Subgroups — #29837664

Order 15616.

506 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21458824	C₂	1	1	yes	maximal, characteristic, complemented
2	#21458671	C₂	1	1	yes	maximal, characteristic
3	#21448048	C₂	1	1	yes	maximal, characteristic, complemented
4	#21448045	C₂	1	2	yes	maximal, complemented
5	#21447960	C₂	1	2	yes	maximal, complemented
6	#21447865	C₂	1	1	yes	maximal, characteristic
7	#21447863	C₂	1	2	yes	maximal
8	#21447721	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄×#14519720	C₂	1	1	yes	maximal, characteristic
10	#21447670	C₂	1	2	yes	maximal
11	C₆₁×#2225	C₂	1	1	yes	maximal, characteristic, complemented
12	#16840490	—	2	1	no	—
13	#16840427	—	2	1	no	—
14	C₂×#14519722	C₂²	1	2	yes	—
15	C₂×#14519720	C₂²	1	1	yes	characteristic
16	#16840166	C₂²	1	2	yes	—
17	#16840142	C₂²	1	1	yes	characteristic
18	#16840140	C₂²	1	2	yes	—
19	#16840041	—	2	2	no	—
20	#16840039	—	2	1	no	—
21	#16839972	—	2	1	no	—
22	#16839971	—	2	2	no	—
23	C₈×Dic₁₂₂	C₂²	1	1	yes	characteristic
24	#16839856	C₂²	1	2	yes	—
25	C₄×D₄₈₈	C₂²	1	1	yes	characteristic
26	#16839844	C₂²	1	2	yes	—
27	#16839589	C₂²	1	1	yes	characteristic
28	#16839580	C₂²	1	2	yes	—
29	#16839567	C₂²	1	2	yes	—
30	#16839566	C₂²	1	1	yes	characteristic
31	#16839565	C₂²	1	2	yes	—
32	#16839565	C₂²	1	2	yes	—
33	#16839562	C₂²	1	1	yes	characteristic
34	C₄×#11758185	C₂²	1	1	yes	characteristic
35	C₁₆×Dic₆₁	C₂²	1	2	yes	—
36	C₆₁×#500	C₂²	1	1	yes	characteristic
37	C₄×Q₁₆×C₆₁	C₂²	1	1	yes	characteristic
38	C₆₁×#367	C₂²	1	2	yes	—
39	C₆₁×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
40	C₆₁×#358	C₂²	1	2	yes	—
41	C₂×Q₈×D₆₁	—	4	1	no	—
42	C₂×Dic₂₄₄	—	2	1	no	—
43	C₂×Dic₂₄₄	C₂³	1	2	yes	—
44	C₂×#11758202	—	2	2	no	—
45	C₂×D₄₈₈	C₂³	1	1	yes	characteristic
46	#14519777	—	2	1	no	—
47	#14519776	—	2	2	no	—
48	#14519775	—	2	2	no	—
49	#14519769	—	4	1	no	—
50	#14519767	—	4	2	no	—
51	#14519755	—	4	2	no	—
52	#14519753	—	4	1	no	—
53	C₄×Dic₁₂₂	—	4	1	no	—
54	C₄×Dic₁₂₂	D₄	1	1	yes	characteristic
55	#14519742	—	2	2	no	—
56	C₄×D₂₄₄	—	2	1	no	—
57	#14519739	—	2	2	no	—
58	#14519722	—	2	4	no	—
59	#14519720	—	2	2	no	—
60	C₂×#11758185	C₂³	1	1	yes	characteristic
61	C₂×#11758185	C₂³	1	1	yes	characteristic
62	#14519711	C₂³	1	1	yes	characteristic
63	#14519710	—	2	2	no	—
64	C₄⋊#92418	D₄	1	1	yes	characteristic
65	#14519706	D₄	1	2	yes	—
66	#14519700	—	2	2	no	—
67	#14519699	—	2	1	no	—
68	#14519697	—	2	2	no	—
69	C₈×Dic₆₁	C₂³	1	2	yes	—
70	C₂×Q₁₆×C₆₁	C₂³	1	1	yes	characteristic
71	C₄⋊Q₈×C₆₁	—	2	1	no	—
72	C₄×Q₈×C₆₁	—	2	1	no	—
73	C₂×C₁₆×C₆₁	C₂³	1	2	yes	—
74	#107×C₆₁	—	2	1	no	—
75	#107×C₆₁	C₂³	1	2	yes	—
76	#107×C₆₁	C₂³	1	1	yes	characteristic
77	#103×C₆₁	—	2	1	no	—
78	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
79	C₂×Q₈×C₆₁	—	2	1	no	—
80	C₂×Q₈×C₆₁	—	4	1	no	—
81	Q₈×D₆₁	—	8	4	no	—
82	C₂×D₂₄₄	—	2	1	no	—
83	C₂×C₄×D₆₁	—	4	1	no	—
84	C₂×C₄×D₆₁	—	4	2	no	—
85	C₂×Dic₁₂₂	—	2	2	no	—
86	C₂×Dic₁₂₂	—	4	2	no	—
87	C₂×Dic₁₂₂	D₈	1	1	yes	characteristic
88	Q₁₆×C₆₁	—	2	1	no	—
89	Q₁₆×C₆₁	—	2	1	no	—
90	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
91	C₂×C₈×C₆₁	C₂⁴	1	1	yes	characteristic
92	C₄⋊C₄×C₆₁	—	2	1	no	—
93	C₄⋊C₄×C₆₁	—	2	2	no	—
94	C₄⋊C₄×C₆₁	—	4	1	no	—
95	C₄⋊C₄×C₆₁	—	4	1	no	—
96	C₄²×C₆₁	—	4	1	no	—
97	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
98	#11758202	—	8	2	no	—
99	#11758198	—	4	1	no	—
100	#11758198	—	4	2	no	—
101	C₄⋊Dic₆₁	—	2	1	no	—
102	C₄⋊Dic₆₁	—	4	1	no	—
103	C₄⋊Dic₆₁	—	4	1	no	—
104	C₄⋊Dic₆₁	D₈	1	1	yes	characteristic
105	#11758196	—	4	2	no	—
106	#11758196	—	4	2	no	—
107	#11758196	—	4	2	no	—
108	#11758196	—	4	2	no	—
109	#11758196	D₈	1	2	yes	—
110	C₄×Dic₆₁	—	4	2	no	—
111	C₄×Dic₆₁	C₂×D₄	1	2	yes	—
112	C₂×#92418	C₂×D₄	1	2	yes	—
113	Dic₂₄₄	—	2	2	no	—
114	Dic₂₄₄	—	2	2	no	—
115	Dic₂₄₄	—	8	1	no	—
116	D₄₈₈	—	2	1	no	—
117	D₄₈₈	#41	1	2	yes	—
118	C₁₆×C₆₁	—	2	2	no	—
119	#11758185	—	2	1	no	—
120	#11758185	—	2	1	no	—
121	C₂²×D₆₁	—	4	1	no	—
122	Q₈×C₆₁	—	4	1	no	—
123	Q₈×C₆₁	—	4	1	no	—
124	Q₈×C₆₁	—	8	2	no	—
125	C₂×C₄×C₆₁	—	4	1	no	—
126	C₂×C₄×C₆₁	—	4	1	no	—
127	C₂×C₄×C₆₁	—	4	2	no	—
128	C₂×C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
129	C₂×C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
130	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
131	C₂×Dic₆₁	—	4	2	no	—
132	C₂×Dic₆₁	—	4	1	no	—
133	C₂×Dic₆₁	C₂×D₈	1	2	yes	—
134	C₂×Dic₆₁	C₂×D₈	1	2	yes	—
135	D₂₄₄	—	2	2	no	—
136	D₂₄₄	—	4	1	no	—
137	C₄×D₆₁	—	8	2	no	—
138	C₄×D₆₁	—	8	4	no	—
139	Dic₁₂₂	—	4	2	no	—
140	Dic₁₂₂	—	4	2	no	—
141	Dic₁₂₂	—	8	4	no	—
142	Dic₁₂₂	SD₃₂	1	4	yes	complemented
143	C₈×C₆₁	—	2	1	no	—
144	C₈×C₆₁	C₂×#41	1	1	yes	characteristic
145	C₈×C₆₁	2₋¹⁺⁴	1	1	yes	characteristic
146	#92418	—	2	2	no	—
147	#33926	—	61	1	no	maximal, Sylow
148	C₂²×C₆₁	C₂²×D₈	1	1	yes	characteristic
149	C₂×D₆₁	—	4	2	no	—
150	C₂×D₆₁	—	8	1	no	—
151	C₂×D₆₁	—	8	1	no	—
152	C₄×C₆₁	—	2	1	no	—
153	C₄×C₆₁	—	8	1	no	—
154	C₄×C₆₁	—	8	2	no	—
155	C₄×C₆₁	—	8	1	no	—
156	C₄×C₆₁	C₂×SD₃₂	1	2	yes	—
157	C₄×C₆₁	#548	1	1	yes	characteristic
158	C₄×C₆₁	#577	1	1	yes	characteristic
159	Dic₆₁	—	2	2	no	—
160	Dic₆₁	—	8	2	no	—
161	Dic₆₁	—	8	1	no	—
162	Dic₆₁	C₂×SD₃₂	1	4	yes	—
163	#3395	—	61	1	no	—
164	Q₈×D₈	—	61	1	no	—
165	#2241	—	61	1	no	—
166	#2241	—	61	2	no	—
167	#2225	—	61	2	no	—
168	#2225	D₆₁	1	1	yes	characteristic, complemented
169	#2208	—	61	1	no	—
170	#2208	—	61	2	no	—
171	Q₈×C₁₆	—	61	1	no	—
172	C₄×SD₃₂	—	61	1	no	—
173	C₄×SD₃₂	—	61	2	no	—
174	C₂×C₆₁	#3358	1	1	yes	characteristic
175	C₂×C₆₁	C₂²×SD₃₂	1	1	yes	characteristic
176	C₂×C₆₁	#3417	1	1	yes	characteristic
177	D₆₁	—	8	2	no	—
178	#557	—	122	1	no	—
179	D₄×Q₈	—	122	1	no	—
180	C₂×SD₃₂	—	61	1	no	—
181	C₂×SD₃₂	—	61	2	no	—
182	#500	—	61	2	no	—
183	#500	C₂×D₆₁	1	1	yes	characteristic
184	#494	—	61	2	no	—
185	#494	—	61	1	no	—
186	#474	—	122	2	no	—
187	#474	—	122	1	no	—
188	#462	—	122	2	no	—
189	#462	—	122	1	no	—
190	C₈×Q₈	—	61	1	no	—
191	C₄×Q₁₆	—	61	2	no	—
192	C₄×Q₁₆	C₂×D₆₁	1	1	yes	characteristic
193	C₄×D₈	—	61	2	no	—
194	C₄×D₈	—	61	1	no	—
195	#367	—	61	1	no	—
196	#367	C₂×D₆₁	1	2	yes	—
197	C₄⋊C₁₆	—	61	2	no	—
198	C₄⋊C₁₆	C₂×D₆₁	1	1	yes	characteristic
199	#358	—	61	1	no	—
200	#358	—	61	2	no	—
201	#358	—	61	2	no	—
202	#358	—	61	2	no	—
203	#358	C₂×D₆₁	1	2	yes	—
204	#357	—	61	1	no	—
205	C₄×C₁₆	—	61	2	no	—
206	C₄×C₁₆	—	61	1	no	—
207	C₆₁	#33926	1	1	yes	characteristic, complemented, Sylow
208	C₂²×Q₈	—	244	1	no	—
209	C₂×Q₁₆	—	61	2	no	—
210	C₂×Q₁₆	—	122	2	no	—
211	C₂×Q₁₆	—	122	1	no	—
212	C₂×Q₁₆	C₂²×D₆₁	1	1	yes	characteristic
213	C₂×D₈	—	61	1	no	—
214	C₄⋊Q₈	—	2	1	no	—
215	C₄⋊Q₈	—	122	2	no	—
216	C₄⋊Q₈	—	122	2	no	—
217	C₄⋊Q₈	—	122	1	no	—
218	#125	—	244	1	no	—
219	#125	—	244	2	no	—
220	C₂²⋊Q₈	—	244	2	no	—
221	C₂²⋊Q₈	—	244	1	no	—
222	C₄×Q₈	—	2	1	no	—
223	C₄×Q₈	—	61	1	no	—
224	C₄×Q₈	—	122	2	no	—
225	C₄×Q₈	—	244	1	no	—
226	C₄×D₄	—	122	2	no	—
227	C₄×D₄	—	122	1	no	—
228	SD₃₂	—	122	4	no	—
229	SD₃₂	—	122	2	no	—
230	C₂×C₁₆	—	61	1	no	—
231	C₂×C₁₆	—	61	1	no	—
232	C₂×C₁₆	C₂²×D₆₁	1	2	yes	—
233	#107	—	2	1	no	—
234	#107	—	61	1	no	—
235	#107	—	122	2	no	—
236	#107	C₂²×D₆₁	1	1	yes	characteristic
237	#107	C₂²×D₆₁	1	2	yes	—
238	C₄⋊C₈	—	61	2	no	—
239	C₄⋊C₈	—	61	1	no	—
240	#103	—	2	1	no	—
241	#103	—	122	2	no	—
242	#102	—	122	2	no	—
243	#102	—	122	1	no	—
244	C₄×C₈	—	61	2	no	—
245	C₄×C₈	C₂²×D₆₁	1	1	yes	characteristic
246	C₂×Q₈	—	2	1	no	—
247	C₂×Q₈	—	4	1	no	—
248	C₂×Q₈	—	61	1	no	—
249	C₂×Q₈	—	122	2	no	—
250	C₂×Q₈	—	488	4	no	—
251	C₂×Q₈	—	244	2	no	—
252	C₂×D₄	—	122	1	no	—
253	C₂²×C₄	—	244	2	no	—
254	C₂²×C₄	—	244	1	no	—
255	Q₁₆	—	2	1	no	—
256	Q₁₆	—	2	1	no	—
257	Q₁₆	—	122	2	no	—
258	Q₁₆	—	122	2	no	—
259	Q₁₆	—	488	1	no	—
260	Q₁₆	—	488	2	no	—
261	D₈	—	61	2	no	—
262	D₈	—	122	1	no	—
263	C₂×C₈	—	61	2	no	—
264	C₂×C₈	D₄×D₆₁	1	1	yes	characteristic
265	C₂×C₈	C₂³×D₆₁	1	1	yes	characteristic
266	C₄⋊C₄	—	2	2	no	—
267	C₄⋊C₄	—	2	1	no	—
268	C₄⋊C₄	—	4	1	no	—
269	C₄⋊C₄	—	4	1	no	—
270	C₄⋊C₄	—	61	2	no	—
271	C₄⋊C₄	—	61	1	no	—
272	C₄⋊C₄	—	122	1	no	—
273	C₄⋊C₄	—	244	2	no	—
274	C₄⋊C₄	—	244	2	no	—
275	C₄⋊C₄	—	244	2	no	—
276	C₄⋊C₄	—	244	2	no	—
277	C₄⋊C₄	—	244	1	no	—
278	C₄⋊C₄	—	244	1	no	—
279	C₂²⋊C₄	—	244	2	no	—
280	C₂²⋊C₄	—	244	1	no	—
281	C₄²	—	4	1	no	—
282	C₄²	—	61	2	no	—
283	C₄²	—	244	2	no	—
284	C₄²	D₄×D₆₁	1	1	yes	characteristic
285	C₁₆	—	2	2	no	—
286	C₁₆	—	122	1	no	—
287	C₁₆	—	122	1	no	—
288	C₂³	—	244	1	no	—
289	Q₈	—	4	1	no	—
290	Q₈	—	4	1	no	—
291	Q₈	—	8	2	no	—
292	Q₈	—	61	4	no	—
293	Q₈	—	488	4	no	—
294	Q₈	—	244	2	no	—
295	Q₈	—	244	2	no	—
296	D₄	—	122	2	no	—
297	D₄	—	244	1	no	—
298	C₂×C₄	—	4	1	no	—
299	C₂×C₄	—	4	1	no	—
300	C₂×C₄	—	4	2	no	—
301	C₂×C₄	—	61	2	no	—
302	C₂×C₄	—	61	2	no	—
303	C₂×C₄	—	488	4	no	—
304	C₂×C₄	—	488	2	no	—
305	C₂×C₄	—	244	2	no	—
306	C₂×C₄	—	244	1	no	—
307	C₂×C₄	D₈×D₆₁	1	1	yes	characteristic
308	C₂×C₄	D₈×D₆₁	1	1	yes	characteristic
309	C₂×C₄	C₂×D₄×D₆₁	1	1	yes	characteristic
310	C₈	—	2	1	no	—
311	C₈	—	122	2	no	—
312	C₈	#41×D₆₁	1	1	yes	characteristic
313	C₈	#14519825	1	1	yes	characteristic
314	C₂²	—	488	1	no	—
315	C₂²	—	488	1	no	—
316	C₂²	—	244	2	no	—
317	C₂²	C₂×D₈×D₆₁	1	1	yes	characteristic
318	C₄	—	2	1	no	—
319	C₄	—	8	1	no	—
320	C₄	—	8	1	no	—
321	C₄	—	8	2	no	—
322	C₄	—	61	4	no	—
323	C₄	—	122	2	no	—
324	C₄	—	488	2	no	—
325	C₄	—	488	1	no	—
326	C₄	SD₃₂×D₆₁	1	2	yes	—
327	C₄	#16840422	1	1	yes	characteristic
328	C₄	#16840571	1	1	yes	characteristic
329	C₂	—	488	2	no	—
330	C₂	#21458453	1	1	yes	characteristic
331	C₂	C₂×SD₃₂×D₆₁	1	1	yes	characteristic
332	C₂	#21458971	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×#2225	C₂
socle	C₂²×C₆₁	C₂²×D₈
prime radical	C₂×D₄₈₈	C₂³
hypercenter	#2225	D₆₁
hypocenter	C₆₁	#33926
LCS	G ▷ C₂×C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2225

Complements

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

N	H ≅ G/N	type
1. #21458824	C₂	semidirect
3. #21448048	C₂	semidirect
4. #21448045	C₂	semidirect
5. #21447960	C₂	semidirect
8. #21447721	C₂	semidirect
11. C₆₁×#2225	C₂	semidirect
142. Dic₁₂₂	SD₃₂	semidirect
168. #2225	D₆₁	semidirect
207. C₆₁	#33926	semidirect

Maximal subgroups

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

subgroup	index
#21458824	2
#21458671	2
#21448048	2
#21448045	2
#21447960	2
#21447865	2
#21447863	2
#21447721	2
C₄×#14519720	2
#21447670	2
C₆₁×#2225	2
#33926	61