Subgroups — C₆₁×#33713 #29537146

Order 15616.

834 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₆₁×#508	C₂	1	4	yes	maximal, complemented
2	C₆₁×#3147	C₂	1	2	yes	maximal, complemented
3	C₆₁×#2249	C₂	1	4	yes	maximal, complemented
4	C₆₁×#2248	C₂	1	2	yes	maximal, complemented
5	C₆₁×#2247	C₂	1	1	yes	maximal, characteristic, complemented
6	C₆₁×#2246	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₄×C₁₆×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#135×C₆₁	—	2	2	no	—
9	C₂×#135×C₆₁	C₂²	1	4	yes	—
10	C₆₁×#532	—	2	2	no	—
11	C₆₁×#508	—	2	16	no	—
12	C₂×Q₃₂×C₆₁	C₂²	1	4	yes	—
13	C₂×SD₃₂×C₆₁	C₂²	1	8	yes	—
14	C₂×D₁₆×C₆₁	C₂²	1	4	yes	—
15	C₂²×C₁₆×C₆₁	C₂²	1	2	yes	—
16	C₆₁×#495	—	2	2	no	—
17	C₆₁×#495	C₂²	1	4	yes	—
18	C₆₁×#494	C₂²	1	2	yes	—
19	C₆₁×#493	C₂²	1	2	yes	—
20	C₆₁×#492	—	2	2	no	—
21	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
22	C₄×C₁₆×C₆₁	C₂²	1	1	yes	characteristic
23	C₄×C₁₆×C₆₁	C₂²	1	1	yes	characteristic
24	C₄×C₁₆×C₆₁	C₂²	1	2	yes	—
25	C₂×#41×C₆₁	—	2	4	no	—
26	#135×C₆₁	—	2	16	no	—
27	#135×C₆₁	—	4	8	no	—
28	C₂×Q₁₆×C₆₁	—	2	2	no	—
29	C₂×Q₁₆×C₆₁	C₂³	1	4	yes	—
30	C₂×SD₁₆×C₆₁	—	2	4	no	—
31	C₂×SD₁₆×C₆₁	—	2	4	no	—
32	C₂×D₈×C₆₁	—	2	2	no	—
33	C₂×D₈×C₆₁	C₂³	1	4	yes	—
34	C₂²×C₈×C₆₁	D₄	1	1	yes	characteristic
35	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
36	C₄⋊Q₈×C₆₁	—	2	2	no	—
37	#127×C₆₁	—	2	2	no	—
38	#124×C₆₁	—	2	4	no	—
39	#121×C₆₁	—	4	4	no	—
40	C₄×D₄×C₆₁	—	4	4	no	—
41	C₂×C₄²×C₆₁	D₄	1	1	yes	characteristic
42	Q₃₂×C₆₁	—	2	8	no	—
43	SD₃₂×C₆₁	—	2	16	no	—
44	D₁₆×C₆₁	—	2	8	no	—
45	C₂×C₁₆×C₆₁	D₄	1	8	yes	complemented
46	C₂×C₁₆×C₆₁	C₂³	1	2	yes	—
47	C₂×C₁₆×C₆₁	C₂³	1	2	yes	—
48	C₄×C₈×C₆₁	D₄	1	1	yes	characteristic
49	C₄×C₈×C₆₁	D₄	1	1	yes	characteristic
50	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
51	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
52	#41×C₆₁	—	4	8	no	—
53	#41×C₆₁	—	4	8	no	—
54	C₂×Q₈×C₆₁	—	2	4	no	—
55	C₂×D₄×C₆₁	—	2	4	no	—
56	C₂×D₄×C₆₁	—	4	4	no	—
57	C₂×D₄×C₆₁	—	4	4	no	—
58	C₂²×C₄×C₆₁	—	4	4	no	—
59	C₂²×C₄×C₆₁	D₈	1	2	yes	—
60	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
61	Q₁₆×C₆₁	—	2	8	no	—
62	Q₁₆×C₆₁	—	4	4	no	—
63	SD₁₆×C₆₁	—	4	8	no	—
64	SD₁₆×C₆₁	—	4	8	no	—
65	D₈×C₆₁	—	2	8	no	—
66	D₈×C₆₁	—	4	4	no	—
67	C₂×C₈×C₆₁	D₈	1	4	yes	—
68	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
69	C₂×C₈×C₆₁	C₂×D₄	1	4	yes	—
70	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
71	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
72	C₂×C₈×C₆₁	C₂⁴	1	1	yes	characteristic
73	C₄⋊C₄×C₆₁	—	4	4	no	—
74	C₂²⋊C₄×C₆₁	—	4	8	no	—
75	C₄²×C₆₁	D₈	1	2	yes	—
76	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
77	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
78	C₁₆×C₆₁	C₂×D₄	1	4	yes	—
79	C₁₆×C₆₁	C₂×D₄	1	4	yes	—
80	C₂³×C₆₁	—	4	4	no	—
81	C₂³×C₆₁	C₂×D₈	1	1	yes	characteristic
82	Q₈×C₆₁	—	4	4	no	—
83	Q₈×C₆₁	—	4	4	no	—
84	D₄×C₆₁	—	4	4	no	—
85	D₄×C₆₁	—	4	4	no	—
86	D₄×C₆₁	—	8	8	no	—
87	D₄×C₆₁	—	8	8	no	—
88	C₂×C₄×C₆₁	—	2	4	no	—
89	C₂×C₄×C₆₁	—	4	4	no	—
90	C₂×C₄×C₆₁	—	8	8	no	—
91	C₂×C₄×C₆₁	#127	1	2	yes	—
92	C₂×C₄×C₆₁	C₂×D₈	1	2	yes	—
93	C₂×C₄×C₆₁	C₂×D₈	1	2	yes	—
94	C₂×C₄×C₆₁	C₂×D₈	1	2	yes	—
95	C₂×C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
96	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
97	C₈×C₆₁	#127	1	2	yes	—
98	C₈×C₆₁	C₂×D₈	1	2	yes	—
99	C₈×C₆₁	C₂×D₈	1	2	yes	—
100	C₈×C₆₁	C₂²×D₄	1	2	yes	—
101	#33713	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
102	C₂²×C₆₁	—	2	2	no	—
103	C₂²×C₆₁	—	8	4	no	—
104	C₂²×C₆₁	—	8	8	no	—
105	C₂²×C₆₁	#493	1	2	yes	—
106	C₂²×C₆₁	C₂²×D₈	1	1	yes	characteristic
107	C₄×C₆₁	—	2	2	no	—
108	C₄×C₆₁	—	8	4	no	—
109	C₄×C₆₁	#493	1	2	yes	—
110	C₄×C₆₁	#508	1	4	yes	complemented
111	C₄×C₆₁	C₂×#127	1	1	yes	characteristic
112	C₄×C₆₁	C₂²×D₈	1	1	yes	characteristic
113	C₂×#508	C₂×C₆₁	1	4	yes	complemented
114	#3147	C₂×C₆₁	1	2	yes	complemented
115	#2249	C₂×C₆₁	1	4	yes	complemented
116	#2248	C₂×C₆₁	1	2	yes	complemented
117	#2247	C₂×C₆₁	1	1	yes	characteristic, complemented
118	#2246	C₂×C₆₁	1	1	yes	characteristic, complemented
119	C₂×C₄×C₁₆	C₂×C₆₁	1	1	yes	characteristic, complemented
120	C₂×C₆₁	—	2	2	no	—
121	C₂×C₆₁	—	16	4	no	—
122	C₂×C₆₁	C₂×#493	1	1	yes	characteristic
123	C₂×C₆₁	C₂×#508	1	2	yes	—
124	C₂×#135	—	2	2	no	—
125	C₂×#135	C₂²×C₆₁	1	4	yes	—
126	#532	—	2	2	no	—
127	#508	—	2	16	no	—
128	C₂×Q₃₂	C₂²×C₆₁	1	4	yes	—
129	C₂×SD₃₂	C₂²×C₆₁	1	8	yes	—
130	C₂×D₁₆	C₂²×C₆₁	1	4	yes	—
131	C₂²×C₁₆	C₂²×C₆₁	1	2	yes	—
132	#495	—	2	2	no	—
133	#495	C₂²×C₆₁	1	4	yes	—
134	#494	C₂²×C₆₁	1	2	yes	—
135	#493	C₂²×C₆₁	1	2	yes	—
136	#492	—	2	2	no	—
137	C₂×C₄×C₈	C₂²×C₆₁	1	1	yes	characteristic
138	C₄×C₁₆	C₂²×C₆₁	1	1	yes	characteristic
139	C₄×C₁₆	C₂²×C₆₁	1	1	yes	characteristic
140	C₄×C₁₆	C₂²×C₆₁	1	2	yes	—
141	C₆₁	#33713	1	1	yes	characteristic, direct product factor, Sylow
142	C₂×#41	—	2	4	no	—
143	#135	—	2	16	no	—
144	#135	—	4	8	no	—
145	C₂×Q₁₆	—	2	2	no	—
146	C₂×Q₁₆	C₂³×C₆₁	1	4	yes	—
147	C₂×SD₁₆	—	2	4	no	—
148	C₂×SD₁₆	—	2	4	no	—
149	C₂×D₈	—	2	2	no	—
150	C₂×D₈	C₂³×C₆₁	1	4	yes	—
151	C₂²×C₈	D₄×C₆₁	1	1	yes	characteristic
152	C₂²×C₈	C₂³×C₆₁	1	1	yes	characteristic
153	C₄⋊Q₈	—	2	2	no	—
154	#127	—	2	2	no	—
155	#124	—	2	4	no	—
156	#121	—	4	4	no	—
157	C₄×D₄	—	4	4	no	—
158	C₂×C₄²	D₄×C₆₁	1	1	yes	characteristic
159	Q₃₂	—	2	8	no	—
160	SD₃₂	—	2	16	no	—
161	D₁₆	—	2	8	no	—
162	C₂×C₁₆	D₄×C₆₁	1	8	yes	complemented
163	C₂×C₁₆	C₂³×C₆₁	1	2	yes	—
164	C₂×C₁₆	C₂³×C₆₁	1	2	yes	—
165	C₄×C₈	D₄×C₆₁	1	1	yes	characteristic
166	C₄×C₈	D₄×C₆₁	1	1	yes	characteristic
167	C₄×C₈	C₂³×C₆₁	1	1	yes	characteristic
168	C₄×C₈	C₂³×C₆₁	1	1	yes	characteristic
169	#41	—	4	8	no	—
170	#41	—	4	8	no	—
171	C₂×Q₈	—	2	4	no	—
172	C₂×D₄	—	2	4	no	—
173	C₂×D₄	—	4	4	no	—
174	C₂×D₄	—	4	4	no	—
175	C₂²×C₄	—	4	4	no	—
176	C₂²×C₄	D₈×C₆₁	1	2	yes	—
177	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
178	Q₁₆	—	2	8	no	—
179	Q₁₆	—	4	4	no	—
180	SD₁₆	—	4	8	no	—
181	SD₁₆	—	4	8	no	—
182	D₈	—	2	8	no	—
183	D₈	—	4	4	no	—
184	C₂×C₈	D₈×C₆₁	1	4	yes	—
185	C₂×C₈	C₂×D₄×C₆₁	1	1	yes	characteristic
186	C₂×C₈	C₂×D₄×C₆₁	1	4	yes	—
187	C₂×C₈	C₂×D₄×C₆₁	1	1	yes	characteristic
188	C₂×C₈	C₂×D₄×C₆₁	1	1	yes	characteristic
189	C₂×C₈	C₂⁴×C₆₁	1	1	yes	characteristic
190	C₄⋊C₄	—	4	4	no	—
191	C₂²⋊C₄	—	4	8	no	—
192	C₄²	D₈×C₆₁	1	2	yes	—
193	C₄²	C₂×D₄×C₆₁	1	1	yes	characteristic
194	C₄²	C₂×D₄×C₆₁	1	1	yes	characteristic
195	C₁₆	C₂×D₄×C₆₁	1	4	yes	—
196	C₁₆	C₂×D₄×C₆₁	1	4	yes	—
197	C₂³	—	4	4	no	—
198	C₂³	C₂×D₈×C₆₁	1	1	yes	characteristic
199	Q₈	—	4	4	no	—
200	Q₈	—	4	4	no	—
201	D₄	—	4	4	no	—
202	D₄	—	4	4	no	—
203	D₄	—	8	8	no	—
204	D₄	—	8	8	no	—
205	C₂×C₄	—	2	4	no	—
206	C₂×C₄	—	4	4	no	—
207	C₂×C₄	—	8	8	no	—
208	C₂×C₄	#127×C₆₁	1	2	yes	—
209	C₂×C₄	C₂×D₈×C₆₁	1	2	yes	—
210	C₂×C₄	C₂×D₈×C₆₁	1	2	yes	—
211	C₂×C₄	C₂×D₈×C₆₁	1	1	yes	characteristic
212	C₂×C₄	C₂×D₈×C₆₁	1	2	yes	—
213	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
214	C₈	#127×C₆₁	1	2	yes	—
215	C₈	C₂×D₈×C₆₁	1	2	yes	—
216	C₈	C₂×D₈×C₆₁	1	2	yes	—
217	C₈	C₂²×D₄×C₆₁	1	2	yes	—
218	C₂²	—	2	2	no	—
219	C₂²	—	8	4	no	—
220	C₂²	—	8	8	no	—
221	C₂²	C₆₁×#493	1	2	yes	—
222	C₂²	C₂²×D₈×C₆₁	1	1	yes	characteristic
223	C₄	—	2	2	no	—
224	C₄	—	8	4	no	—
225	C₄	C₆₁×#493	1	2	yes	—
226	C₄	C₆₁×#508	1	4	yes	complemented
227	C₄	C₂×#127×C₆₁	1	1	yes	characteristic
228	C₄	C₂²×D₈×C₆₁	1	1	yes	characteristic
229	C₂	—	2	2	no	—
230	C₂	—	16	4	no	—
231	C₂	C₂×C₆₁×#493	1	1	yes	characteristic
232	C₂	C₂×C₆₁×#508	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₆₁	C₂×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₆₁
socle	C₂²×C₆₁	C₂²×D₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×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
1. C₂×C₆₁×#508	C₂	semidirect
2. C₆₁×#3147	C₂	semidirect
3. C₆₁×#2249	C₂	semidirect
4. C₆₁×#2248	C₂	semidirect
5. C₆₁×#2247	C₂	semidirect
6. C₆₁×#2246	C₂	semidirect
7. C₂×C₄×C₁₆×C₆₁	C₂	semidirect
45. C₂×C₁₆×C₆₁	D₄	semidirect
101. #33713	C₆₁	direct (N × H)
110. C₄×C₆₁	#508	semidirect
113. C₂×#508	C₂×C₆₁	semidirect
114. #3147	C₂×C₆₁	semidirect
115. #2249	C₂×C₆₁	semidirect
116. #2248	C₂×C₆₁	semidirect
117. #2247	C₂×C₆₁	semidirect
118. #2246	C₂×C₆₁	semidirect
119. C₂×C₄×C₁₆	C₂×C₆₁	semidirect
141. C₆₁	#33713	direct (N × H)
162. C₂×C₁₆	D₄×C₆₁	semidirect
226. C₄	C₆₁×#508	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₆₁×#508	2
C₆₁×#3147	2
C₆₁×#2249	2
C₆₁×#2248	2
C₆₁×#2247	2
C₆₁×#2246	2
C₂×C₄×C₁₆×C₆₁	2
#33713	61

Subgroups — C61×#33713 #29537146

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#33713 #29537146