Subgroups — #29650528

Order 15616.

750 conjugacy classes of subgroups.

The subgroup lattice has 290 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21456398	C₂	1	1	yes	maximal, characteristic
2	#21454530	C₂	1	2	yes	maximal, complemented
3	#21444703	C₂	1	4	yes	maximal, complemented
4	#21444613	C₂	1	4	yes	maximal, complemented
5	#21444269	C₂	1	2	yes	maximal, complemented
6	C₂×C₈×#92418	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₁×#1567	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#14519785	—	2	2	no	—
9	#16840312	C₂²	1	2	yes	—
10	#16840167	C₄	1	1	yes	characteristic
11	#16840139	C₄	1	1	yes	characteristic
12	#16840109	C₄	1	2	yes	—
13	#16839998	C₄	1	4	yes	—
14	C₄×D₄₈₈	C₂²	1	4	yes	—
15	#16839839	—	2	2	no	—
16	#16839839	C₂²	1	4	yes	—
17	#16839827	—	2	2	no	—
18	#16839827	C₂²	1	4	yes	—
19	#16839730	—	2	2	no	—
20	C₂×C₄×#92418	C₂²	1	1	yes	characteristic
21	C₂×C₄×#92418	C₂²	1	1	yes	characteristic
22	#16839489	C₂²	1	8	yes	—
23	C₈×#92418	C₂²	1	2	yes	—
24	C₈×#92418	C₂²	1	2	yes	—
25	C₆₁×#423	C₂²	1	2	yes	—
26	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
27	C₆₁×#335	C₂²	1	4	yes	—
28	C₂×#11758222	—	2	2	no	—
29	C₂×D₄₈₈	C₂×C₄	1	4	yes	—
30	#14519785	—	4	8	no	—
31	C₂×#11758189	—	2	4	no	—
32	C₂²×#92418	—	2	1	no	—
33	C₂²×#92418	D₄	1	1	yes	characteristic
34	C₂²×#92418	D₄	1	1	yes	characteristic
35	C₂×C₈×D₆₁	—	2	4	no	—
36	#14519777	C₂×C₄	1	2	yes	—
37	#14519774	C₂×C₄	1	2	yes	—
38	#14519765	—	2	2	no	—
39	#14519751	—	2	2	no	—
40	#14519751	C₂×C₄	1	4	yes	—
41	C₄×D₂₄₄	—	2	2	no	—
42	C₄×D₂₄₄	C₂³	1	4	yes	—
43	#14519711	C₂×C₄	1	4	yes	—
44	#14519699	C₂×C₄	1	8	yes	—
45	#14519689	—	2	4	no	—
46	#14519689	—	2	4	no	—
47	C₄×#92418	—	2	1	no	—
48	C₄×#92418	D₄	1	1	yes	characteristic
49	C₄×#92418	D₄	1	1	yes	characteristic
50	C₄×#92418	C₂³	1	2	yes	—
51	C₄×#92418	C₂³	1	1	yes	characteristic
52	C₄×#92418	C₂³	1	1	yes	characteristic
53	C₂×M₁₆×C₆₁	—	2	2	no	—
54	C₂²×C₈×C₆₁	C₂×C₄	1	2	yes	—
55	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
56	C₄⋊C₈×C₆₁	—	2	2	no	—
57	C₄⋊C₈×C₆₁	C₂³	1	4	yes	—
58	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
59	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
60	C₄×C₈×C₆₁	C₂³	1	2	yes	—
61	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
62	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₄×C₆₁	#41	1	1	yes	characteristic
64	C₂×C₂²⋊D₆₁	—	2	4	no	—
65	#11758222	—	4	4	no	—
66	#11758222	—	4	4	no	—
67	C₂×D₂₄₄	—	2	2	no	—
68	C₂×D₂₄₄	—	2	2	no	—
69	C₂×D₂₄₄	C₂²×C₄	1	4	yes	—
70	C₂×C₄×D₆₁	—	2	4	no	—
71	C₂×Dic₁₂₂	—	2	2	no	—
72	M₁₆×C₆₁	—	4	4	no	—
73	C₂×C₈×C₆₁	—	2	4	no	—
74	C₂×C₈×C₆₁	—	2	4	no	—
75	C₂×C₈×C₆₁	C₂²×C₄	1	2	yes	—
76	C₂×C₈×C₆₁	C₂²×C₄	1	2	yes	—
77	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
78	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
79	C₄²×C₆₁	#41	1	1	yes	characteristic
80	C₄²×C₆₁	C₂⁴	1	1	yes	characteristic
81	#11758198	—	2	4	no	—
82	#11758198	—	2	4	no	—
83	C₄⋊Dic₆₁	—	2	2	no	—
84	C₄⋊Dic₆₁	C₂²×C₄	1	4	yes	—
85	C₂×#92418	—	2	4	no	—
86	C₂×#92418	—	2	2	no	—
87	C₂×#92418	—	2	2	no	—
88	C₂×#92418	D₈	1	4	yes	complemented
89	C₂×#92418	C₂×D₄	1	1	yes	characteristic
90	C₂×#92418	C₂×D₄	1	1	yes	characteristic
91	C₂×#92418	C₂×D₄	1	1	yes	characteristic
92	C₂×#92418	C₂×D₄	1	1	yes	characteristic
93	D₄₈₈	—	2	8	no	—
94	#11758189	—	4	8	no	—
95	C₈×D₆₁	—	4	8	no	—
96	C₂³×C₆₁	C₄×D₄	1	1	yes	characteristic
97	C₂²×D₆₁	—	2	4	no	—
98	C₂×C₄×C₆₁	—	2	2	no	—
99	C₂×C₄×C₆₁	—	2	2	no	—
100	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
101	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
102	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
103	C₂×C₄×C₆₁	C₂×D₈	1	2	yes	—
104	C₂×C₄×C₆₁	#135	1	2	yes	—
105	C₂×C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
106	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
107	C₂×C₄×C₆₁	C₂×#41	1	1	yes	characteristic
108	C₂²⋊D₆₁	—	4	8	no	—
109	C₂×Dic₆₁	—	2	4	no	—
110	D₂₄₄	—	2	8	no	—
111	D₂₄₄	—	4	4	no	—
112	D₂₄₄	—	4	4	no	—
113	C₄×D₆₁	—	4	4	no	—
114	C₄×D₆₁	—	4	4	no	—
115	Dic₁₂₂	—	4	4	no	—
116	C₈×C₆₁	—	2	2	no	—
117	C₈×C₆₁	—	4	4	no	—
118	C₈×C₆₁	#131	1	4	yes	complemented
119	#92418	—	2	4	no	—
120	#92418	—	2	1	no	—
121	#92418	—	2	1	no	—
122	#92418	C₂×D₈	1	2	yes	—
123	#92418	C₂×D₈	1	2	yes	—
124	#17956	—	61	1	no	maximal, Sylow
125	C₂²×C₆₁	—	2	1	no	—
126	C₂²×C₆₁	—	2	1	no	—
127	C₂²×C₆₁	C₄×D₈	1	2	yes	—
128	C₂²×C₆₁	C₂×C₄×D₄	1	1	yes	characteristic
129	C₂×D₆₁	—	4	4	no	—
130	C₂×D₆₁	—	4	4	no	—
131	C₂×D₆₁	—	4	4	no	—
132	C₄×C₆₁	—	2	1	no	—
133	C₄×C₆₁	—	2	1	no	—
134	C₄×C₆₁	C₄×D₈	1	2	yes	—
135	C₄×C₆₁	#443	1	2	yes	—
136	C₄×C₆₁	C₂×#131	1	2	yes	—
137	C₄×C₆₁	C₂²×D₈	1	1	yes	characteristic
138	C₄×C₆₁	C₂×#135	1	1	yes	characteristic
139	Dic₆₁	—	4	4	no	—
140	#3169	—	61	1	no	—
141	#2931	—	61	2	no	—
142	#1589	—	61	4	no	—
143	C₈×D₈	—	61	4	no	—
144	#1567	D₆₁	1	1	yes	characteristic, complemented
145	#1492	—	61	2	no	—
146	C₂×C₈²	—	61	1	no	—
147	C₂×C₆₁	—	2	2	no	—
148	C₂×C₆₁	#2931	1	1	yes	characteristic
149	C₂×C₆₁	C₂×C₄×D₈	1	1	yes	characteristic
150	C₂×C₆₁	C₂×#443	1	1	yes	characteristic
151	D₆₁	—	8	4	no	—
152	C₂×#131	—	122	2	no	—
153	#532	—	61	2	no	—
154	#500	—	61	1	no	—
155	#493	—	61	1	no	—
156	#486	—	61	2	no	—
157	#466	—	61	4	no	—
158	C₄×D₈	—	61	4	no	—
159	#436	—	61	4	no	—
160	#436	—	122	2	no	—
161	C₈×D₄	—	61	4	no	—
162	C₈×D₄	—	122	2	no	—
163	#423	C₂×D₆₁	1	2	yes	—
164	#408	—	122	2	no	—
165	C₂×C₄×C₈	—	61	1	no	—
166	C₂×C₄×C₈	—	61	1	no	—
167	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
168	#335	C₂×D₆₁	1	4	yes	—
169	D₄⋊C₈	—	61	8	no	—
170	C₈²	—	61	2	no	—
171	C₈²	—	61	2	no	—
172	C₆₁	#17956	1	1	yes	characteristic, complemented, Sylow
173	C₂×#41	—	122	2	no	—
174	C₂×D₈	—	61	4	no	—
175	#131	—	244	8	no	—
176	C₂×M₁₆	—	2	2	no	—
177	C₂×M₁₆	—	122	4	no	—
178	C₂²×C₈	—	61	1	no	—
179	C₂²×C₈	—	61	1	no	—
180	C₂²×C₈	—	122	4	no	—
181	C₂²×C₈	—	122	1	no	—
182	C₂²×C₈	C₄×D₆₁	1	2	yes	—
183	C₄⋊Q₈	—	61	2	no	—
184	#127	—	61	2	no	—
185	#124	—	122	2	no	—
186	#121	—	61	4	no	—
187	#121	—	122	2	no	—
188	C₄×D₄	—	61	4	no	—
189	C₄×D₄	—	122	2	no	—
190	C₂×C₄²	C₂²×D₆₁	1	1	yes	characteristic
191	#107	—	61	4	no	—
192	C₄⋊C₈	—	2	2	no	—
193	C₄⋊C₈	C₂²×D₆₁	1	4	yes	—
194	#102	—	61	8	no	—
195	C₂²⋊C₈	—	122	4	no	—
196	C₂²⋊C₈	—	122	4	no	—
197	C₄×C₈	—	61	1	no	—
198	C₄×C₈	—	61	1	no	—
199	C₄×C₈	—	61	1	no	—
200	C₄×C₈	—	61	1	no	—
201	C₄×C₈	—	61	2	no	—
202	C₄×C₈	—	122	1	no	—
203	C₄×C₈	C₄×D₆₁	1	1	yes	characteristic
204	C₄×C₈	C₄×D₆₁	1	1	yes	characteristic
205	C₄×C₈	C₂²×D₆₁	1	2	yes	—
206	#41	—	244	4	no	—
207	#41	—	244	4	no	—
208	C₂×Q₈	—	122	2	no	—
209	C₂×D₄	—	61	4	no	—
210	C₂×D₄	—	122	4	no	—
211	C₂×D₄	—	122	2	no	—
212	C₂×D₄	—	122	2	no	—
213	C₂²×C₄	—	122	4	no	—
214	C₂²×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
215	C₂²×C₄	D₄×D₆₁	1	1	yes	characteristic
216	C₂²×C₄	#11758226	1	1	yes	characteristic
217	D₈	—	122	8	no	—
218	M₁₆	—	4	4	no	—
219	M₁₆	—	244	8	no	—
220	C₂×C₈	—	2	4	no	—
221	C₂×C₈	—	2	4	no	—
222	C₂×C₈	—	61	1	no	—
223	C₂×C₈	—	61	1	no	—
224	C₂×C₈	—	61	4	no	—
225	C₂×C₈	—	61	1	no	—
226	C₂×C₈	—	61	1	no	—
227	C₂×C₈	—	122	4	no	—
228	C₂×C₈	—	122	2	no	—
229	C₂×C₈	—	122	2	no	—
230	C₂×C₈	—	244	8	no	—
231	C₂×C₈	C₂×C₄×D₆₁	1	2	yes	—
232	C₂×C₈	C₂×C₄×D₆₁	1	2	yes	—
233	C₄⋊C₄	—	61	4	no	—
234	C₄⋊C₄	—	122	2	no	—
235	C₂²⋊C₄	—	122	4	no	—
236	C₂²⋊C₄	—	122	4	no	—
237	C₄²	C₂×C₄×D₆₁	1	1	yes	characteristic
238	C₄²	D₄×D₆₁	1	1	yes	characteristic
239	C₄²	#11758226	1	1	yes	characteristic
240	C₄²	C₂³×D₆₁	1	1	yes	characteristic
241	C₂³	—	122	4	no	—
242	C₂³	#14519739	1	1	yes	characteristic
243	Q₈	—	244	4	no	—
244	D₄	—	122	8	no	—
245	D₄	—	244	8	no	—
246	D₄	—	244	4	no	—
247	D₄	—	244	4	no	—
248	C₂×C₄	—	2	2	no	—
249	C₂×C₄	—	2	2	no	—
250	C₂×C₄	—	122	4	no	—
251	C₂×C₄	—	244	4	no	—
252	C₂×C₄	—	244	4	no	—
253	C₂×C₄	#14519739	1	1	yes	characteristic
254	C₂×C₄	#14519739	1	1	yes	characteristic
255	C₂×C₄	#14519739	1	1	yes	characteristic
256	C₂×C₄	D₈×D₆₁	1	2	yes	—
257	C₂×C₄	#14519800	1	2	yes	—
258	C₂×C₄	C₂²×C₄×D₆₁	1	1	yes	characteristic
259	C₂×C₄	C₂×D₄×D₆₁	1	1	yes	characteristic
260	C₂×C₄	C₂×#11758226	1	1	yes	characteristic
261	C₈	—	2	2	no	—
262	C₈	—	4	4	no	—
263	C₈	—	61	2	no	—
264	C₈	—	61	2	no	—
265	C₈	—	122	4	no	—
266	C₈	—	122	1	no	—
267	C₈	—	122	1	no	—
268	C₈	#14519785	1	4	yes	complemented
269	C₂²	—	2	1	no	—
270	C₂²	—	2	1	no	—
271	C₂²	—	244	4	no	—
272	C₂²	—	244	4	no	—
273	C₂²	—	244	4	no	—
274	C₂²	#16839844	1	2	yes	—
275	C₂²	C₂×#14519739	1	1	yes	characteristic
276	C₄	—	2	1	no	—
277	C₄	—	2	1	no	—
278	C₄	—	244	4	no	—
279	C₄	#16839844	1	2	yes	—
280	C₄	#16839878	1	2	yes	—
281	C₄	C₂×#14519785	1	2	yes	—
282	C₄	C₂×D₈×D₆₁	1	1	yes	characteristic
283	C₄	C₂×#14519800	1	1	yes	characteristic
284	C₂	—	2	2	no	—
285	C₂	—	488	4	no	—
286	C₂	#21454530	1	1	yes	characteristic
287	C₂	C₂×#16839844	1	1	yes	characteristic
288	C₂	C₂×#16839878	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₆₁
commutator	C₂×C₄×C₆₁	C₂³×C₄
Frattini	C₄²	C₂³×D₆₁
Fitting	C₆₁×#1567	C₂
socle	C₂²×C₆₁	C₂×C₄×D₄
prime radical	#21456398	C₂
cosocle	C₄²×C₆₁	C₂⁴
hypercenter	#1567	D₆₁
hypocenter	C₆₁	#17956
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1567

Complements

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

N	H ≅ G/N	type
2. #21454530	C₂	semidirect
3. #21444703	C₂	semidirect
4. #21444613	C₂	semidirect
5. #21444269	C₂	semidirect
6. C₂×C₈×#92418	C₂	semidirect
7. C₆₁×#1567	C₂	semidirect
88. C₂×#92418	D₈	semidirect
118. C₈×C₆₁	#131	semidirect
144. #1567	D₆₁	semidirect
172. C₆₁	#17956	semidirect
268. C₈	#14519785	semidirect

Maximal subgroups

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

subgroup	index
#21456398	2
#21454530	2
#21444703	2
#21444613	2
#21444269	2
C₂×C₈×#92418	2
C₆₁×#1567	2
#17956	61