Subgroups — C₂×#543×D₆₁ #30586733

Order 15616.

9310 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16840388	C₂	1	3	yes	maximal, complemented
2	C₂×#16840387	C₂	1	12	yes	maximal, complemented
3	#543×D₆₁	C₂	1	32	yes	maximal, direct product factor
4	C₂×C₂²⋊Q₈×D₆₁	C₂	1	12	yes	maximal, complemented
5	C₂²×C₂²⋊C₄×D₆₁	C₂	1	3	yes	maximal
6	C₂×C₆₁×#543	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂⁵×D₆₁	C₂²	1	1	yes	characteristic
8	C₂²×Q₈×D₆₁	C₂²	1	4	yes	complemented
9	C₂³×C₄×D₆₁	C₂²	1	6	yes	—
10	#16840388	C₂²	1	48	yes	complemented
11	#16840387	C₂²	1	192	yes	complemented
12	C₂×#14519757	C₂²	1	12	yes	complemented
13	C₂×#14519756	C₂²	1	12	yes	complemented
14	C₂×#14519755	C₂²	1	24	yes	complemented
15	C₂×#14519754	C₂²	1	24	yes	complemented
16	C₂×#14519753	C₂²	1	12	yes	complemented
17	C₂²⋊Q₈×D₆₁	C₂²	1	192	yes	complemented
18	C₂×C₄⋊C₄×D₆₁	C₂²	1	12	yes	complemented
19	C₂²×C₂²⋊Dic₆₁	C₂²	1	3	yes	—
20	C₂²×#11758198	C₂²	1	6	yes	—
21	C₂×C₂²⋊C₄×D₆₁	C₂²	1	48	yes	—
22	C₂×C₂²⋊C₄×D₆₁	C₂²	1	12	yes	—
23	C₂×C₂²⋊C₄×D₆₁	C₂²	1	12	yes	—
24	C₆₁×#543	C₂²	1	16	yes	complemented
25	C₂×C₂²⋊Q₈×C₆₁	C₂²	1	12	yes	complemented
26	C₂²×C₂²⋊C₄×C₆₁	C₂²	1	3	yes	—
27	C₂⁴×D₆₁	—	4	8	no	—
28	C₂⁴×D₆₁	Q₈	1	16	yes	complemented
29	C₂⁴×D₆₁	C₂³	1	3	yes	—
30	C₂⁴×D₆₁	C₂³	1	8	yes	—
31	C₂⁴×D₆₁	C₂³	1	3	yes	—
32	C₂²×Dic₁₂₂	C₂³	1	12	yes	complemented
33	C₂×Q₈×D₆₁	—	4	16	no	—
34	C₂×Q₈×D₆₁	C₂³	1	32	yes	complemented
35	C₂³×Dic₆₁	C₂³	1	6	yes	—
36	C₂²×C₄×D₆₁	—	4	24	no	—
37	C₂²×C₄×D₆₁	C₂³	1	12	yes	—
38	C₂²×C₄×D₆₁	C₂³	1	48	yes	—
39	C₂²×C₄×D₆₁	C₂³	1	12	yes	—
40	#14519757	C₂³	1	96	yes	complemented
41	#14519756	C₂³	1	96	yes	complemented
42	#14519755	C₂³	1	192	yes	complemented
43	#14519754	C₂³	1	192	yes	complemented
44	#14519753	C₂³	1	96	yes	complemented
45	C₂×C₄⋊Dic₆₁	C₂³	1	12	yes	complemented
46	C₂×#11758196	C₂³	1	24	yes	complemented
47	C₄⋊C₄×D₆₁	C₂³	1	96	yes	complemented
48	C₂×C₂²⋊Dic₆₁	C₂³	1	12	yes	—
49	C₂×C₂²⋊Dic₆₁	C₂³	1	24	yes	—
50	C₂×#11758198	C₂³	1	24	yes	—
51	C₂×#11758198	C₂³	1	48	yes	—
52	C₂²⋊C₄×D₆₁	C₂³	1	96	yes	—
53	C₂²⋊C₄×D₆₁	C₂³	1	96	yes	—
54	C₂⁵×C₆₁	C₂³	1	1	yes	characteristic
55	C₂²×Q₈×C₆₁	C₂³	1	4	yes	complemented
56	C₂³×C₄×C₆₁	C₂³	1	6	yes	—
57	C₂²⋊Q₈×C₆₁	C₂³	1	96	yes	complemented
58	C₂×C₄⋊C₄×C₆₁	C₂³	1	12	yes	complemented
59	C₂×C₂²⋊C₄×C₆₁	C₂³	1	24	yes	—
60	C₂×C₂²⋊C₄×C₆₁	C₂³	1	12	yes	—
61	C₂⁴×C₆₁	—	4	4	no	—
62	C₂⁴×C₆₁	C₂×Q₈	1	8	yes	complemented
63	C₂⁴×C₆₁	C₂⁴	1	3	yes	—
64	C₂⁴×C₆₁	C₂⁴	1	4	yes	—
65	C₂³×D₆₁	—	2	24	no	—
66	C₂³×D₆₁	—	4	64	no	—
67	C₂³×D₆₁	—	4	32	no	—
68	C₂³×D₆₁	—	4	12	no	—
69	C₂³×D₆₁	C₂×Q₈	1	24	yes	—
70	C₂³×D₆₁	C₂×Q₈	1	64	yes	complemented
71	C₂³×D₆₁	C₂×Q₈	1	24	yes	complemented
72	C₂³×D₆₁	C₂⁴	1	12	yes	—
73	C₂³×D₆₁	C₂⁴	1	3	yes	—
74	C₂³×D₆₁	C₂⁴	1	12	yes	—
75	C₂³×D₆₁	C₂⁴	1	1	yes	characteristic
76	C₂×Q₈×C₆₁	—	4	8	no	—
77	C₂×Q₈×C₆₁	C₂⁴	1	16	yes	complemented
78	C₂²×C₄×C₆₁	—	4	12	no	—
79	C₂²×C₄×C₆₁	C₂⁴	1	12	yes	—
80	C₂²×C₄×C₆₁	C₂⁴	1	24	yes	—
81	C₂²×Dic₆₁	—	4	12	no	—
82	C₂²×Dic₆₁	C₂⁴	1	12	yes	—
83	C₂²×Dic₆₁	C₂⁴	1	24	yes	—
84	Q₈×D₆₁	—	4	64	no	—
85	C₂×C₄×D₆₁	—	2	48	no	—
86	C₂×C₄×D₆₁	—	4	96	no	—
87	C₂×C₄×D₆₁	—	4	24	no	—
88	C₂×C₄×D₆₁	C₂⁴	1	48	yes	—
89	C₂×C₄×D₆₁	C₂⁴	1	48	yes	—
90	C₂×Dic₁₂₂	—	4	24	no	—
91	C₂×Dic₁₂₂	C₂⁴	1	48	yes	complemented
92	C₄⋊C₄×C₆₁	C₂⁴	1	48	yes	complemented
93	C₂²⋊C₄×C₆₁	C₂⁴	1	48	yes	—
94	C₂²⋊Dic₆₁	C₂⁴	1	48	yes	—
95	#11758198	C₂⁴	1	96	yes	—
96	C₄⋊Dic₆₁	C₂⁴	1	48	yes	complemented
97	#11758196	C₂⁴	1	96	yes	complemented
98	C₂³×C₆₁	—	2	12	no	—
99	C₂³×C₆₁	—	4	16	no	—
100	C₂³×C₆₁	—	4	8	no	—
101	C₂³×C₆₁	C₂²×Q₈	1	12	yes	—
102	C₂³×C₆₁	C₂²×Q₈	1	16	yes	complemented
103	C₂³×C₆₁	C₂⁵	1	6	yes	—
104	C₂³×C₆₁	C₂⁵	1	1	yes	characteristic
105	C₂²×D₆₁	—	2	12	no	—
106	C₂²×D₆₁	—	2	96	no	—
107	C₂²×D₆₁	—	2	48	no	—
108	C₂²×D₆₁	—	4	128	no	—
109	C₂²×D₆₁	—	4	48	no	—
110	C₂²×D₆₁	—	4	24	no	—
111	C₂²×D₆₁	C₂²×Q₈	1	12	yes	—
112	C₂²×D₆₁	C₂²×Q₈	1	48	yes	—
113	C₂²×D₆₁	C₂²×Q₈	1	4	yes	—
114	C₂²×D₆₁	C₂²×Q₈	1	48	yes	complemented
115	C₂²×D₆₁	H(5,𝔽₂)	1	8	yes	—
116	C₂²×D₆₁	C₂⁵	1	6	yes	—
117	C₂²×D₆₁	C₂⁵	1	2	yes	—
118	Q₈×C₆₁	—	4	16	no	—
119	C₂×C₄×C₆₁	—	2	24	no	—
120	C₂×C₄×C₆₁	—	4	24	no	—
121	C₂×C₄×C₆₁	C₂⁵	1	24	yes	—
122	C₂×Dic₆₁	—	2	24	no	—
123	C₂×Dic₆₁	—	4	24	no	—
124	C₂×Dic₆₁	C₂⁵	1	24	yes	—
125	C₄×D₆₁	—	2	96	no	—
126	C₄×D₆₁	—	4	48	no	—
127	Dic₁₂₂	—	4	48	no	—
128	C₂²×#543	—	61	1	no	maximal, Sylow
129	C₂²×C₆₁	—	2	12	no	—
130	C₂²×C₆₁	—	2	24	no	—
131	C₂²×C₆₁	—	4	16	no	—
132	C₂²×C₆₁	C₂³×Q₈	1	12	yes	—
133	C₂²×C₆₁	C₂³×Q₈	1	2	yes	—
134	C₂²×C₆₁	C₂×H(5,𝔽₂)	1	4	yes	—
135	C₂²×C₆₁	C₂⁶	1	1	yes	characteristic
136	C₂×D₆₁	—	2	12	no	—
137	C₂×D₆₁	—	2	24	no	—
138	C₂×D₆₁	—	2	96	no	—
139	C₂×D₆₁	—	4	48	no	—
140	C₂×D₆₁	#543	1	16	yes	direct product factor
141	C₂×D₆₁	C₂³×Q₈	1	12	yes	—
142	C₂×D₆₁	C₂³×Q₈	1	4	yes	—
143	C₂×D₆₁	C₂×H(5,𝔽₂)	1	8	yes	—
144	C₄×C₆₁	—	2	24	no	—
145	Dic₆₁	—	2	24	no	—
146	C₂×#543	—	61	32	no	—
147	C₂×#543	—	61	12	no	—
148	C₂×#543	—	61	3	no	—
149	C₂×#543	D₆₁	1	1	yes	characteristic, direct product factor
150	C₂²×C₂²⋊Q₈	—	61	12	no	—
151	C₂³×C₂²⋊C₄	—	61	3	no	—
152	C₂×C₆₁	—	2	12	no	—
153	C₂×C₆₁	C₂×#543	1	4	yes	complemented
154	C₂×C₆₁	C₂⁴×Q₈	1	1	yes	characteristic
155	C₂×C₆₁	C₂²×H(5,𝔽₂)	1	2	yes	—
156	D₆₁	—	2	12	no	—
157	D₆₁	C₂×#543	1	8	yes	direct product factor
158	C₂⁶	—	61	1	no	—
159	C₂³×Q₈	—	61	4	no	—
160	C₂⁴×C₄	—	61	6	no	—
161	#543	—	61	192	no	—
162	#543	—	61	48	no	—
163	#543	C₂×D₆₁	1	16	yes	direct product factor
164	C₂×C₂²⋊Q₈	—	61	12	no	—
165	C₂×C₂²⋊Q₈	—	61	12	no	—
166	C₂×C₂²⋊Q₈	—	61	192	no	—
167	C₂×C₂²⋊Q₈	—	61	24	no	—
168	C₂×C₂²⋊Q₈	—	61	12	no	—
169	C₂×C₂²⋊Q₈	—	61	24	no	—
170	C₂×C₂²⋊Q₈	C₂×D₆₁	1	12	yes	complemented
171	C₂²×C₄⋊C₄	—	61	12	no	—
172	C₂²×C₂²⋊C₄	—	61	3	no	—
173	C₂²×C₂²⋊C₄	—	61	6	no	—
174	C₂²×C₂²⋊C₄	—	61	12	no	—
175	C₂²×C₂²⋊C₄	—	61	12	no	—
176	C₂²×C₂²⋊C₄	—	61	48	no	—
177	C₂²×C₂²⋊C₄	C₂×D₆₁	1	3	yes	—
178	C₆₁	C₂²×#543	1	1	yes	characteristic, complemented, Sylow
179	C₂⁵	—	61	3	no	—
180	C₂⁵	—	61	16	no	—
181	C₂⁵	—	61	8	no	—
182	C₂⁵	—	61	3	no	—
183	C₂⁵	—	244	8	no	—
184	C₂⁵	C₂²×D₆₁	1	1	yes	characteristic
185	C₂²×Q₈	—	61	32	no	—
186	C₂²×Q₈	—	61	12	no	—
187	C₂²×Q₈	—	244	16	no	—
188	C₂²×Q₈	C₂²×D₆₁	1	4	yes	complemented
189	C₂³×C₄	—	61	6	no	—
190	C₂³×C₄	—	61	12	no	—
191	C₂³×C₄	—	61	12	no	—
192	C₂³×C₄	—	61	48	no	—
193	C₂³×C₄	—	244	24	no	—
194	C₂³×C₄	C₂²×D₆₁	1	6	yes	—
195	C₂²⋊Q₈	—	61	96	no	—
196	C₂²⋊Q₈	—	61	96	no	—
197	C₂²⋊Q₈	—	61	192	no	—
198	C₂²⋊Q₈	—	61	96	no	—
199	C₂²⋊Q₈	—	61	192	no	—
200	C₂²⋊Q₈	C₂²×D₆₁	1	96	yes	complemented
201	C₂×C₄⋊C₄	—	61	24	no	—
202	C₂×C₄⋊C₄	—	61	12	no	—
203	C₂×C₄⋊C₄	—	61	96	no	—
204	C₂×C₄⋊C₄	C₂²×D₆₁	1	12	yes	complemented
205	C₂×C₂²⋊C₄	—	61	12	no	—
206	C₂×C₂²⋊C₄	—	61	24	no	—
207	C₂×C₂²⋊C₄	—	61	24	no	—
208	C₂×C₂²⋊C₄	—	61	48	no	—
209	C₂×C₂²⋊C₄	—	61	96	no	—
210	C₂×C₂²⋊C₄	—	61	96	no	—
211	C₂×C₂²⋊C₄	C₂²×D₆₁	1	12	yes	—
212	C₂×C₂²⋊C₄	C₂²×D₆₁	1	24	yes	—
213	C₂⁴	—	4	4	no	—
214	C₂⁴	—	61	1	no	—
215	C₂⁴	—	61	24	no	—
216	C₂⁴	—	61	12	no	—
217	C₂⁴	—	61	3	no	—
218	C₂⁴	—	61	24	no	—
219	C₂⁴	—	61	64	no	—
220	C₂⁴	—	61	12	no	—
221	C₂⁴	—	122	24	no	—
222	C₂⁴	—	244	12	no	—
223	C₂⁴	—	244	64	no	—
224	C₂⁴	—	244	32	no	—
225	C₂⁴	Q₈×D₆₁	1	8	yes	complemented
226	C₂⁴	C₂³×D₆₁	1	3	yes	—
227	C₂⁴	C₂³×D₆₁	1	4	yes	—
228	C₂×Q₈	—	4	8	no	—
229	C₂×Q₈	—	61	48	no	—
230	C₂×Q₈	—	244	24	no	—
231	C₂×Q₈	—	244	64	no	—
232	C₂×Q₈	C₂³×D₆₁	1	16	yes	complemented
233	C₂²×C₄	—	4	12	no	—
234	C₂²×C₄	—	61	12	no	—
235	C₂²×C₄	—	61	24	no	—
236	C₂²×C₄	—	61	48	no	—
237	C₂²×C₄	—	61	48	no	—
238	C₂²×C₄	—	122	48	no	—
239	C₂²×C₄	—	244	24	no	—
240	C₂²×C₄	—	244	96	no	—
241	C₂²×C₄	—	244	12	no	—
242	C₂²×C₄	C₂³×D₆₁	1	12	yes	—
243	C₂²×C₄	C₂³×D₆₁	1	24	yes	—
244	C₄⋊C₄	—	61	96	no	—
245	C₄⋊C₄	—	61	48	no	—
246	C₄⋊C₄	C₂³×D₆₁	1	48	yes	complemented
247	C₂²⋊C₄	—	61	48	no	—
248	C₂²⋊C₄	—	61	96	no	—
249	C₂²⋊C₄	C₂³×D₆₁	1	48	yes	—
250	C₂³	—	2	12	no	—
251	C₂³	—	4	16	no	—
252	C₂³	—	4	8	no	—
253	C₂³	—	61	4	no	—
254	C₂³	—	61	8	no	—
255	C₂³	—	61	2	no	—
256	C₂³	—	61	48	no	—
257	C₂³	—	61	12	no	—
258	C₂³	—	61	6	no	—
259	C₂³	—	61	48	no	—
260	C₂³	—	122	48	no	—
261	C₂³	—	122	96	no	—
262	C₂³	—	122	12	no	—
263	C₂³	—	244	48	no	—
264	C₂³	—	244	24	no	—
265	C₂³	—	244	128	no	—
266	C₂³	C₂×Q₈×D₆₁	1	12	yes	—
267	C₂³	C₂×Q₈×D₆₁	1	16	yes	complemented
268	C₂³	C₂⁴×D₆₁	1	6	yes	—
269	C₂³	C₂⁴×D₆₁	1	1	yes	characteristic
270	Q₈	—	4	16	no	—
271	Q₈	—	244	48	no	—
272	C₂×C₄	—	2	24	no	—
273	C₂×C₄	—	4	24	no	—
274	C₂×C₄	—	61	24	no	—
275	C₂×C₄	—	122	96	no	—
276	C₂×C₄	—	122	24	no	—
277	C₂×C₄	—	244	48	no	—
278	C₂×C₄	—	244	24	no	—
279	C₂×C₄	C₂⁴×D₆₁	1	24	yes	—
280	C₂²	—	2	24	no	—
281	C₂²	—	2	12	no	—
282	C₂²	—	4	16	no	—
283	C₂²	—	61	16	no	—
284	C₂²	—	61	4	no	—
285	C₂²	—	61	8	no	—
286	C₂²	—	61	12	no	—
287	C₂²	—	122	96	no	—
288	C₂²	—	122	24	no	—
289	C₂²	—	122	12	no	—
290	C₂²	—	244	48	no	—
291	C₂²	C₂²×Q₈×D₆₁	1	12	yes	—
292	C₂²	C₂²×Q₈×D₆₁	1	2	yes	—
293	C₂²	H(5,𝔽₂)×D₆₁	1	4	yes	—
294	C₂²	C₂⁵×D₆₁	1	1	yes	characteristic
295	C₄	—	2	24	no	—
296	C₄	—	122	24	no	—
297	C₂	—	2	12	no	—
298	C₂	—	61	8	no	—
299	C₂	—	122	12	no	—
300	C₂	#543×D₆₁	1	4	yes	direct product factor
301	C₂	C₂³×Q₈×D₆₁	1	1	yes	characteristic
302	C₂	C₂×H(5,𝔽₂)×D₆₁	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂⁴×D₆₁
commutator, cosocle	C₂²×C₆₁	C₂⁶
Frattini	C₂²	C₂⁵×D₆₁
Fitting	C₂×C₆₁×#543	C₂
socle	C₂³×C₆₁	C₂⁵
prime radical	C₂⁵×D₆₁	C₂²
hypercenter	C₂×#543	D₆₁
hypocenter	C₆₁	C₂²×#543
LCS	G ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂³ ◁ C₂×#543

Complements

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

N	H ≅ G/N	type
1. C₂×#16840388	C₂	semidirect
2. C₂×#16840387	C₂	semidirect
3. #543×D₆₁	C₂	direct (N × H)
4. C₂×C₂²⋊Q₈×D₆₁	C₂	semidirect
6. C₂×C₆₁×#543	C₂	semidirect
8. C₂²×Q₈×D₆₁	C₂²	semidirect
10. #16840388	C₂²	semidirect
11. #16840387	C₂²	semidirect
12. C₂×#14519757	C₂²	semidirect
13. C₂×#14519756	C₂²	semidirect
14. C₂×#14519755	C₂²	semidirect
15. C₂×#14519754	C₂²	semidirect
16. C₂×#14519753	C₂²	semidirect
17. C₂²⋊Q₈×D₆₁	C₂²	semidirect
18. C₂×C₄⋊C₄×D₆₁	C₂²	semidirect
24. C₆₁×#543	C₂²	semidirect
25. C₂×C₂²⋊Q₈×C₆₁	C₂²	semidirect
28. C₂⁴×D₆₁	Q₈	semidirect
32. C₂²×Dic₁₂₂	C₂³	semidirect
34. C₂×Q₈×D₆₁	C₂³	semidirect
40. #14519757	C₂³	semidirect
41. #14519756	C₂³	semidirect
42. #14519755	C₂³	semidirect
43. #14519754	C₂³	semidirect
44. #14519753	C₂³	semidirect
45. C₂×C₄⋊Dic₆₁	C₂³	semidirect
46. C₂×#11758196	C₂³	semidirect
47. C₄⋊C₄×D₆₁	C₂³	semidirect
55. C₂²×Q₈×C₆₁	C₂³	semidirect
57. C₂²⋊Q₈×C₆₁	C₂³	semidirect
58. C₂×C₄⋊C₄×C₆₁	C₂³	semidirect
62. C₂⁴×C₆₁	C₂×Q₈	semidirect
70. C₂³×D₆₁	C₂×Q₈	semidirect
71. C₂³×D₆₁	C₂×Q₈	semidirect
77. C₂×Q₈×C₆₁	C₂⁴	semidirect
91. C₂×Dic₁₂₂	C₂⁴	semidirect
92. C₄⋊C₄×C₆₁	C₂⁴	semidirect
96. C₄⋊Dic₆₁	C₂⁴	semidirect
97. #11758196	C₂⁴	semidirect
102. C₂³×C₆₁	C₂²×Q₈	semidirect
114. C₂²×D₆₁	C₂²×Q₈	semidirect
140. C₂×D₆₁	#543	direct (N × H)
149. C₂×#543	D₆₁	direct (N × H)
153. C₂×C₆₁	C₂×#543	semidirect
157. D₆₁	C₂×#543	direct (N × H)
163. #543	C₂×D₆₁	direct (N × H)
170. C₂×C₂²⋊Q₈	C₂×D₆₁	semidirect
178. C₆₁	C₂²×#543	semidirect
188. C₂²×Q₈	C₂²×D₆₁	semidirect
200. C₂²⋊Q₈	C₂²×D₆₁	semidirect
204. C₂×C₄⋊C₄	C₂²×D₆₁	semidirect
225. C₂⁴	Q₈×D₆₁	semidirect
232. C₂×Q₈	C₂³×D₆₁	semidirect
246. C₄⋊C₄	C₂³×D₆₁	semidirect
267. C₂³	C₂×Q₈×D₆₁	semidirect
300. C₂	#543×D₆₁	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#16840388	2
C₂×#16840387	2
#543×D₆₁	2
C₂×C₂²⋊Q₈×D₆₁	2
C₂²×C₂²⋊C₄×D₆₁	2
C₂×C₆₁×#543	2
C₂²×#543	61

Subgroups — C2×#543×D61 #30586733

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#543×D₆₁ #30586733