Subgroups — C₆₁×#33360 #29536793

Order 15616.

1188 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₁×#3499	C₂	1	1	yes	maximal, characteristic
2	C₆₁×#2884	C₂	1	2	yes	maximal, complemented
3	C₆₁×#2129	C₂	1	6	yes	maximal, complemented
4	C₆₁×#2126	C₂	1	6	yes	maximal, complemented
5	C₂×2₋¹⁺⁴×C₆₁	C₄	1	1	yes	characteristic, complemented
6	C₂×2₋¹⁺⁴×C₆₁	C₂²	1	1	yes	characteristic
7	D₄×Q₈×C₆₁	—	2	2	no	—
8	C₆₁×#547	—	2	6	no	—
9	C₆₁×#536	C₄	1	3	yes	complemented
10	C₆₁×#536	C₂²	1	3	yes	—
11	C₆₁×#532	C₂²	1	3	yes	—
12	C₂×C₄⋊Q₈×C₆₁	C₄	1	3	yes	complemented
13	C₆₁×#519	—	2	2	no	—
14	C₄×#41×C₆₁	C₄	1	1	yes	characteristic, complemented
15	C₆₁×#410	—	2	6	no	—
16	C₆₁×#410	C₂²	1	12	yes	—
17	C₆₁×#354	C₂²	1	12	yes	—
18	C₆₁×#353	C₂²	1	4	yes	—
19	2₋¹⁺⁴×C₆₁	—	4	4	no	—
20	2₋¹⁺⁴×C₆₁	—	4	4	no	—
21	C₂×#41×C₆₁	—	2	3	no	—
22	C₂×#41×C₆₁	C₂×C₄	1	3	yes	—
23	C₂×#41×C₆₁	C₂×C₄	1	1	yes	characteristic
24	C₂×#41×C₆₁	D₄	1	6	yes	—
25	C₂×#41×C₆₁	C₂³	1	3	yes	—
26	C₂²×Q₈×C₆₁	—	2	1	no	—
27	C₂²×Q₈×C₆₁	C₂×C₄	1	3	yes	—
28	C₂²×Q₈×C₆₁	C₂×C₄	1	3	yes	—
29	C₂²×Q₈×C₆₁	D₄	1	2	yes	—
30	C₄⋊Q₈×C₆₁	—	2	6	no	—
31	C₄⋊Q₈×C₆₁	C₂×C₄	1	3	yes	—
32	C₄⋊Q₈×C₆₁	C₂×C₄	1	6	yes	—
33	#127×C₆₁	C₂³	1	1	yes	characteristic
34	#124×C₆₁	C₂×C₄	1	3	yes	—
35	#124×C₆₁	C₂³	1	3	yes	—
36	#123×C₆₁	—	2	6	no	—
37	#123×C₆₁	—	2	6	no	—
38	C₂²⋊Q₈×C₆₁	—	2	6	no	—
39	C₂²⋊Q₈×C₆₁	—	2	6	no	—
40	#121×C₆₁	—	2	6	no	—
41	C₄×Q₈×C₆₁	—	2	1	no	—
42	C₄×Q₈×C₆₁	—	4	8	no	—
43	C₄×D₄×C₆₁	—	2	3	no	—
44	C₄×D₄×C₆₁	—	2	6	no	—
45	#117×C₆₁	—	2	12	no	—
46	#117×C₆₁	C₂×C₄	1	3	yes	—
47	C₂×C₄⋊C₄×C₆₁	—	2	6	no	—
48	C₂×C₄²×C₆₁	C₂×C₄	1	3	yes	—
49	#99×C₆₁	—	2	12	no	—
50	#99×C₆₁	C₂³	1	8	yes	—
51	#41×C₆₁	—	2	4	no	—
52	#41×C₆₁	—	4	6	no	—
53	#41×C₆₁	—	4	12	no	—
54	#41×C₆₁	—	4	6	no	—
55	#41×C₆₁	—	4	12	no	—
56	C₂×Q₈×C₆₁	—	2	8	no	—
57	C₂×Q₈×C₆₁	—	2	3	no	—
58	C₂×Q₈×C₆₁	—	4	6	no	—
59	C₂×Q₈×C₆₁	—	4	6	no	—
60	C₂×Q₈×C₆₁	—	4	4	no	—
61	C₂×Q₈×C₆₁	C₂²⋊C₄	1	6	yes	—
62	C₂×Q₈×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
63	C₂×Q₈×C₆₁	C₂²×C₄	1	3	yes	—
64	C₂×Q₈×C₆₁	C₂²×C₄	1	3	yes	—
65	C₂×D₄×C₆₁	—	2	3	no	—
66	C₂×D₄×C₆₁	—	2	2	no	—
67	C₂×D₄×C₆₁	C₂²⋊C₄	1	3	yes	—
68	C₂×D₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
69	C₂×D₄×C₆₁	C₂×D₄	1	6	yes	—
70	C₂×D₄×C₆₁	C₂⁴	1	1	yes	characteristic
71	C₂²×C₄×C₆₁	—	2	3	no	—
72	C₂²×C₄×C₆₁	—	2	3	no	—
73	C₂²×C₄×C₆₁	—	2	3	no	—
74	C₂²×C₄×C₆₁	C₂²⋊C₄	1	6	yes	—
75	C₂²×C₄×C₆₁	C₂²×C₄	1	3	yes	—
76	C₂²×C₄×C₆₁	C₂×D₄	1	6	yes	—
77	C₄⋊C₄×C₆₁	—	2	6	no	—
78	C₄⋊C₄×C₆₁	—	2	12	no	—
79	C₄⋊C₄×C₆₁	—	2	3	no	—
80	C₄⋊C₄×C₆₁	—	4	12	no	—
81	C₂²⋊C₄×C₆₁	—	2	3	no	—
82	C₂²⋊C₄×C₆₁	—	2	6	no	—
83	C₂²⋊C₄×C₆₁	—	2	6	no	—
84	C₂²⋊C₄×C₆₁	—	2	8	no	—
85	C₄²×C₆₁	—	2	3	no	—
86	C₄²×C₆₁	—	4	12	no	—
87	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
88	C₄²×C₆₁	C₂²×C₄	1	3	yes	—
89	C₂³×C₆₁	—	2	1	no	—
90	C₂³×C₆₁	C₂×C₂²⋊C₄	1	3	yes	—
91	C₂³×C₆₁	C₂²×D₄	1	2	yes	—
92	Q₈×C₆₁	—	2	12	no	—
93	Q₈×C₆₁	—	4	3	no	—
94	Q₈×C₆₁	—	4	3	no	—
95	Q₈×C₆₁	—	4	6	no	—
96	Q₈×C₆₁	#99	1	4	yes	—
97	D₄×C₆₁	—	2	6	no	—
98	D₄×C₆₁	—	4	6	no	—
99	D₄×C₆₁	—	4	1	no	—
100	D₄×C₆₁	—	4	1	no	—
101	D₄×C₆₁	—	4	4	no	—
102	D₄×C₆₁	—	4	6	no	—
103	C₂×C₄×C₆₁	—	2	1	no	—
104	C₂×C₄×C₆₁	—	2	12	no	—
105	C₂×C₄×C₆₁	—	2	6	no	—
106	C₂×C₄×C₆₁	—	2	3	no	—
107	C₂×C₄×C₆₁	—	2	3	no	—
108	C₂×C₄×C₆₁	—	4	6	no	—
109	C₂×C₄×C₆₁	—	4	6	no	—
110	C₂×C₄×C₆₁	—	4	6	no	—
111	C₂×C₄×C₆₁	—	4	6	no	—
112	C₂×C₄×C₆₁	—	4	8	no	—
113	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	6	yes	—
114	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	3	yes	—
115	C₂×C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
116	#33360	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
117	C₂²×C₆₁	—	2	2	no	—
118	C₂²×C₆₁	—	2	3	no	—
119	C₂²×C₆₁	—	4	3	no	—
120	C₂²×C₆₁	—	4	2	no	—
121	C₂²×C₆₁	—	4	2	no	—
122	C₂²×C₆₁	—	4	1	no	—
123	C₂²×C₆₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
124	C₄×C₆₁	—	2	2	no	—
125	C₄×C₆₁	—	2	1	no	—
126	C₄×C₆₁	—	2	6	no	—
127	C₄×C₆₁	—	4	3	no	—
128	C₄×C₆₁	—	4	3	no	—
129	C₄×C₆₁	—	8	8	no	—
130	C₄×C₆₁	C₂×#99	1	6	yes	—
131	#3499	C₂×C₆₁	1	1	yes	characteristic
132	#2884	C₂×C₆₁	1	2	yes	complemented
133	#2129	C₂×C₆₁	1	6	yes	complemented
134	#2126	C₂×C₆₁	1	6	yes	complemented
135	C₂×C₆₁	—	2	1	no	—
136	C₂×C₆₁	—	4	2	no	—
137	C₂×C₆₁	—	4	3	no	—
138	C₂×C₆₁	—	8	1	no	—
139	C₂×C₆₁	C₂²×#99	1	1	yes	characteristic
140	C₂×2₋¹⁺⁴	C₄×C₆₁	1	1	yes	characteristic, complemented
141	C₂×2₋¹⁺⁴	C₂²×C₆₁	1	1	yes	characteristic
142	D₄×Q₈	—	2	2	no	—
143	#547	—	2	6	no	—
144	#536	C₄×C₆₁	1	3	yes	complemented
145	#536	C₂²×C₆₁	1	3	yes	—
146	#532	C₂²×C₆₁	1	3	yes	—
147	C₂×C₄⋊Q₈	C₄×C₆₁	1	3	yes	complemented
148	#519	—	2	2	no	—
149	C₄×#41	C₄×C₆₁	1	1	yes	characteristic, complemented
150	#410	—	2	6	no	—
151	#410	C₂²×C₆₁	1	12	yes	—
152	#354	C₂²×C₆₁	1	12	yes	—
153	#353	C₂²×C₆₁	1	4	yes	—
154	C₆₁	#33360	1	1	yes	characteristic, direct product factor, Sylow
155	2₋¹⁺⁴	—	4	4	no	—
156	2₋¹⁺⁴	—	4	4	no	—
157	C₂×#41	—	2	3	no	—
158	C₂×#41	C₂×C₄×C₆₁	1	3	yes	—
159	C₂×#41	C₂×C₄×C₆₁	1	1	yes	characteristic
160	C₂×#41	D₄×C₆₁	1	6	yes	—
161	C₂×#41	C₂³×C₆₁	1	3	yes	—
162	C₂²×Q₈	—	2	1	no	—
163	C₂²×Q₈	C₂×C₄×C₆₁	1	3	yes	—
164	C₂²×Q₈	C₂×C₄×C₆₁	1	3	yes	—
165	C₂²×Q₈	D₄×C₆₁	1	2	yes	—
166	C₄⋊Q₈	—	2	6	no	—
167	C₄⋊Q₈	C₂×C₄×C₆₁	1	3	yes	—
168	C₄⋊Q₈	C₂×C₄×C₆₁	1	6	yes	—
169	#127	C₂³×C₆₁	1	1	yes	characteristic
170	#124	C₂×C₄×C₆₁	1	3	yes	—
171	#124	C₂³×C₆₁	1	3	yes	—
172	#123	—	2	6	no	—
173	#123	—	2	6	no	—
174	C₂²⋊Q₈	—	2	6	no	—
175	C₂²⋊Q₈	—	2	6	no	—
176	#121	—	2	6	no	—
177	C₄×Q₈	—	2	1	no	—
178	C₄×Q₈	—	4	8	no	—
179	C₄×D₄	—	2	3	no	—
180	C₄×D₄	—	2	6	no	—
181	#117	—	2	12	no	—
182	#117	C₂×C₄×C₆₁	1	3	yes	—
183	C₂×C₄⋊C₄	—	2	6	no	—
184	C₂×C₄²	C₂×C₄×C₆₁	1	3	yes	—
185	#99	—	2	12	no	—
186	#99	C₂³×C₆₁	1	8	yes	—
187	#41	—	2	4	no	—
188	#41	—	4	6	no	—
189	#41	—	4	12	no	—
190	#41	—	4	6	no	—
191	#41	—	4	12	no	—
192	C₂×Q₈	—	2	8	no	—
193	C₂×Q₈	—	2	3	no	—
194	C₂×Q₈	—	4	6	no	—
195	C₂×Q₈	—	4	6	no	—
196	C₂×Q₈	—	4	4	no	—
197	C₂×Q₈	C₂²⋊C₄×C₆₁	1	6	yes	—
198	C₂×Q₈	C₂²⋊C₄×C₆₁	1	1	yes	characteristic
199	C₂×Q₈	C₂²×C₄×C₆₁	1	3	yes	—
200	C₂×Q₈	C₂²×C₄×C₆₁	1	3	yes	—
201	C₂×D₄	—	2	3	no	—
202	C₂×D₄	—	2	2	no	—
203	C₂×D₄	C₂²⋊C₄×C₆₁	1	3	yes	—
204	C₂×D₄	C₂²×C₄×C₆₁	1	1	yes	characteristic
205	C₂×D₄	C₂×D₄×C₆₁	1	6	yes	—
206	C₂×D₄	C₂⁴×C₆₁	1	1	yes	characteristic
207	C₂²×C₄	—	2	3	no	—
208	C₂²×C₄	—	2	3	no	—
209	C₂²×C₄	—	2	3	no	—
210	C₂²×C₄	C₂²⋊C₄×C₆₁	1	6	yes	—
211	C₂²×C₄	C₂²×C₄×C₆₁	1	3	yes	—
212	C₂²×C₄	C₂×D₄×C₆₁	1	6	yes	—
213	C₄⋊C₄	—	2	6	no	—
214	C₄⋊C₄	—	2	12	no	—
215	C₄⋊C₄	—	2	3	no	—
216	C₄⋊C₄	—	4	12	no	—
217	C₂²⋊C₄	—	2	3	no	—
218	C₂²⋊C₄	—	2	6	no	—
219	C₂²⋊C₄	—	2	6	no	—
220	C₂²⋊C₄	—	2	8	no	—
221	C₄²	—	2	3	no	—
222	C₄²	—	4	12	no	—
223	C₄²	C₂²×C₄×C₆₁	1	1	yes	characteristic
224	C₄²	C₂²×C₄×C₆₁	1	3	yes	—
225	C₂³	—	2	1	no	—
226	C₂³	C₂×C₂²⋊C₄×C₆₁	1	3	yes	—
227	C₂³	C₂²×D₄×C₆₁	1	2	yes	—
228	Q₈	—	2	12	no	—
229	Q₈	—	4	3	no	—
230	Q₈	—	4	3	no	—
231	Q₈	—	4	6	no	—
232	Q₈	#99×C₆₁	1	4	yes	—
233	D₄	—	2	6	no	—
234	D₄	—	4	6	no	—
235	D₄	—	4	1	no	—
236	D₄	—	4	1	no	—
237	D₄	—	4	4	no	—
238	D₄	—	4	6	no	—
239	C₂×C₄	—	2	1	no	—
240	C₂×C₄	—	2	12	no	—
241	C₂×C₄	—	2	6	no	—
242	C₂×C₄	—	2	3	no	—
243	C₂×C₄	—	2	3	no	—
244	C₂×C₄	—	4	6	no	—
245	C₂×C₄	—	4	6	no	—
246	C₂×C₄	—	4	6	no	—
247	C₂×C₄	—	4	6	no	—
248	C₂×C₄	—	4	8	no	—
249	C₂×C₄	C₂×C₂²⋊C₄×C₆₁	1	6	yes	—
250	C₂×C₄	C₂×C₂²⋊C₄×C₆₁	1	3	yes	—
251	C₂×C₄	C₂³×C₄×C₆₁	1	1	yes	characteristic
252	C₂²	—	2	2	no	—
253	C₂²	—	2	3	no	—
254	C₂²	—	4	3	no	—
255	C₂²	—	4	2	no	—
256	C₂²	—	4	2	no	—
257	C₂²	—	4	1	no	—
258	C₂²	C₂²×C₂²⋊C₄×C₆₁	1	1	yes	characteristic
259	C₄	—	2	2	no	—
260	C₄	—	2	1	no	—
261	C₄	—	2	6	no	—
262	C₄	—	4	3	no	—
263	C₄	—	4	3	no	—
264	C₄	—	8	8	no	—
265	C₄	C₂×#99×C₆₁	1	6	yes	—
266	C₂	—	2	1	no	—
267	C₂	—	4	2	no	—
268	C₂	—	4	3	no	—
269	C₂	—	8	1	no	—
270	C₂	C₂²×#99×C₆₁	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₆₁	C₂²×#99
commutator	C₂×C₄	C₂³×C₄×C₆₁
Frattini, cosocle	C₂×D₄	C₂⁴×C₆₁
prime radical	C₆₁×#3499	C₂
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₆₁ ◁ C₂×Q₈×C₆₁ ◁ C₂×2₋¹⁺⁴×C₆₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₆₁×#2884	C₂	semidirect
3. C₆₁×#2129	C₂	semidirect
4. C₆₁×#2126	C₂	semidirect
5. C₂×2₋¹⁺⁴×C₆₁	C₄	semidirect
9. C₆₁×#536	C₄	semidirect
12. C₂×C₄⋊Q₈×C₆₁	C₄	semidirect
14. C₄×#41×C₆₁	C₄	semidirect
116. #33360	C₆₁	direct (N × H)
132. #2884	C₂×C₆₁	semidirect
133. #2129	C₂×C₆₁	semidirect
134. #2126	C₂×C₆₁	semidirect
140. C₂×2₋¹⁺⁴	C₄×C₆₁	semidirect
144. #536	C₄×C₆₁	semidirect
147. C₂×C₄⋊Q₈	C₄×C₆₁	semidirect
149. C₄×#41	C₄×C₆₁	semidirect
154. C₆₁	#33360	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×#3499	2
C₆₁×#2884	2
C₆₁×#2129	2
C₆₁×#2126	2
#33360	61

Subgroups — C61×#33360 #29536793

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#33360 #29536793