Subgroups — C₂³×#14519737 #30586575

Order 15616.

19998 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×C₂²⋊D₆₁	C₂	1	1	yes	maximal, characteristic
2	C₂⁵×Dic₆₁	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂⁴×C₄×D₆₁	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²×#14519737	C₂	1	56	yes	maximal, direct product factor
5	C₂³×#11758196	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂³×#11758198	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂³×C₄×Dic₆₁	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂³×C₂²⋊C₄×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂⁵×D₆₁	C₂²	1	1	yes	characteristic
10	C₂³×C₂²⋊D₆₁	C₄	1	32	yes	complemented
11	C₂³×C₂²⋊D₆₁	C₂²	1	28	yes	—
12	C₂⁴×Dic₆₁	—	2	16	no	—
13	C₂⁴×Dic₆₁	C₂²	1	1	yes	characteristic
14	C₂⁴×Dic₆₁	C₂²	1	1	yes	characteristic
15	C₂⁴×Dic₆₁	C₂²	1	1	yes	characteristic
16	C₂⁴×Dic₆₁	C₂²	1	28	yes	complemented
17	C₂³×C₄×D₆₁	—	2	16	no	—
18	C₂³×C₄×D₆₁	C₂²	1	28	yes	complemented
19	C₂×#14519737	C₂²	1	448	yes	direct product factor
20	C₂²×#11758196	C₂²	1	28	yes	complemented
21	C₂²×#11758198	C₂²	1	28	yes	complemented
22	C₂²×C₄×Dic₆₁	C₂²	1	28	yes	complemented
23	C₂⁶×C₆₁	C₂²	1	1	yes	characteristic
24	C₂⁴×C₄×C₆₁	C₂²	1	1	yes	characteristic
25	C₂⁴×C₄×C₆₁	C₂²	1	1	yes	characteristic
26	C₂²×C₂²⋊C₄×C₆₁	C₂²	1	28	yes	complemented
27	C₂⁴×D₆₁	—	2	8	no	—
28	C₂⁴×D₆₁	—	2	8	no	—
29	C₂⁴×D₆₁	C₂×C₄	1	16	yes	complemented
30	C₂⁴×D₆₁	C₂³	1	14	yes	—
31	C₂²×C₂²⋊D₆₁	C₂×C₄	1	448	yes	complemented
32	C₂²×C₂²⋊D₆₁	C₂³	1	112	yes	—
33	C₂³×Dic₆₁	—	2	8	no	—
34	C₂³×Dic₆₁	—	2	224	no	—
35	C₂³×Dic₆₁	C₂×C₄	1	16	yes	complemented
36	C₂³×Dic₆₁	D₄	1	16	yes	complemented
37	C₂³×Dic₆₁	C₂³	1	14	yes	—
38	C₂³×Dic₆₁	C₂³	1	14	yes	—
39	C₂³×Dic₆₁	C₂³	1	14	yes	—
40	C₂³×Dic₆₁	C₂³	1	112	yes	complemented
41	C₂²×C₄×D₆₁	—	2	224	no	—
42	C₂²×C₄×D₆₁	C₂³	1	112	yes	complemented
43	#14519737	C₂³	1	512	yes	direct product factor
44	C₂×#11758196	C₂³	1	112	yes	complemented
45	C₂×#11758198	C₂³	1	112	yes	complemented
46	C₂×C₄×Dic₆₁	C₂³	1	112	yes	complemented
47	C₂⁵×C₆₁	—	2	8	no	—
48	C₂⁵×C₆₁	—	2	8	no	—
49	C₂⁵×C₆₁	C₂×C₄	1	16	yes	complemented
50	C₂⁵×C₆₁	C₂³	1	14	yes	—
51	C₂⁵×C₆₁	C₂³	1	1	yes	characteristic
52	C₂³×C₄×C₆₁	—	2	8	no	—
53	C₂³×C₄×C₆₁	—	2	8	no	—
54	C₂³×C₄×C₆₁	C₂³	1	14	yes	—
55	C₂³×C₄×C₆₁	C₂³	1	14	yes	—
56	C₂×C₂²⋊C₄×C₆₁	C₂³	1	112	yes	complemented
57	C₂⁴×C₆₁	—	2	128	no	—
58	C₂⁴×C₆₁	—	2	56	no	—
59	C₂⁴×C₆₁	—	2	56	no	—
60	C₂⁴×C₆₁	C₂²×C₄	1	8	yes	—
61	C₂⁴×C₆₁	C₂²×C₄	1	112	yes	complemented
62	C₂⁴×C₆₁	C₂×D₄	1	8	yes	—
63	C₂⁴×C₆₁	#41	1	8	yes	—
64	C₂⁴×C₆₁	C₂⁴	1	28	yes	—
65	C₂⁴×C₆₁	C₂⁴	1	7	yes	—
66	C₂³×D₆₁	—	2	128	no	—
67	C₂³×D₆₁	—	2	56	no	—
68	C₂³×D₆₁	—	2	56	no	—
69	C₂³×D₆₁	C₂²×C₄	1	112	yes	complemented
70	C₂³×D₆₁	C₂⁴	1	28	yes	—
71	C₂²×C₄×C₆₁	—	2	56	no	—
72	C₂²×C₄×C₆₁	—	2	56	no	—
73	C₂²×C₄×C₆₁	C₂⁴	1	28	yes	—
74	C₂²×C₄×C₆₁	C₂⁴	1	28	yes	—
75	C₂×C₂²⋊D₆₁	C₂²×C₄	1	896	yes	complemented
76	C₂×C₂²⋊D₆₁	C₂⁴	1	64	yes	—
77	C₂²×Dic₆₁	—	2	56	no	—
78	C₂²×Dic₆₁	—	2	448	no	—
79	C₂²×Dic₆₁	C₂²×C₄	1	112	yes	complemented
80	C₂²×Dic₆₁	C₂×D₄	1	112	yes	complemented
81	C₂²×Dic₆₁	C₂⁴	1	28	yes	—
82	C₂²×Dic₆₁	C₂⁴	1	28	yes	—
83	C₂²×Dic₆₁	C₂⁴	1	28	yes	—
84	C₂²×Dic₆₁	C₂⁴	1	64	yes	complemented
85	C₂×C₄×D₆₁	—	2	448	no	—
86	C₂×C₄×D₆₁	C₂⁴	1	64	yes	complemented
87	C₂²⋊C₄×C₆₁	C₂⁴	1	64	yes	complemented
88	#11758198	C₂⁴	1	64	yes	complemented
89	#11758196	C₂⁴	1	64	yes	complemented
90	C₄×Dic₆₁	C₂⁴	1	64	yes	complemented
91	C₂³×C₆₁	—	2	448	no	—
92	C₂³×C₆₁	—	2	56	no	—
93	C₂³×C₆₁	—	2	56	no	—
94	C₂³×C₆₁	C₄×D₄	1	64	yes	complemented
95	C₂³×C₆₁	C₂³×C₄	1	28	yes	—
96	C₂³×C₆₁	C₂³×C₄	1	112	yes	complemented
97	C₂³×C₆₁	C₂²×D₄	1	28	yes	—
98	C₂³×C₆₁	C₂×#41	1	28	yes	—
99	C₂³×C₆₁	C₂⁵	1	8	yes	—
100	C₂³×C₆₁	C₂⁵	1	7	yes	—
101	C₂²×D₆₁	—	2	448	no	—
102	C₂²×D₆₁	—	2	56	no	—
103	C₂²×D₆₁	—	2	56	no	—
104	C₂²×D₆₁	C₂³×C₄	1	112	yes	complemented
105	C₂²×D₆₁	C₂⁵	1	8	yes	—
106	C₂×C₄×C₆₁	—	2	56	no	—
107	C₂×C₄×C₆₁	—	2	56	no	—
108	C₂×C₄×C₆₁	C₂⁵	1	8	yes	—
109	C₂×C₄×C₆₁	C₂⁵	1	8	yes	—
110	C₂²⋊D₆₁	C₂³×C₄	1	256	yes	complemented
111	C₂×Dic₆₁	—	2	56	no	—
112	C₂×Dic₆₁	—	2	128	no	—
113	C₂×Dic₆₁	C₂³×C₄	1	112	yes	complemented
114	C₂×Dic₆₁	C₂²×D₄	1	112	yes	complemented
115	C₂×Dic₆₁	C₂⁵	1	8	yes	—
116	C₂×Dic₆₁	C₂⁵	1	8	yes	—
117	C₂×Dic₆₁	C₂⁵	1	8	yes	—
118	C₄×D₆₁	—	2	128	no	—
119	C₂³×C₄×D₄	—	61	1	no	maximal, Sylow
120	C₂²×C₆₁	—	2	8	no	—
121	C₂²×C₆₁	—	2	8	no	—
122	C₂²×C₆₁	—	2	224	no	—
123	C₂²×C₆₁	C₂×C₄×D₄	1	112	yes	complemented
124	C₂²×C₆₁	C₂⁴×C₄	1	14	yes	—
125	C₂²×C₆₁	C₂⁴×C₄	1	16	yes	complemented
126	C₂²×C₆₁	C₂³×D₄	1	14	yes	—
127	C₂²×C₆₁	C₂²×#41	1	14	yes	—
128	C₂²×C₆₁	C₂⁶	1	1	yes	characteristic
129	C₂×D₆₁	—	2	8	no	—
130	C₂×D₆₁	—	2	8	no	—
131	C₂×D₆₁	—	2	224	no	—
132	C₂×D₆₁	C₂⁴×C₄	1	16	yes	complemented
133	C₄×C₆₁	—	2	8	no	—
134	C₄×C₆₁	—	2	8	no	—
135	Dic₆₁	—	2	8	no	—
136	Dic₆₁	C₂⁴×C₄	1	16	yes	complemented
137	Dic₆₁	C₂³×D₄	1	16	yes	complemented
138	C₂⁴×D₄	—	61	1	no	—
139	C₂⁵×C₄	—	61	1	no	—
140	C₂⁵×C₄	—	61	1	no	—
141	C₂²×C₄×D₄	—	61	56	no	—
142	C₂³×C₄⋊C₄	—	61	1	no	—
143	C₂³×C₂²⋊C₄	—	61	1	no	—
144	C₂³×C₂²⋊C₄	D₆₁	1	1	yes	characteristic, complemented
145	C₂³×C₄²	—	61	1	no	—
146	C₂×C₆₁	—	2	16	no	—
147	C₂×C₆₁	C₂²×C₄×D₄	1	28	yes	complemented
148	C₂×C₆₁	C₂⁵×C₄	1	1	yes	characteristic
149	C₂×C₆₁	C₂⁴×D₄	1	1	yes	characteristic
150	C₂×C₆₁	C₂³×#41	1	1	yes	characteristic
151	D₆₁	—	2	16	no	—
152	C₂⁶	—	61	1	no	—
153	C₂⁶	C₂×D₆₁	1	1	yes	characteristic
154	C₂³×D₄	—	61	32	no	—
155	C₂³×D₄	—	61	28	no	—
156	C₂⁴×C₄	—	61	1	no	—
157	C₂⁴×C₄	—	61	1	no	—
158	C₂⁴×C₄	—	61	28	no	—
159	C₂⁴×C₄	—	61	1	no	—
160	C₂⁴×C₄	—	61	28	no	—
161	C₂⁴×C₄	—	122	16	no	—
162	C₂⁴×C₄	—	122	16	no	—
163	C₂⁴×C₄	C₂×D₆₁	1	1	yes	characteristic
164	C₂⁴×C₄	C₂×D₆₁	1	1	yes	characteristic
165	C₂×C₄×D₄	—	61	448	no	—
166	C₂²×C₄⋊C₄	—	61	28	no	—
167	C₂²×C₂²⋊C₄	—	61	28	no	—
168	C₂²×C₂²⋊C₄	C₂×D₆₁	1	28	yes	complemented
169	C₂²×C₄²	—	61	28	no	—
170	C₆₁	C₂³×C₄×D₄	1	1	yes	characteristic, complemented, Sylow
171	C₂⁵	—	2	8	no	—
172	C₂⁵	—	2	8	no	—
173	C₂⁵	—	61	16	no	—
174	C₂⁵	—	61	14	no	—
175	C₂⁵	—	122	8	no	—
176	C₂⁵	—	122	8	no	—
177	C₂⁵	C₄×D₆₁	1	16	yes	complemented
178	C₂⁵	C₂²×D₆₁	1	14	yes	—
179	C₂⁵	C₂²×D₆₁	1	1	yes	characteristic
180	C₂²×D₄	—	61	448	no	—
181	C₂²×D₄	—	61	112	no	—
182	C₂³×C₄	—	2	8	no	—
183	C₂³×C₄	—	2	8	no	—
184	C₂³×C₄	—	61	16	no	—
185	C₂³×C₄	—	61	16	no	—
186	C₂³×C₄	—	61	14	no	—
187	C₂³×C₄	—	61	14	no	—
188	C₂³×C₄	—	61	14	no	—
189	C₂³×C₄	—	61	112	no	—
190	C₂³×C₄	—	61	112	no	—
191	C₂³×C₄	—	122	224	no	—
192	C₂³×C₄	—	122	224	no	—
193	C₂³×C₄	—	122	8	no	—
194	C₂³×C₄	C₂²×D₆₁	1	14	yes	—
195	C₂³×C₄	C₂²×D₆₁	1	14	yes	—
196	C₄×D₄	—	61	512	no	—
197	C₂×C₄⋊C₄	—	61	112	no	—
198	C₂×C₂²⋊C₄	—	61	112	no	—
199	C₂×C₂²⋊C₄	C₂²×D₆₁	1	112	yes	complemented
200	C₂×C₄²	—	61	112	no	—
201	C₂⁴	—	2	56	no	—
202	C₂⁴	—	2	56	no	—
203	C₂⁴	—	2	128	no	—
204	C₂⁴	—	61	112	no	—
205	C₂⁴	—	61	28	no	—
206	C₂⁴	—	122	56	no	—
207	C₂⁴	—	122	56	no	—
208	C₂⁴	—	122	128	no	—
209	C₂⁴	C₂×C₄×D₆₁	1	8	yes	—
210	C₂⁴	C₂×C₄×D₆₁	1	112	yes	complemented
211	C₂⁴	D₄×D₆₁	1	8	yes	—
212	C₂⁴	#11758224	1	8	yes	—
213	C₂⁴	C₂³×D₆₁	1	28	yes	—
214	C₂⁴	C₂³×D₆₁	1	7	yes	—
215	C₂×D₄	—	61	896	no	—
216	C₂×D₄	—	61	64	no	—
217	C₂²×C₄	—	2	56	no	—
218	C₂²×C₄	—	2	56	no	—
219	C₂²×C₄	—	61	112	no	—
220	C₂²×C₄	—	61	112	no	—
221	C₂²×C₄	—	61	28	no	—
222	C₂²×C₄	—	61	28	no	—
223	C₂²×C₄	—	61	28	no	—
224	C₂²×C₄	—	61	64	no	—
225	C₂²×C₄	—	61	64	no	—
226	C₂²×C₄	—	122	448	no	—
227	C₂²×C₄	—	122	448	no	—
228	C₂²×C₄	—	122	56	no	—
229	C₂²×C₄	C₂³×D₆₁	1	28	yes	—
230	C₂²×C₄	C₂³×D₆₁	1	28	yes	—
231	C₄⋊C₄	—	61	64	no	—
232	C₂²⋊C₄	—	61	64	no	—
233	C₂²⋊C₄	C₂³×D₆₁	1	64	yes	complemented
234	C₄²	—	61	64	no	—
235	C₂³	—	2	448	no	—
236	C₂³	—	2	56	no	—
237	C₂³	—	2	56	no	—
238	C₂³	—	61	112	no	—
239	C₂³	—	61	8	no	—
240	C₂³	—	122	448	no	—
241	C₂³	—	122	56	no	—
242	C₂³	—	122	56	no	—
243	C₂³	#14519737	1	64	yes	direct product factor
244	C₂³	C₂²×C₄×D₆₁	1	28	yes	—
245	C₂³	C₂²×C₄×D₆₁	1	112	yes	complemented
246	C₂³	C₂×D₄×D₆₁	1	28	yes	—
247	C₂³	C₂×#11758224	1	28	yes	—
248	C₂³	C₂⁴×D₆₁	1	8	yes	—
249	C₂³	C₂⁴×D₆₁	1	7	yes	—
250	D₄	—	61	256	no	—
251	C₂×C₄	—	2	56	no	—
252	C₂×C₄	—	2	56	no	—
253	C₂×C₄	—	61	112	no	—
254	C₂×C₄	—	61	112	no	—
255	C₂×C₄	—	61	8	no	—
256	C₂×C₄	—	61	8	no	—
257	C₂×C₄	—	61	8	no	—
258	C₂×C₄	—	122	128	no	—
259	C₂×C₄	—	122	128	no	—
260	C₂×C₄	—	122	56	no	—
261	C₂×C₄	C₂⁴×D₆₁	1	8	yes	—
262	C₂×C₄	C₂⁴×D₆₁	1	8	yes	—
263	C₂²	—	2	224	no	—
264	C₂²	—	2	8	no	—
265	C₂²	—	2	8	no	—
266	C₂²	—	61	16	no	—
267	C₂²	—	122	224	no	—
268	C₂²	—	122	8	no	—
269	C₂²	—	122	8	no	—
270	C₂²	C₂×#14519737	1	112	yes	direct product factor
271	C₂²	C₂³×C₄×D₆₁	1	14	yes	—
272	C₂²	C₂³×C₄×D₆₁	1	16	yes	complemented
273	C₂²	C₂²×D₄×D₆₁	1	14	yes	—
274	C₂²	C₂²×#11758224	1	14	yes	—
275	C₂²	C₂⁵×D₆₁	1	1	yes	characteristic
276	C₄	—	2	8	no	—
277	C₄	—	2	8	no	—
278	C₄	—	61	16	no	—
279	C₄	—	61	16	no	—
280	C₄	—	122	8	no	—
281	C₂	—	2	16	no	—
282	C₂	—	122	16	no	—
283	C₂	C₂²×#14519737	1	28	yes	direct product factor
284	C₂	C₂⁴×C₄×D₆₁	1	1	yes	characteristic
285	C₂	C₂³×D₄×D₆₁	1	1	yes	characteristic
286	C₂	C₂³×#11758224	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁵	C₂²×D₆₁
commutator	C₂×C₆₁	C₂⁵×C₄
Frattini	C₂²	C₂⁵×D₆₁
Fitting	C₂³×C₂²⋊C₄×C₆₁	C₂
socle	C₂⁵×C₆₁	C₂³
prime radical	C₂⁴×C₂²⋊D₆₁	C₂
cosocle	C₂²×C₆₁	C₂⁶
hypercenter	C₂³×C₂²⋊C₄	D₆₁
hypocenter	C₆₁	C₂³×C₄×D₄
LCS	G ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂⁵ ◁ C₂³×C₂²⋊C₄

Complements

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

N	H ≅ G/N	type
2. C₂⁵×Dic₆₁	C₂	semidirect
3. C₂⁴×C₄×D₆₁	C₂	semidirect
4. C₂²×#14519737	C₂	direct (N × H)
5. C₂³×#11758196	C₂	semidirect
6. C₂³×#11758198	C₂	semidirect
7. C₂³×C₄×Dic₆₁	C₂	semidirect
8. C₂³×C₂²⋊C₄×C₆₁	C₂	semidirect
10. C₂³×C₂²⋊D₆₁	C₄	semidirect
16. C₂⁴×Dic₆₁	C₂²	semidirect
18. C₂³×C₄×D₆₁	C₂²	semidirect
19. C₂×#14519737	C₂²	direct (N × H)
20. C₂²×#11758196	C₂²	semidirect
21. C₂²×#11758198	C₂²	semidirect
22. C₂²×C₄×Dic₆₁	C₂²	semidirect
26. C₂²×C₂²⋊C₄×C₆₁	C₂²	semidirect
29. C₂⁴×D₆₁	C₂×C₄	semidirect
31. C₂²×C₂²⋊D₆₁	C₂×C₄	semidirect
35. C₂³×Dic₆₁	C₂×C₄	semidirect
36. C₂³×Dic₆₁	D₄	semidirect
40. C₂³×Dic₆₁	C₂³	semidirect
42. C₂²×C₄×D₆₁	C₂³	semidirect
43. #14519737	C₂³	direct (N × H)
44. C₂×#11758196	C₂³	semidirect
45. C₂×#11758198	C₂³	semidirect
46. C₂×C₄×Dic₆₁	C₂³	semidirect
49. C₂⁵×C₆₁	C₂×C₄	semidirect
56. C₂×C₂²⋊C₄×C₆₁	C₂³	semidirect
61. C₂⁴×C₆₁	C₂²×C₄	semidirect
69. C₂³×D₆₁	C₂²×C₄	semidirect
75. C₂×C₂²⋊D₆₁	C₂²×C₄	semidirect
79. C₂²×Dic₆₁	C₂²×C₄	semidirect
80. C₂²×Dic₆₁	C₂×D₄	semidirect
84. C₂²×Dic₆₁	C₂⁴	semidirect
86. C₂×C₄×D₆₁	C₂⁴	semidirect
87. C₂²⋊C₄×C₆₁	C₂⁴	semidirect
88. #11758198	C₂⁴	semidirect
89. #11758196	C₂⁴	semidirect
90. C₄×Dic₆₁	C₂⁴	semidirect
94. C₂³×C₆₁	C₄×D₄	semidirect
96. C₂³×C₆₁	C₂³×C₄	semidirect
104. C₂²×D₆₁	C₂³×C₄	semidirect
110. C₂²⋊D₆₁	C₂³×C₄	semidirect
113. C₂×Dic₆₁	C₂³×C₄	semidirect
114. C₂×Dic₆₁	C₂²×D₄	semidirect
123. C₂²×C₆₁	C₂×C₄×D₄	semidirect
125. C₂²×C₆₁	C₂⁴×C₄	semidirect
132. C₂×D₆₁	C₂⁴×C₄	semidirect
136. Dic₆₁	C₂⁴×C₄	semidirect
137. Dic₆₁	C₂³×D₄	semidirect
144. C₂³×C₂²⋊C₄	D₆₁	semidirect
147. C₂×C₆₁	C₂²×C₄×D₄	semidirect
168. C₂²×C₂²⋊C₄	C₂×D₆₁	semidirect
170. C₆₁	C₂³×C₄×D₄	semidirect
177. C₂⁵	C₄×D₆₁	semidirect
199. C₂×C₂²⋊C₄	C₂²×D₆₁	semidirect
210. C₂⁴	C₂×C₄×D₆₁	semidirect
233. C₂²⋊C₄	C₂³×D₆₁	semidirect
243. C₂³	#14519737	direct (N × H)
245. C₂³	C₂²×C₄×D₆₁	semidirect
270. C₂²	C₂×#14519737	direct (N × H)
272. C₂²	C₂³×C₄×D₆₁	semidirect
283. C₂	C₂²×#14519737	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁴×C₂²⋊D₆₁	2
C₂⁵×Dic₆₁	2
C₂⁴×C₄×D₆₁	2
C₂²×#14519737	2
C₂³×#11758196	2
C₂³×#11758198	2
C₂³×C₄×Dic₆₁	2
C₂³×C₂²⋊C₄×C₆₁	2
C₂³×C₄×D₄	61

Subgroups — C23×#14519737 #30586575

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#14519737 #30586575