Subgroups — C₆₁×#13265 #29516698

Order 15616.

338 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₁×#2960	C₂	1	1	yes	maximal, characteristic
2	C₂×C₆₁×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic
3	C₄×M₃₂×C₆₁	C₂	1	1	yes	maximal, characteristic
4	C₆₁×#1379	C₂	1	4	yes	maximal
5	C₆₁×#533	C₄	1	1	yes	characteristic
6	C₂×M₃₂×C₆₁	C₂²	1	2	yes	—
7	C₂²×C₁₆×C₆₁	C₂²	1	2	yes	—
8	C₆₁×#446	—	2	1	no	—
9	C₆₁×#446	—	2	1	no	—
10	C₈×Q₈×C₆₁	—	2	1	no	—
11	C₈×Q₈×C₆₁	—	2	1	no	—
12	C₆₁×#424	C₄	1	2	yes	—
13	C₆₁×#423	C₄	1	1	yes	characteristic
14	C₆₁×#405	C₂²	1	2	yes	—
15	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
16	C₆₁×C₄⋊C₁₆	—	2	1	no	—
17	C₆₁×C₄⋊C₁₆	—	2	1	no	—
18	C₆₁×#346	—	2	1	no	—
19	C₄×C₁₆×C₆₁	—	2	1	no	—
20	C₂×M₁₆×C₆₁	C₂×C₄	1	2	yes	—
21	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
23	C₄⋊Q₈×C₆₁	C₈	1	1	yes	characteristic
24	C₄⋊Q₈×C₆₁	C₈	1	1	yes	characteristic
25	#125×C₆₁	C₈	1	2	yes	—
26	C₂²⋊Q₈×C₆₁	—	2	1	no	—
27	C₂²⋊Q₈×C₆₁	—	2	1	no	—
28	C₄×Q₈×C₆₁	—	2	1	no	—
29	C₄×Q₈×C₆₁	—	2	1	no	—
30	#117×C₆₁	C₂×C₄	1	2	yes	—
31	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
32	M₃₂×C₆₁	—	2	2	no	—
33	M₃₂×C₆₁	Q₈	1	4	yes	complemented
34	C₂×C₁₆×C₆₁	—	2	2	no	—
35	C₂×C₁₆×C₆₁	—	2	2	no	—
36	C₂×C₁₆×C₆₁	—	2	4	no	—
37	C₄⋊C₈×C₆₁	—	2	2	no	—
38	C₄⋊C₈×C₆₁	—	2	2	no	—
39	C₄⋊C₈×C₆₁	—	2	1	no	—
40	C₄⋊C₈×C₆₁	—	2	1	no	—
41	#97×C₆₁	—	2	2	no	—
42	C₄×C₈×C₆₁	—	2	2	no	—
43	C₄×C₈×C₆₁	D₄	1	1	yes	characteristic
44	C₄×C₈×C₆₁	D₄	1	1	yes	characteristic
45	C₄×C₈×C₆₁	D₄	1	1	yes	characteristic
46	C₄×C₈×C₆₁	D₄	1	1	yes	characteristic
47	C₂×Q₈×C₆₁	—	2	1	no	—
48	C₂×Q₈×C₆₁	—	2	1	no	—
49	C₂²×C₄×C₆₁	M₁₆	1	1	yes	characteristic
50	C₂²×C₄×C₆₁	M₁₆	1	1	yes	characteristic
51	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
52	M₁₆×C₆₁	—	2	4	no	—
53	C₂×C₈×C₆₁	—	2	2	no	—
54	C₂×C₈×C₆₁	—	2	2	no	—
55	C₂×C₈×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
56	C₂×C₈×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
57	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
58	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
59	C₂×C₈×C₆₁	C₂×Q₈	1	2	yes	—
60	C₂×C₈×C₆₁	#41	1	2	yes	—
61	C₄⋊C₄×C₆₁	—	2	2	no	—
62	C₄⋊C₄×C₆₁	—	2	2	no	—
63	C₄⋊C₄×C₆₁	—	2	1	no	—
64	C₄⋊C₄×C₆₁	—	2	1	no	—
65	C₄⋊C₄×C₆₁	C₂×C₈	1	2	yes	—
66	C₄⋊C₄×C₆₁	C₂×C₈	1	2	yes	—
67	C₂²⋊C₄×C₆₁	—	2	2	no	—
68	C₄²×C₆₁	—	2	2	no	—
69	C₄²×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
70	C₄²×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
71	C₄²×C₆₁	C₂×C₈	1	1	yes	characteristic
72	C₄²×C₆₁	C₂×C₈	1	1	yes	characteristic
73	C₁₆×C₆₁	—	2	4	no	—
74	C₁₆×C₆₁	—	4	2	no	—
75	C₂³×C₆₁	C₂×M₁₆	1	1	yes	characteristic
76	Q₈×C₆₁	—	4	2	no	—
77	Q₈×C₆₁	—	4	2	no	—
78	C₂×C₄×C₆₁	—	2	2	no	—
79	C₂×C₄×C₆₁	—	2	2	no	—
80	C₂×C₄×C₆₁	—	2	2	no	—
81	C₂×C₄×C₆₁	—	4	1	no	—
82	C₂×C₄×C₆₁	C₂²⋊C₈	1	1	yes	characteristic
83	C₂×C₄×C₆₁	C₂²⋊C₈	1	1	yes	characteristic
84	C₂×C₄×C₆₁	C₂²⋊C₈	1	1	yes	characteristic
85	C₂×C₄×C₆₁	C₂²⋊C₈	1	1	yes	characteristic
86	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
87	C₂×C₄×C₆₁	C₄×Q₈	1	2	yes	—
88	C₂×C₄×C₆₁	C₂²×C₈	1	1	yes	characteristic
89	C₂×C₄×C₆₁	#131	1	2	yes	—
90	C₈×C₆₁	—	2	1	no	—
91	C₈×C₆₁	—	2	1	no	—
92	C₈×C₆₁	—	4	2	no	—
93	C₈×C₆₁	C₂²⋊Q₈	1	2	yes	—
94	C₈×C₆₁	#124	1	1	yes	characteristic
95	C₈×C₆₁	C₄⋊Q₈	1	1	yes	characteristic
96	#13265	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
97	C₂²×C₆₁	—	2	1	no	—
98	C₂²×C₆₁	—	2	1	no	—
99	C₂²×C₆₁	C₂×C₂²⋊C₈	1	1	yes	characteristic
100	C₂²×C₆₁	#446	1	2	yes	—
101	C₄×C₆₁	—	2	1	no	—
102	C₄×C₆₁	—	2	1	no	—
103	C₄×C₆₁	—	2	1	no	—
104	C₄×C₆₁	—	4	2	no	—
105	C₄×C₆₁	—	4	2	no	—
106	C₄×C₆₁	#350	1	2	yes	—
107	C₄×C₆₁	#391	1	1	yes	characteristic
108	C₄×C₆₁	#408	1	1	yes	characteristic
109	C₄×C₆₁	C₈×Q₈	1	2	yes	—
110	#2960	C₂×C₆₁	1	1	yes	characteristic
111	C₂×C₄⋊C₁₆	C₂×C₆₁	1	1	yes	characteristic
112	C₄×M₃₂	C₂×C₆₁	1	1	yes	characteristic
113	#1379	C₂×C₆₁	1	4	yes	—
114	C₂×C₆₁	—	2	2	no	—
115	C₂×C₆₁	#1984	1	1	yes	characteristic
116	C₂×C₆₁	C₂×#350	1	1	yes	characteristic
117	C₂×C₆₁	#2117	1	1	yes	characteristic
118	#533	C₄×C₆₁	1	1	yes	characteristic
119	C₂×M₃₂	C₂²×C₆₁	1	2	yes	—
120	C₂²×C₁₆	C₂²×C₆₁	1	2	yes	—
121	#446	—	2	1	no	—
122	#446	—	2	1	no	—
123	C₈×Q₈	—	2	1	no	—
124	C₈×Q₈	—	2	1	no	—
125	#424	C₄×C₆₁	1	2	yes	—
126	#423	C₄×C₆₁	1	1	yes	characteristic
127	#405	C₂²×C₆₁	1	2	yes	—
128	C₂×C₄×C₈	C₂²×C₆₁	1	1	yes	characteristic
129	C₄⋊C₁₆	—	2	1	no	—
130	C₄⋊C₁₆	—	2	1	no	—
131	#346	—	2	1	no	—
132	C₄×C₁₆	—	2	1	no	—
133	C₆₁	#13265	1	1	yes	characteristic, direct product factor, Sylow
134	C₂×M₁₆	C₂×C₄×C₆₁	1	2	yes	—
135	C₂²×C₈	C₂×C₄×C₆₁	1	1	yes	characteristic
136	C₂²×C₈	C₂³×C₆₁	1	1	yes	characteristic
137	C₄⋊Q₈	C₈×C₆₁	1	1	yes	characteristic
138	C₄⋊Q₈	C₈×C₆₁	1	1	yes	characteristic
139	#125	C₈×C₆₁	1	2	yes	—
140	C₂²⋊Q₈	—	2	1	no	—
141	C₂²⋊Q₈	—	2	1	no	—
142	C₄×Q₈	—	2	1	no	—
143	C₄×Q₈	—	2	1	no	—
144	#117	C₂×C₄×C₆₁	1	2	yes	—
145	C₂×C₄²	C₂×C₄×C₆₁	1	1	yes	characteristic
146	M₃₂	—	2	2	no	—
147	M₃₂	Q₈×C₆₁	1	4	yes	complemented
148	C₂×C₁₆	—	2	2	no	—
149	C₂×C₁₆	—	2	2	no	—
150	C₂×C₁₆	—	2	4	no	—
151	C₄⋊C₈	—	2	2	no	—
152	C₄⋊C₈	—	2	2	no	—
153	C₄⋊C₈	—	2	1	no	—
154	C₄⋊C₈	—	2	1	no	—
155	#97	—	2	2	no	—
156	C₄×C₈	—	2	2	no	—
157	C₄×C₈	D₄×C₆₁	1	1	yes	characteristic
158	C₄×C₈	D₄×C₆₁	1	1	yes	characteristic
159	C₄×C₈	D₄×C₆₁	1	1	yes	characteristic
160	C₄×C₈	D₄×C₆₁	1	1	yes	characteristic
161	C₂×Q₈	—	2	1	no	—
162	C₂×Q₈	—	2	1	no	—
163	C₂²×C₄	M₁₆×C₆₁	1	1	yes	characteristic
164	C₂²×C₄	M₁₆×C₆₁	1	1	yes	characteristic
165	C₂²×C₄	C₂²×C₄×C₆₁	1	1	yes	characteristic
166	M₁₆	—	2	4	no	—
167	C₂×C₈	—	2	2	no	—
168	C₂×C₈	—	2	2	no	—
169	C₂×C₈	C₂²⋊C₄×C₆₁	1	1	yes	characteristic
170	C₂×C₈	C₂²⋊C₄×C₆₁	1	1	yes	characteristic
171	C₂×C₈	C₂×D₄×C₆₁	1	1	yes	characteristic
172	C₂×C₈	C₂×D₄×C₆₁	1	1	yes	characteristic
173	C₂×C₈	C₂×Q₈×C₆₁	1	2	yes	—
174	C₂×C₈	#41×C₆₁	1	2	yes	—
175	C₄⋊C₄	—	2	2	no	—
176	C₄⋊C₄	—	2	2	no	—
177	C₄⋊C₄	—	2	1	no	—
178	C₄⋊C₄	—	2	1	no	—
179	C₄⋊C₄	C₂×C₈×C₆₁	1	2	yes	—
180	C₄⋊C₄	C₂×C₈×C₆₁	1	2	yes	—
181	C₂²⋊C₄	—	2	2	no	—
182	C₄²	—	2	2	no	—
183	C₄²	C₂²⋊C₄×C₆₁	1	1	yes	characteristic
184	C₄²	C₂²⋊C₄×C₆₁	1	1	yes	characteristic
185	C₄²	C₂×C₈×C₆₁	1	1	yes	characteristic
186	C₄²	C₂×C₈×C₆₁	1	1	yes	characteristic
187	C₁₆	—	2	4	no	—
188	C₁₆	—	4	2	no	—
189	C₂³	C₂×M₁₆×C₆₁	1	1	yes	characteristic
190	Q₈	—	4	2	no	—
191	Q₈	—	4	2	no	—
192	C₂×C₄	—	2	2	no	—
193	C₂×C₄	—	2	2	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	4	1	no	—
196	C₂×C₄	C₂²⋊C₈×C₆₁	1	1	yes	characteristic
197	C₂×C₄	C₂²⋊C₈×C₆₁	1	1	yes	characteristic
198	C₂×C₄	C₂²⋊C₈×C₆₁	1	1	yes	characteristic
199	C₂×C₄	C₂²⋊C₈×C₆₁	1	1	yes	characteristic
200	C₂×C₄	C₂×C₂²⋊C₄×C₆₁	1	1	yes	characteristic
201	C₂×C₄	C₄×Q₈×C₆₁	1	2	yes	—
202	C₂×C₄	C₂²×C₈×C₆₁	1	1	yes	characteristic
203	C₂×C₄	#131×C₆₁	1	2	yes	—
204	C₈	—	2	1	no	—
205	C₈	—	2	1	no	—
206	C₈	—	4	2	no	—
207	C₈	C₂²⋊Q₈×C₆₁	1	2	yes	—
208	C₈	#124×C₆₁	1	1	yes	characteristic
209	C₈	C₄⋊Q₈×C₆₁	1	1	yes	characteristic
210	C₂²	—	2	1	no	—
211	C₂²	—	2	1	no	—
212	C₂²	C₂×C₂²⋊C₈×C₆₁	1	1	yes	characteristic
213	C₂²	C₆₁×#446	1	2	yes	—
214	C₄	—	2	1	no	—
215	C₄	—	2	1	no	—
216	C₄	—	2	1	no	—
217	C₄	—	4	2	no	—
218	C₄	—	4	2	no	—
219	C₄	C₆₁×#350	1	2	yes	—
220	C₄	C₆₁×#391	1	1	yes	characteristic
221	C₄	C₆₁×#408	1	1	yes	characteristic
222	C₄	C₈×Q₈×C₆₁	1	2	yes	—
223	C₂	—	2	2	no	—
224	C₂	C₆₁×#1984	1	1	yes	characteristic
225	C₂	C₂×C₆₁×#350	1	1	yes	characteristic
226	C₂	C₆₁×#2117	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈×C₆₁	C₂×D₄
commutator	C₂×C₄	C₂²×C₈×C₆₁
Frattini, cosocle	C₂²×C₈	C₂³×C₆₁
socle	C₂²×C₆₁	C₂×C₂²⋊C₈
prime radical	C₂³×C₆₁	C₂×M₁₆
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈×C₆₁ ◁ C₂×C₄×C₈×C₆₁ ◁ G

Complements

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

N	H ≅ G/N	type
33. M₃₂×C₆₁	Q₈	semidirect
96. #13265	C₆₁	direct (N × H)
133. C₆₁	#13265	direct (N × H)
147. M₃₂	Q₈×C₆₁	semidirect

Maximal subgroups

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

subgroup	index
C₆₁×#2960	2
C₂×C₆₁×C₄⋊C₁₆	2
C₄×M₃₂×C₆₁	2
C₆₁×#1379	2
#13265	61

Subgroups — C61×#13265 #29516698

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#13265 #29516698