Subgroups — #29837719

Order 15616.

484 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21458832	C₂	1	1	yes	maximal, characteristic
2	#21448053	C₂	1	2	yes	maximal, complemented
3	#21448044	C₂	1	1	yes	maximal, characteristic
4	#21447973	C₂	1	2	yes	maximal, complemented
5	#21447958	C₂	1	1	yes	maximal, characteristic, complemented
6	#21447881	C₂	1	2	yes	maximal, complemented
7	#21447859	C₂	1	1	yes	maximal, characteristic
8	Q₈×#11758185	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄×#14519723	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×#14519720	C₂	1	2	yes	maximal, complemented
11	C₆₁×#3383	C₂	1	1	yes	maximal, characteristic, complemented
12	#16840497	—	2	1	no	—
13	C₂×#14519723	C₂²	1	1	yes	characteristic
14	C₂×#14519720	C₂²	1	2	yes	—
15	#16840146	C₂²	1	2	yes	—
16	#16840139	C₂²	1	1	yes	characteristic
17	#16839978	—	2	2	no	—
18	#16839963	—	2	1	no	—
19	C₄×Dic₂₄₄	C₂²	1	1	yes	characteristic
20	C₄×D₄₈₈	C₂²	1	2	yes	—
21	#16839589	C₂²	1	2	yes	—
22	#16839587	C₂²	1	1	yes	characteristic
23	C₄⋊#11758185	C₂²	1	1	yes	characteristic
24	C₄⋊#11758185	C₂²	1	2	yes	—
25	#16839566	C₂²	1	1	yes	characteristic
26	#16839565	C₂²	1	2	yes	—
27	#16839565	C₂²	1	1	yes	characteristic
28	#16839562	C₂²	1	2	yes	—
29	#16839562	C₂²	1	2	yes	—
30	#16839562	C₂²	1	2	yes	—
31	C₄×#11758185	C₂²	1	1	yes	characteristic
32	C₄×#11758185	C₂²	1	2	yes	—
33	C₆₁×#557	—	2	1	no	—
34	C₆₁×#500	C₂²	1	1	yes	characteristic
35	C₆₁×#499	C₂²	1	2	yes	—
36	C₆₁×#479	—	2	2	no	—
37	C₆₁×#475	—	2	1	no	—
38	C₈×Q₈×C₆₁	C₂²	1	1	yes	characteristic
39	C₄×Q₁₆×C₆₁	C₂²	1	1	yes	characteristic
40	C₄×Q₁₆×C₆₁	C₂²	1	2	yes	—
41	C₂×Dic₂₄₄	C₂³	1	1	yes	characteristic
42	C₂×#11758190	—	2	2	no	—
43	C₂×D₄₈₈	—	2	1	no	—
44	C₂×D₄₈₈	C₂³	1	2	yes	—
45	#14519774	—	2	1	no	—
46	#14519766	—	4	1	no	—
47	#14519765	—	2	2	no	—
48	#14519762	—	4	2	no	—
49	C₄×Dic₁₂₂	—	2	1	no	—
50	C₄×D₂₄₄	—	2	2	no	—
51	#14519739	—	4	1	no	—
52	#14519723	—	2	2	no	—
53	#14519720	—	2	4	no	—
54	C₂×#11758185	C₂³	1	1	yes	characteristic
55	C₂×#11758185	C₂³	1	1	yes	characteristic
56	C₂×#11758185	C₂³	1	2	yes	—
57	#14519711	C₂³	1	1	yes	characteristic
58	#14519702	—	2	1	no	—
59	#14519699	—	2	2	no	—
60	C₂×Q₁₆×C₆₁	C₂³	1	1	yes	characteristic
61	C₄⋊Q₈×C₆₁	—	2	1	no	—
62	C₄⋊Q₈×C₆₁	—	4	1	no	—
63	#125×C₆₁	—	2	2	no	—
64	#125×C₆₁	—	4	2	no	—
65	C₄×Q₈×C₆₁	—	2	1	no	—
66	C₄×Q₈×C₆₁	—	2	2	no	—
67	C₄×Q₈×C₆₁	—	4	1	no	—
68	C₄×Q₈×C₆₁	D₄	1	1	yes	characteristic
69	#107×C₆₁	—	2	1	no	—
70	#107×C₆₁	C₂³	1	1	yes	characteristic
71	#107×C₆₁	C₂³	1	2	yes	—
72	#106×C₆₁	—	2	2	no	—
73	C₄⋊C₈×C₆₁	D₄	1	1	yes	characteristic
74	C₄⋊C₈×C₆₁	D₄	1	2	yes	—
75	#103×C₆₁	—	2	2	no	—
76	#103×C₆₁	—	2	1	no	—
77	C₄×C₈×C₆₁	C₂³	1	2	yes	—
78	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
79	C₂×Q₈×C₆₁	—	2	1	no	—
80	C₂×Q₈×C₆₁	—	4	1	no	—
81	C₂×Q₈×C₆₁	D₈	1	1	yes	characteristic
82	C₂×D₂₄₄	—	2	2	no	—
83	C₂×D₂₄₄	—	4	1	no	—
84	C₂×D₂₄₄	—	4	1	no	—
85	C₂×C₄×D₆₁	—	4	2	no	—
86	C₂×Dic₁₂₂	—	2	1	no	—
87	Q₁₆×C₆₁	—	2	1	no	—
88	Q₁₆×C₆₁	#41	1	2	yes	—
89	C₂×C₈×C₆₁	C₂×D₄	1	2	yes	—
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	2	no	—
96	C₄⋊C₄×C₆₁	—	4	2	no	—
97	C₄⋊C₄×C₆₁	—	4	2	no	—
98	C₄⋊C₄×C₆₁	—	4	1	no	—
99	C₄⋊C₄×C₆₁	D₈	1	1	yes	characteristic
100	C₄⋊C₄×C₆₁	D₈	1	2	yes	—
101	C₄²×C₆₁	—	4	2	no	—
102	C₄²×C₆₁	—	4	1	no	—
103	C₄²×C₆₁	C₂×D₄	1	2	yes	—
104	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
105	#11758198	—	4	2	no	—
106	#11758198	—	4	2	no	—
107	#11758198	—	4	2	no	—
108	C₄⋊Dic₆₁	—	2	1	no	—
109	#11758196	—	4	1	no	—
110	C₄×Dic₆₁	—	4	1	no	—
111	Dic₂₄₄	—	2	1	no	—
112	Dic₂₄₄	—	2	1	no	—
113	D₄₈₈	—	2	2	no	—
114	D₄₈₈	—	2	2	no	—
115	D₄₈₈	—	8	1	no	—
116	#11758190	—	8	2	no	—
117	#11758185	—	2	2	no	—
118	#11758185	—	2	1	no	—
119	#11758185	—	2	1	no	—
120	C₂²×D₆₁	—	4	2	no	—
121	Q₈×C₆₁	—	2	2	no	—
122	Q₈×C₆₁	—	2	2	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	2	yes	—
130	C₂×C₄×C₆₁	C₂×D₈	1	2	yes	—
131	C₂×C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
132	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
133	C₂×Dic₆₁	—	4	1	no	—
134	C₂×Dic₆₁	—	4	1	no	—
135	D₂₄₄	—	4	2	no	—
136	D₂₄₄	—	4	2	no	—
137	D₂₄₄	—	8	2	no	—
138	D₂₄₄	—	16	1	no	—
139	C₄×D₆₁	—	16	2	no	—
140	Dic₁₂₂	—	4	1	no	—
141	Dic₁₂₂	—	4	1	no	—
142	C₈×C₆₁	—	2	2	no	—
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	#33932	—	61	1	no	maximal, Sylow
147	C₂²×C₆₁	C₂²×D₈	1	1	yes	characteristic
148	C₂×D₆₁	—	8	2	no	—
149	C₂×D₆₁	—	8	2	no	—
150	C₂×D₆₁	—	8	2	no	—
151	C₄×C₆₁	—	2	2	no	—
152	C₄×C₆₁	—	2	2	no	—
153	C₄×C₆₁	—	2	1	no	—
154	C₄×C₆₁	—	4	2	no	—
155	C₄×C₆₁	—	8	2	no	—
156	C₄×C₆₁	—	8	1	no	—
157	C₄×C₆₁	#508	1	2	yes	—
158	C₄×C₆₁	#548	1	1	yes	characteristic
159	C₄×C₆₁	#577	1	1	yes	characteristic
160	Dic₆₁	—	8	1	no	—
161	Dic₆₁	—	8	1	no	—
162	#3396	—	61	1	no	—
163	#3383	D₆₁	1	1	yes	characteristic, complemented
164	#2242	—	61	2	no	—
165	#2240	—	61	1	no	—
166	#2228	—	61	2	no	—
167	#2225	—	61	1	no	—
168	#2211	—	61	2	no	—
169	#2207	—	61	1	no	—
170	Q₈×C₁₆	—	61	1	no	—
171	C₄×Q₃₂	—	61	1	no	—
172	C₄×SD₃₂	—	61	2	no	—
173	C₂×C₆₁	#3358	1	1	yes	characteristic
174	C₂×C₆₁	C₂×#508	1	1	yes	characteristic
175	C₂×C₆₁	#3416	1	1	yes	characteristic
176	D₆₁	—	16	2	no	—
177	#559	—	122	1	no	—
178	#557	—	2	1	no	—
179	C₂×Q₃₂	—	61	1	no	—
180	C₂×SD₃₂	—	61	2	no	—
181	#500	C₂×D₆₁	1	1	yes	characteristic
182	#499	C₂×D₆₁	1	2	yes	—
183	#495	—	61	2	no	—
184	#493	—	61	1	no	—
185	#479	—	2	2	no	—
186	#475	—	2	1	no	—
187	#463	—	122	2	no	—
188	#460	—	122	1	no	—
189	C₈×Q₈	C₂×D₆₁	1	1	yes	characteristic
190	C₄×Q₁₆	—	61	1	no	—
191	C₄×Q₁₆	C₂×D₆₁	1	2	yes	—
192	C₄×Q₁₆	C₂×D₆₁	1	1	yes	characteristic
193	C₄×D₈	—	61	2	no	—
194	#367	—	61	2	no	—
195	#366	—	61	1	no	—
196	C₄⋊C₁₆	—	61	1	no	—
197	C₄⋊C₁₆	—	61	2	no	—
198	#358	—	61	1	no	—
199	#358	—	61	1	no	—
200	#358	—	61	2	no	—
201	#357	—	61	2	no	—
202	#357	—	61	2	no	—
203	#357	—	61	2	no	—
204	C₄×C₁₆	—	61	2	no	—
205	C₄×C₁₆	—	61	1	no	—
206	C₆₁	#33932	1	1	yes	characteristic, complemented, Sylow
207	C₂×Q₁₆	—	61	1	no	—
208	C₂×Q₁₆	C₂²×D₆₁	1	1	yes	characteristic
209	C₂×SD₁₆	—	122	2	no	—
210	C₂×D₈	—	61	2	no	—
211	C₂×D₈	—	122	1	no	—
212	C₄⋊Q₈	—	2	1	no	—
213	C₄⋊Q₈	—	4	1	no	—
214	#127	—	122	1	no	—
215	#125	—	2	2	no	—
216	#125	—	4	2	no	—
217	#124	—	122	2	no	—
218	#124	—	244	1	no	—
219	#123	—	244	2	no	—
220	C₄×Q₈	—	2	1	no	—
221	C₄×Q₈	—	2	2	no	—
222	C₄×Q₈	—	4	1	no	—
223	C₄×Q₈	—	122	1	no	—
224	C₄×Q₈	C₂²⋊D₆₁	1	1	yes	characteristic
225	C₄×D₄	—	122	2	no	—
226	C₄×D₄	—	244	1	no	—
227	Q₃₂	—	122	2	no	—
228	SD₃₂	—	122	4	no	—
229	C₂×C₁₆	—	61	1	no	—
230	C₂×C₁₆	—	61	1	no	—
231	C₂×C₁₆	—	61	2	no	—
232	#107	—	2	1	no	—
233	#107	—	61	1	no	—
234	#107	C₂²×D₆₁	1	1	yes	characteristic
235	#107	C₂²×D₆₁	1	2	yes	—
236	#106	—	2	2	no	—
237	C₄⋊C₈	C₂²⋊D₆₁	1	1	yes	characteristic
238	C₄⋊C₈	C₂²⋊D₆₁	1	2	yes	—
239	#103	—	2	2	no	—
240	#103	—	2	1	no	—
241	#103	—	122	1	no	—
242	#102	—	122	2	no	—
243	C₄×C₈	C₂²×D₆₁	1	2	yes	—
244	C₄×C₈	C₂²×D₆₁	1	1	yes	characteristic
245	C₂×Q₈	—	2	1	no	—
246	C₂×Q₈	—	4	1	no	—
247	C₂×Q₈	—	122	1	no	—
248	C₂×Q₈	#11758199	1	1	yes	characteristic
249	C₂×D₄	—	122	2	no	—
250	C₂×D₄	—	244	1	no	—
251	C₂×D₄	—	244	1	no	—
252	C₂²×C₄	—	244	2	no	—
253	Q₁₆	—	2	1	no	—
254	Q₁₆	—	122	1	no	—
255	Q₁₆	—	122	1	no	—
256	Q₁₆	#11758226	1	2	yes	—
257	SD₁₆	—	488	2	no	—
258	D₈	—	122	2	no	—
259	D₈	—	122	2	no	—
260	D₈	—	488	1	no	—
261	C₂×C₈	C₂×C₂²⋊D₆₁	1	2	yes	—
262	C₂×C₈	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
263	C₂×C₈	C₂³×D₆₁	1	1	yes	characteristic
264	C₄⋊C₄	—	2	1	no	—
265	C₄⋊C₄	—	2	2	no	—
266	C₄⋊C₄	—	4	1	no	—
267	C₄⋊C₄	—	4	2	no	—
268	C₄⋊C₄	—	4	2	no	—
269	C₄⋊C₄	—	4	2	no	—
270	C₄⋊C₄	—	4	1	no	—
271	C₄⋊C₄	—	122	1	no	—
272	C₄⋊C₄	—	244	1	no	—
273	C₄⋊C₄	#11758199	1	1	yes	characteristic
274	C₄⋊C₄	#11758199	1	2	yes	—
275	C₂²⋊C₄	—	244	2	no	—
276	C₂²⋊C₄	—	244	2	no	—
277	C₂²⋊C₄	—	244	2	no	—
278	C₄²	—	4	2	no	—
279	C₄²	—	4	1	no	—
280	C₄²	—	244	1	no	—
281	C₄²	C₂×C₂²⋊D₆₁	1	2	yes	—
282	C₄²	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
283	C₁₆	—	122	2	no	—
284	C₁₆	—	122	1	no	—
285	C₁₆	—	122	1	no	—
286	C₂³	—	244	2	no	—
287	Q₈	—	2	2	no	—
288	Q₈	—	2	2	no	—
289	Q₈	—	4	1	no	—
290	Q₈	—	8	2	no	—
291	Q₈	—	244	1	no	—
292	Q₈	—	244	1	no	—
293	D₄	—	976	1	no	—
294	D₄	—	488	2	no	—
295	D₄	—	244	2	no	—
296	D₄	—	244	2	no	—
297	C₂×C₄	—	4	1	no	—
298	C₂×C₄	—	4	1	no	—
299	C₂×C₄	—	4	2	no	—
300	C₂×C₄	—	976	2	no	—
301	C₂×C₄	—	244	1	no	—
302	C₂×C₄	—	244	1	no	—
303	C₂×C₄	C₂×#11758199	1	2	yes	—
304	C₂×C₄	C₂×#11758199	1	2	yes	—
305	C₂×C₄	C₂×#11758199	1	1	yes	characteristic
306	C₂×C₄	C₂×#11758199	1	1	yes	characteristic
307	C₂×C₄	C₂²×C₂²⋊D₆₁	1	1	yes	characteristic
308	C₈	—	2	1	no	—
309	C₈	—	2	2	no	—
310	C₈	C₂×#11758226	1	1	yes	characteristic
311	C₈	#14519825	1	1	yes	characteristic
312	C₂²	—	488	2	no	—
313	C₂²	—	488	2	no	—
314	C₂²	—	488	2	no	—
315	C₂²	C₂²×#11758199	1	1	yes	characteristic
316	C₄	—	2	2	no	—
317	C₄	—	2	1	no	—
318	C₄	—	2	2	no	—
319	C₄	—	4	2	no	—
320	C₄	—	8	1	no	—
321	C₄	—	8	2	no	—
322	C₄	—	488	1	no	—
323	C₄	—	488	1	no	—
324	C₄	#16840192	1	2	yes	—
325	C₄	#16840423	1	1	yes	characteristic
326	C₄	#16840570	1	1	yes	characteristic
327	C₂	—	976	2	no	—
328	C₂	#21458454	1	1	yes	characteristic
329	C₂	C₂×#16840192	1	1	yes	characteristic
330	C₂	#21458963	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×#11758199
commutator, cosocle	C₂×C₈×C₆₁	C₂⁴
Frattini	C₂×C₈	C₂³×D₆₁
Fitting	C₆₁×#3383	C₂
socle	C₂²×C₆₁	C₂²×D₈
prime radical	#16840139	C₂²
hypercenter	#3383	D₆₁
hypocenter	C₆₁	#33932
LCS	G ▷ C₂×C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ C₈×Q₈ ◁ #3383

Complements

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

N	H ≅ G/N	type
2. #21448053	C₂	semidirect
4. #21447973	C₂	semidirect
5. #21447958	C₂	semidirect
6. #21447881	C₂	semidirect
8. Q₈×#11758185	C₂	semidirect
9. C₄×#14519723	C₂	semidirect
10. C₄×#14519720	C₂	semidirect
11. C₆₁×#3383	C₂	semidirect
163. #3383	D₆₁	semidirect
206. C₆₁	#33932	semidirect

Maximal subgroups

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

subgroup	index
#21458832	2
#21448053	2
#21448044	2
#21447973	2
#21447958	2
#21447881	2
#21447859	2
Q₈×#11758185	2
C₄×#14519723	2
C₄×#14519720	2
C₆₁×#3383	2
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