Subgroups — #29589822

Order 15616.

334 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21454748	C₂	1	1	yes	maximal, characteristic
2	#21447573	C₂	1	1	yes	maximal, characteristic
3	C₄×C₁₆×D₆₁	C₂	1	1	yes	maximal, characteristic
4	#21443799	C₂	1	1	yes	maximal, characteristic, complemented
5	Q₈⋊#11758185	C₂	1	1	yes	maximal, characteristic, complemented
6	#21443797	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₁×Q₈⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
8	#16840316	C₄	1	1	yes	characteristic
9	C₂×#14519716	—	2	1	no	—
10	C₂×C₁₆×D₆₁	—	2	1	no	—
11	#16839893	C₄	1	1	yes	characteristic
12	#16839891	—	2	1	no	—
13	#16839889	C₄	1	1	yes	characteristic
14	C₈×Dic₁₂₂	—	2	1	no	—
15	C₈×Dic₁₂₂	C₂²	1	1	yes	characteristic
16	#16839787	—	2	1	no	—
17	#16839784	C₄	1	1	yes	characteristic
18	#16839716	—	2	1	no	—
19	C₄×C₈×D₆₁	C₂²	1	1	yes	characteristic
20	C₄⋊#11758185	C₂²	1	1	yes	characteristic
21	#16839580	—	2	1	no	—
22	C₄×#11758185	C₂²	1	1	yes	characteristic
23	C₁₆×Dic₆₁	—	2	1	no	—
24	C₈×Q₈×C₆₁	C₂²	1	1	yes	characteristic
25	C₆₁×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
26	C₄×C₁₆×C₆₁	C₂²	1	1	yes	characteristic
27	C₂×#11758189	—	2	1	no	—
28	C₂×C₈×D₆₁	D₄	1	1	yes	characteristic
29	C₂×C₈×D₆₁	D₄	1	1	yes	characteristic
30	#14519776	C₈	1	1	yes	characteristic
31	#14519775	C₈	1	1	yes	characteristic
32	#14519767	—	2	1	no	—
33	#14519757	C₈	1	1	yes	characteristic
34	#14519755	—	2	1	no	—
35	#14519753	C₈	1	1	yes	characteristic
36	C₄×Dic₁₂₂	—	2	1	no	—
37	C₄×Dic₁₂₂	C₂×C₄	1	1	yes	characteristic
38	#14519734	—	2	1	no	—
39	C₄²×D₆₁	C₂×C₄	1	1	yes	characteristic
40	#14519716	—	4	2	no	—
41	C₂×#11758185	—	2	1	no	—
42	C₂×#11758185	—	2	1	no	—
43	C₁₆×D₆₁	—	2	4	no	—
44	C₄⋊#92418	—	2	1	no	—
45	C₄⋊#92418	C₂×C₄	1	1	yes	characteristic
46	#14519706	—	2	1	no	—
47	#14519706	—	2	1	no	—
48	#14519706	—	2	1	no	—
49	#14519688	—	2	1	no	—
50	#14519687	—	2	1	no	—
51	C₄×#92418	C₂×C₄	1	1	yes	characteristic
52	C₈×Dic₆₁	—	2	1	no	—
53	C₈×Dic₆₁	D₄	1	1	yes	characteristic
54	C₈×Dic₆₁	D₄	1	1	yes	characteristic
55	C₄×Q₈×C₆₁	C₂×C₄	1	1	yes	characteristic
56	C₂×C₁₆×C₆₁	—	2	1	no	—
57	C₂×C₁₆×C₆₁	—	2	1	no	—
58	C₄⋊C₈×C₆₁	—	2	1	no	—
59	C₄⋊C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
60	C₄×C₈×C₆₁	—	2	1	no	—
61	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
62	C₂×Q₈×C₆₁	C₂×C₈	1	1	yes	characteristic
63	C₂×C₄×D₆₁	C₂²⋊C₄	1	1	yes	characteristic
64	C₂×C₄×D₆₁	C₂×C₈	1	1	yes	characteristic
65	C₂×C₄×D₆₁	M₁₆	1	1	yes	characteristic
66	C₂×Dic₁₂₂	—	2	1	no	—
67	C₂×Dic₁₂₂	C₂×C₈	1	1	yes	characteristic
68	C₂×C₈×C₆₁	—	2	1	no	—
69	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
70	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
71	C₄⋊C₄×C₆₁	—	2	1	no	—
72	C₄⋊C₄×C₆₁	C₂×C₈	1	1	yes	characteristic
73	C₄²×C₆₁	—	2	1	no	—
74	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
75	#11758198	—	2	1	no	—
76	#11758198	—	2	1	no	—
77	C₄⋊Dic₆₁	—	2	1	no	—
78	C₄⋊Dic₆₁	C₂×C₈	1	1	yes	characteristic
79	#11758196	—	2	1	no	—
80	#11758196	—	2	1	no	—
81	#11758196	—	2	1	no	—
82	#11758196	—	2	1	no	—
83	#11758196	—	2	1	no	—
84	C₄×Dic₆₁	—	2	1	no	—
85	C₄×Dic₆₁	C₂²⋊C₄	1	1	yes	characteristic
86	C₄×Dic₆₁	C₂×C₈	1	1	yes	characteristic
87	C₄×Dic₆₁	M₁₆	1	1	yes	characteristic
88	C₂×#92418	—	2	1	no	—
89	C₂×#92418	C₂²⋊C₄	1	1	yes	characteristic
90	C₂×#92418	C₂²⋊C₄	1	1	yes	characteristic
91	#11758189	—	4	2	no	—
92	C₈×D₆₁	—	2	2	no	—
93	C₈×D₆₁	SD₁₆	1	2	yes	—
94	C₈×D₆₁	Q₁₆	1	2	yes	—
95	C₁₆×C₆₁	—	2	2	no	—
96	C₁₆×C₆₁	—	4	1	no	—
97	#11758185	—	2	2	no	—
98	#11758185	—	4	1	no	—
99	C₂²×D₆₁	C₂²⋊C₈	1	1	yes	characteristic
100	Q₈×C₆₁	—	2	1	no	—
101	Q₈×C₆₁	—	2	1	no	—
102	C₂×C₄×C₆₁	—	2	1	no	—
103	C₂×C₄×C₆₁	—	2	1	no	—
104	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
105	C₂×C₄×C₆₁	C₂²×C₈	1	1	yes	characteristic
106	C₂×C₄×C₆₁	C₂×M₁₆	1	1	yes	characteristic
107	C₂×Dic₆₁	—	2	1	no	—
108	C₂×Dic₆₁	—	2	1	no	—
109	C₂×Dic₆₁	C₂²⋊C₈	1	1	yes	characteristic
110	C₂×Dic₆₁	C₂²⋊C₈	1	1	yes	characteristic
111	C₂×Dic₆₁	C₂²⋊C₈	1	1	yes	characteristic
112	C₄×D₆₁	—	2	1	no	—
113	C₄×D₆₁	—	2	1	no	—
114	C₄×D₆₁	—	2	2	no	—
115	C₄×D₆₁	#103	1	2	yes	—
116	C₄×D₆₁	#104	1	2	yes	—
117	Dic₁₂₂	—	2	1	no	—
118	Dic₁₂₂	—	2	1	no	—
119	Dic₁₂₂	—	4	2	no	—
120	C₈×C₆₁	—	2	1	no	—
121	C₈×C₆₁	—	4	1	no	—
122	C₈×C₆₁	C₂×SD₁₆	1	1	yes	characteristic
123	C₈×C₆₁	C₂×Q₁₆	1	1	yes	characteristic
124	#92418	—	2	1	no	—
125	#92418	—	4	1	no	—
126	#92418	#103	1	2	yes	—
127	#11608	—	61	1	no	maximal, Sylow
128	C₂²×C₆₁	C₂×C₂²⋊C₈	1	1	yes	characteristic
129	C₂×D₆₁	—	2	1	no	—
130	C₂×D₆₁	—	2	1	no	—
131	C₂×D₆₁	Q₈⋊C₈	1	2	yes	—
132	C₄×C₆₁	—	2	1	no	—
133	C₄×C₆₁	—	4	1	no	—
134	C₄×C₆₁	—	4	1	no	—
135	C₄×C₆₁	C₂×#103	1	1	yes	characteristic
136	C₄×C₆₁	C₂×#104	1	1	yes	characteristic
137	C₄×C₆₁	#504	1	1	yes	characteristic
138	C₄×C₆₁	#504	1	1	yes	characteristic
139	Dic₆₁	—	2	1	no	—
140	Dic₆₁	—	4	1	no	—
141	Dic₆₁	—	4	1	no	—
142	Dic₆₁	Q₈⋊C₈	1	2	yes	—
143	Dic₆₁	#350	1	2	yes	—
144	#2960	—	61	1	no	—
145	#2150	—	61	1	no	—
146	C₂×C₄×C₁₆	—	61	1	no	—
147	Q₈⋊C₁₆	—	61	1	no	—
148	Q₈⋊C₁₆	—	61	1	no	—
149	Q₈⋊C₁₆	—	61	1	no	—
150	Q₈⋊C₁₆	D₆₁	1	1	yes	characteristic, complemented
151	C₂×C₆₁	C₂×Q₈⋊C₈	1	1	yes	characteristic
152	C₂×C₆₁	#2113	1	1	yes	characteristic
153	C₂×C₆₁	C₂×#350	1	1	yes	characteristic
154	D₆₁	—	2	2	no	—
155	#533	—	61	1	no	—
156	C₂×M₃₂	—	122	1	no	—
157	C₂²×C₁₆	—	122	1	no	—
158	#446	—	61	1	no	—
159	#446	—	61	1	no	—
160	#446	—	122	1	no	—
161	C₈×Q₈	—	61	1	no	—
162	C₈×Q₈	—	122	1	no	—
163	C₈×Q₈	C₂×D₆₁	1	1	yes	characteristic
164	#424	—	122	1	no	—
165	#423	—	61	1	no	—
166	#405	—	122	1	no	—
167	C₂×C₄×C₈	—	61	1	no	—
168	C₄⋊C₁₆	—	61	1	no	—
169	C₄⋊C₁₆	—	122	1	no	—
170	C₄⋊C₁₆	C₂×D₆₁	1	1	yes	characteristic
171	C₄×C₁₆	—	61	1	no	—
172	C₄×C₁₆	—	122	1	no	—
173	C₄×C₁₆	C₂×D₆₁	1	1	yes	characteristic
174	C₆₁	#11608	1	1	yes	characteristic, complemented, Sylow
175	C₂×M₁₆	—	122	1	no	—
176	C₂²×C₈	—	61	1	no	—
177	C₂²×C₈	—	61	1	no	—
178	C₄⋊Q₈	—	61	1	no	—
179	C₄⋊Q₈	—	61	1	no	—
180	#125	—	122	1	no	—
181	C₂²⋊Q₈	—	61	1	no	—
182	C₂²⋊Q₈	—	61	1	no	—
183	C₂²⋊Q₈	—	122	1	no	—
184	C₄×Q₈	—	61	1	no	—
185	C₄×Q₈	—	122	1	no	—
186	C₄×Q₈	C₄×D₆₁	1	1	yes	characteristic
187	#117	—	122	1	no	—
188	C₂×C₄²	—	61	1	no	—
189	M₃₂	—	244	2	no	—
190	C₂×C₁₆	—	2	1	no	—
191	C₂×C₁₆	—	2	1	no	—
192	C₂×C₁₆	—	122	4	no	—
193	C₂×C₁₆	—	122	1	no	—
194	C₂×C₁₆	—	122	1	no	—
195	C₄⋊C₈	—	2	1	no	—
196	C₄⋊C₈	—	61	1	no	—
197	C₄⋊C₈	—	122	1	no	—
198	C₄⋊C₈	—	122	1	no	—
199	C₄⋊C₈	—	122	1	no	—
200	C₄⋊C₈	—	122	1	no	—
201	C₄⋊C₈	C₄×D₆₁	1	1	yes	characteristic
202	#97	—	122	1	no	—
203	#97	—	122	1	no	—
204	C₄×C₈	—	2	1	no	—
205	C₄×C₈	—	61	1	no	—
206	C₄×C₈	—	61	1	no	—
207	C₄×C₈	—	61	1	no	—
208	C₄×C₈	—	122	1	no	—
209	C₄×C₈	C₂²×D₆₁	1	1	yes	characteristic
210	C₂×Q₈	—	61	1	no	—
211	C₂×Q₈	—	122	1	no	—
212	C₂×Q₈	C₈×D₆₁	1	1	yes	characteristic
213	C₂²×C₄	—	61	1	no	—
214	C₂²×C₄	—	61	1	no	—
215	C₂²×C₄	—	61	1	no	—
216	M₁₆	—	244	2	no	—
217	C₂×C₈	—	2	1	no	—
218	C₂×C₈	—	61	1	no	—
219	C₂×C₈	—	61	2	no	—
220	C₂×C₈	—	61	2	no	—
221	C₂×C₈	—	61	1	no	—
222	C₂×C₈	—	122	2	no	—
223	C₂×C₈	—	122	1	no	—
224	C₂×C₈	D₄×D₆₁	1	1	yes	characteristic
225	C₂×C₈	D₄×D₆₁	1	1	yes	characteristic
226	C₄⋊C₄	—	2	1	no	—
227	C₄⋊C₄	—	61	1	no	—
228	C₄⋊C₄	—	122	1	no	—
229	C₄⋊C₄	—	122	1	no	—
230	C₄⋊C₄	—	122	1	no	—
231	C₄⋊C₄	—	122	1	no	—
232	C₄⋊C₄	—	122	1	no	—
233	C₄⋊C₄	—	122	1	no	—
234	C₄⋊C₄	C₈×D₆₁	1	1	yes	characteristic
235	C₂²⋊C₄	—	122	1	no	—
236	C₂²⋊C₄	—	122	1	no	—
237	C₄²	—	2	1	no	—
238	C₄²	—	61	1	no	—
239	C₄²	—	61	1	no	—
240	C₄²	—	61	1	no	—
241	C₄²	—	122	1	no	—
242	C₄²	C₂×C₄×D₆₁	1	1	yes	characteristic
243	C₁₆	—	2	2	no	—
244	C₁₆	—	4	1	no	—
245	C₁₆	—	122	2	no	—
246	C₁₆	—	244	1	no	—
247	C₂³	—	61	1	no	—
248	Q₈	—	2	1	no	—
249	Q₈	—	2	1	no	—
250	Q₈	—	122	1	no	—
251	Q₈	—	122	1	no	—
252	Q₈	—	244	2	no	—
253	C₂×C₄	—	2	1	no	—
254	C₂×C₄	—	2	1	no	—
255	C₂×C₄	—	61	1	no	—
256	C₂×C₄	—	61	1	no	—
257	C₂×C₄	—	61	2	no	—
258	C₂×C₄	—	61	2	no	—
259	C₂×C₄	—	61	1	no	—
260	C₂×C₄	—	122	2	no	—
261	C₂×C₄	—	122	1	no	—
262	C₂×C₄	—	122	1	no	—
263	C₂×C₄	—	122	1	no	—
264	C₂×C₄	—	122	1	no	—
265	C₂×C₄	C₂²⋊C₄×D₆₁	1	1	yes	characteristic
266	C₂×C₄	C₂×C₈×D₆₁	1	1	yes	characteristic
267	C₂×C₄	M₁₆×D₆₁	1	1	yes	characteristic
268	C₈	—	2	1	no	—
269	C₈	—	4	1	no	—
270	C₈	—	61	2	no	—
271	C₈	—	122	1	no	—
272	C₈	—	244	1	no	—
273	C₈	SD₁₆×D₆₁	1	1	yes	characteristic
274	C₈	Q₁₆×D₆₁	1	1	yes	characteristic
275	C₂²	—	61	2	no	—
276	C₂²	—	122	1	no	—
277	C₂²	—	122	1	no	—
278	C₂²	C₂²⋊C₈×D₆₁	1	1	yes	characteristic
279	C₄	—	2	1	no	—
280	C₄	—	4	1	no	—
281	C₄	—	4	1	no	—
282	C₄	—	61	2	no	—
283	C₄	—	61	2	no	—
284	C₄	—	122	1	no	—
285	C₄	—	244	1	no	—
286	C₄	—	244	1	no	—
287	C₄	#103×D₆₁	1	1	yes	characteristic
288	C₄	#104×D₆₁	1	1	yes	characteristic
289	C₄	#16840178	1	1	yes	characteristic
290	C₄	#16840180	1	1	yes	characteristic
291	C₂	—	122	2	no	—
292	C₂	Q₈⋊C₈×D₆₁	1	1	yes	characteristic
293	C₂	#21447431	1	1	yes	characteristic
294	C₂	#350×D₆₁	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	D₄×D₆₁
commutator	C₂×C₄×C₆₁	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₆₁
Fitting	C₆₁×Q₈⋊C₁₆	C₂
socle	C₂²×C₆₁	C₂×C₂²⋊C₈
prime radical	C₂²×D₆₁	C₂²⋊C₈
cosocle	C₄×C₈×C₆₁	C₂³
hypercenter	Q₈⋊C₁₆	D₆₁
hypocenter	C₆₁	#11608
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₈ ◁ C₄×C₈ ◁ Q₈⋊C₁₆

Complements

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

N	H ≅ G/N	type
4. #21443799	C₂	semidirect
5. Q₈⋊#11758185	C₂	semidirect
6. #21443797	C₂	semidirect
7. C₆₁×Q₈⋊C₁₆	C₂	semidirect
150. Q₈⋊C₁₆	D₆₁	semidirect
174. C₆₁	#11608	semidirect

Maximal subgroups

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

subgroup	index
#21454748	2
#21447573	2
C₄×C₁₆×D₆₁	2
#21443799	2
Q₈⋊#11758185	2
#21443797	2
C₆₁×Q₈⋊C₁₆	2
#11608	61