Subgroups — #29585909

Order 15616.

982 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	C₂²×#14519709	C₂	1	2	yes	maximal
2	C₂×#16839525	C₂	1	2	yes	maximal
3	#21444049	C₂	1	2	yes	maximal
4	C₂²×C₄×C₈×C₆₁	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₄⋊Dic₆₁	C₂²	1	4	yes	—
6	C₂×#14519709	—	2	4	no	—
7	C₂×#14519709	—	2	4	no	—
8	C₂×#14519709	C₄	1	8	yes	complemented
9	C₂×#14519684	—	2	4	no	—
10	#16839525	—	2	4	no	—
11	C₂³×C₈×C₆₁	C₂²	1	1	yes	characteristic
12	C₂³×C₈×C₆₁	C₂²	1	1	yes	characteristic
13	C₂²×C₄²×C₆₁	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₈×C₆₁	—	2	4	no	—
15	C₂×C₄×C₈×C₆₁	C₄	1	4	yes	complemented
16	C₂³×Dic₆₁	—	2	4	no	—
17	C₂×C₄⋊Dic₆₁	—	2	4	no	—
18	C₂×C₄⋊Dic₆₁	—	2	4	no	—
19	C₂×C₄⋊Dic₆₁	—	2	4	no	—
20	C₂×C₄⋊Dic₆₁	—	2	4	no	—
21	C₂×C₄⋊Dic₆₁	—	2	4	no	—
22	C₂×C₄⋊Dic₆₁	C₂×C₄	1	8	yes	—
23	#14519709	—	2	16	no	—
24	#14519684	—	4	8	no	—
25	C₂³×C₄×C₆₁	D₄	1	2	yes	—
26	C₂³×C₄×C₆₁	C₂³	1	1	yes	characteristic
27	C₂²×C₈×C₆₁	—	2	2	no	—
28	C₂²×C₈×C₆₁	—	2	2	no	—
29	C₂²×C₈×C₆₁	C₂×C₄	1	2	yes	—
30	C₂²×C₈×C₆₁	C₂×C₄	1	4	yes	—
31	C₂²×C₈×C₆₁	C₂×C₄	1	2	yes	—
32	C₂²×C₈×C₆₁	D₄	1	2	yes	—
33	C₂²×C₈×C₆₁	D₄	1	2	yes	—
34	C₂²×C₈×C₆₁	D₄	1	4	yes	—
35	C₂²×C₈×C₆₁	Q₈	1	2	yes	—
36	C₂²×C₈×C₆₁	Q₈	1	2	yes	—
37	C₂×C₄²×C₆₁	—	2	2	no	—
38	C₂×C₄²×C₆₁	—	2	2	no	—
39	C₂×C₄²×C₆₁	C₂×C₄	1	2	yes	—
40	C₂×C₄²×C₆₁	Q₈	1	2	yes	—
41	C₄×C₈×C₆₁	—	2	8	no	—
42	C₂⁴×C₆₁	C₂×D₄	1	1	yes	characteristic
43	C₂²×C₄×C₆₁	—	2	4	no	—
44	C₂²×C₄×C₆₁	—	2	4	no	—
45	C₂²×C₄×C₆₁	C₄⋊C₄	1	4	yes	—
46	C₂²×C₄×C₆₁	SD₁₆	1	8	yes	—
47	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
48	C₂²×C₄×C₆₁	C₂²×C₄	1	2	yes	—
49	C₂²×C₄×C₆₁	C₂×D₄	1	2	yes	—
50	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
51	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
52	C₂²×C₄×C₆₁	C₂×Q₈	1	1	yes	characteristic
53	C₂²×C₄×C₆₁	C₂×Q₈	1	1	yes	characteristic
54	C₂²×C₄×C₆₁	C₂×Q₈	1	1	yes	characteristic
55	C₂²×C₄×C₆₁	#41	1	2	yes	—
56	C₂²×C₄×C₆₁	#41	1	2	yes	—
57	C₂²×Dic₆₁	—	2	8	no	—
58	C₂²×Dic₆₁	—	4	4	no	—
59	C₂²×Dic₆₁	—	4	4	no	—
60	C₂²×Dic₆₁	—	4	8	no	—
61	C₂²×Dic₆₁	—	4	8	no	—
62	C₂×C₈×C₆₁	—	2	2	no	—
63	C₂×C₈×C₆₁	—	2	2	no	—
64	C₂×C₈×C₆₁	—	2	4	no	—
65	C₂×C₈×C₆₁	—	2	4	no	—
66	C₂×C₈×C₆₁	C₄²	1	4	yes	—
67	C₂×C₈×C₆₁	C₂²⋊C₄	1	8	yes	—
68	C₂×C₈×C₆₁	C₂²⋊C₄	1	4	yes	—
69	C₂×C₈×C₆₁	C₄⋊C₄	1	4	yes	—
70	C₂×C₈×C₆₁	C₄⋊C₄	1	8	yes	—
71	C₂×C₈×C₆₁	C₄⋊C₄	1	4	yes	—
72	C₄²×C₆₁	—	2	2	no	—
73	C₄²×C₆₁	—	2	2	no	—
74	C₄²×C₆₁	—	2	4	no	—
75	C₄⋊Dic₆₁	—	2	8	no	—
76	C₄⋊Dic₆₁	—	2	8	no	—
77	C₄⋊Dic₆₁	—	4	4	no	—
78	C₄⋊Dic₆₁	—	4	4	no	—
79	C₂³×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
80	C₂³×C₆₁	C₄×D₄	1	2	yes	—
81	C₂³×C₆₁	#121	1	2	yes	—
82	C₂³×C₆₁	#127	1	1	yes	characteristic
83	C₂³×C₆₁	C₄⋊Q₈	1	1	yes	characteristic
84	C₂³×C₆₁	C₂×SD₁₆	1	4	yes	—
85	C₂³×C₆₁	#135	1	4	yes	—
86	C₂×C₄×C₆₁	—	2	4	no	—
87	C₂×C₄×C₆₁	—	2	4	no	—
88	C₂×C₄×C₆₁	—	2	16	no	—
89	C₂×C₄×C₆₁	#106	1	8	yes	complemented
90	C₂×C₄×C₆₁	C₂×C₄²	1	1	yes	characteristic
91	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
92	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	2	yes	—
93	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
94	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	2	yes	—
95	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
96	C₂×C₄×C₆₁	#117	1	1	yes	characteristic
97	C₂×C₄×C₆₁	#117	1	1	yes	characteristic
98	C₂×C₄×C₆₁	C₄×D₄	1	2	yes	—
99	C₂×C₄×C₆₁	C₄×Q₈	1	2	yes	—
100	C₂×C₄×C₆₁	C₄×Q₈	1	2	yes	—
101	C₂×C₄×C₆₁	#121	1	2	yes	—
102	C₂×C₄×C₆₁	C₂²⋊Q₈	1	2	yes	—
103	C₂×C₄×C₆₁	C₂²⋊Q₈	1	2	yes	—
104	C₂×C₄×C₆₁	#124	1	2	yes	—
105	C₂×C₄×C₆₁	#125	1	2	yes	—
106	C₂×C₄×C₆₁	C₄⋊Q₈	1	1	yes	characteristic
107	C₂×C₄×C₆₁	C₄⋊Q₈	1	1	yes	characteristic
108	C₂×Dic₆₁	—	4	8	no	—
109	C₂×Dic₆₁	—	4	16	no	—
110	C₂×Dic₆₁	—	8	8	no	—
111	C₂×Dic₆₁	—	8	4	no	—
112	C₂×Dic₆₁	—	8	4	no	—
113	C₈×C₆₁	—	2	4	no	—
114	C₈×C₆₁	#95	1	8	yes	complemented
115	#10861	—	61	1	no	maximal, Sylow
116	C₂²×C₆₁	C₄×C₄⋊C₄	1	1	yes	characteristic
117	C₂²×C₆₁	#380	1	1	yes	characteristic
118	C₂²×C₆₁	#384	1	1	yes	characteristic
119	C₂²×C₆₁	#389	1	2	yes	—
120	C₂²×C₆₁	#390	1	2	yes	—
121	C₂²×C₆₁	C₂×#106	1	2	yes	—
122	C₂²×C₆₁	#427	1	2	yes	—
123	C₂²×C₆₁	C₄×SD₁₆	1	8	yes	—
124	C₂²×C₆₁	#465	1	8	yes	—
125	C₂²×C₆₁	#492	1	2	yes	—
126	C₂²×C₆₁	#495	1	2	yes	—
127	C₂²×C₆₁	#498	1	2	yes	—
128	C₂²×C₆₁	#499	1	2	yes	—
129	C₄×C₆₁	—	2	8	no	—
130	C₄×C₆₁	C₂×#95	1	1	yes	characteristic
131	C₄×C₆₁	C₄×C₄⋊C₄	1	1	yes	characteristic
132	C₄×C₆₁	#380	1	1	yes	characteristic
133	C₄×C₆₁	#382	1	1	yes	characteristic
134	C₄×C₆₁	#389	1	2	yes	—
135	C₄×C₆₁	#391	1	2	yes	—
136	Dic₆₁	—	8	8	no	—
137	C₂²×#106	—	61	2	no	—
138	C₂²×C₄×C₈	D₆₁	1	1	yes	characteristic
139	C₂×#340	—	61	2	no	—
140	C₄⋊#95	—	61	2	no	—
141	C₂×C₆₁	C₄⋊#95	1	1	yes	characteristic
142	C₂×C₆₁	C₄×#106	1	2	yes	—
143	C₂×C₆₁	#1807	1	2	yes	—
144	C₂×C₆₁	#1844	1	1	yes	characteristic
145	C₂×C₆₁	#1846	1	1	yes	characteristic
146	C₂×C₆₁	#1941	1	4	yes	—
147	C₂×C₆₁	#1968	1	4	yes	—
148	C₂³×C₈	C₂×D₆₁	1	1	yes	characteristic
149	C₂³×C₈	C₂×D₆₁	1	1	yes	characteristic
150	C₂²×C₄⋊C₄	—	61	4	no	—
151	C₂²×C₄²	C₂×D₆₁	1	1	yes	characteristic
152	C₂×#106	—	61	8	no	—
153	C₂×#106	—	122	4	no	—
154	C₂×#106	—	122	4	no	—
155	C₂×C₄×C₈	—	2	4	no	—
156	C₂×C₄×C₈	Dic₆₁	1	4	yes	complemented
157	C₂×#95	—	122	4	no	—
158	#340	—	122	4	no	—
159	C₆₁	#10861	1	1	yes	characteristic, complemented, Sylow
160	C₂³×C₄	—	122	4	no	—
161	C₂³×C₄	D₂₄₄	1	2	yes	—
162	C₂³×C₄	C₂²×D₆₁	1	1	yes	characteristic
163	C₂²×C₈	—	2	2	no	—
164	C₂²×C₈	—	2	2	no	—
165	C₂²×C₈	Dic₁₂₂	1	2	yes	—
166	C₂²×C₈	Dic₁₂₂	1	2	yes	—
167	C₂²×C₈	C₄×D₆₁	1	4	yes	—
168	C₂²×C₈	D₂₄₄	1	2	yes	—
169	C₂²×C₈	D₂₄₄	1	2	yes	—
170	C₂²×C₈	C₂×Dic₆₁	1	2	yes	—
171	C₂²×C₈	C₂×Dic₆₁	1	2	yes	—
172	C₂²×C₈	C₂²⋊D₆₁	1	4	yes	—
173	C₂×C₄⋊C₄	—	61	8	no	—
174	C₂×C₄⋊C₄	—	122	4	no	—
175	C₂×C₄⋊C₄	—	122	4	no	—
176	C₂×C₄⋊C₄	—	122	4	no	—
177	C₂×C₄⋊C₄	—	122	4	no	—
178	C₂×C₄⋊C₄	—	122	4	no	—
179	C₂×C₄²	—	2	2	no	—
180	C₂×C₄²	—	2	2	no	—
181	C₂×C₄²	Dic₁₂₂	1	2	yes	—
182	C₂×C₄²	C₂×Dic₆₁	1	2	yes	—
183	#106	—	122	16	no	—
184	C₄×C₈	—	2	8	no	—
185	#95	—	244	8	no	—
186	C₂⁴	C₂×D₂₄₄	1	1	yes	characteristic
187	C₂²×C₄	—	2	4	no	—
188	C₂²×C₄	—	2	4	no	—
189	C₂²×C₄	—	122	8	no	—
190	C₂²×C₄	—	244	4	no	—
191	C₂²×C₄	—	244	4	no	—
192	C₂²×C₄	—	244	8	no	—
193	C₂²×C₄	—	244	8	no	—
194	C₂²×C₄	#11758190	1	8	yes	—
195	C₂²×C₄	C₄⋊Dic₆₁	1	4	yes	—
196	C₂²×C₄	C₂×Dic₁₂₂	1	1	yes	characteristic
197	C₂²×C₄	C₂×Dic₁₂₂	1	1	yes	characteristic
198	C₂²×C₄	C₂×Dic₁₂₂	1	1	yes	characteristic
199	C₂²×C₄	C₂×C₄×D₆₁	1	2	yes	—
200	C₂²×C₄	C₂×D₂₄₄	1	1	yes	characteristic
201	C₂²×C₄	C₂×D₂₄₄	1	1	yes	characteristic
202	C₂²×C₄	#11758222	1	2	yes	—
203	C₂²×C₄	#11758222	1	2	yes	—
204	C₂²×C₄	C₂²×Dic₆₁	1	1	yes	characteristic
205	C₂²×C₄	C₂×C₂²⋊D₆₁	1	2	yes	—
206	C₂×C₈	—	2	2	no	—
207	C₂×C₈	—	2	2	no	—
208	C₂×C₈	—	2	4	no	—
209	C₂×C₈	—	2	4	no	—
210	C₂×C₈	C₄×Dic₆₁	1	4	yes	—
211	C₂×C₈	#11758196	1	8	yes	—
212	C₂×C₈	C₄⋊Dic₆₁	1	4	yes	—
213	C₂×C₈	C₄⋊Dic₆₁	1	4	yes	—
214	C₂×C₈	#11758198	1	8	yes	—
215	C₂×C₈	C₂²⋊Dic₆₁	1	4	yes	—
216	C₄⋊C₄	—	122	8	no	—
217	C₄⋊C₄	—	122	8	no	—
218	C₄⋊C₄	—	244	4	no	—
219	C₄⋊C₄	—	244	4	no	—
220	C₄²	—	2	2	no	—
221	C₄²	—	2	2	no	—
222	C₄²	—	2	4	no	—
223	C₂³	C₂×C₄⋊Dic₆₁	1	1	yes	characteristic
224	C₂³	C₄×D₂₄₄	1	2	yes	—
225	C₂³	#14519751	1	2	yes	—
226	C₂³	#14519774	1	1	yes	characteristic
227	C₂³	#14519777	1	1	yes	characteristic
228	C₂³	C₂×#11758190	1	4	yes	—
229	C₂³	#14519799	1	4	yes	—
230	C₂×C₄	—	2	16	no	—
231	C₂×C₄	—	2	4	no	—
232	C₂×C₄	—	2	4	no	—
233	C₂×C₄	—	488	8	no	—
234	C₂×C₄	—	488	4	no	—
235	C₂×C₄	—	488	4	no	—
236	C₂×C₄	—	244	16	no	—
237	C₂×C₄	—	244	8	no	—
238	C₂×C₄	#14519709	1	8	yes	complemented
239	C₂×C₄	C₂×C₄×Dic₆₁	1	1	yes	characteristic
240	C₂×C₄	C₂×#11758198	1	2	yes	—
241	C₂×C₄	C₂×C₂²⋊Dic₆₁	1	1	yes	characteristic
242	C₂×C₄	C₂×#11758196	1	2	yes	—
243	C₂×C₄	C₂×C₄⋊Dic₆₁	1	1	yes	characteristic
244	C₂×C₄	C₂×C₄⋊Dic₆₁	1	1	yes	characteristic
245	C₂×C₄	#14519736	1	1	yes	characteristic
246	C₂×C₄	#14519736	1	1	yes	characteristic
247	C₂×C₄	C₄×D₂₄₄	1	2	yes	—
248	C₂×C₄	C₄×Dic₁₂₂	1	2	yes	—
249	C₂×C₄	C₄×Dic₁₂₂	1	2	yes	—
250	C₂×C₄	#14519751	1	2	yes	—
251	C₂×C₄	#14519756	1	2	yes	—
252	C₂×C₄	#14519756	1	2	yes	—
253	C₂×C₄	#14519765	1	2	yes	—
254	C₂×C₄	#14519769	1	2	yes	—
255	C₂×C₄	#14519777	1	1	yes	characteristic
256	C₂×C₄	#14519777	1	1	yes	characteristic
257	C₈	—	2	4	no	—
258	C₈	#14519684	1	8	yes	complemented
259	C₂²	C₄×C₄⋊Dic₆₁	1	1	yes	characteristic
260	C₂²	#16839622	1	1	yes	characteristic
261	C₂²	#16839632	1	1	yes	characteristic
262	C₂²	#16839660	1	2	yes	—
263	C₂²	#16839664	1	2	yes	—
264	C₂²	C₂×#14519709	1	2	yes	—
265	C₂²	#16839800	1	2	yes	—
266	C₂²	C₄×#11758190	1	8	yes	—
267	C₂²	#16839991	1	8	yes	—
268	C₂²	#16840135	1	2	yes	—
269	C₂²	#16840146	1	2	yes	—
270	C₂²	#16840160	1	2	yes	—
271	C₂²	#16840164	1	2	yes	—
272	C₄	—	2	8	no	—
273	C₄	—	488	8	no	—
274	C₄	C₂×#14519684	1	1	yes	characteristic
275	C₄	C₄×C₄⋊Dic₆₁	1	1	yes	characteristic
276	C₄	#16839622	1	1	yes	characteristic
277	C₄	#16839628	1	1	yes	characteristic
278	C₄	#16839660	1	2	yes	—
279	C₄	#16839668	1	2	yes	—
280	C₂	#21444049	1	1	yes	characteristic
281	C₂	C₄×#14519709	1	2	yes	—
282	C₂	#21445844	1	2	yes	—
283	C₂	#21445976	1	1	yes	characteristic
284	C₂	#21445983	1	1	yes	characteristic
285	C₂	#21446547	1	4	yes	—
286	C₂	#21446673	1	4	yes	—

	subgroup	G/H
center	C₂⁴	C₂×D₂₄₄
commutator	C₂×C₄×C₆₁	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₆₁
Fitting	C₂²×C₄×C₈×C₆₁	C₂
socle, prime radical	C₂⁴×C₆₁	C₂×D₄
cosocle	C₂³×C₄×C₆₁	C₂³
hypercenter	C₂²×C₄×C₈	D₆₁
hypocenter	C₆₁	#10861
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈

Complements

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

N	H ≅ G/N	type
8. C₂×#14519709	C₄	semidirect
15. C₂×C₄×C₈×C₆₁	C₄	semidirect
89. C₂×C₄×C₆₁	#106	semidirect
114. C₈×C₆₁	#95	semidirect
156. C₂×C₄×C₈	Dic₆₁	semidirect
159. C₆₁	#10861	semidirect
238. C₂×C₄	#14519709	semidirect
258. C₈	#14519684	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#14519709	2
C₂×#16839525	2
#21444049	2
C₂²×C₄×C₈×C₆₁	2
#10861	61