Subgroups — C₆₁×#11929 #29515362

Order 15616.

414 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₁×#3147	C₂	1	1	yes	maximal, characteristic
2	C₂×C₆₁×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₁×#1564	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₁×#1362	C₂	1	2	yes	maximal, complemented
5	C₆₁×#1361	C₂	1	2	yes	maximal, complemented
6	C₂×#135×C₆₁	—	2	1	no	—
7	C₂×#135×C₆₁	C₄	1	2	yes	—
8	C₆₁×#532	—	2	1	no	—
9	C₂²×C₁₆×C₆₁	—	2	1	no	—
10	C₆₁×#495	—	2	1	no	—
11	C₆₁×#495	C₄	1	2	yes	—
12	C₆₁×#494	C₂²	1	1	yes	characteristic
13	C₆₁×#493	C₂²	1	1	yes	characteristic
14	C₆₁×#492	—	2	1	no	—
15	C₆₁×#431	—	2	1	no	—
16	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
17	C₆₁×C₄⋊C₁₆	—	2	1	no	—
18	C₆₁×C₄⋊C₁₆	C₂²	1	2	yes	—
19	C₆₁×#335	C₂²	1	2	yes	—
20	C₆₁×#334	—	2	1	no	—
21	C₂×#41×C₆₁	—	2	2	no	—
22	#135×C₆₁	—	4	4	no	—
23	#135×C₆₁	—	4	2	no	—
24	#135×C₆₁	—	4	2	no	—
25	C₂×Q₁₆×C₆₁	—	2	1	no	—
26	C₂×Q₁₆×C₆₁	C₂×C₄	1	2	yes	—
27	C₂×SD₁₆×C₆₁	—	2	2	no	—
28	C₂×SD₁₆×C₆₁	—	2	2	no	—
29	C₂×D₈×C₆₁	—	2	1	no	—
30	C₂×D₈×C₆₁	C₂×C₄	1	2	yes	—
31	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₈×C₆₁	D₄	1	1	yes	characteristic
33	C₄⋊Q₈×C₆₁	—	2	1	no	—
34	#127×C₆₁	—	2	1	no	—
35	#124×C₆₁	—	2	2	no	—
36	#121×C₆₁	—	4	1	no	—
37	#121×C₆₁	—	4	1	no	—
38	C₄×D₄×C₆₁	—	4	1	no	—
39	C₄×D₄×C₆₁	—	4	1	no	—
40	C₂×C₄²×C₆₁	D₄	1	1	yes	characteristic
41	C₂×C₁₆×C₆₁	—	2	2	no	—
42	C₂×C₁₆×C₆₁	—	4	2	no	—
43	C₄⋊C₈×C₆₁	—	2	2	no	—
44	C₂²⋊C₈×C₆₁	—	4	1	no	—
45	C₄×C₈×C₆₁	—	4	1	no	—
46	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
47	C₄×C₈×C₆₁	D₄	1	1	yes	characteristic
48	C₄×C₈×C₆₁	D₄	1	1	yes	characteristic
49	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
50	#41×C₆₁	—	4	4	no	—
51	#41×C₆₁	—	4	4	no	—
52	C₂×Q₈×C₆₁	—	2	2	no	—
53	C₂×D₄×C₆₁	—	2	2	no	—
54	C₂×D₄×C₆₁	—	4	2	no	—
55	C₂×D₄×C₆₁	—	4	1	no	—
56	C₂×D₄×C₆₁	—	4	1	no	—
57	C₂²×C₄×C₆₁	—	4	2	no	—
58	C₂²×C₄×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
59	C₂²×C₄×C₆₁	D₈	1	1	yes	characteristic
60	C₂²×C₄×C₆₁	SD₁₆	1	1	yes	characteristic
61	Q₁₆×C₆₁	—	4	2	no	—
62	Q₁₆×C₆₁	—	4	1	no	—
63	Q₁₆×C₆₁	—	4	1	no	—
64	SD₁₆×C₆₁	—	4	4	no	—
65	SD₁₆×C₆₁	—	4	4	no	—
66	D₈×C₆₁	—	4	2	no	—
67	D₈×C₆₁	—	4	1	no	—
68	D₈×C₆₁	—	4	1	no	—
69	C₂×C₈×C₆₁	—	2	2	no	—
70	C₂×C₈×C₆₁	—	4	2	no	—
71	C₂×C₈×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
72	C₂×C₈×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
73	C₂×C₈×C₆₁	D₈	1	2	yes	—
74	C₂×C₈×C₆₁	SD₁₆	1	2	yes	—
75	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
76	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
77	C₄⋊C₄×C₆₁	—	4	1	no	—
78	C₄⋊C₄×C₆₁	—	4	1	no	—
79	C₂²⋊C₄×C₆₁	—	4	2	no	—
80	C₂²⋊C₄×C₆₁	—	4	2	no	—
81	C₄²×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
82	C₄²×C₆₁	D₈	1	1	yes	characteristic
83	C₄²×C₆₁	SD₁₆	1	1	yes	characteristic
84	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
85	C₁₆×C₆₁	—	4	2	no	—
86	C₂³×C₆₁	—	4	2	no	—
87	C₂³×C₆₁	#102	1	1	yes	characteristic
88	Q₈×C₆₁	—	4	2	no	—
89	Q₈×C₆₁	—	4	2	no	—
90	D₄×C₆₁	—	4	2	no	—
91	D₄×C₆₁	—	4	2	no	—
92	D₄×C₆₁	—	8	2	no	—
93	D₄×C₆₁	—	8	4	no	—
94	D₄×C₆₁	—	8	2	no	—
95	C₂×C₄×C₆₁	—	2	2	no	—
96	C₂×C₄×C₆₁	—	4	1	no	—
97	C₂×C₄×C₆₁	—	4	2	no	—
98	C₂×C₄×C₆₁	—	8	4	no	—
99	C₂×C₄×C₆₁	#100	1	2	yes	—
100	C₂×C₄×C₆₁	#102	1	1	yes	characteristic
101	C₂×C₄×C₆₁	#102	1	1	yes	characteristic
102	C₂×C₄×C₆₁	#102	1	2	yes	—
103	C₂×C₄×C₆₁	#102	1	1	yes	characteristic
104	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
105	C₂×C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
106	C₂×C₄×C₆₁	C₂×SD₁₆	1	1	yes	characteristic
107	C₈×C₆₁	—	2	1	no	—
108	C₈×C₆₁	—	2	1	no	—
109	C₈×C₆₁	—	8	2	no	—
110	C₈×C₆₁	#102	1	2	yes	—
111	C₈×C₆₁	C₂×D₈	1	1	yes	characteristic
112	C₈×C₆₁	C₂×SD₁₆	1	1	yes	characteristic
113	#11929	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
114	C₂²×C₆₁	—	2	1	no	—
115	C₂²×C₆₁	—	2	1	no	—
116	C₂²×C₆₁	—	8	2	no	—
117	C₂²×C₆₁	—	8	2	no	—
118	C₂²×C₆₁	—	8	2	no	—
119	C₂²×C₆₁	#331	1	2	yes	—
120	C₂²×C₆₁	C₂×#102	1	1	yes	characteristic
121	C₄×C₆₁	—	2	1	no	—
122	C₄×C₆₁	—	2	1	no	—
123	C₄×C₆₁	—	2	1	no	—
124	C₄×C₆₁	—	8	2	no	—
125	C₄×C₆₁	#331	1	2	yes	—
126	C₄×C₆₁	C₂×#100	1	1	yes	characteristic
127	C₄×C₆₁	C₂×#102	1	1	yes	characteristic
128	C₄×C₆₁	#508	1	2	yes	—
129	#3147	C₂×C₆₁	1	1	yes	characteristic
130	C₂×C₄⋊C₁₆	C₂×C₆₁	1	1	yes	characteristic, complemented
131	#1564	C₂×C₆₁	1	1	yes	characteristic, complemented
132	#1362	C₂×C₆₁	1	2	yes	complemented
133	#1361	C₂×C₆₁	1	2	yes	complemented
134	C₂×C₆₁	—	2	2	no	—
135	C₂×C₆₁	—	16	2	no	—
136	C₂×C₆₁	C₂×#331	1	1	yes	characteristic
137	C₂×C₆₁	#2138	1	1	yes	characteristic
138	C₂×C₆₁	#2148	1	1	yes	characteristic
139	C₂×#135	—	2	1	no	—
140	C₂×#135	C₄×C₆₁	1	2	yes	—
141	#532	—	2	1	no	—
142	C₂²×C₁₆	—	2	1	no	—
143	#495	—	2	1	no	—
144	#495	C₄×C₆₁	1	2	yes	—
145	#494	C₂²×C₆₁	1	1	yes	characteristic
146	#493	C₂²×C₆₁	1	1	yes	characteristic
147	#492	—	2	1	no	—
148	#431	—	2	1	no	—
149	C₂×C₄×C₈	C₂²×C₆₁	1	1	yes	characteristic
150	C₄⋊C₁₆	—	2	1	no	—
151	C₄⋊C₁₆	C₂²×C₆₁	1	2	yes	—
152	#335	C₂²×C₆₁	1	2	yes	—
153	#334	—	2	1	no	—
154	C₆₁	#11929	1	1	yes	characteristic, direct product factor, Sylow
155	C₂×#41	—	2	2	no	—
156	#135	—	4	4	no	—
157	#135	—	4	2	no	—
158	#135	—	4	2	no	—
159	C₂×Q₁₆	—	2	1	no	—
160	C₂×Q₁₆	C₂×C₄×C₆₁	1	2	yes	—
161	C₂×SD₁₆	—	2	2	no	—
162	C₂×SD₁₆	—	2	2	no	—
163	C₂×D₈	—	2	1	no	—
164	C₂×D₈	C₂×C₄×C₆₁	1	2	yes	—
165	C₂²×C₈	C₂×C₄×C₆₁	1	1	yes	characteristic
166	C₂²×C₈	D₄×C₆₁	1	1	yes	characteristic
167	C₄⋊Q₈	—	2	1	no	—
168	#127	—	2	1	no	—
169	#124	—	2	2	no	—
170	#121	—	4	1	no	—
171	#121	—	4	1	no	—
172	C₄×D₄	—	4	1	no	—
173	C₄×D₄	—	4	1	no	—
174	C₂×C₄²	D₄×C₆₁	1	1	yes	characteristic
175	C₂×C₁₆	—	2	2	no	—
176	C₂×C₁₆	—	4	2	no	—
177	C₄⋊C₈	—	2	2	no	—
178	C₂²⋊C₈	—	4	1	no	—
179	C₄×C₈	—	4	1	no	—
180	C₄×C₈	C₂×C₄×C₆₁	1	1	yes	characteristic
181	C₄×C₈	D₄×C₆₁	1	1	yes	characteristic
182	C₄×C₈	D₄×C₆₁	1	1	yes	characteristic
183	C₄×C₈	C₂³×C₆₁	1	1	yes	characteristic
184	#41	—	4	4	no	—
185	#41	—	4	4	no	—
186	C₂×Q₈	—	2	2	no	—
187	C₂×D₄	—	2	2	no	—
188	C₂×D₄	—	4	2	no	—
189	C₂×D₄	—	4	1	no	—
190	C₂×D₄	—	4	1	no	—
191	C₂²×C₄	—	4	2	no	—
192	C₂²×C₄	C₂²⋊C₄×C₆₁	1	1	yes	characteristic
193	C₂²×C₄	D₈×C₆₁	1	1	yes	characteristic
194	C₂²×C₄	SD₁₆×C₆₁	1	1	yes	characteristic
195	Q₁₆	—	4	2	no	—
196	Q₁₆	—	4	1	no	—
197	Q₁₆	—	4	1	no	—
198	SD₁₆	—	4	4	no	—
199	SD₁₆	—	4	4	no	—
200	D₈	—	4	2	no	—
201	D₈	—	4	1	no	—
202	D₈	—	4	1	no	—
203	C₂×C₈	—	2	2	no	—
204	C₂×C₈	—	4	2	no	—
205	C₂×C₈	C₂²⋊C₄×C₆₁	1	1	yes	characteristic
206	C₂×C₈	C₂²⋊C₄×C₆₁	1	1	yes	characteristic
207	C₂×C₈	D₈×C₆₁	1	2	yes	—
208	C₂×C₈	SD₁₆×C₆₁	1	2	yes	—
209	C₂×C₈	C₂²×C₄×C₆₁	1	1	yes	characteristic
210	C₂×C₈	C₂×D₄×C₆₁	1	1	yes	characteristic
211	C₄⋊C₄	—	4	1	no	—
212	C₄⋊C₄	—	4	1	no	—
213	C₂²⋊C₄	—	4	2	no	—
214	C₂²⋊C₄	—	4	2	no	—
215	C₄²	C₂²⋊C₄×C₆₁	1	1	yes	characteristic
216	C₄²	D₈×C₆₁	1	1	yes	characteristic
217	C₄²	SD₁₆×C₆₁	1	1	yes	characteristic
218	C₄²	C₂×D₄×C₆₁	1	1	yes	characteristic
219	C₁₆	—	4	2	no	—
220	C₂³	—	4	2	no	—
221	C₂³	#102×C₆₁	1	1	yes	characteristic
222	Q₈	—	4	2	no	—
223	Q₈	—	4	2	no	—
224	D₄	—	4	2	no	—
225	D₄	—	4	2	no	—
226	D₄	—	8	2	no	—
227	D₄	—	8	4	no	—
228	D₄	—	8	2	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	4	1	no	—
231	C₂×C₄	—	4	2	no	—
232	C₂×C₄	—	8	4	no	—
233	C₂×C₄	#100×C₆₁	1	2	yes	—
234	C₂×C₄	#102×C₆₁	1	1	yes	characteristic
235	C₂×C₄	#102×C₆₁	1	1	yes	characteristic
236	C₂×C₄	#102×C₆₁	1	1	yes	characteristic
237	C₂×C₄	#102×C₆₁	1	2	yes	—
238	C₂×C₄	C₂×C₂²⋊C₄×C₆₁	1	1	yes	characteristic
239	C₂×C₄	C₂×D₈×C₆₁	1	1	yes	characteristic
240	C₂×C₄	C₂×SD₁₆×C₆₁	1	1	yes	characteristic
241	C₈	—	2	1	no	—
242	C₈	—	2	1	no	—
243	C₈	—	8	2	no	—
244	C₈	#102×C₆₁	1	2	yes	—
245	C₈	C₂×D₈×C₆₁	1	1	yes	characteristic
246	C₈	C₂×SD₁₆×C₆₁	1	1	yes	characteristic
247	C₂²	—	2	1	no	—
248	C₂²	—	2	1	no	—
249	C₂²	—	8	2	no	—
250	C₂²	—	8	2	no	—
251	C₂²	—	8	2	no	—
252	C₂²	C₆₁×#331	1	2	yes	—
253	C₂²	C₂×#102×C₆₁	1	1	yes	characteristic
254	C₄	—	2	1	no	—
255	C₄	—	2	1	no	—
256	C₄	—	2	1	no	—
257	C₄	—	8	2	no	—
258	C₄	C₆₁×#331	1	2	yes	—
259	C₄	C₂×#100×C₆₁	1	1	yes	characteristic
260	C₄	C₂×#102×C₆₁	1	1	yes	characteristic
261	C₄	C₆₁×#508	1	2	yes	—
262	C₂	—	2	2	no	—
263	C₂	—	16	2	no	—
264	C₂	C₂×C₆₁×#331	1	1	yes	characteristic
265	C₂	C₆₁×#2138	1	1	yes	characteristic
266	C₂	C₆₁×#2148	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₆₁	#102
commutator	C₂×C₈	C₂²×C₄×C₆₁
Frattini, cosocle	C₄×C₈	C₂³×C₆₁
socle	C₂²×C₆₁	C₂×#102
prime radical	C₆₁×#3147	C₂
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₆₁ ◁ C₂×C₄²×C₆₁ ◁ C₂×C₄×C₈×C₆₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₆₁×C₄⋊C₁₆	C₂	semidirect
3. C₆₁×#1564	C₂	semidirect
4. C₆₁×#1362	C₂	semidirect
5. C₆₁×#1361	C₂	semidirect
113. #11929	C₆₁	direct (N × H)
130. C₂×C₄⋊C₁₆	C₂×C₆₁	semidirect
131. #1564	C₂×C₆₁	semidirect
132. #1362	C₂×C₆₁	semidirect
133. #1361	C₂×C₆₁	semidirect
154. C₆₁	#11929	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×#3147	2
C₂×C₆₁×C₄⋊C₁₆	2
C₆₁×#1564	2
C₆₁×#1362	2
C₆₁×#1361	2
#11929	61

Subgroups — C61×#11929 #29515362

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#11929 #29515362