Subgroups — C₆₁×#11493 #29514926

Order 15616.

342 conjugacy classes of subgroups.

The subgroup lattice has 182 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×Q₈×C₆₁	C₂	1	1	yes	maximal, characteristic
2	C₆₁×#2162	C₂	1	2	yes	maximal
3	C₆₁×#1322	C₂	1	4	yes	maximal
4	C₂×C₄×Q₈×C₆₁	C₄	1	1	yes	characteristic
5	C₈×Q₈×C₆₁	—	2	2	no	—
6	C₈×Q₈×C₆₁	—	2	2	no	—
7	C₂×C₄⋊C₈×C₆₁	C₄	1	1	yes	characteristic
8	C₂×C₄⋊C₈×C₆₁	C₂²	1	2	yes	—
9	C₂×C₄×C₈×C₆₁	C₄	1	2	yes	—
10	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
11	C₆₁×C₄⋊C₁₆	—	2	2	no	—
12	C₆₁×C₂²⋊C₁₆	C₂²	1	4	yes	—
13	C₄×C₁₆×C₆₁	—	2	2	no	—
14	C₂²×Q₈×C₆₁	C₈	1	1	yes	characteristic
15	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
16	C₂²×C₈×C₆₁	C₂×C₄	1	2	yes	—
17	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
18	C₄×Q₈×C₆₁	—	2	2	no	—
19	C₄×Q₈×C₆₁	—	2	2	no	—
20	C₂×C₄⋊C₄×C₆₁	C₈	1	2	yes	—
21	C₂×C₄⋊C₄×C₆₁	C₈	1	1	yes	characteristic
22	C₂×C₄²×C₆₁	C₂×C₄	1	2	yes	—
23	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
24	C₂×C₁₆×C₆₁	—	2	4	no	—
25	C₄⋊C₈×C₆₁	—	2	4	no	—
26	C₄⋊C₈×C₆₁	—	2	2	no	—
27	C₄×C₈×C₆₁	—	2	2	no	—
28	C₄×C₈×C₆₁	—	2	2	no	—
29	C₄×C₈×C₆₁	D₄	1	2	yes	—
30	C₄×C₈×C₆₁	D₄	1	2	yes	—
31	C₂×Q₈×C₆₁	—	2	2	no	—
32	C₂×Q₈×C₆₁	—	2	2	no	—
33	C₂×Q₈×C₆₁	C₁₆	1	4	yes	complemented
34	C₂²×C₄×C₆₁	C₂×C₈	1	2	yes	—
35	C₂²×C₄×C₆₁	C₂×C₈	1	2	yes	—
36	C₂²×C₄×C₆₁	C₂×C₈	1	1	yes	characteristic
37	C₂²×C₄×C₆₁	C₂×C₈	1	1	yes	characteristic
38	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
39	C₂×C₈×C₆₁	—	2	2	no	—
40	C₂×C₈×C₆₁	—	2	2	no	—
41	C₂×C₈×C₆₁	—	4	1	no	—
42	C₂×C₈×C₆₁	—	4	2	no	—
43	C₂×C₈×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
44	C₂×C₈×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
45	C₂×C₈×C₆₁	C₂×D₄	1	2	yes	—
46	C₄⋊C₄×C₆₁	—	2	4	no	—
47	C₄⋊C₄×C₆₁	—	2	2	no	—
48	C₄²×C₆₁	—	2	2	no	—
49	C₄²×C₆₁	—	2	2	no	—
50	C₄²×C₆₁	C₂²⋊C₄	1	2	yes	—
51	C₄²×C₆₁	M₁₆	1	2	yes	—
52	C₁₆×C₆₁	—	4	4	no	—
53	C₂³×C₆₁	C₂²×C₈	1	1	yes	characteristic
54	Q₈×C₆₁	—	2	8	no	—
55	C₂×C₄×C₆₁	—	2	2	no	—
56	C₂×C₄×C₆₁	—	2	2	no	—
57	C₂×C₄×C₆₁	—	2	1	no	—
58	C₂×C₄×C₆₁	—	2	1	no	—
59	C₂×C₄×C₆₁	—	2	2	no	—
60	C₂×C₄×C₆₁	—	4	1	no	—
61	C₂×C₄×C₆₁	—	4	2	no	—
62	C₂×C₄×C₆₁	C₂²⋊C₈	1	2	yes	—
63	C₂×C₄×C₆₁	C₂²⋊C₈	1	2	yes	—
64	C₂×C₄×C₆₁	C₂×C₁₆	1	4	yes	—
65	C₂×C₄×C₆₁	C₂×C₁₆	1	2	yes	—
66	C₂×C₄×C₆₁	M₃₂	1	2	yes	—
67	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
68	C₂×C₄×C₆₁	C₂×M₁₆	1	1	yes	characteristic
69	C₈×C₆₁	—	2	2	no	—
70	C₈×C₆₁	—	2	1	no	—
71	C₈×C₆₁	—	4	2	no	—
72	C₈×C₆₁	#101	1	2	yes	—
73	#11493	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
74	C₂²×C₆₁	—	2	1	no	—
75	C₂²×C₆₁	—	2	1	no	—
76	C₂²×C₆₁	C₂×C₂²⋊C₈	1	1	yes	characteristic
77	C₂²×C₆₁	C₂²×C₁₆	1	1	yes	characteristic
78	C₂²×C₆₁	C₂×M₃₂	1	1	yes	characteristic
79	C₄×C₆₁	—	2	4	no	—
80	C₄×C₆₁	—	2	2	no	—
81	C₄×C₆₁	—	2	1	no	—
82	C₄×C₆₁	—	4	2	no	—
83	C₄×C₆₁	C₂²⋊C₁₆	1	4	yes	—
84	C₄×C₆₁	#410	1	1	yes	characteristic
85	C₄×C₆₁	C₂×#101	1	1	yes	characteristic
86	C₂×C₈×Q₈	C₂×C₆₁	1	1	yes	characteristic
87	#2162	C₂×C₆₁	1	2	yes	—
88	#1322	C₂×C₆₁	1	4	yes	—
89	C₂×C₆₁	—	2	1	no	—
90	C₂×C₆₁	—	2	1	no	—
91	C₂×C₆₁	#1461	1	1	yes	characteristic
92	C₂×C₆₁	C₂×C₂²⋊C₁₆	1	1	yes	characteristic
93	C₂×C₆₁	#2115	1	1	yes	characteristic
94	C₂×C₄×Q₈	C₄×C₆₁	1	1	yes	characteristic
95	C₈×Q₈	—	2	2	no	—
96	C₈×Q₈	—	2	2	no	—
97	C₂×C₄⋊C₈	C₄×C₆₁	1	1	yes	characteristic
98	C₂×C₄⋊C₈	C₂²×C₆₁	1	2	yes	—
99	C₂×C₄×C₈	C₄×C₆₁	1	2	yes	—
100	C₂×C₄×C₈	C₂²×C₆₁	1	1	yes	characteristic
101	C₄⋊C₁₆	—	2	2	no	—
102	C₂²⋊C₁₆	C₂²×C₆₁	1	4	yes	—
103	C₄×C₁₆	—	2	2	no	—
104	C₆₁	#11493	1	1	yes	characteristic, direct product factor, Sylow
105	C₂²×Q₈	C₈×C₆₁	1	1	yes	characteristic
106	C₂²×C₈	C₂×C₄×C₆₁	1	1	yes	characteristic
107	C₂²×C₈	C₂×C₄×C₆₁	1	2	yes	—
108	C₂²×C₈	C₂³×C₆₁	1	1	yes	characteristic
109	C₄×Q₈	—	2	2	no	—
110	C₄×Q₈	—	2	2	no	—
111	C₂×C₄⋊C₄	C₈×C₆₁	1	2	yes	—
112	C₂×C₄⋊C₄	C₈×C₆₁	1	1	yes	characteristic
113	C₂×C₄²	C₂×C₄×C₆₁	1	2	yes	—
114	C₂×C₄²	C₂×C₄×C₆₁	1	1	yes	characteristic
115	C₂×C₁₆	—	2	4	no	—
116	C₄⋊C₈	—	2	4	no	—
117	C₄⋊C₈	—	2	2	no	—
118	C₄×C₈	—	2	2	no	—
119	C₄×C₈	—	2	2	no	—
120	C₄×C₈	D₄×C₆₁	1	2	yes	—
121	C₄×C₈	D₄×C₆₁	1	2	yes	—
122	C₂×Q₈	—	2	2	no	—
123	C₂×Q₈	—	2	2	no	—
124	C₂×Q₈	C₁₆×C₆₁	1	4	yes	complemented
125	C₂²×C₄	C₂×C₈×C₆₁	1	2	yes	—
126	C₂²×C₄	C₂×C₈×C₆₁	1	2	yes	—
127	C₂²×C₄	C₂×C₈×C₆₁	1	1	yes	characteristic
128	C₂²×C₄	C₂×C₈×C₆₁	1	1	yes	characteristic
129	C₂²×C₄	C₂²×C₄×C₆₁	1	1	yes	characteristic
130	C₂×C₈	—	2	2	no	—
131	C₂×C₈	—	2	2	no	—
132	C₂×C₈	—	4	1	no	—
133	C₂×C₈	—	4	2	no	—
134	C₂×C₈	C₂²⋊C₄×C₆₁	1	1	yes	characteristic
135	C₂×C₈	C₂²⋊C₄×C₆₁	1	1	yes	characteristic
136	C₂×C₈	C₂×D₄×C₆₁	1	2	yes	—
137	C₄⋊C₄	—	2	4	no	—
138	C₄⋊C₄	—	2	2	no	—
139	C₄²	—	2	2	no	—
140	C₄²	—	2	2	no	—
141	C₄²	C₂²⋊C₄×C₆₁	1	2	yes	—
142	C₄²	M₁₆×C₆₁	1	2	yes	—
143	C₁₆	—	4	4	no	—
144	C₂³	C₂²×C₈×C₆₁	1	1	yes	characteristic
145	Q₈	—	2	8	no	—
146	C₂×C₄	—	2	2	no	—
147	C₂×C₄	—	2	2	no	—
148	C₂×C₄	—	2	1	no	—
149	C₂×C₄	—	2	1	no	—
150	C₂×C₄	—	2	2	no	—
151	C₂×C₄	—	4	1	no	—
152	C₂×C₄	—	4	2	no	—
153	C₂×C₄	C₂²⋊C₈×C₆₁	1	2	yes	—
154	C₂×C₄	C₂²⋊C₈×C₆₁	1	2	yes	—
155	C₂×C₄	C₂×C₁₆×C₆₁	1	4	yes	—
156	C₂×C₄	C₂×C₁₆×C₆₁	1	2	yes	—
157	C₂×C₄	M₃₂×C₆₁	1	2	yes	—
158	C₂×C₄	C₂×C₂²⋊C₄×C₆₁	1	1	yes	characteristic
159	C₂×C₄	C₂×M₁₆×C₆₁	1	1	yes	characteristic
160	C₈	—	2	2	no	—
161	C₈	—	2	1	no	—
162	C₈	—	4	2	no	—
163	C₈	#101×C₆₁	1	2	yes	—
164	C₂²	—	2	1	no	—
165	C₂²	—	2	1	no	—
166	C₂²	C₂×C₂²⋊C₈×C₆₁	1	1	yes	characteristic
167	C₂²	C₂²×C₁₆×C₆₁	1	1	yes	characteristic
168	C₂²	C₂×M₃₂×C₆₁	1	1	yes	characteristic
169	C₄	—	2	4	no	—
170	C₄	—	2	2	no	—
171	C₄	—	2	1	no	—
172	C₄	—	4	2	no	—
173	C₄	C₆₁×C₂²⋊C₁₆	1	4	yes	—
174	C₄	C₆₁×#410	1	1	yes	characteristic
175	C₄	C₂×#101×C₆₁	1	1	yes	characteristic
176	C₂	—	2	1	no	—
177	C₂	—	2	1	no	—
178	C₂	C₆₁×#1461	1	1	yes	characteristic
179	C₂	C₂×C₆₁×C₂²⋊C₁₆	1	1	yes	characteristic
180	C₂	C₆₁×#2115	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈×C₆₁	C₂²⋊C₄
commutator	C₂²	C₂²×C₁₆×C₆₁
Frattini, cosocle	C₂²×C₈	C₂³×C₆₁
socle	C₂²×C₆₁	C₂×C₂²⋊C₈
prime radical	C₂³×C₆₁	C₂²×C₈
LCS	G ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈×C₆₁ ◁ C₂×C₄×C₈×C₆₁ ◁ G

Complements

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

N	H ≅ G/N	type
33. C₂×Q₈×C₆₁	C₁₆	semidirect
73. #11493	C₆₁	direct (N × H)
104. C₆₁	#11493	direct (N × H)
124. C₂×Q₈	C₁₆×C₆₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₈×Q₈×C₆₁	2
C₆₁×#2162	2
C₆₁×#1322	2
#11493	61

Subgroups — C61×#11493 #29514926

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#11493 #29514926