Subgroups — C₂×C₆₁×#2151 #29536829

Order 15616.

734 conjugacy classes of subgroups.

The subgroup lattice has 212 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×M₃₂×C₆₁	C₂	1	1	yes	maximal, characteristic
2	C₂³×C₁₆×C₆₁	C₂	1	1	yes	maximal, characteristic
3	C₂×C₆₁×#405	C₂	1	1	yes	maximal, characteristic
4	C₆₁×#2151	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₆₁×C₄⋊C₁₆	C₂	1	4	yes	maximal, complemented
6	C₂²×M₁₆×C₆₁	C₄	1	1	yes	characteristic
7	C₂³×C₈×C₆₁	C₂²	1	1	yes	characteristic
8	C₂×#117×C₆₁	C₄	1	1	yes	characteristic
9	C₂×M₃₂×C₆₁	—	2	4	no	—
10	C₂×M₃₂×C₆₁	C₂²	1	4	yes	—
11	C₂²×C₁₆×C₆₁	—	2	4	no	—
12	C₂²×C₁₆×C₆₁	C₂²	1	4	yes	—
13	C₂²×C₁₆×C₆₁	C₂²	1	2	yes	—
14	C₂²×C₁₆×C₆₁	C₂²	1	2	yes	—
15	C₆₁×#405	C₄	1	4	yes	—
16	C₆₁×#405	C₂²	1	4	yes	—
17	C₂×#97×C₆₁	C₄	1	2	yes	—
18	C₂×C₄×C₈×C₆₁	C₂²	1	2	yes	—
19	C₆₁×C₄⋊C₁₆	C₂²	1	16	yes	complemented
20	C₂³×C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
21	C₂×M₁₆×C₆₁	—	2	4	no	—
22	C₂×M₁₆×C₆₁	C₂×C₄	1	4	yes	—
23	C₂²×C₈×C₆₁	C₂×C₄	1	2	yes	—
24	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₈×C₆₁	C₂×C₄	1	2	yes	—
26	C₂²×C₈×C₆₁	D₄	1	4	yes	—
27	C₂²×C₈×C₆₁	Q₈	1	4	yes	—
28	C₂²×C₈×C₆₁	C₂³	1	2	yes	—
29	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
30	#117×C₆₁	C₈	1	4	yes	—
31	#117×C₆₁	C₂×C₄	1	4	yes	—
32	C₂×C₄⋊C₄×C₆₁	C₈	1	2	yes	—
33	C₂×C₂²⋊C₄×C₆₁	C₈	1	2	yes	—
34	C₂×C₄²×C₆₁	C₂×C₄	1	2	yes	—
35	M₃₂×C₆₁	—	2	8	no	—
36	C₂×C₁₆×C₆₁	—	2	4	no	—
37	C₂×C₁₆×C₆₁	—	2	8	no	—
38	C₂×C₁₆×C₆₁	—	2	4	no	—
39	C₂×C₁₆×C₆₁	C₂³	1	4	yes	—
40	C₂×C₁₆×C₆₁	C₂³	1	4	yes	—
41	#97×C₆₁	C₂×C₄	1	8	yes	—
42	C₄×C₈×C₆₁	C₂×C₄	1	4	yes	—
43	C₄×C₈×C₆₁	C₂³	1	4	yes	—
44	C₂⁴×C₆₁	C₂×C₈	1	1	yes	characteristic
45	C₂²×C₄×C₆₁	C₄⋊C₄	1	4	yes	—
46	C₂²×C₄×C₆₁	C₂×C₈	1	4	yes	—
47	C₂²×C₄×C₆₁	C₂×C₈	1	1	yes	characteristic
48	C₂²×C₄×C₆₁	C₂×C₈	1	2	yes	—
49	C₂²×C₄×C₆₁	M₁₆	1	4	yes	—
50	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
51	C₂²×C₄×C₆₁	C₂²×C₄	1	2	yes	—
52	M₁₆×C₆₁	—	2	8	no	—
53	C₂×C₈×C₆₁	—	2	4	no	—
54	C₂×C₈×C₆₁	C₄⋊C₄	1	4	yes	—
55	C₂×C₈×C₆₁	C₄⋊C₄	1	8	yes	—
56	C₂×C₈×C₆₁	C₂²×C₄	1	4	yes	—
57	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₈×C₆₁	C₂²×C₄	1	2	yes	—
59	C₂×C₈×C₆₁	C₂×D₄	1	2	yes	—
60	C₂×C₈×C₆₁	C₂×D₄	1	4	yes	—
61	C₂×C₈×C₆₁	C₂×Q₈	1	4	yes	—
62	C₂×C₈×C₆₁	C₂×Q₈	1	2	yes	—
63	C₂×C₈×C₆₁	C₂⁴	1	1	yes	characteristic
64	C₄⋊C₄×C₆₁	C₂×C₈	1	8	yes	—
65	C₂²⋊C₄×C₆₁	C₂×C₈	1	8	yes	—
66	C₄²×C₆₁	C₂×C₈	1	4	yes	—
67	C₄²×C₆₁	C₂²×C₄	1	4	yes	—
68	C₁₆×C₆₁	—	2	4	no	—
69	C₁₆×C₆₁	—	2	4	no	—
70	C₂³×C₆₁	—	2	4	no	—
71	C₂³×C₆₁	C₄⋊C₈	1	4	yes	—
72	C₂³×C₆₁	C₂²×C₈	1	2	yes	—
73	C₂³×C₆₁	C₂²×C₈	1	1	yes	characteristic
74	C₂×C₄×C₆₁	—	2	8	no	—
75	C₂×C₄×C₆₁	C₄⋊C₈	1	8	yes	—
76	C₂×C₄×C₆₁	C₄⋊C₈	1	4	yes	—
77	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	4	yes	—
78	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	2	yes	—
79	C₂×C₄×C₆₁	C₂²×C₈	1	8	yes	—
80	C₂×C₄×C₆₁	C₂²×C₈	1	1	yes	characteristic
81	C₂×C₄×C₆₁	C₂²×C₈	1	2	yes	—
82	C₂×C₄×C₆₁	C₂×M₁₆	1	2	yes	—
83	C₂×C₄×C₆₁	C₂×M₁₆	1	4	yes	—
84	C₂×C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
85	C₈×C₆₁	—	2	4	no	—
86	C₈×C₆₁	C₂×C₄⋊C₄	1	2	yes	—
87	C₈×C₆₁	C₂×C₄⋊C₄	1	4	yes	—
88	C₈×C₆₁	C₂²×D₄	1	1	yes	characteristic
89	C₈×C₆₁	C₂²×Q₈	1	1	yes	characteristic
90	C₂×#2151	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
91	C₂²×C₆₁	—	2	4	no	—
92	C₂²×C₆₁	—	2	8	no	—
93	C₂²×C₆₁	C₂×C₄⋊C₈	1	4	yes	—
94	C₂²×C₆₁	C₂×C₄⋊C₈	1	2	yes	—
95	C₂²×C₆₁	#504	1	4	yes	—
96	C₂²×C₆₁	C₂³×C₈	1	1	yes	characteristic
97	C₄×C₆₁	—	2	8	no	—
98	C₄×C₆₁	C₂×C₄⋊C₈	1	2	yes	—
99	C₄×C₆₁	C₂×C₄⋊C₈	1	4	yes	—
100	C₄×C₆₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
101	C₄×C₆₁	C₂²×M₁₆	1	1	yes	characteristic
102	C₂²×M₃₂	C₂×C₆₁	1	1	yes	characteristic
103	C₂³×C₁₆	C₂×C₆₁	1	1	yes	characteristic
104	C₂×#405	C₂×C₆₁	1	1	yes	characteristic
105	#2151	C₂×C₆₁	1	8	yes	direct product factor
106	C₂×C₄⋊C₁₆	C₂×C₆₁	1	4	yes	complemented
107	C₂×C₆₁	—	2	4	no	—
108	C₂×C₆₁	#2151	1	4	yes	direct product factor
109	C₂×C₆₁	C₂²×C₄⋊C₈	1	1	yes	characteristic
110	C₂×C₆₁	C₂×#504	1	2	yes	—
111	C₂²×M₁₆	C₄×C₆₁	1	1	yes	characteristic
112	C₂³×C₈	C₂²×C₆₁	1	1	yes	characteristic
113	C₂×#117	C₄×C₆₁	1	1	yes	characteristic
114	C₂×M₃₂	—	2	4	no	—
115	C₂×M₃₂	C₂²×C₆₁	1	4	yes	—
116	C₂²×C₁₆	—	2	4	no	—
117	C₂²×C₁₆	C₂²×C₆₁	1	2	yes	—
118	C₂²×C₁₆	C₂²×C₆₁	1	4	yes	—
119	C₂²×C₁₆	C₂²×C₆₁	1	2	yes	—
120	#405	C₄×C₆₁	1	4	yes	—
121	#405	C₂²×C₆₁	1	4	yes	—
122	C₂×#97	C₄×C₆₁	1	2	yes	—
123	C₂×C₄×C₈	C₂²×C₆₁	1	2	yes	—
124	C₄⋊C₁₆	C₂²×C₆₁	1	16	yes	complemented
125	C₆₁	C₂×#2151	1	1	yes	characteristic, direct product factor, Sylow
126	C₂³×C₄	C₂×C₄×C₆₁	1	1	yes	characteristic
127	C₂×M₁₆	—	2	4	no	—
128	C₂×M₁₆	C₂×C₄×C₆₁	1	4	yes	—
129	C₂²×C₈	C₂×C₄×C₆₁	1	2	yes	—
130	C₂²×C₈	C₂×C₄×C₆₁	1	1	yes	characteristic
131	C₂²×C₈	C₂×C₄×C₆₁	1	2	yes	—
132	C₂²×C₈	D₄×C₆₁	1	4	yes	—
133	C₂²×C₈	Q₈×C₆₁	1	4	yes	—
134	C₂²×C₈	C₂³×C₆₁	1	2	yes	—
135	C₂²×C₈	C₂³×C₆₁	1	1	yes	characteristic
136	#117	C₈×C₆₁	1	4	yes	—
137	#117	C₂×C₄×C₆₁	1	4	yes	—
138	C₂×C₄⋊C₄	C₈×C₆₁	1	2	yes	—
139	C₂×C₂²⋊C₄	C₈×C₆₁	1	2	yes	—
140	C₂×C₄²	C₂×C₄×C₆₁	1	2	yes	—
141	M₃₂	—	2	8	no	—
142	C₂×C₁₆	—	2	4	no	—
143	C₂×C₁₆	—	2	8	no	—
144	C₂×C₁₆	—	2	4	no	—
145	C₂×C₁₆	C₂³×C₆₁	1	4	yes	—
146	C₂×C₁₆	C₂³×C₆₁	1	4	yes	—
147	#97	C₂×C₄×C₆₁	1	8	yes	—
148	C₄×C₈	C₂×C₄×C₆₁	1	4	yes	—
149	C₄×C₈	C₂³×C₆₁	1	4	yes	—
150	C₂⁴	C₂×C₈×C₆₁	1	1	yes	characteristic
151	C₂²×C₄	C₄⋊C₄×C₆₁	1	4	yes	—
152	C₂²×C₄	C₂×C₈×C₆₁	1	4	yes	—
153	C₂²×C₄	C₂×C₈×C₆₁	1	1	yes	characteristic
154	C₂²×C₄	C₂×C₈×C₆₁	1	2	yes	—
155	C₂²×C₄	M₁₆×C₆₁	1	4	yes	—
156	C₂²×C₄	C₂²×C₄×C₆₁	1	1	yes	characteristic
157	C₂²×C₄	C₂²×C₄×C₆₁	1	2	yes	—
158	M₁₆	—	2	8	no	—
159	C₂×C₈	—	2	4	no	—
160	C₂×C₈	C₄⋊C₄×C₆₁	1	4	yes	—
161	C₂×C₈	C₄⋊C₄×C₆₁	1	8	yes	—
162	C₂×C₈	C₂²×C₄×C₆₁	1	4	yes	—
163	C₂×C₈	C₂²×C₄×C₆₁	1	1	yes	characteristic
164	C₂×C₈	C₂²×C₄×C₆₁	1	2	yes	—
165	C₂×C₈	C₂×D₄×C₆₁	1	2	yes	—
166	C₂×C₈	C₂×D₄×C₆₁	1	4	yes	—
167	C₂×C₈	C₂×Q₈×C₆₁	1	4	yes	—
168	C₂×C₈	C₂×Q₈×C₆₁	1	2	yes	—
169	C₂×C₈	C₂⁴×C₆₁	1	1	yes	characteristic
170	C₄⋊C₄	C₂×C₈×C₆₁	1	8	yes	—
171	C₂²⋊C₄	C₂×C₈×C₆₁	1	8	yes	—
172	C₄²	C₂×C₈×C₆₁	1	4	yes	—
173	C₄²	C₂²×C₄×C₆₁	1	4	yes	—
174	C₁₆	—	2	4	no	—
175	C₁₆	—	2	4	no	—
176	C₂³	—	2	4	no	—
177	C₂³	C₄⋊C₈×C₆₁	1	4	yes	—
178	C₂³	C₂²×C₈×C₆₁	1	2	yes	—
179	C₂³	C₂²×C₈×C₆₁	1	1	yes	characteristic
180	C₂×C₄	—	2	8	no	—
181	C₂×C₄	C₄⋊C₈×C₆₁	1	8	yes	—
182	C₂×C₄	C₄⋊C₈×C₆₁	1	4	yes	—
183	C₂×C₄	C₂×C₄⋊C₄×C₆₁	1	4	yes	—
184	C₂×C₄	C₂×C₄⋊C₄×C₆₁	1	2	yes	—
185	C₂×C₄	C₂²×C₈×C₆₁	1	8	yes	—
186	C₂×C₄	C₂²×C₈×C₆₁	1	1	yes	characteristic
187	C₂×C₄	C₂²×C₈×C₆₁	1	2	yes	—
188	C₂×C₄	C₂×M₁₆×C₆₁	1	2	yes	—
189	C₂×C₄	C₂×M₁₆×C₆₁	1	4	yes	—
190	C₂×C₄	C₂³×C₄×C₆₁	1	1	yes	characteristic
191	C₈	—	2	4	no	—
192	C₈	C₂×C₄⋊C₄×C₆₁	1	2	yes	—
193	C₈	C₂×C₄⋊C₄×C₆₁	1	4	yes	—
194	C₈	C₂²×D₄×C₆₁	1	1	yes	characteristic
195	C₈	C₂²×Q₈×C₆₁	1	1	yes	characteristic
196	C₂²	—	2	4	no	—
197	C₂²	—	2	8	no	—
198	C₂²	C₂×C₄⋊C₈×C₆₁	1	4	yes	—
199	C₂²	C₂×C₄⋊C₈×C₆₁	1	2	yes	—
200	C₂²	C₆₁×#504	1	4	yes	—
201	C₂²	C₂³×C₈×C₆₁	1	1	yes	characteristic
202	C₄	—	2	8	no	—
203	C₄	C₂×C₄⋊C₈×C₆₁	1	2	yes	—
204	C₄	C₂×C₄⋊C₈×C₆₁	1	4	yes	—
205	C₄	C₂²×C₄⋊C₄×C₆₁	1	1	yes	characteristic
206	C₄	C₂²×M₁₆×C₆₁	1	1	yes	characteristic
207	C₂	—	2	4	no	—
208	C₂	C₆₁×#2151	1	4	yes	direct product factor
209	C₂	C₂²×C₄⋊C₈×C₆₁	1	1	yes	characteristic
210	C₂	C₂×C₆₁×#504	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
4. C₆₁×#2151	C₂	direct (N × H)
5. C₂×C₆₁×C₄⋊C₁₆	C₂	semidirect
19. C₆₁×C₄⋊C₁₆	C₂²	semidirect
90. C₂×#2151	C₆₁	direct (N × H)
105. #2151	C₂×C₆₁	direct (N × H)
106. C₂×C₄⋊C₁₆	C₂×C₆₁	semidirect
108. C₂×C₆₁	#2151	direct (N × H)
124. C₄⋊C₁₆	C₂²×C₆₁	semidirect
125. C₆₁	C₂×#2151	direct (N × H)
208. C₂	C₆₁×#2151	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×M₃₂×C₆₁	2
C₂³×C₁₆×C₆₁	2
C₂×C₆₁×#405	2
C₆₁×#2151	2
C₂×C₆₁×C₄⋊C₁₆	2
C₂×#2151	61

Subgroups — C2×C61×#2151 #29536829

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₆₁×#2151 #29536829