Subgroups — #30613648

Order 15616.

586 conjugacy classes of subgroups.

The subgroup lattice has 294 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	#21461458	C₂	1	2	yes	maximal, complemented
3	#21461431	C₂	1	4	yes	maximal, complemented
4	#21461407	C₂	1	2	yes	maximal, complemented
5	D₆₁⋊#335	C₂	1	1	yes	maximal, characteristic
6	#21461129	C₂	1	2	yes	maximal
7	C₈×#11758213	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	#16840626	C₂²	1	2	yes	—
18	#16840625	C₂²	1	1	yes	characteristic
19	#16840625	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	C₈×#92420	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	1	yes	characteristic
34	C₂×#11758213	D₄	1	1	yes	characteristic
35	C₂×D₆₁⋊C₈	—	2	2	no	—
36	#14519844	—	2	4	no	—
37	#14519844	—	2	2	no	—
38	C₄⋊#92420	—	2	4	no	—
39	C₄⋊#92420	—	2	1	no	—
40	C₄⋊#92420	C₂³	1	2	yes	—
41	C₄⋊#92420	C₂³	1	2	yes	—
42	#14519834	—	2	1	no	—
43	#14519832	—	2	4	no	—
44	#14519832	D₄	1	2	yes	—
45	C₄×#92420	—	2	4	no	—
46	C₄×#92420	C₂³	1	2	yes	—
47	C₄×#92420	C₂³	1	2	yes	—
48	C₂×Dic₂₄₄	C₂×C₄	1	2	yes	—
49	C₂×Dic₂₄₄	C₂×C₄	1	2	yes	—
50	C₂×C₈×D₆₁	C₂×C₄	1	1	yes	characteristic
51	C₂×C₈×D₆₁	C₂×C₄	1	1	yes	characteristic
52	#14519777	C₂×C₄	1	2	yes	—
53	#14519768	—	2	2	no	—
54	#14519757	C₂×C₄	1	4	yes	—
55	#14519753	—	2	2	no	—
56	Q₈×Dic₆₁	—	2	2	no	—
57	#14519742	C₂³	1	4	yes	—
58	#14519735	—	2	2	no	—
59	C₄²×D₆₁	C₂³	1	1	yes	characteristic
60	#14519700	C₂×C₄	1	4	yes	—
61	#14519700	C₂×C₄	1	4	yes	—
62	C₄×#92418	C₂×C₄	1	1	yes	characteristic
63	C₈×Dic₆₁	C₂³	1	1	yes	characteristic
64	C₈×Dic₆₁	C₂³	1	1	yes	characteristic
65	C₄⋊Q₈×C₆₁	C₂×C₄	1	2	yes	—
66	#107×C₆₁	C₂×C₄	1	2	yes	—
67	#107×C₆₁	C₂×C₄	1	2	yes	—
68	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
69	C₂×Q₈×C₆₁	—	2	2	no	—
70	C₂×C₄×D₆₁	C₂²×C₄	1	1	yes	characteristic
71	C₂×C₄×D₆₁	C₂×D₄	1	1	yes	characteristic
72	C₂×C₄×D₆₁	#41	1	1	yes	characteristic
73	C₂×Dic₁₂₂	C₂²×C₄	1	4	yes	—
74	C₂×#92420	—	2	2	no	—
75	C₂×#92420	—	2	2	no	—
76	C₂×#92420	—	2	2	no	—
77	C₂×#92420	C₂×D₄	1	2	yes	—
78	C₂×#92420	C₂×D₄	1	2	yes	—
79	#11758213	—	2	2	no	—
80	#11758213	Q₁₆	1	4	yes	complemented
81	D₆₁⋊C₈	—	4	4	no	—
82	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
83	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
84	C₄⋊C₄×C₆₁	—	2	2	no	—
85	C₄⋊C₄×C₆₁	C₂²×C₄	1	4	yes	—
86	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
87	#11758198	—	2	4	no	—
88	C₄⋊Dic₆₁	—	2	2	no	—
89	C₄⋊Dic₆₁	—	2	2	no	—
90	C₄⋊Dic₆₁	—	2	4	no	—
91	#11758196	—	2	4	no	—
92	C₄×Dic₆₁	—	2	4	no	—
93	C₄×Dic₆₁	C₂×D₄	1	1	yes	characteristic
94	C₄×Dic₆₁	#41	1	1	yes	characteristic
95	C₄×Dic₆₁	C₂⁴	1	1	yes	characteristic
96	C₂×#92418	C₂²×C₄	1	1	yes	characteristic
97	C₂×#92418	C₂²×C₄	1	1	yes	characteristic
98	Dic₂₄₄	—	2	4	no	—
99	Dic₂₄₄	—	4	2	no	—
100	C₈×D₆₁	—	2	2	no	—
101	C₈×D₆₁	—	4	1	no	—
102	C₂²×D₆₁	C₄×D₄	1	1	yes	characteristic
103	Q₈×C₆₁	—	8	2	no	—
104	C₂×C₄×C₆₁	—	2	4	no	—
105	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
106	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
107	C₂×C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
108	C₂×Dic₆₁	—	2	4	no	—
109	C₂×Dic₆₁	C₄×D₄	1	1	yes	characteristic
110	C₂×Dic₆₁	C₂²×D₄	1	1	yes	characteristic
111	C₂×Dic₆₁	C₂×#41	1	1	yes	characteristic
112	C₄×D₆₁	—	2	1	no	—
113	C₄×D₆₁	—	2	1	no	—
114	C₄×D₆₁	—	4	1	no	—
115	C₄×D₆₁	C₂×Q₁₆	1	2	yes	—
116	C₄×D₆₁	#135	1	2	yes	—
117	Dic₁₂₂	—	2	4	no	—
118	Dic₁₂₂	—	2	4	no	—
119	#92420	—	2	2	no	—
120	#92420	—	4	2	no	—
121	#92420	—	4	2	no	—
122	#92420	—	4	2	no	—
123	#92420	C₂×Q₁₆	1	4	yes	—
124	C₈×C₆₁	—	2	1	no	—
125	C₈×C₆₁	#131	1	2	yes	—
126	#92418	—	2	1	no	—
127	#92418	—	2	1	no	—
128	#18068	—	61	1	no	maximal, Sylow
129	C₂²×C₆₁	C₂×C₄×D₄	1	1	yes	characteristic
130	C₂×D₆₁	—	2	1	no	—
131	C₂×D₆₁	—	2	1	no	—
132	C₂×D₆₁	C₄×Q₁₆	1	2	yes	—
133	C₄×C₆₁	—	2	1	no	—
134	C₄×C₆₁	—	4	4	no	—
135	C₄×C₆₁	C₄×Q₁₆	1	2	yes	—
136	C₄×C₆₁	C₂×#131	1	1	yes	characteristic
137	C₄×C₆₁	#568	1	1	yes	characteristic
138	Dic₆₁	—	2	1	no	—
139	Dic₆₁	—	2	1	no	—
140	Dic₆₁	—	4	4	no	—
141	Dic₆₁	C₂²×Q₁₆	1	1	yes	characteristic
142	Dic₆₁	C₂×#135	1	1	yes	characteristic
143	#3170	—	61	1	no	—
144	#2962	—	61	2	no	—
145	#1591	—	61	2	no	—
146	#1584	—	61	4	no	—
147	C₈×Q₁₆	—	61	2	no	—
148	C₂×#335	—	61	1	no	—
149	#1481	—	61	2	no	—
150	C₈×M₁₆	—	61	1	no	—
151	C₂×C₆₁	#2935	1	1	yes	characteristic
152	C₂×C₆₁	C₂×C₄×Q₁₆	1	1	yes	characteristic
153	C₂×C₆₁	#2957	1	1	yes	characteristic
154	D₆₁	—	2	2	no	—
155	#533	—	61	2	no	—
156	#500	#6923	1	1	yes	characteristic
157	#494	—	61	1	no	—
158	#487	—	61	2	no	—
159	#467	—	61	2	no	—
160	#467	—	61	2	no	—
161	#446	—	61	4	no	—
162	#446	—	122	4	no	—
163	C₈×Q₈	—	61	4	no	—
164	C₄×Q₁₆	—	61	2	no	—
165	C₄×Q₁₆	—	61	2	no	—
166	#431	—	61	1	no	—
167	#424	—	122	2	no	—
168	C₂×C₄⋊C₈	—	61	2	no	—
169	#405	—	122	2	no	—
170	C₄×M₁₆	—	61	1	no	—
171	C₂×C₄×C₈	—	61	1	no	—
172	#335	—	61	1	no	—
173	#335	—	61	2	no	—
174	#335	—	61	1	no	—
175	Q₈⋊C₈	—	61	4	no	—
176	Q₈⋊C₈	—	61	4	no	—
177	#322	—	61	2	no	—
178	C₈²	—	61	2	no	—
179	C₆₁	#18068	1	1	yes	characteristic, complemented, Sylow
180	C₂×Q₁₆	—	61	2	no	—
181	C₂×Q₁₆	—	61	2	no	—
182	C₂×M₁₆	—	61	1	no	—
183	C₂×M₁₆	—	61	1	no	—
184	C₂²×C₈	—	61	1	no	—
185	C₂²×C₈	—	61	1	no	—
186	C₂²×C₈	—	122	2	no	—
187	C₄⋊Q₈	—	61	2	no	—
188	C₄⋊Q₈	C₂×#6923	1	2	yes	—
189	#125	—	122	2	no	—
190	C₂²⋊Q₈	—	61	4	no	—
191	C₂²⋊Q₈	—	122	2	no	—
192	C₄×Q₈	—	61	4	no	—
193	C₄×Q₈	—	122	2	no	—
194	#117	—	122	2	no	—
195	C₂×C₄²	—	61	1	no	—
196	#107	C₂×#6923	1	2	yes	—
197	#107	C₂×#6923	1	2	yes	—
198	C₄⋊C₈	—	61	2	no	—
199	C₄⋊C₈	—	61	2	no	—
200	C₄⋊C₈	—	122	1	no	—
201	C₄⋊C₈	—	122	2	no	—
202	C₄⋊C₈	—	122	4	no	—
203	C₄⋊C₈	—	122	4	no	—
204	#103	—	61	4	no	—
205	#103	—	61	4	no	—
206	C₂²⋊C₈	—	122	1	no	—
207	#97	—	61	2	no	—
208	#97	—	122	4	no	—
209	C₄×C₈	—	61	2	no	—
210	C₄×C₈	—	61	1	no	—
211	C₄×C₈	—	61	1	no	—
212	C₄×C₈	—	61	2	no	—
213	C₄×C₈	—	61	1	no	—
214	C₄×C₈	—	122	4	no	—
215	C₄×C₈	C₂×#6923	1	1	yes	characteristic
216	C₂×Q₈	—	2	2	no	—
217	C₂×Q₈	—	61	4	no	—
218	C₂²×C₄	—	61	1	no	—
219	C₂²×C₄	—	61	1	no	—
220	C₂²×C₄	—	61	1	no	—
221	Q₁₆	—	122	4	no	—
222	Q₁₆	—	244	2	no	—
223	M₁₆	—	61	4	no	—
224	M₁₆	—	122	2	no	—
225	C₂×C₈	—	61	2	no	—
226	C₂×C₈	—	61	2	no	—
227	C₂×C₈	—	61	1	no	—
228	C₂×C₈	—	61	1	no	—
229	C₂×C₈	—	122	2	no	—
230	C₂×C₈	—	122	2	no	—
231	C₂×C₈	—	122	2	no	—
232	C₂×C₈	—	122	2	no	—
233	C₂×C₈	—	244	1	no	—
234	C₂×C₈	—	244	4	no	—
235	C₂×C₈	C₂²×#6923	1	1	yes	characteristic
236	C₂×C₈	C₂²×#6923	1	1	yes	characteristic
237	C₄⋊C₄	—	2	2	no	—
238	C₄⋊C₄	—	122	4	no	—
239	C₄⋊C₄	—	122	4	no	—
240	C₄⋊C₄	—	122	2	no	—
241	C₄⋊C₄	—	122	2	no	—
242	C₄⋊C₄	C₂²×#6923	1	4	yes	—
243	C₂²⋊C₄	—	122	4	no	—
244	C₄²	—	61	1	no	—
245	C₄²	—	61	1	no	—
246	C₄²	—	61	1	no	—
247	C₄²	—	122	4	no	—
248	C₄²	C₂²×#6923	1	1	yes	characteristic
249	C₂³	—	61	1	no	—
250	Q₈	—	8	2	no	—
251	Q₈	—	122	4	no	—
252	Q₈	—	122	4	no	—
253	C₂×C₄	—	2	4	no	—
254	C₂×C₄	—	61	1	no	—
255	C₂×C₄	—	61	2	no	—
256	C₂×C₄	—	61	2	no	—
257	C₂×C₄	—	61	1	no	—
258	C₂×C₄	—	61	1	no	—
259	C₂×C₄	—	122	1	no	—
260	C₂×C₄	—	122	1	no	—
261	C₂×C₄	—	122	4	no	—
262	C₂×C₄	—	244	1	no	—
263	C₂×C₄	D₄×#6923	1	1	yes	characteristic
264	C₂×C₄	D₄×#6923	1	1	yes	characteristic
265	C₂×C₄	C₂³×#6923	1	1	yes	characteristic
266	C₈	—	2	1	no	—
267	C₈	—	61	4	no	—
268	C₈	—	122	2	no	—
269	C₈	—	122	1	no	—
270	C₈	—	122	1	no	—
271	C₈	—	244	2	no	—
272	C₈	—	244	2	no	—
273	C₈	—	244	2	no	—
274	C₈	#14519865	1	2	yes	—
275	C₂²	—	61	2	no	—
276	C₂²	—	122	1	no	—
277	C₂²	—	122	1	no	—
278	C₂²	C₂×D₄×#6923	1	1	yes	characteristic
279	C₄	—	2	1	no	—
280	C₄	—	4	4	no	—
281	C₄	—	61	1	no	—
282	C₄	—	61	1	no	—
283	C₄	—	122	1	no	—
284	C₄	—	122	1	no	—
285	C₄	—	244	4	no	—
286	C₄	Q₁₆×#6923	1	2	yes	—
287	C₄	C₂×#14519865	1	1	yes	characteristic
288	C₄	#16840842	1	1	yes	characteristic
289	C₂	—	122	2	no	—
290	C₂	#21462565	1	1	yes	characteristic
291	C₂	C₂×Q₁₆×#6923	1	1	yes	characteristic
292	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	#18068
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. #21461458	C₂	semidirect
3. #21461431	C₂	semidirect
4. #21461407	C₂	semidirect
80. #11758213	Q₁₆	semidirect
179. C₆₁	#18068	semidirect

Maximal subgroups

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

subgroup	index
#21462615	2
#21461458	2
#21461431	2
#21461407	2
D₆₁⋊#335	2
#21461129	2
C₈×#11758213	2
#21456406	2
#18068	61