Subgroups — #29578701

Order 15616.

370 conjugacy classes of subgroups.

The subgroup lattice has 250 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21446583	C₂	1	2	yes	maximal
2	#21445979	C₂	1	1	yes	maximal, characteristic
3	#21445686	C₂	1	1	yes	maximal, characteristic
4	#21443703	C₂	1	2	yes	maximal
5	C₂×C₆₁×#334	C₂	1	1	yes	maximal, characteristic
6	C₂×#14519711	—	2	1	no	—
7	C₂×#14519711	—	2	1	no	—
8	C₂×#14519709	—	2	2	no	—
9	C₂×C₄⋊#92418	C₂²	1	1	yes	characteristic
10	C₂×#14519687	—	2	1	no	—
11	#16839632	C₂²	1	2	yes	—
12	C₄×C₄⋊Dic₆₁	C₂²	1	1	yes	characteristic
13	#16839525	—	2	2	no	—
14	#16839516	—	2	1	no	—
15	C₂×C₄⋊C₈×C₆₁	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
17	C₆₁×#334	C₄	1	4	yes	complemented
18	C₂²×#92418	—	2	1	no	—
19	C₂×C₄⋊Dic₆₁	—	2	2	no	—
20	C₂×C₄⋊Dic₆₁	—	2	2	no	—
21	C₂×C₄⋊Dic₆₁	—	2	2	no	—
22	C₂×C₄⋊Dic₆₁	D₄	1	1	yes	characteristic
23	C₂×C₄⋊Dic₆₁	D₄	1	1	yes	characteristic
24	C₂×C₄×Dic₆₁	—	2	1	no	—
25	#14519711	—	4	2	no	—
26	#14519711	—	4	2	no	—
27	#14519709	—	4	4	no	—
28	C₄⋊#92418	—	2	1	no	—
29	C₄⋊#92418	Q₈	1	2	yes	—
30	#14519687	—	4	2	no	—
31	#14519684	—	2	1	no	—
32	C₂²×C₈×C₆₁	—	2	2	no	—
33	C₂²×C₈×C₆₁	D₄	1	1	yes	characteristic
34	C₂²×C₈×C₆₁	D₄	1	1	yes	characteristic
35	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
36	C₄⋊C₈×C₆₁	—	2	2	no	—
37	C₄⋊C₈×C₆₁	C₂×C₄	1	4	yes	—
38	C₄×C₈×C₆₁	C₂×C₄	1	2	yes	—
39	C₄×C₈×C₆₁	Q₈	1	2	yes	—
40	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
41	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
42	C₂²×C₄×C₆₁	#41	1	1	yes	characteristic
43	C₂²×Dic₆₁	—	2	1	no	—
44	C₂²×Dic₆₁	—	2	1	no	—
45	C₂²×Dic₆₁	—	4	2	no	—
46	C₂×C₈×C₆₁	—	2	1	no	—
47	C₂×C₈×C₆₁	—	2	1	no	—
48	C₂×C₈×C₆₁	—	2	4	no	—
49	C₂×C₈×C₆₁	—	4	2	no	—
50	C₂×C₈×C₆₁	—	4	2	no	—
51	C₂×C₈×C₆₁	C₄⋊C₄	1	2	yes	—
52	C₂×C₈×C₆₁	C₄⋊C₄	1	2	yes	—
53	C₂×C₈×C₆₁	SD₁₆	1	2	yes	—
54	C₂×C₈×C₆₁	SD₁₆	1	2	yes	—
55	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
56	C₄²×C₆₁	C₂×Q₈	1	1	yes	characteristic
57	C₄²×C₆₁	C₂×Q₈	1	1	yes	characteristic
58	C₄²×C₆₁	#41	1	1	yes	characteristic
59	C₄⋊Dic₆₁	—	2	2	no	—
60	C₄⋊Dic₆₁	—	2	2	no	—
61	C₄⋊Dic₆₁	—	4	2	no	—
62	C₄⋊Dic₆₁	—	4	2	no	—
63	C₄⋊Dic₆₁	—	4	2	no	—
64	C₄⋊Dic₆₁	—	4	2	no	—
65	C₄⋊Dic₆₁	—	8	2	no	—
66	C₄×Dic₆₁	—	4	1	no	—
67	C₄×Dic₆₁	—	4	1	no	—
68	C₂×#92418	—	2	2	no	—
69	C₂×#92418	—	4	1	no	—
70	C₂×#92418	—	4	1	no	—
71	C₂³×C₆₁	#121	1	1	yes	characteristic
72	C₂×C₄×C₆₁	—	2	1	no	—
73	C₂×C₄×C₆₁	—	2	1	no	—
74	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
75	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
76	C₂×C₄×C₆₁	C₄×Q₈	1	1	yes	characteristic
77	C₂×C₄×C₆₁	C₂²⋊Q₈	1	1	yes	characteristic
78	C₂×C₄×C₆₁	#125	1	1	yes	characteristic
79	C₂×C₄×C₆₁	C₄⋊Q₈	1	1	yes	characteristic
80	C₂×C₄×C₆₁	C₂×SD₁₆	1	1	yes	characteristic
81	C₂×C₄×C₆₁	C₂×SD₁₆	1	1	yes	characteristic
82	C₂×C₄×C₆₁	AGL(1,ℤ/8)	1	1	yes	characteristic
83	C₂×C₄×C₆₁	AGL(1,ℤ/8)	1	2	yes	—
84	C₂×C₄×C₆₁	#137	1	1	yes	characteristic
85	C₂×C₄×C₆₁	#137	1	2	yes	—
86	C₂×Dic₆₁	—	4	1	no	—
87	C₂×Dic₆₁	—	4	1	no	—
88	C₂×Dic₆₁	—	4	1	no	—
89	C₂×Dic₆₁	—	4	1	no	—
90	C₂×Dic₆₁	—	4	1	no	—
91	C₂×Dic₆₁	—	4	1	no	—
92	C₂×Dic₆₁	—	8	2	no	—
93	C₂×Dic₆₁	—	8	2	no	—
94	C₂×Dic₆₁	—	8	2	no	—
95	C₈×C₆₁	—	2	2	no	—
96	C₈×C₆₁	—	4	4	no	—
97	C₈×C₆₁	#106	1	4	yes	complemented
98	#92418	—	4	2	no	—
99	#9621	—	61	1	no	maximal, Sylow
100	C₂²×C₆₁	#389	1	1	yes	characteristic
101	C₂²×C₆₁	#460	1	1	yes	characteristic
102	C₂²×C₆₁	#461	1	1	yes	characteristic
103	C₂²×C₆₁	#469	1	1	yes	characteristic
104	C₂²×C₆₁	#470	1	1	yes	characteristic
105	C₂²×C₆₁	#472	1	1	yes	characteristic
106	C₂²×C₆₁	#473	1	1	yes	characteristic
107	C₄×C₆₁	—	2	1	no	—
108	C₄×C₆₁	—	2	1	no	—
109	C₄×C₆₁	C₂×#106	1	1	yes	characteristic
110	C₄×C₆₁	#428	1	1	yes	characteristic
111	C₄×C₆₁	#440	1	2	yes	—
112	C₄×C₆₁	#475	1	1	yes	characteristic
113	C₄×C₆₁	#476	1	1	yes	characteristic
114	C₄×C₆₁	#501	1	2	yes	—
115	Dic₆₁	—	8	1	no	—
116	Dic₆₁	—	8	1	no	—
117	Dic₆₁	—	16	2	no	—
118	#1948	—	61	2	no	—
119	#1845	—	61	1	no	—
120	#1770	—	61	1	no	—
121	C₂×#334	D₆₁	1	1	yes	characteristic
122	#1296	—	61	2	no	—
123	C₂×C₆₁	#1688	1	1	yes	characteristic
124	C₂×C₆₁	#1690	1	1	yes	characteristic
125	C₂×C₆₁	#1693	1	1	yes	characteristic
126	C₂×C₆₁	#1695	1	1	yes	characteristic
127	C₂×C₆₁	#1944	1	1	yes	characteristic
128	C₂×C₆₁	#1948	1	1	yes	characteristic
129	C₂×C₆₁	#1952	1	1	yes	characteristic
130	C₂×#107	—	122	1	no	—
131	C₂×#107	—	122	1	no	—
132	C₂×#106	—	122	2	no	—
133	C₂×C₄⋊C₈	—	61	1	no	—
134	C₂×C₄⋊C₈	C₂×D₆₁	1	2	yes	—
135	C₂×#97	—	122	1	no	—
136	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
137	#384	—	61	2	no	—
138	C₄×C₄⋊C₄	—	61	1	no	—
139	#340	—	122	2	no	—
140	#336	—	122	1	no	—
141	#334	Dic₆₁	1	4	yes	complemented
142	C₆₁	#9621	1	1	yes	characteristic, complemented, Sylow
143	C₂²×C₈	—	2	2	no	—
144	C₂²×C₈	—	122	1	no	—
145	C₂²×C₈	D₂₄₄	1	1	yes	characteristic
146	C₂²×C₈	D₂₄₄	1	1	yes	characteristic
147	C₂×C₄⋊C₄	—	61	1	no	—
148	C₂×C₄⋊C₄	—	61	1	no	—
149	C₂×C₄⋊C₄	—	122	2	no	—
150	C₂×C₄⋊C₄	—	122	2	no	—
151	C₂×C₄⋊C₄	—	122	2	no	—
152	C₂×C₄²	—	122	1	no	—
153	C₂×C₄²	C₂²×D₆₁	1	1	yes	characteristic
154	#107	—	244	2	no	—
155	#107	—	244	2	no	—
156	#106	—	244	4	no	—
157	C₄⋊C₈	—	2	2	no	—
158	C₄⋊C₈	—	61	2	no	—
159	C₄⋊C₈	—	122	1	no	—
160	C₄⋊C₈	C₂×Dic₆₁	1	4	yes	—
161	#97	—	244	2	no	—
162	C₄×C₈	Dic₁₂₂	1	2	yes	—
163	C₄×C₈	C₂×Dic₆₁	1	2	yes	—
164	#95	—	122	1	no	—
165	C₂²×C₄	—	122	1	no	—
166	C₂²×C₄	—	122	1	no	—
167	C₂²×C₄	—	244	2	no	—
168	C₂²×C₄	C₂×D₂₄₄	1	1	yes	characteristic
169	C₂²×C₄	D₄×D₆₁	1	1	yes	characteristic
170	C₂²×C₄	#11758226	1	1	yes	characteristic
171	C₂×C₈	—	2	1	no	—
172	C₂×C₈	—	2	1	no	—
173	C₂×C₈	—	2	4	no	—
174	C₂×C₈	—	4	2	no	—
175	C₂×C₈	—	4	2	no	—
176	C₂×C₈	—	122	2	no	—
177	C₂×C₈	—	244	1	no	—
178	C₂×C₈	—	244	1	no	—
179	C₂×C₈	#11758190	1	2	yes	—
180	C₂×C₈	#11758190	1	2	yes	—
181	C₂×C₈	C₄⋊Dic₆₁	1	2	yes	—
182	C₂×C₈	C₄⋊Dic₆₁	1	2	yes	—
183	C₄⋊C₄	—	122	2	no	—
184	C₄⋊C₄	—	122	2	no	—
185	C₄⋊C₄	—	488	2	no	—
186	C₄⋊C₄	—	244	2	no	—
187	C₄⋊C₄	—	244	2	no	—
188	C₄⋊C₄	—	244	2	no	—
189	C₄⋊C₄	—	244	2	no	—
190	C₄²	—	244	1	no	—
191	C₄²	—	244	1	no	—
192	C₄²	C₂×Dic₁₂₂	1	1	yes	characteristic
193	C₄²	#11758224	1	1	yes	characteristic
194	C₄²	Q₈×D₆₁	1	1	yes	characteristic
195	C₄²	C₂²×Dic₆₁	1	1	yes	characteristic
196	C₂³	#14519752	1	1	yes	characteristic
197	C₂×C₄	—	2	1	no	—
198	C₂×C₄	—	2	1	no	—
199	C₂×C₄	—	488	2	no	—
200	C₂×C₄	—	488	2	no	—
201	C₂×C₄	—	488	2	no	—
202	C₂×C₄	—	244	1	no	—
203	C₂×C₄	—	244	1	no	—
204	C₂×C₄	—	244	1	no	—
205	C₂×C₄	—	244	1	no	—
206	C₂×C₄	—	244	1	no	—
207	C₂×C₄	—	244	1	no	—
208	C₂×C₄	C₂×C₄⋊Dic₆₁	1	1	yes	characteristic
209	C₂×C₄	D₄×Dic₆₁	1	1	yes	characteristic
210	C₂×C₄	Q₈×Dic₆₁	1	1	yes	characteristic
211	C₂×C₄	#14519757	1	1	yes	characteristic
212	C₂×C₄	#14519768	1	1	yes	characteristic
213	C₂×C₄	#14519776	1	1	yes	characteristic
214	C₂×C₄	C₂×#11758190	1	1	yes	characteristic
215	C₂×C₄	C₂×#11758190	1	1	yes	characteristic
216	C₂×C₄	#14519804	1	2	yes	—
217	C₂×C₄	#14519805	1	1	yes	characteristic
218	C₂×C₄	#14519806	1	2	yes	—
219	C₂×C₄	#14519810	1	1	yes	characteristic
220	C₈	—	2	2	no	—
221	C₈	—	4	4	no	—
222	C₈	—	244	2	no	—
223	C₈	#14519709	1	4	yes	complemented
224	C₂²	#16839661	1	1	yes	characteristic
225	C₂²	#16839963	1	1	yes	characteristic
226	C₂²	#16839968	1	1	yes	characteristic
227	C₂²	#16840016	1	1	yes	characteristic
228	C₂²	#16840021	1	1	yes	characteristic
229	C₂²	#16840033	1	1	yes	characteristic
230	C₂²	#16840038	1	1	yes	characteristic
231	C₄	—	2	1	no	—
232	C₄	—	2	1	no	—
233	C₄	—	976	2	no	—
234	C₄	—	488	1	no	—
235	C₄	—	488	1	no	—
236	C₄	C₂×#14519709	1	1	yes	characteristic
237	C₄	#16839804	1	1	yes	characteristic
238	C₄	#16839863	1	2	yes	—
239	C₄	#16840048	1	1	yes	characteristic
240	C₄	#16840053	1	1	yes	characteristic
241	C₄	#16840169	1	2	yes	—
242	C₂	#21445304	1	1	yes	characteristic
243	C₂	#21445314	1	1	yes	characteristic
244	C₂	#21445333	1	1	yes	characteristic
245	C₂	#21445343	1	1	yes	characteristic
246	C₂	#21446563	1	1	yes	characteristic
247	C₂	#21446583	1	1	yes	characteristic
248	C₂	#21446605	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#14519752
commutator	C₄²×C₆₁	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₁
Fitting	C₂×C₆₁×#334	C₂
socle, prime radical	C₂³×C₆₁	#121
cosocle	C₂×C₄²×C₆₁	C₂³
hypercenter	C₂×#334	D₆₁
hypocenter	C₆₁	#9621
LCS	G ▷ C₄²×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#334

Complements

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

N	H ≅ G/N	type
17. C₆₁×#334	C₄	semidirect
97. C₈×C₆₁	#106	semidirect
141. #334	Dic₆₁	semidirect
142. C₆₁	#9621	semidirect
223. C₈	#14519709	semidirect

Maximal subgroups

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

subgroup	index
#21446583	2
#21445979	2
#21445686	2
#21443703	2
C₂×C₆₁×#334	2
#9621	61