Subgroups — #29596498

Order 15616.

384 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	C₂×#16840180	C₂	1	1	yes	maximal, characteristic
2	#21447579	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#16839547	C₂	1	1	yes	maximal, characteristic
4	#21445656	C₂	1	1	yes	maximal, characteristic
5	#21443909	C₂	1	1	yes	maximal, characteristic, complemented
6	#21443897	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₆₁×#364	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#14519783	C₂²	1	1	yes	characteristic
9	#16840180	—	2	2	no	—
10	#16840180	—	2	2	no	—
11	C₂×#14519716	—	2	1	no	—
12	C₂²×#11758185	C₂²	1	1	yes	characteristic
13	C₂²×#11758185	C₂²	1	1	yes	characteristic
14	C₂×C₁₆×D₆₁	—	2	1	no	—
15	C₂×#14519705	C₄	1	1	yes	characteristic
16	#16839759	C₄	1	1	yes	characteristic
17	#16839717	C₂²	1	1	yes	characteristic
18	#16839580	—	2	1	no	—
19	#16839580	—	2	1	no	—
20	#16839547	—	2	2	no	—
21	#16839527	C₄	1	1	yes	characteristic
22	#16839522	C₄	1	1	yes	characteristic
23	C₂×M₃₂×C₆₁	C₂²	1	1	yes	characteristic
24	C₂×M₃₂×C₆₁	C₂²	1	1	yes	characteristic
25	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
26	C₆₁×#364	—	2	2	no	—
27	C₂×#11758222	C₂×C₄	1	1	yes	characteristic
28	#14519783	—	2	1	no	—
29	#14519783	—	2	1	no	—
30	#14519783	D₄	1	2	yes	—
31	#14519783	Q₈	1	2	yes	—
32	C₂×#11758194	C₂×C₄	1	1	yes	characteristic
33	C₂×#11758194	C₂×C₄	1	1	yes	characteristic
34	C₂×#11758189	—	2	1	no	—
35	C₂×C₈×D₆₁	—	2	1	no	—
36	#14519736	C₂×C₄	1	1	yes	characteristic
37	#14519716	—	2	4	no	—
38	C₂×#11758185	—	2	1	no	—
39	C₂×#11758185	—	2	1	no	—
40	C₂×#11758185	—	2	1	no	—
41	C₂×#11758185	—	2	1	no	—
42	C₂×#11758185	—	2	1	no	—
43	C₂×#11758185	D₄	1	1	yes	characteristic
44	C₂×#11758185	D₄	1	1	yes	characteristic
45	C₁₆×D₆₁	—	2	4	no	—
46	#14519705	—	2	2	no	—
47	#14519702	C₈	1	1	yes	characteristic
48	#14519702	C₈	1	1	yes	characteristic
49	#14519699	C₈	1	1	yes	characteristic
50	#14519699	C₈	1	1	yes	characteristic
51	#14519687	—	2	1	no	—
52	C₈×Dic₆₁	—	2	1	no	—
53	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
54	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
55	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
56	M₃₂×C₆₁	—	2	1	no	—
57	M₃₂×C₆₁	—	2	1	no	—
58	M₃₂×C₆₁	—	2	1	no	—
59	M₃₂×C₆₁	D₄	1	2	yes	—
60	C₂×C₁₆×C₆₁	—	2	1	no	—
61	C₂×C₁₆×C₆₁	—	2	1	no	—
62	C₄×C₈×C₆₁	—	2	1	no	—
63	C₄×C₈×C₆₁	—	2	1	no	—
64	C₂²×C₄×C₆₁	—	2	1	no	—
65	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
66	C₂×C₂²⋊D₆₁	—	2	1	no	—
67	#11758222	—	2	1	no	—
68	#11758222	—	2	1	no	—
69	#11758222	C₄⋊C₄	1	2	yes	—
70	#11758222	M₁₆	1	2	yes	—
71	C₂×D₂₄₄	C₂×C₈	1	1	yes	characteristic
72	C₂×C₄×D₆₁	—	2	1	no	—
73	C₂×Dic₁₂₂	C₂×C₈	1	1	yes	characteristic
74	C₂×C₈×C₆₁	—	2	1	no	—
75	C₂×C₈×C₆₁	—	2	1	no	—
76	C₂×C₈×C₆₁	C₂×C₈	1	1	yes	characteristic
77	C₂×C₈×C₆₁	C₂×C₈	1	1	yes	characteristic
78	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
79	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
80	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
81	C₂×C₈×C₆₁	C₂×Q₈	1	1	yes	characteristic
82	C₂×C₈×C₆₁	#41	1	1	yes	characteristic
83	C₂×C₈×C₆₁	#41	1	1	yes	characteristic
84	C₄²×C₆₁	—	2	1	no	—
85	C₄²×C₆₁	—	2	1	no	—
86	C₂²⋊Dic₆₁	—	2	1	no	—
87	C₄⋊Dic₆₁	C₂×C₈	1	1	yes	characteristic
88	C₄⋊Dic₆₁	C₂×C₈	1	1	yes	characteristic
89	C₄×Dic₆₁	—	2	1	no	—
90	#11758194	—	2	1	no	—
91	#11758194	—	2	1	no	—
92	#11758194	—	2	1	no	—
93	#11758194	C₄⋊C₄	1	2	yes	—
94	C₂×#92418	—	2	1	no	—
95	C₂×#92418	—	2	1	no	—
96	#11758189	—	2	2	no	—
97	#11758189	—	2	2	no	—
98	C₈×D₆₁	—	2	2	no	—
99	C₈×D₆₁	—	2	2	no	—
100	C₁₆×C₆₁	—	2	2	no	—
101	C₁₆×C₆₁	—	4	1	no	—
102	#11758185	—	2	1	no	—
103	#11758185	—	2	1	no	—
104	#11758185	—	4	1	no	—
105	C₂³×C₆₁	#131	1	1	yes	characteristic
106	C₂²×D₆₁	—	2	1	no	—
107	C₂×C₄×C₆₁	—	2	2	no	—
108	C₂×C₄×C₆₁	—	4	1	no	—
109	C₂×C₄×C₆₁	—	4	1	no	—
110	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
111	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
112	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
113	C₂×C₄×C₆₁	C₂²×C₈	1	1	yes	characteristic
114	C₂×C₄×C₆₁	C₂×M₁₆	1	1	yes	characteristic
115	C₂×C₄×C₆₁	#131	1	1	yes	characteristic
116	C₂²⋊D₆₁	—	2	2	no	—
117	C₂²⋊D₆₁	—	2	2	no	—
118	C₂×Dic₆₁	—	2	1	no	—
119	C₂×Dic₆₁	—	2	1	no	—
120	C₂×Dic₆₁	—	2	1	no	—
121	D₂₄₄	—	2	1	no	—
122	D₂₄₄	C₄⋊C₈	1	2	yes	—
123	C₄×D₆₁	—	2	2	no	—
124	C₄×D₆₁	—	2	2	no	—
125	Dic₁₂₂	—	2	1	no	—
126	Dic₁₂₂	C₄⋊C₈	1	2	yes	—
127	C₈×C₆₁	—	2	1	no	—
128	C₈×C₆₁	—	2	1	no	—
129	C₈×C₆₁	#120	1	1	yes	characteristic
130	C₈×C₆₁	C₂²⋊Q₈	1	1	yes	characteristic
131	C₈×C₆₁	C₂²⋊Q₈	1	1	yes	characteristic
132	C₈×C₆₁	#123	1	1	yes	characteristic
133	#92418	—	2	2	no	—
134	#92418	—	4	1	no	—
135	#12917	—	61	1	no	maximal, Sylow
136	C₂²×C₆₁	—	2	1	no	—
137	C₂²×C₆₁	—	2	1	no	—
138	C₂²×C₆₁	#424	1	1	yes	characteristic
139	C₂²×C₆₁	C₈×D₄	1	1	yes	characteristic
140	C₂²×C₆₁	#435	1	1	yes	characteristic
141	C₂×D₆₁	—	2	2	no	—
142	C₂×D₆₁	—	4	1	no	—
143	C₂×D₆₁	—	4	1	no	—
144	C₄×C₆₁	—	4	2	no	—
145	C₄×C₆₁	#385	1	1	yes	characteristic
146	C₄×C₆₁	C₂×C₄⋊C₈	1	1	yes	characteristic
147	C₄×C₆₁	C₈×D₄	1	1	yes	characteristic
148	C₄×C₆₁	#435	1	1	yes	characteristic
149	Dic₆₁	—	2	2	no	—
150	Dic₆₁	—	4	1	no	—
151	Dic₆₁	—	4	1	no	—
152	C₂×#504	—	61	1	no	—
153	C₂×#364	D₆₁	1	1	yes	characteristic, complemented
154	#2151	—	61	1	no	—
155	C₂×#350	—	61	1	no	—
156	#1764	—	61	1	no	—
157	#1379	—	61	1	no	—
158	#1374	—	61	1	no	—
159	C₂×C₆₁	—	2	1	no	—
160	C₂×C₆₁	—	2	1	no	—
161	C₂×C₆₁	#1850	1	1	yes	characteristic
162	C₂×C₆₁	#2170	1	1	yes	characteristic
163	C₂×C₆₁	#2171	1	1	yes	characteristic
164	D₆₁	—	4	2	no	—
165	C₂×#131	—	61	1	no	—
166	#504	—	122	2	no	—
167	#504	—	122	2	no	—
168	C₂×M₃₂	—	122	1	no	—
169	C₂×M₃₂	C₂×D₆₁	1	1	yes	characteristic
170	C₂×M₃₂	C₂×D₆₁	1	1	yes	characteristic
171	C₂²×C₁₆	—	61	1	no	—
172	C₂²×C₁₆	—	61	1	no	—
173	C₂²×C₁₆	—	122	1	no	—
174	C₂×#104	—	61	1	no	—
175	#416	—	61	1	no	—
176	#405	—	61	1	no	—
177	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
178	#364	—	2	2	no	—
179	C₄⋊C₁₆	—	122	1	no	—
180	C₄⋊C₁₆	—	122	1	no	—
181	#350	—	122	2	no	—
182	#341	—	61	1	no	—
183	#339	—	61	1	no	—
184	C₆₁	#12917	1	1	yes	characteristic, complemented, Sylow
185	C₂×#41	—	61	1	no	—
186	#131	—	61	2	no	—
187	#131	—	61	2	no	—
188	#131	—	122	1	no	—
189	#131	—	122	1	no	—
190	C₂×M₁₆	—	61	1	no	—
191	C₂×M₁₆	—	61	1	no	—
192	C₂×M₁₆	—	122	1	no	—
193	C₂²×C₈	—	122	1	no	—
194	C₂²×C₈	C₄×D₆₁	1	1	yes	characteristic
195	C₂²×C₈	C₂²×D₆₁	1	1	yes	characteristic
196	#117	—	61	1	no	—
197	C₂×C₄²	C₄×D₆₁	1	1	yes	characteristic
198	M₃₂	—	2	1	no	—
199	M₃₂	—	2	1	no	—
200	M₃₂	—	2	1	no	—
201	M₃₂	—	122	4	no	—
202	M₃₂	C₂²⋊D₆₁	1	2	yes	—
203	C₂×C₁₆	—	2	1	no	—
204	C₂×C₁₆	—	2	1	no	—
205	C₂×C₁₆	—	61	1	no	—
206	C₂×C₁₆	—	61	1	no	—
207	C₂×C₁₆	—	122	4	no	—
208	C₂×C₁₆	—	122	1	no	—
209	C₂×C₁₆	—	122	1	no	—
210	C₂×C₁₆	—	122	1	no	—
211	C₂×C₁₆	—	122	1	no	—
212	C₂×C₁₆	—	122	1	no	—
213	#104	—	122	2	no	—
214	#103	—	61	1	no	—
215	#103	—	61	1	no	—
216	#102	—	61	1	no	—
217	#102	—	61	1	no	—
218	#97	—	122	1	no	—
219	C₄×C₈	—	2	1	no	—
220	C₄×C₈	—	2	1	no	—
221	C₄×C₈	—	122	1	no	—
222	#41	—	61	2	no	—
223	#41	—	61	2	no	—
224	#41	—	122	1	no	—
225	#41	—	122	1	no	—
226	C₂×Q₈	—	61	1	no	—
227	C₂×D₄	—	61	1	no	—
228	C₂×D₄	—	122	1	no	—
229	C₂²×C₄	—	2	1	no	—
230	C₂²×C₄	—	122	1	no	—
231	C₂²×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
232	M₁₆	—	61	2	no	—
233	M₁₆	—	122	1	no	—
234	M₁₆	—	122	2	no	—
235	M₁₆	—	122	2	no	—
236	M₁₆	—	122	1	no	—
237	M₁₆	—	122	1	no	—
238	C₂×C₈	—	2	1	no	—
239	C₂×C₈	—	2	1	no	—
240	C₂×C₈	—	122	1	no	—
241	C₂×C₈	—	122	2	no	—
242	C₂×C₈	—	122	2	no	—
243	C₂×C₈	—	122	1	no	—
244	C₂×C₈	C₈×D₆₁	1	1	yes	characteristic
245	C₂×C₈	C₈×D₆₁	1	1	yes	characteristic
246	C₂×C₈	#11758222	1	1	yes	characteristic
247	C₂×C₈	D₄×D₆₁	1	1	yes	characteristic
248	C₂×C₈	D₄×D₆₁	1	1	yes	characteristic
249	C₂×C₈	Q₈×D₆₁	1	1	yes	characteristic
250	C₂×C₈	#11758226	1	1	yes	characteristic
251	C₂×C₈	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
252	C₄⋊C₄	—	61	1	no	—
253	C₄⋊C₄	—	61	1	no	—
254	C₂²⋊C₄	—	122	1	no	—
255	C₄²	—	2	1	no	—
256	C₄²	—	2	1	no	—
257	C₄²	—	122	1	no	—
258	C₁₆	—	2	2	no	—
259	C₁₆	—	4	1	no	—
260	C₁₆	—	122	1	no	—
261	C₁₆	—	122	1	no	—
262	C₁₆	—	244	1	no	—
263	C₂³	—	122	1	no	—
264	C₂³	#14519785	1	1	yes	characteristic
265	Q₈	—	61	2	no	—
266	Q₈	—	122	1	no	—
267	D₄	—	61	2	no	—
268	D₄	—	122	2	no	—
269	D₄	—	122	2	no	—
270	D₄	—	122	1	no	—
271	C₂×C₄	—	2	2	no	—
272	C₂×C₄	—	4	1	no	—
273	C₂×C₄	—	4	1	no	—
274	C₂×C₄	—	122	1	no	—
275	C₂×C₄	—	122	2	no	—
276	C₂×C₄	—	122	2	no	—
277	C₂×C₄	—	122	1	no	—
278	C₂×C₄	—	122	1	no	—
279	C₂×C₄	C₄⋊C₄×D₆₁	1	1	yes	characteristic
280	C₂×C₄	C₄×C₂²⋊D₆₁	1	1	yes	characteristic
281	C₂×C₄	#14519739	1	1	yes	characteristic
282	C₂×C₄	C₂×C₈×D₆₁	1	1	yes	characteristic
283	C₂×C₄	M₁₆×D₆₁	1	1	yes	characteristic
284	C₂×C₄	#14519783	1	1	yes	characteristic
285	C₈	—	2	1	no	—
286	C₈	—	2	1	no	—
287	C₈	—	122	2	no	—
288	C₈	—	244	1	no	—
289	C₈	#14519745	1	1	yes	characteristic
290	C₈	#14519753	1	1	yes	characteristic
291	C₈	#14519755	1	1	yes	characteristic
292	C₈	#14519762	1	1	yes	characteristic
293	C₂²	—	2	1	no	—
294	C₂²	—	2	1	no	—
295	C₂²	—	122	2	no	—
296	C₂²	—	244	1	no	—
297	C₂²	—	244	1	no	—
298	C₂²	#16839787	1	1	yes	characteristic
299	C₂²	#16839827	1	1	yes	characteristic
300	C₂²	#16839831	1	1	yes	characteristic
301	C₄	—	4	2	no	—
302	C₄	—	122	2	no	—
303	C₄	—	244	1	no	—
304	C₄	—	244	1	no	—
305	C₄	#16839633	1	1	yes	characteristic
306	C₄	C₄⋊C₈×D₆₁	1	1	yes	characteristic
307	C₄	C₈×C₂²⋊D₆₁	1	1	yes	characteristic
308	C₄	#16839832	1	1	yes	characteristic
309	C₂	—	2	1	no	—
310	C₂	—	2	1	no	—
311	C₂	—	244	2	no	—
312	C₂	#21445995	1	1	yes	characteristic
313	C₂	#21447655	1	1	yes	characteristic
314	C₂	#21447660	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	D₄×D₆₁
commutator	C₂×C₄×C₆₁	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₆₁
Fitting	C₂×C₆₁×#364	C₂
socle	C₂²×C₆₁	C₈×D₄
prime radical	C₂×#11758222	C₂×C₄
cosocle	C₂²×C₈×C₆₁	C₂³
hypercenter	C₂×#364	D₆₁
hypocenter	C₆₁	#12917
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ C₂×#364

Complements

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

N	H ≅ G/N	type
2. #21447579	C₂	semidirect
5. #21443909	C₂	semidirect
6. #21443897	C₂	semidirect
7. C₂×C₆₁×#364	C₂	semidirect
153. C₂×#364	D₆₁	semidirect
184. C₆₁	#12917	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16840180	2
#21447579	2
C₂×#16839547	2
#21445656	2
#21443909	2
#21443897	2
C₂×C₆₁×#364	2
#12917	61