Subgroups — C₆₁×#9904 #29513337

Order 15616.

438 conjugacy classes of subgroups.

The subgroup lattice has 204 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₁×#2859	C₂	1	1	yes	maximal, characteristic
2	C₆₁×#1842	C₂	1	1	yes	maximal, characteristic
3	C₆₁×#1767	C₂	1	1	yes	maximal, characteristic
4	C₆₁×#1296	C₂	1	4	yes	maximal
5	C₂×#125×C₆₁	—	2	1	no	—
6	C₂×C₄⋊C₈×C₆₁	—	2	2	no	—
7	C₂×C₄⋊C₈×C₆₁	C₂²	1	2	yes	—
8	C₂×C₄⋊C₈×C₆₁	C₂²	1	2	yes	—
9	C₂×#97×C₆₁	—	2	1	no	—
10	C₆₁×#389	—	2	2	no	—
11	C₆₁×#389	C₄	1	4	yes	—
12	C₆₁×#384	C₂²	1	2	yes	—
13	C₆₁×#382	—	2	1	no	—
14	C₄³×C₆₁	C₂²	1	1	yes	characteristic
15	C₆₁×#336	—	2	1	no	—
16	C₂²×C₈×C₆₁	—	2	2	no	—
17	C₂²×C₈×C₆₁	—	2	2	no	—
18	#125×C₆₁	—	4	2	no	—
19	#125×C₆₁	—	4	2	no	—
20	C₂×C₄⋊C₄×C₆₁	—	2	4	no	—
21	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
22	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
23	C₂×C₄⋊C₄×C₆₁	C₂×C₄	1	4	yes	—
24	C₂×C₄²×C₆₁	C₂×C₄	1	2	yes	—
25	C₂×C₄²×C₆₁	D₄	1	1	yes	characteristic
26	C₂×C₄²×C₆₁	D₄	1	2	yes	—
27	C₂×C₄²×C₆₁	D₄	1	1	yes	characteristic
28	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
29	C₄⋊C₈×C₆₁	—	2	2	no	—
30	C₄⋊C₈×C₆₁	—	2	4	no	—
31	C₄⋊C₈×C₆₁	—	4	4	no	—
32	C₄⋊C₈×C₆₁	Q₈	1	4	yes	—
33	#97×C₆₁	—	4	2	no	—
34	#95×C₆₁	—	2	4	no	—
35	C₂²×C₄×C₆₁	—	2	4	no	—
36	C₂²×C₄×C₆₁	C₂²⋊C₄	1	2	yes	—
37	C₂²×C₄×C₆₁	C₂²⋊C₄	1	2	yes	—
38	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
39	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
40	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
41	C₂×C₈×C₆₁	—	2	4	no	—
42	C₂×C₈×C₆₁	—	4	2	no	—
43	C₂×C₈×C₆₁	—	4	2	no	—
44	C₂×C₈×C₆₁	—	4	2	no	—
45	C₂×C₈×C₆₁	—	4	4	no	—
46	C₄⋊C₄×C₆₁	—	4	4	no	—
47	C₄⋊C₄×C₆₁	—	4	8	no	—
48	C₄⋊C₄×C₆₁	—	4	2	no	—
49	C₄⋊C₄×C₆₁	—	4	2	no	—
50	C₄⋊C₄×C₆₁	—	4	2	no	—
51	C₄⋊C₄×C₆₁	—	4	2	no	—
52	C₄²×C₆₁	—	2	2	no	—
53	C₄²×C₆₁	—	2	2	no	—
54	C₄²×C₆₁	—	2	2	no	—
55	C₄²×C₆₁	—	4	2	no	—
56	C₄²×C₆₁	D₈	1	2	yes	—
57	C₄²×C₆₁	SD₁₆	1	2	yes	—
58	C₄²×C₆₁	C₂×Q₈	1	2	yes	—
59	C₄²×C₆₁	#41	1	2	yes	—
60	C₂³×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
61	C₂×C₄×C₆₁	—	2	2	no	—
62	C₂×C₄×C₆₁	—	2	2	no	—
63	C₂×C₄×C₆₁	—	2	2	no	—
64	C₂×C₄×C₆₁	—	2	2	no	—
65	C₂×C₄×C₆₁	—	2	2	no	—
66	C₂×C₄×C₆₁	—	2	2	no	—
67	C₂×C₄×C₆₁	—	4	4	no	—
68	C₂×C₄×C₆₁	—	4	4	no	—
69	C₂×C₄×C₆₁	—	4	4	no	—
70	C₂×C₄×C₆₁	#102	1	4	yes	—
71	C₂×C₄×C₆₁	C₄×Q₈	1	2	yes	—
72	C₂×C₄×C₆₁	C₂²⋊Q₈	1	2	yes	—
73	C₂×C₄×C₆₁	#124	1	1	yes	characteristic
74	C₂×C₄×C₆₁	C₄⋊Q₈	1	1	yes	characteristic
75	C₂×C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
76	C₂×C₄×C₆₁	C₂×SD₁₆	1	1	yes	characteristic
77	C₂×C₄×C₆₁	AGL(1,ℤ/8)	1	2	yes	—
78	C₂×C₄×C₆₁	#137	1	2	yes	—
79	C₂×C₄×C₆₁	#137	1	2	yes	—
80	C₈×C₆₁	—	4	4	no	—
81	C₈×C₆₁	—	8	2	no	—
82	#9904	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
83	C₂²×C₆₁	#391	1	1	yes	characteristic
84	C₂²×C₆₁	#413	1	2	yes	—
85	C₂²×C₆₁	C₂×#102	1	1	yes	characteristic
86	C₂²×C₆₁	#417	1	2	yes	—
87	C₂²×C₆₁	#419	1	1	yes	characteristic
88	C₄×C₆₁	—	2	4	no	—
89	C₄×C₆₁	—	2	2	no	—
90	C₄×C₆₁	—	4	2	no	—
91	C₄×C₆₁	—	8	4	no	—
92	C₄×C₆₁	#474	1	2	yes	—
93	C₄×C₆₁	#475	1	2	yes	—
94	C₄×C₆₁	#490	1	1	yes	characteristic
95	C₄×C₆₁	#491	1	1	yes	characteristic
96	C₄×C₆₁	#501	1	2	yes	—
97	#2859	C₂×C₆₁	1	1	yes	characteristic
98	#1842	C₂×C₆₁	1	1	yes	characteristic
99	#1767	C₂×C₆₁	1	1	yes	characteristic
100	#1296	C₂×C₆₁	1	4	yes	—
101	C₂×C₆₁	#1547	1	2	yes	—
102	C₂×C₆₁	#1559	1	2	yes	—
103	C₂×C₆₁	#1988	1	1	yes	characteristic
104	C₂×C₆₁	#1992	1	1	yes	characteristic
105	C₂×C₆₁	#1998	1	1	yes	characteristic
106	C₂×#125	—	2	1	no	—
107	C₂×C₄⋊C₈	—	2	2	no	—
108	C₂×C₄⋊C₈	C₂²×C₆₁	1	2	yes	—
109	C₂×C₄⋊C₈	C₂²×C₆₁	1	2	yes	—
110	C₂×#97	—	2	1	no	—
111	#389	—	2	2	no	—
112	#389	C₄×C₆₁	1	4	yes	—
113	#384	C₂²×C₆₁	1	2	yes	—
114	#382	—	2	1	no	—
115	C₄³	C₂²×C₆₁	1	1	yes	characteristic
116	#336	—	2	1	no	—
117	C₆₁	#9904	1	1	yes	characteristic, direct product factor, Sylow
118	C₂²×C₈	—	2	2	no	—
119	C₂²×C₈	—	2	2	no	—
120	#125	—	4	2	no	—
121	#125	—	4	2	no	—
122	C₂×C₄⋊C₄	—	2	4	no	—
123	C₂×C₄⋊C₄	—	2	2	no	—
124	C₂×C₄⋊C₄	—	2	2	no	—
125	C₂×C₄⋊C₄	C₂×C₄×C₆₁	1	4	yes	—
126	C₂×C₄²	C₂×C₄×C₆₁	1	2	yes	—
127	C₂×C₄²	D₄×C₆₁	1	1	yes	characteristic
128	C₂×C₄²	D₄×C₆₁	1	2	yes	—
129	C₂×C₄²	D₄×C₆₁	1	1	yes	characteristic
130	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
131	C₄⋊C₈	—	2	2	no	—
132	C₄⋊C₈	—	2	4	no	—
133	C₄⋊C₈	—	4	4	no	—
134	C₄⋊C₈	Q₈×C₆₁	1	4	yes	—
135	#97	—	4	2	no	—
136	#95	—	2	4	no	—
137	C₂²×C₄	—	2	4	no	—
138	C₂²×C₄	C₂²⋊C₄×C₆₁	1	2	yes	—
139	C₂²×C₄	C₂²⋊C₄×C₆₁	1	2	yes	—
140	C₂²×C₄	C₂²×C₄×C₆₁	1	1	yes	characteristic
141	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
142	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
143	C₂×C₈	—	2	4	no	—
144	C₂×C₈	—	4	2	no	—
145	C₂×C₈	—	4	2	no	—
146	C₂×C₈	—	4	2	no	—
147	C₂×C₈	—	4	4	no	—
148	C₄⋊C₄	—	4	4	no	—
149	C₄⋊C₄	—	4	8	no	—
150	C₄⋊C₄	—	4	2	no	—
151	C₄⋊C₄	—	4	2	no	—
152	C₄⋊C₄	—	4	2	no	—
153	C₄⋊C₄	—	4	2	no	—
154	C₄²	—	2	2	no	—
155	C₄²	—	2	2	no	—
156	C₄²	—	2	2	no	—
157	C₄²	—	4	2	no	—
158	C₄²	D₈×C₆₁	1	2	yes	—
159	C₄²	SD₁₆×C₆₁	1	2	yes	—
160	C₄²	C₂×Q₈×C₆₁	1	2	yes	—
161	C₄²	#41×C₆₁	1	2	yes	—
162	C₂³	C₂×C₂²⋊C₄×C₆₁	1	1	yes	characteristic
163	C₂×C₄	—	2	2	no	—
164	C₂×C₄	—	2	2	no	—
165	C₂×C₄	—	2	2	no	—
166	C₂×C₄	—	2	2	no	—
167	C₂×C₄	—	2	2	no	—
168	C₂×C₄	—	2	2	no	—
169	C₂×C₄	—	4	4	no	—
170	C₂×C₄	—	4	4	no	—
171	C₂×C₄	—	4	4	no	—
172	C₂×C₄	#102×C₆₁	1	4	yes	—
173	C₂×C₄	C₄×Q₈×C₆₁	1	2	yes	—
174	C₂×C₄	C₂²⋊Q₈×C₆₁	1	2	yes	—
175	C₂×C₄	#124×C₆₁	1	1	yes	characteristic
176	C₂×C₄	C₄⋊Q₈×C₆₁	1	1	yes	characteristic
177	C₂×C₄	C₂×D₈×C₆₁	1	1	yes	characteristic
178	C₂×C₄	C₂×SD₁₆×C₆₁	1	1	yes	characteristic
179	C₂×C₄	AGL(1,ℤ/8)×C₆₁	1	2	yes	—
180	C₂×C₄	#137×C₆₁	1	2	yes	—
181	C₂×C₄	#137×C₆₁	1	2	yes	—
182	C₈	—	4	4	no	—
183	C₈	—	8	2	no	—
184	C₂²	C₆₁×#391	1	1	yes	characteristic
185	C₂²	C₆₁×#413	1	2	yes	—
186	C₂²	C₂×#102×C₆₁	1	1	yes	characteristic
187	C₂²	C₆₁×#417	1	2	yes	—
188	C₂²	C₆₁×#419	1	1	yes	characteristic
189	C₄	—	2	4	no	—
190	C₄	—	2	2	no	—
191	C₄	—	4	2	no	—
192	C₄	—	8	4	no	—
193	C₄	C₆₁×#474	1	2	yes	—
194	C₄	C₆₁×#475	1	2	yes	—
195	C₄	C₆₁×#490	1	1	yes	characteristic
196	C₄	C₆₁×#491	1	1	yes	characteristic
197	C₄	C₆₁×#501	1	2	yes	—
198	C₂	C₆₁×#1547	1	2	yes	—
199	C₂	C₆₁×#1559	1	2	yes	—
200	C₂	C₆₁×#1988	1	1	yes	characteristic
201	C₂	C₆₁×#1992	1	1	yes	characteristic
202	C₂	C₆₁×#1998	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₆₁	C₂×C₂²⋊C₄
commutator	C₂²×C₄	C₂²×C₄×C₆₁
Frattini, cosocle	C₂×C₄²	C₂³×C₆₁
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₆₁ ◁ C₄³×C₆₁ ◁ G

Complements

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

N	H ≅ G/N	type
82. #9904	C₆₁	direct (N × H)
117. C₆₁	#9904	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×#2859	2
C₆₁×#1842	2
C₆₁×#1767	2
C₆₁×#1296	2
#9904	61

Subgroups — C61×#9904 #29513337

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#9904 #29513337