Subgroups — #29650694

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	#21456405	C₂	1	1	yes	maximal, characteristic
2	#21454913	C₂	1	2	yes	maximal
3	#21444729	C₂	1	2	yes	maximal, complemented
4	#21444717	C₂	1	2	yes	maximal, complemented
5	#21444639	C₂	1	2	yes	maximal, complemented
6	C₈×Dic₂₄₄	C₂	1	2	yes	maximal, complemented
7	#21444582	C₂	1	1	yes	maximal, characteristic
8	#21444293	C₂	1	2	yes	maximal
9	C₆₁×#1450	C₂	1	1	yes	maximal, characteristic
10	#16840319	C₂²	1	2	yes	—
11	#16840167	C₄	1	1	yes	characteristic
12	#16840142	C₄	1	1	yes	characteristic
13	#16840113	C₄	1	2	yes	—
14	#16840003	C₄	1	2	yes	—
15	#16840003	C₄	1	2	yes	—
16	#16839892	C₂²	1	4	yes	—
17	#16839891	—	2	4	no	—
18	C₈×Dic₁₂₂	C₂²	1	4	yes	—
19	C₄×Dic₂₄₄	C₂²	1	2	yes	—
20	C₄×Dic₂₄₄	C₂²	1	2	yes	—
21	#16839820	—	2	4	no	—
22	#16839819	C₂²	1	2	yes	—
23	#16839515	C₂²	1	2	yes	—
24	#16839513	C₂²	1	2	yes	—
25	#16839494	C₂²	1	4	yes	—
26	#16839494	C₂²	1	4	yes	—
27	C₆₁×#431	C₂²	1	1	yes	characteristic
28	C₆₁×#431	C₂²	1	1	yes	characteristic
29	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
30	C₆₁×#322	C₂²	1	1	yes	characteristic
31	C₆₁×#322	C₂²	1	1	yes	characteristic
32	C₈²×C₆₁	C₂²	1	1	yes	characteristic
33	C₈²×C₆₁	C₂²	1	1	yes	characteristic
34	C₂×Dic₂₄₄	C₂×C₄	1	2	yes	—
35	C₂×Dic₂₄₄	C₂×C₄	1	2	yes	—
36	#14519777	C₂×C₄	1	2	yes	—
37	#14519777	C₂×C₄	1	2	yes	—
38	#14519769	—	2	2	no	—
39	#14519756	—	2	2	no	—
40	#14519756	C₂×C₄	1	4	yes	—
41	C₄×Dic₁₂₂	—	2	2	no	—
42	C₄×Dic₁₂₂	C₂³	1	4	yes	—
43	#14519736	—	2	2	no	—
44	#14519711	C₂×C₄	1	2	yes	—
45	#14519711	C₂×C₄	1	2	yes	—
46	C₄⋊#92418	C₂³	1	4	yes	—
47	#14519706	—	2	4	no	—
48	#14519706	—	2	4	no	—
49	#14519702	C₂×C₄	1	4	yes	—
50	#14519702	C₂×C₄	1	4	yes	—
51	C₂²⋊#92418	—	2	2	no	—
52	#14519688	—	2	2	no	—
53	#14519687	—	2	4	no	—
54	C₈×Dic₆₁	—	2	4	no	—
55	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
56	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
57	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
58	C₄⋊C₈×C₆₁	—	2	1	no	—
59	C₄⋊C₈×C₆₁	D₄	1	2	yes	—
60	C₂²⋊C₈×C₆₁	—	2	1	no	—
61	C₂²⋊C₈×C₆₁	D₄	1	2	yes	—
62	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
63	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
64	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
65	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
66	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
67	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
68	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
69	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
70	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
71	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₆₁	#41	1	1	yes	characteristic
73	C₂×Dic₁₂₂	—	2	2	no	—
74	C₂×Dic₁₂₂	C₂²×C₄	1	4	yes	—
75	C₂×C₈×C₆₁	—	2	2	no	—
76	C₂×C₈×C₆₁	—	2	2	no	—
77	C₂×C₈×C₆₁	—	2	1	no	—
78	C₂×C₈×C₆₁	—	2	1	no	—
79	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
81	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
82	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
83	C₂×C₈×C₆₁	C₂×D₄	1	2	yes	—
84	C₂×C₈×C₆₁	C₂×D₄	1	2	yes	—
85	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
86	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
87	C₄²×C₆₁	#41	1	1	yes	characteristic
88	C₄²×C₆₁	C₂⁴	1	1	yes	characteristic
89	C₂²⋊Dic₆₁	—	2	4	no	—
90	C₄⋊Dic₆₁	—	2	2	no	—
91	C₄⋊Dic₆₁	—	2	2	no	—
92	C₄⋊Dic₆₁	—	2	2	no	—
93	C₄⋊Dic₆₁	C₂²×C₄	1	4	yes	—
94	#11758196	—	2	4	no	—
95	#11758196	—	2	4	no	—
96	C₄×Dic₆₁	—	2	4	no	—
97	C₂×#92418	—	2	4	no	—
98	Dic₂₄₄	—	2	4	no	—
99	Dic₂₄₄	—	2	4	no	—
100	C₂³×C₆₁	C₄×D₄	1	1	yes	characteristic
101	C₂×C₄×C₆₁	—	2	1	no	—
102	C₂×C₄×C₆₁	—	2	1	no	—
103	C₂×C₄×C₆₁	—	2	2	no	—
104	C₂×C₄×C₆₁	—	4	1	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	C₄×D₄	1	1	yes	characteristic
108	C₂×C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
109	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
110	C₂×C₄×C₆₁	C₂×#41	1	1	yes	characteristic
111	C₂×Dic₆₁	—	2	4	no	—
112	C₂×Dic₆₁	—	2	4	no	—
113	Dic₁₂₂	—	2	4	no	—
114	Dic₁₂₂	—	2	4	no	—
115	Dic₁₂₂	—	8	2	no	—
116	C₈×C₆₁	—	2	1	no	—
117	C₈×C₆₁	—	2	1	no	—
118	C₈×C₆₁	—	2	2	no	—
119	C₈×C₆₁	—	2	2	no	—
120	C₈×C₆₁	—	4	1	no	—
121	C₈×C₆₁	—	4	1	no	—
122	C₈×C₆₁	#131	1	2	yes	—
123	C₈×C₆₁	#131	1	2	yes	—
124	#92418	—	4	4	no	—
125	#17972	—	61	1	no	maximal, Sylow
126	C₂²×C₆₁	—	2	1	no	—
127	C₂²×C₆₁	—	2	1	no	—
128	C₂²×C₆₁	—	2	1	no	—
129	C₂²×C₆₁	C₂×C₄×D₄	1	1	yes	characteristic
130	C₄×C₆₁	—	2	1	no	—
131	C₄×C₆₁	—	2	1	no	—
132	C₄×C₆₁	—	2	1	no	—
133	C₄×C₆₁	—	2	1	no	—
134	C₄×C₆₁	C₂×#131	1	1	yes	characteristic
135	C₄×C₆₁	C₂×#131	1	1	yes	characteristic
136	C₄×C₆₁	#578	1	2	yes	—
137	Dic₆₁	—	4	4	no	—
138	Dic₆₁	—	4	4	no	—
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	D₆₁	1	1	yes	characteristic
148	C₂×C₆₁	—	4	1	no	—
149	C₂×C₆₁	#2931	1	1	yes	characteristic
150	C₂×C₆₁	#2950	1	1	yes	characteristic
151	C₂×C₆₁	#2957	1	1	yes	characteristic
152	#533	—	61	2	no	—
153	#500	—	61	1	no	—
154	#494	—	61	1	no	—
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	C₂×D₆₁	1	1	yes	characteristic
166	#431	C₂×D₆₁	1	1	yes	characteristic
167	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
168	#335	—	61	2	no	—
169	#334	—	61	2	no	—
170	Q₈⋊C₈	—	61	4	no	—
171	Q₈⋊C₈	—	61	4	no	—
172	#322	C₂×D₆₁	1	1	yes	characteristic
173	#322	C₂×D₆₁	1	1	yes	characteristic
174	C₈²	C₂×D₆₁	1	1	yes	characteristic
175	C₈²	C₂×D₆₁	1	1	yes	characteristic
176	C₆₁	#17972	1	1	yes	characteristic, complemented, Sylow
177	C₂×Q₁₆	—	61	2	no	—
178	C₂×Q₁₆	—	61	2	no	—
179	C₂²×C₈	C₄×D₆₁	1	1	yes	characteristic
180	C₂²×C₈	C₄×D₆₁	1	1	yes	characteristic
181	C₄⋊Q₈	—	61	2	no	—
182	C₄⋊Q₈	—	61	2	no	—
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	C₂²×D₆₁	1	1	yes	characteristic
190	#107	—	61	2	no	—
191	#107	—	61	2	no	—
192	C₄⋊C₈	—	2	1	no	—
193	C₄⋊C₈	—	61	4	no	—
194	C₄⋊C₈	—	122	4	no	—
195	C₄⋊C₈	—	122	4	no	—
196	C₄⋊C₈	D₂₄₄	1	2	yes	—
197	#103	—	61	4	no	—
198	#103	—	61	4	no	—
199	C₂²⋊C₈	—	2	1	no	—
200	C₂²⋊C₈	—	122	2	no	—
201	C₂²⋊C₈	D₂₄₄	1	2	yes	—
202	#97	—	122	4	no	—
203	#97	—	122	2	no	—
204	C₄×C₈	—	122	4	no	—
205	C₄×C₈	C₄×D₆₁	1	1	yes	characteristic
206	C₄×C₈	C₄×D₆₁	1	1	yes	characteristic
207	C₄×C₈	C₂²×D₆₁	1	1	yes	characteristic
208	C₄×C₈	C₂²×D₆₁	1	1	yes	characteristic
209	C₄×C₈	C₂²×D₆₁	1	1	yes	characteristic
210	C₄×C₈	C₂²×D₆₁	1	1	yes	characteristic
211	C₄×C₈	C₂²×D₆₁	1	1	yes	characteristic
212	C₄×C₈	C₂²×D₆₁	1	1	yes	characteristic
213	C₂×Q₈	—	61	4	no	—
214	C₂×Q₈	—	122	2	no	—
215	C₂²×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
216	C₂²×C₄	C₂×D₂₄₄	1	1	yes	characteristic
217	C₂²×C₄	#11758222	1	1	yes	characteristic
218	Q₁₆	—	122	4	no	—
219	Q₁₆	—	122	4	no	—
220	C₂×C₈	—	2	2	no	—
221	C₂×C₈	—	2	2	no	—
222	C₂×C₈	—	2	1	no	—
223	C₂×C₈	—	2	1	no	—
224	C₂×C₈	—	122	4	no	—
225	C₂×C₈	C₂×C₄×D₆₁	1	1	yes	characteristic
226	C₂×C₈	C₂×C₄×D₆₁	1	1	yes	characteristic
227	C₂×C₈	C₂×C₄×D₆₁	1	1	yes	characteristic
228	C₂×C₈	C₂×C₄×D₆₁	1	1	yes	characteristic
229	C₂×C₈	C₂×D₂₄₄	1	2	yes	—
230	C₂×C₈	C₂×D₂₄₄	1	2	yes	—
231	C₄⋊C₄	—	61	4	no	—
232	C₄⋊C₄	—	122	4	no	—
233	C₄⋊C₄	—	122	4	no	—
234	C₄⋊C₄	—	122	2	no	—
235	C₄⋊C₄	—	122	2	no	—
236	C₄⋊C₄	—	122	2	no	—
237	C₂²⋊C₄	—	122	4	no	—
238	C₄²	—	122	4	no	—
239	C₄²	C₂×C₄×D₆₁	1	1	yes	characteristic
240	C₄²	C₂×D₂₄₄	1	1	yes	characteristic
241	C₄²	#11758222	1	1	yes	characteristic
242	C₄²	C₂³×D₆₁	1	1	yes	characteristic
243	C₂³	C₄×D₂₄₄	1	1	yes	characteristic
244	Q₈	—	122	4	no	—
245	Q₈	—	122	4	no	—
246	Q₈	—	488	2	no	—
247	C₂×C₄	—	2	1	no	—
248	C₂×C₄	—	2	1	no	—
249	C₂×C₄	—	2	2	no	—
250	C₂×C₄	—	4	1	no	—
251	C₂×C₄	—	122	4	no	—
252	C₂×C₄	—	122	4	no	—
253	C₂×C₄	C₄×D₂₄₄	1	1	yes	characteristic
254	C₂×C₄	C₄×D₂₄₄	1	1	yes	characteristic
255	C₂×C₄	C₄×D₂₄₄	1	1	yes	characteristic
256	C₂×C₄	C₂²×C₄×D₆₁	1	1	yes	characteristic
257	C₂×C₄	C₂²×D₂₄₄	1	1	yes	characteristic
258	C₂×C₄	C₂×#11758222	1	1	yes	characteristic
259	C₈	—	2	1	no	—
260	C₈	—	2	1	no	—
261	C₈	—	2	2	no	—
262	C₈	—	2	2	no	—
263	C₈	—	4	1	no	—
264	C₈	—	4	1	no	—
265	C₈	—	244	4	no	—
266	C₈	#14519783	1	2	yes	—
267	C₈	#14519783	1	2	yes	—
268	C₂²	—	2	1	no	—
269	C₂²	—	2	1	no	—
270	C₂²	—	2	1	no	—
271	C₂²	C₂×C₄×D₂₄₄	1	1	yes	characteristic
272	C₄	—	2	1	no	—
273	C₄	—	2	1	no	—
274	C₄	—	2	1	no	—
275	C₄	—	2	1	no	—
276	C₄	—	244	4	no	—
277	C₄	—	244	4	no	—
278	C₄	C₂×#14519783	1	1	yes	characteristic
279	C₄	C₂×#14519783	1	1	yes	characteristic
280	C₄	#16840579	1	2	yes	—
281	C₂	—	4	1	no	—
282	C₂	#21454532	1	1	yes	characteristic
283	C₂	#21454688	1	1	yes	characteristic
284	C₂	#21454737	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₂₄₄
commutator	C₂×C₄×C₆₁	C₂³×C₄
Frattini	C₄²	C₂³×D₆₁
Fitting	C₆₁×#1450	C₂
socle	C₂²×C₆₁	C₂×C₄×D₄
prime radical	C₂³×C₆₁	C₄×D₄
cosocle	C₄²×C₆₁	C₂⁴
hypercenter	#1450	D₆₁
hypocenter	C₆₁	#17972
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₄ ◁ #431 ◁ #1450

Complements

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

N	H ≅ G/N	type
3. #21444729	C₂	semidirect
4. #21444717	C₂	semidirect
5. #21444639	C₂	semidirect
6. C₈×Dic₂₄₄	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
#21456405	2
#21454913	2
#21444729	2
#21444717	2
#21444639	2
C₈×Dic₂₄₄	2
#21444582	2
#21444293	2
C₆₁×#1450	2
#17972	61