Subgroups — #29570543

Order 15616.

506 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16839718	C₂	1	1	yes	maximal, characteristic
2	#21444133	C₂	1	1	yes	maximal, characteristic
3	#21444103	C₂	1	2	yes	maximal, complemented
4	#21443663	C₂	1	2	yes	maximal, complemented
5	C₆₁×#1458	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11758194	C₄	1	1	yes	characteristic
7	C₂³×#92418	C₄	1	1	yes	characteristic
8	#16839815	C₂²	1	2	yes	—
9	#16839718	—	2	2	no	—
10	#16839718	—	2	2	no	—
11	C₂×#14519688	C₄	1	1	yes	characteristic
12	C₂×#14519688	C₂²	1	1	yes	characteristic
13	C₂×C₄×#92418	C₄	1	1	yes	characteristic
14	C₂×C₄×#92418	C₂²	1	1	yes	characteristic
15	#16839488	C₄	1	2	yes	—
16	#16839486	C₄	1	2	yes	—
17	#16839484	—	2	2	no	—
18	C₈×#92418	—	2	2	no	—
19	C₂×#117×C₆₁	C₂²	1	1	yes	characteristic
20	C₆₁×#431	C₂²	1	2	yes	—
21	C₆₁×#324	C₄	1	2	yes	—
22	C₆₁×#323	C₄	1	2	yes	—
23	C₂×#11758194	—	2	2	no	—
24	C₂×#11758194	—	2	2	no	—
25	C₂×#11758194	C₈	1	4	yes	complemented
26	C₂²×#92418	—	2	1	no	—
27	C₂²×#92418	—	2	1	no	—
28	C₂²×#92418	—	2	2	no	—
29	C₂²×#92418	C₈	1	4	yes	complemented
30	C₂²×#92418	C₂×C₄	1	1	yes	characteristic
31	C₂²×#92418	C₂×C₄	1	1	yes	characteristic
32	C₂²×#92418	C₂×C₄	1	1	yes	characteristic
33	C₂²×#92418	C₂×C₄	1	1	yes	characteristic
34	C₄⋊#92418	—	2	2	no	—
35	C₂²⋊#92418	C₂×C₄	1	4	yes	—
36	#14519688	—	2	2	no	—
37	#14519688	—	2	2	no	—
38	C₄×#92418	—	2	2	no	—
39	C₄×#92418	—	2	1	no	—
40	C₄×#92418	—	2	1	no	—
41	C₄×#92418	D₄	1	2	yes	—
42	C₄×#92418	D₄	1	2	yes	—
43	C₂³×C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
44	#117×C₆₁	—	2	2	no	—
45	#117×C₆₁	—	2	2	no	—
46	C₂×C₄⋊C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄⋊C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
48	C₂×C₂²⋊C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
49	C₂×C₂²⋊C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
50	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
51	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
52	C₄⋊C₈×C₆₁	—	2	2	no	—
53	C₂²⋊C₈×C₆₁	C₂×C₄	1	4	yes	—
54	C₄×C₈×C₆₁	—	2	2	no	—
55	C₂⁴×C₆₁	C₄²	1	1	yes	characteristic
56	C₂²×C₄×C₆₁	—	2	1	no	—
57	C₂²×C₄×C₆₁	—	2	1	no	—
58	C₂²×C₄×C₆₁	—	2	1	no	—
59	C₂²×C₄×C₆₁	—	2	1	no	—
60	C₂²×C₄×C₆₁	C₄²	1	1	yes	characteristic
61	C₂²×C₄×C₆₁	C₄²	1	1	yes	characteristic
62	C₂²×C₄×C₆₁	C₄²	1	1	yes	characteristic
63	C₂²×C₄×C₆₁	C₂×C₈	1	2	yes	—
64	C₂²×C₄×C₆₁	C₂×C₈	1	2	yes	—
65	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
66	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
67	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
68	C₂×C₈×C₆₁	—	2	4	no	—
69	C₄⋊C₄×C₆₁	—	2	2	no	—
70	C₄⋊C₄×C₆₁	—	2	2	no	—
71	C₂²⋊C₄×C₆₁	—	2	2	no	—
72	C₂²⋊C₄×C₆₁	—	2	2	no	—
73	C₄²×C₆₁	—	2	1	no	—
74	C₄²×C₆₁	—	2	1	no	—
75	C₄²×C₆₁	C₂×D₄	1	2	yes	—
76	C₄²×C₆₁	#41	1	2	yes	—
77	#11758194	—	2	8	no	—
78	C₂×#92418	—	2	1	no	—
79	C₂×#92418	—	2	1	no	—
80	C₂×#92418	—	2	1	no	—
81	C₂×#92418	—	2	1	no	—
82	C₂×#92418	—	2	4	no	—
83	C₂×#92418	—	2	4	no	—
84	C₂×#92418	—	2	4	no	—
85	C₂×#92418	C₂²⋊C₄	1	1	yes	characteristic
86	C₂×#92418	C₂²⋊C₄	1	1	yes	characteristic
87	C₂×#92418	C₂²⋊C₄	1	2	yes	—
88	C₂×#92418	C₂×C₈	1	2	yes	—
89	C₂×#92418	C₂×C₈	1	2	yes	—
90	C₂×#92418	C₂×C₈	1	2	yes	—
91	C₂×#92418	C₂×C₈	1	2	yes	—
92	C₂×#92418	M₁₆	1	2	yes	—
93	C₂×#92418	M₁₆	1	2	yes	—
94	C₂³×C₆₁	—	2	1	no	—
95	C₂³×C₆₁	—	2	1	no	—
96	C₂³×C₆₁	—	4	1	no	—
97	C₂³×C₆₁	—	4	1	no	—
98	C₂³×C₆₁	C₄×C₈	1	2	yes	—
99	C₂³×C₆₁	C₂×C₄²	1	1	yes	characteristic
100	C₂×C₄×C₆₁	—	2	1	no	—
101	C₂×C₄×C₆₁	—	2	1	no	—
102	C₂×C₄×C₆₁	—	2	2	no	—
103	C₂×C₄×C₆₁	—	2	2	no	—
104	C₂×C₄×C₆₁	—	2	2	no	—
105	C₂×C₄×C₆₁	—	2	2	no	—
106	C₂×C₄×C₆₁	—	2	1	no	—
107	C₂×C₄×C₆₁	—	2	1	no	—
108	C₂×C₄×C₆₁	—	4	1	no	—
109	C₂×C₄×C₆₁	—	4	1	no	—
110	C₂×C₄×C₆₁	—	4	1	no	—
111	C₂×C₄×C₆₁	—	4	1	no	—
112	C₂×C₄×C₆₁	C₄×C₈	1	2	yes	—
113	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
114	C₂×C₄×C₆₁	#117	1	1	yes	characteristic
115	C₂×C₄×C₆₁	C₄×D₄	1	2	yes	—
116	C₂×C₄×C₆₁	C₄×D₄	1	2	yes	—
117	C₂×C₄×C₆₁	C₂²×C₈	1	1	yes	characteristic
118	C₂×C₄×C₆₁	C₂²×C₈	1	1	yes	characteristic
119	C₂×C₄×C₆₁	C₂×M₁₆	1	1	yes	characteristic
120	C₂×C₄×C₆₁	C₂×M₁₆	1	1	yes	characteristic
121	C₈×C₆₁	—	4	4	no	—
122	#92418	—	2	2	no	—
123	#92418	—	2	2	no	—
124	#92418	—	4	4	no	—
125	#92418	C₂²⋊C₈	1	2	yes	—
126	#92418	C₂²⋊C₈	1	2	yes	—
127	#92418	C₂²⋊C₈	1	4	yes	—
128	#8156	—	61	1	no	maximal, Sylow
129	C₂²×C₆₁	—	2	1	no	—
130	C₂²×C₆₁	—	2	1	no	—
131	C₂²×C₆₁	—	2	2	no	—
132	C₂²×C₆₁	—	4	2	no	—
133	C₂²×C₆₁	—	4	2	no	—
134	C₂²×C₆₁	—	4	1	no	—
135	C₂²×C₆₁	—	4	1	no	—
136	C₂²×C₆₁	C₄×C₂²⋊C₄	1	1	yes	characteristic
137	C₂²×C₆₁	C₂×C₄×C₈	1	1	yes	characteristic
138	C₂²×C₆₁	C₄×M₁₆	1	1	yes	characteristic
139	C₄×C₆₁	—	2	2	no	—
140	C₄×C₆₁	—	2	2	no	—
141	C₄×C₆₁	—	2	2	no	—
142	C₄×C₆₁	—	4	1	no	—
143	C₄×C₆₁	—	4	1	no	—
144	C₄×C₆₁	C₂×C₂²⋊C₈	1	1	yes	characteristic
145	C₄×C₆₁	C₂×C₂²⋊C₈	1	1	yes	characteristic
146	C₄×C₆₁	#431	1	1	yes	characteristic
147	C₄×C₆₁	#431	1	1	yes	characteristic
148	C₂×#405	—	61	1	no	—
149	#1458	—	61	1	no	—
150	#1458	D₆₁	1	1	yes	characteristic
151	#1450	—	61	2	no	—
152	#1281	—	61	2	no	—
153	C₂×C₆₁	—	2	1	no	—
154	C₂×C₆₁	—	2	1	no	—
155	C₂×C₆₁	—	4	2	no	—
156	C₂×C₆₁	C₄×C₂²⋊C₈	1	1	yes	characteristic
157	C₂×C₆₁	#1757	1	1	yes	characteristic
158	C₂×C₆₁	#1757	1	1	yes	characteristic
159	C₂²×M₁₆	—	61	1	no	—
160	C₂³×C₈	—	61	1	no	—
161	C₂×#117	C₂×D₆₁	1	1	yes	characteristic
162	#431	—	61	2	no	—
163	#431	C₂×D₆₁	1	2	yes	—
164	#405	—	122	2	no	—
165	#405	—	122	2	no	—
166	C₂×#97	—	61	1	no	—
167	C₂×#97	—	61	1	no	—
168	C₂×C₄×C₈	—	61	1	no	—
169	C₂×C₄×C₈	—	61	1	no	—
170	#324	—	61	2	no	—
171	#324	Dic₆₁	1	2	yes	—
172	#323	—	61	2	no	—
173	#323	Dic₆₁	1	2	yes	—
174	#322	—	122	2	no	—
175	C₈²	—	122	2	no	—
176	C₆₁	#8156	1	1	yes	characteristic, complemented, Sylow
177	C₂³×C₄	C₄×D₆₁	1	1	yes	characteristic
178	C₂×M₁₆	—	61	4	no	—
179	C₂×M₁₆	—	122	2	no	—
180	C₂×M₁₆	—	122	2	no	—
181	C₂²×C₈	—	61	1	no	—
182	C₂²×C₈	—	61	1	no	—
183	C₂²×C₈	—	61	4	no	—
184	C₂²×C₈	—	61	1	no	—
185	C₂²×C₈	—	61	1	no	—
186	C₂²×C₈	—	122	1	no	—
187	C₂²×C₈	—	122	1	no	—
188	C₂²×C₈	—	122	2	no	—
189	#117	—	2	2	no	—
190	#117	—	2	2	no	—
191	C₂×C₄⋊C₄	C₄×D₆₁	1	1	yes	characteristic
192	C₂×C₄⋊C₄	C₂×Dic₆₁	1	1	yes	characteristic
193	C₂×C₂²⋊C₄	C₄×D₆₁	1	1	yes	characteristic
194	C₂×C₂²⋊C₄	C₂×Dic₆₁	1	1	yes	characteristic
195	C₂×C₄²	C₄×D₆₁	1	1	yes	characteristic
196	C₂×C₄²	C₂²×D₆₁	1	1	yes	characteristic
197	C₄⋊C₈	—	2	2	no	—
198	C₄⋊C₈	—	122	2	no	—
199	C₂²⋊C₈	—	61	4	no	—
200	C₂²⋊C₈	C₂×Dic₆₁	1	4	yes	—
201	#97	—	122	2	no	—
202	#97	—	122	2	no	—
203	C₄×C₈	—	2	2	no	—
204	C₄×C₈	—	61	2	no	—
205	C₄×C₈	—	61	2	no	—
206	C₄×C₈	—	122	1	no	—
207	C₄×C₈	—	122	1	no	—
208	C₄×C₈	—	122	2	no	—
209	C₂⁴	C₄×Dic₆₁	1	1	yes	characteristic
210	C₂²×C₄	—	2	1	no	—
211	C₂²×C₄	—	2	1	no	—
212	C₂²×C₄	—	2	1	no	—
213	C₂²×C₄	—	2	1	no	—
214	C₂²×C₄	C₈×D₆₁	1	2	yes	—
215	C₂²×C₄	C₈×D₆₁	1	2	yes	—
216	C₂²×C₄	C₄×Dic₆₁	1	1	yes	characteristic
217	C₂²×C₄	C₄×Dic₆₁	1	1	yes	characteristic
218	C₂²×C₄	C₄×Dic₆₁	1	1	yes	characteristic
219	C₂²×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
220	C₂²×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
221	C₂²×C₄	C₂²×Dic₆₁	1	1	yes	characteristic
222	M₁₆	—	122	8	no	—
223	C₂×C₈	—	2	4	no	—
224	C₂×C₈	—	61	1	no	—
225	C₂×C₈	—	61	1	no	—
226	C₂×C₈	—	61	2	no	—
227	C₂×C₈	—	61	2	no	—
228	C₂×C₈	—	61	2	no	—
229	C₂×C₈	—	61	2	no	—
230	C₂×C₈	—	61	2	no	—
231	C₂×C₈	—	61	2	no	—
232	C₂×C₈	—	61	2	no	—
233	C₂×C₈	—	122	4	no	—
234	C₂×C₈	—	122	4	no	—
235	C₂×C₈	—	122	1	no	—
236	C₂×C₈	—	122	1	no	—
237	C₂×C₈	—	122	1	no	—
238	C₂×C₈	—	122	1	no	—
239	C₂×C₈	—	122	4	no	—
240	C₄⋊C₄	—	2	2	no	—
241	C₄⋊C₄	—	2	2	no	—
242	C₂²⋊C₄	—	2	2	no	—
243	C₂²⋊C₄	—	2	2	no	—
244	C₄²	—	2	1	no	—
245	C₄²	—	2	1	no	—
246	C₄²	D₄×D₆₁	1	2	yes	—
247	C₄²	#11758224	1	2	yes	—
248	C₂³	—	2	1	no	—
249	C₂³	—	2	1	no	—
250	C₂³	—	4	1	no	—
251	C₂³	—	4	1	no	—
252	C₂³	C₈×Dic₆₁	1	2	yes	—
253	C₂³	C₂×C₄×Dic₆₁	1	1	yes	characteristic
254	C₂×C₄	—	2	1	no	—
255	C₂×C₄	—	2	1	no	—
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	2	2	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	2	1	no	—
261	C₂×C₄	—	2	1	no	—
262	C₂×C₄	—	4	1	no	—
263	C₂×C₄	—	4	1	no	—
264	C₂×C₄	—	4	1	no	—
265	C₂×C₄	—	4	1	no	—
266	C₂×C₄	C₈×Dic₆₁	1	2	yes	—
267	C₂×C₄	C₂²⋊C₄×D₆₁	1	1	yes	characteristic
268	C₂×C₄	#14519733	1	1	yes	characteristic
269	C₂×C₄	#14519737	1	2	yes	—
270	C₂×C₄	D₄×Dic₆₁	1	2	yes	—
271	C₂×C₄	C₂×C₈×D₆₁	1	1	yes	characteristic
272	C₂×C₄	C₂×C₈×D₆₁	1	1	yes	characteristic
273	C₂×C₄	M₁₆×D₆₁	1	1	yes	characteristic
274	C₂×C₄	M₁₆×D₆₁	1	1	yes	characteristic
275	C₈	—	4	4	no	—
276	C₈	—	61	2	no	—
277	C₈	—	61	2	no	—
278	C₈	—	61	4	no	—
279	C₈	—	122	2	no	—
280	C₈	—	122	2	no	—
281	C₈	—	244	4	no	—
282	C₂²	—	2	1	no	—
283	C₂²	—	2	1	no	—
284	C₂²	—	2	2	no	—
285	C₂²	—	4	2	no	—
286	C₂²	—	4	2	no	—
287	C₂²	—	4	1	no	—
288	C₂²	—	4	1	no	—
289	C₂²	C₂²⋊C₄×Dic₆₁	1	1	yes	characteristic
290	C₂²	C₂×C₈×Dic₆₁	1	1	yes	characteristic
291	C₂²	M₁₆×Dic₆₁	1	1	yes	characteristic
292	C₄	—	2	2	no	—
293	C₄	—	2	2	no	—
294	C₄	—	2	2	no	—
295	C₄	—	4	1	no	—
296	C₄	—	4	1	no	—
297	C₄	C₂²⋊C₈×D₆₁	1	1	yes	characteristic
298	C₄	C₂²⋊C₈×D₆₁	1	1	yes	characteristic
299	C₄	#16839812	1	1	yes	characteristic
300	C₄	#16839812	1	1	yes	characteristic
301	C₂	—	2	1	no	—
302	C₂	—	2	1	no	—
303	C₂	—	4	2	no	—
304	C₂	C₂²⋊C₈×Dic₆₁	1	1	yes	characteristic
305	C₂	#21445621	1	1	yes	characteristic
306	C₂	#21445622	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²⋊C₄×D₆₁
commutator	C₂²×C₆₁	C₂×C₄×C₈
Frattini	C₂×C₄²	C₂²×D₆₁
Fitting	C₆₁×#1458	C₂
socle	C₂²×C₆₁	C₄×C₂²⋊C₄
prime radical	C₂⁴×C₆₁	C₄²
cosocle	C₂×C₄²×C₆₁	C₂³
hypercenter	#1458	D₆₁
hypocenter	C₆₁	#8156
LCS	G ▷ C₂²×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1458

Complements

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

N	H ≅ G/N	type
3. #21444103	C₂	semidirect
4. #21443663	C₂	semidirect
25. C₂×#11758194	C₈	semidirect
29. C₂²×#92418	C₈	semidirect
176. C₆₁	#8156	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16839718	2
#21444133	2
#21444103	2
#21443663	2
C₆₁×#1458	2
#8156	61