Subgroups — C₂×C₆₁×#1831 #29524183

Order 15616.

1246 conjugacy classes of subgroups.

The subgroup lattice has 246 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄⋊C₈×C₆₁	C₂	1	2	yes	maximal
2	C₂×C₄³×C₆₁	C₂	1	1	yes	maximal, characteristic
3	C₆₁×#1831	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₆₁×#336	C₂	1	4	yes	maximal
5	C₂³×C₈×C₆₁	C₂²	1	4	yes	—
6	C₂²×C₄²×C₆₁	C₄	1	2	yes	—
7	C₂²×C₄²×C₆₁	C₄	1	1	yes	characteristic
8	C₂²×C₄²×C₆₁	C₄	1	1	yes	characteristic
9	C₂²×C₄²×C₆₁	C₂²	1	2	yes	—
10	C₂²×C₄²×C₆₁	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊C₈×C₆₁	—	2	8	no	—
12	C₂×C₄⋊C₈×C₆₁	C₂²	1	8	yes	—
13	C₄³×C₆₁	C₄	1	4	yes	—
14	C₄³×C₆₁	C₂²	1	4	yes	—
15	C₆₁×#336	C₂²	1	16	yes	—
16	C₂³×C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
17	C₂³×C₄×C₆₁	C₂×C₄	1	2	yes	—
18	C₂³×C₄×C₆₁	C₂×C₄	1	2	yes	—
19	C₂³×C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
20	C₂³×C₄×C₆₁	C₂³	1	1	yes	characteristic
21	C₂²×C₈×C₆₁	—	2	8	no	—
22	C₂²×C₈×C₆₁	—	2	16	no	—
23	C₂²×C₈×C₆₁	C₂³	1	8	yes	—
24	C₂×C₄²×C₆₁	—	2	8	no	—
25	C₂×C₄²×C₆₁	—	2	4	no	—
26	C₂×C₄²×C₆₁	—	2	8	no	—
27	C₂×C₄²×C₆₁	C₈	1	8	yes	complemented
28	C₂×C₄²×C₆₁	C₂×C₄	1	4	yes	—
29	C₂×C₄²×C₆₁	C₂×C₄	1	8	yes	—
30	C₂×C₄²×C₆₁	C₂×C₄	1	4	yes	—
31	C₂×C₄²×C₆₁	C₂×C₄	1	2	yes	—
32	C₂×C₄²×C₆₁	C₂×C₄	1	4	yes	—
33	C₂×C₄²×C₆₁	D₄	1	4	yes	—
34	C₂×C₄²×C₆₁	Q₈	1	4	yes	—
35	C₂×C₄²×C₆₁	C₂³	1	4	yes	—
36	C₂×C₄²×C₆₁	C₂³	1	2	yes	—
37	C₄⋊C₈×C₆₁	—	2	16	no	—
38	C₂⁴×C₆₁	C₂²×C₄	1	1	yes	characteristic
39	C₂²×C₄×C₆₁	—	2	4	no	—
40	C₂²×C₄×C₆₁	—	2	4	no	—
41	C₂²×C₄×C₆₁	—	2	4	no	—
42	C₂²×C₄×C₆₁	—	2	4	no	—
43	C₂²×C₄×C₆₁	—	2	4	no	—
44	C₂²×C₄×C₆₁	C₄⋊C₄	1	4	yes	—
45	C₂²×C₄×C₆₁	C₄⋊C₄	1	4	yes	—
46	C₂²×C₄×C₆₁	C₂×C₈	1	8	yes	—
47	C₂²×C₄×C₆₁	C₂×C₈	1	4	yes	—
48	C₂²×C₄×C₆₁	M₁₆	1	8	yes	—
49	C₂²×C₄×C₆₁	M₁₆	1	4	yes	—
50	C₂²×C₄×C₆₁	C₂²×C₄	1	2	yes	—
51	C₂²×C₄×C₆₁	C₂²×C₄	1	4	yes	—
52	C₂²×C₄×C₆₁	C₂²×C₄	1	4	yes	—
53	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
54	C₂²×C₄×C₆₁	C₂²×C₄	1	2	yes	—
55	C₂²×C₄×C₆₁	C₂×D₄	1	2	yes	—
56	C₂²×C₄×C₆₁	C₂×Q₈	1	2	yes	—
57	C₂²×C₄×C₆₁	#41	1	8	yes	—
58	C₂²×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
59	C₂×C₈×C₆₁	—	2	8	no	—
60	C₂×C₈×C₆₁	—	2	16	no	—
61	C₂×C₈×C₆₁	—	4	16	no	—
62	C₄²×C₆₁	—	2	8	no	—
63	C₄²×C₆₁	—	2	16	no	—
64	C₄²×C₆₁	—	2	8	no	—
65	C₄²×C₆₁	—	2	8	no	—
66	C₄²×C₆₁	C₄⋊C₄	1	8	yes	—
67	C₄²×C₆₁	C₂×C₈	1	16	yes	complemented
68	C₄²×C₆₁	C₂×D₄	1	4	yes	—
69	C₄²×C₆₁	C₂×Q₈	1	4	yes	—
70	C₂³×C₆₁	C₂×C₄⋊C₄	1	2	yes	—
71	C₂³×C₆₁	#117	1	4	yes	—
72	C₂³×C₆₁	C₂²×C₈	1	2	yes	—
73	C₂³×C₆₁	C₂×M₁₆	1	2	yes	—
74	C₂³×C₆₁	C₂×M₁₆	1	4	yes	—
75	C₂³×C₆₁	C₂³×C₄	1	1	yes	characteristic
76	C₂×C₄×C₆₁	—	2	4	no	—
77	C₂×C₄×C₆₁	—	2	4	no	—
78	C₂×C₄×C₆₁	—	2	4	no	—
79	C₂×C₄×C₆₁	—	2	4	no	—
80	C₂×C₄×C₆₁	—	2	4	no	—
81	C₂×C₄×C₆₁	—	2	8	no	—
82	C₂×C₄×C₆₁	—	2	16	no	—
83	C₂×C₄×C₆₁	—	2	16	no	—
84	C₂×C₄×C₆₁	C₄⋊C₈	1	16	yes	—
85	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	4	yes	—
86	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	2	yes	—
87	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	4	yes	—
88	C₂×C₄×C₆₁	#117	1	4	yes	—
89	C₂×C₄×C₆₁	#124	1	8	yes	—
90	C₂×C₄×C₆₁	#125	1	8	yes	—
91	C₂×C₄×C₆₁	C₂²×C₈	1	8	yes	—
92	C₂×C₄×C₆₁	C₂²×C₈	1	4	yes	—
93	C₂×C₄×C₆₁	C₂×M₁₆	1	8	yes	—
94	C₂×C₄×C₆₁	C₂×M₁₆	1	4	yes	—
95	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
96	C₂×C₄×C₆₁	C₂²×Q₈	1	1	yes	characteristic
97	C₂×C₄×C₆₁	C₂×#41	1	4	yes	—
98	C₈×C₆₁	—	4	8	no	—
99	C₂×#1831	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
100	C₂²×C₆₁	#382	1	4	yes	—
101	C₂²×C₆₁	C₂×C₄⋊C₈	1	4	yes	—
102	C₂²×C₆₁	#423	1	4	yes	—
103	C₂²×C₆₁	#431	1	8	yes	—
104	C₂²×C₆₁	#432	1	8	yes	—
105	C₂²×C₆₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
106	C₂²×C₆₁	C₂×#117	1	2	yes	—
107	C₂²×C₆₁	C₂³×C₈	1	1	yes	characteristic
108	C₂²×C₆₁	C₂²×M₁₆	1	1	yes	characteristic
109	C₂²×C₆₁	C₂²×M₁₆	1	2	yes	—
110	C₄×C₆₁	—	2	4	no	—
111	C₄×C₆₁	—	2	8	no	—
112	C₄×C₆₁	—	2	8	no	—
113	C₄×C₆₁	#382	1	4	yes	—
114	C₄×C₆₁	C₂×C₄⋊C₈	1	8	yes	—
115	C₄×C₆₁	C₂×#124	1	2	yes	—
116	C₄×C₆₁	C₂×#125	1	2	yes	—
117	C₂²×C₄⋊C₈	C₂×C₆₁	1	2	yes	—
118	C₂×C₄³	C₂×C₆₁	1	1	yes	characteristic
119	#1831	C₂×C₆₁	1	8	yes	direct product factor
120	C₂×#336	C₂×C₆₁	1	4	yes	—
121	C₂×C₆₁	#1831	1	8	yes	direct product factor
122	C₂×C₆₁	C₂×#382	1	1	yes	characteristic
123	C₂×C₆₁	C₂²×C₄⋊C₈	1	1	yes	characteristic
124	C₂×C₆₁	C₂×#423	1	1	yes	characteristic
125	C₂×C₆₁	C₂×#431	1	2	yes	—
126	C₂×C₆₁	C₂×#432	1	2	yes	—
127	C₂³×C₈	C₂²×C₆₁	1	4	yes	—
128	C₂²×C₄²	C₄×C₆₁	1	2	yes	—
129	C₂²×C₄²	C₄×C₆₁	1	1	yes	characteristic
130	C₂²×C₄²	C₄×C₆₁	1	1	yes	characteristic
131	C₂²×C₄²	C₂²×C₆₁	1	2	yes	—
132	C₂²×C₄²	C₂²×C₆₁	1	1	yes	characteristic
133	C₂×C₄⋊C₈	—	2	8	no	—
134	C₂×C₄⋊C₈	C₂²×C₆₁	1	8	yes	—
135	C₄³	C₄×C₆₁	1	4	yes	—
136	C₄³	C₂²×C₆₁	1	4	yes	—
137	#336	C₂²×C₆₁	1	16	yes	—
138	C₆₁	C₂×#1831	1	1	yes	characteristic, direct product factor, Sylow
139	C₂³×C₄	C₂×C₄×C₆₁	1	1	yes	characteristic
140	C₂³×C₄	C₂×C₄×C₆₁	1	2	yes	—
141	C₂³×C₄	C₂×C₄×C₆₁	1	2	yes	—
142	C₂³×C₄	C₂×C₄×C₆₁	1	1	yes	characteristic
143	C₂³×C₄	C₂³×C₆₁	1	1	yes	characteristic
144	C₂²×C₈	—	2	8	no	—
145	C₂²×C₈	—	2	16	no	—
146	C₂²×C₈	C₂³×C₆₁	1	8	yes	—
147	C₂×C₄²	—	2	8	no	—
148	C₂×C₄²	—	2	4	no	—
149	C₂×C₄²	—	2	8	no	—
150	C₂×C₄²	C₈×C₆₁	1	8	yes	complemented
151	C₂×C₄²	C₂×C₄×C₆₁	1	4	yes	—
152	C₂×C₄²	C₂×C₄×C₆₁	1	8	yes	—
153	C₂×C₄²	C₂×C₄×C₆₁	1	4	yes	—
154	C₂×C₄²	C₂×C₄×C₆₁	1	2	yes	—
155	C₂×C₄²	C₂×C₄×C₆₁	1	4	yes	—
156	C₂×C₄²	D₄×C₆₁	1	4	yes	—
157	C₂×C₄²	Q₈×C₆₁	1	4	yes	—
158	C₂×C₄²	C₂³×C₆₁	1	4	yes	—
159	C₂×C₄²	C₂³×C₆₁	1	2	yes	—
160	C₄⋊C₈	—	2	16	no	—
161	C₂⁴	C₂²×C₄×C₆₁	1	1	yes	characteristic
162	C₂²×C₄	—	2	4	no	—
163	C₂²×C₄	—	2	4	no	—
164	C₂²×C₄	—	2	4	no	—
165	C₂²×C₄	—	2	4	no	—
166	C₂²×C₄	—	2	4	no	—
167	C₂²×C₄	C₄⋊C₄×C₆₁	1	4	yes	—
168	C₂²×C₄	C₄⋊C₄×C₆₁	1	4	yes	—
169	C₂²×C₄	C₂×C₈×C₆₁	1	8	yes	—
170	C₂²×C₄	C₂×C₈×C₆₁	1	4	yes	—
171	C₂²×C₄	M₁₆×C₆₁	1	8	yes	—
172	C₂²×C₄	M₁₆×C₆₁	1	4	yes	—
173	C₂²×C₄	C₂²×C₄×C₆₁	1	2	yes	—
174	C₂²×C₄	C₂²×C₄×C₆₁	1	4	yes	—
175	C₂²×C₄	C₂²×C₄×C₆₁	1	4	yes	—
176	C₂²×C₄	C₂²×C₄×C₆₁	1	1	yes	characteristic
177	C₂²×C₄	C₂²×C₄×C₆₁	1	2	yes	—
178	C₂²×C₄	C₂×D₄×C₆₁	1	2	yes	—
179	C₂²×C₄	C₂×Q₈×C₆₁	1	2	yes	—
180	C₂²×C₄	#41×C₆₁	1	8	yes	—
181	C₂²×C₄	C₂⁴×C₆₁	1	1	yes	characteristic
182	C₂×C₈	—	2	8	no	—
183	C₂×C₈	—	2	16	no	—
184	C₂×C₈	—	4	16	no	—
185	C₄²	—	2	8	no	—
186	C₄²	—	2	16	no	—
187	C₄²	—	2	8	no	—
188	C₄²	—	2	8	no	—
189	C₄²	C₄⋊C₄×C₆₁	1	8	yes	—
190	C₄²	C₂×C₈×C₆₁	1	16	yes	complemented
191	C₄²	C₂×D₄×C₆₁	1	4	yes	—
192	C₄²	C₂×Q₈×C₆₁	1	4	yes	—
193	C₂³	C₂×C₄⋊C₄×C₆₁	1	2	yes	—
194	C₂³	#117×C₆₁	1	4	yes	—
195	C₂³	C₂²×C₈×C₆₁	1	2	yes	—
196	C₂³	C₂×M₁₆×C₆₁	1	2	yes	—
197	C₂³	C₂×M₁₆×C₆₁	1	4	yes	—
198	C₂³	C₂³×C₄×C₆₁	1	1	yes	characteristic
199	C₂×C₄	—	2	4	no	—
200	C₂×C₄	—	2	4	no	—
201	C₂×C₄	—	2	4	no	—
202	C₂×C₄	—	2	8	no	—
203	C₂×C₄	—	2	16	no	—
204	C₂×C₄	—	2	16	no	—
205	C₂×C₄	—	2	4	no	—
206	C₂×C₄	—	2	4	no	—
207	C₂×C₄	C₄⋊C₈×C₆₁	1	16	yes	—
208	C₂×C₄	C₂×C₄⋊C₄×C₆₁	1	4	yes	—
209	C₂×C₄	C₂×C₄⋊C₄×C₆₁	1	2	yes	—
210	C₂×C₄	C₂×C₄⋊C₄×C₆₁	1	4	yes	—
211	C₂×C₄	#117×C₆₁	1	4	yes	—
212	C₂×C₄	#124×C₆₁	1	8	yes	—
213	C₂×C₄	#125×C₆₁	1	8	yes	—
214	C₂×C₄	C₂²×C₈×C₆₁	1	8	yes	—
215	C₂×C₄	C₂²×C₈×C₆₁	1	4	yes	—
216	C₂×C₄	C₂×M₁₆×C₆₁	1	8	yes	—
217	C₂×C₄	C₂×M₁₆×C₆₁	1	4	yes	—
218	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
219	C₂×C₄	C₂²×Q₈×C₆₁	1	1	yes	characteristic
220	C₂×C₄	C₂×#41×C₆₁	1	4	yes	—
221	C₈	—	4	8	no	—
222	C₂²	C₆₁×#382	1	4	yes	—
223	C₂²	C₂×C₄⋊C₈×C₆₁	1	4	yes	—
224	C₂²	C₆₁×#423	1	4	yes	—
225	C₂²	C₆₁×#431	1	8	yes	—
226	C₂²	C₆₁×#432	1	8	yes	—
227	C₂²	C₂²×C₄⋊C₄×C₆₁	1	1	yes	characteristic
228	C₂²	C₂×#117×C₆₁	1	2	yes	—
229	C₂²	C₂³×C₈×C₆₁	1	1	yes	characteristic
230	C₂²	C₂²×M₁₆×C₆₁	1	1	yes	characteristic
231	C₂²	C₂²×M₁₆×C₆₁	1	2	yes	—
232	C₄	—	2	4	no	—
233	C₄	—	2	8	no	—
234	C₄	—	2	8	no	—
235	C₄	C₆₁×#382	1	4	yes	—
236	C₄	C₂×C₄⋊C₈×C₆₁	1	8	yes	—
237	C₄	C₂×#124×C₆₁	1	2	yes	—
238	C₄	C₂×#125×C₆₁	1	2	yes	—
239	C₂	C₆₁×#1831	1	8	yes	direct product factor
240	C₂	C₂×C₆₁×#382	1	1	yes	characteristic
241	C₂	C₂²×C₄⋊C₈×C₆₁	1	1	yes	characteristic
242	C₂	C₂×C₆₁×#423	1	1	yes	characteristic
243	C₂	C₂×C₆₁×#431	1	2	yes	—
244	C₂	C₂×C₆₁×#432	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
3. C₆₁×#1831	C₂	direct (N × H)
27. C₂×C₄²×C₆₁	C₈	semidirect
67. C₄²×C₆₁	C₂×C₈	semidirect
99. C₂×#1831	C₆₁	direct (N × H)
119. #1831	C₂×C₆₁	direct (N × H)
121. C₂×C₆₁	#1831	direct (N × H)
138. C₆₁	C₂×#1831	direct (N × H)
150. C₂×C₄²	C₈×C₆₁	semidirect
190. C₄²	C₂×C₈×C₆₁	semidirect
239. C₂	C₆₁×#1831	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×C₄⋊C₈×C₆₁	2
C₂×C₄³×C₆₁	2
C₆₁×#1831	2
C₂×C₆₁×#336	2
C₂×#1831	61

Subgroups — C2×C61×#1831 #29524183

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₆₁×#1831 #29524183