Subgroups — C₆₁×#33768 #29537201

Order 15616.

552 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	C₆₁×#3059	C₂	1	2	yes	maximal
2	C₆₁×#2237	C₂	1	4	yes	maximal, complemented
3	C₆₁×#2219	C₂	1	2	yes	maximal, complemented
4	C₆₁×#2216	C₂	1	2	yes	maximal, complemented
5	C₆₁×#2188	C₂	1	4	yes	maximal, complemented
6	C₆₁×#2113	C₂	1	1	yes	maximal, characteristic
7	C₂²×Q₁₆×C₆₁	C₂²	1	2	yes	—
8	C₂×#131×C₆₁	C₂²	1	1	yes	characteristic
9	C₆₁×#536	—	2	2	no	—
10	C₂×Q₃₂×C₆₁	C₂²	1	2	yes	—
11	C₂×Q₃₂×C₆₁	C₂²	1	2	yes	—
12	C₂×M₃₂×C₆₁	C₂²	1	1	yes	characteristic
13	C₂²×C₁₆×C₆₁	C₂²	1	1	yes	characteristic
14	C₆₁×#470	—	2	2	no	—
15	C₆₁×#470	—	2	2	no	—
16	C₆₁×#467	C₂²	1	4	yes	—
17	C₆₁×#467	C₂²	1	4	yes	—
18	C₆₁×#427	C₂²	1	2	yes	—
19	C₆₁×#419	—	2	2	no	—
20	C₆₁×#367	C₂²	1	2	yes	—
21	C₆₁×#366	C₂²	1	2	yes	—
22	C₆₁×#358	C₂²	1	4	yes	—
23	C₆₁×#358	C₂²	1	4	yes	—
24	C₆₁×C₂²⋊C₁₆	C₂²	1	4	yes	—
25	C₂×#41×C₆₁	D₄	1	1	yes	characteristic
26	C₂²×Q₈×C₆₁	—	2	2	no	—
27	C₂×Q₁₆×C₆₁	—	2	2	no	—
28	C₂×Q₁₆×C₆₁	—	4	2	no	—
29	C₂×Q₁₆×C₆₁	—	4	2	no	—
30	C₂×Q₁₆×C₆₁	C₂³	1	4	yes	—
31	#131×C₆₁	—	4	1	no	—
32	#131×C₆₁	—	4	1	no	—
33	C₂×M₁₆×C₆₁	D₄	1	2	yes	—
34	C₂×M₁₆×C₆₁	D₄	1	1	yes	characteristic
35	C₂²×C₈×C₆₁	C₂³	1	2	yes	—
36	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
37	C₄⋊Q₈×C₆₁	—	4	2	no	—
38	#124×C₆₁	—	4	2	no	—
39	#123×C₆₁	—	4	2	no	—
40	#123×C₆₁	—	4	2	no	—
41	C₂²⋊Q₈×C₆₁	—	2	4	no	—
42	C₂²⋊Q₈×C₆₁	—	2	4	no	—
43	#117×C₆₁	—	2	2	no	—
44	Q₃₂×C₆₁	—	4	2	no	—
45	Q₃₂×C₆₁	—	4	2	no	—
46	M₃₂×C₆₁	—	4	1	no	—
47	C₂×C₁₆×C₆₁	—	4	1	no	—
48	C₂×C₁₆×C₆₁	C₂³	1	2	yes	—
49	C₂×C₁₆×C₆₁	C₂³	1	2	yes	—
50	#107×C₆₁	C₂³	1	4	yes	—
51	#106×C₆₁	—	2	2	no	—
52	#103×C₆₁	—	2	4	no	—
53	#103×C₆₁	—	2	4	no	—
54	#41×C₆₁	—	4	1	no	—
55	#41×C₆₁	—	4	1	no	—
56	C₂×Q₈×C₆₁	—	2	4	no	—
57	C₂×Q₈×C₆₁	—	4	4	no	—
58	C₂×Q₈×C₆₁	—	4	4	no	—
59	C₂×Q₈×C₆₁	—	4	2	no	—
60	C₂×Q₈×C₆₁	D₈	1	1	yes	characteristic
61	C₂×D₄×C₆₁	D₈	1	1	yes	characteristic
62	C₂×D₄×C₆₁	D₈	1	2	yes	—
63	C₂²×C₄×C₆₁	—	4	2	no	—
64	C₂²×C₄×C₆₁	C₂×D₄	1	2	yes	—
65	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
66	Q₁₆×C₆₁	—	2	4	no	—
67	Q₁₆×C₆₁	—	2	4	no	—
68	Q₁₆×C₆₁	—	4	4	no	—
69	M₁₆×C₆₁	—	4	2	no	—
70	M₁₆×C₆₁	—	4	1	no	—
71	C₂×C₈×C₆₁	—	2	1	no	—
72	C₂×C₈×C₆₁	—	2	1	no	—
73	C₂×C₈×C₆₁	—	2	2	no	—
74	C₂×C₈×C₆₁	—	2	2	no	—
75	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
76	C₂×C₈×C₆₁	C₂×D₄	1	2	yes	—
77	C₂×C₈×C₆₁	C₂⁴	1	1	yes	characteristic
78	C₄⋊C₄×C₆₁	—	2	4	no	—
79	C₄⋊C₄×C₆₁	—	4	4	no	—
80	C₄⋊C₄×C₆₁	—	4	4	no	—
81	C₂²⋊C₄×C₆₁	—	4	2	no	—
82	C₂²⋊C₄×C₆₁	—	4	4	no	—
83	C₂²⋊C₄×C₆₁	—	4	4	no	—
84	C₄²×C₆₁	—	4	2	no	—
85	C₁₆×C₆₁	—	2	2	no	—
86	C₁₆×C₆₁	—	2	2	no	—
87	C₂³×C₆₁	C₂×D₈	1	1	yes	characteristic
88	C₂³×C₆₁	C₂×D₈	1	2	yes	—
89	Q₈×C₆₁	—	4	1	no	—
90	Q₈×C₆₁	—	4	4	no	—
91	Q₈×C₆₁	—	4	4	no	—
92	Q₈×C₆₁	—	8	4	no	—
93	D₄×C₆₁	—	4	1	no	—
94	D₄×C₆₁	—	4	2	no	—
95	C₂×C₄×C₆₁	—	2	2	no	—
96	C₂×C₄×C₆₁	—	2	1	no	—
97	C₂×C₄×C₆₁	—	2	2	no	—
98	C₂×C₄×C₆₁	—	2	1	no	—
99	C₂×C₄×C₆₁	—	4	4	no	—
100	C₂×C₄×C₆₁	—	4	4	no	—
101	C₂×C₄×C₆₁	—	8	4	no	—
102	C₂×C₄×C₆₁	C₂×D₈	1	2	yes	—
103	C₂×C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
104	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
105	C₈×C₆₁	—	2	1	no	—
106	C₈×C₆₁	—	2	2	no	—
107	C₈×C₆₁	H(5,𝔽₂)	1	1	yes	characteristic
108	C₈×C₆₁	H(5,𝔽₂)	1	1	yes	characteristic
109	#33768	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
110	C₂²×C₆₁	—	2	2	no	—
111	C₂²×C₆₁	—	2	4	no	—
112	C₂²×C₆₁	—	2	2	no	—
113	C₂²×C₆₁	—	2	1	no	—
114	C₂²×C₆₁	C₂²×D₈	1	1	yes	characteristic
115	C₄×C₆₁	—	2	1	no	—
116	C₄×C₆₁	—	2	2	no	—
117	C₄×C₆₁	—	8	4	no	—
118	C₄×C₆₁	—	8	4	no	—
119	C₄×C₆₁	#534	1	1	yes	characteristic
120	C₄×C₆₁	#575	1	1	yes	characteristic
121	#3059	C₂×C₆₁	1	2	yes	—
122	#2237	C₂×C₆₁	1	4	yes	complemented
123	#2219	C₂×C₆₁	1	2	yes	complemented
124	#2216	C₂×C₆₁	1	2	yes	complemented
125	#2188	C₂×C₆₁	1	4	yes	complemented
126	#2113	C₂×C₆₁	1	1	yes	characteristic
127	C₂×C₆₁	—	4	2	no	—
128	C₂×C₆₁	—	4	1	no	—
129	C₂×C₆₁	#3186	1	1	yes	characteristic
130	C₂×C₆₁	#3417	1	2	yes	—
131	C₂²×Q₁₆	C₂²×C₆₁	1	2	yes	—
132	C₂×#131	C₂²×C₆₁	1	1	yes	characteristic
133	#536	—	2	2	no	—
134	C₂×Q₃₂	C₂²×C₆₁	1	2	yes	—
135	C₂×Q₃₂	C₂²×C₆₁	1	2	yes	—
136	C₂×M₃₂	C₂²×C₆₁	1	1	yes	characteristic
137	C₂²×C₁₆	C₂²×C₆₁	1	1	yes	characteristic
138	#470	—	2	2	no	—
139	#470	—	2	2	no	—
140	#467	C₂²×C₆₁	1	4	yes	—
141	#467	C₂²×C₆₁	1	4	yes	—
142	#427	C₂²×C₆₁	1	2	yes	—
143	#419	—	2	2	no	—
144	#367	C₂²×C₆₁	1	2	yes	—
145	#366	C₂²×C₆₁	1	2	yes	—
146	#358	C₂²×C₆₁	1	4	yes	—
147	#358	C₂²×C₆₁	1	4	yes	—
148	C₂²⋊C₁₆	C₂²×C₆₁	1	4	yes	—
149	C₆₁	#33768	1	1	yes	characteristic, direct product factor, Sylow
150	C₂×#41	D₄×C₆₁	1	1	yes	characteristic
151	C₂²×Q₈	—	2	2	no	—
152	C₂×Q₁₆	—	2	2	no	—
153	C₂×Q₁₆	—	4	2	no	—
154	C₂×Q₁₆	—	4	2	no	—
155	C₂×Q₁₆	C₂³×C₆₁	1	4	yes	—
156	#131	—	4	1	no	—
157	#131	—	4	1	no	—
158	C₂×M₁₆	D₄×C₆₁	1	2	yes	—
159	C₂×M₁₆	D₄×C₆₁	1	1	yes	characteristic
160	C₂²×C₈	C₂³×C₆₁	1	2	yes	—
161	C₂²×C₈	C₂³×C₆₁	1	1	yes	characteristic
162	C₄⋊Q₈	—	4	2	no	—
163	#124	—	4	2	no	—
164	#123	—	4	2	no	—
165	#123	—	4	2	no	—
166	C₂²⋊Q₈	—	2	4	no	—
167	C₂²⋊Q₈	—	2	4	no	—
168	#117	—	2	2	no	—
169	Q₃₂	—	4	2	no	—
170	Q₃₂	—	4	2	no	—
171	M₃₂	—	4	1	no	—
172	C₂×C₁₆	—	4	1	no	—
173	C₂×C₁₆	C₂³×C₆₁	1	2	yes	—
174	C₂×C₁₆	C₂³×C₆₁	1	2	yes	—
175	#107	C₂³×C₆₁	1	4	yes	—
176	#106	—	2	2	no	—
177	#103	—	2	4	no	—
178	#103	—	2	4	no	—
179	#41	—	4	1	no	—
180	#41	—	4	1	no	—
181	C₂×Q₈	—	2	4	no	—
182	C₂×Q₈	—	4	4	no	—
183	C₂×Q₈	—	4	4	no	—
184	C₂×Q₈	—	4	2	no	—
185	C₂×Q₈	D₈×C₆₁	1	1	yes	characteristic
186	C₂×D₄	D₈×C₆₁	1	1	yes	characteristic
187	C₂×D₄	D₈×C₆₁	1	2	yes	—
188	C₂²×C₄	—	4	2	no	—
189	C₂²×C₄	C₂×D₄×C₆₁	1	2	yes	—
190	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
191	Q₁₆	—	2	4	no	—
192	Q₁₆	—	2	4	no	—
193	Q₁₆	—	4	4	no	—
194	M₁₆	—	4	2	no	—
195	M₁₆	—	4	1	no	—
196	C₂×C₈	—	2	1	no	—
197	C₂×C₈	—	2	1	no	—
198	C₂×C₈	—	2	2	no	—
199	C₂×C₈	—	2	2	no	—
200	C₂×C₈	C₂×D₄×C₆₁	1	1	yes	characteristic
201	C₂×C₈	C₂×D₄×C₆₁	1	2	yes	—
202	C₂×C₈	C₂⁴×C₆₁	1	1	yes	characteristic
203	C₄⋊C₄	—	2	4	no	—
204	C₄⋊C₄	—	4	4	no	—
205	C₄⋊C₄	—	4	4	no	—
206	C₂²⋊C₄	—	4	2	no	—
207	C₂²⋊C₄	—	4	4	no	—
208	C₂²⋊C₄	—	4	4	no	—
209	C₄²	—	4	2	no	—
210	C₁₆	—	2	2	no	—
211	C₁₆	—	2	2	no	—
212	C₂³	C₂×D₈×C₆₁	1	1	yes	characteristic
213	C₂³	C₂×D₈×C₆₁	1	2	yes	—
214	Q₈	—	4	1	no	—
215	Q₈	—	4	4	no	—
216	Q₈	—	4	4	no	—
217	Q₈	—	8	4	no	—
218	D₄	—	4	1	no	—
219	D₄	—	4	2	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	2	1	no	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	2	1	no	—
224	C₂×C₄	—	4	4	no	—
225	C₂×C₄	—	4	4	no	—
226	C₂×C₄	—	8	4	no	—
227	C₂×C₄	C₂×D₈×C₆₁	1	2	yes	—
228	C₂×C₄	C₂×D₈×C₆₁	1	1	yes	characteristic
229	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
230	C₈	—	2	1	no	—
231	C₈	—	2	2	no	—
232	C₈	H(5,𝔽₂)×C₆₁	1	1	yes	characteristic
233	C₈	H(5,𝔽₂)×C₆₁	1	1	yes	characteristic
234	C₂²	—	2	2	no	—
235	C₂²	—	2	4	no	—
236	C₂²	—	2	2	no	—
237	C₂²	—	2	1	no	—
238	C₂²	C₂²×D₈×C₆₁	1	1	yes	characteristic
239	C₄	—	2	1	no	—
240	C₄	—	2	2	no	—
241	C₄	—	8	4	no	—
242	C₄	—	8	4	no	—
243	C₄	C₆₁×#534	1	1	yes	characteristic
244	C₄	C₆₁×#575	1	1	yes	characteristic
245	C₂	—	4	2	no	—
246	C₂	—	4	1	no	—
247	C₂	C₆₁×#3186	1	1	yes	characteristic
248	C₂	C₆₁×#3417	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₆₁	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₆₁
prime radical	C₂×#41×C₆₁	D₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₁ ◁ C₂×#41×C₆₁ ◁ C₂×#131×C₆₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₆₁×#2237	C₂	semidirect
3. C₆₁×#2219	C₂	semidirect
4. C₆₁×#2216	C₂	semidirect
5. C₆₁×#2188	C₂	semidirect
109. #33768	C₆₁	direct (N × H)
122. #2237	C₂×C₆₁	semidirect
123. #2219	C₂×C₆₁	semidirect
124. #2216	C₂×C₆₁	semidirect
125. #2188	C₂×C₆₁	semidirect
149. C₆₁	#33768	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×#3059	2
C₆₁×#2237	2
C₆₁×#2219	2
C₆₁×#2216	2
C₆₁×#2188	2
C₆₁×#2113	2
#33768	61

Subgroups — C61×#33768 #29537201

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#33768 #29537201