Subgroups — #29579377

Order 15616.

502 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21445686	C₂	1	1	yes	maximal, characteristic, complemented
2	#21444567	C₂	1	1	yes	maximal, characteristic, complemented
3	#21444314	C₂	1	2	yes	maximal, complemented
4	#21444218	C₂	1	2	yes	maximal, complemented
5	C₆₁×#3147	C₂	1	1	yes	maximal, characteristic
6	#16839815	C₂²	1	2	yes	—
7	C₂×C₄⋊#92418	C₂²	1	1	yes	characteristic
8	C₂×#14519687	—	2	1	no	—
9	C₄×C₄⋊Dic₆₁	C₂²	1	1	yes	characteristic
10	#16839516	—	2	1	no	—
11	#16839515	—	2	1	no	—
12	#16839513	—	2	1	no	—
13	#16839496	—	2	4	no	—
14	D₄⋊#92418	—	2	4	no	—
15	C₂×#135×C₆₁	—	2	1	no	—
16	C₂×#135×C₆₁	—	2	1	no	—
17	C₆₁×#532	C₂²	1	2	yes	—
18	C₆₁×#495	—	2	1	no	—
19	C₆₁×#495	—	2	1	no	—
20	C₆₁×#494	C₄	1	1	yes	characteristic, complemented
21	C₆₁×#493	C₄	1	1	yes	characteristic, complemented
22	C₆₁×#492	C₄	1	2	yes	complemented
23	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
24	C₂²×#92418	—	2	1	no	—
25	C₂×C₄⋊Dic₆₁	—	2	1	no	—
26	C₂×C₄×Dic₆₁	—	2	1	no	—
27	C₄⋊#92418	—	2	1	no	—
28	C₄⋊#92418	—	2	2	no	—
29	C₄⋊#92418	D₄	1	2	yes	—
30	C₂²⋊#92418	—	2	2	no	—
31	#14519687	—	4	2	no	—
32	C₄×#92418	—	2	2	no	—
33	#14519684	D₄	1	2	yes	—
34	C₂×#41×C₆₁	—	2	2	no	—
35	#135×C₆₁	—	4	2	no	—
36	#135×C₆₁	—	4	2	no	—
37	#135×C₆₁	—	4	2	no	—
38	#135×C₆₁	—	4	2	no	—
39	C₂×Q₁₆×C₆₁	—	2	1	no	—
40	C₂×Q₁₆×C₆₁	—	2	1	no	—
41	C₂×SD₁₆×C₆₁	—	2	2	no	—
42	C₂×SD₁₆×C₆₁	—	2	2	no	—
43	C₂×D₈×C₆₁	—	2	1	no	—
44	C₂×D₈×C₆₁	—	2	1	no	—
45	C₂²×C₈×C₆₁	D₄	1	1	yes	characteristic
46	C₂²×C₈×C₆₁	D₄	1	1	yes	characteristic
47	C₄⋊Q₈×C₆₁	C₂×C₄	1	2	yes	—
48	#127×C₆₁	C₂×C₄	1	2	yes	—
49	#124×C₆₁	—	2	2	no	—
50	#121×C₆₁	—	2	2	no	—
51	#121×C₆₁	—	2	2	no	—
52	C₄×D₄×C₆₁	—	2	2	no	—
53	C₄×D₄×C₆₁	—	2	2	no	—
54	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
55	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
56	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
57	C₄×C₈×C₆₁	D₄	1	1	yes	characteristic
58	C₄×C₈×C₆₁	D₄	1	1	yes	characteristic
59	#41×C₆₁	—	4	2	no	—
60	#41×C₆₁	—	4	2	no	—
61	#41×C₆₁	—	4	2	no	—
62	#41×C₆₁	—	4	2	no	—
63	C₂×Q₈×C₆₁	—	2	2	no	—
64	C₂×D₄×C₆₁	—	2	2	no	—
65	C₂×D₄×C₆₁	—	2	2	no	—
66	C₂×D₄×C₆₁	—	2	2	no	—
67	C₂×D₄×C₆₁	—	4	2	no	—
68	C₂²×C₄×C₆₁	—	4	2	no	—
69	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₆₁	#41	1	1	yes	characteristic
72	C₂²×Dic₆₁	—	2	2	no	—
73	Q₁₆×C₆₁	—	4	2	no	—
74	Q₁₆×C₆₁	—	4	2	no	—
75	SD₁₆×C₆₁	—	4	4	no	—
76	SD₁₆×C₆₁	—	4	4	no	—
77	D₈×C₆₁	—	4	2	no	—
78	D₈×C₆₁	—	4	2	no	—
79	C₂×C₈×C₆₁	—	2	1	no	—
80	C₂×C₈×C₆₁	—	2	1	no	—
81	C₂×C₈×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
82	C₂×C₈×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
83	C₂×C₈×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
84	C₂×C₈×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
85	C₂×C₈×C₆₁	D₈	1	2	yes	—
86	C₂×C₈×C₆₁	SD₁₆	1	2	yes	—
87	C₄⋊C₄×C₆₁	—	2	2	no	—
88	C₄⋊C₄×C₆₁	—	2	2	no	—
89	C₂²⋊C₄×C₆₁	—	4	2	no	—
90	C₂²⋊C₄×C₆₁	—	4	2	no	—
91	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
92	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
93	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
94	C₄²×C₆₁	#41	1	1	yes	characteristic
95	C₄⋊Dic₆₁	—	4	2	no	—
96	C₄×Dic₆₁	—	4	1	no	—
97	C₄×Dic₆₁	—	4	1	no	—
98	C₂×#92418	—	2	2	no	—
99	C₂×#92418	—	4	1	no	—
100	C₂×#92418	—	4	1	no	—
101	C₂×#92418	—	4	2	no	—
102	C₂³×C₆₁	—	4	2	no	—
103	C₂³×C₆₁	#121	1	1	yes	characteristic
104	Q₈×C₆₁	—	4	2	no	—
105	Q₈×C₆₁	—	4	2	no	—
106	D₄×C₆₁	—	4	2	no	—
107	D₄×C₆₁	—	4	2	no	—
108	D₄×C₆₁	—	4	4	no	—
109	D₄×C₆₁	—	8	2	no	—
110	D₄×C₆₁	—	8	2	no	—
111	D₄×C₆₁	—	8	2	no	—
112	C₂×C₄×C₆₁	—	2	2	no	—
113	C₂×C₄×C₆₁	—	4	1	no	—
114	C₂×C₄×C₆₁	—	4	2	no	—
115	C₂×C₄×C₆₁	—	8	2	no	—
116	C₂×C₄×C₆₁	—	8	2	no	—
117	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
118	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
119	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
120	C₂×C₄×C₆₁	#121	1	1	yes	characteristic
121	C₂×C₄×C₆₁	#124	1	1	yes	characteristic
122	C₂×C₄×C₆₁	#127	1	1	yes	characteristic
123	C₂×C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
124	C₂×C₄×C₆₁	C₂×SD₁₆	1	1	yes	characteristic
125	C₂×C₄×C₆₁	AGL(1,ℤ/8)	1	1	yes	characteristic
126	C₂×C₄×C₆₁	AGL(1,ℤ/8)	1	1	yes	characteristic
127	C₂×Dic₆₁	—	4	2	no	—
128	C₂×Dic₆₁	—	4	2	no	—
129	C₂×Dic₆₁	—	4	2	no	—
130	C₈×C₆₁	—	2	1	no	—
131	C₈×C₆₁	—	2	1	no	—
132	C₈×C₆₁	#102	1	2	yes	—
133	C₈×C₆₁	#102	1	2	yes	—
134	#92418	—	4	2	no	—
135	#92418	—	8	2	no	—
136	#9753	—	61	1	no	maximal, Sylow
137	C₂²×C₆₁	—	2	1	no	—
138	C₂²×C₆₁	—	2	1	no	—
139	C₂²×C₆₁	—	8	2	no	—
140	C₂²×C₆₁	—	8	2	no	—
141	C₂²×C₆₁	—	8	2	no	—
142	C₂²×C₆₁	#390	1	1	yes	characteristic
143	C₂²×C₆₁	#459	1	1	yes	characteristic
144	C₂²×C₆₁	#460	1	1	yes	characteristic
145	C₄×C₆₁	—	2	1	no	—
146	C₄×C₆₁	—	2	1	no	—
147	C₄×C₆₁	—	2	1	no	—
148	C₄×C₆₁	—	8	2	no	—
149	C₄×C₆₁	C₂×#102	1	1	yes	characteristic
150	C₄×C₆₁	#418	1	1	yes	characteristic
151	C₄×C₆₁	#444	1	2	yes	—
152	C₄×C₆₁	#459	1	1	yes	characteristic
153	C₄×C₆₁	#460	1	1	yes	characteristic
154	Dic₆₁	—	8	2	no	—
155	#3147	D₆₁	1	1	yes	characteristic
156	#1770	—	61	1	no	—
157	#1564	—	61	1	no	—
158	#1502	—	61	2	no	—
159	#1480	—	61	2	no	—
160	C₂×C₆₁	—	2	1	no	—
161	C₂×C₆₁	—	2	1	no	—
162	C₂×C₆₁	—	16	2	no	—
163	C₂×C₆₁	#1975	1	1	yes	characteristic
164	C₂×C₆₁	#1976	1	1	yes	characteristic
165	C₂×C₆₁	#1982	1	1	yes	characteristic
166	C₂×#135	—	2	1	no	—
167	C₂×#135	—	2	1	no	—
168	#532	C₂×D₆₁	1	2	yes	—
169	#495	—	2	1	no	—
170	#495	—	2	1	no	—
171	#494	Dic₆₁	1	1	yes	characteristic, complemented
172	#493	Dic₆₁	1	1	yes	characteristic, complemented
173	#492	Dic₆₁	1	2	yes	complemented
174	#431	—	61	2	no	—
175	C₂×C₄⋊C₈	—	61	1	no	—
176	C₂×#97	—	122	1	no	—
177	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
178	C₄×C₄⋊C₄	—	61	1	no	—
179	#336	—	122	1	no	—
180	#335	—	122	1	no	—
181	#334	—	122	1	no	—
182	#327	—	122	4	no	—
183	D₄⋊C₈	—	122	4	no	—
184	C₆₁	#9753	1	1	yes	characteristic, complemented, Sylow
185	C₂×#41	—	2	2	no	—
186	#135	—	4	2	no	—
187	#135	—	4	2	no	—
188	#135	—	4	2	no	—
189	#135	—	4	2	no	—
190	C₂×Q₁₆	—	2	1	no	—
191	C₂×Q₁₆	—	2	1	no	—
192	C₂×SD₁₆	—	2	2	no	—
193	C₂×SD₁₆	—	2	2	no	—
194	C₂×D₈	—	2	1	no	—
195	C₂×D₈	—	2	1	no	—
196	C₂²×C₈	—	122	1	no	—
197	C₂²×C₈	C₂²⋊D₆₁	1	1	yes	characteristic
198	C₂²×C₈	C₂²⋊D₆₁	1	1	yes	characteristic
199	C₄⋊Q₈	C₂×Dic₆₁	1	2	yes	—
200	#127	C₂×Dic₆₁	1	2	yes	—
201	#124	—	2	2	no	—
202	#121	—	2	2	no	—
203	#121	—	2	2	no	—
204	C₄×D₄	—	2	2	no	—
205	C₄×D₄	—	2	2	no	—
206	C₂×C₄⋊C₄	—	122	1	no	—
207	C₂×C₄²	—	122	1	no	—
208	C₂×C₄²	C₂²×D₆₁	1	1	yes	characteristic
209	C₄⋊C₈	—	61	2	no	—
210	C₄⋊C₈	—	122	1	no	—
211	C₄⋊C₈	—	122	2	no	—
212	C₂²⋊C₈	—	122	2	no	—
213	#97	—	244	2	no	—
214	C₄×C₈	—	122	2	no	—
215	C₄×C₈	C₂×Dic₆₁	1	1	yes	characteristic
216	C₄×C₈	C₂×Dic₆₁	1	1	yes	characteristic
217	C₄×C₈	C₂²⋊D₆₁	1	1	yes	characteristic
218	C₄×C₈	C₂²⋊D₆₁	1	1	yes	characteristic
219	#95	—	61	2	no	—
220	#41	—	4	2	no	—
221	#41	—	4	2	no	—
222	#41	—	4	2	no	—
223	#41	—	4	2	no	—
224	C₂×Q₈	—	2	2	no	—
225	C₂×D₄	—	2	2	no	—
226	C₂×D₄	—	2	2	no	—
227	C₂×D₄	—	2	2	no	—
228	C₂×D₄	—	4	2	no	—
229	C₂²×C₄	—	4	2	no	—
230	C₂²×C₄	—	122	2	no	—
231	C₂²×C₄	D₄×D₆₁	1	1	yes	characteristic
232	C₂²×C₄	#11758224	1	1	yes	characteristic
233	C₂²×C₄	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
234	Q₁₆	—	4	2	no	—
235	Q₁₆	—	4	2	no	—
236	SD₁₆	—	4	4	no	—
237	SD₁₆	—	4	4	no	—
238	D₈	—	4	2	no	—
239	D₈	—	4	2	no	—
240	C₂×C₈	—	2	1	no	—
241	C₂×C₈	—	2	1	no	—
242	C₂×C₈	—	122	2	no	—
243	C₂×C₈	—	244	2	no	—
244	C₂×C₈	—	244	1	no	—
245	C₂×C₈	—	244	1	no	—
246	C₂×C₈	#11758199	1	2	yes	—
247	C₂×C₈	#11758200	1	2	yes	—
248	C₂×C₈	C₂²⋊Dic₆₁	1	1	yes	characteristic
249	C₂×C₈	C₂²⋊Dic₆₁	1	1	yes	characteristic
250	C₂×C₈	C₂²⋊Dic₆₁	1	1	yes	characteristic
251	C₂×C₈	C₂²⋊Dic₆₁	1	1	yes	characteristic
252	C₄⋊C₄	—	2	2	no	—
253	C₄⋊C₄	—	2	2	no	—
254	C₄⋊C₄	—	244	2	no	—
255	C₂²⋊C₄	—	4	2	no	—
256	C₂²⋊C₄	—	4	2	no	—
257	C₄²	—	244	1	no	—
258	C₄²	—	244	1	no	—
259	C₄²	D₄×D₆₁	1	1	yes	characteristic
260	C₄²	#11758224	1	1	yes	characteristic
261	C₄²	C₂²×Dic₆₁	1	1	yes	characteristic
262	C₄²	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
263	C₂³	—	4	2	no	—
264	C₂³	#14519748	1	1	yes	characteristic
265	Q₈	—	4	2	no	—
266	Q₈	—	4	2	no	—
267	D₄	—	4	2	no	—
268	D₄	—	4	2	no	—
269	D₄	—	4	4	no	—
270	D₄	—	8	2	no	—
271	D₄	—	8	2	no	—
272	D₄	—	8	2	no	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	4	1	no	—
275	C₂×C₄	—	4	2	no	—
276	C₂×C₄	—	8	2	no	—
277	C₂×C₄	—	8	2	no	—
278	C₂×C₄	—	244	2	no	—
279	C₂×C₄	—	244	2	no	—
280	C₂×C₄	—	244	2	no	—
281	C₂×C₄	C₂×C₂²⋊Dic₆₁	1	1	yes	characteristic
282	C₂×C₄	D₄×Dic₆₁	1	1	yes	characteristic
283	C₂×C₄	D₄×Dic₆₁	1	1	yes	characteristic
284	C₂×C₄	#14519748	1	1	yes	characteristic
285	C₂×C₄	#14519764	1	1	yes	characteristic
286	C₂×C₄	#14519773	1	1	yes	characteristic
287	C₂×C₄	C₂×#11758199	1	1	yes	characteristic
288	C₂×C₄	C₂×#11758200	1	1	yes	characteristic
289	C₂×C₄	#14519803	1	1	yes	characteristic
290	C₂×C₄	#14519803	1	1	yes	characteristic
291	C₈	—	2	1	no	—
292	C₈	—	2	1	no	—
293	C₈	—	488	2	no	—
294	C₈	—	244	2	no	—
295	C₈	#14519698	1	2	yes	—
296	C₈	#14519698	1	2	yes	—
297	C₂²	—	2	1	no	—
298	C₂²	—	2	1	no	—
299	C₂²	—	8	2	no	—
300	C₂²	—	8	2	no	—
301	C₂²	—	8	2	no	—
302	C₂²	#16839665	1	1	yes	characteristic
303	C₂²	#16839954	1	1	yes	characteristic
304	C₂²	#16839959	1	1	yes	characteristic
305	C₄	—	2	1	no	—
306	C₄	—	2	1	no	—
307	C₄	—	2	1	no	—
308	C₄	—	8	2	no	—
309	C₄	—	488	2	no	—
310	C₄	C₂×#14519698	1	1	yes	characteristic
311	C₄	#16839767	1	1	yes	characteristic
312	C₄	#16839884	1	2	yes	—
313	C₄	#16839954	1	1	yes	characteristic
314	C₄	#16839959	1	1	yes	characteristic
315	C₂	—	2	1	no	—
316	C₂	—	2	1	no	—
317	C₂	—	16	2	no	—
318	C₂	#21446713	1	1	yes	characteristic
319	C₂	#21446718	1	1	yes	characteristic
320	C₂	#21446747	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#14519748
commutator	C₄²×C₆₁	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₁
Fitting, prime radical	C₆₁×#3147	C₂
socle	C₂²×C₆₁	#390
cosocle	C₂×C₄²×C₆₁	C₂³
hypercenter	#3147	D₆₁
hypocenter	C₆₁	#9753
LCS	G ▷ C₄²×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3147

Complements

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

N	H ≅ G/N	type
1. #21445686	C₂	semidirect
2. #21444567	C₂	semidirect
3. #21444314	C₂	semidirect
4. #21444218	C₂	semidirect
20. C₆₁×#494	C₄	semidirect
21. C₆₁×#493	C₄	semidirect
22. C₆₁×#492	C₄	semidirect
171. #494	Dic₆₁	semidirect
172. #493	Dic₆₁	semidirect
173. #492	Dic₆₁	semidirect
184. C₆₁	#9753	semidirect

Maximal subgroups

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

subgroup	index
#21445686	2
#21444567	2
#21444314	2
#21444218	2
C₆₁×#3147	2
#9753	61