Subgroups — C₂×C₆₁×#1374 #29515976

Order 15616.

726 conjugacy classes of subgroups.

The subgroup lattice has 232 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₁₆×C₆₁	C₂	1	2	yes	maximal
2	C₂²×C₄×C₈×C₆₁	C₂	1	1	yes	maximal, characteristic
3	C₆₁×#1374	C₂	1	4	yes	maximal, direct product factor
4	C₂³×C₈×C₆₁	C₄	1	1	yes	characteristic
5	C₂³×C₈×C₆₁	C₂²	1	1	yes	characteristic
6	C₂²×C₄²×C₆₁	C₄	1	1	yes	characteristic
7	C₂²×C₁₆×C₆₁	—	2	4	no	—
8	C₂²×C₁₆×C₆₁	—	2	4	no	—
9	C₂²×C₁₆×C₆₁	C₄	1	8	yes	complemented
10	C₂²×C₁₆×C₆₁	C₂²	1	4	yes	—
11	C₂×C₄×C₈×C₆₁	—	2	4	no	—
12	C₂×C₄×C₈×C₆₁	C₄	1	2	yes	—
13	C₂×C₄×C₈×C₆₁	C₂²	1	2	yes	—
14	C₂³×C₄×C₆₁	C₈	1	2	yes	—
15	C₂³×C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
16	C₂²×C₈×C₆₁	—	2	2	no	—
17	C₂²×C₈×C₆₁	—	2	2	no	—
18	C₂²×C₈×C₆₁	C₈	1	4	yes	—
19	C₂²×C₈×C₆₁	C₂×C₄	1	2	yes	—
20	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₈×C₆₁	C₂×C₄	1	4	yes	—
22	C₂²×C₈×C₆₁	D₄	1	2	yes	—
23	C₂²×C₈×C₆₁	D₄	1	4	yes	—
24	C₂²×C₈×C₆₁	Q₈	1	2	yes	—
25	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
26	C₂×C₄²×C₆₁	—	2	2	no	—
27	C₂×C₄²×C₆₁	—	2	2	no	—
28	C₂×C₄²×C₆₁	C₈	1	2	yes	—
29	C₂×C₄²×C₆₁	C₂×C₄	1	2	yes	—
30	C₂×C₁₆×C₆₁	—	2	16	no	—
31	C₂×C₁₆×C₆₁	—	2	4	no	—
32	C₂×C₁₆×C₆₁	—	2	4	no	—
33	C₂×C₁₆×C₆₁	C₂×C₄	1	8	yes	complemented
34	C₄×C₈×C₆₁	—	2	4	no	—
35	C₄×C₈×C₆₁	—	2	4	no	—
36	C₂⁴×C₆₁	C₂×C₈	1	1	yes	characteristic
37	C₂²×C₄×C₆₁	—	2	4	no	—
38	C₂²×C₄×C₆₁	—	2	4	no	—
39	C₂²×C₄×C₆₁	C₁₆	1	8	yes	complemented
40	C₂²×C₄×C₆₁	C₄²	1	2	yes	—
41	C₂²×C₄×C₆₁	C₂²⋊C₄	1	2	yes	—
42	C₂²×C₄×C₆₁	C₄⋊C₄	1	2	yes	—
43	C₂²×C₄×C₆₁	C₂×C₈	1	4	yes	—
44	C₂²×C₄×C₆₁	C₂×C₈	1	1	yes	characteristic
45	C₂²×C₄×C₆₁	C₂×C₈	1	2	yes	—
46	C₂²×C₄×C₆₁	M₁₆	1	2	yes	—
47	C₂²×C₄×C₆₁	M₁₆	1	2	yes	—
48	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
49	C₂×C₈×C₆₁	—	2	8	no	—
50	C₂×C₈×C₆₁	—	2	2	no	—
51	C₂×C₈×C₆₁	—	2	2	no	—
52	C₂×C₈×C₆₁	C₄²	1	2	yes	—
53	C₂×C₈×C₆₁	C₂²⋊C₄	1	2	yes	—
54	C₂×C₈×C₆₁	C₂²⋊C₄	1	8	yes	—
55	C₂×C₈×C₆₁	C₄⋊C₄	1	2	yes	—
56	C₂×C₈×C₆₁	C₄⋊C₄	1	8	yes	—
57	C₂×C₈×C₆₁	C₂×C₈	1	4	yes	—
58	C₂×C₈×C₆₁	C₂²×C₄	1	2	yes	—
59	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
60	C₂×C₈×C₆₁	C₂×D₄	1	2	yes	—
61	C₂×C₈×C₆₁	C₂×Q₈	1	1	yes	characteristic
62	C₄²×C₆₁	—	2	4	no	—
63	C₄²×C₆₁	—	2	2	no	—
64	C₄²×C₆₁	—	2	2	no	—
65	C₁₆×C₆₁	—	2	8	no	—
66	C₂³×C₆₁	C₄×C₈	1	2	yes	—
67	C₂³×C₆₁	C₂²⋊C₈	1	2	yes	—
68	C₂³×C₆₁	C₄⋊C₈	1	2	yes	—
69	C₂³×C₆₁	C₂×C₁₆	1	4	yes	—
70	C₂³×C₆₁	M₃₂	1	4	yes	—
71	C₂³×C₆₁	C₂²×C₈	1	1	yes	characteristic
72	C₂×C₄×C₆₁	—	2	16	no	—
73	C₂×C₄×C₆₁	—	2	4	no	—
74	C₂×C₄×C₆₁	—	2	4	no	—
75	C₂×C₄×C₆₁	#95	1	4	yes	—
76	C₂×C₄×C₆₁	C₄×C₈	1	2	yes	—
77	C₂×C₄×C₆₁	#97	1	4	yes	—
78	C₂×C₄×C₆₁	C₂²⋊C₈	1	2	yes	—
79	C₂×C₄×C₆₁	C₂²⋊C₈	1	4	yes	—
80	C₂×C₄×C₆₁	C₄⋊C₈	1	2	yes	—
81	C₂×C₄×C₆₁	C₄⋊C₈	1	4	yes	—
82	C₂×C₄×C₆₁	C₂×C₁₆	1	8	yes	complemented
83	C₂×C₄×C₆₁	C₂×C₄²	1	1	yes	characteristic
84	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
85	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
86	C₂×C₄×C₆₁	C₂²×C₈	1	1	yes	characteristic
87	C₂×C₄×C₆₁	C₂×M₁₆	1	1	yes	characteristic
88	C₂×C₄×C₆₁	C₂×M₁₆	1	1	yes	characteristic
89	C₈×C₆₁	—	2	4	no	—
90	C₈×C₆₁	#95	1	4	yes	—
91	C₈×C₆₁	C₂×C₂²⋊C₄	1	2	yes	—
92	C₈×C₆₁	C₂×C₄⋊C₄	1	2	yes	—
93	C₂×#1374	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
94	C₂²×C₆₁	#336	1	4	yes	—
95	C₂²×C₆₁	C₄×C₁₆	1	4	yes	—
96	C₂²×C₆₁	#346	1	4	yes	—
97	C₂²×C₆₁	C₂²⋊C₁₆	1	8	yes	—
98	C₂²×C₆₁	C₄⋊C₁₆	1	8	yes	—
99	C₂²×C₆₁	C₂×C₄×C₈	1	1	yes	characteristic
100	C₂²×C₆₁	C₂×C₂²⋊C₈	1	1	yes	characteristic
101	C₂²×C₆₁	C₂×C₄⋊C₈	1	1	yes	characteristic
102	C₂²×C₆₁	C₂²×C₁₆	1	2	yes	—
103	C₂²×C₆₁	C₂×M₃₂	1	2	yes	—
104	C₄×C₆₁	—	2	8	no	—
105	C₄×C₆₁	#336	1	4	yes	—
106	C₄×C₆₁	C₂×#95	1	1	yes	characteristic
107	C₄×C₆₁	C₂×#97	1	1	yes	characteristic
108	C₄×C₆₁	C₂×C₂²⋊C₈	1	1	yes	characteristic
109	C₄×C₆₁	C₂×C₄⋊C₈	1	1	yes	characteristic
110	C₂³×C₁₆	C₂×C₆₁	1	2	yes	—
111	C₂²×C₄×C₈	C₂×C₆₁	1	1	yes	characteristic
112	#1374	C₂×C₆₁	1	4	yes	direct product factor
113	C₂×C₆₁	#1374	1	8	yes	direct product factor
114	C₂×C₆₁	C₂×#336	1	1	yes	characteristic
115	C₂×C₆₁	C₂×C₄×C₁₆	1	1	yes	characteristic
116	C₂×C₆₁	C₂×#346	1	1	yes	characteristic
117	C₂×C₆₁	C₂×C₂²⋊C₁₆	1	2	yes	—
118	C₂×C₆₁	C₂×C₄⋊C₁₆	1	2	yes	—
119	C₂³×C₈	C₄×C₆₁	1	1	yes	characteristic
120	C₂³×C₈	C₂²×C₆₁	1	1	yes	characteristic
121	C₂²×C₄²	C₄×C₆₁	1	1	yes	characteristic
122	C₂²×C₁₆	—	2	4	no	—
123	C₂²×C₁₆	—	2	4	no	—
124	C₂²×C₁₆	C₄×C₆₁	1	8	yes	complemented
125	C₂²×C₁₆	C₂²×C₆₁	1	4	yes	—
126	C₂×C₄×C₈	—	2	4	no	—
127	C₂×C₄×C₈	C₄×C₆₁	1	2	yes	—
128	C₂×C₄×C₈	C₂²×C₆₁	1	2	yes	—
129	C₆₁	C₂×#1374	1	1	yes	characteristic, direct product factor, Sylow
130	C₂³×C₄	C₈×C₆₁	1	2	yes	—
131	C₂³×C₄	C₂×C₄×C₆₁	1	1	yes	characteristic
132	C₂²×C₈	—	2	2	no	—
133	C₂²×C₈	—	2	2	no	—
134	C₂²×C₈	C₈×C₆₁	1	4	yes	—
135	C₂²×C₈	C₂×C₄×C₆₁	1	2	yes	—
136	C₂²×C₈	C₂×C₄×C₆₁	1	1	yes	characteristic
137	C₂²×C₈	C₂×C₄×C₆₁	1	4	yes	—
138	C₂²×C₈	D₄×C₆₁	1	2	yes	—
139	C₂²×C₈	D₄×C₆₁	1	4	yes	—
140	C₂²×C₈	Q₈×C₆₁	1	2	yes	—
141	C₂²×C₈	C₂³×C₆₁	1	1	yes	characteristic
142	C₂×C₄²	—	2	2	no	—
143	C₂×C₄²	—	2	2	no	—
144	C₂×C₄²	C₈×C₆₁	1	2	yes	—
145	C₂×C₄²	C₂×C₄×C₆₁	1	2	yes	—
146	C₂×C₁₆	—	2	16	no	—
147	C₂×C₁₆	—	2	4	no	—
148	C₂×C₁₆	—	2	4	no	—
149	C₂×C₁₆	C₂×C₄×C₆₁	1	8	yes	complemented
150	C₄×C₈	—	2	4	no	—
151	C₄×C₈	—	2	4	no	—
152	C₂⁴	C₂×C₈×C₆₁	1	1	yes	characteristic
153	C₂²×C₄	—	2	4	no	—
154	C₂²×C₄	—	2	4	no	—
155	C₂²×C₄	C₁₆×C₆₁	1	8	yes	complemented
156	C₂²×C₄	C₄²×C₆₁	1	2	yes	—
157	C₂²×C₄	C₂²⋊C₄×C₆₁	1	2	yes	—
158	C₂²×C₄	C₄⋊C₄×C₆₁	1	2	yes	—
159	C₂²×C₄	C₂×C₈×C₆₁	1	4	yes	—
160	C₂²×C₄	C₂×C₈×C₆₁	1	1	yes	characteristic
161	C₂²×C₄	C₂×C₈×C₆₁	1	2	yes	—
162	C₂²×C₄	M₁₆×C₆₁	1	2	yes	—
163	C₂²×C₄	M₁₆×C₆₁	1	2	yes	—
164	C₂²×C₄	C₂²×C₄×C₆₁	1	1	yes	characteristic
165	C₂×C₈	—	2	8	no	—
166	C₂×C₈	—	2	2	no	—
167	C₂×C₈	—	2	2	no	—
168	C₂×C₈	C₄²×C₆₁	1	2	yes	—
169	C₂×C₈	C₂²⋊C₄×C₆₁	1	2	yes	—
170	C₂×C₈	C₂²⋊C₄×C₆₁	1	8	yes	—
171	C₂×C₈	C₄⋊C₄×C₆₁	1	2	yes	—
172	C₂×C₈	C₄⋊C₄×C₆₁	1	8	yes	—
173	C₂×C₈	C₂×C₈×C₆₁	1	4	yes	—
174	C₂×C₈	C₂²×C₄×C₆₁	1	2	yes	—
175	C₂×C₈	C₂×D₄×C₆₁	1	1	yes	characteristic
176	C₂×C₈	C₂×D₄×C₆₁	1	2	yes	—
177	C₂×C₈	C₂×Q₈×C₆₁	1	1	yes	characteristic
178	C₄²	—	2	4	no	—
179	C₄²	—	2	2	no	—
180	C₄²	—	2	2	no	—
181	C₁₆	—	2	8	no	—
182	C₂³	C₄×C₈×C₆₁	1	2	yes	—
183	C₂³	C₂²⋊C₈×C₆₁	1	2	yes	—
184	C₂³	C₄⋊C₈×C₆₁	1	2	yes	—
185	C₂³	C₂×C₁₆×C₆₁	1	4	yes	—
186	C₂³	M₃₂×C₆₁	1	4	yes	—
187	C₂³	C₂²×C₈×C₆₁	1	1	yes	characteristic
188	C₂×C₄	—	2	16	no	—
189	C₂×C₄	—	2	4	no	—
190	C₂×C₄	—	2	4	no	—
191	C₂×C₄	#95×C₆₁	1	4	yes	—
192	C₂×C₄	C₄×C₈×C₆₁	1	2	yes	—
193	C₂×C₄	#97×C₆₁	1	4	yes	—
194	C₂×C₄	C₂²⋊C₈×C₆₁	1	2	yes	—
195	C₂×C₄	C₂²⋊C₈×C₆₁	1	4	yes	—
196	C₂×C₄	C₄⋊C₈×C₆₁	1	2	yes	—
197	C₂×C₄	C₄⋊C₈×C₆₁	1	4	yes	—
198	C₂×C₄	C₂×C₁₆×C₆₁	1	8	yes	complemented
199	C₂×C₄	C₂×C₄²×C₆₁	1	1	yes	characteristic
200	C₂×C₄	C₂×C₂²⋊C₄×C₆₁	1	1	yes	characteristic
201	C₂×C₄	C₂×C₄⋊C₄×C₆₁	1	1	yes	characteristic
202	C₂×C₄	C₂²×C₈×C₆₁	1	1	yes	characteristic
203	C₂×C₄	C₂×M₁₆×C₆₁	1	1	yes	characteristic
204	C₂×C₄	C₂×M₁₆×C₆₁	1	1	yes	characteristic
205	C₈	—	2	4	no	—
206	C₈	#95×C₆₁	1	4	yes	—
207	C₈	C₂×C₂²⋊C₄×C₆₁	1	2	yes	—
208	C₈	C₂×C₄⋊C₄×C₆₁	1	2	yes	—
209	C₂²	C₆₁×#336	1	4	yes	—
210	C₂²	C₄×C₁₆×C₆₁	1	4	yes	—
211	C₂²	C₆₁×#346	1	4	yes	—
212	C₂²	C₆₁×C₂²⋊C₁₆	1	8	yes	—
213	C₂²	C₆₁×C₄⋊C₁₆	1	8	yes	—
214	C₂²	C₂×C₄×C₈×C₆₁	1	1	yes	characteristic
215	C₂²	C₂×C₂²⋊C₈×C₆₁	1	1	yes	characteristic
216	C₂²	C₂×C₄⋊C₈×C₆₁	1	1	yes	characteristic
217	C₂²	C₂²×C₁₆×C₆₁	1	2	yes	—
218	C₂²	C₂×M₃₂×C₆₁	1	2	yes	—
219	C₄	—	2	8	no	—
220	C₄	C₆₁×#336	1	4	yes	—
221	C₄	C₂×#95×C₆₁	1	1	yes	characteristic
222	C₄	C₂×#97×C₆₁	1	1	yes	characteristic
223	C₄	C₂×C₂²⋊C₈×C₆₁	1	1	yes	characteristic
224	C₄	C₂×C₄⋊C₈×C₆₁	1	1	yes	characteristic
225	C₂	C₆₁×#1374	1	8	yes	direct product factor
226	C₂	C₂×C₆₁×#336	1	1	yes	characteristic
227	C₂	C₂×C₄×C₁₆×C₆₁	1	1	yes	characteristic
228	C₂	C₂×C₆₁×#346	1	1	yes	characteristic
229	C₂	C₂×C₆₁×C₂²⋊C₁₆	1	2	yes	—
230	C₂	C₂×C₆₁×C₄⋊C₁₆	1	2	yes	—

	subgroup	G/H
center	C₂³×C₈×C₆₁	C₂²
commutator	C₂	C₂×C₄×C₁₆×C₆₁
Frattini, cosocle	C₂²×C₈	C₂³×C₆₁
socle, prime radical	C₂⁴×C₆₁	C₂×C₈
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₈×C₆₁ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₆₁×#1374	C₂	direct (N × H)
9. C₂²×C₁₆×C₆₁	C₄	semidirect
33. C₂×C₁₆×C₆₁	C₂×C₄	semidirect
39. C₂²×C₄×C₆₁	C₁₆	semidirect
82. C₂×C₄×C₆₁	C₂×C₁₆	semidirect
93. C₂×#1374	C₆₁	direct (N × H)
112. #1374	C₂×C₆₁	direct (N × H)
113. C₂×C₆₁	#1374	direct (N × H)
124. C₂²×C₁₆	C₄×C₆₁	semidirect
129. C₆₁	C₂×#1374	direct (N × H)
149. C₂×C₁₆	C₂×C₄×C₆₁	semidirect
155. C₂²×C₄	C₁₆×C₆₁	semidirect
198. C₂×C₄	C₂×C₁₆×C₆₁	semidirect
225. C₂	C₆₁×#1374	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₁₆×C₆₁	2
C₂²×C₄×C₈×C₆₁	2
C₆₁×#1374	2
C₂×#1374	61

Subgroups — C2×C61×#1374 #29515976

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₆₁×#1374 #29515976