Subgroups — #29589844

Order 15616.

378 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈×#11758226	C₂	1	1	yes	maximal, characteristic
2	#21447617	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×C₁₆×D₆₁	C₂	1	1	yes	maximal, characteristic, complemented
4	Q₈⋊#11758185	C₂	1	1	yes	maximal, characteristic, complemented
5	#21443775	C₂	1	1	yes	maximal, characteristic, complemented
6	#21443773	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₁×Q₈⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×#11758226	C₄	1	1	yes	characteristic
9	C₂×C₁₆×D₆₁	—	2	1	no	—
10	Q₈×#92418	C₄	1	1	yes	characteristic
11	C₈×D₂₄₄	—	2	1	no	—
12	C₈×D₂₄₄	C₂²	1	1	yes	characteristic
13	#16839827	—	2	1	no	—
14	#16839827	C₄	1	1	yes	characteristic
15	#16839814	—	2	1	no	—
16	#16839814	C₄	1	1	yes	characteristic
17	C₄×C₈×D₆₁	—	2	1	no	—
18	C₄×C₈×D₆₁	C₂²	1	1	yes	characteristic
19	C₄⋊#11758185	C₂²	1	1	yes	characteristic
20	#16839541	—	2	1	no	—
21	C₄×#11758185	C₂²	1	1	yes	characteristic
22	C₁₆×Dic₆₁	—	2	1	no	—
23	C₁₆×Dic₆₁	—	2	1	no	—
24	C₈×Q₈×C₆₁	C₂²	1	1	yes	characteristic
25	C₆₁×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
26	C₄×C₁₆×C₆₁	C₂²	1	1	yes	characteristic
27	C₂×#11758226	C₈	1	1	yes	characteristic
28	C₂×C₈×D₆₁	—	2	1	no	—
29	C₂×C₈×D₆₁	—	2	1	no	—
30	C₂×C₈×D₆₁	D₄	1	1	yes	characteristic
31	C₂×C₈×D₆₁	D₄	1	1	yes	characteristic
32	Q₈×Dic₆₁	C₈	1	1	yes	characteristic
33	C₄×D₂₄₄	—	2	1	no	—
34	C₄×D₂₄₄	C₂×C₄	1	1	yes	characteristic
35	#14519739	—	2	1	no	—
36	#14519739	C₈	1	1	yes	characteristic
37	#14519735	—	2	1	no	—
38	#14519735	C₈	1	1	yes	characteristic
39	C₄²×D₆₁	—	2	1	no	—
40	C₄²×D₆₁	C₂×C₄	1	1	yes	characteristic
41	C₂×#11758185	—	2	1	no	—
42	C₂×#11758185	—	2	1	no	—
43	C₁₆×D₆₁	—	2	4	no	—
44	C₄⋊#92418	—	2	1	no	—
45	C₄⋊#92418	C₂×C₄	1	1	yes	characteristic
46	#14519689	—	2	1	no	—
47	#14519689	—	2	1	no	—
48	#14519689	—	2	1	no	—
49	C₄×#92418	—	2	1	no	—
50	C₄×#92418	C₂×C₄	1	1	yes	characteristic
51	C₈×Dic₆₁	—	2	1	no	—
52	C₈×Dic₆₁	D₄	1	1	yes	characteristic
53	C₈×Dic₆₁	D₄	1	1	yes	characteristic
54	C₄×Q₈×C₆₁	C₂×C₄	1	1	yes	characteristic
55	C₂×C₁₆×C₆₁	—	2	1	no	—
56	C₂×C₁₆×C₆₁	—	2	1	no	—
57	C₄⋊C₈×C₆₁	—	2	1	no	—
58	C₄⋊C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
59	C₄×C₈×C₆₁	—	2	1	no	—
60	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
61	C₂×Q₈×C₆₁	C₂×C₈	1	1	yes	characteristic
62	#11758226	—	2	2	no	—
63	#11758226	C₁₆	1	4	yes	complemented
64	C₂×D₂₄₄	—	2	1	no	—
65	C₂×D₂₄₄	C₂×C₈	1	1	yes	characteristic
66	C₂×C₄×D₆₁	—	2	1	no	—
67	C₂×C₄×D₆₁	—	2	1	no	—
68	C₂×C₄×D₆₁	—	2	1	no	—
69	C₂×C₄×D₆₁	C₂²⋊C₄	1	1	yes	characteristic
70	C₂×C₄×D₆₁	C₂×C₈	1	1	yes	characteristic
71	C₂×C₄×D₆₁	M₁₆	1	1	yes	characteristic
72	C₂×C₈×C₆₁	—	2	1	no	—
73	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
74	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
75	C₄⋊C₄×C₆₁	—	2	1	no	—
76	C₄⋊C₄×C₆₁	C₂×C₈	1	1	yes	characteristic
77	C₄²×C₆₁	—	2	1	no	—
78	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
79	#11758198	—	2	1	no	—
80	#11758198	—	2	1	no	—
81	#11758198	—	2	1	no	—
82	C₄⋊Dic₆₁	—	2	1	no	—
83	C₄⋊Dic₆₁	C₂×C₈	1	1	yes	characteristic
84	C₄×Dic₆₁	—	2	1	no	—
85	C₄×Dic₆₁	—	2	1	no	—
86	C₄×Dic₆₁	C₂²⋊C₄	1	1	yes	characteristic
87	C₄×Dic₆₁	C₂×C₈	1	1	yes	characteristic
88	C₄×Dic₆₁	M₁₆	1	1	yes	characteristic
89	C₂×#92418	—	2	1	no	—
90	C₂×#92418	C₂²⋊C₄	1	1	yes	characteristic
91	C₂×#92418	C₂²⋊C₄	1	1	yes	characteristic
92	C₈×D₆₁	—	2	1	no	—
93	C₈×D₆₁	—	2	1	no	—
94	C₈×D₆₁	—	2	2	no	—
95	C₈×D₆₁	—	4	1	no	—
96	C₈×D₆₁	—	4	1	no	—
97	C₈×D₆₁	—	4	2	no	—
98	C₁₆×C₆₁	—	2	2	no	—
99	C₁₆×C₆₁	—	4	1	no	—
100	#11758185	—	2	2	no	—
101	#11758185	—	4	1	no	—
102	C₂²×D₆₁	—	2	1	no	—
103	C₂²×D₆₁	C₂²⋊C₈	1	1	yes	characteristic
104	Q₈×C₆₁	—	2	1	no	—
105	Q₈×C₆₁	C₂×C₁₆	1	2	yes	complemented
106	C₂×C₄×C₆₁	—	2	1	no	—
107	C₂×C₄×C₆₁	—	2	1	no	—
108	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
109	C₂×C₄×C₆₁	C₂²×C₈	1	1	yes	characteristic
110	C₂×C₄×C₆₁	C₂×M₁₆	1	1	yes	characteristic
111	C₂×Dic₆₁	—	2	1	no	—
112	C₂×Dic₆₁	C₂²⋊C₈	1	1	yes	characteristic
113	C₂×Dic₆₁	C₂²⋊C₈	1	1	yes	characteristic
114	C₂×Dic₆₁	C₂²⋊C₈	1	1	yes	characteristic
115	D₂₄₄	—	2	4	no	—
116	D₂₄₄	—	2	1	no	—
117	D₂₄₄	C₂×C₁₆	1	2	yes	complemented
118	C₄×D₆₁	—	2	1	no	—
119	C₄×D₆₁	—	2	1	no	—
120	C₄×D₆₁	—	2	4	no	—
121	C₄×D₆₁	—	2	2	no	—
122	C₄×D₆₁	—	4	1	no	—
123	C₄×D₆₁	—	4	1	no	—
124	C₄×D₆₁	—	4	2	no	—
125	C₄×D₆₁	C₂×C₁₆	1	2	yes	—
126	C₄×D₆₁	M₃₂	1	2	yes	—
127	C₈×C₆₁	—	2	1	no	—
128	C₈×C₆₁	—	4	1	no	—
129	C₈×C₆₁	#135	1	1	yes	characteristic
130	C₈×C₆₁	#135	1	1	yes	characteristic
131	#92418	—	2	1	no	—
132	#92418	—	4	1	no	—
133	#92418	#104	1	2	yes	—
134	#11613	—	61	1	no	maximal, Sylow
135	C₂²×C₆₁	C₂×C₂²⋊C₈	1	1	yes	characteristic
136	C₂×D₆₁	—	2	1	no	—
137	C₂×D₆₁	—	2	1	no	—
138	C₂×D₆₁	—	2	2	no	—
139	C₂×D₆₁	—	4	1	no	—
140	C₂×D₆₁	—	4	1	no	—
141	C₂×D₆₁	C₂²⋊C₁₆	1	2	yes	—
142	C₄×C₆₁	—	2	2	no	—
143	C₄×C₆₁	—	2	1	no	—
144	C₄×C₆₁	—	4	1	no	—
145	C₄×C₆₁	#416	1	1	yes	characteristic
146	C₄×C₆₁	C₂×#104	1	1	yes	characteristic
147	C₄×C₆₁	C₂²×C₁₆	1	1	yes	characteristic
148	C₄×C₆₁	C₂×M₃₂	1	1	yes	characteristic
149	Dic₆₁	—	2	1	no	—
150	Dic₆₁	—	4	1	no	—
151	Dic₆₁	C₂²⋊C₁₆	1	2	yes	—
152	Dic₆₁	#350	1	2	yes	—
153	C₈×#41	—	61	1	no	—
154	#2162	—	61	1	no	—
155	C₂×C₄×C₁₆	—	61	1	no	—
156	Q₈⋊C₁₆	—	61	1	no	—
157	Q₈⋊C₁₆	D₆₁	1	1	yes	characteristic, complemented
158	D₄⋊C₁₆	—	61	1	no	—
159	D₄⋊C₁₆	—	61	1	no	—
160	C₂×C₆₁	#1476	1	1	yes	characteristic
161	C₂×C₆₁	C₂×C₂²⋊C₁₆	1	1	yes	characteristic
162	C₂×C₆₁	C₂×#350	1	1	yes	characteristic
163	D₆₁	—	2	2	no	—
164	D₆₁	—	4	2	no	—
165	C₄×#41	—	61	1	no	—
166	C₂²×C₁₆	—	122	1	no	—
167	C₈×Q₈	—	61	1	no	—
168	C₈×Q₈	C₂×D₆₁	1	1	yes	characteristic
169	C₈×D₄	—	61	1	no	—
170	C₈×D₄	—	61	1	no	—
171	C₈×D₄	—	122	1	no	—
172	C₈×D₄	—	122	1	no	—
173	#431	—	61	1	no	—
174	#431	—	122	1	no	—
175	C₂×C₄×C₈	—	61	1	no	—
176	C₂×C₄×C₈	—	122	1	no	—
177	C₄⋊C₁₆	—	61	1	no	—
178	C₄⋊C₁₆	C₂×D₆₁	1	1	yes	characteristic
179	C₂²⋊C₁₆	—	122	1	no	—
180	C₄×C₁₆	—	61	1	no	—
181	C₄×C₁₆	—	122	1	no	—
182	C₄×C₁₆	—	122	1	no	—
183	C₄×C₁₆	C₂×D₆₁	1	1	yes	characteristic
184	C₆₁	#11613	1	1	yes	characteristic, complemented, Sylow
185	C₂×#41	—	61	1	no	—
186	C₂²×C₈	—	61	1	no	—
187	C₂²×C₈	—	61	1	no	—
188	C₂²×C₈	—	122	1	no	—
189	C₂²×C₈	—	122	1	no	—
190	C₄×Q₈	—	61	1	no	—
191	C₄×Q₈	C₄×D₆₁	1	1	yes	characteristic
192	C₄×D₄	—	61	1	no	—
193	C₄×D₄	—	61	1	no	—
194	C₄×D₄	—	122	1	no	—
195	C₄×D₄	—	122	1	no	—
196	#117	—	61	1	no	—
197	#117	—	122	1	no	—
198	C₂×C₄²	—	61	1	no	—
199	C₂×C₄²	—	122	1	no	—
200	C₂×C₁₆	—	2	1	no	—
201	C₂×C₁₆	—	2	1	no	—
202	C₂×C₁₆	—	122	4	no	—
203	C₂×C₁₆	—	122	1	no	—
204	C₂×C₁₆	—	122	1	no	—
205	C₄⋊C₈	—	2	1	no	—
206	C₄⋊C₈	—	61	1	no	—
207	C₄⋊C₈	—	122	1	no	—
208	C₄⋊C₈	C₄×D₆₁	1	1	yes	characteristic
209	C₂²⋊C₈	—	122	1	no	—
210	C₂²⋊C₈	—	122	1	no	—
211	C₂²⋊C₈	—	122	1	no	—
212	C₄×C₈	—	2	1	no	—
213	C₄×C₈	—	61	1	no	—
214	C₄×C₈	—	61	1	no	—
215	C₄×C₈	—	61	1	no	—
216	C₄×C₈	—	122	1	no	—
217	C₄×C₈	—	122	1	no	—
218	C₄×C₈	C₂²×D₆₁	1	1	yes	characteristic
219	#41	—	61	4	no	—
220	#41	—	122	2	no	—
221	C₂×Q₈	C₈×D₆₁	1	1	yes	characteristic
222	C₂×D₄	—	61	1	no	—
223	C₂×D₄	—	122	1	no	—
224	C₂²×C₄	—	61	1	no	—
225	C₂²×C₄	—	61	1	no	—
226	C₂²×C₄	—	61	1	no	—
227	C₂²×C₄	—	122	1	no	—
228	C₂²×C₄	—	122	1	no	—
229	C₂²×C₄	—	122	1	no	—
230	C₂×C₈	—	2	1	no	—
231	C₂×C₈	—	61	1	no	—
232	C₂×C₈	—	61	1	no	—
233	C₂×C₈	—	122	2	no	—
234	C₂×C₈	—	122	1	no	—
235	C₂×C₈	—	122	1	no	—
236	C₂×C₈	—	122	1	no	—
237	C₂×C₈	—	244	2	no	—
238	C₂×C₈	—	244	1	no	—
239	C₂×C₈	—	244	1	no	—
240	C₂×C₈	D₄×D₆₁	1	1	yes	characteristic
241	C₂×C₈	D₄×D₆₁	1	1	yes	characteristic
242	C₄⋊C₄	—	2	1	no	—
243	C₄⋊C₄	—	61	1	no	—
244	C₄⋊C₄	—	122	1	no	—
245	C₄⋊C₄	C₈×D₆₁	1	1	yes	characteristic
246	C₂²⋊C₄	—	122	1	no	—
247	C₂²⋊C₄	—	122	1	no	—
248	C₂²⋊C₄	—	122	1	no	—
249	C₄²	—	2	1	no	—
250	C₄²	—	61	1	no	—
251	C₄²	—	61	1	no	—
252	C₄²	—	61	1	no	—
253	C₄²	—	122	1	no	—
254	C₄²	—	122	1	no	—
255	C₄²	C₂×C₄×D₆₁	1	1	yes	characteristic
256	C₁₆	—	2	2	no	—
257	C₁₆	—	4	1	no	—
258	C₁₆	—	122	2	no	—
259	C₁₆	—	244	1	no	—
260	C₂³	—	61	1	no	—
261	C₂³	—	122	1	no	—
262	Q₈	—	2	1	no	—
263	Q₈	C₁₆×D₆₁	1	2	yes	complemented
264	D₄	—	61	2	no	—
265	D₄	—	122	4	no	—
266	D₄	—	122	1	no	—
267	C₂×C₄	—	2	1	no	—
268	C₂×C₄	—	2	1	no	—
269	C₂×C₄	—	61	1	no	—
270	C₂×C₄	—	61	1	no	—
271	C₂×C₄	—	61	2	no	—
272	C₂×C₄	—	61	2	no	—
273	C₂×C₄	—	61	1	no	—
274	C₂×C₄	—	122	2	no	—
275	C₂×C₄	—	122	4	no	—
276	C₂×C₄	—	122	1	no	—
277	C₂×C₄	—	122	1	no	—
278	C₂×C₄	—	122	1	no	—
279	C₂×C₄	—	244	2	no	—
280	C₂×C₄	—	244	1	no	—
281	C₂×C₄	—	244	1	no	—
282	C₂×C₄	C₂²⋊C₄×D₆₁	1	1	yes	characteristic
283	C₂×C₄	C₂×C₈×D₆₁	1	1	yes	characteristic
284	C₂×C₄	M₁₆×D₆₁	1	1	yes	characteristic
285	C₈	—	2	1	no	—
286	C₈	—	4	1	no	—
287	C₈	—	61	2	no	—
288	C₈	—	122	1	no	—
289	C₈	—	244	1	no	—
290	C₈	#14519798	1	1	yes	characteristic
291	C₈	#14519800	1	1	yes	characteristic
292	C₂²	—	61	2	no	—
293	C₂²	—	122	1	no	—
294	C₂²	—	122	1	no	—
295	C₂²	—	122	2	no	—
296	C₂²	—	244	1	no	—
297	C₂²	—	244	1	no	—
298	C₂²	C₂²⋊C₈×D₆₁	1	1	yes	characteristic
299	C₄	—	2	2	no	—
300	C₄	—	2	1	no	—
301	C₄	—	4	1	no	—
302	C₄	—	61	2	no	—
303	C₄	—	61	2	no	—
304	C₄	—	122	1	no	—
305	C₄	—	244	1	no	—
306	C₄	#16839757	1	1	yes	characteristic
307	C₄	#104×D₆₁	1	1	yes	characteristic
308	C₄	C₂×C₁₆×D₆₁	1	1	yes	characteristic
309	C₄	M₃₂×D₆₁	1	1	yes	characteristic
310	C₂	—	122	2	no	—
311	C₂	—	244	2	no	—
312	C₂	#21444200	1	1	yes	characteristic
313	C₂	C₂²⋊C₁₆×D₆₁	1	1	yes	characteristic
314	C₂	#350×D₆₁	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	D₄×D₆₁
commutator	C₄×C₆₁	C₂²×C₁₆
Frattini	C₄×C₈	C₂²×D₆₁
Fitting	C₆₁×Q₈⋊C₁₆	C₂
socle	C₂²×C₆₁	C₂×C₂²⋊C₈
prime radical	C₂×#11758226	C₈
cosocle	C₄×C₈×C₆₁	C₂³
hypercenter	Q₈⋊C₁₆	D₆₁
hypocenter	C₆₁	#11613
LCS	G ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₈ ◁ C₄×C₈ ◁ Q₈⋊C₁₆

Complements

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

N	H ≅ G/N	type
2. #21447617	C₂	semidirect
3. C₄×C₁₆×D₆₁	C₂	semidirect
4. Q₈⋊#11758185	C₂	semidirect
5. #21443775	C₂	semidirect
6. #21443773	C₂	semidirect
7. C₆₁×Q₈⋊C₁₆	C₂	semidirect
63. #11758226	C₁₆	semidirect
105. Q₈×C₆₁	C₂×C₁₆	semidirect
117. D₂₄₄	C₂×C₁₆	semidirect
157. Q₈⋊C₁₆	D₆₁	semidirect
184. C₆₁	#11613	semidirect
263. Q₈	C₁₆×D₆₁	semidirect

Maximal subgroups

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

subgroup	index
C₈×#11758226	2
#21447617	2
C₄×C₁₆×D₆₁	2
Q₈⋊#11758185	2
#21443775	2
#21443773	2
C₆₁×Q₈⋊C₁₆	2
#11613	61