Subgroups — #29680388

Order 15616.

608 conjugacy classes of subgroups.

The subgroup lattice has 310 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	#21445703	C₂	1	2	yes	maximal, complemented
7	#21445619	C₂	1	2	yes	maximal
8	#21445574	C₂	1	2	yes	maximal, complemented
9	C₆₁×#2875	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×#97×C₆₁	C₄	1	2	yes	complemented
30	C₂×#97×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	4	no	—
44	#131×C₆₁	—	2	2	no	—
45	#131×C₆₁	—	2	2	no	—
46	C₂×M₁₆×C₆₁	C₂×C₄	1	2	yes	—
47	C₂×M₁₆×C₆₁	C₂×C₄	1	2	yes	—
48	C₂×M₁₆×C₆₁	C₂³	1	1	yes	characteristic
49	C₂×M₁₆×C₆₁	C₂³	1	1	yes	characteristic
50	C₂²×C₈×C₆₁	C₂×C₄	1	2	yes	—
51	C₂²×C₈×C₆₁	C₂×C₄	1	2	yes	—
52	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
53	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
54	C₄×Q₈×C₆₁	—	2	1	no	—
55	C₄×D₄×C₆₁	—	2	2	no	—
56	C₄×D₄×C₆₁	—	2	1	no	—
57	#117×C₆₁	C₂×C₄	1	2	yes	—
58	#117×C₆₁	C₂³	1	1	yes	characteristic
59	C₂×C₄²×C₆₁	C₂×C₄	1	2	yes	—
60	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
61	#97×C₆₁	C₂×C₄	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	#97×C₆₁	C₂×C₄	1	1	yes	characteristic
66	#97×C₆₁	C₂×C₄	1	1	yes	characteristic
67	#97×C₆₁	C₂×C₄	1	1	yes	characteristic
68	C₄×C₈×C₆₁	C₂×C₄	1	2	yes	—
69	C₄×C₈×C₆₁	C₂×C₄	1	2	yes	—
70	C₄×C₈×C₆₁	C₂×C₄	1	2	yes	—
71	#41×C₆₁	—	2	2	no	—
72	#41×C₆₁	—	2	2	no	—
73	C₂×Q₈×C₆₁	#41	1	1	yes	characteristic
74	C₂×D₄×C₆₁	#41	1	2	yes	—
75	C₂×D₄×C₆₁	#41	1	1	yes	characteristic
76	C₂²×C₄×C₆₁	—	2	2	no	—
77	C₂²×C₄×C₆₁	—	2	1	no	—
78	C₂²×C₄×C₆₁	C₂²×C₄	1	2	yes	—
79	C₂²×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
80	M₁₆×C₆₁	—	2	2	no	—
81	M₁₆×C₆₁	—	2	2	no	—
82	M₁₆×C₆₁	—	2	4	no	—
83	M₁₆×C₆₁	—	2	4	no	—
84	C₂×C₈×C₆₁	—	2	2	no	—
85	C₂×C₈×C₆₁	—	2	2	no	—
86	C₂×C₈×C₆₁	—	2	4	no	—
87	C₂×C₈×C₆₁	C₂²×C₄	1	2	yes	—
88	C₂×C₈×C₆₁	C₂²×C₄	1	2	yes	—
89	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
90	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
91	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
92	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
93	C₂×C₈×C₆₁	#41	1	4	yes	—
94	C₂×C₈×C₆₁	#41	1	2	yes	—
95	C₂×C₈×C₆₁	#41	1	2	yes	—
96	C₄⋊C₄×C₆₁	—	2	2	no	—
97	C₄⋊C₄×C₆₁	—	2	1	no	—
98	C₂²⋊C₄×C₆₁	—	2	2	no	—
99	C₂²⋊C₄×C₆₁	—	2	1	no	—
100	C₄²×C₆₁	—	2	2	no	—
101	C₄²×C₆₁	—	2	1	no	—
102	C₄²×C₆₁	C₂²×C₄	1	2	yes	—
103	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
104	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
105	C₂²⋊Dic₆₁	—	2	4	no	—
106	C₄⋊Dic₆₁	—	2	4	no	—
107	C₄×Dic₆₁	—	2	4	no	—
108	#11758194	—	2	4	no	—
109	#11758194	—	2	4	no	—
110	C₂×#92418	—	2	4	no	—
111	C₂³×C₆₁	#117	1	2	yes	—
112	C₂³×C₆₁	C₂×#41	1	1	yes	characteristic
113	Q₈×C₆₁	—	2	2	no	—
114	D₄×C₆₁	—	2	4	no	—
115	D₄×C₆₁	—	2	2	no	—
116	C₂×C₄×C₆₁	—	2	2	no	—
117	C₂×C₄×C₆₁	—	2	2	no	—
118	C₂×C₄×C₆₁	—	2	1	no	—
119	C₂×C₄×C₆₁	—	2	2	no	—
120	C₂×C₄×C₆₁	—	2	1	no	—
121	C₂×C₄×C₆₁	—	8	1	no	—
122	C₂×C₄×C₆₁	—	8	2	no	—
123	C₂×C₄×C₆₁	#117	1	2	yes	—
124	C₂×C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
125	C₂×C₄×C₆₁	C₂×#41	1	2	yes	—
126	C₂×C₄×C₆₁	C₂×#41	1	2	yes	—
127	C₂×C₄×C₆₁	C₂×#41	1	1	yes	characteristic
128	C₂×Dic₆₁	—	4	4	no	—
129	C₈×C₆₁	—	2	2	no	—
130	C₈×C₆₁	—	2	2	no	—
131	C₈×C₆₁	#117	1	1	yes	characteristic
132	C₈×C₆₁	#117	1	1	yes	characteristic
133	C₈×C₆₁	#117	1	2	yes	—
134	C₈×C₆₁	#117	1	2	yes	—
135	C₈×C₆₁	#117	1	2	yes	—
136	#92418	—	4	4	no	—
137	#20788	—	61	1	no	maximal, Sylow
138	C₂²×C₆₁	—	2	2	no	—
139	C₂²×C₆₁	—	4	1	no	—
140	C₂²×C₆₁	—	4	2	no	—
141	C₂²×C₆₁	C₂×#117	1	1	yes	characteristic
142	C₂²×C₆₁	#529	1	2	yes	—
143	C₄×C₆₁	—	2	2	no	—
144	C₄×C₆₁	—	4	2	no	—
145	C₄×C₆₁	—	4	1	no	—
146	C₄×C₆₁	—	4	1	no	—
147	C₄×C₆₁	C₂×#117	1	1	yes	characteristic
148	C₄×C₆₁	C₂×#117	1	1	yes	characteristic
149	C₄×C₆₁	#529	1	2	yes	—
150	Dic₆₁	—	8	4	no	—
151	#2875	D₆₁	1	1	yes	characteristic
152	#1841	—	61	2	no	—
153	#1833	—	61	2	no	—
154	#1827	—	61	2	no	—
155	#1813	—	61	1	no	—
156	#1812	—	61	1	no	—
157	#1773	—	61	2	no	—
158	#1757	—	61	2	no	—
159	#1747	—	61	2	no	—
160	C₂×C₆₁	—	2	2	no	—
161	C₂×C₆₁	—	2	1	no	—
162	C₂×C₆₁	—	4	2	no	—
163	C₂×C₆₁	#2283	1	1	yes	characteristic
164	C₂×#131	C₂×D₆₁	1	1	yes	characteristic
165	C₂×#131	C₂×D₆₁	1	1	yes	characteristic
166	C₄×#41	C₂×D₆₁	1	1	yes	characteristic
167	C₂×#108	—	61	2	no	—
168	#427	—	61	2	no	—
169	#413	—	61	2	no	—
170	#410	—	61	2	no	—
171	#405	—	61	4	no	—
172	#405	Dic₆₁	1	2	yes	complemented
173	#405	Dic₆₁	1	2	yes	complemented
174	#405	C₂×D₆₁	1	1	yes	characteristic
175	#405	C₂×D₆₁	1	1	yes	characteristic
176	C₄×M₁₆	Dic₆₁	1	2	yes	complemented
177	C₄×M₁₆	C₂×D₆₁	1	1	yes	characteristic
178	C₂×#97	Dic₆₁	1	2	yes	complemented
179	C₂×#97	C₂×D₆₁	1	1	yes	characteristic
180	#344	—	61	4	no	—
181	#339	—	61	4	no	—
182	#338	—	61	2	no	—
183	H(3,ℤ/4)	—	61	2	no	—
184	H(3,ℤ/4)	—	61	4	no	—
185	C₆₁	#20788	1	1	yes	characteristic, complemented, Sylow
186	C₂×#41	C₂²×D₆₁	1	1	yes	characteristic
187	#131	—	2	4	no	—
188	#131	—	2	2	no	—
189	#131	—	2	2	no	—
190	C₂×M₁₆	—	61	4	no	—
191	C₂×M₁₆	C₂×Dic₆₁	1	2	yes	—
192	C₂×M₁₆	C₂×Dic₆₁	1	2	yes	—
193	C₂×M₁₆	C₂²×D₆₁	1	1	yes	characteristic
194	C₂×M₁₆	C₂²×D₆₁	1	1	yes	characteristic
195	C₂²×C₈	C₂×Dic₆₁	1	2	yes	—
196	C₂²×C₈	C₂×Dic₆₁	1	2	yes	—
197	C₂²×C₈	C₂²×D₆₁	1	1	yes	characteristic
198	C₂²×C₈	C₂²×D₆₁	1	1	yes	characteristic
199	C₄×Q₈	—	2	1	no	—
200	C₄×D₄	—	2	1	no	—
201	C₄×D₄	—	2	2	no	—
202	#117	—	61	4	no	—
203	#117	C₂×Dic₆₁	1	2	yes	—
204	#117	C₂²×D₆₁	1	1	yes	characteristic
205	C₂×C₄²	C₂×Dic₆₁	1	2	yes	—
206	C₂×C₄²	C₂²×D₆₁	1	1	yes	characteristic
207	#108	—	122	2	no	—
208	#108	—	122	2	no	—
209	#107	—	122	2	no	—
210	#106	—	122	2	no	—
211	#101	—	122	2	no	—
212	#100	—	122	2	no	—
213	#99	—	122	4	no	—
214	#97	—	122	4	no	—
215	#97	C₂×Dic₆₁	1	1	yes	characteristic
216	#97	C₂×Dic₆₁	1	2	yes	—
217	#97	C₂×Dic₆₁	1	2	yes	—
218	#97	C₂×Dic₆₁	1	2	yes	—
219	#97	C₂×Dic₆₁	1	1	yes	characteristic
220	#97	C₂×Dic₆₁	1	1	yes	characteristic
221	#97	C₂×Dic₆₁	1	1	yes	characteristic
222	C₄×C₈	—	122	4	no	—
223	C₄×C₈	C₂×Dic₆₁	1	2	yes	—
224	C₄×C₈	C₂×Dic₆₁	1	2	yes	—
225	C₄×C₈	C₂×Dic₆₁	1	2	yes	—
226	#41	—	2	2	no	—
227	#41	—	2	2	no	—
228	C₂×Q₈	#11758222	1	1	yes	characteristic
229	C₂×D₄	#11758222	1	1	yes	characteristic
230	C₂×D₄	#11758222	1	2	yes	—
231	C₂²×C₄	—	2	1	no	—
232	C₂²×C₄	—	2	2	no	—
233	C₂²×C₄	C₂²×Dic₆₁	1	2	yes	—
234	C₂²×C₄	C₂³×D₆₁	1	1	yes	characteristic
235	M₁₆	—	2	2	no	—
236	M₁₆	—	2	2	no	—
237	M₁₆	—	2	4	no	—
238	M₁₆	—	2	4	no	—
239	M₁₆	—	122	4	no	—
240	M₁₆	—	122	4	no	—
241	C₂×C₈	—	2	2	no	—
242	C₂×C₈	—	2	2	no	—
243	C₂×C₈	—	2	4	no	—
244	C₂×C₈	—	122	4	no	—
245	C₂×C₈	#11758222	1	4	yes	—
246	C₂×C₈	#11758222	1	2	yes	—
247	C₂×C₈	#11758222	1	2	yes	—
248	C₂×C₈	C₂²×Dic₆₁	1	2	yes	—
249	C₂×C₈	C₂²×Dic₆₁	1	2	yes	—
250	C₂×C₈	C₂²×Dic₆₁	1	1	yes	characteristic
251	C₂×C₈	C₂²×Dic₆₁	1	1	yes	characteristic
252	C₂×C₈	C₂²×Dic₆₁	1	1	yes	characteristic
253	C₂×C₈	C₂²×Dic₆₁	1	1	yes	characteristic
254	C₄⋊C₄	—	2	1	no	—
255	C₄⋊C₄	—	2	2	no	—
256	C₄⋊C₄	—	122	4	no	—
257	C₂²⋊C₄	—	2	1	no	—
258	C₂²⋊C₄	—	2	2	no	—
259	C₂²⋊C₄	—	122	4	no	—
260	C₄²	—	2	1	no	—
261	C₄²	—	2	2	no	—
262	C₄²	—	122	4	no	—
263	C₄²	C₂²×Dic₆₁	1	2	yes	—
264	C₄²	C₂²×Dic₆₁	1	1	yes	characteristic
265	C₄²	C₂²×Dic₆₁	1	1	yes	characteristic
266	C₂³	#14519736	1	2	yes	—
267	C₂³	C₂×#11758222	1	1	yes	characteristic
268	Q₈	—	2	2	no	—
269	D₄	—	2	4	no	—
270	D₄	—	2	2	no	—
271	C₂×C₄	—	2	2	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	2	1	no	—
274	C₂×C₄	—	2	1	no	—
275	C₂×C₄	—	2	2	no	—
276	C₂×C₄	—	8	1	no	—
277	C₂×C₄	—	8	2	no	—
278	C₂×C₄	—	244	4	no	—
279	C₂×C₄	#14519736	1	2	yes	—
280	C₂×C₄	C₂³×Dic₆₁	1	1	yes	characteristic
281	C₂×C₄	C₂×#11758222	1	2	yes	—
282	C₂×C₄	C₂×#11758222	1	2	yes	—
283	C₂×C₄	C₂×#11758222	1	1	yes	characteristic
284	C₈	—	2	2	no	—
285	C₈	—	2	2	no	—
286	C₈	—	244	4	no	—
287	C₈	#14519736	1	1	yes	characteristic
288	C₈	#14519736	1	1	yes	characteristic
289	C₈	#14519736	1	2	yes	—
290	C₈	#14519736	1	2	yes	—
291	C₈	#14519736	1	2	yes	—
292	C₂²	—	2	2	no	—
293	C₂²	—	4	1	no	—
294	C₂²	—	4	2	no	—
295	C₂²	C₂×#14519736	1	1	yes	characteristic
296	C₂²	#16840296	1	2	yes	—
297	C₄	—	2	2	no	—
298	C₄	—	4	1	no	—
299	C₄	—	4	2	no	—
300	C₄	—	4	1	no	—
301	C₄	—	488	4	no	—
302	C₄	C₂×#14519736	1	1	yes	characteristic
303	C₄	C₂×#14519736	1	1	yes	characteristic
304	C₄	#16840296	1	2	yes	—
305	C₂	—	2	1	no	—
306	C₂	—	2	2	no	—
307	C₂	—	4	2	no	—
308	C₂	#21448217	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×#14519736
commutator	C₂×C₄×C₆₁	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₆₁
Fitting	C₆₁×#2875	C₂
socle	C₂×C₆₁	#2283
prime radical	C₂×#41×C₆₁	C₂³
cosocle	C₂²×C₄×C₆₁	C₂⁴
hypercenter	#2875	D₆₁
hypocenter	C₆₁	#20788
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₄ ◁ #405 ◁ #2875

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. #21445703	C₂	semidirect
8. #21445574	C₂	semidirect
23. C₆₁×#405	C₄	semidirect
24. C₆₁×#405	C₄	semidirect
27. C₄×M₁₆×C₆₁	C₄	semidirect
29. C₂×#97×C₆₁	C₄	semidirect
172. #405	Dic₆₁	semidirect
173. #405	Dic₆₁	semidirect
176. C₄×M₁₆	Dic₆₁	semidirect
178. C₂×#97	Dic₆₁	semidirect
185. C₆₁	#20788	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
#21445703	2
#21445619	2
#21445574	2
C₆₁×#2875	2
#20788	61