Subgroups — #29570545

Order 15616.

482 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	#21444133	C₂	1	2	yes	maximal
2	#21444111	C₂	1	2	yes	maximal, complemented
3	#21443664	C₂	1	2	yes	maximal, complemented
4	C₂×C₆₁×#405	C₂	1	1	yes	maximal, characteristic
5	#16839815	C₂²	1	4	yes	—
6	#16839488	C₄	1	4	yes	—
7	#16839486	C₄	1	4	yes	—
8	#16839484	—	2	4	no	—
9	C₂²×M₁₆×C₆₁	C₄	1	1	yes	characteristic
10	C₂³×C₈×C₆₁	C₄	1	1	yes	characteristic
11	C₂×#117×C₆₁	C₂²	1	1	yes	characteristic
12	C₆₁×#405	—	2	4	no	—
13	C₂×#97×C₆₁	C₄	1	1	yes	characteristic
14	C₂×#97×C₆₁	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₈×C₆₁	C₄	1	1	yes	characteristic
16	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
17	C₄⋊#92418	—	2	4	no	—
18	C₂²⋊#92418	C₂×C₄	1	8	yes	—
19	C₄×#92418	—	2	4	no	—
20	C₂³×C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
21	C₂×M₁₆×C₆₁	—	2	2	no	—
22	C₂×M₁₆×C₆₁	—	2	2	no	—
23	C₂×M₁₆×C₆₁	—	2	2	no	—
24	C₂²×C₈×C₆₁	—	2	2	no	—
25	C₂²×C₈×C₆₁	—	2	2	no	—
26	C₂²×C₈×C₆₁	—	2	2	no	—
27	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
31	#117×C₆₁	—	2	2	no	—
32	#117×C₆₁	—	2	2	no	—
33	C₂×C₄⋊C₄×C₆₁	C₂×C₄	1	2	yes	—
34	C₂×C₂²⋊C₄×C₆₁	C₂×C₄	1	2	yes	—
35	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
37	#97×C₆₁	—	2	2	no	—
38	#97×C₆₁	—	2	2	no	—
39	C₄×C₈×C₆₁	—	2	2	no	—
40	C₄×C₈×C₆₁	D₄	1	4	yes	—
41	C₂⁴×C₆₁	C₄²	1	1	yes	characteristic
42	C₂²×C₄×C₆₁	—	2	1	no	—
43	C₂²×C₄×C₆₁	—	2	1	no	—
44	C₂²×C₄×C₆₁	—	2	1	no	—
45	C₂²×C₄×C₆₁	—	2	1	no	—
46	C₂²×C₄×C₆₁	C₄²	1	1	yes	characteristic
47	C₂²×C₄×C₆₁	C₄²	1	2	yes	—
48	C₂²×C₄×C₆₁	M₁₆	1	2	yes	—
49	C₂²×C₄×C₆₁	M₁₆	1	2	yes	—
50	C₂²×C₄×C₆₁	C₂²×C₄	1	2	yes	—
51	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
52	M₁₆×C₆₁	—	2	8	no	—
53	C₂×C₈×C₆₁	—	2	8	no	—
54	C₂×C₈×C₆₁	—	2	1	no	—
55	C₂×C₈×C₆₁	—	2	1	no	—
56	C₂×C₈×C₆₁	—	2	1	no	—
57	C₂×C₈×C₆₁	—	2	1	no	—
58	C₂×C₈×C₆₁	—	2	1	no	—
59	C₂×C₈×C₆₁	—	2	1	no	—
60	C₂×C₈×C₆₁	—	2	1	no	—
61	C₂×C₈×C₆₁	—	2	1	no	—
62	C₂×C₈×C₆₁	C₂²⋊C₄	1	2	yes	—
63	C₂×C₈×C₆₁	C₂²⋊C₄	1	2	yes	—
64	C₂×C₈×C₆₁	M₁₆	1	2	yes	—
65	C₂×C₈×C₆₁	M₁₆	1	2	yes	—
66	C₄⋊C₄×C₆₁	—	2	4	no	—
67	C₂²⋊C₄×C₆₁	—	2	4	no	—
68	C₄²×C₆₁	—	2	1	no	—
69	C₄²×C₆₁	—	2	1	no	—
70	C₄²×C₆₁	C₂×D₄	1	2	yes	—
71	C₄²×C₆₁	#41	1	2	yes	—
72	C₂×#92418	—	2	8	no	—
73	C₂³×C₆₁	—	2	1	no	—
74	C₂³×C₆₁	—	2	1	no	—
75	C₂³×C₆₁	—	4	2	no	—
76	C₂³×C₆₁	#97	1	2	yes	—
77	C₂³×C₆₁	C₂×C₄²	1	1	yes	characteristic
78	C₂×C₄×C₆₁	—	2	1	no	—
79	C₂×C₄×C₆₁	—	2	1	no	—
80	C₂×C₄×C₆₁	—	2	2	no	—
81	C₂×C₄×C₆₁	—	2	2	no	—
82	C₂×C₄×C₆₁	—	2	2	no	—
83	C₂×C₄×C₆₁	—	2	2	no	—
84	C₂×C₄×C₆₁	—	2	2	no	—
85	C₂×C₄×C₆₁	—	4	2	no	—
86	C₂×C₄×C₆₁	—	4	2	no	—
87	C₂×C₄×C₆₁	#97	1	2	yes	—
88	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
89	C₂×C₄×C₆₁	#117	1	1	yes	characteristic
90	C₂×C₄×C₆₁	C₄×D₄	1	4	yes	—
91	C₂×C₄×C₆₁	C₂×M₁₆	1	1	yes	characteristic
92	C₂×C₄×C₆₁	C₂×M₁₆	1	1	yes	characteristic
93	C₂×C₄×C₆₁	C₂×M₁₆	1	1	yes	characteristic
94	C₂×C₄×C₆₁	C₂×M₁₆	1	1	yes	characteristic
95	C₈×C₆₁	—	2	2	no	—
96	C₈×C₆₁	—	2	2	no	—
97	C₈×C₆₁	—	2	2	no	—
98	C₈×C₆₁	—	2	2	no	—
99	#92418	—	4	8	no	—
100	#8157	—	61	1	no	maximal, Sylow
101	C₂²×C₆₁	—	2	2	no	—
102	C₂²×C₆₁	—	2	2	no	—
103	C₂²×C₆₁	—	4	2	no	—
104	C₂²×C₆₁	—	4	2	no	—
105	C₂²×C₆₁	—	4	2	no	—
106	C₂²×C₆₁	C₄×C₂²⋊C₄	1	1	yes	characteristic
107	C₂²×C₆₁	C₂×#97	1	1	yes	characteristic
108	C₂²×C₆₁	C₄×M₁₆	1	1	yes	characteristic
109	C₄×C₆₁	—	2	2	no	—
110	C₄×C₆₁	—	2	4	no	—
111	C₄×C₆₁	—	4	2	no	—
112	C₄×C₆₁	#407	1	1	yes	characteristic
113	C₄×C₆₁	#407	1	1	yes	characteristic
114	C₄×C₆₁	#432	1	1	yes	characteristic
115	C₄×C₆₁	#432	1	1	yes	characteristic
116	C₂×#405	D₆₁	1	1	yes	characteristic
117	#1458	—	61	2	no	—
118	#1452	—	61	2	no	—
119	#1281	—	61	2	no	—
120	C₂×C₆₁	—	2	1	no	—
121	C₂×C₆₁	—	2	1	no	—
122	C₂×C₆₁	—	4	2	no	—
123	C₂×C₆₁	#1749	1	1	yes	characteristic
124	C₂×C₆₁	#1757	1	2	yes	—
125	C₂²×M₁₆	Dic₆₁	1	1	yes	characteristic
126	C₂³×C₈	Dic₆₁	1	1	yes	characteristic
127	C₂×#117	C₂×D₆₁	1	1	yes	characteristic
128	#431	—	61	4	no	—
129	#405	—	2	4	no	—
130	C₂×#97	Dic₆₁	1	1	yes	characteristic
131	C₂×#97	C₂×D₆₁	1	1	yes	characteristic
132	C₂×C₄×C₈	Dic₆₁	1	1	yes	characteristic
133	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
134	#324	—	61	4	no	—
135	#323	—	61	4	no	—
136	#322	—	122	4	no	—
137	C₆₁	#8157	1	1	yes	characteristic, complemented, Sylow
138	C₂³×C₄	C₂×Dic₆₁	1	1	yes	characteristic
139	C₂×M₁₆	—	2	2	no	—
140	C₂×M₁₆	—	2	2	no	—
141	C₂×M₁₆	—	2	2	no	—
142	C₂²×C₈	—	2	2	no	—
143	C₂²×C₈	—	2	2	no	—
144	C₂²×C₈	—	2	2	no	—
145	C₂²×C₈	C₂×Dic₆₁	1	1	yes	characteristic
146	C₂²×C₈	C₂×Dic₆₁	1	1	yes	characteristic
147	C₂²×C₈	C₂×Dic₆₁	1	1	yes	characteristic
148	C₂²×C₈	C₂×Dic₆₁	1	1	yes	characteristic
149	#117	—	2	2	no	—
150	#117	—	2	2	no	—
151	C₂×C₄⋊C₄	C₄×D₆₁	1	2	yes	—
152	C₂×C₂²⋊C₄	C₄×D₆₁	1	2	yes	—
153	C₂×C₄²	C₂×Dic₆₁	1	1	yes	characteristic
154	C₂×C₄²	C₂²×D₆₁	1	1	yes	characteristic
155	C₄⋊C₈	—	122	4	no	—
156	C₂²⋊C₈	—	61	8	no	—
157	#97	—	2	2	no	—
158	#97	—	2	2	no	—
159	C₄×C₈	—	2	2	no	—
160	C₄×C₈	—	122	4	no	—
161	C₄×C₈	C₂²⋊D₆₁	1	4	yes	—
162	C₂⁴	C₄×Dic₆₁	1	1	yes	characteristic
163	C₂²×C₄	—	2	1	no	—
164	C₂²×C₄	—	2	1	no	—
165	C₂²×C₄	—	2	1	no	—
166	C₂²×C₄	—	2	1	no	—
167	C₂²×C₄	#11758194	1	2	yes	—
168	C₂²×C₄	#11758194	1	2	yes	—
169	C₂²×C₄	C₄×Dic₆₁	1	1	yes	characteristic
170	C₂²×C₄	C₄×Dic₆₁	1	2	yes	—
171	C₂²×C₄	C₂×C₄×D₆₁	1	2	yes	—
172	C₂²×C₄	C₂²×Dic₆₁	1	1	yes	characteristic
173	M₁₆	—	2	8	no	—
174	C₂×C₈	—	2	8	no	—
175	C₂×C₈	—	2	1	no	—
176	C₂×C₈	—	2	1	no	—
177	C₂×C₈	—	2	1	no	—
178	C₂×C₈	—	2	1	no	—
179	C₂×C₈	—	2	1	no	—
180	C₂×C₈	—	2	1	no	—
181	C₂×C₈	—	2	1	no	—
182	C₂×C₈	—	2	1	no	—
183	C₂×C₈	—	122	8	no	—
184	C₂×C₈	#11758194	1	2	yes	—
185	C₂×C₈	#11758194	1	2	yes	—
186	C₂×C₈	C₂²⋊Dic₆₁	1	2	yes	—
187	C₂×C₈	C₂²⋊Dic₆₁	1	2	yes	—
188	C₄⋊C₄	—	2	4	no	—
189	C₂²⋊C₄	—	2	4	no	—
190	C₄²	—	2	1	no	—
191	C₄²	—	2	1	no	—
192	C₄²	#11758222	1	2	yes	—
193	C₄²	C₂×C₂²⋊D₆₁	1	2	yes	—
194	C₂³	—	2	1	no	—
195	C₂³	—	2	1	no	—
196	C₂³	—	4	2	no	—
197	C₂³	#14519688	1	2	yes	—
198	C₂³	C₂×C₄×Dic₆₁	1	1	yes	characteristic
199	C₂×C₄	—	2	1	no	—
200	C₂×C₄	—	2	1	no	—
201	C₂×C₄	—	2	2	no	—
202	C₂×C₄	—	2	2	no	—
203	C₂×C₄	—	2	2	no	—
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	4	2	no	—
207	C₂×C₄	—	4	2	no	—
208	C₂×C₄	#14519688	1	2	yes	—
209	C₂×C₄	C₂×C₂²⋊Dic₆₁	1	1	yes	characteristic
210	C₂×C₄	#14519736	1	1	yes	characteristic
211	C₂×C₄	C₄×C₂²⋊D₆₁	1	4	yes	—
212	C₂×C₄	C₂×#11758194	1	1	yes	characteristic
213	C₂×C₄	C₂×#11758194	1	1	yes	characteristic
214	C₂×C₄	C₂×#11758194	1	1	yes	characteristic
215	C₂×C₄	C₂×#11758194	1	1	yes	characteristic
216	C₈	—	2	2	no	—
217	C₈	—	2	2	no	—
218	C₈	—	2	2	no	—
219	C₈	—	2	2	no	—
220	C₈	—	244	8	no	—
221	C₂²	—	2	2	no	—
222	C₂²	—	2	2	no	—
223	C₂²	—	4	2	no	—
224	C₂²	—	4	2	no	—
225	C₂²	—	4	2	no	—
226	C₂²	C₄×C₂²⋊Dic₆₁	1	1	yes	characteristic
227	C₂²	C₂×#14519688	1	1	yes	characteristic
228	C₂²	C₄×#11758194	1	1	yes	characteristic
229	C₄	—	2	2	no	—
230	C₄	—	2	4	no	—
231	C₄	—	4	2	no	—
232	C₄	#16839726	1	1	yes	characteristic
233	C₄	#16839726	1	1	yes	characteristic
234	C₄	#16839819	1	1	yes	characteristic
235	C₄	#16839819	1	1	yes	characteristic
236	C₂	—	2	1	no	—
237	C₂	—	2	1	no	—
238	C₂	—	4	2	no	—
239	C₂	#21445581	1	1	yes	characteristic
240	C₂	#21445619	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊Dic₆₁
commutator	C₂³×C₆₁	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₆₁
Fitting	C₂×C₆₁×#405	C₂
socle	C₂²×C₆₁	C₄×C₂²⋊C₄
prime radical	C₂⁴×C₆₁	C₄²
cosocle	C₂×C₄²×C₆₁	C₂³
hypercenter	C₂×#405	D₆₁
hypocenter	C₆₁	#8157
LCS	G ▷ C₂³×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#405

Complements

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

N	H ≅ G/N	type
2. #21444111	C₂	semidirect
3. #21443664	C₂	semidirect
137. C₆₁	#8157	semidirect

Maximal subgroups

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

subgroup	index
#21444133	2
#21444111	2
#21443664	2
C₂×C₆₁×#405	2
#8157	61