Subgroups — C₆₁×#15201 #29518634

Order 15616.

1714 conjugacy classes of subgroups.

The subgroup lattice has 250 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×Q₈×C₆₁	C₂	1	1	yes	maximal, characteristic
2	C₂×C₆₁×#391	C₂	1	3	yes	maximal
3	C₆₁×#1446	C₂	1	2	yes	maximal
4	C₆₁×#1442	C₂	1	6	yes	maximal
5	C₆₁×C₄⋊#95	C₂	1	3	yes	maximal
6	C₂³×Q₈×C₆₁	C₂²	1	1	yes	characteristic
7	C₂³×Q₈×C₆₁	C₂²	1	1	yes	characteristic
8	C₂×C₄×Q₈×C₆₁	—	2	12	no	—
9	C₂²×C₄⋊C₄×C₆₁	C₂²	1	3	yes	—
10	C₂²×C₄⋊C₄×C₆₁	C₂²	1	3	yes	—
11	C₂²×C₄²×C₆₁	C₂²	1	3	yes	—
12	C₆₁×#391	—	2	12	no	—
13	C₂×#95×C₆₁	C₂²	1	6	yes	—
14	C₂×#95×C₆₁	C₂²	1	6	yes	—
15	C₂×#95×C₆₁	C₂²	1	6	yes	—
16	C₂×#95×C₆₁	C₂²	1	6	yes	—
17	C₂²×Q₈×C₆₁	—	2	6	no	—
18	C₂²×Q₈×C₆₁	—	2	6	no	—
19	C₂²×Q₈×C₆₁	—	2	6	no	—
20	C₂²×Q₈×C₆₁	—	2	6	no	—
21	C₂²×Q₈×C₆₁	Q₈	1	4	yes	complemented
22	C₂²×Q₈×C₆₁	Q₈	1	4	yes	complemented
23	C₂³×C₄×C₆₁	C₂³	1	6	yes	—
24	C₂³×C₄×C₆₁	C₂³	1	3	yes	—
25	C₂³×C₄×C₆₁	C₂³	1	2	yes	—
26	C₂³×C₄×C₆₁	C₂³	1	3	yes	—
27	C₂³×C₄×C₆₁	C₂³	1	1	yes	characteristic
28	C₄×Q₈×C₆₁	—	2	32	no	—
29	C₂×C₄⋊C₄×C₆₁	—	2	12	no	—
30	C₂×C₄⋊C₄×C₆₁	—	2	6	no	—
31	C₂×C₄⋊C₄×C₆₁	—	2	6	no	—
32	C₂×C₄⋊C₄×C₆₁	—	2	12	no	—
33	C₂×C₄²×C₆₁	—	2	12	no	—
34	C₂×C₄²×C₆₁	D₄	1	12	yes	—
35	#95×C₆₁	—	2	12	no	—
36	#95×C₆₁	—	2	12	no	—
37	#95×C₆₁	—	2	12	no	—
38	#95×C₆₁	—	4	6	no	—
39	C₂⁴×C₆₁	C₂⁴	1	1	yes	characteristic
40	C₂×Q₈×C₆₁	—	2	8	no	—
41	C₂×Q₈×C₆₁	—	2	24	no	—
42	C₂×Q₈×C₆₁	—	2	24	no	—
43	C₂×Q₈×C₆₁	—	2	24	no	—
44	C₂×Q₈×C₆₁	—	4	4	no	—
45	C₂×Q₈×C₆₁	—	4	12	no	—
46	C₂²×C₄×C₆₁	—	2	6	no	—
47	C₂²×C₄×C₆₁	—	2	6	no	—
48	C₂²×C₄×C₆₁	—	2	6	no	—
49	C₂²×C₄×C₆₁	—	2	6	no	—
50	C₂²×C₄×C₆₁	—	2	6	no	—
51	C₂²×C₄×C₆₁	—	2	3	no	—
52	C₂²×C₄×C₆₁	—	2	6	no	—
53	C₂²×C₄×C₆₁	—	2	3	no	—
54	C₂²×C₄×C₆₁	—	2	6	no	—
55	C₂²×C₄×C₆₁	—	2	12	no	—
56	C₂²×C₄×C₆₁	—	2	12	no	—
57	C₂²×C₄×C₆₁	—	2	6	no	—
58	C₂²×C₄×C₆₁	—	2	6	no	—
59	C₂²×C₄×C₆₁	C₂×D₄	1	6	yes	—
60	C₂²×C₄×C₆₁	C₂×D₄	1	6	yes	—
61	C₂²×C₄×C₆₁	C₂×D₄	1	6	yes	—
62	C₂²×C₄×C₆₁	C₂×Q₈	1	4	yes	—
63	C₂²×C₄×C₆₁	C₂×Q₈	1	6	yes	—
64	C₂²×C₄×C₆₁	C₂×Q₈	1	2	yes	—
65	C₂²×C₄×C₆₁	#41	1	6	yes	—
66	C₂²×C₄×C₆₁	#41	1	6	yes	—
67	C₄⋊C₄×C₆₁	—	2	24	no	—
68	C₄⋊C₄×C₆₁	—	4	12	no	—
69	C₄²×C₆₁	—	2	24	no	—
70	C₂³×C₆₁	C₂²×D₄	1	3	yes	—
71	C₂³×C₆₁	C₂²×Q₈	1	1	yes	characteristic
72	C₂³×C₆₁	C₂²×Q₈	1	1	yes	characteristic
73	C₂³×C₆₁	C₂×#41	1	3	yes	—
74	C₂³×C₆₁	C₂×#41	1	3	yes	—
75	C₂³×C₆₁	H(5,𝔽₂)	1	1	yes	characteristic
76	C₂³×C₆₁	2₋¹⁺⁴	1	3	yes	—
77	Q₈×C₆₁	—	2	32	no	—
78	Q₈×C₆₁	—	4	16	no	—
79	C₂×C₄×C₆₁	—	2	6	no	—
80	C₂×C₄×C₆₁	—	2	12	no	—
81	C₂×C₄×C₆₁	—	2	6	no	—
82	C₂×C₄×C₆₁	—	2	12	no	—
83	C₂×C₄×C₆₁	—	2	6	no	—
84	C₂×C₄×C₆₁	—	2	12	no	—
85	C₂×C₄×C₆₁	—	2	6	no	—
86	C₂×C₄×C₆₁	—	2	6	no	—
87	C₂×C₄×C₆₁	—	4	2	no	—
88	C₂×C₄×C₆₁	—	4	6	no	—
89	C₂×C₄×C₆₁	—	4	6	no	—
90	C₂×C₄×C₆₁	—	4	6	no	—
91	C₂×C₄×C₆₁	—	4	6	no	—
92	C₂×C₄×C₆₁	—	4	6	no	—
93	C₂×C₄×C₆₁	—	4	6	no	—
94	C₂×C₄×C₆₁	—	4	12	no	—
95	C₂×C₄×C₆₁	—	4	12	no	—
96	C₂×C₄×C₆₁	#120	1	4	yes	—
97	C₂×C₄×C₆₁	C₂²⋊Q₈	1	12	yes	—
98	C₂×C₄×C₆₁	C₂²⋊Q₈	1	12	yes	—
99	C₂×C₄×C₆₁	#124	1	12	yes	—
100	#15201	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
101	C₂²×C₆₁	C₂×#120	1	1	yes	characteristic
102	C₂²×C₆₁	C₂×C₂²⋊Q₈	1	3	yes	—
103	C₂²×C₆₁	C₂×C₂²⋊Q₈	1	3	yes	—
104	C₂²×C₆₁	#525	1	3	yes	—
105	C₂²×C₆₁	C₂×#124	1	3	yes	—
106	C₂²×C₆₁	#536	1	3	yes	—
107	C₂²×C₆₁	#537	1	3	yes	—
108	C₂²×C₆₁	#556	1	6	yes	—
109	C₂²×C₆₁	#557	1	3	yes	—
110	C₂²×C₆₁	#557	1	3	yes	—
111	C₂²×C₆₁	Q₈²	1	1	yes	characteristic
112	C₂²×C₆₁	#559	1	3	yes	—
113	C₄×C₆₁	—	2	12	no	—
114	C₄×C₆₁	—	4	6	no	—
115	C₄×C₆₁	—	4	4	no	—
116	C₄×C₆₁	—	8	6	no	—
117	C₄×C₆₁	#393	1	8	yes	—
118	C₂²×C₄×Q₈	C₂×C₆₁	1	1	yes	characteristic
119	C₂×#391	C₂×C₆₁	1	3	yes	—
120	#1446	C₂×C₆₁	1	2	yes	—
121	#1442	C₂×C₆₁	1	6	yes	—
122	C₄⋊#95	C₂×C₆₁	1	3	yes	—
123	C₂×C₆₁	C₂×#393	1	1	yes	characteristic
124	C₂×C₆₁	#2568	1	3	yes	—
125	C₂×C₆₁	#2581	1	3	yes	—
126	C₂×C₆₁	#2582	1	3	yes	—
127	C₂×C₆₁	#2592	1	3	yes	—
128	C₂×C₆₁	#2616	1	1	yes	characteristic
129	C₂×C₆₁	#2848	1	1	yes	characteristic
130	C₂³×Q₈	C₂²×C₆₁	1	1	yes	characteristic
131	C₂³×Q₈	C₂²×C₆₁	1	1	yes	characteristic
132	C₂×C₄×Q₈	—	2	12	no	—
133	C₂²×C₄⋊C₄	C₂²×C₆₁	1	3	yes	—
134	C₂²×C₄⋊C₄	C₂²×C₆₁	1	3	yes	—
135	C₂²×C₄²	C₂²×C₆₁	1	3	yes	—
136	#391	—	2	12	no	—
137	C₂×#95	C₂²×C₆₁	1	6	yes	—
138	C₂×#95	C₂²×C₆₁	1	6	yes	—
139	C₂×#95	C₂²×C₆₁	1	6	yes	—
140	C₂×#95	C₂²×C₆₁	1	6	yes	—
141	C₆₁	#15201	1	1	yes	characteristic, direct product factor, Sylow
142	C₂²×Q₈	—	2	6	no	—
143	C₂²×Q₈	—	2	6	no	—
144	C₂²×Q₈	—	2	6	no	—
145	C₂²×Q₈	—	2	6	no	—
146	C₂²×Q₈	Q₈×C₆₁	1	4	yes	complemented
147	C₂²×Q₈	Q₈×C₆₁	1	4	yes	complemented
148	C₂³×C₄	C₂³×C₆₁	1	6	yes	—
149	C₂³×C₄	C₂³×C₆₁	1	3	yes	—
150	C₂³×C₄	C₂³×C₆₁	1	2	yes	—
151	C₂³×C₄	C₂³×C₆₁	1	3	yes	—
152	C₂³×C₄	C₂³×C₆₁	1	1	yes	characteristic
153	C₄×Q₈	—	2	32	no	—
154	C₂×C₄⋊C₄	—	2	12	no	—
155	C₂×C₄⋊C₄	—	2	6	no	—
156	C₂×C₄⋊C₄	—	2	6	no	—
157	C₂×C₄⋊C₄	—	2	12	no	—
158	C₂×C₄²	—	2	12	no	—
159	C₂×C₄²	D₄×C₆₁	1	12	yes	—
160	#95	—	2	12	no	—
161	#95	—	2	12	no	—
162	#95	—	2	12	no	—
163	#95	—	4	6	no	—
164	C₂⁴	C₂⁴×C₆₁	1	1	yes	characteristic
165	C₂×Q₈	—	2	8	no	—
166	C₂×Q₈	—	2	24	no	—
167	C₂×Q₈	—	2	24	no	—
168	C₂×Q₈	—	2	24	no	—
169	C₂×Q₈	—	4	4	no	—
170	C₂×Q₈	—	4	12	no	—
171	C₂²×C₄	—	2	6	no	—
172	C₂²×C₄	—	2	6	no	—
173	C₂²×C₄	—	2	6	no	—
174	C₂²×C₄	—	2	6	no	—
175	C₂²×C₄	—	2	6	no	—
176	C₂²×C₄	—	2	3	no	—
177	C₂²×C₄	—	2	6	no	—
178	C₂²×C₄	—	2	3	no	—
179	C₂²×C₄	—	2	6	no	—
180	C₂²×C₄	—	2	12	no	—
181	C₂²×C₄	—	2	12	no	—
182	C₂²×C₄	—	2	6	no	—
183	C₂²×C₄	—	2	6	no	—
184	C₂²×C₄	C₂×D₄×C₆₁	1	6	yes	—
185	C₂²×C₄	C₂×D₄×C₆₁	1	6	yes	—
186	C₂²×C₄	C₂×D₄×C₆₁	1	6	yes	—
187	C₂²×C₄	C₂×Q₈×C₆₁	1	4	yes	—
188	C₂²×C₄	C₂×Q₈×C₆₁	1	6	yes	—
189	C₂²×C₄	C₂×Q₈×C₆₁	1	2	yes	—
190	C₂²×C₄	#41×C₆₁	1	6	yes	—
191	C₂²×C₄	#41×C₆₁	1	6	yes	—
192	C₄⋊C₄	—	2	24	no	—
193	C₄⋊C₄	—	4	12	no	—
194	C₄²	—	2	24	no	—
195	C₂³	C₂²×D₄×C₆₁	1	3	yes	—
196	C₂³	C₂²×Q₈×C₆₁	1	1	yes	characteristic
197	C₂³	C₂²×Q₈×C₆₁	1	1	yes	characteristic
198	C₂³	C₂×#41×C₆₁	1	3	yes	—
199	C₂³	C₂×#41×C₆₁	1	3	yes	—
200	C₂³	H(5,𝔽₂)×C₆₁	1	1	yes	characteristic
201	C₂³	2₋¹⁺⁴×C₆₁	1	3	yes	—
202	Q₈	—	2	32	no	—
203	Q₈	—	4	16	no	—
204	C₂×C₄	—	2	6	no	—
205	C₂×C₄	—	2	12	no	—
206	C₂×C₄	—	2	6	no	—
207	C₂×C₄	—	2	12	no	—
208	C₂×C₄	—	2	6	no	—
209	C₂×C₄	—	2	6	no	—
210	C₂×C₄	—	2	6	no	—
211	C₂×C₄	—	2	12	no	—
212	C₂×C₄	—	4	6	no	—
213	C₂×C₄	—	4	6	no	—
214	C₂×C₄	—	4	2	no	—
215	C₂×C₄	—	4	6	no	—
216	C₂×C₄	—	4	6	no	—
217	C₂×C₄	—	4	12	no	—
218	C₂×C₄	—	4	12	no	—
219	C₂×C₄	—	4	6	no	—
220	C₂×C₄	—	4	6	no	—
221	C₂×C₄	#120×C₆₁	1	4	yes	—
222	C₂×C₄	C₂²⋊Q₈×C₆₁	1	12	yes	—
223	C₂×C₄	C₂²⋊Q₈×C₆₁	1	12	yes	—
224	C₂×C₄	#124×C₆₁	1	12	yes	—
225	C₂²	C₂×#120×C₆₁	1	1	yes	characteristic
226	C₂²	C₂×C₂²⋊Q₈×C₆₁	1	3	yes	—
227	C₂²	C₂×C₂²⋊Q₈×C₆₁	1	3	yes	—
228	C₂²	C₆₁×#525	1	3	yes	—
229	C₂²	C₂×#124×C₆₁	1	3	yes	—
230	C₂²	C₆₁×#536	1	3	yes	—
231	C₂²	C₆₁×#537	1	3	yes	—
232	C₂²	C₆₁×#556	1	6	yes	—
233	C₂²	C₆₁×#557	1	3	yes	—
234	C₂²	C₆₁×#557	1	3	yes	—
235	C₂²	Q₈²×C₆₁	1	1	yes	characteristic
236	C₂²	C₆₁×#559	1	3	yes	—
237	C₄	—	2	12	no	—
238	C₄	—	4	6	no	—
239	C₄	—	4	4	no	—
240	C₄	—	8	6	no	—
241	C₄	C₆₁×#393	1	8	yes	—
242	C₂	C₂×C₆₁×#393	1	1	yes	characteristic
243	C₂	C₆₁×#2568	1	3	yes	—
244	C₂	C₆₁×#2581	1	3	yes	—
245	C₂	C₆₁×#2582	1	3	yes	—
246	C₂	C₆₁×#2592	1	3	yes	—
247	C₂	C₆₁×#2616	1	1	yes	characteristic
248	C₂	C₆₁×#2848	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂⁴×C₆₁	C₂⁴
commutator, Frattini, cosocle	C₂⁴	C₂⁴×C₆₁
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₆₁ ◁ G

Complements

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

N	H ≅ G/N	type
21. C₂²×Q₈×C₆₁	Q₈	semidirect
22. C₂²×Q₈×C₆₁	Q₈	semidirect
100. #15201	C₆₁	direct (N × H)
141. C₆₁	#15201	direct (N × H)
146. C₂²×Q₈	Q₈×C₆₁	semidirect
147. C₂²×Q₈	Q₈×C₆₁	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₄×Q₈×C₆₁	2
C₂×C₆₁×#391	2
C₆₁×#1446	2
C₆₁×#1442	2
C₆₁×C₄⋊#95	2
#15201	61

Subgroups — C61×#15201 #29518634

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#15201 #29518634