Subgroups — C₆₁×#33729 #29537162

Order 15616.

474 conjugacy classes of subgroups.

The subgroup lattice has 220 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₁×#3164	C₂	1	2	yes	maximal
2	C₆₁×#2255	C₂	1	4	yes	maximal, complemented
3	C₆₁×#2253	C₂	1	2	yes	maximal, complemented
4	C₆₁×#2252	C₂	1	2	yes	maximal, complemented
5	C₆₁×#2159	C₂	1	2	yes	maximal
6	C₆₁×#2158	C₂	1	1	yes	maximal, characteristic
7	C₂×C₆₁×#366	C₂	1	1	yes	maximal, characteristic
8	C₆₁×#2108	C₂	1	1	yes	maximal, characteristic
9	C₆₁×#544	—	2	2	no	—
10	C₂×M₃₂×C₆₁	C₂²	1	1	yes	characteristic
11	C₂²×C₁₆×C₆₁	C₂²	1	1	yes	characteristic
12	C₆₁×#501	—	2	4	no	—
13	C₆₁×#500	C₂²	1	4	yes	—
14	C₆₁×#499	C₂²	1	4	yes	—
15	C₆₁×#428	—	2	2	no	—
16	C₆₁×#427	C₂²	1	2	yes	—
17	C₂×#107×C₆₁	C₂²	1	2	yes	—
18	C₆₁×#405	C₂²	1	1	yes	characteristic
19	C₆₁×#367	C₂²	1	2	yes	—
20	C₆₁×#367	C₂²	1	4	yes	—
21	C₆₁×#366	C₂²	1	2	yes	—
22	C₆₁×#366	C₂²	1	2	yes	—
23	C₆₁×#366	C₂²	1	1	yes	characteristic
24	C₆₁×#366	C₂²	1	1	yes	characteristic
25	C₆₁×#366	C₂²	1	4	yes	—
26	C₆₁×#346	C₂²	1	2	yes	—
27	C₄×C₁₆×C₆₁	C₂²	1	2	yes	—
28	C₂×M₁₆×C₆₁	D₄	1	1	yes	characteristic
29	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
30	C₄⋊Q₈×C₆₁	—	2	4	no	—
31	C₄⋊Q₈×C₆₁	—	4	2	no	—
32	#125×C₆₁	—	2	4	no	—
33	#125×C₆₁	—	4	2	no	—
34	C₂²⋊Q₈×C₆₁	—	4	4	no	—
35	#117×C₆₁	—	2	2	no	—
36	#117×C₆₁	D₄	1	1	yes	characteristic
37	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
38	M₃₂×C₆₁	Q₈	1	4	yes	complemented
39	C₂×C₁₆×C₆₁	Q₈	1	4	yes	complemented
40	C₂×C₁₆×C₆₁	C₂³	1	1	yes	characteristic
41	C₂×C₁₆×C₆₁	C₂³	1	2	yes	—
42	C₂×C₁₆×C₆₁	C₂³	1	1	yes	characteristic
43	#107×C₆₁	—	2	2	no	—
44	#107×C₆₁	—	2	4	no	—
45	#107×C₆₁	C₂³	1	4	yes	—
46	#107×C₆₁	C₂³	1	2	yes	—
47	#107×C₆₁	C₂³	1	2	yes	—
48	#106×C₆₁	—	2	2	no	—
49	#106×C₆₁	—	2	4	no	—
50	#97×C₆₁	D₄	1	2	yes	—
51	C₄×C₈×C₆₁	C₂³	1	2	yes	—
52	C₂×Q₈×C₆₁	—	4	4	no	—
53	C₂²×C₄×C₆₁	—	4	2	no	—
54	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
55	M₁₆×C₆₁	Q₁₆	1	4	yes	—
56	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
57	C₂×C₈×C₆₁	C₂×D₄	1	2	yes	—
58	C₂×C₈×C₆₁	C₂×Q₈	1	2	yes	—
59	C₂×C₈×C₆₁	C₂×Q₈	1	2	yes	—
60	C₂×C₈×C₆₁	C₂⁴	1	1	yes	characteristic
61	C₄⋊C₄×C₆₁	—	2	2	no	—
62	C₄⋊C₄×C₆₁	—	2	4	no	—
63	C₄⋊C₄×C₆₁	—	2	2	no	—
64	C₄⋊C₄×C₆₁	—	4	2	no	—
65	C₄⋊C₄×C₆₁	—	4	4	no	—
66	C₄⋊C₄×C₆₁	—	4	4	no	—
67	C₄⋊C₄×C₆₁	—	4	4	no	—
68	C₄⋊C₄×C₆₁	D₈	1	1	yes	characteristic
69	C₄⋊C₄×C₆₁	D₈	1	1	yes	characteristic
70	C₂²⋊C₄×C₆₁	—	4	2	no	—
71	C₂²⋊C₄×C₆₁	D₈	1	2	yes	—
72	C₄²×C₆₁	—	4	2	no	—
73	C₄²×C₆₁	C₂×D₄	1	2	yes	—
74	C₁₆×C₆₁	C₂×Q₈	1	2	yes	—
75	C₁₆×C₆₁	C₂×Q₈	1	2	yes	—
76	C₁₆×C₆₁	C₂×Q₈	1	4	yes	—
77	C₂³×C₆₁	C₂×D₈	1	1	yes	characteristic
78	Q₈×C₆₁	—	16	4	no	—
79	C₂×C₄×C₆₁	—	4	2	no	—
80	C₂×C₄×C₆₁	—	4	2	no	—
81	C₂×C₄×C₆₁	—	4	4	no	—
82	C₂×C₄×C₆₁	—	16	2	no	—
83	C₂×C₄×C₆₁	C₄⋊Q₈	1	2	yes	—
84	C₂×C₄×C₆₁	C₂×D₈	1	2	yes	—
85	C₂×C₄×C₆₁	C₂×D₈	1	2	yes	—
86	C₂×C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
87	C₂×C₄×C₆₁	C₂×Q₁₆	1	2	yes	—
88	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
89	C₈×C₆₁	C₄⋊Q₈	1	2	yes	—
90	C₈×C₆₁	C₂×Q₁₆	1	4	yes	—
91	C₈×C₆₁	C₂²×Q₈	1	1	yes	characteristic
92	C₈×C₆₁	C₂²×Q₈	1	1	yes	characteristic
93	#33729	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
94	C₂²×C₆₁	—	2	1	no	—
95	C₂²×C₆₁	—	2	1	no	—
96	C₂²×C₆₁	#500	1	2	yes	—
97	C₂²×C₆₁	C₂²×D₈	1	1	yes	characteristic
98	C₄×C₆₁	—	2	2	no	—
99	C₄×C₆₁	—	2	2	no	—
100	C₄×C₆₁	—	8	4	no	—
101	C₄×C₆₁	—	8	2	no	—
102	C₄×C₆₁	—	8	2	no	—
103	C₄×C₆₁	#500	1	2	yes	—
104	C₄×C₆₁	C₂×C₄⋊Q₈	1	1	yes	characteristic
105	C₄×C₆₁	C₂²×Q₁₆	1	1	yes	characteristic
106	#3164	C₂×C₆₁	1	2	yes	—
107	#2255	C₂×C₆₁	1	4	yes	complemented
108	#2253	C₂×C₆₁	1	2	yes	complemented
109	#2252	C₂×C₆₁	1	2	yes	complemented
110	#2159	C₂×C₆₁	1	2	yes	—
111	#2158	C₂×C₆₁	1	1	yes	characteristic
112	C₂×#366	C₂×C₆₁	1	1	yes	characteristic
113	#2108	C₂×C₆₁	1	1	yes	characteristic
114	C₂×C₆₁	—	2	2	no	—
115	C₂×C₆₁	C₂×#500	1	1	yes	characteristic
116	C₂×C₆₁	#3415	1	1	yes	characteristic
117	C₂×C₆₁	#3417	1	1	yes	characteristic
118	#544	—	2	2	no	—
119	C₂×M₃₂	C₂²×C₆₁	1	1	yes	characteristic
120	C₂²×C₁₆	C₂²×C₆₁	1	1	yes	characteristic
121	#501	—	2	4	no	—
122	#500	C₂²×C₆₁	1	4	yes	—
123	#499	C₂²×C₆₁	1	4	yes	—
124	#428	—	2	2	no	—
125	#427	C₂²×C₆₁	1	2	yes	—
126	C₂×#107	C₂²×C₆₁	1	2	yes	—
127	#405	C₂²×C₆₁	1	1	yes	characteristic
128	#367	C₂²×C₆₁	1	2	yes	—
129	#367	C₂²×C₆₁	1	4	yes	—
130	#366	C₂²×C₆₁	1	2	yes	—
131	#366	C₂²×C₆₁	1	2	yes	—
132	#366	C₂²×C₆₁	1	1	yes	characteristic
133	#366	C₂²×C₆₁	1	1	yes	characteristic
134	#366	C₂²×C₆₁	1	4	yes	—
135	#346	C₂²×C₆₁	1	2	yes	—
136	C₄×C₁₆	C₂²×C₆₁	1	2	yes	—
137	C₆₁	#33729	1	1	yes	characteristic, direct product factor, Sylow
138	C₂×M₁₆	D₄×C₆₁	1	1	yes	characteristic
139	C₂²×C₈	C₂³×C₆₁	1	1	yes	characteristic
140	C₄⋊Q₈	—	2	4	no	—
141	C₄⋊Q₈	—	4	2	no	—
142	#125	—	2	4	no	—
143	#125	—	4	2	no	—
144	C₂²⋊Q₈	—	4	4	no	—
145	#117	—	2	2	no	—
146	#117	D₄×C₆₁	1	1	yes	characteristic
147	C₂×C₄⋊C₄	—	2	2	no	—
148	M₃₂	Q₈×C₆₁	1	4	yes	complemented
149	C₂×C₁₆	Q₈×C₆₁	1	4	yes	complemented
150	C₂×C₁₆	C₂³×C₆₁	1	1	yes	characteristic
151	C₂×C₁₆	C₂³×C₆₁	1	2	yes	—
152	C₂×C₁₆	C₂³×C₆₁	1	1	yes	characteristic
153	#107	—	2	2	no	—
154	#107	—	2	4	no	—
155	#107	C₂³×C₆₁	1	4	yes	—
156	#107	C₂³×C₆₁	1	2	yes	—
157	#107	C₂³×C₆₁	1	2	yes	—
158	#106	—	2	2	no	—
159	#106	—	2	4	no	—
160	#97	D₄×C₆₁	1	2	yes	—
161	C₄×C₈	C₂³×C₆₁	1	2	yes	—
162	C₂×Q₈	—	4	4	no	—
163	C₂²×C₄	—	4	2	no	—
164	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
165	M₁₆	Q₁₆×C₆₁	1	4	yes	—
166	C₂×C₈	C₂×D₄×C₆₁	1	1	yes	characteristic
167	C₂×C₈	C₂×D₄×C₆₁	1	2	yes	—
168	C₂×C₈	C₂×Q₈×C₆₁	1	2	yes	—
169	C₂×C₈	C₂×Q₈×C₆₁	1	2	yes	—
170	C₂×C₈	C₂⁴×C₆₁	1	1	yes	characteristic
171	C₄⋊C₄	—	2	2	no	—
172	C₄⋊C₄	—	2	4	no	—
173	C₄⋊C₄	—	2	2	no	—
174	C₄⋊C₄	—	4	2	no	—
175	C₄⋊C₄	—	4	4	no	—
176	C₄⋊C₄	—	4	4	no	—
177	C₄⋊C₄	—	4	4	no	—
178	C₄⋊C₄	D₈×C₆₁	1	1	yes	characteristic
179	C₄⋊C₄	D₈×C₆₁	1	1	yes	characteristic
180	C₂²⋊C₄	—	4	2	no	—
181	C₂²⋊C₄	D₈×C₆₁	1	2	yes	—
182	C₄²	—	4	2	no	—
183	C₄²	C₂×D₄×C₆₁	1	2	yes	—
184	C₁₆	C₂×Q₈×C₆₁	1	2	yes	—
185	C₁₆	C₂×Q₈×C₆₁	1	2	yes	—
186	C₁₆	C₂×Q₈×C₆₁	1	4	yes	—
187	C₂³	C₂×D₈×C₆₁	1	1	yes	characteristic
188	Q₈	—	16	4	no	—
189	C₂×C₄	—	4	2	no	—
190	C₂×C₄	—	4	2	no	—
191	C₂×C₄	—	4	4	no	—
192	C₂×C₄	—	16	2	no	—
193	C₂×C₄	C₄⋊Q₈×C₆₁	1	2	yes	—
194	C₂×C₄	C₂×D₈×C₆₁	1	2	yes	—
195	C₂×C₄	C₂×D₈×C₆₁	1	2	yes	—
196	C₂×C₄	C₂×D₈×C₆₁	1	1	yes	characteristic
197	C₂×C₄	C₂×Q₁₆×C₆₁	1	2	yes	—
198	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
199	C₈	C₄⋊Q₈×C₆₁	1	2	yes	—
200	C₈	C₂×Q₁₆×C₆₁	1	4	yes	—
201	C₈	C₂²×Q₈×C₆₁	1	1	yes	characteristic
202	C₈	C₂²×Q₈×C₆₁	1	1	yes	characteristic
203	C₂²	—	2	1	no	—
204	C₂²	—	2	1	no	—
205	C₂²	C₆₁×#500	1	2	yes	—
206	C₂²	C₂²×D₈×C₆₁	1	1	yes	characteristic
207	C₄	—	2	2	no	—
208	C₄	—	2	2	no	—
209	C₄	—	8	4	no	—
210	C₄	—	8	2	no	—
211	C₄	—	8	2	no	—
212	C₄	C₆₁×#500	1	2	yes	—
213	C₄	C₂×C₄⋊Q₈×C₆₁	1	1	yes	characteristic
214	C₄	C₂²×Q₁₆×C₆₁	1	1	yes	characteristic
215	C₂	—	2	2	no	—
216	C₂	C₂×C₆₁×#500	1	1	yes	characteristic
217	C₂	C₆₁×#3415	1	1	yes	characteristic
218	C₂	C₆₁×#3417	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₆₁	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₆₁
prime radical	C₂³×C₆₁	C₂×D₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₁ ◁ #117×C₆₁ ◁ C₆₁×#405 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₆₁×#2255	C₂	semidirect
3. C₆₁×#2253	C₂	semidirect
4. C₆₁×#2252	C₂	semidirect
38. M₃₂×C₆₁	Q₈	semidirect
39. C₂×C₁₆×C₆₁	Q₈	semidirect
93. #33729	C₆₁	direct (N × H)
107. #2255	C₂×C₆₁	semidirect
108. #2253	C₂×C₆₁	semidirect
109. #2252	C₂×C₆₁	semidirect
137. C₆₁	#33729	direct (N × H)
148. M₃₂	Q₈×C₆₁	semidirect
149. C₂×C₁₆	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×#3164	2
C₆₁×#2255	2
C₆₁×#2253	2
C₆₁×#2252	2
C₆₁×#2159	2
C₆₁×#2158	2
C₂×C₆₁×#366	2
C₆₁×#2108	2
#33729	61

Subgroups — C61×#33729 #29537162

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#33729 #29537162