Subgroups — #29571158

Order 15616.

462 conjugacy classes of subgroups.

The subgroup lattice has 282 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#14519706	C₂	1	1	yes	maximal, characteristic
2	C₄×#14519706	C₂	1	1	yes	maximal, characteristic
3	#21445564	C₂	1	1	yes	maximal, characteristic
4	C₂×#16839483	C₂	1	1	yes	maximal, characteristic
5	C₂×C₈×#92418	C₂	1	1	yes	maximal, characteristic
6	#21443646	C₂	1	1	yes	maximal, characteristic
7	C₆₁×#1273	C₂	1	1	yes	maximal, characteristic
8	C₂×#14519706	C₄	1	2	yes	—
9	C₂×#14519706	C₄	1	2	yes	—
10	C₂×C₄×#92418	C₂²	1	1	yes	characteristic
11	C₂×C₄×#92418	C₂²	1	1	yes	characteristic
12	C₂×C₄×#92418	C₂²	1	1	yes	characteristic
13	C₄²×Dic₆₁	C₂²	1	1	yes	characteristic
14	#16839516	C₄	1	2	yes	—
15	#16839516	C₄	1	2	yes	—
16	#16839516	C₄	1	2	yes	—
17	#16839516	C₄	1	2	yes	—
18	#16839483	—	2	2	no	—
19	C₈×#92418	—	2	2	no	—
20	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
21	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
22	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
23	C₂²×#92418	C₂×C₄	1	2	yes	—
24	C₂²×#92418	C₂×C₄	1	2	yes	—
25	C₂²×#92418	C₂×C₄	1	2	yes	—
26	C₂×C₄×Dic₆₁	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄×Dic₆₁	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄×Dic₆₁	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄×Dic₆₁	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄×Dic₆₁	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄×Dic₆₁	C₂×C₄	1	1	yes	characteristic
32	#14519706	—	2	4	no	—
33	#14519706	C₈	1	8	yes	complemented
34	C₄×#92418	—	2	1	no	—
35	C₄×#92418	—	2	1	no	—
36	C₄×#92418	—	2	1	no	—
37	C₄×#92418	—	2	1	no	—
38	C₄×#92418	D₄	1	2	yes	—
39	C₄×#92418	Q₈	1	2	yes	—
40	C₂²×C₈×C₆₁	C₂×C₄	1	2	yes	—
41	C₂²×C₈×C₆₁	C₂×C₄	1	2	yes	—
42	C₂²×C₈×C₆₁	C₂×C₄	1	2	yes	—
43	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
44	C₄×C₈×C₆₁	—	2	1	no	—
45	C₄×C₈×C₆₁	—	2	1	no	—
46	C₄×C₈×C₆₁	—	2	1	no	—
47	C₄×C₈×C₆₁	—	2	1	no	—
48	C₄×C₈×C₆₁	—	2	1	no	—
49	C₄×C₈×C₆₁	—	2	1	no	—
50	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
51	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
52	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
53	C₂²×Dic₆₁	C₄²	1	1	yes	characteristic
54	C₂²×Dic₆₁	C₄²	1	1	yes	characteristic
55	C₂²×Dic₆₁	C₄²	1	1	yes	characteristic
56	C₂²×Dic₆₁	C₄²	1	1	yes	characteristic
57	C₂×C₈×C₆₁	—	2	2	no	—
58	C₂×C₈×C₆₁	—	2	2	no	—
59	C₂×C₈×C₆₁	—	2	2	no	—
60	C₂×C₈×C₆₁	—	2	2	no	—
61	C₂×C₈×C₆₁	—	2	2	no	—
62	C₂×C₈×C₆₁	—	2	2	no	—
63	C₂×C₈×C₆₁	—	2	2	no	—
64	C₂×C₈×C₆₁	C₂×C₈	1	4	yes	—
65	C₂×C₈×C₆₁	M₁₆	1	4	yes	—
66	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
67	C₄²×C₆₁	C₂×Q₈	1	1	yes	characteristic
68	C₄²×C₆₁	#41	1	1	yes	characteristic
69	C₄²×C₆₁	#41	1	1	yes	characteristic
70	C₄×Dic₆₁	—	2	2	no	—
71	C₄×Dic₆₁	—	2	2	no	—
72	C₄×Dic₆₁	—	2	2	no	—
73	C₄×Dic₆₁	—	2	2	no	—
74	C₄×Dic₆₁	—	2	2	no	—
75	C₄×Dic₆₁	C₂×C₈	1	4	yes	—
76	C₂×#92418	—	2	2	no	—
77	C₂×#92418	—	2	2	no	—
78	C₂×#92418	—	2	2	no	—
79	C₂×#92418	—	2	2	no	—
80	C₂×#92418	—	2	2	no	—
81	C₂×#92418	—	2	2	no	—
82	C₂×#92418	—	2	2	no	—
83	C₂×#92418	C₄⋊C₄	1	4	yes	—
84	C₂×#92418	C₂×C₈	1	4	yes	—
85	C₂³×C₆₁	C₂×C₄²	1	1	yes	characteristic
86	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
87	C₂×C₄×C₆₁	#117	1	1	yes	characteristic
88	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
89	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
90	C₂×C₄×C₆₁	C₄×Q₈	1	1	yes	characteristic
91	C₂×C₄×C₆₁	C₄×Q₈	1	1	yes	characteristic
92	C₂×C₄×C₆₁	C₂²×C₈	1	2	yes	—
93	C₂×C₄×C₆₁	C₂×M₁₆	1	2	yes	—
94	C₂×C₄×C₆₁	#131	1	2	yes	—
95	C₂×C₄×C₆₁	#131	1	2	yes	—
96	C₂×C₄×C₆₁	#131	1	2	yes	—
97	C₂×C₄×C₆₁	#131	1	2	yes	—
98	C₂×Dic₆₁	—	2	1	no	—
99	C₂×Dic₆₁	—	2	1	no	—
100	C₂×Dic₆₁	—	2	1	no	—
101	C₂×Dic₆₁	—	2	1	no	—
102	C₂×Dic₆₁	—	2	1	no	—
103	C₂×Dic₆₁	—	2	1	no	—
104	C₂×Dic₆₁	—	2	1	no	—
105	C₂×Dic₆₁	—	2	1	no	—
106	C₂×Dic₆₁	—	2	1	no	—
107	C₂×Dic₆₁	—	2	1	no	—
108	C₂×Dic₆₁	C₄×C₈	1	2	yes	—
109	C₂×Dic₆₁	C₄×C₈	1	2	yes	—
110	C₈×C₆₁	—	2	4	no	—
111	C₈×C₆₁	—	2	4	no	—
112	C₈×C₆₁	—	4	2	no	—
113	#92418	—	2	4	no	—
114	#92418	—	2	4	no	—
115	#92418	—	4	2	no	—
116	#8264	—	61	1	no	maximal, Sylow
117	C₂²×C₆₁	C₄×C₄⋊C₄	1	1	yes	characteristic
118	C₂²×C₆₁	C₂×C₄×C₈	1	1	yes	characteristic
119	C₂²×C₆₁	C₄×M₁₆	1	1	yes	characteristic
120	C₂²×C₆₁	#405	1	1	yes	characteristic
121	C₂²×C₆₁	#405	1	1	yes	characteristic
122	C₂²×C₆₁	#405	1	1	yes	characteristic
123	C₂²×C₆₁	#405	1	1	yes	characteristic
124	C₄×C₆₁	#424	1	2	yes	—
125	C₄×C₆₁	#433	1	2	yes	—
126	C₄×C₆₁	C₈×D₄	1	2	yes	—
127	C₄×C₆₁	#435	1	2	yes	—
128	C₄×C₆₁	C₈×Q₈	1	2	yes	—
129	C₄×C₆₁	#446	1	2	yes	—
130	Dic₆₁	—	2	2	no	—
131	Dic₆₁	—	2	2	no	—
132	Dic₆₁	—	4	1	no	—
133	Dic₆₁	—	4	1	no	—
134	C₄×C₄⋊C₈	—	61	1	no	—
135	C₄×C₄⋊C₈	—	61	1	no	—
136	#1744	—	61	1	no	—
137	C₂×#322	—	61	1	no	—
138	C₂×C₈²	—	61	1	no	—
139	#1273	—	61	1	no	—
140	#1273	D₆₁	1	1	yes	characteristic
141	C₂×C₆₁	C₈×M₁₆	1	1	yes	characteristic
142	C₂×C₆₁	#1450	1	1	yes	characteristic
143	C₂×C₆₁	#1453	1	1	yes	characteristic
144	C₂×C₆₁	#1454	1	1	yes	characteristic
145	C₂×C₆₁	#1768	1	1	yes	characteristic
146	C₂×C₆₁	C₈×C₄⋊C₄	1	1	yes	characteristic
147	C₂×C₆₁	#1771	1	1	yes	characteristic
148	C₂×C₄⋊C₈	—	61	2	no	—
149	C₂×C₄⋊C₈	—	61	2	no	—
150	C₂×C₄×C₈	—	61	1	no	—
151	C₂×C₄×C₈	—	61	1	no	—
152	C₂×C₄×C₈	—	61	1	no	—
153	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
154	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
155	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
156	C₄³	—	61	1	no	—
157	#336	—	61	2	no	—
158	#336	—	61	2	no	—
159	#336	—	61	2	no	—
160	#336	—	61	2	no	—
161	#322	—	122	2	no	—
162	C₈²	—	122	2	no	—
163	C₆₁	#8264	1	1	yes	characteristic, complemented, Sylow
164	C₂²×C₈	—	61	2	no	—
165	C₂²×C₈	—	61	2	no	—
166	C₂²×C₈	—	61	2	no	—
167	C₂²×C₈	C₄×D₆₁	1	2	yes	—
168	C₂²×C₈	C₄×D₆₁	1	2	yes	—
169	C₂²×C₈	C₄×D₆₁	1	2	yes	—
170	C₂×C₄²	—	61	1	no	—
171	C₂×C₄²	—	61	1	no	—
172	C₂×C₄²	—	61	1	no	—
173	C₂×C₄²	—	61	1	no	—
174	C₂×C₄²	—	61	1	no	—
175	C₂×C₄²	—	61	1	no	—
176	C₂×C₄²	C₂²×D₆₁	1	1	yes	characteristic
177	C₄⋊C₈	—	61	8	no	—
178	C₄⋊C₈	—	122	4	no	—
179	C₄×C₈	—	2	1	no	—
180	C₄×C₈	—	2	1	no	—
181	C₄×C₈	—	2	1	no	—
182	C₄×C₈	—	2	1	no	—
183	C₄×C₈	—	2	1	no	—
184	C₄×C₈	—	2	1	no	—
185	C₄×C₈	—	61	2	no	—
186	C₄×C₈	—	61	2	no	—
187	C₄×C₈	—	122	1	no	—
188	C₄×C₈	—	122	1	no	—
189	C₄×C₈	—	122	1	no	—
190	C₄×C₈	—	122	1	no	—
191	C₂²×C₄	—	61	1	no	—
192	C₂²×C₄	—	61	1	no	—
193	C₂²×C₄	—	61	1	no	—
194	C₂²×C₄	—	61	1	no	—
195	C₂²×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
196	C₂²×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
197	C₂²×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
198	C₂×C₈	—	2	2	no	—
199	C₂×C₈	—	2	2	no	—
200	C₂×C₈	—	2	2	no	—
201	C₂×C₈	—	2	2	no	—
202	C₂×C₈	—	2	2	no	—
203	C₂×C₈	—	2	2	no	—
204	C₂×C₈	—	2	2	no	—
205	C₂×C₈	—	61	4	no	—
206	C₂×C₈	—	61	4	no	—
207	C₂×C₈	—	122	2	no	—
208	C₂×C₈	—	122	2	no	—
209	C₂×C₈	—	122	2	no	—
210	C₂×C₈	—	122	2	no	—
211	C₂×C₈	—	122	2	no	—
212	C₂×C₈	—	122	2	no	—
213	C₂×C₈	—	122	2	no	—
214	C₂×C₈	C₈×D₆₁	1	4	yes	—
215	C₂×C₈	#11758189	1	4	yes	—
216	C₄²	—	61	4	no	—
217	C₄²	—	122	2	no	—
218	C₄²	—	122	2	no	—
219	C₄²	—	122	2	no	—
220	C₄²	—	122	2	no	—
221	C₄²	—	122	2	no	—
222	C₄²	D₄×D₆₁	1	1	yes	characteristic
223	C₄²	#11758224	1	1	yes	characteristic
224	C₄²	#11758224	1	1	yes	characteristic
225	C₄²	Q₈×D₆₁	1	1	yes	characteristic
226	C₂³	C₄²×D₆₁	1	1	yes	characteristic
227	C₂×C₄	—	61	2	no	—
228	C₂×C₄	—	61	2	no	—
229	C₂×C₄	—	122	1	no	—
230	C₂×C₄	—	122	1	no	—
231	C₂×C₄	—	122	1	no	—
232	C₂×C₄	—	122	1	no	—
233	C₂×C₄	—	122	1	no	—
234	C₂×C₄	—	122	1	no	—
235	C₂×C₄	—	122	1	no	—
236	C₂×C₄	—	122	1	no	—
237	C₂×C₄	—	122	1	no	—
238	C₂×C₄	—	122	1	no	—
239	C₂×C₄	C₄⋊C₄×D₆₁	1	1	yes	characteristic
240	C₂×C₄	#14519733	1	1	yes	characteristic
241	C₂×C₄	#14519737	1	1	yes	characteristic
242	C₂×C₄	#14519737	1	1	yes	characteristic
243	C₂×C₄	#14519742	1	1	yes	characteristic
244	C₂×C₄	#14519742	1	1	yes	characteristic
245	C₂×C₄	C₂×C₈×D₆₁	1	2	yes	—
246	C₂×C₄	C₂×#11758189	1	2	yes	—
247	C₂×C₄	#14519783	1	2	yes	—
248	C₂×C₄	#14519785	1	2	yes	—
249	C₂×C₄	#14519785	1	2	yes	—
250	C₂×C₄	#14519785	1	2	yes	—
251	C₈	—	2	4	no	—
252	C₈	—	2	4	no	—
253	C₈	—	4	2	no	—
254	C₈	—	122	4	no	—
255	C₈	—	122	4	no	—
256	C₈	—	244	2	no	—
257	C₂²	#16839613	1	1	yes	characteristic
258	C₂²	C₄×C₈×D₆₁	1	1	yes	characteristic
259	C₂²	C₄×#11758189	1	1	yes	characteristic
260	C₂²	#16839716	1	1	yes	characteristic
261	C₂²	#16839719	1	1	yes	characteristic
262	C₂²	#16839719	1	1	yes	characteristic
263	C₂²	#16839719	1	1	yes	characteristic
264	C₄	—	122	2	no	—
265	C₄	—	122	2	no	—
266	C₄	—	244	1	no	—
267	C₄	—	244	1	no	—
268	C₄	#16839787	1	2	yes	—
269	C₄	#16839820	1	2	yes	—
270	C₄	#16839825	1	2	yes	—
271	C₄	#16839830	1	2	yes	—
272	C₄	#16839886	1	2	yes	—
273	C₄	#16839889	1	2	yes	—
274	C₂	C₈×#11758189	1	1	yes	characteristic
275	C₂	#21444104	1	1	yes	characteristic
276	C₂	#21444117	1	1	yes	characteristic
277	C₂	#21444120	1	1	yes	characteristic
278	C₂	#21445675	1	1	yes	characteristic
279	C₂	#21445680	1	1	yes	characteristic
280	C₂	#21445690	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄²	C₂²×D₆₁
commutator	C₂²×C₆₁	C₂×C₄×C₈
Fitting	C₆₁×#1273	C₂
socle, prime radical	C₂³×C₆₁	C₂×C₄²
cosocle	C₂×C₄²×C₆₁	C₂³
hypercenter	#1273	D₆₁
hypocenter	C₆₁	#8264
LCS	G ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₄² ◁ #1273

Complements

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

N	H ≅ G/N	type
33. #14519706	C₈	semidirect
163. C₆₁	#8264	semidirect

Maximal subgroups

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

subgroup	index
C₄×#14519706	2
#21445564	2
C₂×#16839483	2
C₂×C₈×#92418	2
#21443646	2
C₆₁×#1273	2
#8264	61