Subgroups — C₆₁×#29126 #29532559

Order 15616.

686 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₁×#2864	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊Q₈×C₆₁	C₂	1	1	yes	maximal, characteristic
3	C₆₁×#2099	C₂	1	2	yes	maximal
4	C₆₁×#2043	C₂	1	2	yes	maximal
5	C₆₁×#1933	C₂	1	4	yes	maximal
6	C₆₁×#1805	C₂	1	2	yes	maximal
7	C₆₁×#1805	C₂	1	2	yes	maximal
8	C₄×C₄⋊C₈×C₆₁	C₂	1	1	yes	maximal, characteristic
9	C₂×C₄⋊Q₈×C₆₁	C₂²	1	1	yes	characteristic
10	C₂×C₄×Q₈×C₆₁	C₂²	1	2	yes	—
11	C₂×C₄⋊C₈×C₆₁	C₂²	1	2	yes	—
12	C₂×#103×C₆₁	C₂²	1	4	yes	—
13	C₂×#103×C₆₁	C₂²	1	4	yes	—
14	C₂×C₄×C₈×C₆₁	C₂²	1	2	yes	—
15	C₆₁×#391	C₂²	1	2	yes	—
16	C₆₁×#389	—	2	2	no	—
17	C₆₁×#389	C₂²	1	2	yes	—
18	C₆₁×#389	C₂²	1	2	yes	—
19	C₆₁×#387	—	2	2	no	—
20	C₆₁×#387	—	2	2	no	—
21	C₆₁×#384	C₂²	1	1	yes	characteristic
22	C₆₁×#382	C₂²	1	1	yes	characteristic
23	C₆₁×#382	C₂²	1	1	yes	characteristic
24	C₄×C₄⋊C₄×C₆₁	—	2	2	no	—
25	C₄³×C₆₁	C₂²	1	1	yes	characteristic
26	C₆₁×#340	C₂²	1	4	yes	—
27	C₆₁×#340	C₂²	1	4	yes	—
28	C₆₁×#336	C₂²	1	2	yes	—
29	C₂²×Q₈×C₆₁	C₂³	1	2	yes	—
30	C₂²×C₈×C₆₁	C₂³	1	4	yes	—
31	C₄⋊Q₈×C₆₁	—	2	4	no	—
32	C₄×Q₈×C₆₁	—	2	4	no	—
33	C₄×Q₈×C₆₁	—	2	4	no	—
34	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
35	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
36	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
37	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
38	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
39	C₂×C₄⋊C₄×C₆₁	C₂³	1	2	yes	—
40	C₂×C₄⋊C₄×C₆₁	C₂³	1	2	yes	—
41	C₂×C₄⋊C₄×C₆₁	C₂³	1	2	yes	—
42	C₂×C₄²×C₆₁	—	2	2	no	—
43	C₂×C₄²×C₆₁	D₄	1	2	yes	—
44	C₂×C₄²×C₆₁	D₄	1	2	yes	—
45	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
46	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
47	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
48	C₄⋊C₈×C₆₁	—	2	4	no	—
49	#103×C₆₁	—	2	8	no	—
50	#103×C₆₁	—	4	4	no	—
51	C₄×C₈×C₆₁	—	4	2	no	—
52	#95×C₆₁	—	2	4	no	—
53	#95×C₆₁	—	2	1	no	—
54	#95×C₆₁	—	2	1	no	—
55	#95×C₆₁	—	2	1	no	—
56	#95×C₆₁	—	2	1	no	—
57	#95×C₆₁	—	2	2	no	—
58	#95×C₆₁	—	2	2	no	—
59	C₂×Q₈×C₆₁	—	2	2	no	—
60	C₂×Q₈×C₆₁	—	2	2	no	—
61	C₂×Q₈×C₆₁	—	2	2	no	—
62	C₂×Q₈×C₆₁	—	2	2	no	—
63	C₂×Q₈×C₆₁	—	2	2	no	—
64	C₂×Q₈×C₆₁	#41	1	4	yes	—
65	C₂²×C₄×C₆₁	—	2	2	no	—
66	C₂²×C₄×C₆₁	—	2	2	no	—
67	C₂²×C₄×C₆₁	—	2	2	no	—
68	C₂²×C₄×C₆₁	—	2	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	C₂×D₄	1	2	yes	—
72	C₂²×C₄×C₆₁	C₂×D₄	1	2	yes	—
73	C₂²×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
74	C₂×C₈×C₆₁	—	2	4	no	—
75	C₂×C₈×C₆₁	—	2	4	no	—
76	C₂×C₈×C₆₁	—	2	4	no	—
77	C₄⋊C₄×C₆₁	—	2	8	no	—
78	C₄⋊C₄×C₆₁	—	2	4	no	—
79	C₄⋊C₄×C₆₁	—	2	2	no	—
80	C₄⋊C₄×C₆₁	—	2	2	no	—
81	C₄⋊C₄×C₆₁	—	2	2	no	—
82	C₄⋊C₄×C₆₁	—	2	2	no	—
83	C₄⋊C₄×C₆₁	—	4	4	no	—
84	C₄⋊C₄×C₆₁	—	4	4	no	—
85	C₄⋊C₄×C₆₁	—	4	4	no	—
86	C₄⋊C₄×C₆₁	—	4	4	no	—
87	C₄²×C₆₁	—	2	4	no	—
88	C₄²×C₆₁	—	2	4	no	—
89	C₄²×C₆₁	—	2	2	no	—
90	C₄²×C₆₁	—	2	1	no	—
91	C₄²×C₆₁	—	2	1	no	—
92	C₄²×C₆₁	—	4	4	no	—
93	C₄²×C₆₁	#41	1	4	yes	—
94	C₂³×C₆₁	C₂²×D₄	1	1	yes	characteristic
95	Q₈×C₆₁	—	2	4	no	—
96	Q₈×C₆₁	—	2	4	no	—
97	Q₈×C₆₁	—	4	2	no	—
98	Q₈×C₆₁	—	4	2	no	—
99	C₂×C₄×C₆₁	—	2	2	no	—
100	C₂×C₄×C₆₁	—	2	2	no	—
101	C₂×C₄×C₆₁	—	2	2	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	4	no	—
106	C₂×C₄×C₆₁	—	4	2	no	—
107	C₂×C₄×C₆₁	—	4	2	no	—
108	C₂×C₄×C₆₁	—	4	2	no	—
109	C₂×C₄×C₆₁	—	4	2	no	—
110	C₂×C₄×C₆₁	—	4	2	no	—
111	C₂×C₄×C₆₁	—	4	2	no	—
112	C₂×C₄×C₆₁	—	4	4	no	—
113	C₂×C₄×C₆₁	—	4	2	no	—
114	C₂×C₄×C₆₁	—	4	2	no	—
115	C₂×C₄×C₆₁	—	4	2	no	—
116	C₂×C₄×C₆₁	#124	1	2	yes	—
117	C₂×C₄×C₆₁	#124	1	2	yes	—
118	C₂×C₄×C₆₁	#135	1	4	yes	—
119	C₂×C₄×C₆₁	#137	1	4	yes	—
120	C₂×C₄×C₆₁	C₂×#41	1	2	yes	—
121	C₂×C₄×C₆₁	C₂×#41	1	2	yes	—
122	C₂×C₄×C₆₁	H(5,𝔽₂)	1	1	yes	characteristic
123	C₂×C₄×C₆₁	2₋¹⁺⁴	1	1	yes	characteristic
124	C₈×C₆₁	—	4	4	no	—
125	#29126	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
126	C₂²×C₆₁	#525	1	1	yes	characteristic
127	C₂²×C₆₁	C₂×#124	1	1	yes	characteristic
128	C₂²×C₆₁	#537	1	1	yes	characteristic
129	C₂²×C₆₁	C₂×#135	1	2	yes	—
130	C₂²×C₆₁	C₂×#137	1	2	yes	—
131	C₄×C₆₁	—	2	4	no	—
132	C₄×C₆₁	—	2	2	no	—
133	C₄×C₆₁	—	2	2	no	—
134	C₄×C₆₁	—	2	4	no	—
135	C₄×C₆₁	—	4	4	no	—
136	C₄×C₆₁	—	8	2	no	—
137	C₄×C₆₁	—	8	2	no	—
138	C₄×C₆₁	—	8	2	no	—
139	C₄×C₆₁	#552	1	2	yes	—
140	C₄×C₆₁	#553	1	2	yes	—
141	#2864	C₂×C₆₁	1	1	yes	characteristic
142	C₄×C₄⋊Q₈	C₂×C₆₁	1	1	yes	characteristic
143	#2099	C₂×C₆₁	1	2	yes	—
144	#2043	C₂×C₆₁	1	2	yes	—
145	#1933	C₂×C₆₁	1	4	yes	—
146	#1805	C₂×C₆₁	1	2	yes	—
147	#1805	C₂×C₆₁	1	2	yes	—
148	C₄×C₄⋊C₈	C₂×C₆₁	1	1	yes	characteristic
149	C₂×C₆₁	#2562	1	1	yes	characteristic
150	C₂×C₆₁	#3108	1	2	yes	—
151	C₂×C₆₁	#3131	1	1	yes	characteristic
152	C₂×C₆₁	#3139	1	1	yes	characteristic
153	C₂×C₆₁	#3248	1	2	yes	—
154	C₂×C₄⋊Q₈	C₂²×C₆₁	1	1	yes	characteristic
155	C₂×C₄×Q₈	C₂²×C₆₁	1	2	yes	—
156	C₂×C₄⋊C₈	C₂²×C₆₁	1	2	yes	—
157	C₂×#103	C₂²×C₆₁	1	4	yes	—
158	C₂×#103	C₂²×C₆₁	1	4	yes	—
159	C₂×C₄×C₈	C₂²×C₆₁	1	2	yes	—
160	#391	C₂²×C₆₁	1	2	yes	—
161	#389	—	2	2	no	—
162	#389	C₂²×C₆₁	1	2	yes	—
163	#389	C₂²×C₆₁	1	2	yes	—
164	#387	—	2	2	no	—
165	#387	—	2	2	no	—
166	#384	C₂²×C₆₁	1	1	yes	characteristic
167	#382	C₂²×C₆₁	1	1	yes	characteristic
168	#382	C₂²×C₆₁	1	1	yes	characteristic
169	C₄×C₄⋊C₄	—	2	2	no	—
170	C₄³	C₂²×C₆₁	1	1	yes	characteristic
171	#340	C₂²×C₆₁	1	4	yes	—
172	#340	C₂²×C₆₁	1	4	yes	—
173	#336	C₂²×C₆₁	1	2	yes	—
174	C₆₁	#29126	1	1	yes	characteristic, direct product factor, Sylow
175	C₂²×Q₈	C₂³×C₆₁	1	2	yes	—
176	C₂²×C₈	C₂³×C₆₁	1	4	yes	—
177	C₄⋊Q₈	—	2	4	no	—
178	C₄×Q₈	—	2	4	no	—
179	C₄×Q₈	—	2	4	no	—
180	C₂×C₄⋊C₄	—	2	2	no	—
181	C₂×C₄⋊C₄	—	2	2	no	—
182	C₂×C₄⋊C₄	—	2	2	no	—
183	C₂×C₄⋊C₄	—	2	2	no	—
184	C₂×C₄⋊C₄	—	2	2	no	—
185	C₂×C₄⋊C₄	C₂³×C₆₁	1	2	yes	—
186	C₂×C₄⋊C₄	C₂³×C₆₁	1	2	yes	—
187	C₂×C₄⋊C₄	C₂³×C₆₁	1	2	yes	—
188	C₂×C₄²	—	2	2	no	—
189	C₂×C₄²	D₄×C₆₁	1	2	yes	—
190	C₂×C₄²	D₄×C₆₁	1	2	yes	—
191	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
192	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
193	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
194	C₄⋊C₈	—	2	4	no	—
195	#103	—	2	8	no	—
196	#103	—	4	4	no	—
197	C₄×C₈	—	4	2	no	—
198	#95	—	2	4	no	—
199	#95	—	2	1	no	—
200	#95	—	2	1	no	—
201	#95	—	2	1	no	—
202	#95	—	2	1	no	—
203	#95	—	2	2	no	—
204	#95	—	2	2	no	—
205	C₂×Q₈	—	2	2	no	—
206	C₂×Q₈	—	2	2	no	—
207	C₂×Q₈	—	2	2	no	—
208	C₂×Q₈	—	2	2	no	—
209	C₂×Q₈	—	2	2	no	—
210	C₂×Q₈	#41×C₆₁	1	4	yes	—
211	C₂²×C₄	—	2	2	no	—
212	C₂²×C₄	—	2	2	no	—
213	C₂²×C₄	—	2	2	no	—
214	C₂²×C₄	—	2	2	no	—
215	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
216	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
217	C₂²×C₄	C₂×D₄×C₆₁	1	2	yes	—
218	C₂²×C₄	C₂×D₄×C₆₁	1	2	yes	—
219	C₂²×C₄	C₂⁴×C₆₁	1	1	yes	characteristic
220	C₂×C₈	—	2	4	no	—
221	C₂×C₈	—	2	4	no	—
222	C₂×C₈	—	2	4	no	—
223	C₄⋊C₄	—	2	8	no	—
224	C₄⋊C₄	—	2	4	no	—
225	C₄⋊C₄	—	2	2	no	—
226	C₄⋊C₄	—	2	2	no	—
227	C₄⋊C₄	—	2	2	no	—
228	C₄⋊C₄	—	2	2	no	—
229	C₄⋊C₄	—	4	4	no	—
230	C₄⋊C₄	—	4	4	no	—
231	C₄⋊C₄	—	4	4	no	—
232	C₄⋊C₄	—	4	4	no	—
233	C₄²	—	2	4	no	—
234	C₄²	—	2	4	no	—
235	C₄²	—	2	2	no	—
236	C₄²	—	2	1	no	—
237	C₄²	—	2	1	no	—
238	C₄²	—	4	4	no	—
239	C₄²	#41×C₆₁	1	4	yes	—
240	C₂³	C₂²×D₄×C₆₁	1	1	yes	characteristic
241	Q₈	—	2	4	no	—
242	Q₈	—	2	4	no	—
243	Q₈	—	4	2	no	—
244	Q₈	—	4	2	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	2	2	no	—
248	C₂×C₄	—	2	2	no	—
249	C₂×C₄	—	2	2	no	—
250	C₂×C₄	—	2	2	no	—
251	C₂×C₄	—	2	4	no	—
252	C₂×C₄	—	4	2	no	—
253	C₂×C₄	—	4	2	no	—
254	C₂×C₄	—	4	2	no	—
255	C₂×C₄	—	4	2	no	—
256	C₂×C₄	—	4	4	no	—
257	C₂×C₄	—	4	2	no	—
258	C₂×C₄	—	4	2	no	—
259	C₂×C₄	—	4	2	no	—
260	C₂×C₄	—	4	2	no	—
261	C₂×C₄	—	4	2	no	—
262	C₂×C₄	#124×C₆₁	1	2	yes	—
263	C₂×C₄	#124×C₆₁	1	2	yes	—
264	C₂×C₄	#135×C₆₁	1	4	yes	—
265	C₂×C₄	#137×C₆₁	1	4	yes	—
266	C₂×C₄	C₂×#41×C₆₁	1	2	yes	—
267	C₂×C₄	C₂×#41×C₆₁	1	2	yes	—
268	C₂×C₄	H(5,𝔽₂)×C₆₁	1	1	yes	characteristic
269	C₂×C₄	2₋¹⁺⁴×C₆₁	1	1	yes	characteristic
270	C₈	—	4	4	no	—
271	C₂²	C₆₁×#525	1	1	yes	characteristic
272	C₂²	C₂×#124×C₆₁	1	1	yes	characteristic
273	C₂²	C₆₁×#537	1	1	yes	characteristic
274	C₂²	C₂×#135×C₆₁	1	2	yes	—
275	C₂²	C₂×#137×C₆₁	1	2	yes	—
276	C₄	—	2	4	no	—
277	C₄	—	2	2	no	—
278	C₄	—	2	2	no	—
279	C₄	—	2	4	no	—
280	C₄	—	4	4	no	—
281	C₄	—	8	2	no	—
282	C₄	—	8	2	no	—
283	C₄	—	8	2	no	—
284	C₄	C₆₁×#552	1	2	yes	—
285	C₄	C₆₁×#553	1	2	yes	—
286	C₂	C₆₁×#2562	1	1	yes	characteristic
287	C₂	C₆₁×#3108	1	2	yes	—
288	C₂	C₆₁×#3131	1	1	yes	characteristic
289	C₂	C₆₁×#3139	1	1	yes	characteristic
290	C₂	C₆₁×#3248	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₆₁	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₁
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₁ ◁ C₄³×C₆₁ ◁ G

Complements

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

N	H ≅ G/N	type
125. #29126	C₆₁	direct (N × H)
174. C₆₁	#29126	direct (N × H)

Maximal subgroups

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

subgroup	index
C₆₁×#2864	2
C₄×C₄⋊Q₈×C₆₁	2
C₆₁×#2099	2
C₆₁×#2043	2
C₆₁×#1933	2
C₆₁×#1805	2
C₄×C₄⋊C₈×C₆₁	2
#29126	61

Subgroups — C61×#29126 #29532559

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#29126 #29532559