Subgroups — C₆₁×#11930 #29515363

Order 15616.

418 conjugacy classes of subgroups.

The subgroup lattice has 276 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	C₆₁×#2150	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₆₁×#335	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	C₂×M₃₂×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	C₂×C₄⋊C₈×C₆₁	—	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	—	2	1	no	—
20	C₆₁×#335	C₂²	1	2	yes	—
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	—	4	1	no	—
32	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₈×C₆₁	D₄	1	1	yes	characteristic
34	C₄⋊Q₈×C₆₁	—	2	1	no	—
35	#127×C₆₁	—	2	1	no	—
36	#124×C₆₁	—	2	2	no	—
37	#121×C₆₁	—	4	1	no	—
38	#121×C₆₁	—	4	1	no	—
39	C₄×D₄×C₆₁	—	4	1	no	—
40	C₄×D₄×C₆₁	—	4	1	no	—
41	C₂×C₄²×C₆₁	D₄	1	1	yes	characteristic
42	M₃₂×C₆₁	—	4	2	no	—
43	C₂×C₁₆×C₆₁	—	2	2	no	—
44	C₄⋊C₈×C₆₁	—	2	2	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	—	8	2	no	—
72	C₂×C₈×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
73	C₂×C₈×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
74	C₂×C₈×C₆₁	D₈	1	2	yes	—
75	C₂×C₈×C₆₁	SD₁₆	1	2	yes	—
76	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
78	C₄⋊C₄×C₆₁	—	4	1	no	—
79	C₄⋊C₄×C₆₁	—	4	1	no	—
80	C₂²⋊C₄×C₆₁	—	4	2	no	—
81	C₂²⋊C₄×C₆₁	—	4	2	no	—
82	C₄²×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
83	C₄²×C₆₁	D₈	1	1	yes	characteristic
84	C₄²×C₆₁	SD₁₆	1	1	yes	characteristic
85	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
86	C₁₆×C₆₁	—	4	2	no	—
87	C₂³×C₆₁	—	4	2	no	—
88	C₂³×C₆₁	#102	1	1	yes	characteristic
89	Q₈×C₆₁	—	4	2	no	—
90	Q₈×C₆₁	—	4	2	no	—
91	D₄×C₆₁	—	4	2	no	—
92	D₄×C₆₁	—	4	2	no	—
93	D₄×C₆₁	—	8	2	no	—
94	D₄×C₆₁	—	8	4	no	—
95	D₄×C₆₁	—	8	1	no	—
96	D₄×C₆₁	—	8	1	no	—
97	C₂×C₄×C₆₁	—	2	1	no	—
98	C₂×C₄×C₆₁	—	2	1	no	—
99	C₂×C₄×C₆₁	—	4	1	no	—
100	C₂×C₄×C₆₁	—	4	2	no	—
101	C₂×C₄×C₆₁	—	8	4	no	—
102	C₂×C₄×C₆₁	#100	1	2	yes	—
103	C₂×C₄×C₆₁	#102	1	1	yes	characteristic
104	C₂×C₄×C₆₁	#102	1	1	yes	characteristic
105	C₂×C₄×C₆₁	#102	1	1	yes	characteristic
106	C₂×C₄×C₆₁	#102	1	2	yes	—
107	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
108	C₂×C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
109	C₂×C₄×C₆₁	C₂×SD₁₆	1	1	yes	characteristic
110	C₈×C₆₁	—	2	1	no	—
111	C₈×C₆₁	—	2	1	no	—
112	C₈×C₆₁	—	8	2	no	—
113	C₈×C₆₁	#102	1	2	yes	—
114	C₈×C₆₁	C₂×D₈	1	1	yes	characteristic
115	C₈×C₆₁	C₂×SD₁₆	1	1	yes	characteristic
116	#11930	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
117	C₂²×C₆₁	—	2	1	no	—
118	C₂²×C₆₁	—	2	1	no	—
119	C₂²×C₆₁	—	8	2	no	—
120	C₂²×C₆₁	—	8	2	no	—
121	C₂²×C₆₁	—	8	2	no	—
122	C₂²×C₆₁	#331	1	2	yes	—
123	C₂²×C₆₁	C₂×#102	1	1	yes	characteristic
124	C₄×C₆₁	—	2	1	no	—
125	C₄×C₆₁	—	2	1	no	—
126	C₄×C₆₁	—	2	1	no	—
127	C₄×C₆₁	—	8	2	no	—
128	C₄×C₆₁	#331	1	2	yes	—
129	C₄×C₆₁	C₂×#100	1	1	yes	characteristic
130	C₄×C₆₁	C₂×#102	1	1	yes	characteristic
131	C₄×C₆₁	#509	1	1	yes	characteristic
132	C₄×C₆₁	#510	1	1	yes	characteristic
133	#3147	C₂×C₆₁	1	1	yes	characteristic
134	#2150	C₂×C₆₁	1	1	yes	characteristic, complemented
135	C₂×#335	C₂×C₆₁	1	1	yes	characteristic, complemented
136	#1362	C₂×C₆₁	1	2	yes	complemented
137	#1361	C₂×C₆₁	1	2	yes	complemented
138	C₂×C₆₁	—	2	2	no	—
139	C₂×C₆₁	—	16	2	no	—
140	C₂×C₆₁	C₂×#331	1	1	yes	characteristic
141	C₂×C₆₁	#2139	1	1	yes	characteristic
142	C₂×C₆₁	#2148	1	1	yes	characteristic
143	C₂×#135	—	2	1	no	—
144	C₂×#135	C₄×C₆₁	1	2	yes	—
145	#532	—	2	1	no	—
146	C₂×M₃₂	—	2	1	no	—
147	#495	—	2	1	no	—
148	#495	C₄×C₆₁	1	2	yes	—
149	#494	C₂²×C₆₁	1	1	yes	characteristic
150	#493	C₂²×C₆₁	1	1	yes	characteristic
151	#492	—	2	1	no	—
152	C₂×C₄⋊C₈	—	2	1	no	—
153	C₂×C₄×C₈	C₂²×C₆₁	1	1	yes	characteristic
154	C₄⋊C₁₆	—	2	1	no	—
155	C₄⋊C₁₆	C₂²×C₆₁	1	2	yes	—
156	#335	—	2	1	no	—
157	#335	C₂²×C₆₁	1	2	yes	—
158	C₆₁	#11930	1	1	yes	characteristic, direct product factor, Sylow
159	C₂×#41	—	2	2	no	—
160	#135	—	4	4	no	—
161	#135	—	4	2	no	—
162	#135	—	4	2	no	—
163	C₂×Q₁₆	—	2	1	no	—
164	C₂×Q₁₆	C₂×C₄×C₆₁	1	2	yes	—
165	C₂×SD₁₆	—	2	2	no	—
166	C₂×SD₁₆	—	2	2	no	—
167	C₂×D₈	—	2	1	no	—
168	C₂×D₈	C₂×C₄×C₆₁	1	2	yes	—
169	C₂²×C₈	—	4	1	no	—
170	C₂²×C₈	C₂×C₄×C₆₁	1	1	yes	characteristic
171	C₂²×C₈	D₄×C₆₁	1	1	yes	characteristic
172	C₄⋊Q₈	—	2	1	no	—
173	#127	—	2	1	no	—
174	#124	—	2	2	no	—
175	#121	—	4	1	no	—
176	#121	—	4	1	no	—
177	C₄×D₄	—	4	1	no	—
178	C₄×D₄	—	4	1	no	—
179	C₂×C₄²	D₄×C₆₁	1	1	yes	characteristic
180	M₃₂	—	4	2	no	—
181	C₂×C₁₆	—	2	2	no	—
182	C₄⋊C₈	—	2	2	no	—
183	C₄⋊C₈	—	4	1	no	—
184	C₄×C₈	C₂×C₄×C₆₁	1	1	yes	characteristic
185	C₄×C₈	D₄×C₆₁	1	1	yes	characteristic
186	C₄×C₈	D₄×C₆₁	1	1	yes	characteristic
187	C₄×C₈	C₂³×C₆₁	1	1	yes	characteristic
188	#41	—	4	4	no	—
189	#41	—	4	4	no	—
190	C₂×Q₈	—	2	2	no	—
191	C₂×D₄	—	2	2	no	—
192	C₂×D₄	—	4	2	no	—
193	C₂×D₄	—	4	1	no	—
194	C₂×D₄	—	4	1	no	—
195	C₂²×C₄	—	4	2	no	—
196	C₂²×C₄	C₂²⋊C₄×C₆₁	1	1	yes	characteristic
197	C₂²×C₄	D₈×C₆₁	1	1	yes	characteristic
198	C₂²×C₄	SD₁₆×C₆₁	1	1	yes	characteristic
199	Q₁₆	—	4	2	no	—
200	Q₁₆	—	4	1	no	—
201	Q₁₆	—	4	1	no	—
202	SD₁₆	—	4	4	no	—
203	SD₁₆	—	4	4	no	—
204	D₈	—	4	2	no	—
205	D₈	—	4	1	no	—
206	D₈	—	4	1	no	—
207	C₂×C₈	—	2	2	no	—
208	C₂×C₈	—	4	2	no	—
209	C₂×C₈	—	8	2	no	—
210	C₂×C₈	C₂²⋊C₄×C₆₁	1	1	yes	characteristic
211	C₂×C₈	C₂²⋊C₄×C₆₁	1	1	yes	characteristic
212	C₂×C₈	D₈×C₆₁	1	2	yes	—
213	C₂×C₈	SD₁₆×C₆₁	1	2	yes	—
214	C₂×C₈	C₂²×C₄×C₆₁	1	1	yes	characteristic
215	C₂×C₈	C₂×D₄×C₆₁	1	1	yes	characteristic
216	C₄⋊C₄	—	4	1	no	—
217	C₄⋊C₄	—	4	1	no	—
218	C₂²⋊C₄	—	4	2	no	—
219	C₂²⋊C₄	—	4	2	no	—
220	C₄²	C₂²⋊C₄×C₆₁	1	1	yes	characteristic
221	C₄²	D₈×C₆₁	1	1	yes	characteristic
222	C₄²	SD₁₆×C₆₁	1	1	yes	characteristic
223	C₄²	C₂×D₄×C₆₁	1	1	yes	characteristic
224	C₁₆	—	4	2	no	—
225	C₂³	—	4	2	no	—
226	C₂³	#102×C₆₁	1	1	yes	characteristic
227	Q₈	—	4	2	no	—
228	Q₈	—	4	2	no	—
229	D₄	—	4	2	no	—
230	D₄	—	4	2	no	—
231	D₄	—	8	2	no	—
232	D₄	—	8	4	no	—
233	D₄	—	8	1	no	—
234	D₄	—	8	1	no	—
235	C₂×C₄	—	2	1	no	—
236	C₂×C₄	—	2	1	no	—
237	C₂×C₄	—	4	1	no	—
238	C₂×C₄	—	4	2	no	—
239	C₂×C₄	—	8	4	no	—
240	C₂×C₄	#100×C₆₁	1	2	yes	—
241	C₂×C₄	#102×C₆₁	1	1	yes	characteristic
242	C₂×C₄	#102×C₆₁	1	1	yes	characteristic
243	C₂×C₄	#102×C₆₁	1	1	yes	characteristic
244	C₂×C₄	#102×C₆₁	1	2	yes	—
245	C₂×C₄	C₂×C₂²⋊C₄×C₆₁	1	1	yes	characteristic
246	C₂×C₄	C₂×D₈×C₆₁	1	1	yes	characteristic
247	C₂×C₄	C₂×SD₁₆×C₆₁	1	1	yes	characteristic
248	C₈	—	2	1	no	—
249	C₈	—	2	1	no	—
250	C₈	—	8	2	no	—
251	C₈	#102×C₆₁	1	2	yes	—
252	C₈	C₂×D₈×C₆₁	1	1	yes	characteristic
253	C₈	C₂×SD₁₆×C₆₁	1	1	yes	characteristic
254	C₂²	—	2	1	no	—
255	C₂²	—	2	1	no	—
256	C₂²	—	8	2	no	—
257	C₂²	—	8	2	no	—
258	C₂²	—	8	2	no	—
259	C₂²	C₆₁×#331	1	2	yes	—
260	C₂²	C₂×#102×C₆₁	1	1	yes	characteristic
261	C₄	—	2	1	no	—
262	C₄	—	2	1	no	—
263	C₄	—	2	1	no	—
264	C₄	—	8	2	no	—
265	C₄	C₆₁×#331	1	2	yes	—
266	C₄	C₂×#100×C₆₁	1	1	yes	characteristic
267	C₄	C₂×#102×C₆₁	1	1	yes	characteristic
268	C₄	C₆₁×#509	1	1	yes	characteristic
269	C₄	C₆₁×#510	1	1	yes	characteristic
270	C₂	—	2	2	no	—
271	C₂	—	16	2	no	—
272	C₂	C₂×C₆₁×#331	1	1	yes	characteristic
273	C₂	C₆₁×#2139	1	1	yes	characteristic
274	C₂	C₆₁×#2148	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₆₁	C₂×#102
commutator	C₂×C₈	C₂²×C₄×C₆₁
Frattini, cosocle	C₄×C₈	C₂³×C₆₁
prime radical	C₆₁×#3147	C₂
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×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. C₆₁×#2150	C₂	semidirect
3. C₂×C₆₁×#335	C₂	semidirect
4. C₆₁×#1362	C₂	semidirect
5. C₆₁×#1361	C₂	semidirect
116. #11930	C₆₁	direct (N × H)
134. #2150	C₂×C₆₁	semidirect
135. C₂×#335	C₂×C₆₁	semidirect
136. #1362	C₂×C₆₁	semidirect
137. #1361	C₂×C₆₁	semidirect
158. C₆₁	#11930	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
C₆₁×#2150	2
C₂×C₆₁×#335	2
C₆₁×#1362	2
C₆₁×#1361	2
#11930	61

Subgroups — C61×#11930 #29515363

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#11930 #29515363