Subgroups — #29571189

Order 15616.

342 conjugacy classes of subgroups.

The subgroup lattice has 236 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21445564	C₂	1	1	yes	maximal, characteristic
2	#21444565	C₂	1	1	yes	maximal, characteristic
3	#21444565	C₂	1	1	yes	maximal, characteristic
4	#21444111	C₂	1	1	yes	maximal, characteristic
5	#21443651	C₂	1	1	yes	maximal, characteristic
6	#21443651	C₂	1	1	yes	maximal, characteristic
7	C₆₁×#1450	C₂	1	1	yes	maximal, characteristic
8	#16839815	C₂²	1	1	yes	characteristic
9	#16839815	C₂²	1	1	yes	characteristic
10	C₂×C₄×#92418	C₂²	1	1	yes	characteristic
11	C₄²×Dic₆₁	C₂²	1	1	yes	characteristic
12	#16839516	C₄	1	2	yes	—
13	#16839516	C₄	1	2	yes	—
14	#16839514	C₄	1	2	yes	complemented
15	#16839514	C₄	1	2	yes	complemented
16	#16839512	C₄	1	2	yes	complemented
17	#16839512	C₄	1	2	yes	complemented
18	#16839484	—	2	1	no	—
19	#16839484	—	2	1	no	—
20	C₆₁×#431	C₂²	1	1	yes	characteristic
21	C₆₁×#431	C₂²	1	1	yes	characteristic
22	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
23	C₆₁×#322	—	2	1	no	—
24	C₈²×C₆₁	—	2	1	no	—
25	C₂²×#92418	C₂×C₄	1	1	yes	characteristic
26	C₂²×#92418	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄×Dic₆₁	—	2	1	no	—
28	C₂×C₄×Dic₆₁	—	2	1	no	—
29	C₂×C₄×Dic₆₁	C₂×C₄	1	2	yes	—
30	C₄⋊#92418	C₂×C₄	1	2	yes	—
31	C₄⋊#92418	C₂×C₄	1	2	yes	—
32	C₂²⋊#92418	—	2	1	no	—
33	C₂²⋊#92418	—	2	1	no	—
34	C₄×#92418	—	2	1	no	—
35	C₄×#92418	—	2	1	no	—
36	C₄×#92418	C₂×C₄	1	1	yes	characteristic
37	C₄×#92418	C₂×C₄	1	1	yes	characteristic
38	C₄×#92418	C₂×C₄	1	1	yes	characteristic
39	C₄×#92418	C₂×C₄	1	1	yes	characteristic
40	C₂²×C₈×C₆₁	C₂×C₄	1	2	yes	—
41	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
42	C₄⋊C₈×C₆₁	C₂×C₄	1	2	yes	—
43	C₄⋊C₈×C₆₁	C₂×C₄	1	2	yes	—
44	C₂²⋊C₈×C₆₁	—	2	1	no	—
45	C₂²⋊C₈×C₆₁	—	2	1	no	—
46	C₄×C₈×C₆₁	—	2	1	no	—
47	C₄×C₈×C₆₁	—	2	1	no	—
48	C₄×C₈×C₆₁	D₄	1	1	yes	characteristic
49	C₄×C₈×C₆₁	D₄	1	1	yes	characteristic
50	C₄×C₈×C₆₁	Q₈	1	2	yes	—
51	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
52	C₂²×C₄×C₆₁	#41	1	1	yes	characteristic
53	C₂²×C₄×C₆₁	#41	1	1	yes	characteristic
54	C₂²×Dic₆₁	—	2	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	—	2	2	no	—
59	C₂×C₈×C₆₁	C₄⋊C₄	1	2	yes	—
60	C₂×C₈×C₆₁	C₄⋊C₄	1	2	yes	—
61	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
62	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
63	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
64	C₄²×C₆₁	C₂×Q₈	1	1	yes	characteristic
65	C₄×Dic₆₁	—	2	4	no	—
66	C₄×Dic₆₁	—	2	2	no	—
67	C₄×Dic₆₁	—	2	2	no	—
68	C₄×Dic₆₁	—	4	2	no	—
69	C₂×#92418	—	2	2	no	—
70	C₂×#92418	—	2	2	no	—
71	C₂×#92418	—	2	2	no	—
72	C₂×#92418	—	2	2	no	—
73	C₂×#92418	C₄²	1	1	yes	characteristic
74	C₂×#92418	C₄²	1	1	yes	characteristic
75	C₂×#92418	C₄²	1	1	yes	characteristic
76	C₂×#92418	C₄²	1	1	yes	characteristic
77	C₂³×C₆₁	#117	1	1	yes	characteristic
78	C₂×C₄×C₆₁	—	2	2	no	—
79	C₂×C₄×C₆₁	—	2	2	no	—
80	C₂×C₄×C₆₁	C₂×C₄²	1	1	yes	characteristic
81	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
82	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
83	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
84	C₂×C₄×C₆₁	C₄×Q₈	1	1	yes	characteristic
85	C₂×C₄×C₆₁	C₄×Q₈	1	1	yes	characteristic
86	C₂×C₄×C₆₁	#131	1	2	yes	—
87	C₂×C₄×C₆₁	#131	1	2	yes	—
88	C₂×Dic₆₁	—	2	4	no	—
89	C₂×Dic₆₁	—	4	2	no	—
90	C₂×Dic₆₁	—	4	2	no	—
91	C₈×C₆₁	—	2	2	no	—
92	C₈×C₆₁	—	2	2	no	—
93	C₈×C₆₁	—	4	2	no	—
94	C₈×C₆₁	—	4	2	no	—
95	#92418	—	2	1	no	—
96	#92418	—	2	1	no	—
97	#92418	—	2	1	no	—
98	#92418	—	2	1	no	—
99	#92418	—	4	2	no	—
100	#92418	—	4	2	no	—
101	#8270	—	61	1	no	maximal, Sylow
102	C₂²×C₆₁	—	2	1	no	—
103	C₂²×C₆₁	—	2	1	no	—
104	C₂²×C₆₁	C₄×C₄⋊C₄	1	1	yes	characteristic
105	C₂²×C₆₁	#433	1	2	yes	—
106	C₄×C₆₁	—	2	1	no	—
107	C₄×C₆₁	—	2	1	no	—
108	C₄×C₆₁	#405	1	1	yes	characteristic
109	C₄×C₆₁	#405	1	1	yes	characteristic
110	C₄×C₆₁	#424	1	2	yes	—
111	C₄×C₆₁	#443	1	2	yes	—
112	C₄×C₆₁	#444	1	2	yes	—
113	Dic₆₁	—	4	4	no	—
114	#1744	—	61	1	no	—
115	#1564	—	61	1	no	—
116	#1564	—	61	1	no	—
117	#1452	—	61	1	no	—
118	#1450	D₆₁	1	1	yes	characteristic
119	#1276	—	61	1	no	—
120	#1276	—	61	1	no	—
121	C₂×C₆₁	—	2	2	no	—
122	C₂×C₆₁	#1768	1	1	yes	characteristic
123	C₂×C₆₁	#1772	1	1	yes	characteristic
124	C₂×C₆₁	#1773	1	1	yes	characteristic
125	#431	—	61	1	no	—
126	#431	—	61	1	no	—
127	#431	C₂×D₆₁	1	1	yes	characteristic
128	#431	C₂×D₆₁	1	1	yes	characteristic
129	C₂×C₄×C₈	—	61	1	no	—
130	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
131	C₄³	—	61	1	no	—
132	#336	—	61	2	no	—
133	#336	—	61	2	no	—
134	#335	—	61	2	no	—
135	#335	—	61	2	no	—
136	#334	—	61	2	no	—
137	#334	—	61	2	no	—
138	#322	—	2	1	no	—
139	#322	—	122	1	no	—
140	#322	—	122	1	no	—
141	C₈²	—	2	1	no	—
142	C₆₁	#8270	1	1	yes	characteristic, complemented, Sylow
143	C₂²×C₈	—	61	1	no	—
144	C₂²×C₈	—	61	1	no	—
145	C₂²×C₈	C₄×D₆₁	1	2	yes	—
146	C₂×C₄²	—	61	2	no	—
147	C₂×C₄²	—	122	1	no	—
148	C₂×C₄²	—	122	1	no	—
149	C₂×C₄²	C₂²×D₆₁	1	1	yes	characteristic
150	C₄⋊C₈	—	61	2	no	—
151	C₄⋊C₈	—	61	2	no	—
152	C₄⋊C₈	C₄×D₆₁	1	2	yes	—
153	C₄⋊C₈	C₄×D₆₁	1	2	yes	—
154	C₂²⋊C₈	—	2	1	no	—
155	C₂²⋊C₈	—	2	1	no	—
156	C₂²⋊C₈	—	122	1	no	—
157	C₂²⋊C₈	—	122	1	no	—
158	C₄×C₈	—	2	1	no	—
159	C₄×C₈	—	2	1	no	—
160	C₄×C₈	—	61	1	no	—
161	C₄×C₈	—	61	1	no	—
162	C₄×C₈	—	61	1	no	—
163	C₄×C₈	—	61	1	no	—
164	C₄×C₈	—	122	1	no	—
165	C₄×C₈	—	122	1	no	—
166	C₄×C₈	Dic₁₂₂	1	2	yes	—
167	C₄×C₈	C₂²⋊D₆₁	1	1	yes	characteristic
168	C₄×C₈	C₂²⋊D₆₁	1	1	yes	characteristic
169	C₂²×C₄	—	122	2	no	—
170	C₂²×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
171	C₂²×C₄	#11758222	1	1	yes	characteristic
172	C₂²×C₄	#11758222	1	1	yes	characteristic
173	C₂×C₈	—	2	2	no	—
174	C₂×C₈	—	2	2	no	—
175	C₂×C₈	—	2	2	no	—
176	C₂×C₈	—	2	2	no	—
177	C₂×C₈	—	61	1	no	—
178	C₂×C₈	—	61	1	no	—
179	C₂×C₈	—	61	1	no	—
180	C₂×C₈	—	61	1	no	—
181	C₂×C₈	—	122	2	no	—
182	C₂×C₈	—	122	2	no	—
183	C₂×C₈	—	122	2	no	—
184	C₂×C₈	—	122	2	no	—
185	C₂×C₈	#11758196	1	2	yes	—
186	C₂×C₈	#11758196	1	2	yes	—
187	C₄²	—	122	4	no	—
188	C₄²	—	122	2	no	—
189	C₄²	—	122	2	no	—
190	C₄²	—	244	2	no	—
191	C₄²	C₂×Dic₁₂₂	1	1	yes	characteristic
192	C₄²	C₂×C₄×D₆₁	1	1	yes	characteristic
193	C₄²	C₂×C₄×D₆₁	1	1	yes	characteristic
194	C₄²	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
195	C₂³	#14519734	1	1	yes	characteristic
196	C₂×C₄	—	2	2	no	—
197	C₂×C₄	—	2	2	no	—
198	C₂×C₄	—	122	4	no	—
199	C₂×C₄	—	244	2	no	—
200	C₂×C₄	—	244	2	no	—
201	C₂×C₄	C₄²×D₆₁	1	1	yes	characteristic
202	C₂×C₄	C₂×#11758196	1	1	yes	characteristic
203	C₂×C₄	C₄×C₂²⋊D₆₁	1	1	yes	characteristic
204	C₂×C₄	C₄×C₂²⋊D₆₁	1	1	yes	characteristic
205	C₂×C₄	C₄×Dic₁₂₂	1	1	yes	characteristic
206	C₂×C₄	C₄×Dic₁₂₂	1	1	yes	characteristic
207	C₂×C₄	#14519783	1	2	yes	—
208	C₂×C₄	#14519783	1	2	yes	—
209	C₈	—	2	2	no	—
210	C₈	—	2	2	no	—
211	C₈	—	4	2	no	—
212	C₈	—	4	2	no	—
213	C₈	—	122	1	no	—
214	C₈	—	122	1	no	—
215	C₈	—	122	1	no	—
216	C₈	—	122	1	no	—
217	C₈	—	244	2	no	—
218	C₈	—	244	2	no	—
219	C₂²	—	2	1	no	—
220	C₂²	—	2	1	no	—
221	C₂²	C₄×#11758196	1	1	yes	characteristic
222	C₂²	#16839821	1	2	yes	—
223	C₄	—	2	1	no	—
224	C₄	—	2	1	no	—
225	C₄	—	244	4	no	—
226	C₄	#16839716	1	1	yes	characteristic
227	C₄	#16839716	1	1	yes	characteristic
228	C₄	#16839788	1	2	yes	—
229	C₄	#16839877	1	2	yes	—
230	C₄	#16839882	1	2	yes	—
231	C₂	—	2	2	no	—
232	C₂	#21445677	1	1	yes	characteristic
233	C₂	#21445699	1	1	yes	characteristic
234	C₂	#21445704	1	1	yes	characteristic

	subgroup	G/H
center	C₄²	C₂×C₂²⋊D₆₁
commutator	C₂×C₄×C₆₁	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₆₁
Fitting	C₆₁×#1450	C₂
socle	C₂²×C₆₁	C₄×C₄⋊C₄
prime radical	C₂³×C₆₁	#117
cosocle	C₂×C₄²×C₆₁	C₂³
hypercenter	#1450	D₆₁
hypocenter	C₆₁	#8270
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₄² ◁ C₂×C₄×C₈ ◁ #1450

Complements

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

N	H ≅ G/N	type
14. #16839514	C₄	semidirect
15. #16839514	C₄	semidirect
16. #16839512	C₄	semidirect
17. #16839512	C₄	semidirect
142. C₆₁	#8270	semidirect

Maximal subgroups

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

subgroup	index
#21445564	2
#21444565	2
#21444111	2
#21443651	2
C₆₁×#1450	2
#8270	61