Subgroups — #29833699

Order 15616.

536 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈×#11758222	C₂	1	1	yes	maximal, characteristic
2	#21447724	C₂	1	2	yes	maximal, complemented
3	#21447642	C₂	1	4	yes	maximal, complemented
4	#21447638	C₂	1	2	yes	maximal, complemented
5	#21447621	C₂	1	2	yes	maximal, complemented
6	#21447618	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄⋊C₁₆×D₆₁	C₂	1	2	yes	maximal, complemented
8	C₆₁×#2150	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄×#11758222	C₄	1	1	yes	characteristic
10	C₂×C₁₆×D₆₁	C₂²	1	4	yes	—
11	C₈×Dic₁₂₂	C₄	1	1	yes	characteristic
12	C₈×Dic₁₂₂	C₂²	1	1	yes	characteristic
13	C₈×D₂₄₄	C₄	1	1	yes	characteristic
14	C₈×D₂₄₄	C₂²	1	1	yes	characteristic
15	C₈×C₂²⋊D₆₁	C₄	1	2	yes	—
16	C₈×C₂²⋊D₆₁	C₂²	1	2	yes	—
17	#16839815	C₄	1	1	yes	characteristic
18	#16839813	C₄	1	2	yes	—
19	C₄×C₈×D₆₁	C₂²	1	2	yes	—
20	C₄⋊#11758185	C₂²	1	1	yes	characteristic
21	C₄⋊#11758185	C₂²	1	1	yes	characteristic
22	#16839580	C₂²	1	4	yes	—
23	C₂²⋊#11758185	C₂²	1	2	yes	—
24	#16839541	C₂²	1	4	yes	—
25	#16839537	C₂²	1	4	yes	—
26	C₄×#11758185	C₂²	1	2	yes	—
27	C₂×M₃₂×C₆₁	C₂²	1	2	yes	—
28	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
29	C₆₁×C₄⋊C₁₆	C₂²	1	2	yes	—
30	C₆₁×C₄⋊C₁₆	C₂²	1	2	yes	—
31	C₂×#11758222	C₈	1	1	yes	characteristic
32	C₂×C₈×D₆₁	C₂×C₄	1	2	yes	—
33	C₂×C₈×D₆₁	C₂³	1	2	yes	—
34	C₄×Dic₁₂₂	C₈	1	1	yes	characteristic
35	C₄×Dic₁₂₂	C₂×C₄	1	1	yes	characteristic
36	C₄×D₂₄₄	C₈	1	1	yes	characteristic
37	C₄×D₂₄₄	C₂×C₄	1	1	yes	characteristic
38	C₄×C₂²⋊D₆₁	C₈	1	2	yes	—
39	C₄×C₂²⋊D₆₁	C₂×C₄	1	2	yes	—
40	#14519736	C₈	1	1	yes	characteristic
41	#14519734	C₈	1	2	yes	—
42	C₄²×D₆₁	C₂×C₄	1	2	yes	—
43	C₂×#11758185	C₂³	1	2	yes	—
44	C₂×#11758185	C₂³	1	2	yes	—
45	C₁₆×D₆₁	—	4	4	no	—
46	C₄⋊#92418	C₂×C₄	1	1	yes	characteristic
47	C₄⋊#92418	C₂×C₄	1	1	yes	characteristic
48	#14519706	C₂×C₄	1	2	yes	—
49	#14519706	C₂×C₄	1	2	yes	—
50	C₂²⋊#92418	C₂×C₄	1	2	yes	—
51	#14519689	C₂×C₄	1	2	yes	—
52	#14519689	C₂×C₄	1	2	yes	—
53	C₄×#92418	C₂×C₄	1	2	yes	—
54	C₈×Dic₆₁	C₂×C₄	1	2	yes	—
55	C₈×Dic₆₁	C₂³	1	2	yes	—
56	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
57	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
58	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
59	M₃₂×C₆₁	—	4	2	no	—
60	C₂×C₁₆×C₆₁	C₂³	1	4	yes	—
61	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
62	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
63	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
64	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
65	C₂²×C₄×C₆₁	C₂×C₈	1	1	yes	characteristic
66	C₂²×C₄×C₆₁	C₂×C₈	1	1	yes	characteristic
67	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
68	C₂×C₂²⋊D₆₁	C₂×C₈	1	2	yes	—
69	#11758222	—	2	4	no	—
70	C₂×D₂₄₄	C₂×C₈	1	1	yes	characteristic
71	C₂×C₄×D₆₁	C₂×C₈	1	2	yes	—
72	C₂×C₄×D₆₁	C₂×C₈	1	2	yes	—
73	C₂×C₄×D₆₁	C₂²×C₄	1	2	yes	—
74	C₂×Dic₁₂₂	C₂×C₈	1	1	yes	characteristic
75	C₂×C₈×C₆₁	—	2	1	no	—
76	C₂×C₈×C₆₁	—	2	1	no	—
77	C₂×C₈×C₆₁	—	4	1	no	—
78	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
81	C₂×C₈×C₆₁	C₂⁴	1	1	yes	characteristic
82	C₄²×C₆₁	C₂×C₈	1	1	yes	characteristic
83	C₄²×C₆₁	C₂×C₈	1	1	yes	characteristic
84	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
85	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
86	C₂²⋊Dic₆₁	C₂×C₈	1	2	yes	—
87	#11758198	C₂×C₈	1	2	yes	—
88	#11758198	C₂×C₈	1	2	yes	—
89	C₄⋊Dic₆₁	C₂×C₈	1	1	yes	characteristic
90	C₄⋊Dic₆₁	C₂×C₈	1	1	yes	characteristic
91	#11758196	C₂×C₈	1	2	yes	—
92	#11758196	C₂×C₈	1	2	yes	—
93	C₄×Dic₆₁	C₂×C₈	1	2	yes	—
94	C₄×Dic₆₁	C₂×C₈	1	2	yes	—
95	C₄×Dic₆₁	C₂²×C₄	1	2	yes	—
96	C₂×#92418	C₂²×C₄	1	2	yes	—
97	C₂×#92418	C₂²×C₄	1	2	yes	—
98	C₈×D₆₁	—	2	2	no	—
99	C₈×D₆₁	—	2	2	no	—
100	C₈×D₆₁	—	4	2	no	—
101	C₁₆×C₆₁	—	2	4	no	—
102	#11758185	—	2	2	no	—
103	#11758185	—	2	2	no	—
104	C₂³×C₆₁	C₂²×C₈	1	1	yes	characteristic
105	C₂²×D₆₁	C₂²×C₈	1	2	yes	—
106	C₂×C₄×C₆₁	—	2	1	no	—
107	C₂×C₄×C₆₁	—	2	1	no	—
108	C₂×C₄×C₆₁	—	2	2	no	—
109	C₂×C₄×C₆₁	—	2	2	no	—
110	C₂×C₄×C₆₁	C₂²×C₈	1	1	yes	characteristic
111	C₂×C₄×C₆₁	C₂²×C₈	1	1	yes	characteristic
112	C₂×C₄×C₆₁	C₂²×C₈	1	1	yes	characteristic
113	C₂×C₄×C₆₁	C₂²×C₈	1	1	yes	characteristic
114	C₂×C₄×C₆₁	C₂²×C₈	1	1	yes	characteristic
115	C₂×C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
116	C₂²⋊D₆₁	—	2	4	no	—
117	C₂×Dic₆₁	C₂²×C₈	1	2	yes	—
118	C₂×Dic₆₁	C₂²×C₈	1	2	yes	—
119	C₂×Dic₆₁	C₂²×C₈	1	2	yes	—
120	D₂₄₄	M₃₂	1	4	yes	complemented
121	C₄×D₆₁	—	2	4	no	—
122	C₄×D₆₁	—	2	2	no	—
123	C₄×D₆₁	—	2	2	no	—
124	C₄×D₆₁	—	2	4	no	—
125	Dic₁₂₂	—	2	2	no	—
126	C₈×C₆₁	—	2	1	no	—
127	C₈×C₆₁	—	2	1	no	—
128	C₈×C₆₁	—	2	1	no	—
129	C₈×C₆₁	H(5,𝔽₂)	1	1	yes	characteristic
130	C₈×C₆₁	2₋¹⁺⁴	1	1	yes	characteristic
131	#92418	—	2	2	no	—
132	#92418	—	2	2	no	—
133	#33504	—	61	1	no	maximal, Sylow
134	C₂²×C₆₁	—	2	1	no	—
135	C₂²×C₆₁	—	2	1	no	—
136	C₂²×C₆₁	—	2	1	no	—
137	C₂²×C₆₁	C₂³×C₈	1	1	yes	characteristic
138	C₂×D₆₁	—	2	2	no	—
139	C₂×D₆₁	—	2	2	no	—
140	C₂×D₆₁	C₂×M₃₂	1	4	yes	—
141	C₄×C₆₁	—	2	1	no	—
142	C₄×C₆₁	—	2	1	no	—
143	C₄×C₆₁	—	2	1	no	—
144	C₄×C₆₁	C₂×M₃₂	1	2	yes	—
145	C₄×C₆₁	#504	1	2	yes	—
146	C₄×C₆₁	#520	1	1	yes	characteristic
147	C₄×C₆₁	#568	1	1	yes	characteristic
148	Dic₆₁	—	2	2	no	—
149	Dic₆₁	—	2	2	no	—
150	Dic₆₁	—	2	2	no	—
151	C₈×#41	—	61	1	no	—
152	#2183	—	61	2	no	—
153	#2168	—	61	4	no	—
154	D₄×C₁₆	—	61	2	no	—
155	#2163	—	61	2	no	—
156	#2162	—	61	1	no	—
157	#2150	D₆₁	1	1	yes	characteristic, complemented
158	C₂×C₄⋊C₁₆	—	61	2	no	—
159	C₂×C₆₁	—	4	1	no	—
160	C₂×C₆₁	#2988	1	1	yes	characteristic
161	C₂×C₆₁	C₂²×M₃₂	1	1	yes	characteristic
162	C₂×C₆₁	C₂×#504	1	1	yes	characteristic
163	D₆₁	—	2	4	no	—
164	C₄×#41	—	61	1	no	—
165	C₂×M₃₂	C₂×D₆₁	1	2	yes	—
166	C₂²×C₁₆	—	61	4	no	—
167	C₈×Q₈	—	61	1	no	—
168	C₈×Q₈	—	61	1	no	—
169	C₈×D₄	—	61	2	no	—
170	C₈×D₄	—	61	2	no	—
171	C₈×D₄	—	61	1	no	—
172	C₈×D₄	—	61	1	no	—
173	#431	—	61	1	no	—
174	#431	—	61	2	no	—
175	C₂×C₄×C₈	—	61	2	no	—
176	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
177	C₄⋊C₁₆	—	61	4	no	—
178	C₄⋊C₁₆	—	61	1	no	—
179	C₄⋊C₁₆	—	61	1	no	—
180	C₄⋊C₁₆	C₂×D₆₁	1	2	yes	—
181	C₄⋊C₁₆	C₂×D₆₁	1	2	yes	—
182	C₂²⋊C₁₆	—	61	2	no	—
183	C₂²⋊C₁₆	—	61	4	no	—
184	#346	—	61	4	no	—
185	C₄×C₁₆	—	61	2	no	—
186	C₆₁	#33504	1	1	yes	characteristic, complemented, Sylow
187	C₂×#41	—	61	1	no	—
188	C₂²×C₈	—	61	2	no	—
189	C₂²×C₈	—	61	2	no	—
190	C₂²×C₈	C₄×D₆₁	1	1	yes	characteristic
191	C₂²×C₈	C₂²×D₆₁	1	1	yes	characteristic
192	C₄×Q₈	—	61	1	no	—
193	C₄×Q₈	—	61	1	no	—
194	C₄×D₄	—	61	2	no	—
195	C₄×D₄	—	61	2	no	—
196	C₄×D₄	—	61	1	no	—
197	C₄×D₄	—	61	1	no	—
198	#117	—	61	1	no	—
199	#117	—	61	2	no	—
200	C₂×C₄²	—	61	2	no	—
201	C₂×C₄²	C₄×D₆₁	1	1	yes	characteristic
202	M₃₂	—	4	2	no	—
203	C₂×C₁₆	—	61	2	no	—
204	C₂×C₁₆	—	61	2	no	—
205	C₂×C₁₆	—	244	4	no	—
206	C₂×C₁₆	C₂²×D₆₁	1	4	yes	—
207	C₄⋊C₈	—	61	2	no	—
208	C₄⋊C₈	—	61	2	no	—
209	C₄⋊C₈	—	61	1	no	—
210	C₄⋊C₈	—	61	1	no	—
211	C₂²⋊C₈	—	61	2	no	—
212	C₂²⋊C₈	—	61	2	no	—
213	C₂²⋊C₈	—	61	2	no	—
214	C₄×C₈	—	61	2	no	—
215	C₄×C₈	—	61	2	no	—
216	C₄×C₈	—	61	2	no	—
217	C₄×C₈	C₄×D₆₁	1	1	yes	characteristic
218	C₄×C₈	C₄×D₆₁	1	1	yes	characteristic
219	C₄×C₈	C₂²×D₆₁	1	1	yes	characteristic
220	C₄×C₈	C₂²×D₆₁	1	1	yes	characteristic
221	#41	—	122	4	no	—
222	C₂×Q₈	—	61	1	no	—
223	C₂×D₄	—	61	2	no	—
224	C₂×D₄	—	61	1	no	—
225	C₂²×C₄	—	61	2	no	—
226	C₂²×C₄	—	61	2	no	—
227	C₂²×C₄	—	61	2	no	—
228	C₂²×C₄	C₈×D₆₁	1	1	yes	characteristic
229	C₂²×C₄	C₈×D₆₁	1	1	yes	characteristic
230	C₂²×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
231	C₂×C₈	—	2	1	no	—
232	C₂×C₈	—	2	1	no	—
233	C₂×C₈	—	4	1	no	—
234	C₂×C₈	—	61	2	no	—
235	C₂×C₈	—	61	2	no	—
236	C₂×C₈	—	122	2	no	—
237	C₂×C₈	—	122	2	no	—
238	C₂×C₈	—	244	2	no	—
239	C₂×C₈	C₂×C₄×D₆₁	1	1	yes	characteristic
240	C₂×C₈	C₂×C₄×D₆₁	1	1	yes	characteristic
241	C₂×C₈	C₂×C₄×D₆₁	1	1	yes	characteristic
242	C₂×C₈	C₂³×D₆₁	1	1	yes	characteristic
243	C₄⋊C₄	—	61	2	no	—
244	C₄⋊C₄	—	61	2	no	—
245	C₄⋊C₄	—	61	1	no	—
246	C₄⋊C₄	—	61	1	no	—
247	C₂²⋊C₄	—	61	2	no	—
248	C₂²⋊C₄	—	61	2	no	—
249	C₂²⋊C₄	—	61	2	no	—
250	C₄²	—	61	2	no	—
251	C₄²	—	61	2	no	—
252	C₄²	—	61	2	no	—
253	C₄²	C₈×D₆₁	1	1	yes	characteristic
254	C₄²	C₈×D₆₁	1	1	yes	characteristic
255	C₄²	C₂×C₄×D₆₁	1	1	yes	characteristic
256	C₄²	C₂×C₄×D₆₁	1	1	yes	characteristic
257	C₁₆	—	2	4	no	—
258	C₁₆	—	122	2	no	—
259	C₁₆	—	122	2	no	—
260	C₂³	—	61	2	no	—
261	C₂³	C₂×C₈×D₆₁	1	1	yes	characteristic
262	Q₈	—	122	2	no	—
263	D₄	—	61	4	no	—
264	D₄	—	122	4	no	—
265	C₂×C₄	—	2	1	no	—
266	C₂×C₄	—	2	1	no	—
267	C₂×C₄	—	2	2	no	—
268	C₂×C₄	—	2	2	no	—
269	C₂×C₄	—	61	2	no	—
270	C₂×C₄	—	61	2	no	—
271	C₂×C₄	—	61	2	no	—
272	C₂×C₄	—	122	4	no	—
273	C₂×C₄	—	122	2	no	—
274	C₂×C₄	—	122	2	no	—
275	C₂×C₄	—	122	4	no	—
276	C₂×C₄	C₂×C₈×D₆₁	1	1	yes	characteristic
277	C₂×C₄	C₂×C₈×D₆₁	1	1	yes	characteristic
278	C₂×C₄	C₂×C₈×D₆₁	1	1	yes	characteristic
279	C₂×C₄	C₂×C₈×D₆₁	1	1	yes	characteristic
280	C₂×C₄	C₂×C₈×D₆₁	1	1	yes	characteristic
281	C₂×C₄	C₂²×C₄×D₆₁	1	1	yes	characteristic
282	C₈	—	2	1	no	—
283	C₈	—	2	1	no	—
284	C₈	—	2	1	no	—
285	C₈	—	122	2	no	—
286	C₈	—	122	2	no	—
287	C₈	#14519822	1	1	yes	characteristic
288	C₈	#14519825	1	1	yes	characteristic
289	C₂²	—	2	1	no	—
290	C₂²	—	2	1	no	—
291	C₂²	—	2	1	no	—
292	C₂²	—	61	4	no	—
293	C₂²	—	122	2	no	—
294	C₂²	—	122	2	no	—
295	C₂²	C₂²×C₈×D₆₁	1	1	yes	characteristic
296	C₄	—	2	1	no	—
297	C₄	—	2	1	no	—
298	C₄	—	2	1	no	—
299	C₄	—	122	2	no	—
300	C₄	—	122	2	no	—
301	C₄	—	122	2	no	—
302	C₄	M₃₂×D₆₁	1	2	yes	—
303	C₄	#16840180	1	2	yes	—
304	C₄	#16840243	1	1	yes	characteristic
305	C₄	#16840528	1	1	yes	characteristic
306	C₂	—	4	1	no	—
307	C₂	—	122	4	no	—
308	C₂	#21454928	1	1	yes	characteristic
309	C₂	C₂×M₃₂×D₆₁	1	1	yes	characteristic
310	C₂	C₂×#16840180	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₆₁
commutator, socle	C₂²×C₆₁	C₂³×C₈
Fitting	C₆₁×#2150	C₂
prime radical	C₂×#11758222	C₈
cosocle	C₂×C₈×C₆₁	C₂⁴
hypercenter	#2150	D₆₁
hypocenter	C₆₁	#33504
LCS	G ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₈ ◁ #2150

Complements

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

N	H ≅ G/N	type
2. #21447724	C₂	semidirect
3. #21447642	C₂	semidirect
4. #21447638	C₂	semidirect
5. #21447621	C₂	semidirect
6. #21447618	C₂	semidirect
7. C₄⋊C₁₆×D₆₁	C₂	semidirect
8. C₆₁×#2150	C₂	semidirect
120. D₂₄₄	M₃₂	semidirect
157. #2150	D₆₁	semidirect
186. C₆₁	#33504	semidirect

Maximal subgroups

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

subgroup	index
C₈×#11758222	2
#21447724	2
#21447642	2
#21447638	2
#21447621	2
#21447618	2
C₄⋊C₁₆×D₆₁	2
C₆₁×#2150	2
#33504	61