Subgroups — #29578667

Order 15616.

382 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	#21446583	C₂	1	2	yes	maximal
2	#21445979	C₂	1	1	yes	maximal, characteristic
3	C₈×C₄⋊Dic₆₁	C₂	1	1	yes	maximal, characteristic
4	#21443703	C₂	1	2	yes	maximal
5	C₂×C₆₁×#335	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×C₈×Dic₆₁	—	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×#335	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	C₈×Dic₆₁	—	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	1	no	—
60	C₄⋊Dic₆₁	—	2	1	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	D₈	1	2	yes	—
67	C₄⋊Dic₆₁	Q₁₆	1	2	yes	—
68	C₄×Dic₆₁	—	4	1	no	—
69	C₄×Dic₆₁	—	4	1	no	—
70	C₂×#92418	—	2	2	no	—
71	C₂×#92418	—	4	1	no	—
72	C₂×#92418	—	4	1	no	—
73	C₂³×C₆₁	#121	1	1	yes	characteristic
74	C₂×C₄×C₆₁	—	2	1	no	—
75	C₂×C₄×C₆₁	—	2	1	no	—
76	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
77	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
78	C₂×C₄×C₆₁	C₄×Q₈	1	1	yes	characteristic
79	C₂×C₄×C₆₁	C₂²⋊Q₈	1	1	yes	characteristic
80	C₂×C₄×C₆₁	#125	1	1	yes	characteristic
81	C₂×C₄×C₆₁	C₄⋊Q₈	1	1	yes	characteristic
82	C₂×C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
83	C₂×C₄×C₆₁	C₂×SD₁₆	1	1	yes	characteristic
84	C₂×C₄×C₆₁	C₂×SD₁₆	1	1	yes	characteristic
85	C₂×C₄×C₆₁	C₂×Q₁₆	1	1	yes	characteristic
86	C₂×C₄×C₆₁	#135	1	1	yes	characteristic
87	C₂×C₄×C₆₁	#135	1	1	yes	characteristic
88	C₂×C₄×C₆₁	AGL(1,ℤ/8)	1	1	yes	characteristic
89	C₂×C₄×C₆₁	#137	1	1	yes	characteristic
90	C₂×Dic₆₁	—	2	2	no	—
91	C₂×Dic₆₁	—	2	2	no	—
92	C₂×Dic₆₁	—	4	1	no	—
93	C₂×Dic₆₁	—	4	1	no	—
94	C₂×Dic₆₁	—	4	1	no	—
95	C₂×Dic₆₁	—	4	1	no	—
96	C₂×Dic₆₁	—	8	2	no	—
97	C₂×Dic₆₁	—	8	2	no	—
98	C₂×Dic₆₁	—	8	2	no	—
99	C₈×C₆₁	—	2	2	no	—
100	C₈×C₆₁	—	4	4	no	—
101	C₈×C₆₁	#106	1	4	yes	complemented
102	#92418	—	4	2	no	—
103	#9615	—	61	1	no	maximal, Sylow
104	C₂²×C₆₁	#389	1	1	yes	characteristic
105	C₂²×C₆₁	#460	1	1	yes	characteristic
106	C₂²×C₆₁	#461	1	1	yes	characteristic
107	C₂²×C₆₁	#466	1	1	yes	characteristic
108	C₂²×C₆₁	#467	1	1	yes	characteristic
109	C₂²×C₆₁	#472	1	1	yes	characteristic
110	C₂²×C₆₁	#473	1	1	yes	characteristic
111	C₄×C₆₁	—	2	1	no	—
112	C₄×C₆₁	—	2	1	no	—
113	C₄×C₆₁	C₂×#106	1	1	yes	characteristic
114	C₄×C₆₁	#428	1	1	yes	characteristic
115	C₄×C₆₁	C₄×D₈	1	1	yes	characteristic
116	C₄×C₆₁	C₄×Q₁₆	1	1	yes	characteristic
117	C₄×C₆₁	#475	1	1	yes	characteristic
118	C₄×C₆₁	#476	1	1	yes	characteristic
119	C₄×C₆₁	#499	1	1	yes	characteristic
120	C₄×C₆₁	#500	1	1	yes	characteristic
121	Dic₆₁	—	4	2	no	—
122	Dic₆₁	—	4	2	no	—
123	Dic₆₁	—	16	2	no	—
124	#1948	—	61	2	no	—
125	#1845	—	61	1	no	—
126	C₈×C₄⋊C₄	—	61	1	no	—
127	C₂×#335	D₆₁	1	1	yes	characteristic
128	#1296	—	61	2	no	—
129	C₂×C₆₁	#1668	1	1	yes	characteristic
130	C₂×C₆₁	#1670	1	1	yes	characteristic
131	C₂×C₆₁	#1673	1	1	yes	characteristic
132	C₂×C₆₁	#1675	1	1	yes	characteristic
133	C₂×C₆₁	#1942	1	1	yes	characteristic
134	C₂×C₆₁	#1948	1	1	yes	characteristic
135	C₂×C₆₁	#1952	1	1	yes	characteristic
136	C₂×#107	—	122	1	no	—
137	C₂×#107	—	122	1	no	—
138	C₂×#106	—	122	2	no	—
139	C₂×C₄⋊C₈	—	61	1	no	—
140	C₂×C₄⋊C₈	C₂×D₆₁	1	2	yes	—
141	C₂×C₄×C₈	—	122	1	no	—
142	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
143	#384	—	61	2	no	—
144	C₄×C₄⋊C₄	—	61	1	no	—
145	#340	—	122	2	no	—
146	#336	—	122	1	no	—
147	#335	Dic₆₁	1	4	yes	complemented
148	C₆₁	#9615	1	1	yes	characteristic, complemented, Sylow
149	C₂²×C₈	—	2	2	no	—
150	C₂²×C₈	—	122	1	no	—
151	C₂²×C₈	D₂₄₄	1	1	yes	characteristic
152	C₂²×C₈	D₂₄₄	1	1	yes	characteristic
153	C₂×C₄⋊C₄	—	61	1	no	—
154	C₂×C₄⋊C₄	—	61	1	no	—
155	C₂×C₄⋊C₄	—	122	2	no	—
156	C₂×C₄⋊C₄	—	122	2	no	—
157	C₂×C₄⋊C₄	—	122	2	no	—
158	C₂×C₄²	—	122	1	no	—
159	C₂×C₄²	C₂²×D₆₁	1	1	yes	characteristic
160	#107	—	244	2	no	—
161	#107	—	244	2	no	—
162	#106	—	244	4	no	—
163	C₄⋊C₈	—	2	2	no	—
164	C₄⋊C₈	—	61	2	no	—
165	C₄⋊C₈	—	122	1	no	—
166	C₄⋊C₈	C₂×Dic₆₁	1	4	yes	—
167	C₄×C₈	—	244	2	no	—
168	C₄×C₈	Dic₁₂₂	1	2	yes	—
169	C₄×C₈	C₂×Dic₆₁	1	2	yes	—
170	#95	—	122	1	no	—
171	C₂²×C₄	—	122	1	no	—
172	C₂²×C₄	—	122	1	no	—
173	C₂²×C₄	—	244	2	no	—
174	C₂²×C₄	C₂×D₂₄₄	1	1	yes	characteristic
175	C₂²×C₄	D₄×D₆₁	1	1	yes	characteristic
176	C₂²×C₄	#11758226	1	1	yes	characteristic
177	C₂×C₈	—	2	1	no	—
178	C₂×C₈	—	2	1	no	—
179	C₂×C₈	—	2	4	no	—
180	C₂×C₈	—	4	2	no	—
181	C₂×C₈	—	4	2	no	—
182	C₂×C₈	—	122	2	no	—
183	C₂×C₈	—	244	1	no	—
184	C₂×C₈	—	244	1	no	—
185	C₂×C₈	#11758190	1	2	yes	—
186	C₂×C₈	#11758190	1	2	yes	—
187	C₂×C₈	C₄⋊Dic₆₁	1	2	yes	—
188	C₂×C₈	C₄⋊Dic₆₁	1	2	yes	—
189	C₄⋊C₄	—	61	2	no	—
190	C₄⋊C₄	—	61	2	no	—
191	C₄⋊C₄	—	122	1	no	—
192	C₄⋊C₄	—	122	1	no	—
193	C₄⋊C₄	—	488	2	no	—
194	C₄⋊C₄	—	244	2	no	—
195	C₄⋊C₄	—	244	2	no	—
196	C₄⋊C₄	—	244	2	no	—
197	C₄⋊C₄	—	244	2	no	—
198	C₄²	—	244	1	no	—
199	C₄²	—	244	1	no	—
200	C₄²	C₂×Dic₁₂₂	1	1	yes	characteristic
201	C₄²	#11758224	1	1	yes	characteristic
202	C₄²	Q₈×D₆₁	1	1	yes	characteristic
203	C₄²	C₂²×Dic₆₁	1	1	yes	characteristic
204	C₂³	#14519752	1	1	yes	characteristic
205	C₂×C₄	—	2	1	no	—
206	C₂×C₄	—	2	1	no	—
207	C₂×C₄	—	122	2	no	—
208	C₂×C₄	—	122	2	no	—
209	C₂×C₄	—	488	2	no	—
210	C₂×C₄	—	488	2	no	—
211	C₂×C₄	—	488	2	no	—
212	C₂×C₄	—	244	1	no	—
213	C₂×C₄	—	244	1	no	—
214	C₂×C₄	—	244	1	no	—
215	C₂×C₄	—	244	1	no	—
216	C₂×C₄	C₂×C₄⋊Dic₆₁	1	1	yes	characteristic
217	C₂×C₄	D₄×Dic₆₁	1	1	yes	characteristic
218	C₂×C₄	Q₈×Dic₆₁	1	1	yes	characteristic
219	C₂×C₄	#14519757	1	1	yes	characteristic
220	C₂×C₄	#14519768	1	1	yes	characteristic
221	C₂×C₄	#14519776	1	1	yes	characteristic
222	C₂×C₄	D₈×D₆₁	1	1	yes	characteristic
223	C₂×C₄	C₂×#11758190	1	1	yes	characteristic
224	C₂×C₄	C₂×#11758190	1	1	yes	characteristic
225	C₂×C₄	Q₁₆×D₆₁	1	1	yes	characteristic
226	C₂×C₄	#14519796	1	1	yes	characteristic
227	C₂×C₄	#14519800	1	1	yes	characteristic
228	C₂×C₄	#14519805	1	1	yes	characteristic
229	C₂×C₄	#14519810	1	1	yes	characteristic
230	C₈	—	2	2	no	—
231	C₈	—	4	4	no	—
232	C₈	—	244	2	no	—
233	C₈	#14519709	1	4	yes	complemented
234	C₂²	#16839661	1	1	yes	characteristic
235	C₂²	#16839963	1	1	yes	characteristic
236	C₂²	#16839968	1	1	yes	characteristic
237	C₂²	#16839999	1	1	yes	characteristic
238	C₂²	#16840004	1	1	yes	characteristic
239	C₂²	#16840033	1	1	yes	characteristic
240	C₂²	#16840038	1	1	yes	characteristic
241	C₄	—	2	1	no	—
242	C₄	—	2	1	no	—
243	C₄	—	976	2	no	—
244	C₄	—	244	2	no	—
245	C₄	—	244	2	no	—
246	C₄	C₂×#14519709	1	1	yes	characteristic
247	C₄	#16839804	1	1	yes	characteristic
248	C₄	D₈×Dic₆₁	1	1	yes	characteristic
249	C₄	Q₁₆×Dic₆₁	1	1	yes	characteristic
250	C₄	#16840048	1	1	yes	characteristic
251	C₄	#16840053	1	1	yes	characteristic
252	C₄	#16840163	1	1	yes	characteristic
253	C₄	#16840166	1	1	yes	characteristic
254	C₂	#21445180	1	1	yes	characteristic
255	C₂	#21445190	1	1	yes	characteristic
256	C₂	#21445209	1	1	yes	characteristic
257	C₂	#21445219	1	1	yes	characteristic
258	C₂	#21446553	1	1	yes	characteristic
259	C₂	#21446583	1	1	yes	characteristic
260	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×#335	C₂
socle, prime radical	C₂³×C₆₁	#121
cosocle	C₂×C₄²×C₆₁	C₂³
hypercenter	C₂×#335	D₆₁
hypocenter	C₆₁	#9615
LCS	G ▷ C₄²×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#335

Complements

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

N	H ≅ G/N	type
17. C₆₁×#335	C₄	semidirect
101. C₈×C₆₁	#106	semidirect
147. #335	Dic₆₁	semidirect
148. C₆₁	#9615	semidirect
233. 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
C₈×C₄⋊Dic₆₁	2
#21443703	2
C₂×C₆₁×#335	2
#9615	61