Subgroups — C₂×C₆₁×#3253 #29564336

Order 15616.

2426 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₆₁×#532	C₂	1	2	yes	maximal, complemented
2	C₆₁×#3253	C₂	1	16	yes	maximal, direct product factor
3	C₂×C₆₁×#476	C₂	1	4	yes	maximal, complemented
4	C₂×C₆₁×#469	C₂	1	4	yes	maximal, complemented
5	C₂×C₆₁×#459	C₂	1	4	yes	maximal, complemented
6	C₂×C₆₁×#423	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×#41×C₆₁	—	2	2	no	—
8	C₂²×D₈×C₆₁	C₂²	1	4	yes	—
9	C₂²×M₁₆×C₆₁	C₂²	1	2	yes	—
10	C₆₁×#532	—	2	8	no	—
11	C₆₁×#532	C₂²	1	16	yes	complemented
12	C₂×C₄⋊Q₈×C₆₁	C₂²	1	2	yes	—
13	C₂×#127×C₆₁	C₂²	1	2	yes	—
14	C₂×#124×C₆₁	—	2	2	no	—
15	C₂×#121×C₆₁	—	2	2	no	—
16	C₂×#121×C₆₁	C₂²	1	4	yes	—
17	C₂×C₄×D₄×C₆₁	—	2	2	no	—
18	C₂×C₄×D₄×C₆₁	C₂²	1	4	yes	—
19	C₂²×C₄²×C₆₁	C₂²	1	1	yes	characteristic
20	C₆₁×#476	C₂²	1	32	yes	complemented
21	C₆₁×#469	C₂²	1	32	yes	complemented
22	C₆₁×#459	C₂²	1	32	yes	complemented
23	C₂×#106×C₆₁	C₂²	1	4	yes	—
24	C₆₁×#423	C₂²	1	8	yes	complemented
25	C₂×C₄⋊C₈×C₆₁	C₂²	1	4	yes	—
26	C₂×#102×C₆₁	C₂²	1	8	yes	—
27	C₂×#41×C₆₁	—	2	16	no	—
28	C₂×#41×C₆₁	—	4	16	no	—
29	C₂²×Q₈×C₆₁	—	2	2	no	—
30	C₂²×D₄×C₆₁	—	2	4	no	—
31	C₂²×D₄×C₆₁	—	2	2	no	—
32	C₂²×D₄×C₆₁	—	2	2	no	—
33	C₂²×D₄×C₆₁	C₂³	1	4	yes	—
34	C₂³×C₄×C₆₁	—	2	4	no	—
35	C₂³×C₄×C₆₁	D₄	1	2	yes	—
36	C₂³×C₄×C₆₁	C₂³	1	1	yes	characteristic
37	C₂×D₈×C₆₁	—	4	8	no	—
38	C₂×D₈×C₆₁	C₂³	1	16	yes	—
39	C₂×M₁₆×C₆₁	—	4	4	no	—
40	C₂×M₁₆×C₆₁	C₂³	1	8	yes	—
41	C₂²×C₈×C₆₁	C₂³	1	4	yes	—
42	C₄⋊Q₈×C₆₁	—	2	4	no	—
43	C₄⋊Q₈×C₆₁	C₂³	1	8	yes	—
44	#127×C₆₁	—	2	4	no	—
45	#127×C₆₁	C₂³	1	8	yes	—
46	#124×C₆₁	—	2	16	no	—
47	#121×C₆₁	—	2	8	no	—
48	#121×C₆₁	—	2	16	no	—
49	#121×C₆₁	C₂³	1	16	yes	—
50	C₄×D₄×C₆₁	—	2	8	no	—
51	C₄×D₄×C₆₁	—	2	16	no	—
52	C₄×D₄×C₆₁	C₂³	1	16	yes	—
53	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
54	C₂×C₄⋊C₄×C₆₁	C₂³	1	4	yes	—
55	C₂×C₂²⋊C₄×C₆₁	—	2	4	no	—
56	C₂×C₂²⋊C₄×C₆₁	—	2	4	no	—
57	C₂×C₄²×C₆₁	D₄	1	4	yes	—
58	C₂×C₄²×C₆₁	D₄	1	1	yes	characteristic
59	C₂×C₄²×C₆₁	D₄	1	1	yes	characteristic
60	C₂×C₄²×C₆₁	C₂³	1	2	yes	—
61	C₂×C₄²×C₆₁	C₂³	1	4	yes	—
62	#106×C₆₁	C₂³	1	16	yes	—
63	C₄⋊C₈×C₆₁	C₂³	1	16	yes	—
64	#102×C₆₁	C₂³	1	32	yes	—
65	C₂⁴×C₆₁	—	2	4	no	—
66	C₂⁴×C₆₁	C₂×D₄	1	1	yes	characteristic
67	#41×C₆₁	—	4	64	no	—
68	C₂×Q₈×C₆₁	—	2	8	no	—
69	C₂×Q₈×C₆₁	—	4	8	no	—
70	C₂×D₄×C₆₁	—	2	16	no	—
71	C₂×D₄×C₆₁	—	2	4	no	—
72	C₂×D₄×C₆₁	—	2	8	no	—
73	C₂×D₄×C₆₁	—	2	8	no	—
74	C₂×D₄×C₆₁	—	2	8	no	—
75	C₂×D₄×C₆₁	—	2	8	no	—
76	C₂×D₄×C₆₁	—	4	8	no	—
77	C₂×D₄×C₆₁	—	4	16	no	—
78	C₂×D₄×C₆₁	—	4	8	no	—
79	C₂×D₄×C₆₁	C₂⁴	1	8	yes	—
80	C₂²×C₄×C₆₁	—	2	8	no	—
81	C₂²×C₄×C₆₁	—	2	2	no	—
82	C₂²×C₄×C₆₁	—	2	2	no	—
83	C₂²×C₄×C₆₁	—	2	4	no	—
84	C₂²×C₄×C₆₁	—	2	16	no	—
85	C₂²×C₄×C₆₁	—	4	16	no	—
86	C₂²×C₄×C₆₁	C₂×D₄	1	2	yes	—
87	C₂²×C₄×C₆₁	C₂×D₄	1	8	yes	—
88	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
89	C₂²×C₄×C₆₁	C₂×D₄	1	2	yes	—
90	C₂²×C₄×C₆₁	C₂×D₄	1	2	yes	—
91	C₂²×C₄×C₆₁	C₂⁴	1	2	yes	—
92	C₂²×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
93	D₈×C₆₁	—	4	16	no	—
94	M₁₆×C₆₁	—	4	8	no	—
95	C₂×C₈×C₆₁	—	2	8	no	—
96	C₂×C₈×C₆₁	C₂⁴	1	8	yes	—
97	C₄⋊C₄×C₆₁	—	2	8	no	—
98	C₄⋊C₄×C₆₁	—	2	4	no	—
99	C₄⋊C₄×C₆₁	C₂⁴	1	8	yes	—
100	C₂²⋊C₄×C₆₁	—	2	16	no	—
101	C₂²⋊C₄×C₆₁	—	2	16	no	—
102	C₄²×C₆₁	C₂×D₄	1	4	yes	—
103	C₄²×C₆₁	C₂×D₄	1	4	yes	—
104	C₄²×C₆₁	C₂×D₄	1	4	yes	—
105	C₄²×C₆₁	C₂⁴	1	4	yes	—
106	C₂³×C₆₁	—	2	2	no	—
107	C₂³×C₆₁	—	2	4	no	—
108	C₂³×C₆₁	—	2	8	no	—
109	C₂³×C₆₁	—	4	8	no	—
110	C₂³×C₆₁	—	4	8	no	—
111	C₂³×C₆₁	—	4	8	no	—
112	C₂³×C₆₁	C₂²×D₄	1	2	yes	—
113	C₂³×C₆₁	C₂²×D₄	1	1	yes	characteristic
114	Q₈×C₆₁	—	4	16	no	—
115	D₄×C₆₁	—	2	8	no	—
116	D₄×C₆₁	—	2	8	no	—
117	D₄×C₆₁	—	4	16	no	—
118	D₄×C₆₁	—	4	32	no	—
119	D₄×C₆₁	—	4	8	no	—
120	D₄×C₆₁	—	4	16	no	—
121	C₂×C₄×C₆₁	—	2	2	no	—
122	C₂×C₄×C₆₁	—	2	4	no	—
123	C₂×C₄×C₆₁	—	2	4	no	—
124	C₂×C₄×C₆₁	—	2	16	no	—
125	C₂×C₄×C₆₁	—	2	2	no	—
126	C₂×C₄×C₆₁	—	2	2	no	—
127	C₂×C₄×C₆₁	—	2	8	no	—
128	C₂×C₄×C₆₁	—	4	8	no	—
129	C₂×C₄×C₆₁	—	4	32	no	—
130	C₂×C₄×C₆₁	AGL(1,ℤ/8)	1	8	yes	—
131	C₂×C₄×C₆₁	C₂²×D₄	1	2	yes	—
132	C₂×C₄×C₆₁	C₂²×D₄	1	4	yes	—
133	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
134	C₂×C₄×C₆₁	C₂²×D₄	1	4	yes	—
135	C₂×C₄×C₆₁	H(5,𝔽₂)	1	2	yes	—
136	C₂×C₄×C₆₁	2₋¹⁺⁴	1	2	yes	—
137	C₂×C₄×C₆₁	C₂⁵	1	1	yes	characteristic
138	C₈×C₆₁	—	2	8	no	—
139	C₂×#3253	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
140	C₂²×C₆₁	—	2	2	no	—
141	C₂²×C₆₁	—	2	4	no	—
142	C₂²×C₆₁	—	4	4	no	—
143	C₂²×C₆₁	—	4	8	no	—
144	C₂²×C₆₁	—	4	8	no	—
145	C₂²×C₆₁	—	4	8	no	—
146	C₂²×C₆₁	—	8	16	no	—
147	C₂²×C₆₁	#537	1	2	yes	—
148	C₂²×C₆₁	C₂×AGL(1,ℤ/8)	1	4	yes	—
149	C₂²×C₆₁	C₂³×D₄	1	1	yes	characteristic
150	C₄×C₆₁	—	2	2	no	—
151	C₄×C₆₁	—	2	4	no	—
152	C₄×C₆₁	—	4	8	no	—
153	C₄×C₆₁	#537	1	2	yes	—
154	C₄×C₆₁	C₂×AGL(1,ℤ/8)	1	8	yes	—
155	C₄×C₆₁	C₂×H(5,𝔽₂)	1	1	yes	characteristic
156	C₄×C₆₁	C₂×2₋¹⁺⁴	1	1	yes	characteristic
157	C₂×#532	C₂×C₆₁	1	2	yes	complemented
158	#3253	C₂×C₆₁	1	16	yes	direct product factor
159	C₂×#476	C₂×C₆₁	1	4	yes	complemented
160	C₂×#469	C₂×C₆₁	1	4	yes	complemented
161	C₂×#459	C₂×C₆₁	1	4	yes	complemented
162	C₂×#423	C₂×C₆₁	1	1	yes	characteristic, complemented
163	C₂×C₆₁	—	4	2	no	—
164	C₂×C₆₁	—	8	8	no	—
165	C₂×C₆₁	#3253	1	4	yes	direct product factor
166	C₂×C₆₁	C₂×#537	1	1	yes	characteristic
167	C₂×C₆₁	C₂²×AGL(1,ℤ/8)	1	2	yes	—
168	C₂²×#41	—	2	2	no	—
169	C₂²×D₈	C₂²×C₆₁	1	4	yes	—
170	C₂²×M₁₆	C₂²×C₆₁	1	2	yes	—
171	#532	—	2	8	no	—
172	#532	C₂²×C₆₁	1	16	yes	complemented
173	C₂×C₄⋊Q₈	C₂²×C₆₁	1	2	yes	—
174	C₂×#127	C₂²×C₆₁	1	2	yes	—
175	C₂×#124	—	2	2	no	—
176	C₂×#121	—	2	2	no	—
177	C₂×#121	C₂²×C₆₁	1	4	yes	—
178	C₂×C₄×D₄	—	2	2	no	—
179	C₂×C₄×D₄	C₂²×C₆₁	1	4	yes	—
180	C₂²×C₄²	C₂²×C₆₁	1	1	yes	characteristic
181	#476	C₂²×C₆₁	1	32	yes	complemented
182	#469	C₂²×C₆₁	1	32	yes	complemented
183	#459	C₂²×C₆₁	1	32	yes	complemented
184	C₂×#106	C₂²×C₆₁	1	4	yes	—
185	#423	C₂²×C₆₁	1	8	yes	complemented
186	C₂×C₄⋊C₈	C₂²×C₆₁	1	4	yes	—
187	C₂×#102	C₂²×C₆₁	1	8	yes	—
188	C₆₁	C₂×#3253	1	1	yes	characteristic, direct product factor, Sylow
189	C₂×#41	—	2	16	no	—
190	C₂×#41	—	4	16	no	—
191	C₂²×Q₈	—	2	2	no	—
192	C₂²×D₄	—	2	4	no	—
193	C₂²×D₄	—	2	2	no	—
194	C₂²×D₄	—	2	2	no	—
195	C₂²×D₄	C₂³×C₆₁	1	4	yes	—
196	C₂³×C₄	—	2	4	no	—
197	C₂³×C₄	D₄×C₆₁	1	2	yes	—
198	C₂³×C₄	C₂³×C₆₁	1	1	yes	characteristic
199	C₂×D₈	—	4	8	no	—
200	C₂×D₈	C₂³×C₆₁	1	16	yes	—
201	C₂×M₁₆	—	4	4	no	—
202	C₂×M₁₆	C₂³×C₆₁	1	8	yes	—
203	C₂²×C₈	C₂³×C₆₁	1	4	yes	—
204	C₄⋊Q₈	—	2	4	no	—
205	C₄⋊Q₈	C₂³×C₆₁	1	8	yes	—
206	#127	—	2	4	no	—
207	#127	C₂³×C₆₁	1	8	yes	—
208	#124	—	2	16	no	—
209	#121	—	2	16	no	—
210	#121	—	2	8	no	—
211	#121	C₂³×C₆₁	1	16	yes	—
212	C₄×D₄	—	2	16	no	—
213	C₄×D₄	—	2	8	no	—
214	C₄×D₄	C₂³×C₆₁	1	16	yes	—
215	C₂×C₄⋊C₄	—	2	2	no	—
216	C₂×C₄⋊C₄	C₂³×C₆₁	1	4	yes	—
217	C₂×C₂²⋊C₄	—	2	4	no	—
218	C₂×C₂²⋊C₄	—	2	4	no	—
219	C₂×C₄²	D₄×C₆₁	1	1	yes	characteristic
220	C₂×C₄²	D₄×C₆₁	1	1	yes	characteristic
221	C₂×C₄²	D₄×C₆₁	1	4	yes	—
222	C₂×C₄²	C₂³×C₆₁	1	2	yes	—
223	C₂×C₄²	C₂³×C₆₁	1	4	yes	—
224	#106	C₂³×C₆₁	1	16	yes	—
225	C₄⋊C₈	C₂³×C₆₁	1	16	yes	—
226	#102	C₂³×C₆₁	1	32	yes	—
227	C₂⁴	—	2	4	no	—
228	C₂⁴	C₂×D₄×C₆₁	1	1	yes	characteristic
229	#41	—	4	64	no	—
230	C₂×Q₈	—	2	8	no	—
231	C₂×Q₈	—	4	8	no	—
232	C₂×D₄	—	2	16	no	—
233	C₂×D₄	—	2	4	no	—
234	C₂×D₄	—	2	8	no	—
235	C₂×D₄	—	2	8	no	—
236	C₂×D₄	—	2	8	no	—
237	C₂×D₄	—	2	8	no	—
238	C₂×D₄	—	4	8	no	—
239	C₂×D₄	—	4	16	no	—
240	C₂×D₄	—	4	8	no	—
241	C₂×D₄	C₂⁴×C₆₁	1	8	yes	—
242	C₂²×C₄	—	2	8	no	—
243	C₂²×C₄	—	2	2	no	—
244	C₂²×C₄	—	2	2	no	—
245	C₂²×C₄	—	2	4	no	—
246	C₂²×C₄	—	2	16	no	—
247	C₂²×C₄	—	4	16	no	—
248	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
249	C₂²×C₄	C₂×D₄×C₆₁	1	2	yes	—
250	C₂²×C₄	C₂×D₄×C₆₁	1	2	yes	—
251	C₂²×C₄	C₂×D₄×C₆₁	1	8	yes	—
252	C₂²×C₄	C₂×D₄×C₆₁	1	2	yes	—
253	C₂²×C₄	C₂⁴×C₆₁	1	2	yes	—
254	C₂²×C₄	C₂⁴×C₆₁	1	1	yes	characteristic
255	D₈	—	4	16	no	—
256	M₁₆	—	4	8	no	—
257	C₂×C₈	—	2	8	no	—
258	C₂×C₈	C₂⁴×C₆₁	1	8	yes	—
259	C₄⋊C₄	—	2	8	no	—
260	C₄⋊C₄	—	2	4	no	—
261	C₄⋊C₄	C₂⁴×C₆₁	1	8	yes	—
262	C₂²⋊C₄	—	2	16	no	—
263	C₂²⋊C₄	—	2	16	no	—
264	C₄²	C₂×D₄×C₆₁	1	4	yes	—
265	C₄²	C₂×D₄×C₆₁	1	4	yes	—
266	C₄²	C₂×D₄×C₆₁	1	4	yes	—
267	C₄²	C₂⁴×C₆₁	1	4	yes	—
268	C₂³	—	2	2	no	—
269	C₂³	—	2	4	no	—
270	C₂³	—	2	8	no	—
271	C₂³	—	4	8	no	—
272	C₂³	—	4	8	no	—
273	C₂³	—	4	8	no	—
274	C₂³	C₂²×D₄×C₆₁	1	2	yes	—
275	C₂³	C₂²×D₄×C₆₁	1	1	yes	characteristic
276	Q₈	—	4	16	no	—
277	D₄	—	2	8	no	—
278	D₄	—	2	8	no	—
279	D₄	—	4	16	no	—
280	D₄	—	4	32	no	—
281	D₄	—	4	8	no	—
282	D₄	—	4	16	no	—
283	C₂×C₄	—	2	2	no	—
284	C₂×C₄	—	2	4	no	—
285	C₂×C₄	—	2	4	no	—
286	C₂×C₄	—	2	16	no	—
287	C₂×C₄	—	2	2	no	—
288	C₂×C₄	—	2	2	no	—
289	C₂×C₄	—	2	8	no	—
290	C₂×C₄	—	4	8	no	—
291	C₂×C₄	—	4	32	no	—
292	C₂×C₄	AGL(1,ℤ/8)×C₆₁	1	8	yes	—
293	C₂×C₄	C₂²×D₄×C₆₁	1	2	yes	—
294	C₂×C₄	C₂²×D₄×C₆₁	1	4	yes	—
295	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
296	C₂×C₄	C₂²×D₄×C₆₁	1	4	yes	—
297	C₂×C₄	H(5,𝔽₂)×C₆₁	1	2	yes	—
298	C₂×C₄	2₋¹⁺⁴×C₆₁	1	2	yes	—
299	C₂×C₄	C₂⁵×C₆₁	1	1	yes	characteristic
300	C₈	—	2	8	no	—
301	C₂²	—	2	4	no	—
302	C₂²	—	2	2	no	—
303	C₂²	—	4	4	no	—
304	C₂²	—	4	8	no	—
305	C₂²	—	4	8	no	—
306	C₂²	—	4	8	no	—
307	C₂²	—	8	16	no	—
308	C₂²	C₆₁×#537	1	2	yes	—
309	C₂²	C₂×AGL(1,ℤ/8)×C₆₁	1	4	yes	—
310	C₂²	C₂³×D₄×C₆₁	1	1	yes	characteristic
311	C₄	—	2	4	no	—
312	C₄	—	2	2	no	—
313	C₄	—	4	8	no	—
314	C₄	C₆₁×#537	1	2	yes	—
315	C₄	C₂×AGL(1,ℤ/8)×C₆₁	1	8	yes	—
316	C₄	C₂×H(5,𝔽₂)×C₆₁	1	1	yes	characteristic
317	C₄	C₂×2₋¹⁺⁴×C₆₁	1	1	yes	characteristic
318	C₂	—	4	2	no	—
319	C₂	—	8	8	no	—
320	C₂	C₆₁×#3253	1	4	yes	direct product factor
321	C₂	C₂×C₆₁×#537	1	1	yes	characteristic
322	C₂	C₂²×AGL(1,ℤ/8)×C₆₁	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₆₁	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₆₁
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₁ ◁ C₂²×C₄²×C₆₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₆₁×#532	C₂	semidirect
2. C₆₁×#3253	C₂	direct (N × H)
3. C₂×C₆₁×#476	C₂	semidirect
4. C₂×C₆₁×#469	C₂	semidirect
5. C₂×C₆₁×#459	C₂	semidirect
6. C₂×C₆₁×#423	C₂	semidirect
11. C₆₁×#532	C₂²	semidirect
20. C₆₁×#476	C₂²	semidirect
21. C₆₁×#469	C₂²	semidirect
22. C₆₁×#459	C₂²	semidirect
24. C₆₁×#423	C₂²	semidirect
139. C₂×#3253	C₆₁	direct (N × H)
157. C₂×#532	C₂×C₆₁	semidirect
158. #3253	C₂×C₆₁	direct (N × H)
159. C₂×#476	C₂×C₆₁	semidirect
160. C₂×#469	C₂×C₆₁	semidirect
161. C₂×#459	C₂×C₆₁	semidirect
162. C₂×#423	C₂×C₆₁	semidirect
165. C₂×C₆₁	#3253	direct (N × H)
172. #532	C₂²×C₆₁	semidirect
181. #476	C₂²×C₆₁	semidirect
182. #469	C₂²×C₆₁	semidirect
183. #459	C₂²×C₆₁	semidirect
185. #423	C₂²×C₆₁	semidirect
188. C₆₁	C₂×#3253	direct (N × H)
320. C₂	C₆₁×#3253	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₆₁×#532	2
C₆₁×#3253	2
C₂×C₆₁×#476	2
C₂×C₆₁×#469	2
C₂×C₆₁×#459	2
C₂×C₆₁×#423	2
C₂×#3253	61

Subgroups — C2×C61×#3253 #29564336

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₆₁×#3253 #29564336