Subgroups — C₆₁×#23836 #29527269

Order 15616.

674 conjugacy classes of subgroups.

The subgroup lattice has 198 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#125×C₆₁	C₂	1	2	yes	maximal
2	C₆₁×#2085	C₂	1	4	yes	maximal
3	C₆₁×#2059	C₂	1	4	yes	maximal
4	C₄×#107×C₆₁	C₂	1	2	yes	maximal
5	C₄×#106×C₆₁	C₂	1	2	yes	maximal
6	C₆₁×#1744	C₂	1	1	yes	maximal, characteristic
7	C₂×#125×C₆₁	C₂²	1	2	yes	—
8	C₂×#107×C₆₁	C₂²	1	4	yes	—
9	C₂×#106×C₆₁	C₂²	1	4	yes	—
10	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
12	C₆₁×#389	C₂²	1	4	yes	—
13	C₆₁×#387	—	2	4	no	—
14	C₆₁×#382	C₂²	1	2	yes	—
15	C₄×C₄⋊C₄×C₆₁	—	2	4	no	—
16	C₄×C₄⋊C₄×C₆₁	C₂²	1	4	yes	—
17	C₄³×C₆₁	C₂²	1	1	yes	characteristic
18	C₆₁×#340	C₂²	1	2	yes	—
19	C₆₁×#340	C₂²	1	2	yes	—
20	C₆₁×#340	C₂²	1	2	yes	—
21	C₆₁×#340	C₂²	1	2	yes	—
22	C₆₁×#336	C₂²	1	4	yes	—
23	C₂²×C₈×C₆₁	C₂³	1	2	yes	—
24	C₂²×C₈×C₆₁	C₂³	1	2	yes	—
25	#125×C₆₁	—	2	8	no	—
26	C₂×C₄⋊C₄×C₆₁	—	2	4	no	—
27	C₂×C₄⋊C₄×C₆₁	—	2	4	no	—
28	C₂×C₄⋊C₄×C₆₁	C₂³	1	4	yes	—
29	C₂×C₄⋊C₄×C₆₁	C₂³	1	4	yes	—
30	C₂×C₄²×C₆₁	—	2	4	no	—
31	C₂×C₄²×C₆₁	D₄	1	2	yes	—
32	C₂×C₄²×C₆₁	D₄	1	2	yes	—
33	C₂×C₄²×C₆₁	C₂³	1	2	yes	—
34	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
35	#107×C₆₁	—	2	8	no	—
36	#106×C₆₁	—	2	8	no	—
37	C₄×C₈×C₆₁	Q₈	1	4	yes	—
38	C₄×C₈×C₆₁	Q₈	1	4	yes	—
39	#95×C₆₁	—	2	4	no	—
40	#95×C₆₁	—	2	4	no	—
41	#95×C₆₁	—	2	4	no	—
42	C₂²×C₄×C₆₁	—	2	4	no	—
43	C₂²×C₄×C₆₁	—	2	4	no	—
44	C₂²×C₄×C₆₁	C₂×D₄	1	4	yes	—
45	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
46	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
47	C₂²×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
48	C₂×C₈×C₆₁	—	2	4	no	—
49	C₂×C₈×C₆₁	—	2	4	no	—
50	C₂×C₈×C₆₁	C₂×Q₈	1	4	yes	—
51	C₂×C₈×C₆₁	C₂×Q₈	1	4	yes	—
52	C₄⋊C₄×C₆₁	—	2	16	no	—
53	C₄⋊C₄×C₆₁	—	2	4	no	—
54	C₄⋊C₄×C₆₁	—	2	4	no	—
55	C₄⋊C₄×C₆₁	—	2	4	no	—
56	C₄⋊C₄×C₆₁	—	4	8	no	—
57	C₄⋊C₄×C₆₁	#41	1	8	yes	—
58	C₄²×C₆₁	—	2	4	no	—
59	C₄²×C₆₁	—	2	4	no	—
60	C₄²×C₆₁	—	4	2	no	—
61	C₄²×C₆₁	—	4	8	no	—
62	C₄²×C₆₁	C₂×Q₈	1	2	yes	—
63	C₄²×C₆₁	C₂×Q₈	1	2	yes	—
64	C₂³×C₆₁	C₂²×D₄	1	1	yes	characteristic
65	C₂×C₄×C₆₁	—	2	2	no	—
66	C₂×C₄×C₆₁	—	2	4	no	—
67	C₂×C₄×C₆₁	—	2	8	no	—
68	C₂×C₄×C₆₁	—	2	4	no	—
69	C₂×C₄×C₆₁	—	2	4	no	—
70	C₂×C₄×C₆₁	—	4	4	no	—
71	C₂×C₄×C₆₁	—	4	4	no	—
72	C₂×C₄×C₆₁	—	4	4	no	—
73	C₂×C₄×C₆₁	—	4	4	no	—
74	C₂×C₄×C₆₁	—	4	4	no	—
75	C₂×C₄×C₆₁	C₄⋊Q₈	1	2	yes	—
76	C₂×C₄×C₆₁	C₄⋊Q₈	1	2	yes	—
77	C₂×C₄×C₆₁	#135	1	4	yes	—
78	C₂×C₄×C₆₁	C₂²×Q₈	1	1	yes	characteristic
79	C₂×C₄×C₆₁	C₂²×Q₈	1	1	yes	characteristic
80	C₂×C₄×C₆₁	C₂×#41	1	4	yes	—
81	C₈×C₆₁	—	2	4	no	—
82	C₈×C₆₁	—	2	4	no	—
83	#23836	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
84	C₂²×C₆₁	#525	1	2	yes	—
85	C₂²×C₆₁	C₂×C₄⋊Q₈	1	1	yes	characteristic
86	C₂²×C₆₁	C₂×#135	1	2	yes	—
87	C₂²×C₆₁	#575	1	2	yes	—
88	C₄×C₆₁	—	2	2	no	—
89	C₄×C₆₁	—	4	4	no	—
90	C₄×C₆₁	—	4	8	no	—
91	C₄×C₆₁	—	8	4	no	—
92	C₄×C₆₁	#499	1	4	yes	—
93	C₄×C₆₁	#533	1	2	yes	—
94	C₄×C₆₁	#533	1	2	yes	—
95	C₄×#125	C₂×C₆₁	1	2	yes	—
96	#2085	C₂×C₆₁	1	4	yes	—
97	#2059	C₂×C₆₁	1	4	yes	—
98	C₄×#107	C₂×C₆₁	1	2	yes	—
99	C₄×#106	C₂×C₆₁	1	2	yes	—
100	#1744	C₂×C₆₁	1	1	yes	characteristic
101	C₂×C₆₁	#2399	1	1	yes	characteristic
102	C₂×C₆₁	#3102	1	4	yes	—
103	C₂×C₆₁	C₂×#499	1	1	yes	characteristic
104	C₂×C₆₁	#3162	1	1	yes	characteristic
105	C₂×#125	C₂²×C₆₁	1	2	yes	—
106	C₂×#107	C₂²×C₆₁	1	4	yes	—
107	C₂×#106	C₂²×C₆₁	1	4	yes	—
108	C₂×C₄×C₈	C₂²×C₆₁	1	1	yes	characteristic
109	C₂×C₄×C₈	C₂²×C₆₁	1	1	yes	characteristic
110	#389	C₂²×C₆₁	1	4	yes	—
111	#387	—	2	4	no	—
112	#382	C₂²×C₆₁	1	2	yes	—
113	C₄×C₄⋊C₄	—	2	4	no	—
114	C₄×C₄⋊C₄	C₂²×C₆₁	1	4	yes	—
115	C₄³	C₂²×C₆₁	1	1	yes	characteristic
116	#340	C₂²×C₆₁	1	2	yes	—
117	#340	C₂²×C₆₁	1	2	yes	—
118	#340	C₂²×C₆₁	1	2	yes	—
119	#340	C₂²×C₆₁	1	2	yes	—
120	#336	C₂²×C₆₁	1	4	yes	—
121	C₆₁	#23836	1	1	yes	characteristic, direct product factor, Sylow
122	C₂²×C₈	C₂³×C₆₁	1	2	yes	—
123	C₂²×C₈	C₂³×C₆₁	1	2	yes	—
124	#125	—	2	8	no	—
125	C₂×C₄⋊C₄	—	2	4	no	—
126	C₂×C₄⋊C₄	—	2	4	no	—
127	C₂×C₄⋊C₄	C₂³×C₆₁	1	4	yes	—
128	C₂×C₄⋊C₄	C₂³×C₆₁	1	4	yes	—
129	C₂×C₄²	—	2	4	no	—
130	C₂×C₄²	D₄×C₆₁	1	2	yes	—
131	C₂×C₄²	D₄×C₆₁	1	2	yes	—
132	C₂×C₄²	C₂³×C₆₁	1	2	yes	—
133	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
134	#107	—	2	8	no	—
135	#106	—	2	8	no	—
136	C₄×C₈	Q₈×C₆₁	1	4	yes	—
137	C₄×C₈	Q₈×C₆₁	1	4	yes	—
138	#95	—	2	4	no	—
139	#95	—	2	4	no	—
140	#95	—	2	4	no	—
141	C₂²×C₄	—	2	4	no	—
142	C₂²×C₄	—	2	4	no	—
143	C₂²×C₄	C₂×D₄×C₆₁	1	4	yes	—
144	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
145	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
146	C₂²×C₄	C₂⁴×C₆₁	1	1	yes	characteristic
147	C₂×C₈	—	2	4	no	—
148	C₂×C₈	—	2	4	no	—
149	C₂×C₈	C₂×Q₈×C₆₁	1	4	yes	—
150	C₂×C₈	C₂×Q₈×C₆₁	1	4	yes	—
151	C₄⋊C₄	—	2	16	no	—
152	C₄⋊C₄	—	2	4	no	—
153	C₄⋊C₄	—	2	4	no	—
154	C₄⋊C₄	—	2	4	no	—
155	C₄⋊C₄	—	4	8	no	—
156	C₄⋊C₄	#41×C₆₁	1	8	yes	—
157	C₄²	—	2	4	no	—
158	C₄²	—	2	4	no	—
159	C₄²	—	4	2	no	—
160	C₄²	—	4	8	no	—
161	C₄²	C₂×Q₈×C₆₁	1	2	yes	—
162	C₄²	C₂×Q₈×C₆₁	1	2	yes	—
163	C₂³	C₂²×D₄×C₆₁	1	1	yes	characteristic
164	C₂×C₄	—	2	2	no	—
165	C₂×C₄	—	2	4	no	—
166	C₂×C₄	—	2	4	no	—
167	C₂×C₄	—	2	4	no	—
168	C₂×C₄	—	2	8	no	—
169	C₂×C₄	—	4	4	no	—
170	C₂×C₄	—	4	4	no	—
171	C₂×C₄	—	4	4	no	—
172	C₂×C₄	—	4	4	no	—
173	C₂×C₄	—	4	4	no	—
174	C₂×C₄	C₄⋊Q₈×C₆₁	1	2	yes	—
175	C₂×C₄	C₄⋊Q₈×C₆₁	1	2	yes	—
176	C₂×C₄	#135×C₆₁	1	4	yes	—
177	C₂×C₄	C₂²×Q₈×C₆₁	1	1	yes	characteristic
178	C₂×C₄	C₂²×Q₈×C₆₁	1	1	yes	characteristic
179	C₂×C₄	C₂×#41×C₆₁	1	4	yes	—
180	C₈	—	2	4	no	—
181	C₈	—	2	4	no	—
182	C₂²	C₆₁×#525	1	2	yes	—
183	C₂²	C₂×C₄⋊Q₈×C₆₁	1	1	yes	characteristic
184	C₂²	C₂×#135×C₆₁	1	2	yes	—
185	C₂²	C₆₁×#575	1	2	yes	—
186	C₄	—	2	2	no	—
187	C₄	—	4	4	no	—
188	C₄	—	4	8	no	—
189	C₄	—	8	4	no	—
190	C₄	C₆₁×#499	1	4	yes	—
191	C₄	C₆₁×#533	1	2	yes	—
192	C₄	C₆₁×#533	1	2	yes	—
193	C₂	C₆₁×#2399	1	1	yes	characteristic
194	C₂	C₆₁×#3102	1	4	yes	—
195	C₂	C₂×C₆₁×#499	1	1	yes	characteristic
196	C₂	C₆₁×#3162	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₆₁	C₂×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₁
socle, prime radical	C₂³×C₆₁	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₆₁ ◁ C₄³×C₆₁ ◁ G

Complements

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

N	H ≅ G/N	type
83. #23836	C₆₁	direct (N × H)
121. C₆₁	#23836	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄×#125×C₆₁	2
C₆₁×#2085	2
C₆₁×#2059	2
C₄×#107×C₆₁	2
C₄×#106×C₆₁	2
C₆₁×#1744	2
#23836	61

Subgroups — C61×#23836 #29527269

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#23836 #29527269