Subgroups — #29573122

Order 15616.

418 conjugacy classes of subgroups.

The subgroup lattice has 258 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21445947	C₂	1	1	yes	maximal, characteristic
2	#21445824	C₂	1	2	yes	maximal
3	C₂×#16839515	C₂	1	1	yes	maximal, characteristic
4	#21443703	C₂	1	2	yes	maximal
5	C₂×C₆₁×#446	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊#92418	C₂²	1	2	yes	—
7	C₂×#14519701	—	2	2	no	—
8	#16839632	C₂²	1	2	yes	—
9	#16839525	—	2	1	no	—
10	#16839525	—	2	1	no	—
11	#16839524	—	2	2	no	—
12	#16839515	—	2	2	no	—
13	C₂×C₄×Q₈×C₆₁	C₂²	1	1	yes	characteristic
14	C₆₁×#446	—	2	2	no	—
15	C₆₁×#446	C₄	1	4	yes	complemented
16	C₂×C₄⋊C₈×C₆₁	—	2	1	no	—
17	C₂×C₄⋊C₈×C₆₁	C₂²	1	1	yes	characteristic
18	C₂×#97×C₆₁	—	2	1	no	—
19	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
20	C₂²×#92418	—	2	2	no	—
21	C₂×C₄⋊Dic₆₁	—	2	2	no	—
22	C₂×C₄⋊Dic₆₁	—	2	2	no	—
23	C₂×C₄⋊Dic₆₁	—	2	2	no	—
24	C₄⋊#92418	—	2	2	no	—
25	C₄⋊#92418	—	2	2	no	—
26	#14519701	—	4	4	no	—
27	C₂²×Q₈×C₆₁	D₄	1	1	yes	characteristic
28	C₂²×C₈×C₆₁	—	2	1	no	—
29	C₂²×C₈×C₆₁	D₄	1	1	yes	characteristic
30	C₂²×C₈×C₆₁	D₄	1	1	yes	characteristic
31	C₄×Q₈×C₆₁	—	2	1	no	—
32	C₄×Q₈×C₆₁	—	2	1	no	—
33	C₄×Q₈×C₆₁	C₂×C₄	1	2	yes	—
34	C₄×Q₈×C₆₁	Q₈	1	2	yes	—
35	C₂×C₄⋊C₄×C₆₁	—	2	1	no	—
36	C₂×C₄⋊C₄×C₆₁	D₄	1	1	yes	characteristic
37	C₂×C₄²×C₆₁	—	2	1	no	—
38	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
39	C₄⋊C₈×C₆₁	—	2	4	no	—
40	C₄⋊C₈×C₆₁	—	2	1	no	—
41	C₄⋊C₈×C₆₁	C₂×C₄	1	2	yes	—
42	#97×C₆₁	—	2	4	no	—
43	C₄×C₈×C₆₁	C₂×C₄	1	2	yes	—
44	C₄×C₈×C₆₁	D₄	1	2	yes	—
45	C₂×Q₈×C₆₁	—	2	1	no	—
46	C₂×Q₈×C₆₁	—	2	2	no	—
47	C₂×Q₈×C₆₁	—	2	2	no	—
48	C₂×Q₈×C₆₁	C₄⋊C₄	1	2	yes	—
49	C₂²×C₄×C₆₁	—	2	1	no	—
50	C₂²×C₄×C₆₁	—	2	1	no	—
51	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
52	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
53	C₂²×C₄×C₆₁	#41	1	1	yes	characteristic
54	C₂²×Dic₆₁	—	4	2	no	—
55	C₂×C₈×C₆₁	—	2	2	no	—
56	C₂×C₈×C₆₁	—	2	2	no	—
57	C₂×C₈×C₆₁	—	2	2	no	—
58	C₂×C₈×C₆₁	—	4	2	no	—
59	C₂×C₈×C₆₁	C₂²⋊C₄	1	2	yes	—
60	C₂×C₈×C₆₁	C₂²⋊C₄	1	2	yes	—
61	C₄⋊C₄×C₆₁	—	2	2	no	—
62	C₄⋊C₄×C₆₁	—	2	1	no	—
63	C₄⋊C₄×C₆₁	—	2	2	no	—
64	C₄⋊C₄×C₆₁	C₄⋊C₄	1	2	yes	—
65	C₄²×C₆₁	—	2	2	no	—
66	C₄²×C₆₁	—	2	2	no	—
67	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
68	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
69	C₄²×C₆₁	C₂×Q₈	1	1	yes	characteristic
70	C₄²×C₆₁	#41	1	1	yes	characteristic
71	C₄⋊Dic₆₁	—	4	2	no	—
72	C₄⋊Dic₆₁	—	4	2	no	—
73	C₄⋊Dic₆₁	—	4	2	no	—
74	C₄⋊Dic₆₁	—	4	2	no	—
75	C₄⋊Dic₆₁	—	8	2	no	—
76	C₂×#92418	—	4	2	no	—
77	C₂×#92418	—	4	2	no	—
78	C₂×#92418	—	4	2	no	—
79	C₂³×C₆₁	#121	1	1	yes	characteristic
80	Q₈×C₆₁	—	2	4	no	—
81	Q₈×C₆₁	—	4	1	no	—
82	Q₈×C₆₁	—	4	1	no	—
83	C₂×C₄×C₆₁	—	2	1	no	—
84	C₂×C₄×C₆₁	—	2	1	no	—
85	C₂×C₄×C₆₁	—	2	2	no	—
86	C₂×C₄×C₆₁	—	2	2	no	—
87	C₂×C₄×C₆₁	—	4	2	no	—
88	C₂×C₄×C₆₁	—	4	1	no	—
89	C₂×C₄×C₆₁	—	4	1	no	—
90	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
91	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
92	C₂×C₄×C₆₁	#117	1	1	yes	characteristic
93	C₂×C₄×C₆₁	#121	1	1	yes	characteristic
94	C₂×C₄×C₆₁	C₂²⋊Q₈	1	1	yes	characteristic
95	C₂×C₄×C₆₁	C₂²⋊Q₈	1	1	yes	characteristic
96	C₂×C₄×C₆₁	#135	1	2	yes	—
97	C₂×C₄×C₆₁	AGL(1,ℤ/8)	1	2	yes	—
98	C₂×C₄×C₆₁	#137	1	2	yes	—
99	C₂×C₄×C₆₁	#137	1	2	yes	—
100	C₂×Dic₆₁	—	8	2	no	—
101	C₂×Dic₆₁	—	8	2	no	—
102	C₂×Dic₆₁	—	8	2	no	—
103	C₈×C₆₁	—	2	2	no	—
104	C₈×C₆₁	—	2	2	no	—
105	C₈×C₆₁	—	4	2	no	—
106	#92418	—	8	2	no	—
107	#8655	—	61	1	no	maximal, Sylow
108	C₂²×C₆₁	#380	1	1	yes	characteristic
109	C₂²×C₆₁	#464	1	2	yes	—
110	C₂²×C₆₁	#468	1	2	yes	—
111	C₂²×C₆₁	#472	1	1	yes	characteristic
112	C₂²×C₆₁	#473	1	1	yes	characteristic
113	C₄×C₆₁	—	2	1	no	—
114	C₄×C₆₁	—	2	1	no	—
115	C₄×C₆₁	—	4	2	no	—
116	C₄×C₆₁	—	4	2	no	—
117	C₄×C₆₁	#416	1	1	yes	characteristic
118	C₄×C₆₁	#419	1	1	yes	characteristic
119	C₄×C₆₁	#428	1	2	yes	—
120	C₄×C₆₁	#469	1	1	yes	characteristic
121	C₄×C₆₁	#470	1	1	yes	characteristic
122	C₄×C₆₁	#479	1	2	yes	—
123	Dic₆₁	—	16	2	no	—
124	C₂×#446	D₆₁	1	1	yes	characteristic
125	#1834	—	61	1	no	—
126	#1803	—	61	2	no	—
127	C₂×#335	—	61	1	no	—
128	#1296	—	61	2	no	—
129	C₂×C₆₁	#1698	1	2	yes	—
130	C₂×C₆₁	#1700	1	2	yes	—
131	C₂×C₆₁	#1805	1	1	yes	characteristic
132	C₂×C₆₁	#1809	1	1	yes	characteristic
133	C₂×C₆₁	#1816	1	1	yes	characteristic
134	C₂×C₄×Q₈	C₂×D₆₁	1	1	yes	characteristic
135	#446	—	2	2	no	—
136	#446	Dic₆₁	1	4	yes	complemented
137	C₂×C₄⋊C₈	—	2	1	no	—
138	C₂×C₄⋊C₈	—	61	2	no	—
139	C₂×C₄⋊C₈	C₂×D₆₁	1	1	yes	characteristic
140	C₂×#103	—	122	2	no	—
141	C₂×#97	—	2	1	no	—
142	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
143	#384	—	61	2	no	—
144	#340	—	122	1	no	—
145	#340	—	122	1	no	—
146	#340	—	122	2	no	—
147	#335	—	122	2	no	—
148	C₆₁	#8655	1	1	yes	characteristic, complemented, Sylow
149	C₂²×Q₈	D₂₄₄	1	1	yes	characteristic
150	C₂²×C₈	—	2	1	no	—
151	C₂²×C₈	—	122	2	no	—
152	C₂²×C₈	C₂²⋊D₆₁	1	1	yes	characteristic
153	C₂²×C₈	C₂²⋊D₆₁	1	1	yes	characteristic
154	C₄×Q₈	—	2	1	no	—
155	C₄×Q₈	—	2	1	no	—
156	C₄×Q₈	Dic₁₂₂	1	2	yes	—
157	C₄×Q₈	C₂×Dic₆₁	1	2	yes	—
158	C₂×C₄⋊C₄	—	2	1	no	—
159	C₂×C₄⋊C₄	—	122	2	no	—
160	C₂×C₄⋊C₄	—	122	2	no	—
161	C₂×C₄⋊C₄	—	122	2	no	—
162	C₂×C₄⋊C₄	D₂₄₄	1	1	yes	characteristic
163	C₂×C₄²	—	2	1	no	—
164	C₂×C₄²	C₂²×D₆₁	1	1	yes	characteristic
165	C₄⋊C₈	—	2	4	no	—
166	C₄⋊C₈	—	2	1	no	—
167	C₄⋊C₈	—	122	2	no	—
168	C₄⋊C₈	—	122	2	no	—
169	C₄⋊C₈	C₂×Dic₆₁	1	2	yes	—
170	#103	—	244	4	no	—
171	#97	—	2	4	no	—
172	C₄×C₈	C₂×Dic₆₁	1	2	yes	—
173	C₄×C₈	C₂²⋊D₆₁	1	2	yes	—
174	C₂×Q₈	—	2	1	no	—
175	C₂×Q₈	—	2	2	no	—
176	C₂×Q₈	—	2	2	no	—
177	C₂×Q₈	C₄⋊Dic₆₁	1	2	yes	—
178	C₂²×C₄	—	2	1	no	—
179	C₂²×C₄	—	2	1	no	—
180	C₂²×C₄	—	244	2	no	—
181	C₂²×C₄	C₂×D₂₄₄	1	1	yes	characteristic
182	C₂²×C₄	#11758222	1	1	yes	characteristic
183	C₂²×C₄	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
184	C₂×C₈	—	2	2	no	—
185	C₂×C₈	—	2	2	no	—
186	C₂×C₈	—	2	2	no	—
187	C₂×C₈	—	4	2	no	—
188	C₂×C₈	—	244	2	no	—
189	C₂×C₈	—	244	2	no	—
190	C₂×C₈	—	244	2	no	—
191	C₂×C₈	C₂²⋊Dic₆₁	1	2	yes	—
192	C₂×C₈	C₂²⋊Dic₆₁	1	2	yes	—
193	C₄⋊C₄	—	2	2	no	—
194	C₄⋊C₄	—	2	1	no	—
195	C₄⋊C₄	—	2	2	no	—
196	C₄⋊C₄	—	488	2	no	—
197	C₄⋊C₄	—	244	2	no	—
198	C₄⋊C₄	—	244	2	no	—
199	C₄⋊C₄	—	244	2	no	—
200	C₄⋊C₄	—	244	2	no	—
201	C₄⋊C₄	C₄⋊Dic₆₁	1	2	yes	—
202	C₄²	—	2	2	no	—
203	C₄²	—	2	2	no	—
204	C₄²	C₂×Dic₁₂₂	1	1	yes	characteristic
205	C₄²	#11758222	1	1	yes	characteristic
206	C₄²	C₂²×Dic₆₁	1	1	yes	characteristic
207	C₄²	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
208	C₂³	#14519751	1	1	yes	characteristic
209	Q₈	—	2	4	no	—
210	Q₈	—	4	1	no	—
211	Q₈	—	4	1	no	—
212	C₂×C₄	—	2	1	no	—
213	C₂×C₄	—	2	1	no	—
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	—	2	2	no	—
216	C₂×C₄	—	4	2	no	—
217	C₂×C₄	—	4	1	no	—
218	C₂×C₄	—	4	1	no	—
219	C₂×C₄	—	488	2	no	—
220	C₂×C₄	—	488	2	no	—
221	C₂×C₄	—	488	2	no	—
222	C₂×C₄	C₂×C₂²⋊Dic₆₁	1	1	yes	characteristic
223	C₂×C₄	C₂×C₄⋊Dic₆₁	1	1	yes	characteristic
224	C₂×C₄	#14519736	1	1	yes	characteristic
225	C₂×C₄	#14519751	1	1	yes	characteristic
226	C₂×C₄	#14519756	1	1	yes	characteristic
227	C₂×C₄	#14519756	1	1	yes	characteristic
228	C₂×C₄	#14519797	1	2	yes	—
229	C₂×C₄	#14519805	1	2	yes	—
230	C₂×C₄	#14519808	1	2	yes	—
231	C₂×C₄	#14519810	1	2	yes	—
232	C₈	—	2	2	no	—
233	C₈	—	2	2	no	—
234	C₈	—	4	2	no	—
235	C₈	—	488	2	no	—
236	C₂²	#16839622	1	1	yes	characteristic
237	C₂²	#16839984	1	2	yes	—
238	C₂²	#16840008	1	2	yes	—
239	C₂²	#16840032	1	1	yes	characteristic
240	C₂²	#16840037	1	1	yes	characteristic
241	C₄	—	2	1	no	—
242	C₄	—	2	1	no	—
243	C₄	—	4	2	no	—
244	C₄	—	4	2	no	—
245	C₄	—	976	2	no	—
246	C₄	#16839758	1	1	yes	characteristic
247	C₄	#16839771	1	1	yes	characteristic
248	C₄	#16839804	1	2	yes	—
249	C₄	#16840015	1	1	yes	characteristic
250	C₄	#16840020	1	1	yes	characteristic
251	C₄	#16840069	1	2	yes	—
252	C₂	#21445359	1	2	yes	—
253	C₂	#21445373	1	2	yes	—
254	C₂	#21445834	1	1	yes	characteristic
255	C₂	#21445852	1	1	yes	characteristic
256	C₂	#21445883	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#14519751
commutator	C₄²×C₆₁	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₁
Fitting	C₂×C₆₁×#446	C₂
socle, prime radical	C₂³×C₆₁	#121
cosocle	C₂×C₄²×C₆₁	C₂³
hypercenter	C₂×#446	D₆₁
hypocenter	C₆₁	#8655
LCS	G ▷ C₄²×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#446

Complements

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

N	H ≅ G/N	type
15. C₆₁×#446	C₄	semidirect
136. #446	Dic₆₁	semidirect
148. C₆₁	#8655	semidirect

Maximal subgroups

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

subgroup	index
#21445947	2
#21445824	2
C₂×#16839515	2
#21443703	2
C₂×C₆₁×#446	2
#8655	61