Subgroups — #29651789

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	#21456405	C₂	1	1	yes	maximal, characteristic
2	#21454762	C₂	1	2	yes	maximal
3	#21444715	C₂	1	2	yes	maximal, complemented
4	#21444668	C₂	1	4	yes	maximal, complemented
5	#21444625	C₂	1	2	yes	maximal, complemented
6	#21444221	C₂	1	2	yes	maximal
7	C₈×#11758194	C₂	1	1	yes	maximal, characteristic
8	C₂×C₆₁×#335	C₂	1	1	yes	maximal, characteristic
9	#16840319	C₂²	1	2	yes	—
10	#16840167	C₄	1	1	yes	characteristic
11	#16840142	C₄	1	1	yes	characteristic
12	#16840113	C₄	1	2	yes	—
13	#16840003	C₄	1	2	yes	—
14	#16840003	C₄	1	2	yes	—
15	#16839891	—	2	4	no	—
16	#16839889	C₂²	1	4	yes	—
17	#16839886	C₂²	1	4	yes	—
18	C₄×Dic₂₄₄	C₂²	1	2	yes	—
19	C₄×Dic₂₄₄	C₂²	1	2	yes	—
20	#16839815	C₂²	1	1	yes	characteristic
21	#16839788	—	2	2	no	—
22	#16839717	—	2	2	no	—
23	C₄×#11758194	C₂²	1	1	yes	characteristic
24	#16839492	C₂²	1	4	yes	—
25	#16839492	C₂²	1	4	yes	—
26	#16839484	C₂²	1	2	yes	—
27	C₈×#92418	C₂²	1	2	yes	—
28	C₂×C₄⋊C₈×C₆₁	C₂²	1	2	yes	—
29	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
30	C₆₁×#335	C₂²	1	1	yes	characteristic
31	C₆₁×#335	C₂²	1	2	yes	—
32	C₆₁×#335	C₂²	1	1	yes	characteristic
33	C₂×Dic₂₄₄	C₂×C₄	1	2	yes	—
34	C₂×Dic₂₄₄	C₂×C₄	1	2	yes	—
35	C₂×#11758194	D₄	1	1	yes	characteristic
36	C₂×#11758194	D₄	1	1	yes	characteristic
37	#14519777	C₂×C₄	1	2	yes	—
38	#14519777	C₂×C₄	1	2	yes	—
39	#14519769	—	2	2	no	—
40	#14519756	—	2	2	no	—
41	#14519756	C₂×C₄	1	4	yes	—
42	C₄×Dic₁₂₂	—	2	2	no	—
43	C₄×Dic₁₂₂	C₂³	1	4	yes	—
44	#14519736	—	2	2	no	—
45	#14519711	C₂×C₄	1	2	yes	—
46	#14519711	C₂×C₄	1	2	yes	—
47	C₄⋊#92418	—	2	1	no	—
48	#14519706	—	2	4	no	—
49	#14519706	—	2	4	no	—
50	#14519702	C₂×C₄	1	4	yes	—
51	#14519702	C₂×C₄	1	4	yes	—
52	C₂²⋊#92418	—	2	1	no	—
53	#14519688	D₄	1	2	yes	—
54	#14519687	—	2	4	no	—
55	C₄×#92418	C₂³	1	2	yes	—
56	C₄×#92418	C₂³	1	2	yes	—
57	C₈×Dic₆₁	—	2	4	no	—
58	C₂²×C₈×C₆₁	—	2	2	no	—
59	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
60	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
61	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
62	C₄⋊C₈×C₆₁	—	2	2	no	—
63	C₄⋊C₈×C₆₁	C₂³	1	2	yes	—
64	C₄⋊C₈×C₆₁	C₂³	1	2	yes	—
65	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
66	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
67	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
68	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
69	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
70	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₆₁	#41	1	1	yes	characteristic
72	C₂×Dic₁₂₂	—	2	2	no	—
73	C₂×Dic₁₂₂	C₂²×C₄	1	4	yes	—
74	C₂×C₈×C₆₁	—	2	2	no	—
75	C₂×C₈×C₆₁	—	2	2	no	—
76	C₂×C₈×C₆₁	—	2	2	no	—
77	C₂×C₈×C₆₁	—	4	1	no	—
78	C₂×C₈×C₆₁	—	4	4	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	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
84	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
85	C₄²×C₆₁	#41	1	1	yes	characteristic
86	C₄²×C₆₁	C₂⁴	1	1	yes	characteristic
87	C₂²⋊Dic₆₁	—	2	4	no	—
88	C₄⋊Dic₆₁	—	2	2	no	—
89	C₄⋊Dic₆₁	—	2	2	no	—
90	C₄⋊Dic₆₁	—	2	2	no	—
91	C₄⋊Dic₆₁	C₂²×C₄	1	4	yes	—
92	#11758196	—	2	4	no	—
93	#11758196	—	2	4	no	—
94	C₄×Dic₆₁	—	2	4	no	—
95	#11758194	—	2	2	no	—
96	#11758194	Q₁₆	1	4	yes	complemented
97	C₂×#92418	—	2	2	no	—
98	C₂×#92418	C₂×D₄	1	2	yes	—
99	C₂×#92418	C₂×D₄	1	2	yes	—
100	Dic₂₄₄	—	2	4	no	—
101	Dic₂₄₄	—	4	2	no	—
102	C₂³×C₆₁	C₄×D₄	1	1	yes	characteristic
103	C₂×C₄×C₆₁	—	2	1	no	—
104	C₂×C₄×C₆₁	—	2	1	no	—
105	C₂×C₄×C₆₁	—	4	1	no	—
106	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
107	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
108	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
109	C₂×C₄×C₆₁	C₂×Q₁₆	1	2	yes	—
110	C₂×C₄×C₆₁	#135	1	2	yes	—
111	C₂×C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
112	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
113	C₂×C₄×C₆₁	C₂×#41	1	1	yes	characteristic
114	C₂×Dic₆₁	—	2	4	no	—
115	C₂×Dic₆₁	—	2	4	no	—
116	Dic₁₂₂	—	2	4	no	—
117	Dic₁₂₂	—	2	4	no	—
118	Dic₁₂₂	—	8	2	no	—
119	C₈×C₆₁	—	2	1	no	—
120	C₈×C₆₁	—	2	1	no	—
121	C₈×C₆₁	—	2	1	no	—
122	C₈×C₆₁	—	4	2	no	—
123	C₈×C₆₁	—	4	2	no	—
124	C₈×C₆₁	#131	1	2	yes	—
125	#92418	—	2	2	no	—
126	#92418	—	4	2	no	—
127	#92418	C₂×Q₁₆	1	4	yes	—
128	#18068	—	61	1	no	maximal, Sylow
129	C₂²×C₆₁	—	2	1	no	—
130	C₂²×C₆₁	—	2	1	no	—
131	C₂²×C₆₁	C₄×Q₁₆	1	2	yes	—
132	C₂²×C₆₁	C₂×C₄×D₄	1	1	yes	characteristic
133	C₄×C₆₁	—	2	1	no	—
134	C₄×C₆₁	—	2	1	no	—
135	C₄×C₆₁	—	2	1	no	—
136	C₄×C₆₁	C₄×Q₁₆	1	2	yes	—
137	C₄×C₆₁	C₂×#131	1	1	yes	characteristic
138	C₄×C₆₁	#568	1	1	yes	characteristic
139	C₄×C₆₁	C₂²×Q₁₆	1	1	yes	characteristic
140	C₄×C₆₁	C₂×#135	1	1	yes	characteristic
141	Dic₆₁	—	4	4	no	—
142	Dic₆₁	—	4	4	no	—
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	D₆₁	1	1	yes	characteristic
149	#1481	—	61	2	no	—
150	C₈×M₁₆	—	61	1	no	—
151	C₂×C₆₁	—	2	2	no	—
152	C₂×C₆₁	#2935	1	1	yes	characteristic
153	C₂×C₆₁	C₂×C₄×Q₁₆	1	1	yes	characteristic
154	C₂×C₆₁	#2957	1	1	yes	characteristic
155	#533	—	61	2	no	—
156	#500	—	61	1	no	—
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	C₂×D₆₁	1	2	yes	—
169	#405	—	122	2	no	—
170	C₄×M₁₆	—	61	1	no	—
171	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
172	#335	C₂×D₆₁	1	1	yes	characteristic
173	#335	C₂×D₆₁	1	2	yes	—
174	#335	C₂×D₆₁	1	1	yes	characteristic
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	—	2	2	no	—
185	C₂²×C₈	C₄×D₆₁	1	1	yes	characteristic
186	C₂²×C₈	C₄×D₆₁	1	1	yes	characteristic
187	C₄⋊Q₈	—	61	2	no	—
188	C₄⋊Q₈	—	61	2	no	—
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	C₂²×D₆₁	1	1	yes	characteristic
196	#107	—	61	2	no	—
197	#107	—	61	2	no	—
198	C₄⋊C₈	—	2	2	no	—
199	C₄⋊C₈	—	122	4	no	—
200	C₄⋊C₈	—	122	4	no	—
201	C₄⋊C₈	—	122	1	no	—
202	C₄⋊C₈	C₂²×D₆₁	1	2	yes	—
203	C₄⋊C₈	C₂²×D₆₁	1	2	yes	—
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	2	no	—
211	C₄×C₈	—	122	4	no	—
212	C₄×C₈	C₄×D₆₁	1	1	yes	characteristic
213	C₄×C₈	C₄×D₆₁	1	1	yes	characteristic
214	C₄×C₈	C₂²×D₆₁	1	1	yes	characteristic
215	C₄×C₈	C₂²×D₆₁	1	1	yes	characteristic
216	C₂×Q₈	—	61	4	no	—
217	C₂×Q₈	—	122	2	no	—
218	C₂²×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
219	C₂²×C₄	D₄×D₆₁	1	1	yes	characteristic
220	C₂²×C₄	#11758226	1	1	yes	characteristic
221	Q₁₆	—	122	4	no	—
222	Q₁₆	—	244	2	no	—
223	M₁₆	—	61	4	no	—
224	M₁₆	—	122	2	no	—
225	C₂×C₈	—	2	2	no	—
226	C₂×C₈	—	2	2	no	—
227	C₂×C₈	—	2	2	no	—
228	C₂×C₈	—	4	1	no	—
229	C₂×C₈	—	4	4	no	—
230	C₂×C₈	—	61	2	no	—
231	C₂×C₈	—	61	2	no	—
232	C₂×C₈	—	122	2	no	—
233	C₂×C₈	C₂×C₄×D₆₁	1	1	yes	characteristic
234	C₂×C₈	C₂×C₄×D₆₁	1	1	yes	characteristic
235	C₂×C₈	C₂×C₄×D₆₁	1	1	yes	characteristic
236	C₂×C₈	C₂×C₄×D₆₁	1	1	yes	characteristic
237	C₄⋊C₄	—	61	4	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	—	122	2	no	—
243	C₂²⋊C₄	—	122	4	no	—
244	C₄²	—	122	4	no	—
245	C₄²	C₂×C₄×D₆₁	1	1	yes	characteristic
246	C₄²	D₄×D₆₁	1	1	yes	characteristic
247	C₄²	#11758226	1	1	yes	characteristic
248	C₄²	C₂³×D₆₁	1	1	yes	characteristic
249	C₂³	#14519739	1	1	yes	characteristic
250	Q₈	—	122	4	no	—
251	Q₈	—	122	4	no	—
252	Q₈	—	488	2	no	—
253	C₂×C₄	—	2	1	no	—
254	C₂×C₄	—	2	1	no	—
255	C₂×C₄	—	4	1	no	—
256	C₂×C₄	—	122	4	no	—
257	C₂×C₄	—	122	4	no	—
258	C₂×C₄	#14519739	1	1	yes	characteristic
259	C₂×C₄	#14519739	1	1	yes	characteristic
260	C₂×C₄	#14519739	1	1	yes	characteristic
261	C₂×C₄	Q₁₆×D₆₁	1	2	yes	—
262	C₂×C₄	#14519796	1	2	yes	—
263	C₂×C₄	C₂²×C₄×D₆₁	1	1	yes	characteristic
264	C₂×C₄	C₂×D₄×D₆₁	1	1	yes	characteristic
265	C₂×C₄	C₂×#11758226	1	1	yes	characteristic
266	C₈	—	2	1	no	—
267	C₈	—	2	1	no	—
268	C₈	—	2	1	no	—
269	C₈	—	4	2	no	—
270	C₈	—	4	2	no	—
271	C₈	—	61	4	no	—
272	C₈	—	122	2	no	—
273	C₈	—	244	2	no	—
274	C₈	#14519783	1	2	yes	—
275	C₂²	—	2	1	no	—
276	C₂²	—	2	1	no	—
277	C₂²	#16839856	1	2	yes	—
278	C₂²	C₂×#14519739	1	1	yes	characteristic
279	C₄	—	2	1	no	—
280	C₄	—	2	1	no	—
281	C₄	—	2	1	no	—
282	C₄	—	244	4	no	—
283	C₄	—	244	4	no	—
284	C₄	#16839856	1	2	yes	—
285	C₄	C₂×#14519783	1	1	yes	characteristic
286	C₄	#16840528	1	1	yes	characteristic
287	C₄	C₂×Q₁₆×D₆₁	1	1	yes	characteristic
288	C₄	C₂×#14519796	1	1	yes	characteristic
289	C₂	—	2	2	no	—
290	C₂	#21454562	1	1	yes	characteristic
291	C₂	C₂×#16839856	1	1	yes	characteristic
292	C₂	#21454735	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₆₁
commutator	C₂×C₄×C₆₁	C₂³×C₄
Frattini	C₄²	C₂³×D₆₁
Fitting	C₂×C₆₁×#335	C₂
socle	C₂²×C₆₁	C₂×C₄×D₄
prime radical	C₂³×C₆₁	C₄×D₄
cosocle	C₄²×C₆₁	C₂⁴
hypercenter	C₂×#335	D₆₁
hypocenter	C₆₁	#18068
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×#335

Complements

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

N	H ≅ G/N	type
3. #21444715	C₂	semidirect
4. #21444668	C₂	semidirect
5. #21444625	C₂	semidirect
96. #11758194	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
#21456405	2
#21454762	2
#21444715	2
#21444668	2
#21444625	2
#21444221	2
C₈×#11758194	2
C₂×C₆₁×#335	2
#18068	61