Subgroups — #29573117

Order 15616.

494 conjugacy classes of subgroups.

The subgroup lattice has 286 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21445947	C₂	1	1	yes	maximal, characteristic, complemented
2	#21445819	C₂	1	2	yes	maximal
3	C₂×#16839515	C₂	1	1	yes	maximal, characteristic, complemented
4	#21443703	C₂	1	2	yes	maximal, complemented
5	C₂×C₆₁×#436	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊#92418	C₂²	1	2	yes	—
7	C₂×#14519698	—	2	2	no	—
8	#16839632	C₂²	1	2	yes	—
9	#16839525	—	2	1	no	—
10	#16839525	—	2	1	no	—
11	#16839524	—	2	2	no	—
12	#16839515	—	2	2	no	—
13	C₂²×M₁₆×C₆₁	—	2	1	no	—
14	C₂×C₄×D₄×C₆₁	C₂²	1	1	yes	characteristic
15	C₆₁×#436	—	2	2	no	—
16	C₆₁×#436	C₄	1	4	yes	complemented
17	C₂×C₄⋊C₈×C₆₁	C₂²	1	1	yes	characteristic
18	C₂×C₂²⋊C₈×C₆₁	—	2	1	no	—
19	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
20	C₂²×#92418	—	2	2	no	—
21	C₂×C₄⋊Dic₆₁	—	2	2	no	—
22	C₂×C₄⋊Dic₆₁	—	2	2	no	—
23	C₂×C₄⋊Dic₆₁	—	2	2	no	—
24	C₄⋊#92418	—	2	2	no	—
25	C₄⋊#92418	—	2	2	no	—
26	#14519698	—	4	4	no	—
27	C₂²×D₄×C₆₁	D₄	1	1	yes	characteristic
28	C₂³×C₄×C₆₁	—	2	1	no	—
29	C₂×M₁₆×C₆₁	—	2	4	no	—
30	C₂×M₁₆×C₆₁	—	4	4	no	—
31	C₂²×C₈×C₆₁	—	2	1	no	—
32	C₂²×C₈×C₆₁	D₄	1	1	yes	characteristic
33	C₂²×C₈×C₆₁	D₄	1	1	yes	characteristic
34	C₄×D₄×C₆₁	—	2	1	no	—
35	C₄×D₄×C₆₁	—	2	1	no	—
36	C₄×D₄×C₆₁	C₂×C₄	1	2	yes	—
37	C₄×D₄×C₆₁	Q₈	1	2	yes	—
38	C₂×C₄⋊C₄×C₆₁	D₄	1	1	yes	characteristic
39	C₂×C₂²⋊C₄×C₆₁	—	2	1	no	—
40	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
41	C₄⋊C₈×C₆₁	—	2	1	no	—
42	C₄⋊C₈×C₆₁	C₂×C₄	1	2	yes	—
43	C₂²⋊C₈×C₆₁	—	2	4	no	—
44	C₄×C₈×C₆₁	C₂×C₄	1	2	yes	—
45	C₄×C₈×C₆₁	D₄	1	2	yes	—
46	C₂⁴×C₆₁	—	2	1	no	—
47	C₂×D₄×C₆₁	—	2	1	no	—
48	C₂×D₄×C₆₁	—	2	2	no	—
49	C₂×D₄×C₆₁	—	2	2	no	—
50	C₂×D₄×C₆₁	C₄⋊C₄	1	2	yes	—
51	C₂²×C₄×C₆₁	—	2	1	no	—
52	C₂²×C₄×C₆₁	—	2	2	no	—
53	C₂²×C₄×C₆₁	—	2	2	no	—
54	C₂²×C₄×C₆₁	—	4	2	no	—
55	C₂²×C₄×C₆₁	—	4	2	no	—
56	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
57	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄×C₆₁	#41	1	1	yes	characteristic
59	C₂²×Dic₆₁	—	4	2	no	—
60	M₁₆×C₆₁	—	4	8	no	—
61	C₂×C₈×C₆₁	—	2	2	no	—
62	C₂×C₈×C₆₁	—	2	2	no	—
63	C₂×C₈×C₆₁	—	2	2	no	—
64	C₂×C₈×C₆₁	—	4	2	no	—
65	C₂×C₈×C₆₁	C₂²⋊C₄	1	2	yes	—
66	C₂×C₈×C₆₁	C₂²⋊C₄	1	2	yes	—
67	C₄⋊C₄×C₆₁	—	2	1	no	—
68	C₄⋊C₄×C₆₁	C₄⋊C₄	1	2	yes	—
69	C₂²⋊C₄×C₆₁	—	2	2	no	—
70	C₂²⋊C₄×C₆₁	—	2	2	no	—
71	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
72	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
73	C₄²×C₆₁	C₂×Q₈	1	1	yes	characteristic
74	C₄²×C₆₁	#41	1	1	yes	characteristic
75	C₄⋊Dic₆₁	—	4	2	no	—
76	C₄⋊Dic₆₁	—	4	2	no	—
77	C₄⋊Dic₆₁	—	4	2	no	—
78	C₄⋊Dic₆₁	—	4	2	no	—
79	C₄⋊Dic₆₁	—	8	2	no	—
80	C₂×#92418	—	4	2	no	—
81	C₂×#92418	—	4	2	no	—
82	C₂×#92418	—	4	2	no	—
83	C₂³×C₆₁	—	2	2	no	—
84	C₂³×C₆₁	—	4	2	no	—
85	C₂³×C₆₁	—	4	2	no	—
86	C₂³×C₆₁	—	4	1	no	—
87	C₂³×C₆₁	—	4	1	no	—
88	C₂³×C₆₁	#121	1	1	yes	characteristic
89	D₄×C₆₁	—	2	4	no	—
90	D₄×C₆₁	—	4	1	no	—
91	D₄×C₆₁	—	4	1	no	—
92	C₂×C₄×C₆₁	—	2	1	no	—
93	C₂×C₄×C₆₁	—	2	1	no	—
94	C₂×C₄×C₆₁	—	2	2	no	—
95	C₂×C₄×C₆₁	—	4	2	no	—
96	C₂×C₄×C₆₁	—	4	2	no	—
97	C₂×C₄×C₆₁	—	4	2	no	—
98	C₂×C₄×C₆₁	—	4	4	no	—
99	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
100	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
101	C₂×C₄×C₆₁	#117	1	1	yes	characteristic
102	C₂×C₄×C₆₁	#121	1	1	yes	characteristic
103	C₂×C₄×C₆₁	C₂²⋊Q₈	1	1	yes	characteristic
104	C₂×C₄×C₆₁	C₂²⋊Q₈	1	1	yes	characteristic
105	C₂×C₄×C₆₁	#135	1	2	yes	—
106	C₂×C₄×C₆₁	AGL(1,ℤ/8)	1	2	yes	—
107	C₂×C₄×C₆₁	AGL(1,ℤ/8)	1	2	yes	—
108	C₂×C₄×C₆₁	#137	1	2	yes	—
109	C₂×Dic₆₁	—	8	2	no	—
110	C₂×Dic₆₁	—	8	2	no	—
111	C₂×Dic₆₁	—	8	2	no	—
112	C₈×C₆₁	—	2	2	no	—
113	C₈×C₆₁	—	2	2	no	—
114	C₈×C₆₁	—	4	2	no	—
115	#92418	—	8	2	no	—
116	#8654	—	61	1	no	maximal, Sylow
117	C₂²×C₆₁	—	4	2	no	—
118	C₂²×C₆₁	—	4	2	no	—
119	C₂²×C₆₁	—	4	2	no	—
120	C₂²×C₆₁	—	8	2	no	—
121	C₂²×C₆₁	—	8	2	no	—
122	C₂²×C₆₁	#380	1	1	yes	characteristic
123	C₂²×C₆₁	#463	1	2	yes	—
124	C₂²×C₆₁	#468	1	2	yes	—
125	C₂²×C₆₁	#472	1	1	yes	characteristic
126	C₂²×C₆₁	#473	1	1	yes	characteristic
127	C₄×C₆₁	—	2	1	no	—
128	C₄×C₆₁	—	2	1	no	—
129	C₄×C₆₁	—	4	2	no	—
130	C₄×C₆₁	#416	1	1	yes	characteristic
131	C₄×C₆₁	#418	1	1	yes	characteristic
132	C₄×C₆₁	#428	1	2	yes	—
133	C₄×C₆₁	#469	1	1	yes	characteristic
134	C₄×C₆₁	#470	1	1	yes	characteristic
135	C₄×C₆₁	#478	1	2	yes	—
136	Dic₆₁	—	16	2	no	—
137	C₂×#436	D₆₁	1	1	yes	characteristic
138	#1834	—	61	1	no	—
139	#1802	—	61	2	no	—
140	C₂×#335	—	61	1	no	—
141	#1296	—	61	2	no	—
142	C₂×C₆₁	—	8	2	no	—
143	C₂×C₆₁	#1697	1	2	yes	—
144	C₂×C₆₁	#1699	1	2	yes	—
145	C₂×C₆₁	#1804	1	1	yes	characteristic
146	C₂×C₆₁	#1809	1	1	yes	characteristic
147	C₂×C₆₁	#1816	1	1	yes	characteristic
148	C₂²×M₁₆	—	2	1	no	—
149	C₂×C₄×D₄	C₂×D₆₁	1	1	yes	characteristic
150	#436	—	2	2	no	—
151	#436	Dic₆₁	1	4	yes	complemented
152	C₂×C₄⋊C₈	—	61	2	no	—
153	C₂×C₄⋊C₈	C₂×D₆₁	1	1	yes	characteristic
154	C₂×#102	—	122	2	no	—
155	C₂×C₂²⋊C₈	—	2	1	no	—
156	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
157	#384	—	61	2	no	—
158	#340	—	122	1	no	—
159	#340	—	122	1	no	—
160	#340	—	122	2	no	—
161	#335	—	122	2	no	—
162	C₆₁	#8654	1	1	yes	characteristic, complemented, Sylow
163	C₂²×D₄	D₂₄₄	1	1	yes	characteristic
164	C₂³×C₄	—	2	1	no	—
165	C₂×M₁₆	—	2	4	no	—
166	C₂×M₁₆	—	4	4	no	—
167	C₂²×C₈	—	2	1	no	—
168	C₂²×C₈	—	122	2	no	—
169	C₂²×C₈	C₂²⋊D₆₁	1	1	yes	characteristic
170	C₂²×C₈	C₂²⋊D₆₁	1	1	yes	characteristic
171	C₄×D₄	—	2	1	no	—
172	C₄×D₄	—	2	1	no	—
173	C₄×D₄	Dic₁₂₂	1	2	yes	—
174	C₄×D₄	C₂×Dic₆₁	1	2	yes	—
175	C₂×C₄⋊C₄	—	122	2	no	—
176	C₂×C₄⋊C₄	—	122	2	no	—
177	C₂×C₄⋊C₄	—	122	2	no	—
178	C₂×C₄⋊C₄	D₂₄₄	1	1	yes	characteristic
179	C₂×C₂²⋊C₄	—	2	1	no	—
180	C₂×C₄²	C₂²×D₆₁	1	1	yes	characteristic
181	C₄⋊C₈	—	2	1	no	—
182	C₄⋊C₈	—	122	2	no	—
183	C₄⋊C₈	—	122	2	no	—
184	C₄⋊C₈	C₂×Dic₆₁	1	2	yes	—
185	#102	—	244	4	no	—
186	C₂²⋊C₈	—	2	4	no	—
187	C₄×C₈	C₂×Dic₆₁	1	2	yes	—
188	C₄×C₈	C₂²⋊D₆₁	1	2	yes	—
189	C₂⁴	—	2	1	no	—
190	C₂×D₄	—	2	2	no	—
191	C₂×D₄	—	2	1	no	—
192	C₂×D₄	—	2	2	no	—
193	C₂×D₄	C₄⋊Dic₆₁	1	2	yes	—
194	C₂²×C₄	—	2	1	no	—
195	C₂²×C₄	—	2	2	no	—
196	C₂²×C₄	—	2	2	no	—
197	C₂²×C₄	—	4	2	no	—
198	C₂²×C₄	—	4	2	no	—
199	C₂²×C₄	—	244	2	no	—
200	C₂²×C₄	C₂×D₂₄₄	1	1	yes	characteristic
201	C₂²×C₄	#11758222	1	1	yes	characteristic
202	C₂²×C₄	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
203	M₁₆	—	4	8	no	—
204	C₂×C₈	—	2	2	no	—
205	C₂×C₈	—	2	2	no	—
206	C₂×C₈	—	2	2	no	—
207	C₂×C₈	—	4	2	no	—
208	C₂×C₈	—	244	2	no	—
209	C₂×C₈	—	244	2	no	—
210	C₂×C₈	—	244	2	no	—
211	C₂×C₈	C₂²⋊Dic₆₁	1	2	yes	—
212	C₂×C₈	C₂²⋊Dic₆₁	1	2	yes	—
213	C₄⋊C₄	—	2	1	no	—
214	C₄⋊C₄	—	488	2	no	—
215	C₄⋊C₄	—	244	2	no	—
216	C₄⋊C₄	—	244	2	no	—
217	C₄⋊C₄	—	244	2	no	—
218	C₄⋊C₄	—	244	2	no	—
219	C₄⋊C₄	C₄⋊Dic₆₁	1	2	yes	—
220	C₂²⋊C₄	—	2	2	no	—
221	C₂²⋊C₄	—	2	2	no	—
222	C₄²	C₂×Dic₁₂₂	1	1	yes	characteristic
223	C₄²	#11758222	1	1	yes	characteristic
224	C₄²	C₂²×Dic₆₁	1	1	yes	characteristic
225	C₄²	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
226	C₂³	—	2	2	no	—
227	C₂³	—	4	2	no	—
228	C₂³	—	4	2	no	—
229	C₂³	—	4	1	no	—
230	C₂³	—	4	1	no	—
231	C₂³	#14519751	1	1	yes	characteristic
232	D₄	—	2	4	no	—
233	D₄	—	4	1	no	—
234	D₄	—	4	1	no	—
235	C₂×C₄	—	2	1	no	—
236	C₂×C₄	—	2	1	no	—
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	4	2	no	—
239	C₂×C₄	—	4	2	no	—
240	C₂×C₄	—	4	2	no	—
241	C₂×C₄	—	4	4	no	—
242	C₂×C₄	—	488	2	no	—
243	C₂×C₄	—	488	2	no	—
244	C₂×C₄	—	488	2	no	—
245	C₂×C₄	C₂×C₂²⋊Dic₆₁	1	1	yes	characteristic
246	C₂×C₄	C₂×C₄⋊Dic₆₁	1	1	yes	characteristic
247	C₂×C₄	#14519736	1	1	yes	characteristic
248	C₂×C₄	#14519751	1	1	yes	characteristic
249	C₂×C₄	#14519756	1	1	yes	characteristic
250	C₂×C₄	#14519756	1	1	yes	characteristic
251	C₂×C₄	#14519797	1	2	yes	—
252	C₂×C₄	#14519803	1	2	yes	—
253	C₂×C₄	#14519805	1	2	yes	—
254	C₂×C₄	#14519810	1	2	yes	—
255	C₈	—	2	2	no	—
256	C₈	—	2	2	no	—
257	C₈	—	4	2	no	—
258	C₈	—	488	2	no	—
259	C₂²	—	4	2	no	—
260	C₂²	—	4	2	no	—
261	C₂²	—	4	2	no	—
262	C₂²	—	8	2	no	—
263	C₂²	—	8	2	no	—
264	C₂²	#16839622	1	1	yes	characteristic
265	C₂²	#16839977	1	2	yes	—
266	C₂²	#16840008	1	2	yes	—
267	C₂²	#16840032	1	1	yes	characteristic
268	C₂²	#16840037	1	1	yes	characteristic
269	C₄	—	2	1	no	—
270	C₄	—	2	1	no	—
271	C₄	—	4	2	no	—
272	C₄	—	976	2	no	—
273	C₄	#16839758	1	1	yes	characteristic
274	C₄	#16839767	1	1	yes	characteristic
275	C₄	#16839804	1	2	yes	—
276	C₄	#16840015	1	1	yes	characteristic
277	C₄	#16840020	1	1	yes	characteristic
278	C₄	#16840062	1	2	yes	—
279	C₂	—	8	2	no	—
280	C₂	#21445352	1	2	yes	—
281	C₂	#21445366	1	2	yes	—
282	C₂	#21445829	1	1	yes	characteristic
283	C₂	#21445852	1	1	yes	characteristic
284	C₂	#21445883	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#14519751
commutator	C₄²×C₆₁	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₁
Fitting	C₂×C₆₁×#436	C₂
socle	C₂³×C₆₁	#121
prime radical	C₂²×D₄×C₆₁	D₄
cosocle	C₂×C₄²×C₆₁	C₂³
hypercenter	C₂×#436	D₆₁
hypocenter	C₆₁	#8654
LCS	G ▷ C₄²×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#436

Complements

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

N	H ≅ G/N	type
1. #21445947	C₂	semidirect
3. C₂×#16839515	C₂	semidirect
4. #21443703	C₂	semidirect
16. C₆₁×#436	C₄	semidirect
151. #436	Dic₆₁	semidirect
162. C₆₁	#8654	semidirect

Maximal subgroups

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

subgroup	index
#21445947	2
#21445819	2
C₂×#16839515	2
#21443703	2
C₂×C₆₁×#436	2
#8654	61