	1	1	yes	characteristic
300	C₂²	—	2	2	no	—
301	C₂²	—	4	1	no	—
302	C₂²	—	4	2	no	—
303	C₂²	C₂×#14519736	1	1	yes	characteristic
304	C₂²	#16840296	1	2	yes	—
305	C₄	—	2	2	no	—
306	C₄	—	2	2	no	—
307	C₄	—	4	1	no	—
308	C₄	—	4	2	no	—
309	C₄	—	488	4	no	—
310	C₄	C₂×#14519736	1	1	yes	characteristic
311	C₄	C₂×#14519736	1	1	yes	characteristic
312	C₄	#16840296	1	2	yes	—
313	C₂	—	2	1	no	—
314	C₂	—	2	2	no	—
315	C₂	—	4	2	no	—
316	C₂	#21448217	1	1	yes	characteristic

	subgroup	G/H
center	C₈	#14519736
commutator	C₂×C₄×C₆₁	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₆₁
Fitting	C₄×#131×C₆₁	C₂
socle	C₂×C₆₁	#2283
prime radical	C₂×#41×C₆₁	C₂³
cosocle	C₂²×C₄×C₆₁	C₂⁴
hypercenter	C₄×#131	D₆₁
hypocenter	C₆₁	#20787
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₈ ◁ #405 ◁ C₄×#131

Complements

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

N	H ≅ G/N	type
1. #21445968	C₂	semidirect
2. #21445943	C₂	semidirect
6. #21445698	C₂	semidirect
8. #21445574	C₂	semidirect
23. C₆₁×#405	C₄	semidirect
24. C₆₁×#405	C₄	semidirect
27. C₄×M₁₆×C₆₁	C₄	semidirect
29. C₂×C₄×C₈×C₆₁	C₄	semidirect
176. #405	Dic₆₁	semidirect
177. #405	Dic₆₁	semidirect
180. C₄×M₁₆	Dic₆₁	semidirect
182. C₂×C₄×C₈	Dic₆₁	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
#21445968	2
#21445943	2
#21445923	2
#21445869	2
#21445865	2
#21445698	2
#21445619	2
#21445574	2
C₄×#131×C₆₁	2
#20787	61