Subgroups — C₂²×C₆₁×#462 #29563840

Order 15616.

2790 conjugacy classes of subgroups.

The subgroup lattice has 274 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×Q₁₆×C₆₁	C₂	1	2	yes	maximal
2	C₂²×C₄⋊Q₈×C₆₁	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₄×Q₈×C₆₁	C₂	1	1	yes	maximal, characteristic
4	C₂×C₆₁×#462	C₂	1	24	yes	maximal, direct product factor
5	C₂²×C₄⋊C₈×C₆₁	C₂	1	1	yes	maximal, characteristic
6	C₂²×#103×C₆₁	C₂	1	2	yes	maximal
7	C₂³×Q₈×C₆₁	C₂²	1	2	yes	—
8	C₂³×Q₈×C₆₁	C₂²	1	1	yes	characteristic
9	C₂²×Q₁₆×C₆₁	—	2	16	no	—
10	C₂²×Q₁₆×C₆₁	C₂²	1	24	yes	—
11	C₂³×C₈×C₆₁	C₂²	1	2	yes	—
12	C₂×C₄⋊Q₈×C₆₁	—	2	6	no	—
13	C₂×C₄⋊Q₈×C₆₁	C₂²	1	12	yes	—
14	C₂×C₄×Q₈×C₆₁	—	2	6	no	—
15	C₂×C₄×Q₈×C₆₁	C₂²	1	12	yes	—
16	C₂²×C₄⋊C₄×C₆₁	—	2	1	no	—
17	C₂²×C₄⋊C₄×C₆₁	—	2	1	no	—
18	C₂²×C₄⋊C₄×C₆₁	—	2	1	no	—
19	C₂²×C₄⋊C₄×C₆₁	C₂²	1	1	yes	characteristic
20	C₂²×C₄²×C₆₁	—	2	1	no	—
21	C₂²×C₄²×C₆₁	C₂²	1	1	yes	characteristic
22	C₆₁×#462	C₂²	1	64	yes	direct product factor
23	C₂×C₄⋊C₈×C₆₁	C₂²	1	12	yes	—
24	C₂×#103×C₆₁	C₂²	1	24	yes	—
25	C₂²×Q₈×C₆₁	—	2	3	no	—
26	C₂²×Q₈×C₆₁	—	2	6	no	—
27	C₂²×Q₈×C₆₁	—	2	4	no	—
28	C₂²×Q₈×C₆₁	—	2	8	no	—
29	C₂²×Q₈×C₆₁	—	2	8	no	—
30	C₂²×Q₈×C₆₁	D₄	1	8	yes	—
31	C₂²×Q₈×C₆₁	C₂³	1	12	yes	—
32	C₂²×Q₈×C₆₁	C₂³	1	6	yes	—
33	C₂³×C₄×C₆₁	—	2	1	no	—
34	C₂³×C₄×C₆₁	—	2	2	no	—
35	C₂³×C₄×C₆₁	—	2	1	no	—
36	C₂³×C₄×C₆₁	D₄	1	1	yes	characteristic
37	C₂³×C₄×C₆₁	D₄	1	1	yes	characteristic
38	C₂³×C₄×C₆₁	C₂³	1	1	yes	characteristic
39	C₂×Q₁₆×C₆₁	—	2	96	no	—
40	C₂×Q₁₆×C₆₁	C₂³	1	32	yes	—
41	C₂²×C₈×C₆₁	—	2	8	no	—
42	C₂²×C₈×C₆₁	C₂³	1	12	yes	—
43	C₄⋊Q₈×C₆₁	—	2	24	no	—
44	C₄⋊Q₈×C₆₁	C₂³	1	16	yes	—
45	C₄×Q₈×C₆₁	—	2	24	no	—
46	C₄×Q₈×C₆₁	C₂³	1	16	yes	—
47	C₂×C₄⋊C₄×C₆₁	—	2	12	no	—
48	C₂×C₄⋊C₄×C₆₁	—	2	12	no	—
49	C₂×C₄⋊C₄×C₆₁	—	2	12	no	—
50	C₂×C₄⋊C₄×C₆₁	—	2	3	no	—
51	C₂×C₄⋊C₄×C₆₁	C₂³	1	6	yes	—
52	C₂×C₄²×C₆₁	—	2	12	no	—
53	C₂×C₄²×C₆₁	D₄	1	6	yes	—
54	C₂×C₄²×C₆₁	C₂³	1	6	yes	—
55	C₄⋊C₈×C₆₁	C₂³	1	16	yes	—
56	#103×C₆₁	C₂³	1	32	yes	—
57	C₂⁴×C₆₁	C₂×D₄	1	1	yes	characteristic
58	C₂×Q₈×C₆₁	—	2	24	no	—
59	C₂×Q₈×C₆₁	—	2	6	no	—
60	C₂×Q₈×C₆₁	—	2	12	no	—
61	C₂×Q₈×C₆₁	—	2	12	no	—
62	C₂×Q₈×C₆₁	—	2	24	no	—
63	C₂×Q₈×C₆₁	—	2	24	no	—
64	C₂×Q₈×C₆₁	—	4	24	no	—
65	C₂×Q₈×C₆₁	C₂×D₄	1	24	yes	—
66	C₂×Q₈×C₆₁	C₂⁴	1	8	yes	—
67	C₂×Q₈×C₆₁	C₂⁴	1	4	yes	—
68	C₂²×C₄×C₆₁	—	2	8	no	—
69	C₂²×C₄×C₆₁	—	2	4	no	—
70	C₂²×C₄×C₆₁	—	2	6	no	—
71	C₂²×C₄×C₆₁	—	2	6	no	—
72	C₂²×C₄×C₆₁	—	2	12	no	—
73	C₂²×C₄×C₆₁	—	4	8	no	—
74	C₂²×C₄×C₆₁	—	4	4	no	—
75	C₂²×C₄×C₆₁	Q₁₆	1	8	yes	complemented
76	C₂²×C₄×C₆₁	C₂×D₄	1	6	yes	—
77	C₂²×C₄×C₆₁	C₂×D₄	1	3	yes	—
78	C₂²×C₄×C₆₁	C₂×D₄	1	4	yes	—
79	C₂²×C₄×C₆₁	C₂×D₄	1	6	yes	—
80	C₂²×C₄×C₆₁	#41	1	4	yes	—
81	C₂²×C₄×C₆₁	C₂⁴	1	3	yes	—
82	Q₁₆×C₆₁	—	2	64	no	—
83	C₂×C₈×C₆₁	—	2	24	no	—
84	C₂×C₈×C₆₁	C₂⁴	1	8	yes	—
85	C₄⋊C₄×C₆₁	—	2	16	no	—
86	C₄⋊C₄×C₆₁	—	2	16	no	—
87	C₄⋊C₄×C₆₁	—	2	16	no	—
88	C₄⋊C₄×C₆₁	—	2	6	no	—
89	C₄⋊C₄×C₆₁	C₂⁴	1	4	yes	—
90	C₄²×C₆₁	—	2	16	no	—
91	C₄²×C₆₁	C₂×D₄	1	12	yes	—
92	C₄²×C₆₁	C₂⁴	1	4	yes	—
93	C₂³×C₆₁	#121	1	4	yes	—
94	C₂³×C₆₁	C₂×Q₁₆	1	4	yes	—
95	C₂³×C₆₁	#137	1	4	yes	—
96	C₂³×C₆₁	C₂²×D₄	1	3	yes	—
97	Q₈×C₆₁	—	2	24	no	—
98	Q₈×C₆₁	—	2	4	no	—
99	Q₈×C₆₁	—	2	8	no	—
100	Q₈×C₆₁	—	2	8	no	—
101	Q₈×C₆₁	—	4	24	no	—
102	Q₈×C₆₁	C₂²×D₄	1	8	yes	—
103	C₂×C₄×C₆₁	—	2	24	no	—
104	C₂×C₄×C₆₁	—	2	12	no	—
105	C₂×C₄×C₆₁	—	2	4	no	—
106	C₂×C₄×C₆₁	—	2	4	no	—
107	C₂×C₄×C₆₁	—	2	8	no	—
108	C₂×C₄×C₆₁	—	4	24	no	—
109	C₂×C₄×C₆₁	—	4	12	no	—
110	C₂×C₄×C₆₁	#121	1	12	yes	—
111	C₂×C₄×C₆₁	C₂×Q₁₆	1	24	yes	complemented
112	C₂×C₄×C₆₁	C₂²×D₄	1	4	yes	—
113	C₂×C₄×C₆₁	C₂²×D₄	1	3	yes	—
114	C₂×C₄×C₆₁	C₂²×D₄	1	6	yes	—
115	C₂×C₄×C₆₁	C₂²×D₄	1	4	yes	—
116	C₂×C₄×C₆₁	C₂×#41	1	6	yes	—
117	C₂×C₄×C₆₁	C₂⁵	1	1	yes	characteristic
118	C₈×C₆₁	—	2	8	no	—
119	C₂²×#462	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
120	C₂²×C₆₁	#462	1	16	yes	direct product factor
121	C₂²×C₆₁	C₂×#121	1	6	yes	—
122	C₂²×C₆₁	C₂²×Q₁₆	1	6	yes	—
123	C₂²×C₆₁	C₂×#137	1	6	yes	—
124	C₂²×C₆₁	C₂³×D₄	1	1	yes	characteristic
125	C₄×C₆₁	—	2	8	no	—
126	C₄×C₆₁	—	2	4	no	—
127	C₄×C₆₁	—	4	8	no	—
128	C₄×C₆₁	—	4	4	no	—
129	C₄×C₆₁	C₂×#121	1	6	yes	—
130	C₄×C₆₁	C₂²×Q₁₆	1	8	yes	complemented
131	C₄×C₆₁	C₂³×D₄	1	1	yes	characteristic
132	C₄×C₆₁	C₂²×#41	1	1	yes	characteristic
133	C₂³×Q₁₆	C₂×C₆₁	1	2	yes	—
134	C₂²×C₄⋊Q₈	C₂×C₆₁	1	1	yes	characteristic
135	C₂²×C₄×Q₈	C₂×C₆₁	1	1	yes	characteristic
136	C₂×#462	C₂×C₆₁	1	24	yes	direct product factor
137	C₂²×C₄⋊C₈	C₂×C₆₁	1	1	yes	characteristic
138	C₂²×#103	C₂×C₆₁	1	2	yes	—
139	C₂×C₆₁	C₂×#462	1	12	yes	direct product factor
140	C₂×C₆₁	C₂²×#121	1	1	yes	characteristic
141	C₂×C₆₁	C₂³×Q₁₆	1	1	yes	characteristic
142	C₂×C₆₁	C₂²×#137	1	1	yes	characteristic
143	C₂³×Q₈	C₂²×C₆₁	1	2	yes	—
144	C₂³×Q₈	C₂²×C₆₁	1	1	yes	characteristic
145	C₂²×Q₁₆	—	2	16	no	—
146	C₂²×Q₁₆	C₂²×C₆₁	1	24	yes	—
147	C₂³×C₈	C₂²×C₆₁	1	2	yes	—
148	C₂×C₄⋊Q₈	—	2	6	no	—
149	C₂×C₄⋊Q₈	C₂²×C₆₁	1	12	yes	—
150	C₂×C₄×Q₈	—	2	6	no	—
151	C₂×C₄×Q₈	C₂²×C₆₁	1	12	yes	—
152	C₂²×C₄⋊C₄	—	2	1	no	—
153	C₂²×C₄⋊C₄	—	2	1	no	—
154	C₂²×C₄⋊C₄	—	2	1	no	—
155	C₂²×C₄⋊C₄	C₂²×C₆₁	1	1	yes	characteristic
156	C₂²×C₄²	—	2	1	no	—
157	C₂²×C₄²	C₂²×C₆₁	1	1	yes	characteristic
158	#462	C₂²×C₆₁	1	64	yes	direct product factor
159	C₂×C₄⋊C₈	C₂²×C₆₁	1	12	yes	—
160	C₂×#103	C₂²×C₆₁	1	24	yes	—
161	C₆₁	C₂²×#462	1	1	yes	characteristic, direct product factor, Sylow
162	C₂²×Q₈	—	2	3	no	—
163	C₂²×Q₈	—	2	6	no	—
164	C₂²×Q₈	—	2	4	no	—
165	C₂²×Q₈	—	2	8	no	—
166	C₂²×Q₈	—	2	8	no	—
167	C₂²×Q₈	D₄×C₆₁	1	8	yes	—
168	C₂²×Q₈	C₂³×C₆₁	1	12	yes	—
169	C₂²×Q₈	C₂³×C₆₁	1	6	yes	—
170	C₂³×C₄	—	2	1	no	—
171	C₂³×C₄	—	2	2	no	—
172	C₂³×C₄	—	2	1	no	—
173	C₂³×C₄	D₄×C₆₁	1	1	yes	characteristic
174	C₂³×C₄	D₄×C₆₁	1	1	yes	characteristic
175	C₂³×C₄	C₂³×C₆₁	1	1	yes	characteristic
176	C₂×Q₁₆	—	2	96	no	—
177	C₂×Q₁₆	C₂³×C₆₁	1	32	yes	—
178	C₂²×C₈	—	2	8	no	—
179	C₂²×C₈	C₂³×C₆₁	1	12	yes	—
180	C₄⋊Q₈	—	2	24	no	—
181	C₄⋊Q₈	C₂³×C₆₁	1	16	yes	—
182	C₄×Q₈	—	2	24	no	—
183	C₄×Q₈	C₂³×C₆₁	1	16	yes	—
184	C₂×C₄⋊C₄	—	2	12	no	—
185	C₂×C₄⋊C₄	—	2	12	no	—
186	C₂×C₄⋊C₄	—	2	12	no	—
187	C₂×C₄⋊C₄	—	2	3	no	—
188	C₂×C₄⋊C₄	C₂³×C₆₁	1	6	yes	—
189	C₂×C₄²	—	2	12	no	—
190	C₂×C₄²	D₄×C₆₁	1	6	yes	—
191	C₂×C₄²	C₂³×C₆₁	1	6	yes	—
192	C₄⋊C₈	C₂³×C₆₁	1	16	yes	—
193	#103	C₂³×C₆₁	1	32	yes	—
194	C₂⁴	C₂×D₄×C₆₁	1	1	yes	characteristic
195	C₂×Q₈	—	2	24	no	—
196	C₂×Q₈	—	2	6	no	—
197	C₂×Q₈	—	2	12	no	—
198	C₂×Q₈	—	2	12	no	—
199	C₂×Q₈	—	2	24	no	—
200	C₂×Q₈	—	2	24	no	—
201	C₂×Q₈	—	4	24	no	—
202	C₂×Q₈	C₂×D₄×C₆₁	1	24	yes	—
203	C₂×Q₈	C₂⁴×C₆₁	1	8	yes	—
204	C₂×Q₈	C₂⁴×C₆₁	1	4	yes	—
205	C₂²×C₄	—	2	8	no	—
206	C₂²×C₄	—	2	4	no	—
207	C₂²×C₄	—	2	6	no	—
208	C₂²×C₄	—	2	6	no	—
209	C₂²×C₄	—	2	12	no	—
210	C₂²×C₄	—	4	8	no	—
211	C₂²×C₄	—	4	4	no	—
212	C₂²×C₄	Q₁₆×C₆₁	1	8	yes	complemented
213	C₂²×C₄	C₂×D₄×C₆₁	1	6	yes	—
214	C₂²×C₄	C₂×D₄×C₆₁	1	3	yes	—
215	C₂²×C₄	C₂×D₄×C₆₁	1	4	yes	—
216	C₂²×C₄	C₂×D₄×C₆₁	1	6	yes	—
217	C₂²×C₄	#41×C₆₁	1	4	yes	—
218	C₂²×C₄	C₂⁴×C₆₁	1	3	yes	—
219	Q₁₆	—	2	64	no	—
220	C₂×C₈	—	2	24	no	—
221	C₂×C₈	C₂⁴×C₆₁	1	8	yes	—
222	C₄⋊C₄	—	2	16	no	—
223	C₄⋊C₄	—	2	16	no	—
224	C₄⋊C₄	—	2	16	no	—
225	C₄⋊C₄	—	2	6	no	—
226	C₄⋊C₄	C₂⁴×C₆₁	1	4	yes	—
227	C₄²	—	2	16	no	—
228	C₄²	C₂×D₄×C₆₁	1	12	yes	—
229	C₄²	C₂⁴×C₆₁	1	4	yes	—
230	C₂³	#121×C₆₁	1	4	yes	—
231	C₂³	C₂×Q₁₆×C₆₁	1	4	yes	—
232	C₂³	#137×C₆₁	1	4	yes	—
233	C₂³	C₂²×D₄×C₆₁	1	3	yes	—
234	Q₈	—	2	4	no	—
235	Q₈	—	2	24	no	—
236	Q₈	—	2	8	no	—
237	Q₈	—	2	8	no	—
238	Q₈	—	4	24	no	—
239	Q₈	C₂²×D₄×C₆₁	1	8	yes	—
240	C₂×C₄	—	2	12	no	—
241	C₂×C₄	—	2	4	no	—
242	C₂×C₄	—	2	24	no	—
243	C₂×C₄	—	2	4	no	—
244	C₂×C₄	—	2	8	no	—
245	C₂×C₄	—	4	12	no	—
246	C₂×C₄	—	4	24	no	—
247	C₂×C₄	#121×C₆₁	1	12	yes	—
248	C₂×C₄	C₂×Q₁₆×C₆₁	1	24	yes	complemented
249	C₂×C₄	C₂²×D₄×C₆₁	1	4	yes	—
250	C₂×C₄	C₂²×D₄×C₆₁	1	4	yes	—
251	C₂×C₄	C₂²×D₄×C₆₁	1	6	yes	—
252	C₂×C₄	C₂²×D₄×C₆₁	1	3	yes	—
253	C₂×C₄	C₂×#41×C₆₁	1	6	yes	—
254	C₂×C₄	C₂⁵×C₆₁	1	1	yes	characteristic
255	C₈	—	2	8	no	—
256	C₂²	C₆₁×#462	1	16	yes	direct product factor
257	C₂²	C₂×#121×C₆₁	1	6	yes	—
258	C₂²	C₂²×Q₁₆×C₆₁	1	6	yes	—
259	C₂²	C₂×#137×C₆₁	1	6	yes	—
260	C₂²	C₂³×D₄×C₆₁	1	1	yes	characteristic
261	C₄	—	2	4	no	—
262	C₄	—	2	8	no	—
263	C₄	—	4	4	no	—
264	C₄	—	4	8	no	—
265	C₄	C₂×#121×C₆₁	1	6	yes	—
266	C₄	C₂²×Q₁₆×C₆₁	1	8	yes	complemented
267	C₄	C₂³×D₄×C₆₁	1	1	yes	characteristic
268	C₄	C₂²×#41×C₆₁	1	1	yes	characteristic
269	C₂	C₂×C₆₁×#462	1	12	yes	direct product factor
270	C₂	C₂²×#121×C₆₁	1	1	yes	characteristic
271	C₂	C₂³×Q₁₆×C₆₁	1	1	yes	characteristic
272	C₂	C₂²×#137×C₆₁	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂⁴×C₆₁	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₆₁
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₆₁ ◁ C₂²×C₄²×C₆₁ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₂×C₆₁×#462	C₂	direct (N × H)
22. C₆₁×#462	C₂²	direct (N × H)
75. C₂²×C₄×C₆₁	Q₁₆	semidirect
111. C₂×C₄×C₆₁	C₂×Q₁₆	semidirect
119. C₂²×#462	C₆₁	direct (N × H)
120. C₂²×C₆₁	#462	direct (N × H)
130. C₄×C₆₁	C₂²×Q₁₆	semidirect
136. C₂×#462	C₂×C₆₁	direct (N × H)
139. C₂×C₆₁	C₂×#462	direct (N × H)
158. #462	C₂²×C₆₁	direct (N × H)
161. C₆₁	C₂²×#462	direct (N × H)
212. C₂²×C₄	Q₁₆×C₆₁	semidirect
248. C₂×C₄	C₂×Q₁₆×C₆₁	semidirect
256. C₂²	C₆₁×#462	direct (N × H)
266. C₄	C₂²×Q₁₆×C₆₁	semidirect
269. C₂	C₂×C₆₁×#462	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×Q₁₆×C₆₁	2
C₂²×C₄⋊Q₈×C₆₁	2
C₂²×C₄×Q₈×C₆₁	2
C₂×C₆₁×#462	2
C₂²×C₄⋊C₈×C₆₁	2
C₂²×#103×C₆₁	2
C₂²×#462	61

Subgroups — C22×C61×#462 #29563840

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₆₁×#462 #29563840