Subgroups — C₆₁×#13201 #29516634

Order 15616.

420 conjugacy classes of subgroups.

The subgroup lattice has 278 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₁×#2930	C₂	1	1	yes	maximal, characteristic
2	C₆₁×#2153	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₁×#2117	C₂	1	2	yes	maximal, complemented
4	C₂×C₆₁×#349	C₂	1	2	yes	maximal, complemented
5	C₆₁×#2110	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#131×C₆₁	C₂²	1	1	yes	characteristic
7	C₂²×M₁₆×C₆₁	C₂²	1	1	yes	characteristic
8	C₆₁×#525	C₄	1	1	yes	characteristic
9	C₂×M₃₂×C₆₁	C₂²	1	2	yes	—
10	C₂×M₃₂×C₆₁	C₂²	1	2	yes	—
11	C₆₁×#436	—	2	2	no	—
12	C₆₁×#435	—	2	2	no	—
13	C₆₁×#424	C₄	1	1	yes	characteristic
14	C₆₁×#408	C₄	1	1	yes	characteristic
15	C₆₁×#407	C₄	1	1	yes	characteristic
16	C₆₁×#405	C₂²	1	1	yes	characteristic
17	C₆₁×#364	—	2	2	no	—
18	C₆₁×#350	—	2	4	no	—
19	C₆₁×#349	—	2	4	no	—
20	C₆₁×#347	—	2	2	no	—
21	C₂×#41×C₆₁	C₂×C₄	1	1	yes	characteristic
22	C₂³×C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
23	#131×C₆₁	—	2	2	no	—
24	#131×C₆₁	—	2	2	no	—
25	C₂×M₁₆×C₆₁	—	2	2	no	—
26	C₂×M₁₆×C₆₁	—	2	1	no	—
27	C₂×M₁₆×C₆₁	—	2	1	no	—
28	C₂×M₁₆×C₆₁	—	2	1	no	—
29	C₂×M₁₆×C₆₁	—	2	2	no	—
30	C₂×M₁₆×C₆₁	C₂×C₄	1	1	yes	characteristic
31	C₂×M₁₆×C₆₁	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₈×C₆₁	—	2	1	no	—
33	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
35	#123×C₆₁	—	2	1	no	—
36	C₂²⋊Q₈×C₆₁	—	2	1	no	—
37	#121×C₆₁	—	2	1	no	—
38	#120×C₆₁	—	2	1	no	—
39	C₄×D₄×C₆₁	—	2	2	no	—
40	#117×C₆₁	C₂×C₄	1	1	yes	characteristic
41	M₃₂×C₆₁	—	2	2	no	—
42	M₃₂×C₆₁	—	2	2	no	—
43	M₃₂×C₆₁	—	2	2	no	—
44	M₃₂×C₆₁	D₄	1	4	yes	complemented
45	C₂×C₁₆×C₆₁	—	2	2	no	—
46	C₂×C₁₆×C₆₁	—	2	2	no	—
47	C₄⋊C₈×C₆₁	—	2	2	no	—
48	C₂²⋊C₈×C₆₁	—	2	2	no	—
49	C₂²⋊C₈×C₆₁	—	2	2	no	—
50	#97×C₆₁	D₄	1	2	yes	—
51	C₄×C₈×C₆₁	D₄	1	2	yes	—
52	C₂⁴×C₆₁	M₁₆	1	1	yes	characteristic
53	#41×C₆₁	—	2	2	no	—
54	#41×C₆₁	—	2	2	no	—
55	C₂×Q₈×C₆₁	M₁₆	1	1	yes	characteristic
56	C₂×D₄×C₆₁	—	2	1	no	—
57	C₂×D₄×C₆₁	—	2	2	no	—
58	C₂×D₄×C₆₁	M₁₆	1	1	yes	characteristic
59	C₂²×C₄×C₆₁	—	2	1	no	—
60	C₂²×C₄×C₆₁	—	2	1	no	—
61	C₂²×C₄×C₆₁	—	2	2	no	—
62	C₂²×C₄×C₆₁	—	2	1	no	—
63	C₂²×C₄×C₆₁	—	2	2	no	—
64	C₂²×C₄×C₆₁	M₁₆	1	1	yes	characteristic
65	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
66	M₁₆×C₆₁	—	2	2	no	—
67	M₁₆×C₆₁	—	2	2	no	—
68	M₁₆×C₆₁	—	4	2	no	—
69	M₁₆×C₆₁	—	4	1	no	—
70	M₁₆×C₆₁	—	4	1	no	—
71	M₁₆×C₆₁	—	4	1	no	—
72	M₁₆×C₆₁	—	4	1	no	—
73	C₂×C₈×C₆₁	—	2	1	no	—
74	C₂×C₈×C₆₁	—	2	1	no	—
75	C₂×C₈×C₆₁	—	2	2	no	—
76	C₂×C₈×C₆₁	—	4	2	no	—
77	C₂×C₈×C₆₁	C₂²⋊C₄	1	2	yes	—
78	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
79	C₂×C₈×C₆₁	C₂×D₄	1	2	yes	—
80	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
81	C₂×C₈×C₆₁	#41	1	2	yes	—
82	C₄⋊C₄×C₆₁	—	2	2	no	—
83	C₄⋊C₄×C₆₁	—	2	1	no	—
84	C₂²⋊C₄×C₆₁	—	2	2	no	—
85	C₂²⋊C₄×C₆₁	—	2	2	no	—
86	C₂²⋊C₄×C₆₁	—	2	1	no	—
87	C₄²×C₆₁	C₂²⋊C₄	1	2	yes	—
88	C₁₆×C₆₁	—	2	4	no	—
89	C₁₆×C₆₁	—	4	2	no	—
90	C₂³×C₆₁	—	2	1	no	—
91	C₂³×C₆₁	—	2	1	no	—
92	C₂³×C₆₁	—	2	2	no	—
93	C₂³×C₆₁	—	8	1	no	—
94	C₂³×C₆₁	C₂×M₁₆	1	1	yes	characteristic
95	Q₈×C₆₁	—	2	2	no	—
96	D₄×C₆₁	—	2	2	no	—
97	D₄×C₆₁	—	4	2	no	—
98	D₄×C₆₁	—	8	2	no	—
99	C₂×C₄×C₆₁	—	2	2	no	—
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	—	4	1	no	—
104	C₂×C₄×C₆₁	—	4	1	no	—
105	C₂×C₄×C₆₁	—	4	1	no	—
106	C₂×C₄×C₆₁	—	4	1	no	—
107	C₂×C₄×C₆₁	—	4	2	no	—
108	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
109	C₂×C₄×C₆₁	C₄×D₄	1	2	yes	—
110	C₂×C₄×C₆₁	C₂×M₁₆	1	1	yes	characteristic
111	C₂×C₄×C₆₁	#131	1	2	yes	—
112	C₈×C₆₁	—	2	2	no	—
113	C₈×C₆₁	—	4	1	no	—
114	C₈×C₆₁	—	4	1	no	—
115	C₈×C₆₁	#121	1	2	yes	—
116	C₈×C₆₁	#124	1	1	yes	characteristic
117	C₈×C₆₁	#127	1	1	yes	characteristic
118	#13201	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
119	C₂²×C₆₁	—	4	1	no	—
120	C₂²×C₆₁	—	4	1	no	—
121	C₂²×C₆₁	—	4	2	no	—
122	C₂²×C₆₁	—	4	1	no	—
123	C₂²×C₆₁	—	4	1	no	—
124	C₂²×C₆₁	—	4	1	no	—
125	C₂²×C₆₁	—	8	2	no	—
126	C₂²×C₆₁	#407	1	1	yes	characteristic
127	C₂²×C₆₁	#435	1	2	yes	—
128	C₄×C₆₁	—	4	2	no	—
129	C₄×C₆₁	—	4	1	no	—
130	C₄×C₆₁	—	4	1	no	—
131	C₄×C₆₁	#390	1	1	yes	characteristic
132	C₄×C₆₁	#408	1	1	yes	characteristic
133	C₄×C₆₁	#435	1	2	yes	—
134	#2930	C₂×C₆₁	1	1	yes	characteristic
135	#2153	C₂×C₆₁	1	1	yes	characteristic, complemented
136	#2117	C₂×C₆₁	1	2	yes	complemented
137	C₂×#349	C₂×C₆₁	1	2	yes	complemented
138	#2110	C₂×C₆₁	1	1	yes	characteristic, complemented
139	C₂×C₆₁	—	2	1	no	—
140	C₂×C₆₁	—	2	2	no	—
141	C₂×C₆₁	—	8	1	no	—
142	C₂×C₆₁	—	8	1	no	—
143	C₂×C₆₁	#1955	1	1	yes	characteristic
144	C₂×#131	C₂²×C₆₁	1	1	yes	characteristic
145	C₂²×M₁₆	C₂²×C₆₁	1	1	yes	characteristic
146	#525	C₄×C₆₁	1	1	yes	characteristic
147	C₂×M₃₂	C₂²×C₆₁	1	2	yes	—
148	C₂×M₃₂	C₂²×C₆₁	1	2	yes	—
149	#436	—	2	2	no	—
150	#435	—	2	2	no	—
151	#424	C₄×C₆₁	1	1	yes	characteristic
152	#408	C₄×C₆₁	1	1	yes	characteristic
153	#407	C₄×C₆₁	1	1	yes	characteristic
154	#405	C₂²×C₆₁	1	1	yes	characteristic
155	#364	—	2	2	no	—
156	#350	—	2	4	no	—
157	#349	—	2	4	no	—
158	#347	—	2	2	no	—
159	C₆₁	#13201	1	1	yes	characteristic, direct product factor, Sylow
160	C₂×#41	C₂×C₄×C₆₁	1	1	yes	characteristic
161	C₂³×C₄	C₂×C₄×C₆₁	1	1	yes	characteristic
162	#131	—	2	2	no	—
163	#131	—	2	2	no	—
164	C₂×M₁₆	—	2	2	no	—
165	C₂×M₁₆	—	2	1	no	—
166	C₂×M₁₆	—	2	1	no	—
167	C₂×M₁₆	—	2	1	no	—
168	C₂×M₁₆	—	2	2	no	—
169	C₂×M₁₆	C₂×C₄×C₆₁	1	1	yes	characteristic
170	C₂×M₁₆	C₂×C₄×C₆₁	1	1	yes	characteristic
171	C₂²×C₈	—	2	1	no	—
172	C₂²×C₈	C₂×C₄×C₆₁	1	1	yes	characteristic
173	C₂²×C₈	C₂³×C₆₁	1	1	yes	characteristic
174	#123	—	2	1	no	—
175	C₂²⋊Q₈	—	2	1	no	—
176	#121	—	2	1	no	—
177	#120	—	2	1	no	—
178	C₄×D₄	—	2	2	no	—
179	#117	C₂×C₄×C₆₁	1	1	yes	characteristic
180	M₃₂	—	2	2	no	—
181	M₃₂	—	2	2	no	—
182	M₃₂	—	2	2	no	—
183	M₃₂	D₄×C₆₁	1	4	yes	complemented
184	C₂×C₁₆	—	2	2	no	—
185	C₂×C₁₆	—	2	2	no	—
186	C₄⋊C₈	—	2	2	no	—
187	C₂²⋊C₈	—	2	2	no	—
188	C₂²⋊C₈	—	2	2	no	—
189	#97	D₄×C₆₁	1	2	yes	—
190	C₄×C₈	D₄×C₆₁	1	2	yes	—
191	C₂⁴	M₁₆×C₆₁	1	1	yes	characteristic
192	#41	—	2	2	no	—
193	#41	—	2	2	no	—
194	C₂×Q₈	M₁₆×C₆₁	1	1	yes	characteristic
195	C₂×D₄	—	2	1	no	—
196	C₂×D₄	—	2	2	no	—
197	C₂×D₄	M₁₆×C₆₁	1	1	yes	characteristic
198	C₂²×C₄	—	2	1	no	—
199	C₂²×C₄	—	2	1	no	—
200	C₂²×C₄	—	2	2	no	—
201	C₂²×C₄	—	2	1	no	—
202	C₂²×C₄	—	2	2	no	—
203	C₂²×C₄	M₁₆×C₆₁	1	1	yes	characteristic
204	C₂²×C₄	C₂²×C₄×C₆₁	1	1	yes	characteristic
205	M₁₆	—	2	2	no	—
206	M₁₆	—	2	2	no	—
207	M₁₆	—	4	2	no	—
208	M₁₆	—	4	1	no	—
209	M₁₆	—	4	1	no	—
210	M₁₆	—	4	1	no	—
211	M₁₆	—	4	1	no	—
212	C₂×C₈	—	2	1	no	—
213	C₂×C₈	—	2	1	no	—
214	C₂×C₈	—	2	2	no	—
215	C₂×C₈	—	4	2	no	—
216	C₂×C₈	C₂²⋊C₄×C₆₁	1	2	yes	—
217	C₂×C₈	C₂×D₄×C₆₁	1	1	yes	characteristic
218	C₂×C₈	C₂×D₄×C₆₁	1	2	yes	—
219	C₂×C₈	C₂×D₄×C₆₁	1	1	yes	characteristic
220	C₂×C₈	#41×C₆₁	1	2	yes	—
221	C₄⋊C₄	—	2	2	no	—
222	C₄⋊C₄	—	2	1	no	—
223	C₂²⋊C₄	—	2	2	no	—
224	C₂²⋊C₄	—	2	2	no	—
225	C₂²⋊C₄	—	2	1	no	—
226	C₄²	C₂²⋊C₄×C₆₁	1	2	yes	—
227	C₁₆	—	2	4	no	—
228	C₁₆	—	4	2	no	—
229	C₂³	—	2	1	no	—
230	C₂³	—	2	1	no	—
231	C₂³	—	2	2	no	—
232	C₂³	—	8	1	no	—
233	C₂³	C₂×M₁₆×C₆₁	1	1	yes	characteristic
234	Q₈	—	2	2	no	—
235	D₄	—	2	2	no	—
236	D₄	—	4	2	no	—
237	D₄	—	8	2	no	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	1	no	—
240	C₂×C₄	—	2	1	no	—
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	4	1	no	—
243	C₂×C₄	—	4	1	no	—
244	C₂×C₄	—	4	1	no	—
245	C₂×C₄	—	4	1	no	—
246	C₂×C₄	—	4	2	no	—
247	C₂×C₄	C₂×C₂²⋊C₄×C₆₁	1	1	yes	characteristic
248	C₂×C₄	C₄×D₄×C₆₁	1	2	yes	—
249	C₂×C₄	C₂×M₁₆×C₆₁	1	1	yes	characteristic
250	C₂×C₄	#131×C₆₁	1	2	yes	—
251	C₈	—	2	2	no	—
252	C₈	—	4	1	no	—
253	C₈	—	4	1	no	—
254	C₈	#121×C₆₁	1	2	yes	—
255	C₈	#124×C₆₁	1	1	yes	characteristic
256	C₈	#127×C₆₁	1	1	yes	characteristic
257	C₂²	—	4	1	no	—
258	C₂²	—	4	1	no	—
259	C₂²	—	4	2	no	—
260	C₂²	—	4	1	no	—
261	C₂²	—	4	1	no	—
262	C₂²	—	4	1	no	—
263	C₂²	—	8	2	no	—
264	C₂²	C₆₁×#407	1	1	yes	characteristic
265	C₂²	C₆₁×#435	1	2	yes	—
266	C₄	—	4	2	no	—
267	C₄	—	4	1	no	—
268	C₄	—	4	1	no	—
269	C₄	C₆₁×#390	1	1	yes	characteristic
270	C₄	C₆₁×#408	1	1	yes	characteristic
271	C₄	C₆₁×#435	1	2	yes	—
272	C₂	—	2	1	no	—
273	C₂	—	2	2	no	—
274	C₂	—	8	1	no	—
275	C₂	—	8	1	no	—
276	C₂	C₆₁×#1955	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₆₁	#390
commutator	C₂²×C₄	C₂²×C₄×C₆₁
Frattini, cosocle	C₂²×C₈	C₂³×C₆₁
socle	C₂×C₆₁	#1955
prime radical	C₆₁×#525	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₆₁ ◁ C₂²×C₈×C₆₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₆₁×#2153	C₂	semidirect
3. C₆₁×#2117	C₂	semidirect
4. C₂×C₆₁×#349	C₂	semidirect
5. C₆₁×#2110	C₂	semidirect
44. M₃₂×C₆₁	D₄	semidirect
118. #13201	C₆₁	direct (N × H)
135. #2153	C₂×C₆₁	semidirect
136. #2117	C₂×C₆₁	semidirect
137. C₂×#349	C₂×C₆₁	semidirect
138. #2110	C₂×C₆₁	semidirect
159. C₆₁	#13201	direct (N × H)
183. M₃₂	D₄×C₆₁	semidirect

Maximal subgroups

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

subgroup	index
C₆₁×#2930	2
C₆₁×#2153	2
C₆₁×#2117	2
C₂×C₆₁×#349	2
C₆₁×#2110	2
#13201	61

Subgroups — C61×#13201 #29516634

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#13201 #29516634