Subgroups — #29588440

Order 15616.

622 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21454409	C₂	1	1	yes	maximal, characteristic
2	#21445557	C₂	1	2	yes	maximal
3	C₂×#16839522	C₂	1	2	yes	maximal
4	C₂×#16839520	C₂	1	1	yes	maximal, characteristic
5	C₆₁×#1728	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11758194	C₂²	1	2	yes	—
7	C₂×#14519736	C₂²	1	2	yes	—
8	#16839804	—	2	4	no	—
9	#16839800	—	2	2	no	—
10	#16839530	—	2	4	no	—
11	#16839522	—	2	4	no	—
12	#16839520	C₄	1	4	yes	complemented
13	C₂²×M₁₆×C₆₁	C₂²	1	1	yes	characteristic
14	C₂²×M₁₆×C₆₁	C₂²	1	1	yes	characteristic
15	C₂²×C₄²×C₆₁	C₂²	1	1	yes	characteristic
16	C₆₁×#336	—	2	2	no	—
17	C₂×#11758194	—	2	2	no	—
18	C₂×#11758194	—	2	2	no	—
19	C₂×#11758194	—	2	2	no	—
20	C₂×#11758194	—	2	2	no	—
21	C₂×#11758194	—	2	2	no	—
22	C₂×#11758194	C₂×C₄	1	4	yes	—
23	C₂²×#92418	—	2	2	no	—
24	#14519736	—	2	2	no	—
25	#14519736	—	2	2	no	—
26	#14519736	—	2	2	no	—
27	#14519736	—	2	2	no	—
28	C₂×C₄⋊Dic₆₁	D₄	1	4	yes	—
29	C₂×C₂²⋊Dic₆₁	—	2	2	no	—
30	C₂×C₄×Dic₆₁	—	2	2	no	—
31	#14519711	—	2	4	no	—
32	#14519709	—	2	4	no	—
33	C₂³×C₄×C₆₁	—	2	1	no	—
34	C₂³×C₄×C₆₁	C₂³	1	1	yes	characteristic
35	C₂×M₁₆×C₆₁	—	2	2	no	—
36	C₂×M₁₆×C₆₁	—	2	2	no	—
37	C₂×M₁₆×C₆₁	—	2	1	no	—
38	C₂×M₁₆×C₆₁	—	2	2	no	—
39	C₂×M₁₆×C₆₁	—	2	2	no	—
40	C₂×M₁₆×C₆₁	C₂×C₄	1	2	yes	—
41	C₂×M₁₆×C₆₁	Q₈	1	2	yes	—
42	C₂×M₁₆×C₆₁	Q₈	1	2	yes	—
43	C₂²×C₈×C₆₁	—	2	1	no	—
44	C₂²×C₈×C₆₁	D₄	1	2	yes	—
45	C₂×C₄²×C₆₁	—	2	2	no	—
46	C₂×C₄²×C₆₁	—	2	2	no	—
47	C₂×C₄²×C₆₁	—	2	1	no	—
48	C₂×C₄²×C₆₁	D₄	1	2	yes	—
49	C₂⁴×C₆₁	#41	1	1	yes	characteristic
50	C₂²×C₄×C₆₁	—	2	2	no	—
51	C₂²×C₄×C₆₁	—	4	4	no	—
52	C₂²×C₄×C₆₁	—	4	2	no	—
53	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
54	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
55	C₂²×C₄×C₆₁	C₂×D₄	1	2	yes	—
56	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
57	C₂²×C₄×C₆₁	C₂×Q₈	1	1	yes	characteristic
58	C₂²×C₄×C₆₁	C₂×Q₈	1	1	yes	characteristic
59	C₂²×C₄×C₆₁	#41	1	2	yes	—
60	C₂²×C₄×C₆₁	#41	1	2	yes	—
61	C₂²×C₄×C₆₁	#41	1	2	yes	—
62	C₂²×C₄×C₆₁	#41	1	1	yes	characteristic
63	C₂²×Dic₆₁	—	2	4	no	—
64	M₁₆×C₆₁	—	2	4	no	—
65	M₁₆×C₆₁	—	2	4	no	—
66	M₁₆×C₆₁	—	2	4	no	—
67	M₁₆×C₆₁	—	4	2	no	—
68	C₂×C₈×C₆₁	—	2	2	no	—
69	C₂×C₈×C₆₁	—	2	2	no	—
70	C₂×C₈×C₆₁	—	2	2	no	—
71	C₂×C₈×C₆₁	—	4	1	no	—
72	C₂×C₈×C₆₁	—	4	1	no	—
73	C₂×C₈×C₆₁	C₄⋊C₄	1	4	yes	—
74	C₄²×C₆₁	—	2	4	no	—
75	C₄²×C₆₁	—	4	1	no	—
76	C₄²×C₆₁	—	4	1	no	—
77	C₂²⋊Dic₆₁	—	4	2	no	—
78	C₂²⋊Dic₆₁	—	4	2	no	—
79	C₄⋊Dic₆₁	—	2	4	no	—
80	C₄⋊Dic₆₁	—	2	4	no	—
81	C₄×Dic₆₁	—	4	2	no	—
82	C₄×Dic₆₁	—	4	2	no	—
83	#11758194	—	2	4	no	—
84	#11758194	—	2	4	no	—
85	#11758194	—	4	2	no	—
86	#11758194	—	4	2	no	—
87	C₂×#92418	—	2	4	no	—
88	C₂×#92418	—	4	2	no	—
89	C₂×#92418	—	4	2	no	—
90	C₂³×C₆₁	—	4	2	no	—
91	C₂³×C₆₁	C₄×Q₈	1	1	yes	characteristic
92	C₂³×C₆₁	#121	1	1	yes	characteristic
93	C₂³×C₆₁	#123	1	2	yes	—
94	C₂³×C₆₁	#125	1	1	yes	characteristic
95	C₂³×C₆₁	#126	1	2	yes	—
96	C₂×C₄×C₆₁	—	2	4	no	—
97	C₂×C₄×C₆₁	—	4	2	no	—
98	C₂×C₄×C₆₁	—	4	8	no	—
99	C₂×C₄×C₆₁	—	4	2	no	—
100	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
101	C₂×C₄×C₆₁	#117	1	2	yes	—
102	C₂×C₄×C₆₁	C₄×D₄	1	2	yes	—
103	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
104	C₂×C₄×C₆₁	#120	1	1	yes	characteristic
105	C₂×C₄×C₆₁	#121	1	2	yes	—
106	C₂×C₄×C₆₁	C₂²⋊Q₈	1	2	yes	—
107	C₂×C₄×C₆₁	C₂²⋊Q₈	1	2	yes	—
108	C₂×C₄×C₆₁	C₂²⋊Q₈	1	1	yes	characteristic
109	C₂×C₄×C₆₁	#123	1	1	yes	characteristic
110	C₂×C₄×C₆₁	#123	1	2	yes	—
111	C₂×C₄×C₆₁	#124	1	1	yes	characteristic
112	C₂×C₄×C₆₁	#124	1	1	yes	characteristic
113	C₂×C₄×C₆₁	#124	1	2	yes	—
114	C₂×C₄×C₆₁	#125	1	2	yes	—
115	C₂×C₄×C₆₁	#125	1	2	yes	—
116	C₂×C₄×C₆₁	#126	1	2	yes	—
117	C₂×C₄×C₆₁	C₄⋊Q₈	1	1	yes	characteristic
118	C₂×Dic₆₁	—	4	4	no	—
119	C₂×Dic₆₁	—	8	4	no	—
120	C₈×C₆₁	—	2	4	no	—
121	C₈×C₆₁	—	4	2	no	—
122	#92418	—	4	4	no	—
123	#11294	—	61	1	no	maximal, Sylow
124	C₂²×C₆₁	—	2	4	no	—
125	C₂²×C₆₁	—	2	2	no	—
126	C₂²×C₆₁	—	4	2	no	—
127	C₂²×C₆₁	—	4	1	no	—
128	C₂²×C₆₁	—	4	1	no	—
129	C₂²×C₆₁	#387	1	2	yes	—
130	C₂²×C₆₁	#389	1	1	yes	characteristic
131	C₂²×C₆₁	#394	1	1	yes	characteristic
132	C₂²×C₆₁	#397	1	1	yes	characteristic
133	C₂²×C₆₁	#400	1	2	yes	—
134	C₄×C₆₁	—	4	4	no	—
135	C₄×C₆₁	#382	1	1	yes	characteristic
136	C₄×C₆₁	#385	1	1	yes	characteristic
137	C₄×C₆₁	#388	1	2	yes	—
138	C₄×C₆₁	#393	1	1	yes	characteristic
139	C₄×C₆₁	#398	1	2	yes	—
140	C₄×C₆₁	#400	1	1	yes	characteristic
141	Dic₆₁	—	8	4	no	—
142	#2911	—	61	1	no	—
143	#1742	—	61	2	no	—
144	C₂×#339	—	61	2	no	—
145	C₂×#338	—	61	1	no	—
146	#1728	D₆₁	1	1	yes	characteristic
147	C₂×C₆₁	—	2	1	no	—
148	C₂×C₆₁	—	2	2	no	—
149	C₂×C₆₁	—	2	1	no	—
150	C₂×C₆₁	—	2	2	no	—
151	C₂×C₆₁	#1442	1	1	yes	characteristic
152	C₂×C₆₁	#2050	1	2	yes	—
153	C₂²×M₁₆	—	61	2	no	—
154	C₂²×M₁₆	C₂×D₆₁	1	1	yes	characteristic
155	C₂²×M₁₆	C₂×D₆₁	1	1	yes	characteristic
156	C₂×#117	—	61	2	no	—
157	C₂²×C₄²	C₂×D₆₁	1	1	yes	characteristic
158	#428	—	122	4	no	—
159	#427	—	122	2	no	—
160	#343	—	122	4	no	—
161	#339	—	122	4	no	—
162	#338	—	61	4	no	—
163	#336	—	2	2	no	—
164	C₆₁	#11294	1	1	yes	characteristic, complemented, Sylow
165	C₂³×C₄	—	2	1	no	—
166	C₂³×C₄	C₂²×D₆₁	1	1	yes	characteristic
167	C₂×M₁₆	—	2	2	no	—
168	C₂×M₁₆	—	2	2	no	—
169	C₂×M₁₆	—	2	1	no	—
170	C₂×M₁₆	—	2	2	no	—
171	C₂×M₁₆	—	2	2	no	—
172	C₂×M₁₆	—	61	4	no	—
173	C₂×M₁₆	—	122	2	no	—
174	C₂×M₁₆	—	122	2	no	—
175	C₂×M₁₆	—	122	2	no	—
176	C₂×M₁₆	—	122	2	no	—
177	C₂×M₁₆	—	122	2	no	—
178	C₂×M₁₆	Dic₁₂₂	1	2	yes	—
179	C₂×M₁₆	Dic₁₂₂	1	2	yes	—
180	C₂×M₁₆	C₄×D₆₁	1	2	yes	—
181	C₂²×C₈	—	2	1	no	—
182	C₂²×C₈	—	122	2	no	—
183	C₂²×C₈	C₂²⋊D₆₁	1	2	yes	—
184	#117	—	122	2	no	—
185	#117	—	122	2	no	—
186	#117	—	122	2	no	—
187	#117	—	122	2	no	—
188	C₂×C₄⋊C₄	—	61	4	no	—
189	C₂×C₂²⋊C₄	—	122	2	no	—
190	C₂×C₄²	—	2	2	no	—
191	C₂×C₄²	—	2	2	no	—
192	C₂×C₄²	—	2	1	no	—
193	C₂×C₄²	—	122	2	no	—
194	C₂×C₄²	D₂₄₄	1	2	yes	—
195	#107	—	122	4	no	—
196	#106	—	122	4	no	—
197	C₂⁴	#11758222	1	1	yes	characteristic
198	C₂²×C₄	—	2	2	no	—
199	C₂²×C₄	—	4	4	no	—
200	C₂²×C₄	—	4	2	no	—
201	C₂²×C₄	—	122	4	no	—
202	C₂²×C₄	C₂×Dic₁₂₂	1	1	yes	characteristic
203	C₂²×C₄	C₂×Dic₁₂₂	1	1	yes	characteristic
204	C₂²×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
205	C₂²×C₄	C₂×D₂₄₄	1	1	yes	characteristic
206	C₂²×C₄	#11758222	1	1	yes	characteristic
207	C₂²×C₄	D₄×D₆₁	1	2	yes	—
208	C₂²×C₄	#11758224	1	2	yes	—
209	C₂²×C₄	#11758224	1	2	yes	—
210	C₂²×C₄	#11758226	1	2	yes	—
211	C₂²×C₄	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
212	M₁₆	—	2	4	no	—
213	M₁₆	—	2	4	no	—
214	M₁₆	—	2	4	no	—
215	M₁₆	—	4	2	no	—
216	M₁₆	—	122	4	no	—
217	M₁₆	—	122	4	no	—
218	M₁₆	—	244	2	no	—
219	M₁₆	—	244	2	no	—
220	C₂×C₈	—	2	2	no	—
221	C₂×C₈	—	2	2	no	—
222	C₂×C₈	—	2	2	no	—
223	C₂×C₈	—	4	1	no	—
224	C₂×C₈	—	4	1	no	—
225	C₂×C₈	—	122	4	no	—
226	C₂×C₈	—	244	2	no	—
227	C₂×C₈	—	244	2	no	—
228	C₂×C₈	#11758196	1	4	yes	—
229	C₄⋊C₄	—	122	4	no	—
230	C₄⋊C₄	—	122	4	no	—
231	C₂²⋊C₄	—	244	2	no	—
232	C₂²⋊C₄	—	244	2	no	—
233	C₄²	—	2	4	no	—
234	C₄²	—	4	1	no	—
235	C₄²	—	4	1	no	—
236	C₄²	—	244	2	no	—
237	C₄²	—	244	2	no	—
238	C₂³	—	4	2	no	—
239	C₂³	C₄×Dic₁₂₂	1	1	yes	characteristic
240	C₂³	#14519751	1	1	yes	characteristic
241	C₂³	#14519762	1	2	yes	—
242	C₂³	#14519769	1	1	yes	characteristic
243	C₂³	#14519770	1	2	yes	—
244	C₂×C₄	—	2	4	no	—
245	C₂×C₄	—	4	2	no	—
246	C₂×C₄	—	4	8	no	—
247	C₂×C₄	—	4	2	no	—
248	C₂×C₄	—	488	4	no	—
249	C₂×C₄	—	244	4	no	—
250	C₂×C₄	C₂×#11758196	1	1	yes	characteristic
251	C₂×C₄	#14519735	1	2	yes	—
252	C₂×C₄	#14519737	1	2	yes	—
253	C₂×C₄	C₄×D₂₄₄	1	1	yes	characteristic
254	C₂×C₄	#14519745	1	1	yes	characteristic
255	C₂×C₄	#14519752	1	2	yes	—
256	C₂×C₄	#14519754	1	2	yes	—
257	C₂×C₄	#14519754	1	2	yes	—
258	C₂×C₄	#14519756	1	1	yes	characteristic
259	C₂×C₄	#14519758	1	1	yes	characteristic
260	C₂×C₄	#14519761	1	2	yes	—
261	C₂×C₄	#14519763	1	2	yes	—
262	C₂×C₄	#14519764	1	1	yes	characteristic
263	C₂×C₄	#14519766	1	1	yes	characteristic
264	C₂×C₄	#14519768	1	2	yes	—
265	C₂×C₄	#14519768	1	2	yes	—
266	C₂×C₄	#14519772	1	2	yes	—
267	C₂×C₄	#14519777	1	1	yes	characteristic
268	C₈	—	2	4	no	—
269	C₈	—	4	2	no	—
270	C₈	—	244	4	no	—
271	C₂²	—	2	2	no	—
272	C₂²	—	2	4	no	—
273	C₂²	—	4	2	no	—
274	C₂²	—	4	1	no	—
275	C₂²	—	4	1	no	—
276	C₂²	#16839649	1	2	yes	—
277	C₂²	#16839660	1	1	yes	characteristic
278	C₂²	#16839679	1	1	yes	characteristic
279	C₂²	#16839687	1	1	yes	characteristic
280	C₂²	#16839704	1	2	yes	—
281	C₄	—	4	4	no	—
282	C₄	—	488	4	no	—
283	C₄	#16839627	1	1	yes	characteristic
284	C₄	#16839634	1	1	yes	characteristic
285	C₄	#16839654	1	2	yes	—
286	C₄	#16839674	1	1	yes	characteristic
287	C₄	#16839694	1	2	yes	—
288	C₄	#16839701	1	1	yes	characteristic
289	C₂	—	2	1	no	—
290	C₂	—	2	2	no	—
291	C₂	—	2	2	no	—
292	C₂	—	2	1	no	—
293	C₂	#21444075	1	1	yes	characteristic
294	C₂	#21447106	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#14519766
commutator	C₂²×C₄×C₆₁	C₂²×C₄
Frattini	C₂³×C₄	C₂²×D₆₁
Fitting	C₆₁×#1728	C₂
socle	C₂²×C₆₁	#394
prime radical	C₂⁴×C₆₁	#41
cosocle	C₂³×C₄×C₆₁	C₂³
hypercenter	#1728	D₆₁
hypocenter	C₆₁	#11294
LCS	G ▷ C₂²×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ #1728

Complements

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

N	H ≅ G/N	type
12. #16839520	C₄	semidirect
164. C₆₁	#11294	semidirect

Maximal subgroups

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

subgroup	index
#21454409	2
#21445557	2
C₂×#16839522	2
C₂×#16839520	2
C₆₁×#1728	2
#11294	61