Subgroups — #29579185

Order 15616.

498 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21446739	C₂	1	2	yes	maximal, complemented
2	C₄×#14519709	C₂	1	1	yes	maximal, characteristic, complemented
3	#21444314	C₂	1	2	yes	maximal, complemented
4	#21444108	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₁×#3169	C₂	1	1	yes	maximal, characteristic
6	#16839815	C₂²	1	1	yes	characteristic
7	C₂×#14519709	—	2	1	no	—
8	C₂×#14519704	—	2	4	no	—
9	C₄×#11758194	C₂²	1	1	yes	characteristic
10	C₄×C₄⋊Dic₆₁	C₂²	1	2	yes	—
11	#16839525	—	2	1	no	—
12	#16839496	—	2	4	no	—
13	#16839484	—	2	1	no	—
14	#16839483	—	2	1	no	—
15	C₆₁×#532	C₂²	1	2	yes	—
16	C₆₁×#500	C₄	1	1	yes	characteristic, complemented
17	C₆₁×#493	C₄	1	1	yes	characteristic, complemented
18	C₆₁×#486	C₄	1	2	yes	complemented
19	C₆₁×#466	—	2	1	no	—
20	C₆₁×#466	—	2	1	no	—
21	C₄×D₈×C₆₁	—	2	1	no	—
22	C₄×D₈×C₆₁	—	2	1	no	—
23	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
24	C₂×#11758194	D₄	1	2	yes	—
25	C₂×C₄⋊Dic₆₁	—	2	2	no	—
26	C₂×C₄×Dic₆₁	—	2	2	no	—
27	#14519709	—	2	4	no	—
28	C₄⋊#92418	—	2	1	no	—
29	#14519704	—	4	8	no	—
30	C₂²⋊#92418	D₄	1	2	yes	—
31	#14519688	—	2	1	no	—
32	C₄×#92418	—	2	1	no	—
33	C₄×#92418	—	2	1	no	—
34	#14519684	—	2	2	no	—
35	C₂×#41×C₆₁	—	2	2	no	—
36	C₂×D₈×C₆₁	—	2	1	no	—
37	C₂×D₈×C₆₁	—	2	1	no	—
38	C₂²×C₈×C₆₁	D₄	1	1	yes	characteristic
39	C₂²×C₈×C₆₁	D₄	1	1	yes	characteristic
40	C₄⋊Q₈×C₆₁	C₂×C₄	1	2	yes	—
41	#127×C₆₁	C₂×C₄	1	2	yes	—
42	#124×C₆₁	—	2	2	no	—
43	#121×C₆₁	—	2	2	no	—
44	#121×C₆₁	—	2	2	no	—
45	C₄×D₄×C₆₁	—	2	2	no	—
46	C₄×D₄×C₆₁	—	2	2	no	—
47	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
48	#107×C₆₁	—	2	1	no	—
49	#107×C₆₁	—	2	1	no	—
50	#102×C₆₁	—	2	2	no	—
51	#102×C₆₁	—	2	2	no	—
52	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
53	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
54	C₄×C₈×C₆₁	D₄	1	1	yes	characteristic
55	C₄×C₈×C₆₁	D₄	1	1	yes	characteristic
56	#41×C₆₁	—	4	4	no	—
57	#41×C₆₁	—	4	4	no	—
58	C₂×Q₈×C₆₁	—	2	2	no	—
59	C₂×D₄×C₆₁	—	2	2	no	—
60	C₂×D₄×C₆₁	—	2	2	no	—
61	C₂×D₄×C₆₁	—	2	2	no	—
62	C₂×D₄×C₆₁	—	4	2	no	—
63	C₂²×C₄×C₆₁	—	4	2	no	—
64	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
67	C₂²×Dic₆₁	—	4	2	no	—
68	D₈×C₆₁	—	4	2	no	—
69	D₈×C₆₁	—	4	2	no	—
70	C₂×C₈×C₆₁	—	2	2	no	—
71	C₂×C₈×C₆₁	—	4	1	no	—
72	C₂×C₈×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
73	C₂×C₈×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
74	C₂×C₈×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
75	C₂×C₈×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
76	C₄⋊C₄×C₆₁	—	2	2	no	—
77	C₄⋊C₄×C₆₁	—	2	2	no	—
78	C₂²⋊C₄×C₆₁	—	4	2	no	—
79	C₂²⋊C₄×C₆₁	—	4	2	no	—
80	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
81	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
82	C₄²×C₆₁	#41	1	1	yes	characteristic
83	C₄²×C₆₁	#41	1	1	yes	characteristic
84	C₄⋊Dic₆₁	—	2	4	no	—
85	C₄⋊Dic₆₁	—	4	2	no	—
86	C₄×Dic₆₁	—	4	4	no	—
87	#11758194	—	2	4	no	—
88	C₂×#92418	—	2	2	no	—
89	C₂×#92418	—	2	2	no	—
90	C₂³×C₆₁	—	4	2	no	—
91	C₂³×C₆₁	#127	1	1	yes	characteristic
92	Q₈×C₆₁	—	4	4	no	—
93	D₄×C₆₁	—	4	2	no	—
94	D₄×C₆₁	—	4	2	no	—
95	D₄×C₆₁	—	4	4	no	—
96	D₄×C₆₁	—	8	2	no	—
97	D₄×C₆₁	—	8	2	no	—
98	D₄×C₆₁	—	8	2	no	—
99	C₂×C₄×C₆₁	—	2	1	no	—
100	C₂×C₄×C₆₁	—	2	1	no	—
101	C₂×C₄×C₆₁	—	4	1	no	—
102	C₂×C₄×C₆₁	—	4	2	no	—
103	C₂×C₄×C₆₁	—	8	4	no	—
104	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
105	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
106	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
107	C₂×C₄×C₆₁	#121	1	1	yes	characteristic
108	C₂×C₄×C₆₁	#121	1	1	yes	characteristic
109	C₂×C₄×C₆₁	#124	1	1	yes	characteristic
110	C₂×C₄×C₆₁	AGL(1,ℤ/8)	1	4	yes	—
111	C₂×Dic₆₁	—	4	4	no	—
112	C₂×Dic₆₁	—	8	2	no	—
113	C₂×Dic₆₁	—	8	2	no	—
114	C₈×C₆₁	—	2	1	no	—
115	C₈×C₆₁	—	2	1	no	—
116	C₈×C₆₁	—	2	1	no	—
117	C₈×C₆₁	#104	1	2	yes	complemented
118	#92418	—	4	2	no	—
119	#92418	—	4	2	no	—
120	#9721	—	61	1	no	maximal, Sylow
121	C₂²×C₆₁	—	2	1	no	—
122	C₂²×C₆₁	—	2	1	no	—
123	C₂²×C₆₁	—	8	2	no	—
124	C₂²×C₆₁	—	8	2	no	—
125	C₂²×C₆₁	—	8	2	no	—
126	C₂²×C₆₁	#390	1	1	yes	characteristic
127	C₂²×C₆₁	#496	1	2	yes	—
128	C₄×C₆₁	—	2	1	no	—
129	C₄×C₆₁	—	2	1	no	—
130	C₄×C₆₁	—	2	1	no	—
131	C₄×C₆₁	—	8	2	no	—
132	C₄×C₆₁	C₂×#104	1	1	yes	characteristic
133	C₄×C₆₁	#421	1	1	yes	characteristic
134	C₄×C₆₁	#442	1	2	yes	—
135	C₄×C₆₁	#469	1	2	yes	—
136	Dic₆₁	—	8	4	no	—
137	#3169	D₆₁	1	1	yes	characteristic
138	#1980	—	61	2	no	—
139	C₄×#106	—	61	1	no	—
140	#1502	—	61	2	no	—
141	#1451	—	61	1	no	—
142	C₂×C₆₁	—	2	2	no	—
143	C₂×C₆₁	—	16	2	no	—
144	C₂×C₆₁	#1972	1	1	yes	characteristic
145	C₂×C₆₁	#1976	1	1	yes	characteristic
146	C₂×C₆₁	#1980	1	1	yes	characteristic
147	#532	C₂×D₆₁	1	2	yes	—
148	#500	Dic₆₁	1	1	yes	characteristic, complemented
149	#493	Dic₆₁	1	1	yes	characteristic, complemented
150	#486	Dic₆₁	1	2	yes	complemented
151	#466	—	2	1	no	—
152	#466	—	2	1	no	—
153	C₄×D₈	—	2	1	no	—
154	C₄×D₈	—	2	1	no	—
155	#431	—	61	1	no	—
156	C₂×#106	—	122	1	no	—
157	C₂×#104	—	122	4	no	—
158	C₄×M₁₆	—	61	1	no	—
159	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
160	C₄×C₄⋊C₄	—	61	2	no	—
161	#340	—	122	1	no	—
162	#327	—	122	4	no	—
163	#322	—	122	1	no	—
164	#322	—	122	1	no	—
165	C₆₁	#9721	1	1	yes	characteristic, complemented, Sylow
166	C₂×#41	—	2	2	no	—
167	C₂×D₈	—	2	1	no	—
168	C₂×D₈	—	2	1	no	—
169	C₂×M₁₆	—	61	2	no	—
170	C₂²×C₈	C₂²⋊D₆₁	1	1	yes	characteristic
171	C₂²×C₈	C₂²⋊D₆₁	1	1	yes	characteristic
172	C₄⋊Q₈	C₂×Dic₆₁	1	2	yes	—
173	#127	C₂×Dic₆₁	1	2	yes	—
174	#124	—	2	2	no	—
175	#121	—	2	2	no	—
176	#121	—	2	2	no	—
177	C₄×D₄	—	2	2	no	—
178	C₄×D₄	—	2	2	no	—
179	C₂×C₄⋊C₄	—	122	2	no	—
180	C₂×C₄²	—	122	2	no	—
181	C₂×C₄²	C₂²×D₆₁	1	1	yes	characteristic
182	#107	—	2	1	no	—
183	#107	—	2	1	no	—
184	#106	—	122	4	no	—
185	C₄⋊C₈	—	122	1	no	—
186	#104	—	244	8	no	—
187	#102	—	2	2	no	—
188	#102	—	2	2	no	—
189	C₂²⋊C₈	—	61	2	no	—
190	#97	—	122	1	no	—
191	C₄×C₈	—	122	1	no	—
192	C₄×C₈	—	122	1	no	—
193	C₄×C₈	C₂×Dic₆₁	1	1	yes	characteristic
194	C₄×C₈	C₂×Dic₆₁	1	1	yes	characteristic
195	C₄×C₈	C₂²⋊D₆₁	1	1	yes	characteristic
196	C₄×C₈	C₂²⋊D₆₁	1	1	yes	characteristic
197	#95	—	122	2	no	—
198	#41	—	4	4	no	—
199	#41	—	4	4	no	—
200	C₂×Q₈	—	2	2	no	—
201	C₂×D₄	—	2	2	no	—
202	C₂×D₄	—	2	2	no	—
203	C₂×D₄	—	2	2	no	—
204	C₂×D₄	—	4	2	no	—
205	C₂²×C₄	—	4	2	no	—
206	C₂²×C₄	—	244	2	no	—
207	C₂²×C₄	D₄×D₆₁	1	1	yes	characteristic
208	C₂²×C₄	D₄×D₆₁	1	1	yes	characteristic
209	C₂²×C₄	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
210	D₈	—	4	2	no	—
211	D₈	—	4	2	no	—
212	M₁₆	—	122	4	no	—
213	C₂×C₈	—	2	2	no	—
214	C₂×C₈	—	4	1	no	—
215	C₂×C₈	—	122	2	no	—
216	C₂×C₈	—	122	2	no	—
217	C₂×C₈	C₂²⋊Dic₆₁	1	1	yes	characteristic
218	C₂×C₈	C₂²⋊Dic₆₁	1	1	yes	characteristic
219	C₂×C₈	C₂²⋊Dic₆₁	1	1	yes	characteristic
220	C₂×C₈	C₂²⋊Dic₆₁	1	1	yes	characteristic
221	C₄⋊C₄	—	2	2	no	—
222	C₄⋊C₄	—	2	2	no	—
223	C₄⋊C₄	—	122	4	no	—
224	C₄⋊C₄	—	244	2	no	—
225	C₂²⋊C₄	—	4	2	no	—
226	C₂²⋊C₄	—	4	2	no	—
227	C₄²	—	244	4	no	—
228	C₄²	#11758224	1	1	yes	characteristic
229	C₄²	#11758224	1	1	yes	characteristic
230	C₄²	C₂²×Dic₆₁	1	1	yes	characteristic
231	C₄²	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
232	C₂³	—	4	2	no	—
233	C₂³	#14519773	1	1	yes	characteristic
234	Q₈	—	4	4	no	—
235	D₄	—	4	2	no	—
236	D₄	—	4	2	no	—
237	D₄	—	4	4	no	—
238	D₄	—	8	2	no	—
239	D₄	—	8	2	no	—
240	D₄	—	8	2	no	—
241	C₂×C₄	—	2	1	no	—
242	C₂×C₄	—	2	1	no	—
243	C₂×C₄	—	4	1	no	—
244	C₂×C₄	—	4	2	no	—
245	C₂×C₄	—	8	4	no	—
246	C₂×C₄	—	488	2	no	—
247	C₂×C₄	—	488	2	no	—
248	C₂×C₄	—	244	4	no	—
249	C₂×C₄	C₂×C₂²⋊Dic₆₁	1	1	yes	characteristic
250	C₂×C₄	D₄×Dic₆₁	1	1	yes	characteristic
251	C₂×C₄	D₄×Dic₆₁	1	1	yes	characteristic
252	C₂×C₄	#14519748	1	1	yes	characteristic
253	C₂×C₄	#14519748	1	1	yes	characteristic
254	C₂×C₄	#14519764	1	1	yes	characteristic
255	C₂×C₄	#14519801	1	4	yes	—
256	C₈	—	2	1	no	—
257	C₈	—	2	1	no	—
258	C₈	—	2	1	no	—
259	C₈	—	244	2	no	—
260	C₈	—	244	2	no	—
261	C₈	#14519704	1	2	yes	complemented
262	C₂²	—	2	1	no	—
263	C₂²	—	2	1	no	—
264	C₂²	—	8	2	no	—
265	C₂²	—	8	2	no	—
266	C₂²	—	8	2	no	—
267	C₂²	#16839665	1	1	yes	characteristic
268	C₂²	#16840150	1	2	yes	—
269	C₄	—	2	1	no	—
270	C₄	—	2	1	no	—
271	C₄	—	2	1	no	—
272	C₄	—	8	2	no	—
273	C₄	—	488	4	no	—
274	C₄	C₂×#14519704	1	1	yes	characteristic
275	C₄	#16839779	1	1	yes	characteristic
276	C₄	#16839874	1	2	yes	—
277	C₄	#16840012	1	2	yes	—
278	C₂	—	2	2	no	—
279	C₂	—	16	2	no	—
280	C₂	#21446696	1	1	yes	characteristic
281	C₂	#21446718	1	1	yes	characteristic
282	C₂	#21446739	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#14519748
commutator	C₄²×C₆₁	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₁
Fitting, prime radical	C₆₁×#3169	C₂
socle	C₂²×C₆₁	#390
cosocle	C₂×C₄²×C₆₁	C₂³
hypercenter	#3169	D₆₁
hypocenter	C₆₁	#9721
LCS	G ▷ C₄²×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3169

Complements

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

N	H ≅ G/N	type
1. #21446739	C₂	semidirect
2. C₄×#14519709	C₂	semidirect
3. #21444314	C₂	semidirect
4. #21444108	C₂	semidirect
16. C₆₁×#500	C₄	semidirect
17. C₆₁×#493	C₄	semidirect
18. C₆₁×#486	C₄	semidirect
117. C₈×C₆₁	#104	semidirect
148. #500	Dic₆₁	semidirect
149. #493	Dic₆₁	semidirect
150. #486	Dic₆₁	semidirect
165. C₆₁	#9721	semidirect
261. C₈	#14519704	semidirect

Maximal subgroups

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

subgroup	index
#21446739	2
C₄×#14519709	2
#21444314	2
#21444108	2
C₆₁×#3169	2
#9721	61