Subgroups — C₂²×C₆₁×#467 #29563869

Order 15616.

3006 conjugacy classes of subgroups.

The subgroup lattice has 272 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	1	yes	maximal, characteristic, complemented
2	C₂⁴×C₈×C₆₁	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₂²⋊Q₈×C₆₁	C₂	1	2	yes	maximal
4	C₂×C₆₁×#467	C₂	1	24	yes	maximal, direct product factor
5	C₂²×#107×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×#103×C₆₁	C₂	1	2	yes	maximal, complemented
7	C₂³×Q₈×C₆₁	C₂²	1	2	yes	—
8	C₂⁴×C₄×C₆₁	C₂²	1	1	yes	characteristic
9	C₂²×Q₁₆×C₆₁	—	2	8	no	—
10	C₂²×Q₁₆×C₆₁	C₂²	1	12	yes	complemented
11	C₂³×C₈×C₆₁	—	2	8	no	—
12	C₂³×C₈×C₆₁	C₂²	1	12	yes	—
13	C₂³×C₈×C₆₁	C₂²	1	1	yes	characteristic
14	C₂³×C₈×C₆₁	C₂²	1	1	yes	characteristic
15	C₂×C₂²⋊Q₈×C₆₁	—	2	12	no	—
16	C₂×C₂²⋊Q₈×C₆₁	C₂²	1	24	yes	—
17	C₂²×C₄⋊C₄×C₆₁	—	2	2	no	—
18	C₂²×C₄⋊C₄×C₆₁	C₂²	1	2	yes	—
19	C₂²×C₂²⋊C₄×C₆₁	—	2	2	no	—
20	C₆₁×#467	C₂²	1	64	yes	direct product factor
21	C₂×#107×C₆₁	C₂²	1	12	yes	complemented
22	C₂×#103×C₆₁	C₂²	1	24	yes	complemented
23	C₂⁵×C₆₁	D₄	1	1	yes	characteristic
24	C₂²×Q₈×C₆₁	—	2	6	no	—
25	C₂²×Q₈×C₆₁	—	2	8	no	—
26	C₂²×Q₈×C₆₁	—	2	8	no	—
27	C₂²×Q₈×C₆₁	C₂³	1	12	yes	—
28	C₂³×C₄×C₆₁	—	2	4	no	—
29	C₂³×C₄×C₆₁	—	2	4	no	—
30	C₂³×C₄×C₆₁	—	2	2	no	—
31	C₂³×C₄×C₆₁	—	2	2	no	—
32	C₂³×C₄×C₆₁	D₄	1	6	yes	—
33	C₂³×C₄×C₆₁	D₄	1	1	yes	characteristic
34	C₂³×C₄×C₆₁	C₂³	1	6	yes	—
35	C₂³×C₄×C₆₁	C₂³	1	1	yes	characteristic
36	C₂×Q₁₆×C₆₁	—	2	48	no	—
37	C₂×Q₁₆×C₆₁	C₂³	1	16	yes	complemented
38	C₂²×C₈×C₆₁	—	2	4	no	—
39	C₂²×C₈×C₆₁	—	2	48	no	—
40	C₂²×C₈×C₆₁	D₄	1	8	yes	—
41	C₂²×C₈×C₆₁	C₂³	1	6	yes	—
42	C₂²×C₈×C₆₁	C₂³	1	16	yes	—
43	C₂²×C₈×C₆₁	C₂³	1	6	yes	—
44	C₂²⋊Q₈×C₆₁	—	2	48	no	—
45	C₂²⋊Q₈×C₆₁	C₂³	1	32	yes	—
46	C₂×C₄⋊C₄×C₆₁	—	2	24	no	—
47	C₂×C₄⋊C₄×C₆₁	—	2	6	no	—
48	C₂×C₄⋊C₄×C₆₁	C₂³	1	12	yes	—
49	C₂×C₂²⋊C₄×C₆₁	—	2	24	no	—
50	#107×C₆₁	C₂³	1	16	yes	complemented
51	#103×C₆₁	C₂³	1	32	yes	complemented
52	C₂⁴×C₆₁	—	2	4	no	—
53	C₂⁴×C₆₁	—	2	4	no	—
54	C₂⁴×C₆₁	Q₁₆	1	8	yes	complemented
55	C₂⁴×C₆₁	C₂×D₄	1	6	yes	—
56	C₂⁴×C₆₁	C₂×D₄	1	1	yes	characteristic
57	C₂×Q₈×C₆₁	—	2	12	no	—
58	C₂×Q₈×C₆₁	—	2	24	no	—
59	C₂×Q₈×C₆₁	—	2	24	no	—
60	C₂×Q₈×C₆₁	—	4	24	no	—
61	C₂×Q₈×C₆₁	C₂⁴	1	8	yes	—
62	C₂²×C₄×C₆₁	—	2	4	no	—
63	C₂²×C₄×C₆₁	—	2	24	no	—
64	C₂²×C₄×C₆₁	—	2	12	no	—
65	C₂²×C₄×C₆₁	—	2	12	no	—
66	C₂²×C₄×C₆₁	—	2	12	no	—
67	C₂²×C₄×C₆₁	—	2	12	no	—
68	C₂²×C₄×C₆₁	—	4	8	no	—
69	C₂²×C₄×C₆₁	—	4	8	no	—
70	C₂²×C₄×C₆₁	C₂×D₄	1	6	yes	—
71	C₂²×C₄×C₆₁	C₂×D₄	1	12	yes	—
72	C₂²×C₄×C₆₁	C₂×D₄	1	3	yes	—
73	C₂²×C₄×C₆₁	C₂×D₄	1	4	yes	—
74	C₂²×C₄×C₆₁	#41	1	4	yes	—
75	C₂²×C₄×C₆₁	C₂⁴	1	4	yes	—
76	C₂²×C₄×C₆₁	C₂⁴	1	3	yes	—
77	Q₁₆×C₆₁	—	2	32	no	—
78	C₂×C₈×C₆₁	—	2	12	no	—
79	C₂×C₈×C₆₁	—	2	32	no	—
80	C₂×C₈×C₆₁	C₂×D₄	1	24	yes	—
81	C₂×C₈×C₆₁	C₂⁴	1	4	yes	—
82	C₂×C₈×C₆₁	C₂⁴	1	4	yes	—
83	C₄⋊C₄×C₆₁	—	2	32	no	—
84	C₄⋊C₄×C₆₁	—	2	12	no	—
85	C₄⋊C₄×C₆₁	C₂⁴	1	8	yes	—
86	C₂²⋊C₄×C₆₁	—	2	32	no	—
87	C₂³×C₆₁	—	2	12	no	—
88	C₂³×C₆₁	—	2	12	no	—
89	C₂³×C₆₁	—	2	32	no	—
90	C₂³×C₆₁	#121	1	4	yes	—
91	C₂³×C₆₁	C₂×Q₁₆	1	4	yes	—
92	C₂³×C₆₁	C₂×Q₁₆	1	24	yes	complemented
93	C₂³×C₆₁	#135	1	4	yes	—
94	C₂³×C₆₁	C₂²×D₄	1	4	yes	—
95	C₂³×C₆₁	C₂²×D₄	1	3	yes	—
96	Q₈×C₆₁	—	2	8	no	—
97	Q₈×C₆₁	—	2	8	no	—
98	Q₈×C₆₁	—	4	24	no	—
99	C₂×C₄×C₆₁	—	2	12	no	—
100	C₂×C₄×C₆₁	—	2	24	no	—
101	C₂×C₄×C₆₁	—	2	4	no	—
102	C₂×C₄×C₆₁	—	2	4	no	—
103	C₂×C₄×C₆₁	—	2	8	no	—
104	C₂×C₄×C₆₁	—	2	8	no	—
105	C₂×C₄×C₆₁	—	4	24	no	—
106	C₂×C₄×C₆₁	—	4	24	no	—
107	C₂×C₄×C₆₁	#121	1	12	yes	—
108	C₂×C₄×C₆₁	C₂²×D₄	1	4	yes	—
109	C₂×C₄×C₆₁	C₂²×D₄	1	3	yes	—
110	C₂×C₄×C₆₁	C₂²×D₄	1	6	yes	—
111	C₂×C₄×C₆₁	C₂×#41	1	6	yes	—
112	C₂×C₄×C₆₁	C₂⁵	1	1	yes	characteristic
113	C₈×C₆₁	—	2	4	no	—
114	C₈×C₆₁	C₂²×D₄	1	8	yes	—
115	C₂²×#467	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
116	C₂²×C₆₁	—	2	4	no	—
117	C₂²×C₆₁	—	2	48	no	—
118	C₂²×C₆₁	—	2	4	no	—
119	C₂²×C₆₁	#467	1	16	yes	direct product factor
120	C₂²×C₆₁	C₂×#121	1	6	yes	—
121	C₂²×C₆₁	C₂²×Q₁₆	1	6	yes	—
122	C₂²×C₆₁	C₂²×Q₁₆	1	8	yes	complemented
123	C₂²×C₆₁	C₂×#135	1	6	yes	—
124	C₂²×C₆₁	C₂³×D₄	1	1	yes	characteristic
125	C₄×C₆₁	—	2	4	no	—
126	C₄×C₆₁	—	4	8	no	—
127	C₄×C₆₁	—	4	8	no	—
128	C₄×C₆₁	C₂×#121	1	6	yes	—
129	C₄×C₆₁	C₂³×D₄	1	1	yes	characteristic
130	C₄×C₆₁	C₂²×#41	1	1	yes	characteristic
131	C₂³×Q₁₆	C₂×C₆₁	1	1	yes	characteristic, complemented
132	C₂⁴×C₈	C₂×C₆₁	1	1	yes	characteristic
133	C₂²×C₂²⋊Q₈	C₂×C₆₁	1	2	yes	—
134	C₂×#467	C₂×C₆₁	1	24	yes	direct product factor
135	C₂²×#107	C₂×C₆₁	1	1	yes	characteristic, complemented
136	C₂²×#103	C₂×C₆₁	1	2	yes	complemented
137	C₂×C₆₁	—	2	8	no	—
138	C₂×C₆₁	C₂×#467	1	12	yes	direct product factor
139	C₂×C₆₁	C₂²×#121	1	1	yes	characteristic
140	C₂×C₆₁	C₂³×Q₁₆	1	1	yes	characteristic
141	C₂×C₆₁	C₂²×#135	1	1	yes	characteristic
142	C₂³×Q₈	C₂²×C₆₁	1	2	yes	—
143	C₂⁴×C₄	C₂²×C₆₁	1	1	yes	characteristic
144	C₂²×Q₁₆	—	2	8	no	—
145	C₂²×Q₁₆	C₂²×C₆₁	1	12	yes	complemented
146	C₂³×C₈	—	2	8	no	—
147	C₂³×C₈	C₂²×C₆₁	1	1	yes	characteristic
148	C₂³×C₈	C₂²×C₆₁	1	12	yes	—
149	C₂³×C₈	C₂²×C₆₁	1	1	yes	characteristic
150	C₂×C₂²⋊Q₈	—	2	12	no	—
151	C₂×C₂²⋊Q₈	C₂²×C₆₁	1	24	yes	—
152	C₂²×C₄⋊C₄	—	2	2	no	—
153	C₂²×C₄⋊C₄	C₂²×C₆₁	1	2	yes	—
154	C₂²×C₂²⋊C₄	—	2	2	no	—
155	#467	C₂²×C₆₁	1	64	yes	direct product factor
156	C₂×#107	C₂²×C₆₁	1	12	yes	complemented
157	C₂×#103	C₂²×C₆₁	1	24	yes	complemented
158	C₆₁	C₂²×#467	1	1	yes	characteristic, direct product factor, Sylow
159	C₂⁵	D₄×C₆₁	1	1	yes	characteristic
160	C₂²×Q₈	—	2	6	no	—
161	C₂²×Q₈	—	2	8	no	—
162	C₂²×Q₈	—	2	8	no	—
163	C₂²×Q₈	C₂³×C₆₁	1	12	yes	—
164	C₂³×C₄	—	2	4	no	—
165	C₂³×C₄	—	2	4	no	—
166	C₂³×C₄	—	2	2	no	—
167	C₂³×C₄	—	2	2	no	—
168	C₂³×C₄	D₄×C₆₁	1	1	yes	characteristic
169	C₂³×C₄	D₄×C₆₁	1	6	yes	—
170	C₂³×C₄	C₂³×C₆₁	1	6	yes	—
171	C₂³×C₄	C₂³×C₆₁	1	1	yes	characteristic
172	C₂×Q₁₆	—	2	48	no	—
173	C₂×Q₁₆	C₂³×C₆₁	1	16	yes	complemented
174	C₂²×C₈	—	2	4	no	—
175	C₂²×C₈	—	2	48	no	—
176	C₂²×C₈	D₄×C₆₁	1	8	yes	—
177	C₂²×C₈	C₂³×C₆₁	1	6	yes	—
178	C₂²×C₈	C₂³×C₆₁	1	16	yes	—
179	C₂²×C₈	C₂³×C₆₁	1	6	yes	—
180	C₂²⋊Q₈	—	2	48	no	—
181	C₂²⋊Q₈	C₂³×C₆₁	1	32	yes	—
182	C₂×C₄⋊C₄	—	2	24	no	—
183	C₂×C₄⋊C₄	—	2	6	no	—
184	C₂×C₄⋊C₄	C₂³×C₆₁	1	12	yes	—
185	C₂×C₂²⋊C₄	—	2	24	no	—
186	#107	C₂³×C₆₁	1	16	yes	complemented
187	#103	C₂³×C₆₁	1	32	yes	complemented
188	C₂⁴	—	2	4	no	—
189	C₂⁴	—	2	4	no	—
190	C₂⁴	Q₁₆×C₆₁	1	8	yes	complemented
191	C₂⁴	C₂×D₄×C₆₁	1	6	yes	—
192	C₂⁴	C₂×D₄×C₆₁	1	1	yes	characteristic
193	C₂×Q₈	—	2	12	no	—
194	C₂×Q₈	—	2	24	no	—
195	C₂×Q₈	—	2	24	no	—
196	C₂×Q₈	—	4	24	no	—
197	C₂×Q₈	C₂⁴×C₆₁	1	8	yes	—
198	C₂²×C₄	—	2	4	no	—
199	C₂²×C₄	—	2	24	no	—
200	C₂²×C₄	—	2	12	no	—
201	C₂²×C₄	—	2	12	no	—
202	C₂²×C₄	—	2	12	no	—
203	C₂²×C₄	—	2	12	no	—
204	C₂²×C₄	—	4	8	no	—
205	C₂²×C₄	—	4	8	no	—
206	C₂²×C₄	C₂×D₄×C₆₁	1	6	yes	—
207	C₂²×C₄	C₂×D₄×C₆₁	1	12	yes	—
208	C₂²×C₄	C₂×D₄×C₆₁	1	3	yes	—
209	C₂²×C₄	C₂×D₄×C₆₁	1	4	yes	—
210	C₂²×C₄	#41×C₆₁	1	4	yes	—
211	C₂²×C₄	C₂⁴×C₆₁	1	4	yes	—
212	C₂²×C₄	C₂⁴×C₆₁	1	3	yes	—
213	Q₁₆	—	2	32	no	—
214	C₂×C₈	—	2	12	no	—
215	C₂×C₈	—	2	32	no	—
216	C₂×C₈	C₂×D₄×C₆₁	1	24	yes	—
217	C₂×C₈	C₂⁴×C₆₁	1	4	yes	—
218	C₂×C₈	C₂⁴×C₆₁	1	4	yes	—
219	C₄⋊C₄	—	2	32	no	—
220	C₄⋊C₄	—	2	12	no	—
221	C₄⋊C₄	C₂⁴×C₆₁	1	8	yes	—
222	C₂²⋊C₄	—	2	32	no	—
223	C₂³	—	2	12	no	—
224	C₂³	—	2	32	no	—
225	C₂³	—	2	12	no	—
226	C₂³	#121×C₆₁	1	4	yes	—
227	C₂³	C₂×Q₁₆×C₆₁	1	4	yes	—
228	C₂³	C₂×Q₁₆×C₆₁	1	24	yes	complemented
229	C₂³	#135×C₆₁	1	4	yes	—
230	C₂³	C₂²×D₄×C₆₁	1	4	yes	—
231	C₂³	C₂²×D₄×C₆₁	1	3	yes	—
232	Q₈	—	2	8	no	—
233	Q₈	—	2	8	no	—
234	Q₈	—	4	24	no	—
235	C₂×C₄	—	2	12	no	—
236	C₂×C₄	—	2	4	no	—
237	C₂×C₄	—	2	4	no	—
238	C₂×C₄	—	2	24	no	—
239	C₂×C₄	—	2	8	no	—
240	C₂×C₄	—	2	8	no	—
241	C₂×C₄	—	4	24	no	—
242	C₂×C₄	—	4	24	no	—
243	C₂×C₄	#121×C₆₁	1	12	yes	—
244	C₂×C₄	C₂²×D₄×C₆₁	1	4	yes	—
245	C₂×C₄	C₂²×D₄×C₆₁	1	6	yes	—
246	C₂×C₄	C₂²×D₄×C₆₁	1	3	yes	—
247	C₂×C₄	C₂×#41×C₆₁	1	6	yes	—
248	C₂×C₄	C₂⁵×C₆₁	1	1	yes	characteristic
249	C₈	—	2	4	no	—
250	C₈	C₂²×D₄×C₆₁	1	8	yes	—
251	C₂²	—	2	48	no	—
252	C₂²	—	2	4	no	—
253	C₂²	—	2	4	no	—
254	C₂²	C₆₁×#467	1	16	yes	direct product factor
255	C₂²	C₂×#121×C₆₁	1	6	yes	—
256	C₂²	C₂²×Q₁₆×C₆₁	1	6	yes	—
257	C₂²	C₂²×Q₁₆×C₆₁	1	8	yes	complemented
258	C₂²	C₂×#135×C₆₁	1	6	yes	—
259	C₂²	C₂³×D₄×C₆₁	1	1	yes	characteristic
260	C₄	—	2	4	no	—
261	C₄	—	4	8	no	—
262	C₄	—	4	8	no	—
263	C₄	C₂×#121×C₆₁	1	6	yes	—
264	C₄	C₂³×D₄×C₆₁	1	1	yes	characteristic
265	C₄	C₂²×#41×C₆₁	1	1	yes	characteristic
266	C₂	—	2	8	no	—
267	C₂	C₂×C₆₁×#467	1	12	yes	direct product factor
268	C₂	C₂²×#121×C₆₁	1	1	yes	characteristic
269	C₂	C₂³×Q₁₆×C₆₁	1	1	yes	characteristic
270	C₂	C₂²×#135×C₆₁	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
1. C₂³×Q₁₆×C₆₁	C₂	semidirect
4. C₂×C₆₁×#467	C₂	direct (N × H)
5. C₂²×#107×C₆₁	C₂	semidirect
6. C₂²×#103×C₆₁	C₂	semidirect
10. C₂²×Q₁₆×C₆₁	C₂²	semidirect
20. C₆₁×#467	C₂²	direct (N × H)
21. C₂×#107×C₆₁	C₂²	semidirect
22. C₂×#103×C₆₁	C₂²	semidirect
37. C₂×Q₁₆×C₆₁	C₂³	semidirect
50. #107×C₆₁	C₂³	semidirect
51. #103×C₆₁	C₂³	semidirect
54. C₂⁴×C₆₁	Q₁₆	semidirect
92. C₂³×C₆₁	C₂×Q₁₆	semidirect
115. C₂²×#467	C₆₁	direct (N × H)
119. C₂²×C₆₁	#467	direct (N × H)
122. C₂²×C₆₁	C₂²×Q₁₆	semidirect
131. C₂³×Q₁₆	C₂×C₆₁	semidirect
134. C₂×#467	C₂×C₆₁	direct (N × H)
135. C₂²×#107	C₂×C₆₁	semidirect
136. C₂²×#103	C₂×C₆₁	semidirect
138. C₂×C₆₁	C₂×#467	direct (N × H)
145. C₂²×Q₁₆	C₂²×C₆₁	semidirect
155. #467	C₂²×C₆₁	direct (N × H)
156. C₂×#107	C₂²×C₆₁	semidirect
157. C₂×#103	C₂²×C₆₁	semidirect
158. C₆₁	C₂²×#467	direct (N × H)
173. C₂×Q₁₆	C₂³×C₆₁	semidirect
186. #107	C₂³×C₆₁	semidirect
187. #103	C₂³×C₆₁	semidirect
190. C₂⁴	Q₁₆×C₆₁	semidirect
228. C₂³	C₂×Q₁₆×C₆₁	semidirect
254. C₂²	C₆₁×#467	direct (N × H)
257. C₂²	C₂²×Q₁₆×C₆₁	semidirect
267. C₂	C₂×C₆₁×#467	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
C₂⁴×C₈×C₆₁	2
C₂²×C₂²⋊Q₈×C₆₁	2
C₂×C₆₁×#467	2
C₂²×#107×C₆₁	2
C₂²×#103×C₆₁	2
C₂²×#467	61

Subgroups — C22×C61×#467 #29563869

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₆₁×#467 #29563869