Subgroups — #29699775

Order 15616.

766 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21448784	C₂	1	2	yes	maximal
2	#21446681	C₂	1	2	yes	maximal
3	#21446455	C₂	1	4	yes	maximal
4	#21445983	C₂	1	1	yes	maximal, characteristic
5	#21445834	C₂	1	2	yes	maximal
6	#107×Dic₆₁	C₂	1	2	yes	maximal
7	C₈×C₄⋊Dic₆₁	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄×Q₁₆×C₆₁	C₂	1	1	yes	maximal, characteristic
9	C₂×#14519711	C₂²	1	2	yes	—
10	C₂×#14519709	C₂²	1	2	yes	—
11	C₂×C₄⋊#92418	C₂²	1	1	yes	characteristic
12	C₂×#14519701	C₂²	1	4	yes	—
13	C₂×C₈×Dic₆₁	C₂²	1	2	yes	—
14	#16839669	C₂²	1	4	yes	—
15	#16839648	—	2	4	no	—
16	#16839630	—	2	2	no	—
17	#16839628	C₂²	1	2	yes	—
18	C₄⋊C₄×Dic₆₁	C₂²	1	4	yes	—
19	C₄×C₄⋊Dic₆₁	C₂²	1	1	yes	characteristic
20	#16839608	—	2	2	no	—
21	#16839524	C₂²	1	4	yes	—
22	#16839516	C₂²	1	2	yes	—
23	C₂²×Q₁₆×C₆₁	C₂²	1	1	yes	characteristic
24	C₂×C₄×Q₈×C₆₁	C₂²	1	2	yes	—
25	C₄×Q₁₆×C₆₁	C₄	1	8	yes	complemented
26	C₂×#107×C₆₁	C₂²	1	1	yes	characteristic
27	C₂×#103×C₆₁	C₂²	1	2	yes	—
28	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
29	C₂²×#92418	C₂³	1	2	yes	—
30	C₂×C₄⋊Dic₆₁	D₄	1	2	yes	—
31	C₂×C₄⋊Dic₆₁	C₂³	1	4	yes	—
32	C₂×C₄×Dic₆₁	—	2	4	no	—
33	C₂×C₄×Dic₆₁	C₂³	1	2	yes	—
34	#14519711	—	2	4	no	—
35	#14519709	—	4	2	no	—
36	C₄⋊#92418	—	2	2	no	—
37	#14519701	—	4	4	no	—
38	C₈×Dic₆₁	—	2	4	no	—
39	#14519684	—	2	2	no	—
40	#14519684	—	2	2	no	—
41	#14519684	—	2	4	no	—
42	#14519684	—	2	4	no	—
43	#14519684	—	2	4	no	—
44	#14519684	D₄	1	1	yes	characteristic
45	#14519684	D₄	1	1	yes	characteristic
46	C₂²×Q₈×C₆₁	C₂³	1	2	yes	—
47	C₂×Q₁₆×C₆₁	—	2	2	no	—
48	C₂×Q₁₆×C₆₁	—	2	2	no	—
49	C₂×Q₁₆×C₆₁	C₂×C₄	1	4	yes	—
50	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
51	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
52	C₄×Q₈×C₆₁	—	2	4	no	—
53	C₄×Q₈×C₆₁	C₂×C₄	1	8	yes	—
54	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
55	C₂×C₄⋊C₄×C₆₁	C₂³	1	2	yes	—
56	C₂×C₄²×C₆₁	—	2	2	no	—
57	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
58	#107×C₆₁	C₂×C₄	1	4	yes	—
59	#103×C₆₁	C₂×C₄	1	8	yes	—
60	C₄×C₈×C₆₁	C₂×C₄	1	4	yes	—
61	C₂×Q₈×C₆₁	—	2	2	no	—
62	C₂×Q₈×C₆₁	—	2	2	no	—
63	C₂×Q₈×C₆₁	—	2	2	no	—
64	C₂×Q₈×C₆₁	—	2	2	no	—
65	C₂×Q₈×C₆₁	—	2	2	no	—
66	C₂×Q₈×C₆₁	C₂²×C₄	1	4	yes	—
67	C₂²×C₄×C₆₁	—	2	2	no	—
68	C₂²×C₄×C₆₁	—	2	2	no	—
69	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
72	C₂²×Dic₆₁	—	2	4	no	—
73	C₂²×Dic₆₁	C₂×D₄	1	2	yes	—
74	C₂²×Dic₆₁	C₂×D₄	1	2	yes	—
75	Q₁₆×C₆₁	—	2	8	no	—
76	C₂×C₈×C₆₁	—	2	1	no	—
77	C₂×C₈×C₆₁	—	2	1	no	—
78	C₂×C₈×C₆₁	C₂²×C₄	1	2	yes	—
79	C₂×C₈×C₆₁	C₂²×C₄	1	2	yes	—
80	C₂×C₈×C₆₁	#41	1	2	yes	—
81	C₂×C₈×C₆₁	#41	1	2	yes	—
82	C₄⋊C₄×C₆₁	—	2	8	no	—
83	C₄⋊C₄×C₆₁	—	2	2	no	—
84	C₄⋊C₄×C₆₁	C₂²×C₄	1	4	yes	—
85	C₄²×C₆₁	—	2	8	no	—
86	C₄²×C₆₁	C₂²×C₄	1	2	yes	—
87	C₄²×C₆₁	#41	1	2	yes	—
88	C₄⋊Dic₆₁	—	2	4	no	—
89	C₄⋊Dic₆₁	—	2	4	no	—
90	C₄⋊Dic₆₁	—	2	4	no	—
91	C₄×Dic₆₁	—	2	2	no	—
92	C₄×Dic₆₁	—	2	2	no	—
93	C₄×Dic₆₁	—	4	8	no	—
94	C₂×#92418	—	2	2	no	—
95	C₂×#92418	—	2	2	no	—
96	C₂×#92418	—	2	2	no	—
97	C₂³×C₆₁	C₂²×D₄	1	1	yes	characteristic
98	Q₈×C₆₁	—	2	4	no	—
99	Q₈×C₆₁	—	2	4	no	—
100	Q₈×C₆₁	—	4	2	no	—
101	Q₈×C₆₁	—	4	2	no	—
102	C₂×C₄×C₆₁	—	2	2	no	—
103	C₂×C₄×C₆₁	—	2	2	no	—
104	C₂×C₄×C₆₁	—	2	4	no	—
105	C₂×C₄×C₆₁	—	2	4	no	—
106	C₂×C₄×C₆₁	—	4	2	no	—
107	C₂×C₄×C₆₁	—	4	2	no	—
108	C₂×C₄×C₆₁	—	4	4	no	—
109	C₂×C₄×C₆₁	C₄×D₄	1	2	yes	—
110	C₂×C₄×C₆₁	C₄×D₄	1	2	yes	—
111	C₂×C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
112	C₂×C₄×C₆₁	C₂×#41	1	1	yes	characteristic
113	C₂×C₄×C₆₁	C₂×#41	1	1	yes	characteristic
114	C₂×C₄×C₆₁	C₂×#41	1	1	yes	characteristic
115	C₂×C₄×C₆₁	H(5,𝔽₂)	1	1	yes	characteristic
116	C₂×C₄×C₆₁	2₋¹⁺⁴	1	1	yes	characteristic
117	C₂×Dic₆₁	—	2	2	no	—
118	C₂×Dic₆₁	—	2	2	no	—
119	C₂×Dic₆₁	—	2	2	no	—
120	C₂×Dic₆₁	—	2	2	no	—
121	C₂×Dic₆₁	—	2	2	no	—
122	C₂×Dic₆₁	—	4	4	no	—
123	C₂×Dic₆₁	—	4	4	no	—
124	C₂×Dic₆₁	—	4	4	no	—
125	C₂×Dic₆₁	#135	1	4	yes	—
126	C₈×C₆₁	—	2	2	no	—
127	C₈×C₆₁	#117	1	4	yes	—
128	#92418	—	4	2	no	—
129	#22809	—	61	1	no	maximal, Sylow
130	C₂²×C₆₁	C₂×C₄×D₄	1	1	yes	characteristic
131	C₂²×C₆₁	#547	1	1	yes	characteristic
132	C₂²×C₆₁	#550	1	1	yes	characteristic
133	C₂²×C₆₁	C₂×#135	1	2	yes	—
134	C₂²×C₆₁	#578	1	2	yes	—
135	C₄×C₆₁	—	2	2	no	—
136	C₄×C₆₁	—	2	2	no	—
137	C₄×C₆₁	—	4	4	no	—
138	C₄×C₆₁	—	4	4	no	—
139	C₄×C₆₁	C₂×#117	1	1	yes	characteristic
140	C₄×C₆₁	#520	1	1	yes	characteristic
141	C₄×C₆₁	#555	1	1	yes	characteristic
142	C₄×C₆₁	#556	1	1	yes	characteristic
143	Dic₆₁	—	2	4	no	—
144	Dic₆₁	—	4	2	no	—
145	Dic₆₁	—	8	4	no	—
146	C₂×C₄×Q₁₆	D₆₁	1	1	yes	characteristic
147	#2356	—	61	2	no	—
148	#1969	—	61	2	no	—
149	#1927	—	61	4	no	—
150	#1846	—	61	1	no	—
151	#1805	—	61	2	no	—
152	C₄×#107	—	61	2	no	—
153	C₈×C₄⋊C₄	—	61	1	no	—
154	C₂×C₆₁	#2354	1	1	yes	characteristic
155	C₂×C₆₁	C₄×#135	1	1	yes	characteristic
156	C₂×C₆₁	#2950	1	1	yes	characteristic
157	C₂×C₆₁	#3336	1	2	yes	—
158	C₂×C₆₁	#3393	1	2	yes	—
159	C₂²×Q₁₆	C₂×D₆₁	1	1	yes	characteristic
160	C₂×C₄×Q₈	C₂×D₆₁	1	2	yes	—
161	C₄×Q₁₆	Dic₆₁	1	8	yes	complemented
162	C₂×#107	—	61	2	no	—
163	C₂×#107	C₂×D₆₁	1	1	yes	characteristic
164	C₂×#106	—	61	2	no	—
165	C₂×C₄⋊C₈	—	61	1	no	—
166	C₂×#103	—	61	4	no	—
167	C₂×#103	C₂×D₆₁	1	2	yes	—
168	C₂×C₄×C₈	—	61	2	no	—
169	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
170	#391	—	61	4	no	—
171	#387	—	122	4	no	—
172	#383	—	122	2	no	—
173	#382	—	61	2	no	—
174	C₄×C₄⋊C₄	—	61	1	no	—
175	C₄×C₄⋊C₄	—	61	4	no	—
176	#376	—	122	2	no	—
177	#340	—	61	4	no	—
178	#336	—	61	2	no	—
179	C₆₁	#22809	1	1	yes	characteristic, complemented, Sylow
180	C₂²×Q₈	C₂²×D₆₁	1	2	yes	—
181	C₂×Q₁₆	—	2	2	no	—
182	C₂×Q₁₆	—	2	2	no	—
183	C₂×Q₁₆	C₂×Dic₆₁	1	4	yes	—
184	C₂²×C₈	—	61	2	no	—
185	C₂²×C₈	C₂²×D₆₁	1	1	yes	characteristic
186	C₂²×C₈	C₂²×D₆₁	1	1	yes	characteristic
187	C₄×Q₈	—	2	4	no	—
188	C₄×Q₈	C₂×Dic₆₁	1	8	yes	—
189	C₂×C₄⋊C₄	—	2	2	no	—
190	C₂×C₄⋊C₄	—	61	2	no	—
191	C₂×C₄⋊C₄	—	61	4	no	—
192	C₂×C₄⋊C₄	C₂²×D₆₁	1	2	yes	—
193	C₂×C₄²	—	2	2	no	—
194	C₂×C₄²	—	61	2	no	—
195	C₂×C₄²	—	122	4	no	—
196	C₂×C₄²	C₂²×D₆₁	1	1	yes	characteristic
197	#107	—	122	4	no	—
198	#107	C₂×Dic₆₁	1	4	yes	—
199	#106	—	244	2	no	—
200	C₄⋊C₈	—	122	2	no	—
201	#103	—	244	4	no	—
202	#103	C₂×Dic₆₁	1	8	yes	—
203	C₄×C₈	—	122	4	no	—
204	C₄×C₈	C₂×Dic₆₁	1	4	yes	—
205	#95	—	61	1	no	—
206	#95	—	61	1	no	—
207	#95	—	122	4	no	—
208	#95	—	122	4	no	—
209	#95	—	122	4	no	—
210	#95	—	122	2	no	—
211	#95	—	122	2	no	—
212	C₂×Q₈	—	2	2	no	—
213	C₂×Q₈	—	2	2	no	—
214	C₂×Q₈	—	2	2	no	—
215	C₂×Q₈	—	2	2	no	—
216	C₂×Q₈	—	2	2	no	—
217	C₂×Q₈	C₂²×Dic₆₁	1	4	yes	—
218	C₂²×C₄	—	2	2	no	—
219	C₂²×C₄	—	2	2	no	—
220	C₂²×C₄	—	61	2	no	—
221	C₂²×C₄	—	61	2	no	—
222	C₂²×C₄	—	122	4	no	—
223	C₂²×C₄	D₄×D₆₁	1	1	yes	characteristic
224	C₂²×C₄	D₄×D₆₁	1	1	yes	characteristic
225	C₂²×C₄	C₂³×D₆₁	1	1	yes	characteristic
226	Q₁₆	—	2	8	no	—
227	C₂×C₈	—	2	1	no	—
228	C₂×C₈	—	2	1	no	—
229	C₂×C₈	—	122	2	no	—
230	C₂×C₈	—	122	2	no	—
231	C₂×C₈	—	122	2	no	—
232	C₂×C₈	#11758222	1	2	yes	—
233	C₂×C₈	#11758222	1	2	yes	—
234	C₂×C₈	C₂²×Dic₆₁	1	2	yes	—
235	C₂×C₈	C₂²×Dic₆₁	1	2	yes	—
236	C₄⋊C₄	—	2	2	no	—
237	C₄⋊C₄	—	2	8	no	—
238	C₄⋊C₄	—	122	4	no	—
239	C₄⋊C₄	—	122	4	no	—
240	C₄⋊C₄	—	122	4	no	—
241	C₄⋊C₄	C₂²×Dic₆₁	1	4	yes	—
242	C₄²	—	2	8	no	—
243	C₄²	—	122	2	no	—
244	C₄²	—	122	2	no	—
245	C₄²	—	244	8	no	—
246	C₄²	#11758224	1	2	yes	—
247	C₄²	C₂²×Dic₆₁	1	2	yes	—
248	C₂³	C₂×D₄×D₆₁	1	1	yes	characteristic
249	Q₈	—	2	4	no	—
250	Q₈	—	2	4	no	—
251	Q₈	—	4	2	no	—
252	Q₈	—	4	2	no	—
253	C₂×C₄	—	2	2	no	—
254	C₂×C₄	—	2	2	no	—
255	C₂×C₄	—	2	4	no	—
256	C₂×C₄	—	2	4	no	—
257	C₂×C₄	—	4	4	no	—
258	C₂×C₄	—	4	2	no	—
259	C₂×C₄	—	4	2	no	—
260	C₂×C₄	—	61	4	no	—
261	C₂×C₄	—	122	2	no	—
262	C₂×C₄	—	122	2	no	—
263	C₂×C₄	—	122	2	no	—
264	C₂×C₄	—	122	2	no	—
265	C₂×C₄	—	122	2	no	—
266	C₂×C₄	—	244	4	no	—
267	C₂×C₄	—	244	4	no	—
268	C₂×C₄	—	244	4	no	—
269	C₂×C₄	D₄×Dic₆₁	1	2	yes	—
270	C₂×C₄	D₄×Dic₆₁	1	2	yes	—
271	C₂×C₄	C₂³×Dic₆₁	1	1	yes	characteristic
272	C₂×C₄	C₂×#11758224	1	1	yes	characteristic
273	C₂×C₄	C₂×#11758222	1	1	yes	characteristic
274	C₂×C₄	C₂×#11758222	1	1	yes	characteristic
275	C₂×C₄	#14519823	1	1	yes	characteristic
276	C₂×C₄	#14519824	1	1	yes	characteristic
277	C₈	—	2	2	no	—
278	C₈	—	244	2	no	—
279	C₈	#14519736	1	4	yes	—
280	C₂²	C₂×D₄×Dic₆₁	1	1	yes	characteristic
281	C₂²	#16840417	1	1	yes	characteristic
282	C₂²	#16840433	1	1	yes	characteristic
283	C₂²	#135×D₆₁	1	2	yes	—
284	C₂²	#16840575	1	2	yes	—
285	C₄	—	2	2	no	—
286	C₄	—	2	2	no	—
287	C₄	—	4	4	no	—
288	C₄	—	4	4	no	—
289	C₄	—	122	4	no	—
290	C₄	—	488	4	no	—
291	C₄	—	244	2	no	—
292	C₄	C₂×#14519736	1	1	yes	characteristic
293	C₄	#16840245	1	1	yes	characteristic
294	C₄	#16840477	1	1	yes	characteristic
295	C₄	#16840482	1	1	yes	characteristic
296	C₂	#21448768	1	1	yes	characteristic
297	C₂	#135×Dic₆₁	1	1	yes	characteristic
298	C₂	#21454687	1	1	yes	characteristic
299	C₂	#21458189	1	2	yes	—
300	C₂	#21458807	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₄×D₆₁
commutator	C₂×C₄×C₆₁	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₆₁
Fitting	C₂×C₄×Q₁₆×C₆₁	C₂
socle, prime radical	C₂³×C₆₁	C₂²×D₄
cosocle	C₂²×C₄×C₆₁	C₂⁴
hypercenter	C₂×C₄×Q₁₆	D₆₁
hypocenter	C₆₁	#22809
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×Q₁₆

Complements

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

N	H ≅ G/N	type
25. C₄×Q₁₆×C₆₁	C₄	semidirect
161. C₄×Q₁₆	Dic₆₁	semidirect
179. C₆₁	#22809	semidirect

Maximal subgroups

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

subgroup	index
#21448784	2
#21446681	2
#21446455	2
#21445983	2
#21445834	2
#107×Dic₆₁	2
C₈×C₄⋊Dic₆₁	2
C₂×C₄×Q₁₆×C₆₁	2
#22809	61