Subgroups — #30612999

Order 15616.

560 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	#21462724	C₂	1	2	yes	maximal
2	#21461466	C₂	1	2	yes	maximal, complemented
3	#21461459	C₂	1	2	yes	maximal, complemented
4	#21461412	C₂	1	2	yes	maximal, complemented
5	Q₁₆×#92420	C₂	1	2	yes	maximal, complemented
6	#21461380	C₂	1	1	yes	maximal, characteristic
7	#21461182	C₂	1	2	yes	maximal
8	#21461023	C₂	1	1	yes	maximal, characteristic
9	#21456403	C₂	1	1	yes	maximal, characteristic
10	#16840814	—	2	4	no	—
11	#16840813	C₂²	1	4	yes	—
12	Q₈×#92420	C₂²	1	4	yes	—
13	#16840787	—	2	4	no	—
14	#16840784	C₂²	1	2	yes	—
15	#16840780	C₂²	1	1	yes	characteristic
16	#16840780	C₂²	1	1	yes	characteristic
17	#16840625	C₂²	1	2	yes	—
18	#16840623	C₂²	1	2	yes	—
19	Q₈⋊#92420	C₂²	1	4	yes	—
20	Q₈⋊#92420	C₂²	1	4	yes	—
21	#16840603	C₂²	1	1	yes	characteristic
22	#16840603	C₂²	1	1	yes	characteristic
23	C₈×#92420	C₂²	1	1	yes	characteristic
24	C₈×#92420	C₂²	1	1	yes	characteristic
25	#16840317	C₂²	1	2	yes	—
26	#16840167	C₄	1	1	yes	characteristic
27	#16840112	C₄	1	2	yes	—
28	#16840000	C₄	1	2	yes	—
29	#16840000	C₄	1	2	yes	—
30	Q₁₆×Dic₆₁	C₂²	1	2	yes	—
31	Q₁₆×Dic₆₁	C₂²	1	2	yes	—
32	C₄×C₈×D₆₁	C₂²	1	1	yes	characteristic
33	C₆₁×#494	C₄	1	1	yes	characteristic
34	#14519844	—	2	4	no	—
35	#14519844	—	2	1	no	—
36	#14519844	D₄	1	2	yes	—
37	C₄⋊#92420	—	2	4	no	—
38	C₄⋊#92420	C₂³	1	4	yes	—
39	#14519834	—	2	2	no	—
40	#14519834	—	2	1	no	—
41	#14519834	D₄	1	2	yes	—
42	#14519832	—	2	4	no	—
43	#14519832	—	2	2	no	—
44	C₄×#92420	—	2	4	no	—
45	C₄×#92420	C₂³	1	1	yes	characteristic
46	C₄×#92420	C₂³	1	1	yes	characteristic
47	C₄×#92420	C₂³	1	1	yes	characteristic
48	C₄×#92420	C₂³	1	1	yes	characteristic
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	—	2	2	no	—
54	#14519753	C₂×C₄	1	4	yes	—
55	Q₈×Dic₆₁	C₂³	1	4	yes	—
56	#14519742	—	2	2	no	—
57	#14519735	—	2	2	no	—
58	C₄²×D₆₁	C₂³	1	1	yes	characteristic
59	#14519711	C₂×C₄	1	2	yes	—
60	#14519711	C₂×C₄	1	2	yes	—
61	#14519701	C₂×C₄	1	4	yes	—
62	#14519701	C₂×C₄	1	4	yes	—
63	C₄×#92418	C₂×C₄	1	1	yes	characteristic
64	C₈×Dic₆₁	C₂³	1	1	yes	characteristic
65	C₈×Dic₆₁	C₂³	1	1	yes	characteristic
66	C₂×Q₁₆×C₆₁	C₂×C₄	1	2	yes	—
67	C₂×Q₁₆×C₆₁	C₂×C₄	1	2	yes	—
68	C₄⋊Q₈×C₆₁	C₂×C₄	1	2	yes	—
69	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
70	C₂×Q₈×C₆₁	C₂²×C₄	1	4	yes	—
71	C₂×C₄×D₆₁	C₂²×C₄	1	1	yes	characteristic
72	C₂×C₄×D₆₁	C₂×D₄	1	1	yes	characteristic
73	C₂×C₄×D₆₁	#41	1	1	yes	characteristic
74	C₂×Dic₁₂₂	—	2	2	no	—
75	C₂×#92420	—	2	4	no	—
76	C₂×#92420	—	2	1	no	—
77	C₂×#92420	—	2	1	no	—
78	C₂×#92420	C₂×D₄	1	2	yes	—
79	C₂×#92420	C₂×D₄	1	2	yes	—
80	Q₁₆×C₆₁	—	2	4	no	—
81	Q₁₆×C₆₁	—	2	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	—	2	2	no	—
86	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
87	#11758198	—	2	4	no	—
88	C₄⋊Dic₆₁	—	2	2	no	—
89	C₄⋊Dic₆₁	—	2	4	no	—
90	C₄⋊Dic₆₁	C₂²×C₄	1	4	yes	—
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	C₈×D₆₁	—	2	2	no	—
99	C₈×D₆₁	—	2	2	no	—
100	C₂²×D₆₁	C₄×D₄	1	1	yes	characteristic
101	Q₈×C₆₁	—	2	4	no	—
102	Q₈×C₆₁	—	2	4	no	—
103	C₂×C₄×C₆₁	—	2	4	no	—
104	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
105	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
106	C₂×C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
107	C₂×Dic₆₁	—	2	4	no	—
108	C₂×Dic₆₁	C₄×D₄	1	1	yes	characteristic
109	C₂×Dic₆₁	C₂²×D₄	1	1	yes	characteristic
110	C₂×Dic₆₁	C₂×#41	1	1	yes	characteristic
111	C₄×D₆₁	—	2	1	no	—
112	C₄×D₆₁	—	2	1	no	—
113	C₄×D₆₁	—	2	2	no	—
114	C₄×D₆₁	—	4	1	no	—
115	Dic₁₂₂	—	8	2	no	—
116	#92420	—	2	2	no	—
117	#92420	—	2	2	no	—
118	#92420	—	4	1	no	—
119	#92420	—	4	1	no	—
120	#92420	—	4	4	no	—
121	C₈×C₆₁	#131	1	2	yes	—
122	C₈×C₆₁	#131	1	2	yes	—
123	#92418	—	2	1	no	—
124	#92418	—	2	1	no	—
125	#17972	—	61	1	no	maximal, Sylow
126	C₂²×C₆₁	C₂×C₄×D₄	1	1	yes	characteristic
127	C₂×D₆₁	—	2	1	no	—
128	C₂×D₆₁	—	2	1	no	—
129	C₂×D₆₁	—	2	1	no	—
130	C₄×C₆₁	—	2	1	no	—
131	C₄×C₆₁	—	2	1	no	—
132	C₄×C₆₁	—	4	4	no	—
133	C₄×C₆₁	C₂×#131	1	1	yes	characteristic
134	C₄×C₆₁	C₂×#131	1	1	yes	characteristic
135	Dic₆₁	—	2	1	no	—
136	Dic₆₁	—	2	1	no	—
137	Dic₆₁	—	4	4	no	—
138	Dic₆₁	#578	1	2	yes	—
139	#3170	—	61	1	no	—
140	#2985	—	61	2	no	—
141	#1593	—	61	2	no	—
142	#1591	—	61	2	no	—
143	#1579	—	61	2	no	—
144	C₈×Q₁₆	—	61	2	no	—
145	#1568	—	61	1	no	—
146	#1497	—	61	2	no	—
147	#1450	—	61	1	no	—
148	C₂×C₆₁	#2931	1	1	yes	characteristic
149	C₂×C₆₁	#2950	1	1	yes	characteristic
150	C₂×C₆₁	#2957	1	1	yes	characteristic
151	D₆₁	—	4	1	no	—
152	#533	—	61	2	no	—
153	#500	—	61	1	no	—
154	#494	#6923	1	1	yes	characteristic
155	#487	—	61	2	no	—
156	#467	—	61	2	no	—
157	#467	—	61	2	no	—
158	#446	—	61	4	no	—
159	#446	—	122	4	no	—
160	C₈×Q₈	—	61	4	no	—
161	C₄×Q₁₆	—	61	2	no	—
162	C₄×Q₁₆	—	61	2	no	—
163	#433	—	122	4	no	—
164	#432	—	61	2	no	—
165	#431	—	61	1	no	—
166	#431	—	61	1	no	—
167	C₂×C₄×C₈	—	61	1	no	—
168	#335	—	61	2	no	—
169	#334	—	61	2	no	—
170	Q₈⋊C₈	—	61	4	no	—
171	Q₈⋊C₈	—	61	4	no	—
172	#322	—	61	1	no	—
173	#322	—	61	1	no	—
174	C₈²	—	61	1	no	—
175	C₈²	—	61	1	no	—
176	C₆₁	#17972	1	1	yes	characteristic, complemented, Sylow
177	C₂×Q₁₆	C₂×#6923	1	2	yes	—
178	C₂×Q₁₆	C₂×#6923	1	2	yes	—
179	C₂²×C₈	—	61	1	no	—
180	C₂²×C₈	—	61	1	no	—
181	C₄⋊Q₈	—	61	2	no	—
182	C₄⋊Q₈	C₂×#6923	1	2	yes	—
183	#125	—	122	2	no	—
184	C₂²⋊Q₈	—	61	4	no	—
185	C₂²⋊Q₈	—	122	2	no	—
186	C₄×Q₈	—	61	4	no	—
187	C₄×Q₈	—	122	2	no	—
188	#117	—	122	2	no	—
189	C₂×C₄²	—	61	1	no	—
190	#107	—	61	2	no	—
191	#107	—	61	2	no	—
192	C₄⋊C₈	—	61	2	no	—
193	C₄⋊C₈	—	61	4	no	—
194	C₄⋊C₈	—	122	1	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	—	61	2	no	—
200	C₂²⋊C₈	—	122	1	no	—
201	C₂²⋊C₈	—	122	2	no	—
202	#97	—	122	2	no	—
203	#97	—	122	4	no	—
204	C₄×C₈	—	61	1	no	—
205	C₄×C₈	—	61	1	no	—
206	C₄×C₈	—	61	1	no	—
207	C₄×C₈	—	61	1	no	—
208	C₄×C₈	—	61	1	no	—
209	C₄×C₈	—	61	1	no	—
210	C₄×C₈	—	61	1	no	—
211	C₄×C₈	—	122	4	no	—
212	C₄×C₈	C₂×#6923	1	1	yes	characteristic
213	C₂×Q₈	—	122	2	no	—
214	C₂×Q₈	C₂²×#6923	1	4	yes	—
215	C₂²×C₄	—	61	1	no	—
216	C₂²×C₄	—	61	1	no	—
217	C₂²×C₄	—	61	1	no	—
218	Q₁₆	—	2	4	no	—
219	Q₁₆	—	2	4	no	—
220	C₂×C₈	—	61	2	no	—
221	C₂×C₈	—	61	2	no	—
222	C₂×C₈	—	61	1	no	—
223	C₂×C₈	—	61	1	no	—
224	C₂×C₈	—	122	2	no	—
225	C₂×C₈	—	122	2	no	—
226	C₂×C₈	—	122	1	no	—
227	C₂×C₈	—	122	1	no	—
228	C₂×C₈	—	122	4	no	—
229	C₂×C₈	C₂²×#6923	1	1	yes	characteristic
230	C₂×C₈	C₂²×#6923	1	1	yes	characteristic
231	C₄⋊C₄	—	2	2	no	—
232	C₄⋊C₄	—	2	2	no	—
233	C₄⋊C₄	—	61	4	no	—
234	C₄⋊C₄	—	122	4	no	—
235	C₄⋊C₄	—	122	4	no	—
236	C₄⋊C₄	—	122	2	no	—
237	C₂²⋊C₄	—	122	4	no	—
238	C₄²	—	61	1	no	—
239	C₄²	—	61	1	no	—
240	C₄²	—	61	1	no	—
241	C₄²	—	122	4	no	—
242	C₄²	C₂²×#6923	1	1	yes	characteristic
243	C₂³	—	61	1	no	—
244	Q₈	—	2	4	no	—
245	Q₈	—	2	4	no	—
246	Q₈	—	488	2	no	—
247	C₂×C₄	—	2	4	no	—
248	C₂×C₄	—	61	1	no	—
249	C₂×C₄	—	61	1	no	—
250	C₂×C₄	—	61	1	no	—
251	C₂×C₄	—	122	2	no	—
252	C₂×C₄	—	122	1	no	—
253	C₂×C₄	—	122	1	no	—
254	C₂×C₄	—	122	4	no	—
255	C₂×C₄	—	244	1	no	—
256	C₂×C₄	D₄×#6923	1	1	yes	characteristic
257	C₂×C₄	D₄×#6923	1	1	yes	characteristic
258	C₂×C₄	C₂³×#6923	1	1	yes	characteristic
259	C₈	—	122	2	no	—
260	C₈	—	122	2	no	—
261	C₈	—	122	1	no	—
262	C₈	—	122	1	no	—
263	C₈	—	244	4	no	—
264	C₈	—	244	1	no	—
265	C₈	—	244	1	no	—
266	C₈	#14519864	1	2	yes	—
267	C₈	#14519864	1	2	yes	—
268	C₂²	—	122	1	no	—
269	C₂²	—	122	1	no	—
270	C₂²	—	122	1	no	—
271	C₂²	C₂×D₄×#6923	1	1	yes	characteristic
272	C₄	—	2	1	no	—
273	C₄	—	2	1	no	—
274	C₄	—	4	4	no	—
275	C₄	—	61	2	no	—
276	C₄	—	122	1	no	—
277	C₄	—	122	1	no	—
278	C₄	—	244	4	no	—
279	C₄	C₂×#14519864	1	1	yes	characteristic
280	C₄	C₂×#14519864	1	1	yes	characteristic
281	C₂	—	244	1	no	—
282	C₂	#21462541	1	1	yes	characteristic
283	C₂	#21462580	1	1	yes	characteristic
284	C₂	#21462599	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×#494	C₄
socle	C₂²×C₆₁	C₂×C₄×D₄
prime radical	C₂²×D₆₁	C₄×D₄
cosocle	C₄×Dic₆₁	C₂⁴
hypercenter	#494	#6923
hypocenter	C₆₁	#17972
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #494

Complements

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

N	H ≅ G/N	type
2. #21461466	C₂	semidirect
3. #21461459	C₂	semidirect
4. #21461412	C₂	semidirect
5. Q₁₆×#92420	C₂	semidirect
176. C₆₁	#17972	semidirect

Maximal subgroups

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

subgroup	index
#21462724	2
#21461466	2
#21461459	2
#21461412	2
Q₁₆×#92420	2
#21461380	2
#21461182	2
#21461023	2
#21456403	2
#17972	61