Subgroups — C₆₁×#21918 #29525351

Order 15616.

694 conjugacy classes of subgroups.

The subgroup lattice has 258 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₁×#2285	C₂	1	1	yes	maximal, characteristic
2	C₆₁×#1917	C₂	1	4	yes	maximal, complemented
3	C₆₁×#1916	C₂	1	4	yes	maximal, complemented
4	C₆₁×#1817	C₂	1	2	yes	maximal
5	C₆₁×#1798	C₂	1	2	yes	maximal
6	C₆₁×#1750	C₂	1	2	yes	maximal
7	C₂×#117×C₆₁	C₂²	1	1	yes	characteristic
8	C₂×#117×C₆₁	C₂²	1	1	yes	characteristic
9	C₂²×C₄²×C₆₁	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊C₈×C₆₁	C₂²	1	4	yes	—
11	C₂×C₂²⋊C₈×C₆₁	C₂²	1	4	yes	—
12	C₂×#97×C₆₁	C₂²	1	2	yes	—
13	C₂×C₄×C₈×C₆₁	C₂²	1	2	yes	—
14	C₆₁×#384	C₄	1	2	yes	—
15	C₆₁×#382	C₄	1	2	yes	—
16	C₆₁×#380	C₄	1	4	yes	—
17	C₄×C₄⋊C₄×C₆₁	C₂²	1	4	yes	—
18	C₆₁×#336	C₂²	1	2	yes	—
19	C₆₁×#336	C₂²	1	2	yes	—
20	C₆₁×#336	C₂²	1	2	yes	—
21	C₆₁×#336	C₂²	1	4	yes	—
22	C₆₁×#336	C₂²	1	2	yes	—
23	C₆₁×#336	C₂²	1	4	yes	—
24	C₂³×C₄×C₆₁	C₂×C₄	1	2	yes	—
25	C₂³×C₄×C₆₁	C₂³	1	1	yes	characteristic
26	C₂²×C₈×C₆₁	C₂³	1	4	yes	—
27	C₂²×C₈×C₆₁	C₂³	1	4	yes	—
28	#117×C₆₁	—	4	2	no	—
29	#117×C₆₁	—	4	2	no	—
30	C₂×C₄⋊C₄×C₆₁	C₂×C₄	1	4	yes	—
31	C₂×C₄⋊C₄×C₆₁	C₂×C₄	1	4	yes	—
32	C₂×C₄⋊C₄×C₆₁	C₂×C₄	1	2	yes	—
33	C₂×C₄⋊C₄×C₆₁	C₂×C₄	1	2	yes	—
34	C₂×C₂²⋊C₄×C₆₁	C₂×C₄	1	2	yes	—
35	C₂×C₂²⋊C₄×C₆₁	C₂×C₄	1	2	yes	—
36	C₂×C₄²×C₆₁	—	2	4	no	—
37	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄²×C₆₁	C₂³	1	2	yes	—
40	C₂×C₄²×C₆₁	C₂³	1	2	yes	—
41	C₂×C₄²×C₆₁	C₂³	1	2	yes	—
42	C₄⋊C₈×C₆₁	—	4	4	no	—
43	C₂²⋊C₈×C₆₁	—	2	8	no	—
44	#97×C₆₁	—	4	2	no	—
45	C₄×C₈×C₆₁	—	4	2	no	—
46	#95×C₆₁	C₂×C₄	1	8	yes	—
47	C₂⁴×C₆₁	C₂²×C₄	1	1	yes	characteristic
48	C₂²×C₄×C₆₁	—	2	2	no	—
49	C₂²×C₄×C₆₁	—	2	2	no	—
50	C₂²×C₄×C₆₁	—	2	4	no	—
51	C₂²×C₄×C₆₁	—	2	4	no	—
52	C₂²×C₄×C₆₁	—	4	2	no	—
53	C₂²×C₄×C₆₁	C₂²×C₄	1	2	yes	—
54	C₂²×C₄×C₆₁	C₂²×C₄	1	4	yes	—
55	C₂²×C₄×C₆₁	C₂²×C₄	1	4	yes	—
56	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
57	C₂²×C₄×C₆₁	C₂²×C₄	1	2	yes	—
58	C₂²×C₄×C₆₁	#41	1	4	yes	—
59	C₂²×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
60	C₂×C₈×C₆₁	—	2	4	no	—
61	C₂×C₈×C₆₁	—	2	4	no	—
62	C₂×C₈×C₆₁	—	2	4	no	—
63	C₂×C₈×C₆₁	—	2	4	no	—
64	C₂×C₈×C₆₁	—	2	4	no	—
65	C₂×C₈×C₆₁	—	2	4	no	—
66	C₄⋊C₄×C₆₁	—	2	8	no	—
67	C₄⋊C₄×C₆₁	—	4	4	no	—
68	C₄⋊C₄×C₆₁	—	4	2	no	—
69	C₄⋊C₄×C₆₁	—	4	2	no	—
70	C₂²⋊C₄×C₆₁	—	4	2	no	—
71	C₂²⋊C₄×C₆₁	—	4	2	no	—
72	C₄²×C₆₁	—	2	2	no	—
73	C₄²×C₆₁	—	2	2	no	—
74	C₄²×C₆₁	—	2	4	no	—
75	C₄²×C₆₁	—	2	2	no	—
76	C₄²×C₆₁	—	2	2	no	—
77	C₄²×C₆₁	—	2	4	no	—
78	C₂³×C₆₁	—	2	1	no	—
79	C₂³×C₆₁	—	2	1	no	—
80	C₂³×C₆₁	—	2	2	no	—
81	C₂³×C₆₁	—	2	2	no	—
82	C₂³×C₆₁	#117	1	2	yes	—
83	C₂³×C₆₁	C₂³×C₄	1	1	yes	characteristic
84	C₂×C₄×C₆₁	—	2	2	no	—
85	C₂×C₄×C₆₁	—	2	1	no	—
86	C₂×C₄×C₆₁	—	2	1	no	—
87	C₂×C₄×C₆₁	—	2	2	no	—
88	C₂×C₄×C₆₁	—	2	8	no	—
89	C₂×C₄×C₆₁	—	2	2	no	—
90	C₂×C₄×C₆₁	—	2	4	no	—
91	C₂×C₄×C₆₁	—	2	4	no	—
92	C₂×C₄×C₆₁	—	2	4	no	—
93	C₂×C₄×C₆₁	—	2	4	no	—
94	C₂×C₄×C₆₁	—	2	2	no	—
95	C₂×C₄×C₆₁	—	2	4	no	—
96	C₂×C₄×C₆₁	—	2	4	no	—
97	C₂×C₄×C₆₁	—	2	2	no	—
98	C₂×C₄×C₆₁	—	4	8	no	—
99	C₂×C₄×C₆₁	#117	1	2	yes	—
100	C₂×C₄×C₆₁	#131	1	4	yes	—
101	C₂×C₄×C₆₁	C₂×#41	1	2	yes	—
102	C₂×C₄×C₆₁	H(5,𝔽₂)	1	2	yes	—
103	C₂×C₄×C₆₁	2₋¹⁺⁴	1	2	yes	—
104	C₈×C₆₁	—	4	4	no	—
105	C₈×C₆₁	—	4	4	no	—
106	#21918	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
107	C₂²×C₆₁	—	2	2	no	—
108	C₂²×C₆₁	—	2	4	no	—
109	C₂²×C₆₁	—	4	2	no	—
110	C₂²×C₆₁	—	4	1	no	—
111	C₂²×C₆₁	—	4	1	no	—
112	C₂²×C₆₁	C₂×#117	1	1	yes	characteristic
113	C₂²×C₆₁	#518	1	1	yes	characteristic
114	C₂²×C₆₁	#519	1	1	yes	characteristic
115	C₂²×C₆₁	C₂×#131	1	2	yes	—
116	C₂²×C₆₁	#568	1	2	yes	—
117	C₄×C₆₁	—	2	2	no	—
118	C₄×C₆₁	—	2	4	no	—
119	C₄×C₆₁	—	4	2	no	—
120	C₄×C₆₁	—	4	4	no	—
121	C₄×C₆₁	—	4	4	no	—
122	C₄×C₆₁	#539	1	4	yes	—
123	#2285	C₂×C₆₁	1	1	yes	characteristic
124	#1917	C₂×C₆₁	1	4	yes	complemented
125	#1916	C₂×C₆₁	1	4	yes	complemented
126	#1817	C₂×C₆₁	1	2	yes	—
127	#1798	C₂×C₆₁	1	2	yes	—
128	#1750	C₂×C₆₁	1	2	yes	—
129	C₂×C₆₁	—	4	2	no	—
130	C₂×C₆₁	#2313	1	1	yes	characteristic
131	C₂×C₆₁	#2924	1	2	yes	—
132	C₂×C₆₁	#2981	1	2	yes	—
133	C₂×C₆₁	#2986	1	1	yes	characteristic
134	C₂×C₆₁	#2987	1	1	yes	characteristic
135	C₂×#117	C₂²×C₆₁	1	1	yes	characteristic
136	C₂×#117	C₂²×C₆₁	1	1	yes	characteristic
137	C₂²×C₄²	C₂²×C₆₁	1	1	yes	characteristic
138	C₂×C₄⋊C₈	C₂²×C₆₁	1	4	yes	—
139	C₂×C₂²⋊C₈	C₂²×C₆₁	1	4	yes	—
140	C₂×#97	C₂²×C₆₁	1	2	yes	—
141	C₂×C₄×C₈	C₂²×C₆₁	1	2	yes	—
142	#384	C₄×C₆₁	1	2	yes	—
143	#382	C₄×C₆₁	1	2	yes	—
144	#380	C₄×C₆₁	1	4	yes	—
145	C₄×C₄⋊C₄	C₂²×C₆₁	1	4	yes	—
146	#336	C₂²×C₆₁	1	2	yes	—
147	#336	C₂²×C₆₁	1	2	yes	—
148	#336	C₂²×C₆₁	1	2	yes	—
149	#336	C₂²×C₆₁	1	4	yes	—
150	#336	C₂²×C₆₁	1	2	yes	—
151	#336	C₂²×C₆₁	1	4	yes	—
152	C₆₁	#21918	1	1	yes	characteristic, direct product factor, Sylow
153	C₂³×C₄	C₂×C₄×C₆₁	1	2	yes	—
154	C₂³×C₄	C₂³×C₆₁	1	1	yes	characteristic
155	C₂²×C₈	C₂³×C₆₁	1	4	yes	—
156	C₂²×C₈	C₂³×C₆₁	1	4	yes	—
157	#117	—	4	2	no	—
158	#117	—	4	2	no	—
159	C₂×C₄⋊C₄	C₂×C₄×C₆₁	1	4	yes	—
160	C₂×C₄⋊C₄	C₂×C₄×C₆₁	1	2	yes	—
161	C₂×C₄⋊C₄	C₂×C₄×C₆₁	1	2	yes	—
162	C₂×C₄⋊C₄	C₂×C₄×C₆₁	1	4	yes	—
163	C₂×C₂²⋊C₄	C₂×C₄×C₆₁	1	2	yes	—
164	C₂×C₂²⋊C₄	C₂×C₄×C₆₁	1	2	yes	—
165	C₂×C₄²	—	2	4	no	—
166	C₂×C₄²	C₂×C₄×C₆₁	1	1	yes	characteristic
167	C₂×C₄²	C₂×C₄×C₆₁	1	1	yes	characteristic
168	C₂×C₄²	C₂³×C₆₁	1	2	yes	—
169	C₂×C₄²	C₂³×C₆₁	1	2	yes	—
170	C₂×C₄²	C₂³×C₆₁	1	2	yes	—
171	C₄⋊C₈	—	4	4	no	—
172	C₂²⋊C₈	—	2	8	no	—
173	#97	—	4	2	no	—
174	C₄×C₈	—	4	2	no	—
175	#95	C₂×C₄×C₆₁	1	8	yes	—
176	C₂⁴	C₂²×C₄×C₆₁	1	1	yes	characteristic
177	C₂²×C₄	—	2	2	no	—
178	C₂²×C₄	—	2	2	no	—
179	C₂²×C₄	—	2	4	no	—
180	C₂²×C₄	—	2	4	no	—
181	C₂²×C₄	—	4	2	no	—
182	C₂²×C₄	C₂²×C₄×C₆₁	1	2	yes	—
183	C₂²×C₄	C₂²×C₄×C₆₁	1	4	yes	—
184	C₂²×C₄	C₂²×C₄×C₆₁	1	4	yes	—
185	C₂²×C₄	C₂²×C₄×C₆₁	1	1	yes	characteristic
186	C₂²×C₄	C₂²×C₄×C₆₁	1	2	yes	—
187	C₂²×C₄	#41×C₆₁	1	4	yes	—
188	C₂²×C₄	C₂⁴×C₆₁	1	1	yes	characteristic
189	C₂×C₈	—	2	4	no	—
190	C₂×C₈	—	2	4	no	—
191	C₂×C₈	—	2	4	no	—
192	C₂×C₈	—	2	4	no	—
193	C₂×C₈	—	2	4	no	—
194	C₂×C₈	—	2	4	no	—
195	C₄⋊C₄	—	2	8	no	—
196	C₄⋊C₄	—	4	2	no	—
197	C₄⋊C₄	—	4	2	no	—
198	C₄⋊C₄	—	4	4	no	—
199	C₂²⋊C₄	—	4	2	no	—
200	C₂²⋊C₄	—	4	2	no	—
201	C₄²	—	2	2	no	—
202	C₄²	—	2	2	no	—
203	C₄²	—	2	4	no	—
204	C₄²	—	2	2	no	—
205	C₄²	—	2	2	no	—
206	C₄²	—	2	4	no	—
207	C₂³	—	2	2	no	—
208	C₂³	—	2	2	no	—
209	C₂³	—	2	1	no	—
210	C₂³	—	2	1	no	—
211	C₂³	#117×C₆₁	1	2	yes	—
212	C₂³	C₂³×C₄×C₆₁	1	1	yes	characteristic
213	C₂×C₄	—	2	2	no	—
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	—	2	1	no	—
216	C₂×C₄	—	2	1	no	—
217	C₂×C₄	—	2	8	no	—
218	C₂×C₄	—	2	4	no	—
219	C₂×C₄	—	2	4	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	2	4	no	—
222	C₂×C₄	—	2	4	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	2	2	no	—
225	C₂×C₄	—	2	4	no	—
226	C₂×C₄	—	2	4	no	—
227	C₂×C₄	—	4	8	no	—
228	C₂×C₄	#117×C₆₁	1	2	yes	—
229	C₂×C₄	#131×C₆₁	1	4	yes	—
230	C₂×C₄	C₂×#41×C₆₁	1	2	yes	—
231	C₂×C₄	H(5,𝔽₂)×C₆₁	1	2	yes	—
232	C₂×C₄	2₋¹⁺⁴×C₆₁	1	2	yes	—
233	C₈	—	4	4	no	—
234	C₈	—	4	4	no	—
235	C₂²	—	2	2	no	—
236	C₂²	—	2	4	no	—
237	C₂²	—	4	2	no	—
238	C₂²	—	4	1	no	—
239	C₂²	—	4	1	no	—
240	C₂²	C₂×#117×C₆₁	1	1	yes	characteristic
241	C₂²	C₆₁×#518	1	1	yes	characteristic
242	C₂²	C₆₁×#519	1	1	yes	characteristic
243	C₂²	C₂×#131×C₆₁	1	2	yes	—
244	C₂²	C₆₁×#568	1	2	yes	—
245	C₄	—	2	2	no	—
246	C₄	—	2	4	no	—
247	C₄	—	4	2	no	—
248	C₄	—	4	4	no	—
249	C₄	—	4	4	no	—
250	C₄	C₆₁×#539	1	4	yes	—
251	C₂	—	4	2	no	—
252	C₂	C₆₁×#2313	1	1	yes	characteristic
253	C₂	C₆₁×#2924	1	2	yes	—
254	C₂	C₆₁×#2981	1	2	yes	—
255	C₂	C₆₁×#2986	1	1	yes	characteristic
256	C₂	C₆₁×#2987	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₆₁	C₂⁴
commutator	C₂³	C₂³×C₄×C₆₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₁
socle	C₂³×C₆₁	C₂³×C₄
prime radical	C₂⁴×C₆₁	C₂²×C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₆₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₆₁×#1917	C₂	semidirect
3. C₆₁×#1916	C₂	semidirect
106. #21918	C₆₁	direct (N × H)
124. #1917	C₂×C₆₁	semidirect
125. #1916	C₂×C₆₁	semidirect
152. C₆₁	#21918	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×#2285	2
C₆₁×#1917	2
C₆₁×#1916	2
C₆₁×#1817	2
C₆₁×#1798	2
C₆₁×#1750	2
#21918	61

Subgroups — C61×#21918 #29525351

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#21918 #29525351