Subgroups — #29573072

Order 15616.

454 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21445979	C₂	1	1	yes	maximal, characteristic
2	#21445824	C₂	1	2	yes	maximal
3	C₂×#16839515	C₂	1	1	yes	maximal, characteristic
4	#21443703	C₂	1	2	yes	maximal
5	C₂×C₈×Q₈×C₆₁	C₂	1	1	yes	maximal, characteristic
6	C₂×#14519711	—	2	1	no	—
7	C₂×#14519711	—	2	1	no	—
8	C₂×C₄⋊#92418	C₂²	1	2	yes	—
9	C₂×#14519701	—	2	2	no	—
10	#16839632	C₂²	1	2	yes	—
11	#16839524	—	2	2	no	—
12	#16839515	—	2	2	no	—
13	C₂×C₄×Q₈×C₆₁	C₂²	1	1	yes	characteristic
14	C₈×Q₈×C₆₁	—	2	2	no	—
15	C₈×Q₈×C₆₁	C₄	1	4	yes	complemented
16	C₂×C₄⋊C₈×C₆₁	—	2	1	no	—
17	C₂×C₄⋊C₈×C₆₁	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₈×C₆₁	—	2	1	no	—
19	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
20	C₂²×#92418	—	2	2	no	—
21	C₂×C₄⋊Dic₆₁	—	2	2	no	—
22	C₂×C₄⋊Dic₆₁	—	2	2	no	—
23	C₂×C₄⋊Dic₆₁	—	2	2	no	—
24	#14519711	—	4	2	no	—
25	#14519711	—	4	2	no	—
26	C₄⋊#92418	—	2	2	no	—
27	C₄⋊#92418	—	2	2	no	—
28	#14519701	—	4	4	no	—
29	C₂²×Q₈×C₆₁	D₄	1	1	yes	characteristic
30	C₂²×C₈×C₆₁	—	2	1	no	—
31	C₂²×C₈×C₆₁	D₄	1	1	yes	characteristic
32	C₂²×C₈×C₆₁	D₄	1	1	yes	characteristic
33	C₄×Q₈×C₆₁	—	2	1	no	—
34	C₄×Q₈×C₆₁	—	2	1	no	—
35	C₄×Q₈×C₆₁	C₂×C₄	1	2	yes	—
36	C₄×Q₈×C₆₁	Q₈	1	2	yes	—
37	C₂×C₄⋊C₄×C₆₁	—	2	1	no	—
38	C₂×C₄⋊C₄×C₆₁	D₄	1	1	yes	characteristic
39	C₂×C₄²×C₆₁	—	2	1	no	—
40	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
41	C₄⋊C₈×C₆₁	—	2	4	no	—
42	C₄⋊C₈×C₆₁	—	2	1	no	—
43	C₄⋊C₈×C₆₁	C₂×C₄	1	2	yes	—
44	C₄×C₈×C₆₁	—	2	4	no	—
45	C₄×C₈×C₆₁	C₂×C₄	1	2	yes	—
46	C₄×C₈×C₆₁	D₄	1	2	yes	—
47	C₂×Q₈×C₆₁	—	2	1	no	—
48	C₂×Q₈×C₆₁	—	2	1	no	—
49	C₂×Q₈×C₆₁	—	2	1	no	—
50	C₂×Q₈×C₆₁	C₄⋊C₄	1	2	yes	—
51	C₂×Q₈×C₆₁	D₈	1	2	yes	—
52	C₂×Q₈×C₆₁	Q₁₆	1	2	yes	—
53	C₂²×C₄×C₆₁	—	2	1	no	—
54	C₂²×C₄×C₆₁	—	2	1	no	—
55	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
56	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
57	C₂²×C₄×C₆₁	#41	1	1	yes	characteristic
58	C₂²×Dic₆₁	—	4	2	no	—
59	C₂×C₈×C₆₁	—	2	1	no	—
60	C₂×C₈×C₆₁	—	2	1	no	—
61	C₂×C₈×C₆₁	—	2	2	no	—
62	C₂×C₈×C₆₁	—	4	1	no	—
63	C₂×C₈×C₆₁	—	4	1	no	—
64	C₂×C₈×C₆₁	C₂²⋊C₄	1	2	yes	—
65	C₂×C₈×C₆₁	C₂²⋊C₄	1	2	yes	—
66	C₂×C₈×C₆₁	SD₁₆	1	2	yes	—
67	C₂×C₈×C₆₁	Q₁₆	1	2	yes	—
68	C₄⋊C₄×C₆₁	—	2	2	no	—
69	C₄⋊C₄×C₆₁	—	2	1	no	—
70	C₄⋊C₄×C₆₁	—	2	2	no	—
71	C₄⋊C₄×C₆₁	C₄⋊C₄	1	2	yes	—
72	C₄²×C₆₁	—	2	2	no	—
73	C₄²×C₆₁	—	2	2	no	—
74	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
75	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
76	C₄²×C₆₁	C₂×Q₈	1	1	yes	characteristic
77	C₄²×C₆₁	#41	1	1	yes	characteristic
78	C₄⋊Dic₆₁	—	4	2	no	—
79	C₄⋊Dic₆₁	—	4	2	no	—
80	C₄⋊Dic₆₁	—	4	2	no	—
81	C₄⋊Dic₆₁	—	4	2	no	—
82	C₄⋊Dic₆₁	—	8	2	no	—
83	C₂×#92418	—	4	2	no	—
84	C₂×#92418	—	4	2	no	—
85	C₂×#92418	—	4	2	no	—
86	C₂³×C₆₁	#121	1	1	yes	characteristic
87	Q₈×C₆₁	—	2	2	no	—
88	Q₈×C₆₁	—	2	2	no	—
89	Q₈×C₆₁	—	2	2	no	—
90	Q₈×C₆₁	#107	1	4	yes	complemented
91	C₂×C₄×C₆₁	—	2	2	no	—
92	C₂×C₄×C₆₁	—	2	2	no	—
93	C₂×C₄×C₆₁	—	2	2	no	—
94	C₂×C₄×C₆₁	—	2	1	no	—
95	C₂×C₄×C₆₁	—	2	1	no	—
96	C₂×C₄×C₆₁	—	2	2	no	—
97	C₂×C₄×C₆₁	—	4	1	no	—
98	C₂×C₄×C₆₁	—	4	1	no	—
99	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
100	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
101	C₂×C₄×C₆₁	#117	1	1	yes	characteristic
102	C₂×C₄×C₆₁	#121	1	1	yes	characteristic
103	C₂×C₄×C₆₁	C₂²⋊Q₈	1	1	yes	characteristic
104	C₂×C₄×C₆₁	C₂²⋊Q₈	1	1	yes	characteristic
105	C₂×C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
106	C₂×C₄×C₆₁	C₂×SD₁₆	1	1	yes	characteristic
107	C₂×C₄×C₆₁	C₂×Q₁₆	1	1	yes	characteristic
108	C₂×C₄×C₆₁	C₂×Q₁₆	1	1	yes	characteristic
109	C₂×C₄×C₆₁	#135	1	1	yes	characteristic
110	C₂×C₄×C₆₁	#135	1	1	yes	characteristic
111	C₂×C₄×C₆₁	AGL(1,ℤ/8)	1	1	yes	characteristic
112	C₂×C₄×C₆₁	#137	1	1	yes	characteristic
113	C₂×Dic₆₁	—	8	2	no	—
114	C₂×Dic₆₁	—	8	2	no	—
115	C₂×Dic₆₁	—	8	2	no	—
116	C₈×C₆₁	—	2	2	no	—
117	C₈×C₆₁	—	4	2	no	—
118	C₈×C₆₁	#103	1	4	yes	complemented
119	#92418	—	8	2	no	—
120	#8645	—	61	1	no	maximal, Sylow
121	C₂²×C₆₁	#380	1	1	yes	characteristic
122	C₂²×C₆₁	#460	1	1	yes	characteristic
123	C₂²×C₆₁	#462	1	1	yes	characteristic
124	C₂²×C₆₁	#466	1	1	yes	characteristic
125	C₂²×C₆₁	#467	1	1	yes	characteristic
126	C₂²×C₆₁	#472	1	1	yes	characteristic
127	C₂²×C₆₁	#473	1	1	yes	characteristic
128	C₄×C₆₁	—	2	4	no	—
129	C₄×C₆₁	—	2	1	no	—
130	C₄×C₆₁	—	2	1	no	—
131	C₄×C₆₁	—	4	2	no	—
132	C₄×C₆₁	C₂×#103	1	1	yes	characteristic
133	C₄×C₆₁	#417	1	1	yes	characteristic
134	C₄×C₆₁	C₂×#107	1	1	yes	characteristic
135	C₄×C₆₁	#427	1	1	yes	characteristic
136	C₄×C₆₁	#466	1	1	yes	characteristic
137	C₄×C₆₁	#467	1	1	yes	characteristic
138	C₄×C₆₁	#475	1	1	yes	characteristic
139	C₄×C₆₁	#477	1	1	yes	characteristic
140	Dic₆₁	—	16	2	no	—
141	C₂×C₈×Q₈	D₆₁	1	1	yes	characteristic
142	#1845	—	61	1	no	—
143	#1803	—	61	2	no	—
144	C₂×#335	—	61	1	no	—
145	#1296	—	61	2	no	—
146	C₂×C₆₁	#1668	1	1	yes	characteristic
147	C₂×C₆₁	#1670	1	1	yes	characteristic
148	C₂×C₆₁	#1674	1	1	yes	characteristic
149	C₂×C₆₁	#1676	1	1	yes	characteristic
150	C₂×C₆₁	#1803	1	1	yes	characteristic
151	C₂×C₆₁	#1808	1	1	yes	characteristic
152	C₂×C₆₁	#1816	1	1	yes	characteristic
153	C₂×C₄×Q₈	C₂×D₆₁	1	1	yes	characteristic
154	C₈×Q₈	—	2	2	no	—
155	C₈×Q₈	Dic₆₁	1	4	yes	complemented
156	C₂×#107	—	122	1	no	—
157	C₂×#107	—	122	1	no	—
158	C₂×C₄⋊C₈	—	2	1	no	—
159	C₂×C₄⋊C₈	—	61	2	no	—
160	C₂×C₄⋊C₈	C₂×D₆₁	1	1	yes	characteristic
161	C₂×#103	—	122	2	no	—
162	C₂×C₄×C₈	—	2	1	no	—
163	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
164	#384	—	61	2	no	—
165	#340	—	122	2	no	—
166	#335	—	122	2	no	—
167	C₆₁	#8645	1	1	yes	characteristic, complemented, Sylow
168	C₂²×Q₈	D₂₄₄	1	1	yes	characteristic
169	C₂²×C₈	—	2	1	no	—
170	C₂²×C₈	—	122	2	no	—
171	C₂²×C₈	C₂²⋊D₆₁	1	1	yes	characteristic
172	C₂²×C₈	C₂²⋊D₆₁	1	1	yes	characteristic
173	C₄×Q₈	—	2	1	no	—
174	C₄×Q₈	—	2	1	no	—
175	C₄×Q₈	Dic₁₂₂	1	2	yes	—
176	C₄×Q₈	C₂×Dic₆₁	1	2	yes	—
177	C₂×C₄⋊C₄	—	2	1	no	—
178	C₂×C₄⋊C₄	—	122	2	no	—
179	C₂×C₄⋊C₄	—	122	2	no	—
180	C₂×C₄⋊C₄	—	122	2	no	—
181	C₂×C₄⋊C₄	D₂₄₄	1	1	yes	characteristic
182	C₂×C₄²	—	2	1	no	—
183	C₂×C₄²	C₂²×D₆₁	1	1	yes	characteristic
184	#107	—	244	2	no	—
185	#107	—	244	2	no	—
186	C₄⋊C₈	—	2	4	no	—
187	C₄⋊C₈	—	2	1	no	—
188	C₄⋊C₈	—	122	2	no	—
189	C₄⋊C₈	—	122	2	no	—
190	C₄⋊C₈	C₂×Dic₆₁	1	2	yes	—
191	#103	—	244	4	no	—
192	C₄×C₈	—	2	4	no	—
193	C₄×C₈	C₂×Dic₆₁	1	2	yes	—
194	C₄×C₈	C₂²⋊D₆₁	1	2	yes	—
195	C₂×Q₈	—	2	1	no	—
196	C₂×Q₈	—	2	1	no	—
197	C₂×Q₈	—	2	1	no	—
198	C₂×Q₈	D₄₈₈	1	2	yes	—
199	C₂×Q₈	Dic₂₄₄	1	2	yes	—
200	C₂×Q₈	C₄⋊Dic₆₁	1	2	yes	—
201	C₂²×C₄	—	2	1	no	—
202	C₂²×C₄	—	2	1	no	—
203	C₂²×C₄	—	244	2	no	—
204	C₂²×C₄	C₂×D₂₄₄	1	1	yes	characteristic
205	C₂²×C₄	#11758222	1	1	yes	characteristic
206	C₂²×C₄	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
207	C₂×C₈	—	2	1	no	—
208	C₂×C₈	—	2	1	no	—
209	C₂×C₈	—	2	2	no	—
210	C₂×C₈	—	4	1	no	—
211	C₂×C₈	—	4	1	no	—
212	C₂×C₈	—	244	2	no	—
213	C₂×C₈	—	244	2	no	—
214	C₂×C₈	—	244	2	no	—
215	C₂×C₈	#11758201	1	2	yes	—
216	C₂×C₈	#11758202	1	2	yes	—
217	C₂×C₈	C₂²⋊Dic₆₁	1	2	yes	—
218	C₂×C₈	C₂²⋊Dic₆₁	1	2	yes	—
219	C₄⋊C₄	—	2	2	no	—
220	C₄⋊C₄	—	2	1	no	—
221	C₄⋊C₄	—	2	2	no	—
222	C₄⋊C₄	—	488	2	no	—
223	C₄⋊C₄	—	244	2	no	—
224	C₄⋊C₄	—	244	2	no	—
225	C₄⋊C₄	—	244	2	no	—
226	C₄⋊C₄	—	244	2	no	—
227	C₄⋊C₄	C₄⋊Dic₆₁	1	2	yes	—
228	C₄²	—	2	2	no	—
229	C₄²	—	2	2	no	—
230	C₄²	C₂×Dic₁₂₂	1	1	yes	characteristic
231	C₄²	#11758222	1	1	yes	characteristic
232	C₄²	C₂²×Dic₆₁	1	1	yes	characteristic
233	C₄²	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
234	C₂³	#14519751	1	1	yes	characteristic
235	Q₈	—	2	2	no	—
236	Q₈	—	2	2	no	—
237	Q₈	—	2	2	no	—
238	Q₈	#14519711	1	4	yes	complemented
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	2	1	no	—
242	C₂×C₄	—	2	1	no	—
243	C₂×C₄	—	2	2	no	—
244	C₂×C₄	—	2	2	no	—
245	C₂×C₄	—	4	1	no	—
246	C₂×C₄	—	4	1	no	—
247	C₂×C₄	—	488	2	no	—
248	C₂×C₄	—	488	2	no	—
249	C₂×C₄	—	488	2	no	—
250	C₂×C₄	C₂×C₂²⋊Dic₆₁	1	1	yes	characteristic
251	C₂×C₄	C₂×C₄⋊Dic₆₁	1	1	yes	characteristic
252	C₂×C₄	#14519736	1	1	yes	characteristic
253	C₂×C₄	#14519751	1	1	yes	characteristic
254	C₂×C₄	#14519756	1	1	yes	characteristic
255	C₂×C₄	#14519756	1	1	yes	characteristic
256	C₂×C₄	C₂×D₄₈₈	1	1	yes	characteristic
257	C₂×C₄	C₂×#11758201	1	1	yes	characteristic
258	C₂×C₄	C₂×#11758202	1	1	yes	characteristic
259	C₂×C₄	C₂×Dic₂₄₄	1	1	yes	characteristic
260	C₂×C₄	#14519799	1	1	yes	characteristic
261	C₂×C₄	#14519799	1	1	yes	characteristic
262	C₂×C₄	#14519802	1	1	yes	characteristic
263	C₂×C₄	#14519807	1	1	yes	characteristic
264	C₈	—	2	2	no	—
265	C₈	—	4	2	no	—
266	C₈	—	488	2	no	—
267	C₈	#14519701	1	4	yes	complemented
268	C₂²	#16839622	1	1	yes	characteristic
269	C₂²	#16839962	1	1	yes	characteristic
270	C₂²	#16839972	1	1	yes	characteristic
271	C₂²	#16839998	1	1	yes	characteristic
272	C₂²	#16840003	1	1	yes	characteristic
273	C₂²	#16840032	1	1	yes	characteristic
274	C₂²	#16840037	1	1	yes	characteristic
275	C₄	—	2	4	no	—
276	C₄	—	2	1	no	—
277	C₄	—	2	1	no	—
278	C₄	—	4	2	no	—
279	C₄	—	976	2	no	—
280	C₄	C₂×#14519701	1	1	yes	characteristic
281	C₄	#16839763	1	1	yes	characteristic
282	C₄	C₂×#14519711	1	1	yes	characteristic
283	C₄	#16839800	1	1	yes	characteristic
284	C₄	#16839998	1	1	yes	characteristic
285	C₄	#16840003	1	1	yes	characteristic
286	C₄	#16840047	1	1	yes	characteristic
287	C₄	#16840057	1	1	yes	characteristic
288	C₂	#21445179	1	1	yes	characteristic
289	C₂	#21445189	1	1	yes	characteristic
290	C₂	#21445213	1	1	yes	characteristic
291	C₂	#21445223	1	1	yes	characteristic
292	C₂	#21445824	1	1	yes	characteristic
293	C₂	#21445848	1	1	yes	characteristic
294	C₂	#21445883	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#14519751
commutator	C₄²×C₆₁	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₁
Fitting	C₂×C₈×Q₈×C₆₁	C₂
socle, prime radical	C₂³×C₆₁	#121
cosocle	C₂×C₄²×C₆₁	C₂³
hypercenter	C₂×C₈×Q₈	D₆₁
hypocenter	C₆₁	#8645
LCS	G ▷ C₄²×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₈×Q₈

Complements

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

N	H ≅ G/N	type
15. C₈×Q₈×C₆₁	C₄	semidirect
90. Q₈×C₆₁	#107	semidirect
118. C₈×C₆₁	#103	semidirect
155. C₈×Q₈	Dic₆₁	semidirect
167. C₆₁	#8645	semidirect
238. Q₈	#14519711	semidirect
267. C₈	#14519701	semidirect

Maximal subgroups

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

subgroup	index
#21445979	2
#21445824	2
C₂×#16839515	2
#21443703	2
C₂×C₈×Q₈×C₆₁	2
#8645	61