Subgroups — C₂²×C₆₁×#324 #29520777

Order 15616.

1806 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₄⋊C₄×C₆₁	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₂²⋊C₈×C₆₁	C₂	1	2	yes	maximal
3	C₂×C₆₁×#324	C₂	1	12	yes	maximal, direct product factor
4	C₂⁴×C₄×C₆₁	C₄	1	1	yes	characteristic
5	C₂⁴×C₄×C₆₁	C₄	1	1	yes	characteristic
6	C₂⁴×C₄×C₆₁	C₂²	1	1	yes	characteristic
7	C₂³×C₈×C₆₁	—	2	2	no	—
8	C₂²×C₄⋊C₄×C₆₁	—	2	6	no	—
9	C₂²×C₄⋊C₄×C₆₁	—	2	2	no	—
10	C₂²×C₄⋊C₄×C₆₁	C₄	1	6	yes	—
11	C₂²×C₄⋊C₄×C₆₁	C₂²	1	6	yes	—
12	C₂×C₂²⋊C₈×C₆₁	—	2	6	no	—
13	C₂×C₂²⋊C₈×C₆₁	C₂²	1	12	yes	—
14	C₆₁×#324	C₂²	1	16	yes	direct product factor
15	C₂⁵×C₆₁	C₂×C₄	1	1	yes	characteristic
16	C₂³×C₄×C₆₁	—	2	3	no	—
17	C₂³×C₄×C₆₁	—	2	1	no	—
18	C₂³×C₄×C₆₁	—	2	4	no	—
19	C₂³×C₄×C₆₁	—	2	3	no	—
20	C₂³×C₄×C₆₁	—	2	1	no	—
21	C₂³×C₄×C₆₁	—	4	4	no	—
22	C₂³×C₄×C₆₁	—	4	4	no	—
23	C₂³×C₄×C₆₁	C₈	1	8	yes	complemented
24	C₂³×C₄×C₆₁	C₂×C₄	1	6	yes	—
25	C₂³×C₄×C₆₁	C₂×C₄	1	6	yes	—
26	C₂³×C₄×C₆₁	C₂×C₄	1	3	yes	—
27	C₂³×C₄×C₆₁	D₄	1	2	yes	—
28	C₂³×C₄×C₆₁	D₄	1	6	yes	—
29	C₂³×C₄×C₆₁	C₂³	1	3	yes	—
30	C₂²×C₈×C₆₁	—	2	12	no	—
31	C₂²×C₈×C₆₁	—	4	8	no	—
32	C₂×C₄⋊C₄×C₆₁	—	2	12	no	—
33	C₂×C₄⋊C₄×C₆₁	—	2	12	no	—
34	C₂×C₄⋊C₄×C₆₁	—	2	12	no	—
35	C₂×C₄⋊C₄×C₆₁	—	2	12	no	—
36	C₂×C₄⋊C₄×C₆₁	C₂×C₄	1	12	yes	—
37	C₂×C₄⋊C₄×C₆₁	C₂³	1	4	yes	—
38	C₂²⋊C₈×C₆₁	—	2	12	no	—
39	C₂²⋊C₈×C₆₁	C₂³	1	8	yes	—
40	C₂⁴×C₆₁	—	2	4	no	—
41	C₂⁴×C₆₁	—	2	4	no	—
42	C₂⁴×C₆₁	C₂²⋊C₄	1	3	yes	—
43	C₂⁴×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
44	C₂⁴×C₆₁	C₂×C₈	1	4	yes	—
45	C₂⁴×C₆₁	M₁₆	1	4	yes	—
46	C₂⁴×C₆₁	C₂²×C₄	1	3	yes	—
47	C₂²×C₄×C₆₁	—	2	24	no	—
48	C₂²×C₄×C₆₁	—	2	8	no	—
49	C₂²×C₄×C₆₁	—	2	8	no	—
50	C₂²×C₄×C₆₁	—	2	6	no	—
51	C₂²×C₄×C₆₁	—	2	6	no	—
52	C₂²×C₄×C₆₁	—	2	12	no	—
53	C₂²×C₄×C₆₁	—	2	6	no	—
54	C₂²×C₄×C₆₁	—	2	6	no	—
55	C₂²×C₄×C₆₁	—	4	6	no	—
56	C₂²×C₄×C₆₁	—	4	12	no	—
57	C₂²×C₄×C₆₁	—	4	6	no	—
58	C₂²×C₄×C₆₁	—	4	12	no	—
59	C₂²×C₄×C₆₁	—	4	4	no	—
60	C₂²×C₄×C₆₁	—	4	12	no	—
61	C₂²×C₄×C₆₁	C₂²⋊C₄	1	6	yes	—
62	C₂²×C₄×C₆₁	C₂²⋊C₄	1	6	yes	—
63	C₂²×C₄×C₆₁	C₂×C₈	1	24	yes	complemented
64	C₂²×C₄×C₆₁	C₂²×C₄	1	4	yes	—
65	C₂²×C₄×C₆₁	C₂²×C₄	1	4	yes	—
66	C₂²×C₄×C₆₁	C₂²×C₄	1	3	yes	—
67	C₂²×C₄×C₆₁	C₂×D₄	1	6	yes	—
68	C₂²×C₄×C₆₁	C₂×D₄	1	6	yes	—
69	C₂²×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
70	C₂×C₈×C₆₁	—	2	8	no	—
71	C₂×C₈×C₆₁	—	4	24	no	—
72	C₄⋊C₄×C₆₁	—	2	24	no	—
73	C₄⋊C₄×C₆₁	—	2	8	no	—
74	C₂³×C₆₁	—	2	16	no	—
75	C₂³×C₆₁	—	2	16	no	—
76	C₂³×C₆₁	—	2	6	no	—
77	C₂³×C₆₁	—	2	6	no	—
78	C₂³×C₆₁	—	2	6	no	—
79	C₂³×C₆₁	—	2	6	no	—
80	C₂³×C₆₁	C₂²⋊C₈	1	12	yes	—
81	C₂³×C₆₁	C₂²⋊C₈	1	4	yes	—
82	C₂³×C₆₁	#99	1	4	yes	—
83	C₂³×C₆₁	#101	1	4	yes	—
84	C₂³×C₆₁	C₂×C₂²⋊C₄	1	3	yes	—
85	C₂³×C₆₁	C₂×C₂²⋊C₄	1	3	yes	—
86	C₂³×C₆₁	C₂²×C₈	1	6	yes	—
87	C₂³×C₆₁	C₂×M₁₆	1	6	yes	—
88	C₂³×C₆₁	C₂³×C₄	1	1	yes	characteristic
89	C₂×C₄×C₆₁	—	2	4	no	—
90	C₂×C₄×C₆₁	—	2	4	no	—
91	C₂×C₄×C₆₁	—	2	24	no	—
92	C₂×C₄×C₆₁	—	2	24	no	—
93	C₂×C₄×C₆₁	—	2	12	no	—
94	C₂×C₄×C₆₁	—	2	12	no	—
95	C₂×C₄×C₆₁	—	2	4	no	—
96	C₂×C₄×C₆₁	—	4	6	no	—
97	C₂×C₄×C₆₁	—	4	3	no	—
98	C₂×C₄×C₆₁	—	4	6	no	—
99	C₂×C₄×C₆₁	—	4	1	no	—
100	C₂×C₄×C₆₁	—	4	12	no	—
101	C₂×C₄×C₆₁	—	4	12	no	—
102	C₂×C₄×C₆₁	—	4	4	no	—
103	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	6	yes	—
104	C₂×C₄×C₆₁	C₂²×C₈	1	8	yes	complemented
105	C₂×C₄×C₆₁	C₂²×D₄	1	2	yes	—
106	C₈×C₆₁	—	4	8	no	—
107	C₂²×#324	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
108	C₂²×C₆₁	—	2	3	no	—
109	C₂²×C₆₁	—	2	1	no	—
110	C₂²×C₆₁	—	2	3	no	—
111	C₂²×C₆₁	—	2	1	no	—
112	C₂²×C₆₁	—	2	24	no	—
113	C₂²×C₆₁	—	2	12	no	—
114	C₂²×C₆₁	—	2	12	no	—
115	C₂²×C₆₁	#324	1	16	yes	direct product factor
116	C₂²×C₆₁	C₂×C₂²⋊C₈	1	6	yes	—
117	C₂²×C₆₁	C₂×C₂²⋊C₈	1	6	yes	—
118	C₂²×C₆₁	C₂×#99	1	6	yes	—
119	C₂²×C₆₁	C₂×#101	1	6	yes	—
120	C₂²×C₆₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
121	C₂²×C₆₁	C₂³×C₈	1	1	yes	characteristic
122	C₂²×C₆₁	C₂²×M₁₆	1	1	yes	characteristic
123	C₄×C₆₁	—	2	8	no	—
124	C₄×C₆₁	—	2	2	no	—
125	C₄×C₆₁	—	2	6	no	—
126	C₄×C₆₁	—	4	4	no	—
127	C₂³×C₄⋊C₄	C₂×C₆₁	1	1	yes	characteristic
128	C₂²×C₂²⋊C₈	C₂×C₆₁	1	2	yes	—
129	C₂×#324	C₂×C₆₁	1	12	yes	direct product factor
130	C₂×C₆₁	—	2	1	no	—
131	C₂×C₆₁	—	2	1	no	—
132	C₂×C₆₁	—	2	6	no	—
133	C₂×C₆₁	C₂×#324	1	12	yes	direct product factor
134	C₂×C₆₁	C₂²×C₂²⋊C₈	1	1	yes	characteristic
135	C₂×C₆₁	C₂²×#99	1	1	yes	characteristic
136	C₂×C₆₁	C₂²×#101	1	1	yes	characteristic
137	C₂⁴×C₄	C₄×C₆₁	1	1	yes	characteristic
138	C₂⁴×C₄	C₄×C₆₁	1	1	yes	characteristic
139	C₂⁴×C₄	C₂²×C₆₁	1	1	yes	characteristic
140	C₂³×C₈	—	2	2	no	—
141	C₂²×C₄⋊C₄	—	2	2	no	—
142	C₂²×C₄⋊C₄	—	2	6	no	—
143	C₂²×C₄⋊C₄	C₄×C₆₁	1	6	yes	—
144	C₂²×C₄⋊C₄	C₂²×C₆₁	1	6	yes	—
145	C₂×C₂²⋊C₈	—	2	6	no	—
146	C₂×C₂²⋊C₈	C₂²×C₆₁	1	12	yes	—
147	#324	C₂²×C₆₁	1	16	yes	direct product factor
148	C₆₁	C₂²×#324	1	1	yes	characteristic, direct product factor, Sylow
149	C₂⁵	C₂×C₄×C₆₁	1	1	yes	characteristic
150	C₂³×C₄	—	2	3	no	—
151	C₂³×C₄	—	2	1	no	—
152	C₂³×C₄	—	2	4	no	—
153	C₂³×C₄	—	2	3	no	—
154	C₂³×C₄	—	2	1	no	—
155	C₂³×C₄	—	4	4	no	—
156	C₂³×C₄	—	4	4	no	—
157	C₂³×C₄	C₈×C₆₁	1	8	yes	complemented
158	C₂³×C₄	C₂×C₄×C₆₁	1	6	yes	—
159	C₂³×C₄	C₂×C₄×C₆₁	1	6	yes	—
160	C₂³×C₄	C₂×C₄×C₆₁	1	3	yes	—
161	C₂³×C₄	D₄×C₆₁	1	2	yes	—
162	C₂³×C₄	D₄×C₆₁	1	6	yes	—
163	C₂³×C₄	C₂³×C₆₁	1	3	yes	—
164	C₂²×C₈	—	2	12	no	—
165	C₂²×C₈	—	4	8	no	—
166	C₂×C₄⋊C₄	—	2	12	no	—
167	C₂×C₄⋊C₄	—	2	12	no	—
168	C₂×C₄⋊C₄	—	2	12	no	—
169	C₂×C₄⋊C₄	—	2	12	no	—
170	C₂×C₄⋊C₄	C₂×C₄×C₆₁	1	12	yes	—
171	C₂×C₄⋊C₄	C₂³×C₆₁	1	4	yes	—
172	C₂²⋊C₈	—	2	12	no	—
173	C₂²⋊C₈	C₂³×C₆₁	1	8	yes	—
174	C₂⁴	—	2	4	no	—
175	C₂⁴	—	2	4	no	—
176	C₂⁴	C₂²⋊C₄×C₆₁	1	3	yes	—
177	C₂⁴	C₂²⋊C₄×C₆₁	1	1	yes	characteristic
178	C₂⁴	C₂×C₈×C₆₁	1	4	yes	—
179	C₂⁴	M₁₆×C₆₁	1	4	yes	—
180	C₂⁴	C₂²×C₄×C₆₁	1	3	yes	—
181	C₂²×C₄	—	2	24	no	—
182	C₂²×C₄	—	2	8	no	—
183	C₂²×C₄	—	2	8	no	—
184	C₂²×C₄	—	2	6	no	—
185	C₂²×C₄	—	2	6	no	—
186	C₂²×C₄	—	2	12	no	—
187	C₂²×C₄	—	2	6	no	—
188	C₂²×C₄	—	2	6	no	—
189	C₂²×C₄	—	4	6	no	—
190	C₂²×C₄	—	4	12	no	—
191	C₂²×C₄	—	4	6	no	—
192	C₂²×C₄	—	4	12	no	—
193	C₂²×C₄	—	4	4	no	—
194	C₂²×C₄	—	4	12	no	—
195	C₂²×C₄	C₂²⋊C₄×C₆₁	1	6	yes	—
196	C₂²×C₄	C₂²⋊C₄×C₆₁	1	6	yes	—
197	C₂²×C₄	C₂×C₈×C₆₁	1	24	yes	complemented
198	C₂²×C₄	C₂²×C₄×C₆₁	1	4	yes	—
199	C₂²×C₄	C₂²×C₄×C₆₁	1	4	yes	—
200	C₂²×C₄	C₂²×C₄×C₆₁	1	3	yes	—
201	C₂²×C₄	C₂×D₄×C₆₁	1	6	yes	—
202	C₂²×C₄	C₂×D₄×C₆₁	1	6	yes	—
203	C₂²×C₄	C₂⁴×C₆₁	1	1	yes	characteristic
204	C₂×C₈	—	2	8	no	—
205	C₂×C₈	—	4	24	no	—
206	C₄⋊C₄	—	2	24	no	—
207	C₄⋊C₄	—	2	8	no	—
208	C₂³	—	2	16	no	—
209	C₂³	—	2	16	no	—
210	C₂³	—	2	6	no	—
211	C₂³	—	2	6	no	—
212	C₂³	—	2	6	no	—
213	C₂³	—	2	6	no	—
214	C₂³	C₂²⋊C₈×C₆₁	1	12	yes	—
215	C₂³	C₂²⋊C₈×C₆₁	1	4	yes	—
216	C₂³	#99×C₆₁	1	4	yes	—
217	C₂³	#101×C₆₁	1	4	yes	—
218	C₂³	C₂×C₂²⋊C₄×C₆₁	1	3	yes	—
219	C₂³	C₂×C₂²⋊C₄×C₆₁	1	3	yes	—
220	C₂³	C₂²×C₈×C₆₁	1	6	yes	—
221	C₂³	C₂×M₁₆×C₆₁	1	6	yes	—
222	C₂³	C₂³×C₄×C₆₁	1	1	yes	characteristic
223	C₂×C₄	—	2	4	no	—
224	C₂×C₄	—	2	24	no	—
225	C₂×C₄	—	2	24	no	—
226	C₂×C₄	—	2	12	no	—
227	C₂×C₄	—	2	12	no	—
228	C₂×C₄	—	2	4	no	—
229	C₂×C₄	—	2	4	no	—
230	C₂×C₄	—	4	6	no	—
231	C₂×C₄	—	4	3	no	—
232	C₂×C₄	—	4	6	no	—
233	C₂×C₄	—	4	1	no	—
234	C₂×C₄	—	4	12	no	—
235	C₂×C₄	—	4	12	no	—
236	C₂×C₄	—	4	4	no	—
237	C₂×C₄	C₂×C₂²⋊C₄×C₆₁	1	6	yes	—
238	C₂×C₄	C₂²×C₈×C₆₁	1	8	yes	complemented
239	C₂×C₄	C₂²×D₄×C₆₁	1	2	yes	—
240	C₈	—	4	8	no	—
241	C₂²	—	2	3	no	—
242	C₂²	—	2	1	no	—
243	C₂²	—	2	3	no	—
244	C₂²	—	2	1	no	—
245	C₂²	—	2	24	no	—
246	C₂²	—	2	12	no	—
247	C₂²	—	2	12	no	—
248	C₂²	C₆₁×#324	1	16	yes	direct product factor
249	C₂²	C₂×C₂²⋊C₈×C₆₁	1	6	yes	—
250	C₂²	C₂×C₂²⋊C₈×C₆₁	1	6	yes	—
251	C₂²	C₂×#99×C₆₁	1	6	yes	—
252	C₂²	C₂×#101×C₆₁	1	6	yes	—
253	C₂²	C₂²×C₂²⋊C₄×C₆₁	1	1	yes	characteristic
254	C₂²	C₂³×C₈×C₆₁	1	1	yes	characteristic
255	C₂²	C₂²×M₁₆×C₆₁	1	1	yes	characteristic
256	C₄	—	2	8	no	—
257	C₄	—	2	2	no	—
258	C₄	—	2	6	no	—
259	C₄	—	4	4	no	—
260	C₂	—	2	1	no	—
261	C₂	—	2	1	no	—
262	C₂	—	2	6	no	—
263	C₂	C₂×C₆₁×#324	1	12	yes	direct product factor
264	C₂	C₂²×C₂²⋊C₈×C₆₁	1	1	yes	characteristic
265	C₂	C₂²×#99×C₆₁	1	1	yes	characteristic
266	C₂	C₂²×#101×C₆₁	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂⁴×C₆₁	C₂²⋊C₄
commutator	C₂²	C₂³×C₈×C₆₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₁
prime radical	C₂⁵×C₆₁	C₂×C₄
LCS	G ▷ C₂² ▷ 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
3. C₂×C₆₁×#324	C₂	direct (N × H)
14. C₆₁×#324	C₂²	direct (N × H)
23. C₂³×C₄×C₆₁	C₈	semidirect
63. C₂²×C₄×C₆₁	C₂×C₈	semidirect
104. C₂×C₄×C₆₁	C₂²×C₈	semidirect
107. C₂²×#324	C₆₁	direct (N × H)
115. C₂²×C₆₁	#324	direct (N × H)
129. C₂×#324	C₂×C₆₁	direct (N × H)
133. C₂×C₆₁	C₂×#324	direct (N × H)
147. #324	C₂²×C₆₁	direct (N × H)
148. C₆₁	C₂²×#324	direct (N × H)
157. C₂³×C₄	C₈×C₆₁	semidirect
197. C₂²×C₄	C₂×C₈×C₆₁	semidirect
238. C₂×C₄	C₂²×C₈×C₆₁	semidirect
248. C₂²	C₆₁×#324	direct (N × H)
263. C₂	C₂×C₆₁×#324	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×C₄⋊C₄×C₆₁	2
C₂²×C₂²⋊C₈×C₆₁	2
C₂×C₆₁×#324	2
C₂²×#324	61

Subgroups — C22×C61×#324 #29520777

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₆₁×#324 #29520777