Subgroups — C₆₁×#2426 #21442467

Order 7808.

632 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#124×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₂²⋊Q₈×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#121×C₆₁	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄×Q₈×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₁×#396	C₂	1	2	yes	maximal, complemented
6	C₆₁×#394	C₂	1	2	yes	maximal, complemented
7	C₆₁×#390	C₂	1	1	yes	maximal, characteristic, complemented
8	C₆₁×#390	C₂	1	2	yes	maximal, complemented
9	C₆₁×#389	C₂	1	1	yes	maximal, characteristic, complemented
10	C₆₁×#388	C₂	1	2	yes	maximal, complemented
11	C₄×C₂²⋊C₄×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×Q₈×C₆₁	C₂²	1	1	yes	characteristic
13	C₂²×D₄×C₆₁	C₂²	1	1	yes	characteristic
14	C₂²×D₄×C₆₁	C₂²	1	2	yes	—
15	C₂³×C₄×C₆₁	C₂²	1	1	yes	characteristic
16	#124×C₆₁	—	2	4	no	—
17	C₂²⋊Q₈×C₆₁	—	2	4	no	—
18	#121×C₆₁	—	2	4	no	—
19	C₄×Q₈×C₆₁	—	2	4	no	—
20	C₂×C₄⋊C₄×C₆₁	C₂²	1	2	yes	—
21	C₂×C₄⋊C₄×C₆₁	C₂²	1	1	yes	characteristic
22	C₂×C₄⋊C₄×C₆₁	C₂²	1	2	yes	—
23	C₂×C₄⋊C₄×C₆₁	C₂²	1	2	yes	—
24	C₂×C₄⋊C₄×C₆₁	C₂²	1	1	yes	characteristic
25	C₂×C₂²⋊C₄×C₆₁	C₂²	1	1	yes	characteristic
26	C₂×C₂²⋊C₄×C₆₁	C₂²	1	1	yes	characteristic
27	C₂×C₂²⋊C₄×C₆₁	C₂²	1	2	yes	—
28	C₂×C₂²⋊C₄×C₆₁	C₂²	1	2	yes	—
29	C₂×C₂²⋊C₄×C₆₁	C₂²	1	2	yes	—
30	C₂×C₂²⋊C₄×C₆₁	C₂²	1	2	yes	—
31	C₂×C₂²⋊C₄×C₆₁	C₂²	1	2	yes	—
32	C₂×C₂²⋊C₄×C₆₁	C₂²	1	2	yes	—
33	C₂×C₄²×C₆₁	C₂²	1	1	yes	characteristic
34	C₂×C₄²×C₆₁	C₂²	1	1	yes	characteristic
35	C₂×C₄²×C₆₁	C₂²	1	2	yes	—
36	#95×C₆₁	C₂²	1	1	yes	characteristic
37	#95×C₆₁	C₂²	1	1	yes	characteristic
38	#95×C₆₁	C₂²	1	2	yes	—
39	C₂⁴×C₆₁	C₂³	1	2	yes	—
40	C₂⁴×C₆₁	C₂³	1	1	yes	characteristic
41	C₂×Q₈×C₆₁	—	2	2	no	—
42	C₂×Q₈×C₆₁	—	2	2	no	—
43	C₂×Q₈×C₆₁	D₄	1	4	yes	complemented
44	C₂×D₄×C₆₁	—	2	4	no	—
45	C₂×D₄×C₆₁	—	2	4	no	—
46	C₂×D₄×C₆₁	—	2	2	no	—
47	C₂×D₄×C₆₁	—	2	2	no	—
48	C₂×D₄×C₆₁	—	2	2	no	—
49	C₂×D₄×C₆₁	—	4	2	no	—
50	C₂²×C₄×C₆₁	—	2	2	no	—
51	C₂²×C₄×C₆₁	—	2	2	no	—
52	C₂²×C₄×C₆₁	—	2	2	no	—
53	C₂²×C₄×C₆₁	C₂³	1	1	yes	characteristic
54	C₂²×C₄×C₆₁	C₂³	1	1	yes	characteristic
55	C₂²×C₄×C₆₁	C₂³	1	1	yes	characteristic
56	C₂²×C₄×C₆₁	C₂³	1	1	yes	characteristic
57	C₂²×C₄×C₆₁	C₂³	1	2	yes	—
58	C₂²×C₄×C₆₁	C₂³	1	2	yes	—
59	C₂²×C₄×C₆₁	C₂³	1	2	yes	—
60	C₂²×C₄×C₆₁	C₂³	1	1	yes	characteristic
61	C₂²×C₄×C₆₁	C₂³	1	1	yes	characteristic
62	C₄⋊C₄×C₆₁	—	2	4	no	—
63	C₄⋊C₄×C₆₁	—	2	2	no	—
64	C₄⋊C₄×C₆₁	—	2	4	no	—
65	C₄⋊C₄×C₆₁	—	4	2	no	—
66	C₄⋊C₄×C₆₁	D₄	1	4	yes	—
67	C₂²⋊C₄×C₆₁	—	2	2	no	—
68	C₂²⋊C₄×C₆₁	—	2	4	no	—
69	C₂²⋊C₄×C₆₁	—	2	4	no	—
70	C₂²⋊C₄×C₆₁	—	2	4	no	—
71	C₂²⋊C₄×C₆₁	—	2	4	no	—
72	C₂²⋊C₄×C₆₁	—	2	2	no	—
73	C₂²⋊C₄×C₆₁	—	2	4	no	—
74	C₂²⋊C₄×C₆₁	—	4	2	no	—
75	C₄²×C₆₁	—	2	4	no	—
76	C₄²×C₆₁	—	2	2	no	—
77	C₄²×C₆₁	—	2	2	no	—
78	C₂³×C₆₁	—	2	1	no	—
79	C₂³×C₆₁	—	2	1	no	—
80	C₂³×C₆₁	—	2	1	no	—
81	C₂³×C₆₁	—	2	1	no	—
82	C₂³×C₆₁	—	2	1	no	—
83	C₂³×C₆₁	—	2	1	no	—
84	C₂³×C₆₁	—	2	2	no	—
85	C₂³×C₆₁	—	2	2	no	—
86	C₂³×C₆₁	—	2	2	no	—
87	C₂³×C₆₁	—	2	2	no	—
88	C₂³×C₆₁	—	2	2	no	—
89	C₂³×C₆₁	—	2	2	no	—
90	C₂³×C₆₁	—	2	2	no	—
91	C₂³×C₆₁	#41	1	2	yes	—
92	C₂³×C₆₁	C₂⁴	1	1	yes	characteristic
93	Q₈×C₆₁	—	2	8	no	—
94	D₄×C₆₁	—	4	8	no	—
95	D₄×C₆₁	—	4	4	no	—
96	C₂×C₄×C₆₁	—	2	1	no	—
97	C₂×C₄×C₆₁	—	2	2	no	—
98	C₂×C₄×C₆₁	—	2	2	no	—
99	C₂×C₄×C₆₁	—	2	2	no	—
100	C₂×C₄×C₆₁	—	2	2	no	—
101	C₂×C₄×C₆₁	—	2	1	no	—
102	C₂×C₄×C₆₁	—	2	1	no	—
103	C₂×C₄×C₆₁	—	2	1	no	—
104	C₂×C₄×C₆₁	—	2	1	no	—
105	C₂×C₄×C₆₁	—	2	1	no	—
106	C₂×C₄×C₆₁	—	2	2	no	—
107	C₂×C₄×C₆₁	—	2	1	no	—
108	C₂×C₄×C₆₁	—	2	1	no	—
109	C₂×C₄×C₆₁	—	2	1	no	—
110	C₂×C₄×C₆₁	—	2	1	no	—
111	C₂×C₄×C₆₁	—	2	2	no	—
112	C₂×C₄×C₆₁	—	2	1	no	—
113	C₂×C₄×C₆₁	—	2	1	no	—
114	C₂×C₄×C₆₁	—	2	2	no	—
115	C₂×C₄×C₆₁	—	2	1	no	—
116	C₂×C₄×C₆₁	—	2	1	no	—
117	C₂×C₄×C₆₁	—	4	4	no	—
118	C₂×C₄×C₆₁	C₂×D₄	1	4	yes	—
119	C₂×C₄×C₆₁	C₂×D₄	1	4	yes	—
120	C₂×C₄×C₆₁	C₂×D₄	1	2	yes	—
121	C₂×C₄×C₆₁	C₂×D₄	1	2	yes	—
122	C₂×C₄×C₆₁	#41	1	2	yes	—
123	C₂×C₄×C₆₁	#41	1	2	yes	—
124	C₂²×C₆₁	—	2	2	no	—
125	C₂²×C₆₁	—	2	2	no	—
126	C₂²×C₆₁	—	2	2	no	—
127	C₂²×C₆₁	—	4	1	no	—
128	C₂²×C₆₁	—	4	1	no	—
129	C₂²×C₆₁	—	4	1	no	—
130	C₂²×C₆₁	—	4	1	no	—
131	C₂²×C₆₁	—	4	2	no	—
132	C₂²×C₆₁	—	4	2	no	—
133	C₂²×C₆₁	—	4	2	no	—
134	C₂²×C₆₁	—	4	2	no	—
135	C₂²×C₆₁	—	4	2	no	—
136	C₂²×C₆₁	—	4	2	no	—
137	C₂²×C₆₁	—	4	2	no	—
138	C₂²×C₆₁	C₂²×D₄	1	1	yes	characteristic
139	C₂²×C₆₁	C₂²×D₄	1	1	yes	characteristic
140	C₂²×C₆₁	C₂×#41	1	1	yes	characteristic
141	C₂²×C₆₁	C₂×#41	1	1	yes	characteristic
142	C₂²×C₆₁	C₂×#41	1	1	yes	characteristic
143	C₂²×C₆₁	H(5,𝔽₂)	1	1	yes	characteristic
144	C₂²×C₆₁	2₋¹⁺⁴	1	1	yes	characteristic
145	C₄×C₆₁	—	2	4	no	—
146	C₄×C₆₁	—	2	4	no	—
147	C₄×C₆₁	—	2	2	no	—
148	C₄×C₆₁	—	4	1	no	—
149	C₄×C₆₁	—	4	1	no	—
150	C₄×C₆₁	—	4	1	no	—
151	C₄×C₆₁	—	4	1	no	—
152	C₄×C₆₁	—	4	2	no	—
153	C₄×C₆₁	#121	1	4	yes	—
154	#2426	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
155	C₂×C₆₁	—	4	2	no	—
156	C₂×C₆₁	—	8	2	no	—
157	C₂×C₆₁	C₂×#121	1	1	yes	characteristic
158	C₂×C₆₁	#529	1	1	yes	characteristic
159	C₂×C₆₁	#542	1	1	yes	characteristic
160	C₂×C₆₁	#546	1	1	yes	characteristic
161	C₂×C₆₁	#548	1	1	yes	characteristic
162	C₂×C₆₁	#548	1	1	yes	characteristic
163	C₂×C₆₁	#550	1	1	yes	characteristic
164	C₂×#124	C₂×C₆₁	1	1	yes	characteristic, complemented
165	C₂×C₂²⋊Q₈	C₂×C₆₁	1	1	yes	characteristic, complemented
166	C₂×#121	C₂×C₆₁	1	1	yes	characteristic
167	C₂×C₄×Q₈	C₂×C₆₁	1	1	yes	characteristic, complemented
168	#396	C₂×C₆₁	1	2	yes	complemented
169	#394	C₂×C₆₁	1	2	yes	complemented
170	#390	C₂×C₆₁	1	1	yes	characteristic, complemented
171	#390	C₂×C₆₁	1	2	yes	complemented
172	#389	C₂×C₆₁	1	1	yes	characteristic, complemented
173	#388	C₂×C₆₁	1	2	yes	complemented
174	C₄×C₂²⋊C₄	C₂×C₆₁	1	1	yes	characteristic, complemented
175	C₆₁	#2426	1	1	yes	characteristic, direct product factor, Sylow
176	C₂²×Q₈	C₂²×C₆₁	1	1	yes	characteristic
177	C₂²×D₄	C₂²×C₆₁	1	1	yes	characteristic
178	C₂²×D₄	C₂²×C₆₁	1	2	yes	—
179	C₂³×C₄	C₂²×C₆₁	1	1	yes	characteristic
180	#124	—	2	4	no	—
181	C₂²⋊Q₈	—	2	4	no	—
182	#121	—	2	4	no	—
183	C₄×Q₈	—	2	4	no	—
184	C₂×C₄⋊C₄	C₂²×C₆₁	1	2	yes	—
185	C₂×C₄⋊C₄	C₂²×C₆₁	1	2	yes	—
186	C₂×C₄⋊C₄	C₂²×C₆₁	1	1	yes	characteristic
187	C₂×C₄⋊C₄	C₂²×C₆₁	1	2	yes	—
188	C₂×C₄⋊C₄	C₂²×C₆₁	1	1	yes	characteristic
189	C₂×C₂²⋊C₄	C₂²×C₆₁	1	2	yes	—
190	C₂×C₂²⋊C₄	C₂²×C₆₁	1	2	yes	—
191	C₂×C₂²⋊C₄	C₂²×C₆₁	1	2	yes	—
192	C₂×C₂²⋊C₄	C₂²×C₆₁	1	2	yes	—
193	C₂×C₂²⋊C₄	C₂²×C₆₁	1	1	yes	characteristic
194	C₂×C₂²⋊C₄	C₂²×C₆₁	1	1	yes	characteristic
195	C₂×C₂²⋊C₄	C₂²×C₆₁	1	2	yes	—
196	C₂×C₂²⋊C₄	C₂²×C₆₁	1	2	yes	—
197	C₂×C₄²	C₂²×C₆₁	1	1	yes	characteristic
198	C₂×C₄²	C₂²×C₆₁	1	2	yes	—
199	C₂×C₄²	C₂²×C₆₁	1	1	yes	characteristic
200	#95	C₂²×C₆₁	1	1	yes	characteristic
201	#95	C₂²×C₆₁	1	1	yes	characteristic
202	#95	C₂²×C₆₁	1	2	yes	—
203	C₂⁴	C₂³×C₆₁	1	2	yes	—
204	C₂⁴	C₂³×C₆₁	1	1	yes	characteristic
205	C₂×Q₈	—	2	2	no	—
206	C₂×Q₈	—	2	2	no	—
207	C₂×Q₈	D₄×C₆₁	1	4	yes	complemented
208	C₂×D₄	—	2	4	no	—
209	C₂×D₄	—	2	4	no	—
210	C₂×D₄	—	2	2	no	—
211	C₂×D₄	—	2	2	no	—
212	C₂×D₄	—	2	2	no	—
213	C₂×D₄	—	4	2	no	—
214	C₂²×C₄	—	2	2	no	—
215	C₂²×C₄	—	2	2	no	—
216	C₂²×C₄	—	2	2	no	—
217	C₂²×C₄	C₂³×C₆₁	1	1	yes	characteristic
218	C₂²×C₄	C₂³×C₆₁	1	1	yes	characteristic
219	C₂²×C₄	C₂³×C₆₁	1	1	yes	characteristic
220	C₂²×C₄	C₂³×C₆₁	1	1	yes	characteristic
221	C₂²×C₄	C₂³×C₆₁	1	2	yes	—
222	C₂²×C₄	C₂³×C₆₁	1	2	yes	—
223	C₂²×C₄	C₂³×C₆₁	1	2	yes	—
224	C₂²×C₄	C₂³×C₆₁	1	1	yes	characteristic
225	C₂²×C₄	C₂³×C₆₁	1	1	yes	characteristic
226	C₄⋊C₄	—	2	4	no	—
227	C₄⋊C₄	—	2	2	no	—
228	C₄⋊C₄	—	2	4	no	—
229	C₄⋊C₄	—	4	2	no	—
230	C₄⋊C₄	D₄×C₆₁	1	4	yes	—
231	C₂²⋊C₄	—	2	2	no	—
232	C₂²⋊C₄	—	2	4	no	—
233	C₂²⋊C₄	—	2	4	no	—
234	C₂²⋊C₄	—	2	4	no	—
235	C₂²⋊C₄	—	2	4	no	—
236	C₂²⋊C₄	—	2	2	no	—
237	C₂²⋊C₄	—	2	4	no	—
238	C₂²⋊C₄	—	4	2	no	—
239	C₄²	—	2	4	no	—
240	C₄²	—	2	2	no	—
241	C₄²	—	2	2	no	—
242	C₂³	—	2	1	no	—
243	C₂³	—	2	1	no	—
244	C₂³	—	2	1	no	—
245	C₂³	—	2	1	no	—
246	C₂³	—	2	1	no	—
247	C₂³	—	2	1	no	—
248	C₂³	—	2	2	no	—
249	C₂³	—	2	2	no	—
250	C₂³	—	2	2	no	—
251	C₂³	—	2	2	no	—
252	C₂³	—	2	2	no	—
253	C₂³	—	2	2	no	—
254	C₂³	—	2	2	no	—
255	C₂³	#41×C₆₁	1	2	yes	—
256	C₂³	C₂⁴×C₆₁	1	1	yes	characteristic
257	Q₈	—	2	8	no	—
258	D₄	—	4	8	no	—
259	D₄	—	4	4	no	—
260	C₂×C₄	—	2	1	no	—
261	C₂×C₄	—	2	2	no	—
262	C₂×C₄	—	2	2	no	—
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	—	2	2	no	—
265	C₂×C₄	—	2	1	no	—
266	C₂×C₄	—	2	1	no	—
267	C₂×C₄	—	2	1	no	—
268	C₂×C₄	—	2	1	no	—
269	C₂×C₄	—	2	1	no	—
270	C₂×C₄	—	2	2	no	—
271	C₂×C₄	—	2	1	no	—
272	C₂×C₄	—	2	1	no	—
273	C₂×C₄	—	2	1	no	—
274	C₂×C₄	—	2	1	no	—
275	C₂×C₄	—	2	2	no	—
276	C₂×C₄	—	2	1	no	—
277	C₂×C₄	—	2	1	no	—
278	C₂×C₄	—	2	2	no	—
279	C₂×C₄	—	2	1	no	—
280	C₂×C₄	—	2	1	no	—
281	C₂×C₄	—	4	4	no	—
282	C₂×C₄	C₂×D₄×C₆₁	1	4	yes	—
283	C₂×C₄	C₂×D₄×C₆₁	1	4	yes	—
284	C₂×C₄	C₂×D₄×C₆₁	1	2	yes	—
285	C₂×C₄	C₂×D₄×C₆₁	1	2	yes	—
286	C₂×C₄	#41×C₆₁	1	2	yes	—
287	C₂×C₄	#41×C₆₁	1	2	yes	—
288	C₂²	—	2	2	no	—
289	C₂²	—	2	2	no	—
290	C₂²	—	2	2	no	—
291	C₂²	—	4	1	no	—
292	C₂²	—	4	1	no	—
293	C₂²	—	4	1	no	—
294	C₂²	—	4	1	no	—
295	C₂²	—	4	2	no	—
296	C₂²	—	4	2	no	—
297	C₂²	—	4	2	no	—
298	C₂²	—	4	2	no	—
299	C₂²	—	4	2	no	—
300	C₂²	—	4	2	no	—
301	C₂²	—	4	2	no	—
302	C₂²	C₂²×D₄×C₆₁	1	1	yes	characteristic
303	C₂²	C₂²×D₄×C₆₁	1	1	yes	characteristic
304	C₂²	C₂×#41×C₆₁	1	1	yes	characteristic
305	C₂²	C₂×#41×C₆₁	1	1	yes	characteristic
306	C₂²	C₂×#41×C₆₁	1	1	yes	characteristic
307	C₂²	H(5,𝔽₂)×C₆₁	1	1	yes	characteristic
308	C₂²	2₋¹⁺⁴×C₆₁	1	1	yes	characteristic
309	C₄	—	2	4	no	—
310	C₄	—	2	2	no	—
311	C₄	—	2	4	no	—
312	C₄	—	4	1	no	—
313	C₄	—	4	1	no	—
314	C₄	—	4	1	no	—
315	C₄	—	4	2	no	—
316	C₄	—	4	1	no	—
317	C₄	#121×C₆₁	1	4	yes	—
318	C₂	—	4	2	no	—
319	C₂	—	8	2	no	—
320	C₂	C₂×#121×C₆₁	1	1	yes	characteristic
321	C₂	C₆₁×#529	1	1	yes	characteristic
322	C₂	C₆₁×#542	1	1	yes	characteristic
323	C₂	C₆₁×#546	1	1	yes	characteristic
324	C₂	C₆₁×#548	1	1	yes	characteristic
325	C₂	C₆₁×#548	1	1	yes	characteristic
326	C₂	C₆₁×#550	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₆₁	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₆₁
prime radical	C₂×#121×C₆₁	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×#124×C₆₁	C₂	semidirect
2. C₂×C₂²⋊Q₈×C₆₁	C₂	semidirect
4. C₂×C₄×Q₈×C₆₁	C₂	semidirect
5. C₆₁×#396	C₂	semidirect
6. C₆₁×#394	C₂	semidirect
7. C₆₁×#390	C₂	semidirect
8. C₆₁×#390	C₂	semidirect
9. C₆₁×#389	C₂	semidirect
10. C₆₁×#388	C₂	semidirect
11. C₄×C₂²⋊C₄×C₆₁	C₂	semidirect
43. C₂×Q₈×C₆₁	D₄	semidirect
154. #2426	C₆₁	direct (N × H)
164. C₂×#124	C₂×C₆₁	semidirect
165. C₂×C₂²⋊Q₈	C₂×C₆₁	semidirect
167. C₂×C₄×Q₈	C₂×C₆₁	semidirect
168. #396	C₂×C₆₁	semidirect
169. #394	C₂×C₆₁	semidirect
170. #390	C₂×C₆₁	semidirect
171. #390	C₂×C₆₁	semidirect
172. #389	C₂×C₆₁	semidirect
173. #388	C₂×C₆₁	semidirect
174. C₄×C₂²⋊C₄	C₂×C₆₁	semidirect
175. C₆₁	#2426	direct (N × H)
207. C₂×Q₈	D₄×C₆₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×#124×C₆₁	2
C₂×C₂²⋊Q₈×C₆₁	2
C₂×#121×C₆₁	2
C₂×C₄×Q₈×C₆₁	2
C₆₁×#396	2
C₆₁×#394	2
C₆₁×#390	2
C₆₁×#389	2
C₆₁×#388	2
C₄×C₂²⋊C₄×C₆₁	2
#2426	61

Subgroups — C61×#2426 #21442467

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#2426 #21442467