Subgroups — #30613671

Order 15616.

578 conjugacy classes of subgroups.

The subgroup lattice has 282 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21462615	C₂	1	2	yes	maximal
2	#21461465	C₂	1	2	yes	maximal, complemented
3	#21461443	C₂	1	4	yes	maximal, complemented
4	#21461414	C₂	1	2	yes	maximal, complemented
5	D₆₁⋊#334	C₂	1	1	yes	maximal, characteristic
6	#21461129	C₂	1	2	yes	maximal
7	#21461029	C₂	1	1	yes	maximal, characteristic
8	#21456406	C₂	1	1	yes	maximal, characteristic
9	#16840813	—	2	4	no	—
10	#16840812	C₂²	1	4	yes	—
11	#16840811	C₂²	1	4	yes	—
12	#16840781	C₂²	1	1	yes	characteristic
13	#16840769	—	2	2	no	—
14	D₆₁⋊(C₄⋊C₈)	C₂²	1	2	yes	—
15	#16840727	—	2	2	no	—
16	C₄×#11758213	C₂²	1	1	yes	characteristic
17	#16840624	C₂²	1	2	yes	—
18	#16840623	C₂²	1	1	yes	characteristic
19	#16840623	C₂²	1	1	yes	characteristic
20	#16840612	C₂²	1	4	yes	—
21	#16840612	C₂²	1	4	yes	—
22	#92420⋊C₈	C₂²	1	2	yes	—
23	#16840603	C₂²	1	2	yes	—
24	#16840317	C₂²	1	2	yes	—
25	#16840142	C₄	1	1	yes	characteristic
26	#16840114	C₄	1	2	yes	—
27	#16840004	C₄	1	2	yes	—
28	#16840004	C₄	1	2	yes	—
29	#16839856	C₂²	1	2	yes	—
30	#16839856	C₂²	1	2	yes	—
31	C₄×C₈×D₆₁	C₂²	1	1	yes	characteristic
32	C₆₁×#500	C₄	1	1	yes	characteristic
33	C₂×#11758213	D₄	1	2	yes	—
34	C₂×D₆₁⋊C₈	—	2	2	no	—
35	#14519844	—	2	4	no	—
36	#14519844	—	2	2	no	—
37	C₄⋊#92420	—	2	4	no	—
38	C₄⋊#92420	—	2	1	no	—
39	C₄⋊#92420	C₂³	1	2	yes	—
40	C₄⋊#92420	C₂³	1	2	yes	—
41	#14519834	—	2	1	no	—
42	#14519832	—	2	4	no	—
43	#14519832	D₄	1	2	yes	—
44	C₄×#92420	—	2	4	no	—
45	C₄×#92420	C₂³	1	2	yes	—
46	C₄×#92420	C₂³	1	2	yes	—
47	C₂×Dic₂₄₄	C₂×C₄	1	2	yes	—
48	C₂×Dic₂₄₄	C₂×C₄	1	2	yes	—
49	C₂×C₈×D₆₁	C₂×C₄	1	1	yes	characteristic
50	C₂×C₈×D₆₁	C₂×C₄	1	1	yes	characteristic
51	#14519777	C₂×C₄	1	2	yes	—
52	#14519768	—	2	2	no	—
53	#14519757	C₂×C₄	1	4	yes	—
54	#14519753	—	2	2	no	—
55	Q₈×Dic₆₁	—	2	2	no	—
56	#14519742	C₂³	1	4	yes	—
57	#14519735	—	2	2	no	—
58	C₄²×D₆₁	C₂³	1	1	yes	characteristic
59	#14519700	C₂×C₄	1	4	yes	—
60	#14519700	C₂×C₄	1	4	yes	—
61	C₄×#92418	C₂×C₄	1	1	yes	characteristic
62	C₈×Dic₆₁	C₂³	1	1	yes	characteristic
63	C₈×Dic₆₁	C₂³	1	1	yes	characteristic
64	C₄⋊Q₈×C₆₁	C₂×C₄	1	2	yes	—
65	#107×C₆₁	C₂×C₄	1	2	yes	—
66	#107×C₆₁	C₂×C₄	1	2	yes	—
67	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
68	C₂×Q₈×C₆₁	—	2	2	no	—
69	C₂×C₄×D₆₁	C₂²×C₄	1	1	yes	characteristic
70	C₂×C₄×D₆₁	C₂×D₄	1	1	yes	characteristic
71	C₂×C₄×D₆₁	#41	1	1	yes	characteristic
72	C₂×Dic₁₂₂	C₂²×C₄	1	4	yes	—
73	C₂×#92420	—	2	2	no	—
74	C₂×#92420	—	2	2	no	—
75	C₂×#92420	—	2	2	no	—
76	C₂×#92420	C₂×D₄	1	4	yes	—
77	#11758213	—	2	4	no	—
78	D₆₁⋊C₈	—	4	4	no	—
79	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
81	C₄⋊C₄×C₆₁	—	2	2	no	—
82	C₄⋊C₄×C₆₁	C₂²×C₄	1	4	yes	—
83	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
84	#11758198	—	2	4	no	—
85	C₄⋊Dic₆₁	—	2	2	no	—
86	C₄⋊Dic₆₁	—	2	2	no	—
87	C₄⋊Dic₆₁	—	2	4	no	—
88	#11758196	—	2	4	no	—
89	C₄×Dic₆₁	—	2	4	no	—
90	C₄×Dic₆₁	C₂×D₄	1	1	yes	characteristic
91	C₄×Dic₆₁	#41	1	1	yes	characteristic
92	C₄×Dic₆₁	C₂⁴	1	1	yes	characteristic
93	C₂×#92418	C₂²×C₄	1	1	yes	characteristic
94	C₂×#92418	C₂²×C₄	1	1	yes	characteristic
95	Dic₂₄₄	—	2	4	no	—
96	Dic₂₄₄	—	4	2	no	—
97	C₈×D₆₁	—	2	2	no	—
98	C₈×D₆₁	—	4	1	no	—
99	C₂²×D₆₁	C₄×D₄	1	1	yes	characteristic
100	Q₈×C₆₁	—	8	2	no	—
101	C₂×C₄×C₆₁	—	2	4	no	—
102	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
103	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
104	C₂×C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
105	C₂×Dic₆₁	—	2	4	no	—
106	C₂×Dic₆₁	C₄×D₄	1	1	yes	characteristic
107	C₂×Dic₆₁	C₂²×D₄	1	1	yes	characteristic
108	C₂×Dic₆₁	C₂×#41	1	1	yes	characteristic
109	C₄×D₆₁	—	2	1	no	—
110	C₄×D₆₁	—	2	1	no	—
111	C₄×D₆₁	—	4	1	no	—
112	C₄×D₆₁	#137	1	4	yes	—
113	Dic₁₂₂	—	2	4	no	—
114	Dic₁₂₂	—	2	4	no	—
115	#92420	—	2	4	no	—
116	#92420	—	4	2	no	—
117	#92420	—	4	2	no	—
118	#92420	—	4	2	no	—
119	C₈×C₆₁	—	2	1	no	—
120	C₈×C₆₁	#131	1	2	yes	—
121	#92418	—	2	1	no	—
122	#92418	—	2	1	no	—
123	#18072	—	61	1	no	maximal, Sylow
124	C₂²×C₆₁	C₂×C₄×D₄	1	1	yes	characteristic
125	C₂×D₆₁	—	2	1	no	—
126	C₂×D₆₁	—	2	1	no	—
127	C₂×D₆₁	#441	1	2	yes	—
128	C₄×C₆₁	—	2	1	no	—
129	C₄×C₆₁	—	4	4	no	—
130	C₄×C₆₁	#441	1	2	yes	—
131	C₄×C₆₁	C₂×#131	1	1	yes	characteristic
132	C₄×C₆₁	#568	1	1	yes	characteristic
133	Dic₆₁	—	2	1	no	—
134	Dic₆₁	—	2	1	no	—
135	Dic₆₁	—	4	4	no	—
136	Dic₆₁	C₂×#137	1	2	yes	—
137	#3170	—	61	1	no	—
138	#2962	—	61	2	no	—
139	#1593	—	61	2	no	—
140	#1587	—	61	4	no	—
141	#1579	—	61	2	no	—
142	C₂×#334	—	61	1	no	—
143	#1481	—	61	2	no	—
144	#1451	—	61	1	no	—
145	C₂×C₆₁	#2935	1	1	yes	characteristic
146	C₂×C₆₁	C₂×#441	1	1	yes	characteristic
147	C₂×C₆₁	#2957	1	1	yes	characteristic
148	D₆₁	—	2	2	no	—
149	#533	—	61	2	no	—
150	#500	#6923	1	1	yes	characteristic
151	#494	—	61	1	no	—
152	#487	—	61	2	no	—
153	#467	—	61	2	no	—
154	#467	—	61	2	no	—
155	#446	—	61	4	no	—
156	#446	—	122	4	no	—
157	C₈×Q₈	—	61	4	no	—
158	C₄×Q₁₆	—	61	2	no	—
159	C₄×Q₁₆	—	61	2	no	—
160	#431	—	61	1	no	—
161	#424	—	122	2	no	—
162	C₂×C₄⋊C₈	—	61	2	no	—
163	#405	—	122	2	no	—
164	C₄×M₁₆	—	61	1	no	—
165	C₂×C₄×C₈	—	61	1	no	—
166	#334	—	61	1	no	—
167	#334	—	61	2	no	—
168	#334	—	61	1	no	—
169	Q₈⋊C₈	—	61	4	no	—
170	Q₈⋊C₈	—	61	4	no	—
171	#322	—	61	2	no	—
172	#322	—	61	2	no	—
173	C₆₁	#18072	1	1	yes	characteristic, complemented, Sylow
174	C₂×Q₁₆	—	61	2	no	—
175	C₂×Q₁₆	—	61	2	no	—
176	C₂×M₁₆	—	61	2	no	—
177	C₂²×C₈	—	61	1	no	—
178	C₂²×C₈	—	61	1	no	—
179	C₂²×C₈	—	122	2	no	—
180	C₄⋊Q₈	—	61	2	no	—
181	C₄⋊Q₈	C₂×#6923	1	2	yes	—
182	#125	—	122	2	no	—
183	C₂²⋊Q₈	—	61	4	no	—
184	C₂²⋊Q₈	—	122	2	no	—
185	C₄×Q₈	—	61	4	no	—
186	C₄×Q₈	—	122	2	no	—
187	#117	—	122	2	no	—
188	C₂×C₄²	—	61	1	no	—
189	#107	C₂×#6923	1	2	yes	—
190	#107	C₂×#6923	1	2	yes	—
191	C₄⋊C₈	—	61	2	no	—
192	C₄⋊C₈	—	61	2	no	—
193	C₄⋊C₈	—	122	1	no	—
194	C₄⋊C₈	—	122	2	no	—
195	C₄⋊C₈	—	122	4	no	—
196	C₄⋊C₈	—	122	4	no	—
197	#103	—	61	4	no	—
198	#103	—	61	4	no	—
199	C₂²⋊C₈	—	122	1	no	—
200	#97	—	61	2	no	—
201	#97	—	122	4	no	—
202	C₄×C₈	—	61	2	no	—
203	C₄×C₈	—	61	1	no	—
204	C₄×C₈	—	61	1	no	—
205	C₄×C₈	—	61	2	no	—
206	C₄×C₈	—	61	1	no	—
207	C₄×C₈	—	122	4	no	—
208	C₄×C₈	C₂×#6923	1	1	yes	characteristic
209	C₂×Q₈	—	2	2	no	—
210	C₂×Q₈	—	61	4	no	—
211	C₂²×C₄	—	61	1	no	—
212	C₂²×C₄	—	61	1	no	—
213	C₂²×C₄	—	61	1	no	—
214	Q₁₆	—	122	4	no	—
215	Q₁₆	—	244	2	no	—
216	M₁₆	—	122	4	no	—
217	C₂×C₈	—	61	4	no	—
218	C₂×C₈	—	61	1	no	—
219	C₂×C₈	—	61	1	no	—
220	C₂×C₈	—	122	2	no	—
221	C₂×C₈	—	122	2	no	—
222	C₂×C₈	—	122	2	no	—
223	C₂×C₈	—	122	2	no	—
224	C₂×C₈	—	244	1	no	—
225	C₂×C₈	—	244	4	no	—
226	C₂×C₈	C₂²×#6923	1	1	yes	characteristic
227	C₂×C₈	C₂²×#6923	1	1	yes	characteristic
228	C₄⋊C₄	—	2	2	no	—
229	C₄⋊C₄	—	122	4	no	—
230	C₄⋊C₄	—	122	4	no	—
231	C₄⋊C₄	—	122	2	no	—
232	C₄⋊C₄	—	122	2	no	—
233	C₄⋊C₄	C₂²×#6923	1	4	yes	—
234	C₂²⋊C₄	—	122	4	no	—
235	C₄²	—	61	1	no	—
236	C₄²	—	61	1	no	—
237	C₄²	—	61	1	no	—
238	C₄²	—	122	4	no	—
239	C₄²	C₂²×#6923	1	1	yes	characteristic
240	C₂³	—	61	1	no	—
241	Q₈	—	8	2	no	—
242	Q₈	—	122	4	no	—
243	Q₈	—	122	4	no	—
244	C₂×C₄	—	2	4	no	—
245	C₂×C₄	—	61	1	no	—
246	C₂×C₄	—	61	4	no	—
247	C₂×C₄	—	61	1	no	—
248	C₂×C₄	—	61	1	no	—
249	C₂×C₄	—	122	1	no	—
250	C₂×C₄	—	122	1	no	—
251	C₂×C₄	—	122	4	no	—
252	C₂×C₄	—	244	1	no	—
253	C₂×C₄	D₄×#6923	1	1	yes	characteristic
254	C₂×C₄	D₄×#6923	1	1	yes	characteristic
255	C₂×C₄	C₂³×#6923	1	1	yes	characteristic
256	C₈	—	2	1	no	—
257	C₈	—	122	4	no	—
258	C₈	—	122	1	no	—
259	C₈	—	122	1	no	—
260	C₈	—	244	2	no	—
261	C₈	—	244	2	no	—
262	C₈	—	244	2	no	—
263	C₈	#14519865	1	2	yes	—
264	C₂²	—	61	2	no	—
265	C₂²	—	122	1	no	—
266	C₂²	—	122	1	no	—
267	C₂²	C₂×D₄×#6923	1	1	yes	characteristic
268	C₄	—	2	1	no	—
269	C₄	—	4	4	no	—
270	C₄	—	61	2	no	—
271	C₄	—	122	1	no	—
272	C₄	—	122	1	no	—
273	C₄	—	244	4	no	—
274	C₄	#16840802	1	2	yes	—
275	C₄	C₂×#14519865	1	1	yes	characteristic
276	C₄	#16840842	1	1	yes	characteristic
277	C₂	—	122	2	no	—
278	C₂	#21462565	1	1	yes	characteristic
279	C₂	C₂×#16840802	1	1	yes	characteristic
280	C₂	#21462595	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×#6923
commutator	C₂×C₄×C₆₁	C₂³×C₄
Frattini	C₂×C₄	C₂³×#6923
Fitting	C₆₁×#500	C₄
socle	C₂²×C₆₁	C₂×C₄×D₄
prime radical	C₂²×D₆₁	C₄×D₄
cosocle	C₄×Dic₆₁	C₂⁴
hypercenter	#500	#6923
hypocenter	C₆₁	#18072
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #500

Complements

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

N	H ≅ G/N	type
2. #21461465	C₂	semidirect
3. #21461443	C₂	semidirect
4. #21461414	C₂	semidirect
173. C₆₁	#18072	semidirect

Maximal subgroups

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

subgroup	index
#21462615	2
#21461465	2
#21461443	2
#21461414	2
D₆₁⋊#334	2
#21461129	2
#21461029	2
#21456406	2
#18072	61