Subgroups — #30601786

Order 15616.

570 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21461752	C₂	1	1	yes	maximal, characteristic
2	C₄×#14519844	C₂	1	1	yes	maximal, characteristic
3	C₂×#16840621	C₂	1	4	yes	maximal
4	#21454216	C₂	1	1	yes	maximal, characteristic
5	C₂×#14519844	—	2	1	no	—
6	C₂×#14519844	C₂²	1	2	yes	—
7	C₂×#14519844	C₂²	1	2	yes	—
8	C₂×C₄⋊#92420	—	2	1	no	—
9	C₂×C₄×#92420	—	2	1	no	—
10	#16840628	—	2	1	no	—
11	#16840621	—	2	8	no	—
12	C₂×#14519776	—	2	1	no	—
13	C₂×#14519776	—	2	1	no	—
14	C₂×#14519776	C₂²	1	2	yes	—
15	#16839632	C₄	1	1	yes	characteristic
16	#16839631	—	2	1	no	—
17	#16839631	—	2	1	no	—
18	#16839631	C₄	1	2	yes	—
19	C₄²×Dic₆₁	C₂²	1	1	yes	characteristic
20	C₆₁×#384	C₄	1	1	yes	characteristic
21	C₂²×#92420	—	2	2	no	—
22	C₂²×#92420	—	2	2	no	—
23	#14519844	—	2	4	no	—
24	#14519844	—	2	2	no	—
25	#14519844	—	4	2	no	—
26	#14519844	D₄	1	4	yes	—
27	C₄⋊#92420	—	4	2	no	—
28	C₄×#92420	—	4	2	no	—
29	C₂²×Dic₁₂₂	—	2	2	no	—
30	#14519776	—	2	4	no	—
31	#14519776	—	2	4	no	—
32	#14519776	—	4	2	no	—
33	#14519776	—	4	2	no	—
34	#14519776	—	4	2	no	—
35	#14519776	—	4	2	no	—
36	C₂×C₄⋊Dic₆₁	—	2	1	no	—
37	C₂×C₄⋊Dic₆₁	—	2	1	no	—
38	C₂×C₄⋊Dic₆₁	C₂×C₄	1	2	yes	—
39	C₂×#11758196	—	2	2	no	—
40	C₂×#11758196	—	2	2	no	—
41	C₂×#11758196	—	2	2	no	—
42	C₂×C₄×Dic₆₁	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄×Dic₆₁	D₄	1	1	yes	characteristic
44	C₂×C₄×Dic₆₁	D₄	1	1	yes	characteristic
45	C₂×C₄×Dic₆₁	D₄	1	1	yes	characteristic
46	C₂×C₄×Dic₆₁	D₄	1	1	yes	characteristic
47	C₂×C₄×Dic₆₁	C₂³	1	1	yes	characteristic
48	C₂×C₄⋊C₄×C₆₁	—	2	1	no	—
49	C₂×C₄⋊C₄×C₆₁	—	2	1	no	—
50	C₂×C₄⋊C₄×C₆₁	C₂×C₄	1	2	yes	—
51	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
52	C₂²×C₄×C₆₁	—	2	2	no	—
53	C₂²×C₄×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
54	C₂²×C₄×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
55	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
56	C₂²×Dic₆₁	—	2	2	no	—
57	C₂²×Dic₆₁	C₂²⋊C₄	1	1	yes	characteristic
58	C₂²×Dic₆₁	C₂²⋊C₄	1	1	yes	characteristic
59	C₂²×Dic₆₁	C₂×D₄	1	1	yes	characteristic
60	C₂²×Dic₆₁	C₂×D₄	1	1	yes	characteristic
61	C₂×Dic₁₂₂	—	4	2	no	—
62	C₂×Dic₁₂₂	—	4	2	no	—
63	C₂×Dic₁₂₂	—	4	2	no	—
64	C₂×Dic₁₂₂	—	4	2	no	—
65	C₂×Dic₁₂₂	—	4	2	no	—
66	C₂×Dic₁₂₂	—	4	2	no	—
67	C₂×#92420	—	2	4	no	—
68	C₂×#92420	—	4	2	no	—
69	C₂×#92420	—	4	2	no	—
70	C₂×#92420	—	4	2	no	—
71	C₂×#92420	—	4	2	no	—
72	C₂×#92420	—	4	2	no	—
73	C₄⋊C₄×C₆₁	—	2	4	no	—
74	C₄⋊C₄×C₆₁	—	4	2	no	—
75	C₄⋊C₄×C₆₁	—	4	2	no	—
76	C₄²×C₆₁	—	2	2	no	—
77	C₄⋊Dic₆₁	—	2	4	no	—
78	C₄⋊Dic₆₁	—	4	2	no	—
79	C₄⋊Dic₆₁	—	4	2	no	—
80	#11758196	—	4	4	no	—
81	#11758196	—	4	4	no	—
82	#11758196	—	4	4	no	—
83	C₄×Dic₆₁	—	2	2	no	—
84	C₄×Dic₆₁	—	2	2	no	—
85	C₄×Dic₆₁	D₈	1	2	yes	—
86	C₄×Dic₆₁	D₈	1	2	yes	—
87	C₄×Dic₆₁	SD₁₆	1	2	yes	—
88	C₄×Dic₆₁	SD₁₆	1	2	yes	—
89	C₄×Dic₆₁	SD₁₆	1	2	yes	—
90	C₄×Dic₆₁	Q₁₆	1	2	yes	—
91	C₄×Dic₆₁	C₂×D₄	1	2	yes	—
92	C₄×Dic₆₁	#41	1	2	yes	—
93	C₂³×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
94	C₂×C₄×C₆₁	—	2	1	no	—
95	C₂×C₄×C₆₁	—	2	1	no	—
96	C₂×C₄×C₆₁	—	2	1	no	—
97	C₂×C₄×C₆₁	—	2	1	no	—
98	C₂×C₄×C₆₁	—	4	2	no	—
99	C₂×C₄×C₆₁	—	4	2	no	—
100	C₂×C₄×C₆₁	—	4	2	no	—
101	C₂×C₄×C₆₁	#101	1	2	yes	—
102	C₂×C₄×C₆₁	#102	1	2	yes	—
103	C₂×C₄×C₆₁	#102	1	2	yes	—
104	C₂×C₄×C₆₁	#103	1	2	yes	—
105	C₂×C₄×C₆₁	C₄×D₄	1	2	yes	—
106	C₂×Dic₆₁	—	2	1	no	—
107	C₂×Dic₆₁	—	2	1	no	—
108	C₂×Dic₆₁	—	4	2	no	—
109	C₂×Dic₆₁	—	4	2	no	—
110	C₂×Dic₆₁	—	4	2	no	—
111	C₂×Dic₆₁	#101	1	2	yes	—
112	C₂×Dic₆₁	#102	1	2	yes	—
113	C₂×Dic₆₁	#102	1	2	yes	—
114	C₂×Dic₆₁	#103	1	2	yes	—
115	C₂×Dic₆₁	#121	1	2	yes	—
116	C₂×Dic₆₁	#124	1	1	yes	characteristic
117	C₂×Dic₆₁	#127	1	1	yes	characteristic
118	C₂×Dic₆₁	C₂×D₈	1	1	yes	characteristic
119	C₂×Dic₆₁	C₂×D₈	1	1	yes	characteristic
120	C₂×Dic₆₁	C₂×SD₁₆	1	1	yes	characteristic
121	C₂×Dic₆₁	C₂×SD₁₆	1	1	yes	characteristic
122	C₂×Dic₆₁	C₂×SD₁₆	1	1	yes	characteristic
123	C₂×Dic₆₁	C₂×Q₁₆	1	1	yes	characteristic
124	C₂×Dic₆₁	#137	1	2	yes	—
125	Dic₁₂₂	—	8	2	no	—
126	Dic₁₂₂	—	8	2	no	—
127	Dic₁₂₂	—	8	2	no	—
128	Dic₁₂₂	—	8	2	no	—
129	#92420	—	4	4	no	—
130	#92420	—	8	2	no	—
131	#9692	—	61	1	no	maximal, Sylow
132	C₂²×C₆₁	#390	1	1	yes	characteristic
133	C₂²×C₆₁	C₂×#101	1	1	yes	characteristic
134	C₂²×C₆₁	C₂×#101	1	1	yes	characteristic
135	C₂²×C₆₁	C₂×#102	1	1	yes	characteristic
136	C₂²×C₆₁	C₂×#102	1	1	yes	characteristic
137	C₂²×C₆₁	C₂×#103	1	1	yes	characteristic
138	C₂²×C₆₁	#419	1	1	yes	characteristic
139	C₄×C₆₁	—	2	2	no	—
140	C₄×C₆₁	—	2	2	no	—
141	C₄×C₆₁	—	8	2	no	—
142	C₄×C₆₁	#332	1	4	yes	complemented
143	Dic₆₁	—	2	2	no	—
144	Dic₆₁	—	8	2	no	—
145	Dic₆₁	#332	1	4	yes	complemented
146	Dic₆₁	#461	1	2	yes	—
147	Dic₆₁	#462	1	2	yes	—
148	Dic₆₁	#486	1	1	yes	characteristic
149	Dic₆₁	#486	1	1	yes	characteristic
150	Dic₆₁	#492	1	1	yes	characteristic
151	Dic₆₁	#493	1	1	yes	characteristic
152	#2868	—	61	1	no	—
153	#1842	—	61	1	no	—
154	C₄×C₄⋊C₈	—	61	1	no	—
155	C₂×#332	—	61	4	no	—
156	C₂×C₆₁	C₂×#332	1	1	yes	characteristic
157	C₂×C₆₁	C₂×#332	1	1	yes	characteristic
158	C₂×C₆₁	#1546	1	1	yes	characteristic
159	C₂×C₆₁	#1546	1	1	yes	characteristic
160	C₂×C₆₁	#1957	1	1	yes	characteristic
161	C₂×C₆₁	#1966	1	1	yes	characteristic
162	C₂×C₆₁	#1981	1	1	yes	characteristic
163	C₂×C₄⋊Q₈	—	61	2	no	—
164	C₂×C₄⋊Q₈	—	122	1	no	—
165	C₂×C₄⋊Q₈	—	122	1	no	—
166	C₂×C₄⋊C₈	—	61	2	no	—
167	C₂×C₄⋊C₈	—	61	2	no	—
168	C₂×C₄⋊C₈	—	122	1	no	—
169	C₂×C₄⋊C₈	—	122	1	no	—
170	C₂×C₄×C₈	—	122	1	no	—
171	#384	—	61	2	no	—
172	#384	—	61	1	no	—
173	#384	—	122	1	no	—
174	#384	—	122	1	no	—
175	#384	#6923	1	1	yes	characteristic
176	C₄³	—	61	1	no	—
177	#336	—	122	1	no	—
178	#332	—	122	8	no	—
179	C₆₁	#9692	1	1	yes	characteristic, complemented, Sylow
180	C₂²×Q₈	—	122	2	no	—
181	C₂²×C₈	—	122	2	no	—
182	C₂²×C₈	—	122	2	no	—
183	C₄⋊Q₈	—	122	4	no	—
184	C₄⋊Q₈	—	122	4	no	—
185	C₄⋊Q₈	—	244	2	no	—
186	C₄⋊Q₈	—	244	2	no	—
187	C₄⋊Q₈	—	244	2	no	—
188	C₄⋊Q₈	—	244	2	no	—
189	C₂×C₄⋊C₄	—	2	1	no	—
190	C₂×C₄⋊C₄	—	2	1	no	—
191	C₂×C₄⋊C₄	—	61	2	no	—
192	C₂×C₄⋊C₄	—	122	2	no	—
193	C₂×C₄⋊C₄	—	122	2	no	—
194	C₂×C₄⋊C₄	—	122	2	no	—
195	C₂×C₄⋊C₄	—	122	1	no	—
196	C₂×C₄⋊C₄	—	122	1	no	—
197	C₂×C₄⋊C₄	C₂×#6923	1	2	yes	—
198	C₂×C₄²	—	61	1	no	—
199	C₂×C₄²	—	61	1	no	—
200	C₂×C₄²	—	61	1	no	—
201	C₂×C₄²	—	61	1	no	—
202	C₂×C₄²	—	61	1	no	—
203	C₂×C₄²	—	61	1	no	—
204	C₂×C₄²	C₂×#6923	1	1	yes	characteristic
205	C₄⋊C₈	—	61	4	no	—
206	C₄⋊C₈	—	122	2	no	—
207	C₄⋊C₈	—	122	4	no	—
208	C₄⋊C₈	—	244	2	no	—
209	C₄⋊C₈	—	244	2	no	—
210	C₄×C₈	—	244	2	no	—
211	C₂×Q₈	—	244	2	no	—
212	C₂×Q₈	—	244	2	no	—
213	C₂×Q₈	—	244	2	no	—
214	C₂×Q₈	—	244	2	no	—
215	C₂×Q₈	—	244	2	no	—
216	C₂×Q₈	—	244	2	no	—
217	C₂²×C₄	—	2	2	no	—
218	C₂²×C₄	—	61	1	no	—
219	C₂²×C₄	—	61	1	no	—
220	C₂²×C₄	—	61	1	no	—
221	C₂²×C₄	—	61	1	no	—
222	C₂²×C₄	—	122	2	no	—
223	C₂²×C₄	C₂²⋊#6923	1	1	yes	characteristic
224	C₂²×C₄	C₂²⋊#6923	1	1	yes	characteristic
225	C₂²×C₄	C₂²×#6923	1	1	yes	characteristic
226	C₂×C₈	—	122	4	no	—
227	C₂×C₈	—	244	2	no	—
228	C₂×C₈	—	244	2	no	—
229	C₂×C₈	—	244	2	no	—
230	C₂×C₈	—	244	2	no	—
231	C₂×C₈	—	244	2	no	—
232	C₄⋊C₄	—	2	4	no	—
233	C₄⋊C₄	—	4	2	no	—
234	C₄⋊C₄	—	4	2	no	—
235	C₄⋊C₄	—	122	4	no	—
236	C₄⋊C₄	—	244	4	no	—
237	C₄⋊C₄	—	244	4	no	—
238	C₄⋊C₄	—	244	4	no	—
239	C₄⋊C₄	—	244	2	no	—
240	C₄⋊C₄	—	244	2	no	—
241	C₄²	—	2	2	no	—
242	C₄²	—	61	2	no	—
243	C₄²	—	61	2	no	—
244	C₄²	—	61	2	no	—
245	C₄²	—	61	2	no	—
246	C₄²	—	61	2	no	—
247	C₄²	—	61	2	no	—
248	C₄²	—	61	2	no	—
249	C₄²	—	61	2	no	—
250	C₄²	—	122	2	no	—
251	C₄²	—	122	2	no	—
252	C₂³	C₂×C₂²⋊#6923	1	1	yes	characteristic
253	Q₈	—	488	2	no	—
254	Q₈	—	488	2	no	—
255	Q₈	—	488	2	no	—
256	Q₈	—	488	2	no	—
257	C₂×C₄	—	2	1	no	—
258	C₂×C₄	—	2	1	no	—
259	C₂×C₄	—	2	1	no	—
260	C₂×C₄	—	2	1	no	—
261	C₂×C₄	—	4	2	no	—
262	C₂×C₄	—	4	2	no	—
263	C₂×C₄	—	4	2	no	—
264	C₂×C₄	—	61	2	no	—
265	C₂×C₄	—	61	1	no	—
266	C₂×C₄	—	61	1	no	—
267	C₂×C₄	—	61	2	no	—
268	C₂×C₄	—	61	2	no	—
269	C₂×C₄	—	61	1	no	—
270	C₂×C₄	—	61	1	no	—
271	C₂×C₄	—	61	1	no	—
272	C₂×C₄	—	61	1	no	—
273	C₂×C₄	—	61	2	no	—
274	C₂×C₄	—	61	1	no	—
275	C₂×C₄	—	61	1	no	—
276	C₂×C₄	—	61	2	no	—
277	C₂×C₄	—	61	2	no	—
278	C₂×C₄	—	122	1	no	—
279	C₂×C₄	—	122	1	no	—
280	C₂×C₄	—	244	2	no	—
281	C₂×C₄	—	244	2	no	—
282	C₂×C₄	—	244	2	no	—
283	C₂×C₄	#14519838	1	2	yes	—
284	C₂×C₄	#14519839	1	2	yes	—
285	C₂×C₄	#14519839	1	2	yes	—
286	C₂×C₄	#14519840	1	2	yes	—
287	C₂×C₄	D₄×#6923	1	2	yes	—
288	C₈	—	488	2	no	—
289	C₈	—	244	4	no	—
290	C₂²	#16840714	1	1	yes	characteristic
291	C₂²	C₂×#14519838	1	1	yes	characteristic
292	C₂²	D₆₁⋊#101	1	1	yes	characteristic
293	C₂²	C₂×#14519839	1	1	yes	characteristic
294	C₂²	C₂×#14519839	1	1	yes	characteristic
295	C₂²	C₂×#14519840	1	1	yes	characteristic
296	C₂²	#16840755	1	1	yes	characteristic
297	C₄	—	2	2	no	—
298	C₄	—	2	2	no	—
299	C₄	—	8	2	no	—
300	C₄	—	61	4	no	—
301	C₄	—	61	1	no	—
302	C₄	—	61	1	no	—
303	C₄	—	61	1	no	—
304	C₄	—	61	1	no	—
305	C₄	—	61	2	no	—
306	C₄	—	61	2	no	—
307	C₄	—	122	2	no	—
308	C₄	—	488	2	no	—
309	C₄	#16840621	1	4	yes	complemented
310	C₂	C₂×#16840621	1	1	yes	characteristic
311	C₂	D₆₁⋊#332	1	1	yes	characteristic
312	C₂	#21461328	1	1	yes	characteristic
313	C₂	#21461329	1	1	yes	characteristic
314	C₂	#21461947	1	1	yes	characteristic
315	C₂	#21461962	1	1	yes	characteristic
316	C₂	#21461985	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊#6923
commutator	C₂²×C₄×C₆₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×#6923
Fitting	C₆₁×#384	C₄
socle, prime radical	C₂³×C₆₁	C₂×C₂²⋊C₄
cosocle	C₂×C₄×Dic₆₁	C₂³
hypercenter	#384	#6923
hypocenter	C₆₁	#9692
LCS	G ▷ C₂²×C₄×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #384

Complements

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

N	H ≅ G/N	type
142. C₄×C₆₁	#332	semidirect
145. Dic₆₁	#332	semidirect
179. C₆₁	#9692	semidirect
309. C₄	#16840621	semidirect

Maximal subgroups

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

subgroup	index
#21461752	2
C₄×#14519844	2
C₂×#16840621	2
#21454216	2
#9692	61