Subgroups — #29837637

Order 15616.

470 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21458249	C₂	1	1	yes	maximal, characteristic
2	#21458211	C₂	1	1	yes	maximal, characteristic
3	#21448030	C₂	1	2	yes	maximal, complemented
4	#21448018	C₂	1	2	yes	maximal, complemented
5	#21447968	C₂	1	1	yes	maximal, characteristic, complemented
6	#21447958	C₂	1	1	yes	maximal, characteristic, complemented
7	#21447928	C₂	1	2	yes	maximal, complemented
8	C₄×Dic₄₈₈	C₂	1	1	yes	maximal, characteristic, complemented
9	#21447652	C₂	1	1	yes	maximal, characteristic
10	#21447606	C₂	1	2	yes	maximal, complemented
11	C₆₁×#2254	C₂	1	1	yes	maximal, characteristic, complemented
12	#16840486	—	2	1	no	—
13	#16840473	—	2	1	no	—
14	C₂×Dic₄₈₈	C₂²	1	1	yes	characteristic
15	C₂×#14519716	C₂²	1	2	yes	—
16	#16840105	—	2	2	no	—
17	#16840099	—	2	2	no	—
18	#16840054	—	2	1	no	—
19	#16840044	—	2	1	no	—
20	#16840004	C₂²	1	2	yes	—
21	#16840004	C₂²	1	2	yes	—
22	C₄×Dic₂₄₄	C₂²	1	1	yes	characteristic
23	C₄×Dic₂₄₄	C₂²	1	1	yes	characteristic
24	C₈×D₂₄₄	C₂²	1	1	yes	characteristic
25	#16839799	C₂²	1	2	yes	—
26	#107×D₆₁	C₂²	1	2	yes	—
27	#16839589	C₂²	1	2	yes	—
28	#16839588	C₂²	1	1	yes	characteristic
29	#16839587	C₂²	1	2	yes	—
30	C₄⋊#11758185	C₂²	1	1	yes	characteristic
31	#16839567	C₂²	1	1	yes	characteristic
32	#16839567	C₂²	1	1	yes	characteristic
33	#16839565	C₂²	1	2	yes	—
34	#16839565	C₂²	1	2	yes	—
35	#16839541	C₂²	1	2	yes	—
36	C₆₁×#500	C₂²	1	1	yes	characteristic
37	C₆₁×#500	C₂²	1	1	yes	characteristic
38	C₆₁×#367	C₂²	1	2	yes	—
39	C₆₁×#367	C₂²	1	2	yes	—
40	C₄×C₁₆×C₆₁	C₂²	1	1	yes	characteristic
41	C₂×Dic₂₄₄	C₂³	1	1	yes	characteristic
42	C₂×Dic₂₄₄	C₂³	1	1	yes	characteristic
43	C₂×C₈×D₆₁	C₂³	1	2	yes	—
44	#14519772	—	4	2	no	—
45	#14519768	—	4	1	no	—
46	#14519766	—	4	1	no	—
47	#14519757	—	2	2	no	—
48	#14519757	—	2	2	no	—
49	#14519753	—	4	2	no	—
50	Q₈×Dic₆₁	—	4	1	no	—
51	C₄×Dic₁₂₂	—	2	1	no	—
52	C₄×Dic₁₂₂	—	2	1	no	—
53	C₄×D₂₄₄	D₄	1	1	yes	characteristic
54	#14519739	—	4	1	no	—
55	#14519735	—	2	2	no	—
56	C₄⋊C₄×D₆₁	—	2	2	no	—
57	Dic₄₈₈	—	2	2	no	—
58	#14519716	—	2	4	no	—
59	C₂×#11758185	C₂³	1	2	yes	—
60	#14519711	C₂³	1	1	yes	characteristic
61	#14519711	C₂³	1	1	yes	characteristic
62	#14519710	—	2	2	no	—
63	#14519708	—	2	2	no	—
64	C₄⋊#92418	D₄	1	1	yes	characteristic
65	#14519702	—	2	1	no	—
66	#14519702	—	2	1	no	—
67	#14519700	—	2	2	no	—
68	#14519700	—	2	2	no	—
69	#14519689	D₄	1	2	yes	—
70	C₄⋊Q₈×C₆₁	—	2	1	no	—
71	C₄⋊Q₈×C₆₁	—	2	1	no	—
72	C₂×C₁₆×C₆₁	C₂³	1	1	yes	characteristic
73	C₂×C₁₆×C₆₁	C₂³	1	1	yes	characteristic
74	#107×C₆₁	—	2	1	no	—
75	#107×C₆₁	—	2	1	no	—
76	#107×C₆₁	C₂³	1	2	yes	—
77	#107×C₆₁	C₂³	1	2	yes	—
78	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
79	C₂×Q₈×C₆₁	—	4	1	no	—
80	C₂×Q₈×C₆₁	—	4	1	no	—
81	C₂×D₂₄₄	D₈	1	1	yes	characteristic
82	C₂×C₄×D₆₁	—	4	2	no	—
83	C₂×C₄×D₆₁	C₂×D₄	1	2	yes	—
84	C₂×Dic₁₂₂	—	2	1	no	—
85	C₂×Dic₁₂₂	—	2	1	no	—
86	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
87	C₂×C₈×C₆₁	C₂⁴	1	1	yes	characteristic
88	C₄⋊C₄×C₆₁	—	2	2	no	—
89	C₄⋊C₄×C₆₁	—	2	2	no	—
90	C₄⋊C₄×C₆₁	—	4	1	no	—
91	C₄⋊C₄×C₆₁	—	4	1	no	—
92	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
93	#11758198	—	4	2	no	—
94	#11758198	—	4	2	no	—
95	#11758198	—	4	2	no	—
96	#11758198	D₈	1	2	yes	—
97	C₄⋊Dic₆₁	—	2	1	no	—
98	C₄⋊Dic₆₁	—	2	1	no	—
99	C₄⋊Dic₆₁	—	4	2	no	—
100	C₄⋊Dic₆₁	—	4	2	no	—
101	C₄⋊Dic₆₁	D₈	1	1	yes	characteristic
102	#11758196	—	4	1	no	—
103	#11758196	—	4	1	no	—
104	#11758196	—	4	2	no	—
105	C₄×Dic₆₁	—	4	1	no	—
106	C₄×Dic₆₁	—	4	1	no	—
107	C₄×Dic₆₁	—	4	2	no	—
108	C₂×#92418	C₂×D₄	1	2	yes	—
109	Dic₂₄₄	—	2	1	no	—
110	Dic₂₄₄	—	2	1	no	—
111	Dic₂₄₄	—	2	1	no	—
112	Dic₂₄₄	—	2	1	no	—
113	C₈×D₆₁	—	2	2	no	—
114	C₈×D₆₁	—	2	2	no	—
115	C₁₆×C₆₁	—	2	1	no	—
116	C₁₆×C₆₁	#41	1	2	yes	—
117	#11758185	—	2	2	no	—
118	C₂²×D₆₁	C₂×D₈	1	2	yes	—
119	Q₈×C₆₁	—	8	2	no	—
120	Q₈×C₆₁	—	16	1	no	—
121	C₂×C₄×C₆₁	—	4	2	no	—
122	C₂×C₄×C₆₁	—	4	2	no	—
123	C₂×C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
124	C₂×C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
125	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
126	C₂×Dic₆₁	—	4	1	no	—
127	C₂×Dic₆₁	—	4	1	no	—
128	C₂×Dic₆₁	—	4	1	no	—
129	C₂×Dic₆₁	—	4	1	no	—
130	C₂×Dic₆₁	C₂×D₈	1	2	yes	—
131	D₂₄₄	—	2	2	no	—
132	C₄×D₆₁	—	2	2	no	—
133	C₄×D₆₁	—	2	2	no	—
134	C₄×D₆₁	—	16	2	no	—
135	Dic₁₂₂	—	4	1	no	—
136	Dic₁₂₂	—	4	1	no	—
137	Dic₁₂₂	—	4	1	no	—
138	Dic₁₂₂	—	4	1	no	—
139	C₈×C₆₁	—	2	1	no	—
140	C₈×C₆₁	C₂×#41	1	1	yes	characteristic
141	C₈×C₆₁	2₋¹⁺⁴	1	1	yes	characteristic
142	#92418	—	2	2	no	—
143	#33923	—	61	1	no	maximal, Sylow
144	C₂²×C₆₁	C₂²×D₈	1	1	yes	characteristic
145	C₂×D₆₁	—	2	2	no	—
146	C₂×D₆₁	—	2	2	no	—
147	C₂×D₆₁	—	2	2	no	—
148	C₄×C₆₁	—	2	1	no	—
149	C₄×C₆₁	—	8	2	no	—
150	C₄×C₆₁	—	8	2	no	—
151	C₄×C₆₁	#510	1	2	yes	—
152	C₄×C₆₁	#547	1	1	yes	characteristic
153	C₄×C₆₁	#578	1	1	yes	characteristic
154	Dic₆₁	—	2	2	no	—
155	Dic₆₁	—	8	1	no	—
156	Dic₆₁	—	8	1	no	—
157	Dic₆₁	—	8	1	no	—
158	Dic₆₁	—	8	1	no	—
159	#3342	—	61	1	no	—
160	#3338	—	61	1	no	—
161	#2254	D₆₁	1	1	yes	characteristic, complemented
162	#2237	—	61	2	no	—
163	#2235	—	61	2	no	—
164	#2227	—	61	1	no	—
165	#2225	—	61	1	no	—
166	#2219	—	61	2	no	—
167	C₄×Q₃₂	—	61	1	no	—
168	#2169	—	61	1	no	—
169	#2159	—	61	2	no	—
170	C₂×C₆₁	#3334	1	1	yes	characteristic
171	C₂×C₆₁	C₂×#510	1	1	yes	characteristic
172	C₂×C₆₁	#3416	1	1	yes	characteristic
173	D₆₁	—	4	2	no	—
174	#556	—	122	1	no	—
175	#554	—	122	1	no	—
176	C₂×Q₃₂	—	61	1	no	—
177	C₂×M₃₂	—	61	2	no	—
178	#500	C₂×D₆₁	1	1	yes	characteristic
179	#500	C₂×D₆₁	1	1	yes	characteristic
180	#485	—	122	2	no	—
181	#484	—	122	2	no	—
182	#477	—	122	1	no	—
183	#475	—	122	1	no	—
184	#467	—	61	2	no	—
185	#467	—	61	2	no	—
186	C₄×Q₁₆	—	61	1	no	—
187	C₄×Q₁₆	—	61	1	no	—
188	C₈×D₄	—	61	1	no	—
189	#427	—	61	2	no	—
190	C₂×#107	—	61	2	no	—
191	#367	—	61	2	no	—
192	#367	C₂×D₆₁	1	2	yes	—
193	#367	C₂×D₆₁	1	2	yes	—
194	#366	—	61	1	no	—
195	#366	—	61	2	no	—
196	C₄⋊C₁₆	—	61	1	no	—
197	#358	—	61	1	no	—
198	#358	—	61	1	no	—
199	#358	—	61	2	no	—
200	#358	—	61	2	no	—
201	C₂²⋊C₁₆	—	61	2	no	—
202	C₄×C₁₆	C₂×D₆₁	1	1	yes	characteristic
203	C₆₁	#33923	1	1	yes	characteristic, complemented, Sylow
204	C₂×Q₁₆	—	61	1	no	—
205	C₂×Q₁₆	—	61	1	no	—
206	C₂²×C₈	—	61	2	no	—
207	C₄⋊Q₈	—	2	1	no	—
208	C₄⋊Q₈	—	2	1	no	—
209	#126	—	244	2	no	—
210	#125	—	244	1	no	—
211	#124	—	244	1	no	—
212	C₂²⋊Q₈	—	122	2	no	—
213	C₂²⋊Q₈	—	122	2	no	—
214	C₂²⋊Q₈	—	244	2	no	—
215	C₄×Q₈	—	122	1	no	—
216	C₄×Q₈	—	122	1	no	—
217	C₄×Q₈	—	244	1	no	—
218	C₄×D₄	—	61	1	no	—
219	C₄×D₄	—	244	1	no	—
220	#117	—	122	2	no	—
221	C₂×C₄⋊C₄	—	122	2	no	—
222	Q₃₂	—	122	2	no	—
223	M₃₂	—	122	4	no	—
224	C₂×C₁₆	—	61	2	no	—
225	C₂×C₁₆	C₂²×D₆₁	1	1	yes	characteristic
226	C₂×C₁₆	C₂²×D₆₁	1	1	yes	characteristic
227	#107	—	2	1	no	—
228	#107	—	2	1	no	—
229	#107	—	61	1	no	—
230	#107	—	61	1	no	—
231	#107	—	122	2	no	—
232	#107	C₂²×D₆₁	1	2	yes	—
233	#107	C₂²×D₆₁	1	2	yes	—
234	#106	—	122	2	no	—
235	C₄⋊C₈	—	61	1	no	—
236	#103	—	122	2	no	—
237	#103	—	122	2	no	—
238	#103	—	122	1	no	—
239	#103	—	122	1	no	—
240	C₂²⋊C₈	—	61	2	no	—
241	C₄×C₈	C₂²×D₆₁	1	1	yes	characteristic
242	C₂×Q₈	—	4	1	no	—
243	C₂×Q₈	—	4	1	no	—
244	C₂×Q₈	—	122	1	no	—
245	C₂×Q₈	—	122	1	no	—
246	C₂×D₄	—	61	1	no	—
247	C₂²×C₄	—	61	2	no	—
248	C₂²×C₄	—	244	2	no	—
249	Q₁₆	—	122	1	no	—
250	Q₁₆	—	122	1	no	—
251	Q₁₆	—	122	1	no	—
252	Q₁₆	—	122	1	no	—
253	C₂×C₈	—	61	2	no	—
254	C₂×C₈	—	122	2	no	—
255	C₂×C₈	—	122	2	no	—
256	C₂×C₈	D₄×D₆₁	1	1	yes	characteristic
257	C₂×C₈	C₂³×D₆₁	1	1	yes	characteristic
258	C₄⋊C₄	—	2	2	no	—
259	C₄⋊C₄	—	2	2	no	—
260	C₄⋊C₄	—	4	1	no	—
261	C₄⋊C₄	—	4	1	no	—
262	C₄⋊C₄	—	61	1	no	—
263	C₄⋊C₄	—	122	1	no	—
264	C₄⋊C₄	—	122	1	no	—
265	C₄⋊C₄	—	244	2	no	—
266	C₄⋊C₄	—	244	2	no	—
267	C₄⋊C₄	—	244	2	no	—
268	C₄⋊C₄	—	244	1	no	—
269	C₄⋊C₄	—	244	1	no	—
270	C₂²⋊C₄	—	61	2	no	—
271	C₂²⋊C₄	—	244	2	no	—
272	C₂²⋊C₄	—	244	2	no	—
273	C₂²⋊C₄	—	244	2	no	—
274	C₄²	—	244	2	no	—
275	C₄²	—	244	1	no	—
276	C₄²	—	244	1	no	—
277	C₄²	D₄×D₆₁	1	1	yes	characteristic
278	C₁₆	—	2	1	no	—
279	C₁₆	—	122	2	no	—
280	C₁₆	#11758226	1	2	yes	—
281	C₂³	—	61	2	no	—
282	Q₈	—	8	2	no	—
283	Q₈	—	16	1	no	—
284	Q₈	—	244	1	no	—
285	Q₈	—	244	1	no	—
286	Q₈	—	244	1	no	—
287	Q₈	—	244	1	no	—
288	D₄	—	122	2	no	—
289	C₂×C₄	—	4	2	no	—
290	C₂×C₄	—	4	2	no	—
291	C₂×C₄	—	61	2	no	—
292	C₂×C₄	—	122	2	no	—
293	C₂×C₄	—	122	2	no	—
294	C₂×C₄	—	976	2	no	—
295	C₂×C₄	—	244	1	no	—
296	C₂×C₄	—	244	1	no	—
297	C₂×C₄	—	244	1	no	—
298	C₂×C₄	—	244	1	no	—
299	C₂×C₄	D₈×D₆₁	1	1	yes	characteristic
300	C₂×C₄	D₈×D₆₁	1	1	yes	characteristic
301	C₂×C₄	C₂×D₄×D₆₁	1	1	yes	characteristic
302	C₈	—	2	1	no	—
303	C₈	—	122	2	no	—
304	C₈	C₂×#11758226	1	1	yes	characteristic
305	C₈	#14519825	1	1	yes	characteristic
306	C₂²	—	122	2	no	—
307	C₂²	—	122	2	no	—
308	C₂²	—	122	2	no	—
309	C₂²	C₂×D₈×D₆₁	1	1	yes	characteristic
310	C₄	—	2	1	no	—
311	C₄	—	8	2	no	—
312	C₄	—	8	2	no	—
313	C₄	—	122	2	no	—
314	C₄	—	488	1	no	—
315	C₄	—	488	1	no	—
316	C₄	—	488	1	no	—
317	C₄	—	488	1	no	—
318	C₄	#16840201	1	2	yes	—
319	C₄	#16840415	1	1	yes	characteristic
320	C₄	#16840575	1	1	yes	characteristic
321	C₂	—	244	2	no	—
322	C₂	#21458166	1	1	yes	characteristic
323	C₂	C₂×#16840201	1	1	yes	characteristic
324	C₂	#21458961	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₈×D₆₁
commutator, cosocle	C₂×C₈×C₆₁	C₂⁴
Frattini	C₂×C₈	C₂³×D₆₁
Fitting	C₆₁×#2254	C₂
socle	C₂²×C₆₁	C₂²×D₈
prime radical	C₂×D₂₄₄	D₈
hypercenter	#2254	D₆₁
hypocenter	C₆₁	#33923
LCS	G ▷ C₂×C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2254

Complements

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

N	H ≅ G/N	type
3. #21448030	C₂	semidirect
4. #21448018	C₂	semidirect
5. #21447968	C₂	semidirect
6. #21447958	C₂	semidirect
7. #21447928	C₂	semidirect
8. C₄×Dic₄₈₈	C₂	semidirect
10. #21447606	C₂	semidirect
11. C₆₁×#2254	C₂	semidirect
161. #2254	D₆₁	semidirect
203. C₆₁	#33923	semidirect

Maximal subgroups

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

subgroup	index
#21458249	2
#21458211	2
#21448030	2
#21448018	2
#21447968	2
#21447958	2
#21447928	2
C₄×Dic₄₈₈	2
#21447652	2
#21447606	2
C₆₁×#2254	2
#33923	61