Subgroups — #29688817

Order 15616.

750 conjugacy classes of subgroups.

The subgroup lattice has 262 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#125×Dic₆₁	C₂	1	2	yes	maximal
2	#21446785	C₂	1	2	yes	maximal
3	#21446563	C₂	1	2	yes	maximal
4	#21446558	C₂	1	2	yes	maximal
5	#21445983	C₂	1	1	yes	maximal, characteristic
6	#21445719	C₂	1	2	yes	maximal
7	#21445718	C₂	1	2	yes	maximal
8	C₄×#14519687	C₂	1	1	yes	maximal, characteristic
9	C₂×C₆₁×#499	C₂	1	1	yes	maximal, characteristic
10	C₂×#14519711	C₂²	1	2	yes	—
11	C₂×#14519709	C₂²	1	2	yes	—
12	C₂×#14519687	C₂²	1	1	yes	characteristic
13	C₂×#14519687	C₂²	1	1	yes	characteristic
14	C₂×#14519687	C₂²	1	1	yes	characteristic
15	C₂×#14519687	C₂²	1	1	yes	characteristic
16	C₂×C₄×#92418	C₂²	1	1	yes	characteristic
17	#16839661	C₂²	1	4	yes	—
18	#16839648	—	2	4	no	—
19	#16839628	C₂²	1	2	yes	—
20	C₄⋊C₄×Dic₆₁	—	2	4	no	—
21	C₄⋊C₄×Dic₆₁	C₂²	1	4	yes	—
22	C₄²×Dic₆₁	C₂²	1	1	yes	characteristic
23	#16839524	C₂²	1	4	yes	—
24	#16839524	C₂²	1	4	yes	—
25	C₂×#125×C₆₁	C₂²	1	2	yes	—
26	C₆₁×#499	C₄	1	8	yes	complemented
27	C₂×#107×C₆₁	C₂²	1	2	yes	—
28	C₂×#106×C₆₁	C₂²	1	2	yes	—
29	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
30	C₂²×#92418	C₂³	1	1	yes	characteristic
31	C₂²×#92418	C₂³	1	1	yes	characteristic
32	C₂×C₄⋊Dic₆₁	—	2	4	no	—
33	C₂×C₄⋊Dic₆₁	C₂³	1	4	yes	—
34	C₂×C₄×Dic₆₁	—	2	4	no	—
35	C₂×C₄×Dic₆₁	D₄	1	4	yes	—
36	C₂×C₄×Dic₆₁	C₂³	1	1	yes	characteristic
37	C₂×C₄×Dic₆₁	C₂³	1	1	yes	characteristic
38	#14519711	—	2	4	no	—
39	#14519709	—	2	4	no	—
40	#14519687	—	2	2	no	—
41	#14519687	—	2	2	no	—
42	#14519687	Q₈	1	4	yes	—
43	#14519687	Q₈	1	4	yes	—
44	C₄×#92418	—	2	2	no	—
45	#14519684	—	2	4	no	—
46	#14519684	—	2	4	no	—
47	#14519684	—	2	4	no	—
48	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
49	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
50	#125×C₆₁	—	2	4	no	—
51	#125×C₆₁	C₂×C₄	1	8	yes	—
52	C₂×C₄⋊C₄×C₆₁	—	2	4	no	—
53	C₂×C₄⋊C₄×C₆₁	C₂³	1	4	yes	—
54	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
55	#107×C₆₁	C₂×C₄	1	8	yes	—
56	#106×C₆₁	C₂×C₄	1	8	yes	—
57	C₄×C₈×C₆₁	C₂×C₄	1	4	yes	—
58	C₂²×C₄×C₆₁	—	2	4	no	—
59	C₂²×C₄×C₆₁	C₂×D₄	1	2	yes	—
60	C₂²×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
61	C₂²×Dic₆₁	—	2	4	no	—
62	C₂²×Dic₆₁	C₂×D₄	1	2	yes	—
63	C₂²×Dic₆₁	C₂×D₄	1	2	yes	—
64	C₂×C₈×C₆₁	C₂²×C₄	1	2	yes	—
65	C₂×C₈×C₆₁	C₂²×C₄	1	2	yes	—
66	C₂×C₈×C₆₁	C₂×Q₈	1	2	yes	—
67	C₂×C₈×C₆₁	C₂×Q₈	1	2	yes	—
68	C₂×C₈×C₆₁	#41	1	2	yes	—
69	C₂×C₈×C₆₁	#41	1	2	yes	—
70	C₄⋊C₄×C₆₁	—	2	16	no	—
71	C₄⋊C₄×C₆₁	—	2	4	no	—
72	C₄⋊C₄×C₆₁	C₂²×C₄	1	8	yes	—
73	C₄²×C₆₁	C₂²×C₄	1	2	yes	—
74	C₄²×C₆₁	#41	1	2	yes	—
75	C₄⋊Dic₆₁	—	2	4	no	—
76	C₄⋊Dic₆₁	—	2	4	no	—
77	C₄⋊Dic₆₁	—	4	8	no	—
78	C₄×Dic₆₁	—	2	1	no	—
79	C₄×Dic₆₁	—	2	1	no	—
80	C₄×Dic₆₁	—	4	4	no	—
81	C₄×Dic₆₁	—	4	8	no	—
82	C₄×Dic₆₁	C₂×Q₈	1	2	yes	—
83	C₄×Dic₆₁	C₂×Q₈	1	2	yes	—
84	C₂×#92418	—	2	1	no	—
85	C₂×#92418	—	2	1	no	—
86	C₂×#92418	—	2	1	no	—
87	C₂×#92418	—	2	1	no	—
88	C₂×#92418	C₂×Q₈	1	2	yes	—
89	C₂×#92418	C₂×Q₈	1	2	yes	—
90	C₂³×C₆₁	C₂²×D₄	1	1	yes	characteristic
91	C₂×C₄×C₆₁	—	2	4	no	—
92	C₂×C₄×C₆₁	—	2	8	no	—
93	C₂×C₄×C₆₁	—	4	4	no	—
94	C₂×C₄×C₆₁	—	4	4	no	—
95	C₂×C₄×C₆₁	C₄×D₄	1	4	yes	—
96	C₂×C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
97	C₂×C₄×C₆₁	C₂²×Q₈	1	1	yes	characteristic
98	C₂×C₄×C₆₁	C₂²×Q₈	1	1	yes	characteristic
99	C₂×C₄×C₆₁	C₂×#41	1	1	yes	characteristic
100	C₂×C₄×C₆₁	C₂×#41	1	1	yes	characteristic
101	C₂×C₄×C₆₁	C₂×#41	1	1	yes	characteristic
102	C₂×Dic₆₁	—	2	4	no	—
103	C₂×Dic₆₁	—	2	2	no	—
104	C₂×Dic₆₁	—	2	4	no	—
105	C₂×Dic₆₁	—	4	4	no	—
106	C₂×Dic₆₁	—	4	4	no	—
107	C₂×Dic₆₁	—	4	4	no	—
108	C₂×Dic₆₁	C₄⋊Q₈	1	4	yes	—
109	C₈×C₆₁	C₄×Q₈	1	4	yes	—
110	C₈×C₆₁	C₄×Q₈	1	4	yes	—
111	#92418	—	2	2	no	—
112	#92418	—	2	2	no	—
113	#21728	—	61	1	no	maximal, Sylow
114	C₂²×C₆₁	C₂×C₄×D₄	1	1	yes	characteristic
115	C₂²×C₆₁	C₂×C₄⋊Q₈	1	1	yes	characteristic
116	C₂²×C₆₁	#532	1	1	yes	characteristic
117	C₂²×C₆₁	#575	1	4	yes	—
118	C₄×C₆₁	—	2	4	no	—
119	C₄×C₆₁	—	4	8	no	—
120	C₄×C₆₁	C₂×C₄×Q₈	1	1	yes	characteristic
121	C₄×C₆₁	C₂×C₄×Q₈	1	1	yes	characteristic
122	C₄×C₆₁	#533	1	1	yes	characteristic
123	C₄×C₆₁	#533	1	1	yes	characteristic
124	Dic₆₁	—	2	4	no	—
125	Dic₆₁	—	4	2	no	—
126	Dic₆₁	—	8	4	no	—
127	C₂×#499	D₆₁	1	1	yes	characteristic
128	C₄×#125	—	61	2	no	—
129	#1991	—	61	2	no	—
130	#1944	—	61	2	no	—
131	#1943	—	61	2	no	—
132	#1846	—	61	1	no	—
133	#1776	—	61	2	no	—
134	#1776	—	61	2	no	—
135	C₄×#97	—	61	1	no	—
136	C₂×C₆₁	C₄×C₄⋊Q₈	1	1	yes	characteristic
137	C₂×C₆₁	#2943	1	2	yes	—
138	C₂×C₆₁	#3166	1	2	yes	—
139	C₂×C₆₁	#3183	1	2	yes	—
140	C₂×#125	C₂×D₆₁	1	2	yes	—
141	#499	Dic₆₁	1	8	yes	complemented
142	C₂×#107	—	61	2	no	—
143	C₂×#107	C₂×D₆₁	1	2	yes	—
144	C₂×#106	—	61	2	no	—
145	C₂×#106	C₂×D₆₁	1	2	yes	—
146	C₂×#97	—	61	1	no	—
147	C₂×#97	—	61	1	no	—
148	C₂×#97	—	61	1	no	—
149	C₂×#97	—	61	1	no	—
150	C₂×C₄×C₈	—	61	1	no	—
151	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
152	#389	—	61	4	no	—
153	#387	—	122	4	no	—
154	#382	—	61	2	no	—
155	C₄×C₄⋊C₄	—	61	4	no	—
156	C₄×C₄⋊C₄	—	122	4	no	—
157	C₄³	—	61	1	no	—
158	#340	—	61	4	no	—
159	#340	—	61	4	no	—
160	C₆₁	#21728	1	1	yes	characteristic, complemented, Sylow
161	C₂²×C₈	—	61	1	no	—
162	C₂²×C₈	—	61	1	no	—
163	C₂²×C₈	C₂²×D₆₁	1	1	yes	characteristic
164	C₂²×C₈	C₂²×D₆₁	1	1	yes	characteristic
165	#125	—	2	4	no	—
166	#125	C₂×Dic₆₁	1	8	yes	—
167	C₂×C₄⋊C₄	—	2	4	no	—
168	C₂×C₄⋊C₄	—	61	4	no	—
169	C₂×C₄⋊C₄	—	122	4	no	—
170	C₂×C₄⋊C₄	C₂²×D₆₁	1	4	yes	—
171	C₂×C₄²	—	61	1	no	—
172	C₂×C₄²	—	61	1	no	—
173	C₂×C₄²	—	61	4	no	—
174	C₂×C₄²	—	122	4	no	—
175	C₂×C₄²	C₂²×D₆₁	1	1	yes	characteristic
176	#107	—	122	4	no	—
177	#107	C₂×Dic₆₁	1	8	yes	—
178	#106	—	122	4	no	—
179	#106	C₂×Dic₆₁	1	8	yes	—
180	#97	—	61	4	no	—
181	#97	—	61	4	no	—
182	#97	—	122	2	no	—
183	#97	—	122	2	no	—
184	C₄×C₈	—	122	2	no	—
185	C₄×C₈	C₂×Dic₆₁	1	4	yes	—
186	#95	—	122	4	no	—
187	#95	—	122	4	no	—
188	#95	—	122	4	no	—
189	C₂²×C₄	—	2	4	no	—
190	C₂²×C₄	—	61	2	no	—
191	C₂²×C₄	—	61	2	no	—
192	C₂²×C₄	—	122	4	no	—
193	C₂²×C₄	D₄×D₆₁	1	2	yes	—
194	C₂²×C₄	C₂³×D₆₁	1	1	yes	characteristic
195	C₂×C₈	—	61	2	no	—
196	C₂×C₈	—	61	2	no	—
197	C₂×C₈	—	122	1	no	—
198	C₂×C₈	—	122	1	no	—
199	C₂×C₈	—	122	1	no	—
200	C₂×C₈	—	122	1	no	—
201	C₂×C₈	Q₈×D₆₁	1	2	yes	—
202	C₂×C₈	Q₈×D₆₁	1	2	yes	—
203	C₂×C₈	#11758226	1	2	yes	—
204	C₂×C₈	#11758226	1	2	yes	—
205	C₂×C₈	C₂²×Dic₆₁	1	2	yes	—
206	C₂×C₈	C₂²×Dic₆₁	1	2	yes	—
207	C₄⋊C₄	—	2	4	no	—
208	C₄⋊C₄	—	2	16	no	—
209	C₄⋊C₄	—	122	4	no	—
210	C₄⋊C₄	—	122	4	no	—
211	C₄⋊C₄	—	244	8	no	—
212	C₄⋊C₄	C₂²×Dic₆₁	1	8	yes	—
213	C₄²	—	61	2	no	—
214	C₄²	—	61	2	no	—
215	C₄²	—	122	1	no	—
216	C₄²	—	122	1	no	—
217	C₄²	—	244	4	no	—
218	C₄²	—	244	8	no	—
219	C₄²	#11758224	1	2	yes	—
220	C₄²	C₂²×Dic₆₁	1	2	yes	—
221	C₂³	C₂×D₄×D₆₁	1	1	yes	characteristic
222	C₂×C₄	—	2	4	no	—
223	C₂×C₄	—	2	8	no	—
224	C₂×C₄	—	4	4	no	—
225	C₂×C₄	—	4	4	no	—
226	C₂×C₄	—	61	4	no	—
227	C₂×C₄	—	122	4	no	—
228	C₂×C₄	—	122	4	no	—
229	C₂×C₄	—	122	2	no	—
230	C₂×C₄	—	244	4	no	—
231	C₂×C₄	—	244	4	no	—
232	C₂×C₄	—	244	4	no	—
233	C₂×C₄	D₄×Dic₆₁	1	4	yes	—
234	C₂×C₄	C₂³×Dic₆₁	1	1	yes	characteristic
235	C₂×C₄	C₂×Q₈×D₆₁	1	1	yes	characteristic
236	C₂×C₄	C₂×Q₈×D₆₁	1	1	yes	characteristic
237	C₂×C₄	C₂×#11758224	1	1	yes	characteristic
238	C₂×C₄	C₂×#11758226	1	1	yes	characteristic
239	C₂×C₄	C₂×#11758226	1	1	yes	characteristic
240	C₈	—	122	2	no	—
241	C₈	—	122	2	no	—
242	C₈	Q₈×Dic₆₁	1	4	yes	—
243	C₈	Q₈×Dic₆₁	1	4	yes	—
244	C₂²	C₂×D₄×Dic₆₁	1	1	yes	characteristic
245	C₂²	C₄⋊Q₈×D₆₁	1	1	yes	characteristic
246	C₂²	#16840313	1	1	yes	characteristic
247	C₂²	#16840559	1	4	yes	—
248	C₄	—	2	4	no	—
249	C₄	—	4	8	no	—
250	C₄	—	122	4	no	—
251	C₄	—	488	4	no	—
252	C₄	—	244	2	no	—
253	C₄	C₂×Q₈×Dic₆₁	1	1	yes	characteristic
254	C₄	C₂×Q₈×Dic₆₁	1	1	yes	characteristic
255	C₄	#16840317	1	1	yes	characteristic
256	C₄	#16840317	1	1	yes	characteristic
257	C₂	C₄⋊Q₈×Dic₆₁	1	1	yes	characteristic
258	C₂	#21454620	1	2	yes	—
259	C₂	#21456367	1	2	yes	—
260	C₂	#21456550	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₄×D₆₁
commutator	C₂×C₄×C₆₁	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₆₁
Fitting	C₂×C₆₁×#499	C₂
socle, prime radical	C₂³×C₆₁	C₂²×D₄
cosocle	C₂²×C₄×C₆₁	C₂⁴
hypercenter	C₂×#499	D₆₁
hypocenter	C₆₁	#21728
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#499

Complements

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

N	H ≅ G/N	type
26. C₆₁×#499	C₄	semidirect
141. #499	Dic₆₁	semidirect
160. C₆₁	#21728	semidirect

Maximal subgroups

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

subgroup	index
#125×Dic₆₁	2
#21446785	2
#21446563	2
#21446558	2
#21445983	2
#21445719	2
#21445718	2
C₄×#14519687	2
C₂×C₆₁×#499	2
#21728	61