Subgroups — C₆₁×#7274 #29510707

Order 15616.

262 conjugacy classes of subgroups.

The subgroup lattice has 226 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₆₁×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic
2	C₆₁×#1845	C₂	1	1	yes	maximal, characteristic
3	C₂×C₆₁×#335	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₁₆×C₆₁	—	2	1	no	—
5	C₂×#107×C₆₁	—	2	1	no	—
6	C₂×#107×C₆₁	C₄	1	2	yes	—
7	C₂×C₄⋊C₈×C₆₁	—	2	1	no	—
8	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
9	C₆₁×#384	—	2	1	no	—
10	C₆₁×C₄⋊C₁₆	—	2	1	no	—
11	C₆₁×C₄⋊C₁₆	C₄	1	2	yes	complemented
12	C₆₁×#335	—	2	1	no	—
13	C₆₁×#335	C₄	1	2	yes	complemented
14	C₂²×C₈×C₆₁	—	4	1	no	—
15	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
16	C₂²×C₈×C₆₁	D₄	1	1	yes	characteristic
17	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
18	C₂×C₄⋊C₄×C₆₁	—	4	1	no	—
19	C₂×C₄⋊C₄×C₆₁	—	4	1	no	—
20	C₂×C₄²×C₆₁	D₄	1	1	yes	characteristic
21	C₂×C₁₆×C₆₁	—	2	2	no	—
22	C₂×C₁₆×C₆₁	—	4	1	no	—
23	C₂×C₁₆×C₆₁	—	4	1	no	—
24	#107×C₆₁	—	4	1	no	—
25	#107×C₆₁	—	4	1	no	—
26	#107×C₆₁	—	4	2	no	—
27	C₄⋊C₈×C₆₁	—	2	2	no	—
28	C₄⋊C₈×C₆₁	—	4	1	no	—
29	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
30	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
31	C₄×C₈×C₆₁	D₄	1	1	yes	characteristic
32	C₄×C₈×C₆₁	Q₈	1	1	yes	characteristic
33	C₂²×C₄×C₆₁	—	4	2	no	—
34	C₂²×C₄×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
35	C₂²×C₄×C₆₁	D₈	1	1	yes	characteristic
36	C₂²×C₄×C₆₁	SD₁₆	1	1	yes	characteristic
37	C₂×C₈×C₆₁	—	2	1	no	—
38	C₂×C₈×C₆₁	—	2	1	no	—
39	C₂×C₈×C₆₁	—	4	2	no	—
40	C₂×C₈×C₆₁	—	8	1	no	—
41	C₂×C₈×C₆₁	—	8	1	no	—
42	C₂×C₈×C₆₁	C₄²	1	1	yes	characteristic
43	C₂×C₈×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
44	C₂×C₈×C₆₁	C₄⋊C₄	1	1	yes	characteristic
45	C₂×C₈×C₆₁	C₄⋊C₄	1	1	yes	characteristic
46	C₂×C₈×C₆₁	D₈	1	1	yes	characteristic
47	C₂×C₈×C₆₁	SD₁₆	1	1	yes	characteristic
48	C₂×C₈×C₆₁	SD₁₆	1	1	yes	characteristic
49	C₂×C₈×C₆₁	Q₁₆	1	1	yes	characteristic
50	C₄⋊C₄×C₆₁	—	4	2	no	—
51	C₄⋊C₄×C₆₁	—	4	2	no	—
52	C₄⋊C₄×C₆₁	—	8	2	no	—
53	C₄⋊C₄×C₆₁	—	8	1	no	—
54	C₄⋊C₄×C₆₁	—	8	1	no	—
55	C₄²×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
56	C₄²×C₆₁	C₄⋊C₄	1	1	yes	characteristic
57	C₄²×C₆₁	SD₁₆	1	1	yes	characteristic
58	C₄²×C₆₁	Q₁₆	1	1	yes	characteristic
59	C₁₆×C₆₁	—	4	2	no	—
60	C₂³×C₆₁	#102	1	1	yes	characteristic
61	C₂×C₄×C₆₁	—	2	1	no	—
62	C₂×C₄×C₆₁	—	2	1	no	—
63	C₂×C₄×C₆₁	—	8	2	no	—
64	C₂×C₄×C₆₁	—	8	2	no	—
65	C₂×C₄×C₆₁	—	8	2	no	—
66	C₂×C₄×C₆₁	#95	1	1	yes	characteristic
67	C₂×C₄×C₆₁	#100	1	1	yes	characteristic
68	C₂×C₄×C₆₁	#101	1	1	yes	characteristic
69	C₂×C₄×C₆₁	#102	1	1	yes	characteristic
70	C₂×C₄×C₆₁	#102	1	1	yes	characteristic
71	C₂×C₄×C₆₁	#103	1	1	yes	characteristic
72	C₂×C₄×C₆₁	#103	1	1	yes	characteristic
73	C₂×C₄×C₆₁	#103	1	1	yes	characteristic
74	C₂×C₄×C₆₁	#106	1	1	yes	characteristic
75	C₂×C₄×C₆₁	#107	1	1	yes	characteristic
76	C₂×C₄×C₆₁	D₁₆	1	1	yes	characteristic
77	C₂×C₄×C₆₁	SD₃₂	1	1	yes	characteristic
78	C₂×C₄×C₆₁	SD₃₂	1	1	yes	characteristic
79	C₂×C₄×C₆₁	Q₃₂	1	1	yes	characteristic
80	C₈×C₆₁	—	2	1	no	—
81	C₈×C₆₁	—	2	1	no	—
82	C₈×C₆₁	—	8	2	no	—
83	C₈×C₆₁	#102	1	1	yes	characteristic
84	C₈×C₆₁	#103	1	1	yes	characteristic
85	C₈×C₆₁	#106	1	1	yes	characteristic
86	C₈×C₆₁	#107	1	1	yes	characteristic
87	#7274	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
88	C₂²×C₆₁	#331	1	1	yes	characteristic
89	C₂²×C₆₁	#332	1	1	yes	characteristic
90	C₂²×C₆₁	#340	1	1	yes	characteristic
91	C₂²×C₆₁	#357	1	1	yes	characteristic
92	C₂²×C₆₁	#358	1	1	yes	characteristic
93	C₂²×C₆₁	#361	1	1	yes	characteristic
94	C₂²×C₆₁	#362	1	1	yes	characteristic
95	C₄×C₆₁	—	2	1	no	—
96	C₄×C₆₁	—	2	1	no	—
97	C₄×C₆₁	—	16	2	no	—
98	C₄×C₆₁	#332	1	1	yes	characteristic
99	C₄×C₆₁	#333	1	1	yes	characteristic
100	C₄×C₆₁	#340	1	1	yes	characteristic
101	C₄×C₆₁	#343	1	1	yes	characteristic
102	C₄×C₆₁	#357	1	1	yes	characteristic
103	C₄×C₆₁	#358	1	1	yes	characteristic
104	C₄×C₆₁	#366	1	1	yes	characteristic
105	C₄×C₆₁	#367	1	1	yes	characteristic
106	C₂×C₄⋊C₁₆	C₂×C₆₁	1	1	yes	characteristic
107	#1845	C₂×C₆₁	1	1	yes	characteristic
108	C₂×#335	C₂×C₆₁	1	1	yes	characteristic
109	C₂×C₆₁	#1296	1	1	yes	characteristic
110	C₂×C₆₁	#1361	1	1	yes	characteristic
111	C₂×C₆₁	#1362	1	1	yes	characteristic
112	C₂×C₆₁	#1364	1	1	yes	characteristic
113	C₂×C₆₁	#1365	1	1	yes	characteristic
114	C₂×C₆₁	#1380	1	1	yes	characteristic
115	C₂×C₆₁	#1387	1	1	yes	characteristic
116	C₂²×C₁₆	—	2	1	no	—
117	C₂×#107	—	2	1	no	—
118	C₂×#107	C₄×C₆₁	1	2	yes	—
119	C₂×C₄⋊C₈	—	2	1	no	—
120	C₂×C₄×C₈	C₂²×C₆₁	1	1	yes	characteristic
121	#384	—	2	1	no	—
122	C₄⋊C₁₆	—	2	1	no	—
123	C₄⋊C₁₆	C₄×C₆₁	1	2	yes	complemented
124	#335	—	2	1	no	—
125	#335	C₄×C₆₁	1	2	yes	complemented
126	C₆₁	#7274	1	1	yes	characteristic, direct product factor, Sylow
127	C₂²×C₈	—	4	1	no	—
128	C₂²×C₈	C₂×C₄×C₆₁	1	1	yes	characteristic
129	C₂²×C₈	D₄×C₆₁	1	1	yes	characteristic
130	C₂×C₄⋊C₄	—	2	2	no	—
131	C₂×C₄⋊C₄	—	4	1	no	—
132	C₂×C₄⋊C₄	—	4	1	no	—
133	C₂×C₄²	D₄×C₆₁	1	1	yes	characteristic
134	C₂×C₁₆	—	2	2	no	—
135	C₂×C₁₆	—	4	1	no	—
136	C₂×C₁₆	—	4	1	no	—
137	#107	—	4	1	no	—
138	#107	—	4	1	no	—
139	#107	—	4	2	no	—
140	C₄⋊C₈	—	2	2	no	—
141	C₄⋊C₈	—	4	1	no	—
142	C₄×C₈	C₂×C₄×C₆₁	1	1	yes	characteristic
143	C₄×C₈	C₂×C₄×C₆₁	1	1	yes	characteristic
144	C₄×C₈	D₄×C₆₁	1	1	yes	characteristic
145	C₄×C₈	Q₈×C₆₁	1	1	yes	characteristic
146	C₂²×C₄	—	4	2	no	—
147	C₂²×C₄	C₂²⋊C₄×C₆₁	1	1	yes	characteristic
148	C₂²×C₄	D₈×C₆₁	1	1	yes	characteristic
149	C₂²×C₄	SD₁₆×C₆₁	1	1	yes	characteristic
150	C₂×C₈	—	2	1	no	—
151	C₂×C₈	—	2	1	no	—
152	C₂×C₈	—	4	2	no	—
153	C₂×C₈	—	8	1	no	—
154	C₂×C₈	—	8	1	no	—
155	C₂×C₈	C₄²×C₆₁	1	1	yes	characteristic
156	C₂×C₈	C₂²⋊C₄×C₆₁	1	1	yes	characteristic
157	C₂×C₈	C₄⋊C₄×C₆₁	1	1	yes	characteristic
158	C₂×C₈	C₄⋊C₄×C₆₁	1	1	yes	characteristic
159	C₂×C₈	D₈×C₆₁	1	1	yes	characteristic
160	C₂×C₈	SD₁₆×C₆₁	1	1	yes	characteristic
161	C₂×C₈	SD₁₆×C₆₁	1	1	yes	characteristic
162	C₂×C₈	Q₁₆×C₆₁	1	1	yes	characteristic
163	C₄⋊C₄	—	4	2	no	—
164	C₄⋊C₄	—	4	2	no	—
165	C₄⋊C₄	—	8	2	no	—
166	C₄⋊C₄	—	8	1	no	—
167	C₄⋊C₄	—	8	1	no	—
168	C₄²	C₂²⋊C₄×C₆₁	1	1	yes	characteristic
169	C₄²	C₄⋊C₄×C₆₁	1	1	yes	characteristic
170	C₄²	SD₁₆×C₆₁	1	1	yes	characteristic
171	C₄²	Q₁₆×C₆₁	1	1	yes	characteristic
172	C₁₆	—	4	2	no	—
173	C₂³	#102×C₆₁	1	1	yes	characteristic
174	C₂×C₄	—	2	1	no	—
175	C₂×C₄	—	2	1	no	—
176	C₂×C₄	—	8	2	no	—
177	C₂×C₄	—	8	2	no	—
178	C₂×C₄	—	8	2	no	—
179	C₂×C₄	#95×C₆₁	1	1	yes	characteristic
180	C₂×C₄	#100×C₆₁	1	1	yes	characteristic
181	C₂×C₄	#101×C₆₁	1	1	yes	characteristic
182	C₂×C₄	#102×C₆₁	1	1	yes	characteristic
183	C₂×C₄	#102×C₆₁	1	1	yes	characteristic
184	C₂×C₄	#103×C₆₁	1	1	yes	characteristic
185	C₂×C₄	#103×C₆₁	1	1	yes	characteristic
186	C₂×C₄	#103×C₆₁	1	1	yes	characteristic
187	C₂×C₄	#106×C₆₁	1	1	yes	characteristic
188	C₂×C₄	#107×C₆₁	1	1	yes	characteristic
189	C₂×C₄	D₁₆×C₆₁	1	1	yes	characteristic
190	C₂×C₄	SD₃₂×C₆₁	1	1	yes	characteristic
191	C₂×C₄	SD₃₂×C₆₁	1	1	yes	characteristic
192	C₂×C₄	Q₃₂×C₆₁	1	1	yes	characteristic
193	C₈	—	2	1	no	—
194	C₈	—	2	1	no	—
195	C₈	—	8	2	no	—
196	C₈	#102×C₆₁	1	1	yes	characteristic
197	C₈	#103×C₆₁	1	1	yes	characteristic
198	C₈	#106×C₆₁	1	1	yes	characteristic
199	C₈	#107×C₆₁	1	1	yes	characteristic
200	C₂²	C₆₁×#331	1	1	yes	characteristic
201	C₂²	C₆₁×#332	1	1	yes	characteristic
202	C₂²	C₆₁×#340	1	1	yes	characteristic
203	C₂²	C₆₁×#357	1	1	yes	characteristic
204	C₂²	C₆₁×#358	1	1	yes	characteristic
205	C₂²	C₆₁×#361	1	1	yes	characteristic
206	C₂²	C₆₁×#362	1	1	yes	characteristic
207	C₄	—	2	1	no	—
208	C₄	—	2	1	no	—
209	C₄	—	16	2	no	—
210	C₄	C₆₁×#332	1	1	yes	characteristic
211	C₄	C₆₁×#333	1	1	yes	characteristic
212	C₄	C₆₁×#340	1	1	yes	characteristic
213	C₄	C₆₁×#343	1	1	yes	characteristic
214	C₄	C₆₁×#357	1	1	yes	characteristic
215	C₄	C₆₁×#358	1	1	yes	characteristic
216	C₄	C₆₁×#366	1	1	yes	characteristic
217	C₄	C₆₁×#367	1	1	yes	characteristic
218	C₂	C₆₁×#1296	1	1	yes	characteristic
219	C₂	C₆₁×#1361	1	1	yes	characteristic
220	C₂	C₆₁×#1362	1	1	yes	characteristic
221	C₂	C₆₁×#1364	1	1	yes	characteristic
222	C₂	C₆₁×#1365	1	1	yes	characteristic
223	C₂	C₆₁×#1380	1	1	yes	characteristic
224	C₂	C₆₁×#1387	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₆₁	#102
commutator	C₂×C₈	C₄²×C₆₁
Frattini, cosocle	C₂×C₄×C₈	C₂²×C₆₁
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×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
11. C₆₁×C₄⋊C₁₆	C₄	semidirect
13. C₆₁×#335	C₄	semidirect
87. #7274	C₆₁	direct (N × H)
123. C₄⋊C₁₆	C₄×C₆₁	semidirect
125. #335	C₄×C₆₁	semidirect
126. C₆₁	#7274	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₆₁×C₄⋊C₁₆	2
C₆₁×#1845	2
C₂×C₆₁×#335	2
#7274	61

Subgroups — C61×#7274 #29510707

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#7274 #29510707