Subgroups — #29573658

Order 15616.

310 conjugacy classes of subgroups.

The subgroup lattice has 242 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21445505	C₂	1	1	yes	maximal, characteristic
2	#21444567	C₂	1	1	yes	maximal, characteristic
3	#21443663	C₂	1	2	yes	maximal
4	#21443647	C₂	1	2	yes	maximal
5	C₆₁×#1564	C₂	1	1	yes	maximal, characteristic
6	#16839815	C₂²	1	2	yes	—
7	C₂×#14519688	C₂²	1	1	yes	characteristic
8	C₂×#14519687	C₄	1	1	yes	characteristic
9	C₂×#14519687	C₄	1	1	yes	characteristic
10	C₂×C₈×Dic₆₁	C₄	1	1	yes	characteristic
11	C₂×C₈×Dic₆₁	C₄	1	1	yes	characteristic
12	#16839608	C₂²	1	1	yes	characteristic
13	#16839515	—	2	1	no	—
14	#16839513	—	2	1	no	—
15	C₆₁×#431	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
17	C₆₁×#335	—	2	1	no	—
18	C₆₁×#334	—	2	1	no	—
19	C₂²×#92418	C₂×C₄	1	1	yes	characteristic
20	C₂²×#92418	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄×Dic₆₁	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄×Dic₆₁	C₂×C₄	1	1	yes	characteristic
23	C₄⋊#92418	—	2	2	no	—
24	C₂²⋊#92418	—	2	2	no	—
25	#14519688	—	2	1	no	—
26	#14519688	—	2	1	no	—
27	#14519687	—	2	2	no	—
28	#14519687	—	2	2	no	—
29	C₄×#92418	—	2	2	no	—
30	C₈×Dic₆₁	—	2	2	no	—
31	C₈×Dic₆₁	C₈	1	4	yes	complemented
32	#14519684	—	2	1	no	—
33	#14519684	—	2	1	no	—
34	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
37	C₄⋊C₈×C₆₁	—	2	2	no	—
38	C₂²⋊C₈×C₆₁	—	2	2	no	—
39	C₄×C₈×C₆₁	—	2	2	no	—
40	C₄×C₈×C₆₁	D₄	1	1	yes	characteristic
41	C₄×C₈×C₆₁	D₄	1	1	yes	characteristic
42	C₄×C₈×C₆₁	Q₈	1	1	yes	characteristic
43	C₄×C₈×C₆₁	Q₈	1	1	yes	characteristic
44	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
45	C₂²×C₄×C₆₁	#41	1	1	yes	characteristic
46	C₂²×C₄×C₆₁	#41	1	1	yes	characteristic
47	C₂²×Dic₆₁	—	2	1	no	—
48	C₂²×Dic₆₁	—	2	1	no	—
49	C₂×C₈×C₆₁	—	2	1	no	—
50	C₂×C₈×C₆₁	—	2	1	no	—
51	C₂×C₈×C₆₁	—	4	2	no	—
52	C₂×C₈×C₆₁	C₄⋊C₄	1	1	yes	characteristic
53	C₂×C₈×C₆₁	C₄⋊C₄	1	1	yes	characteristic
54	C₂×C₈×C₆₁	C₄⋊C₄	1	1	yes	characteristic
55	C₂×C₈×C₆₁	C₄⋊C₄	1	1	yes	characteristic
56	C₂×C₈×C₆₁	C₂×C₈	1	2	yes	—
57	C₂×C₈×C₆₁	M₁₆	1	2	yes	—
58	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
59	C₄²×C₆₁	C₂×Q₈	1	1	yes	characteristic
60	C₄²×C₆₁	#41	1	1	yes	characteristic
61	C₄²×C₆₁	#41	1	1	yes	characteristic
62	C₄×Dic₆₁	—	2	1	no	—
63	C₄×Dic₆₁	—	2	1	no	—
64	C₄×Dic₆₁	C₂×C₈	1	2	yes	—
65	C₄×Dic₆₁	M₁₆	1	2	yes	—
66	C₂×#92418	—	2	1	no	—
67	C₂×#92418	—	2	1	no	—
68	C₂×#92418	—	2	1	no	—
69	C₂×#92418	—	2	1	no	—
70	C₂×#92418	—	4	2	no	—
71	C₂×#92418	C₂×C₈	1	2	yes	—
72	C₂×#92418	M₁₆	1	2	yes	—
73	C₂³×C₆₁	#117	1	1	yes	characteristic
74	C₂×C₄×C₆₁	—	4	1	no	—
75	C₂×C₄×C₆₁	—	4	1	no	—
76	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
77	C₂×C₄×C₆₁	#117	1	1	yes	characteristic
78	C₂×C₄×C₆₁	#124	1	1	yes	characteristic
79	C₂×C₄×C₆₁	#124	1	1	yes	characteristic
80	C₂×C₄×C₆₁	#125	1	1	yes	characteristic
81	C₂×C₄×C₆₁	#125	1	1	yes	characteristic
82	C₂×C₄×C₆₁	C₂²×C₈	1	1	yes	characteristic
83	C₂×C₄×C₆₁	C₂×M₁₆	1	1	yes	characteristic
84	C₂×C₄×C₆₁	C₂×M₁₆	1	1	yes	characteristic
85	C₂×C₄×C₆₁	C₂×M₁₆	1	1	yes	characteristic
86	C₂×Dic₆₁	—	2	2	no	—
87	C₂×Dic₆₁	—	2	2	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	C₈×C₆₁	—	2	1	no	—
93	C₈×C₆₁	—	2	1	no	—
94	C₈×C₆₁	—	8	2	no	—
95	C₈×C₆₁	C₄⋊C₈	1	2	yes	—
96	C₈×C₆₁	C₄⋊C₈	1	2	yes	—
97	#92418	—	2	2	no	—
98	#92418	—	2	2	no	—
99	#92418	—	8	2	no	—
100	#8780	—	61	1	no	maximal, Sylow
101	C₂²×C₆₁	—	2	1	no	—
102	C₂²×C₆₁	—	2	1	no	—
103	C₂²×C₆₁	#382	1	1	yes	characteristic
104	C₂²×C₆₁	#431	1	1	yes	characteristic
105	C₂²×C₆₁	#432	1	1	yes	characteristic
106	C₄×C₆₁	—	2	1	no	—
107	C₄×C₆₁	—	2	1	no	—
108	C₄×C₆₁	—	2	1	no	—
109	C₄×C₆₁	—	2	1	no	—
110	C₄×C₆₁	C₂×C₄⋊C₈	1	1	yes	characteristic
111	C₄×C₆₁	#423	1	1	yes	characteristic
112	C₄×C₆₁	#431	1	1	yes	characteristic
113	C₄×C₆₁	#432	1	1	yes	characteristic
114	Dic₆₁	—	4	2	no	—
115	Dic₆₁	—	4	2	no	—
116	#1726	—	61	1	no	—
117	#1564	—	61	1	no	—
118	#1564	D₆₁	1	1	yes	characteristic
119	#1281	—	61	2	no	—
120	#1274	—	61	2	no	—
121	C₂×C₆₁	—	2	1	no	—
122	C₂×C₆₁	—	2	1	no	—
123	C₂×C₆₁	#1831	1	1	yes	characteristic
124	C₂×C₆₁	#1833	1	1	yes	characteristic
125	C₂×C₆₁	#1833	1	1	yes	characteristic
126	#431	—	61	2	no	—
127	#431	C₂×D₆₁	1	2	yes	—
128	C₂×#97	—	61	1	no	—
129	C₂×#97	—	61	1	no	—
130	C₂×#97	—	61	1	no	—
131	C₂×C₄×C₈	—	61	1	no	—
132	C₂×C₄×C₈	—	61	1	no	—
133	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
134	#376	—	61	1	no	—
135	#335	—	2	1	no	—
136	#335	—	122	1	no	—
137	#334	—	2	1	no	—
138	#334	—	122	1	no	—
139	C₆₁	#8780	1	1	yes	characteristic, complemented, Sylow
140	C₂²×C₈	—	61	1	no	—
141	C₂²×C₈	—	61	1	no	—
142	C₂²×C₈	C₄×D₆₁	1	1	yes	characteristic
143	C₂²×C₈	C₄×D₆₁	1	1	yes	characteristic
144	C₂×C₄²	—	61	1	no	—
145	C₂×C₄²	—	61	1	no	—
146	C₂×C₄²	C₂²×D₆₁	1	1	yes	characteristic
147	C₄⋊C₈	—	2	2	no	—
148	C₄⋊C₈	—	122	2	no	—
149	C₂²⋊C₈	—	2	2	no	—
150	C₂²⋊C₈	—	122	2	no	—
151	#97	—	122	2	no	—
152	#97	—	122	2	no	—
153	#97	—	122	1	no	—
154	#97	—	122	1	no	—
155	C₄×C₈	—	2	2	no	—
156	C₄×C₈	—	61	4	no	—
157	C₄×C₈	—	122	2	no	—
158	C₄×C₈	—	122	2	no	—
159	C₄×C₈	Dic₁₂₂	1	1	yes	characteristic
160	C₄×C₈	Dic₁₂₂	1	1	yes	characteristic
161	C₄×C₈	C₂²⋊D₆₁	1	1	yes	characteristic
162	C₄×C₈	C₂²⋊D₆₁	1	1	yes	characteristic
163	#95	—	122	1	no	—
164	#95	—	122	1	no	—
165	C₂²×C₄	—	122	1	no	—
166	C₂²×C₄	—	122	1	no	—
167	C₂²×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
168	C₂²×C₄	#11758224	1	1	yes	characteristic
169	C₂²×C₄	#11758226	1	1	yes	characteristic
170	C₂×C₈	—	2	1	no	—
171	C₂×C₈	—	2	1	no	—
172	C₂×C₈	—	4	2	no	—
173	C₂×C₈	—	61	2	no	—
174	C₂×C₈	—	61	2	no	—
175	C₂×C₈	—	122	1	no	—
176	C₂×C₈	—	122	1	no	—
177	C₂×C₈	—	122	1	no	—
178	C₂×C₈	—	122	1	no	—
179	C₂×C₈	—	244	2	no	—
180	C₂×C₈	C₈×D₆₁	1	2	yes	—
181	C₂×C₈	#11758189	1	2	yes	—
182	C₂×C₈	#11758196	1	1	yes	characteristic
183	C₂×C₈	#11758196	1	1	yes	characteristic
184	C₂×C₈	#11758196	1	1	yes	characteristic
185	C₂×C₈	#11758196	1	1	yes	characteristic
186	C₄²	—	61	2	no	—
187	C₄²	—	61	2	no	—
188	C₄²	—	122	1	no	—
189	C₄²	—	122	1	no	—
190	C₄²	C₂×Dic₁₂₂	1	1	yes	characteristic
191	C₄²	#11758224	1	1	yes	characteristic
192	C₄²	#11758226	1	1	yes	characteristic
193	C₄²	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
194	C₂³	#14519735	1	1	yes	characteristic
195	C₂×C₄	—	4	1	no	—
196	C₂×C₄	—	4	1	no	—
197	C₂×C₄	—	122	2	no	—
198	C₂×C₄	—	122	2	no	—
199	C₂×C₄	—	244	1	no	—
200	C₂×C₄	—	244	1	no	—
201	C₂×C₄	—	244	1	no	—
202	C₂×C₄	—	244	1	no	—
203	C₂×C₄	C₂×#11758196	1	1	yes	characteristic
204	C₂×C₄	#14519735	1	1	yes	characteristic
205	C₂×C₄	#14519764	1	1	yes	characteristic
206	C₂×C₄	#14519766	1	1	yes	characteristic
207	C₂×C₄	#14519768	1	1	yes	characteristic
208	C₂×C₄	#14519768	1	1	yes	characteristic
209	C₂×C₄	C₂×C₈×D₆₁	1	1	yes	characteristic
210	C₂×C₄	M₁₆×D₆₁	1	1	yes	characteristic
211	C₂×C₄	M₁₆×D₆₁	1	1	yes	characteristic
212	C₂×C₄	C₂×#11758189	1	1	yes	characteristic
213	C₈	—	2	1	no	—
214	C₈	—	2	1	no	—
215	C₈	—	8	2	no	—
216	C₈	—	122	2	no	—
217	C₈	—	122	2	no	—
218	C₈	—	488	2	no	—
219	C₈	#14519706	1	2	yes	—
220	C₈	#14519706	1	2	yes	—
221	C₂²	—	2	1	no	—
222	C₂²	—	2	1	no	—
223	C₂²	#16839627	1	1	yes	characteristic
224	C₂²	#16839814	1	1	yes	characteristic
225	C₂²	#16839818	1	1	yes	characteristic
226	C₄	—	2	1	no	—
227	C₄	—	2	1	no	—
228	C₄	—	2	1	no	—
229	C₄	—	2	1	no	—
230	C₄	—	244	2	no	—
231	C₄	—	244	2	no	—
232	C₄	C₂×#14519706	1	1	yes	characteristic
233	C₄	#16839785	1	1	yes	characteristic
234	C₄	#16839814	1	1	yes	characteristic
235	C₄	#16839818	1	1	yes	characteristic
236	C₂	—	2	1	no	—
237	C₂	—	2	1	no	—
238	C₂	#21445935	1	1	yes	characteristic
239	C₂	#21445942	1	1	yes	characteristic
240	C₂	#21445944	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#14519735
commutator	C₂×C₄×C₆₁	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₆₁
Fitting	C₆₁×#1564	C₂
socle	C₂²×C₆₁	#382
prime radical	C₂³×C₆₁	#117
cosocle	C₂×C₄²×C₆₁	C₂³
hypercenter	#1564	D₆₁
hypocenter	C₆₁	#8780
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1564

Complements

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

N	H ≅ G/N	type
31. C₈×Dic₆₁	C₈	semidirect
139. C₆₁	#8780	semidirect

Maximal subgroups

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

subgroup	index
#21445505	2
#21444567	2
#21443663	2
#21443647	2
C₆₁×#1564	2
#8780	61