Subgroups — #29579773

Order 15616.

526 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21446781	C₂	1	2	yes	maximal
2	#21446622	C₂	1	2	yes	maximal
3	#21445979	C₂	1	1	yes	maximal, characteristic
4	C₂×C₈×#92418	C₂	1	1	yes	maximal, characteristic
5	C₂×C₆₁×#500	C₂	1	1	yes	maximal, characteristic
6	C₂×#14519711	—	2	2	no	—
7	C₂×#14519701	—	2	4	no	—
8	C₂×C₄×#92418	C₂²	1	1	yes	characteristic
9	C₂×C₄×#92418	C₂²	1	1	yes	characteristic
10	#16839632	C₂²	1	2	yes	—
11	#16839524	—	2	4	no	—
12	C₈×#92418	—	2	2	no	—
13	C₂×C₄⋊Q₈×C₆₁	C₂²	1	2	yes	—
14	C₆₁×#500	—	2	2	no	—
15	C₆₁×#500	C₄	1	4	yes	complemented
16	C₂×#107×C₆₁	—	2	2	no	—
17	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
18	C₂²×#92418	D₄	1	2	yes	—
19	C₂²×#92418	D₄	1	2	yes	—
20	C₂×C₄⋊Dic₆₁	—	2	2	no	—
21	C₂×C₄⋊Dic₆₁	—	2	2	no	—
22	C₂×C₄⋊Dic₆₁	—	2	2	no	—
23	#14519711	—	4	4	no	—
24	#14519701	—	4	8	no	—
25	C₄×#92418	—	2	1	no	—
26	C₄×#92418	—	2	1	no	—
27	C₄×#92418	—	2	1	no	—
28	C₄×#92418	—	2	1	no	—
29	C₂²×Q₈×C₆₁	—	2	2	no	—
30	C₂²×C₈×C₆₁	D₄	1	2	yes	—
31	C₄⋊Q₈×C₆₁	—	2	2	no	—
32	C₄⋊Q₈×C₆₁	—	2	2	no	—
33	C₄⋊Q₈×C₆₁	—	2	2	no	—
34	C₄⋊Q₈×C₆₁	C₂×C₄	1	4	yes	—
35	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
36	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
37	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
38	#107×C₆₁	—	2	8	no	—
39	C₄×C₈×C₆₁	C₂×C₄	1	2	yes	—
40	C₄×C₈×C₆₁	D₄	1	2	yes	—
41	C₂×Q₈×C₆₁	—	2	4	no	—
42	C₂×Q₈×C₆₁	—	4	2	no	—
43	C₂×Q₈×C₆₁	—	4	4	no	—
44	C₂×Q₈×C₆₁	—	4	4	no	—
45	C₂²×C₄×C₆₁	—	4	2	no	—
46	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
47	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
48	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
49	C₂²×Dic₆₁	—	4	2	no	—
50	C₂×C₈×C₆₁	C₂²⋊C₄	1	4	yes	—
51	C₂×C₈×C₆₁	SD₁₆	1	4	yes	—
52	C₂×C₈×C₆₁	Q₁₆	1	4	yes	—
53	C₄⋊C₄×C₆₁	—	2	4	no	—
54	C₄⋊C₄×C₆₁	—	2	4	no	—
55	C₄⋊C₄×C₆₁	—	4	2	no	—
56	C₄⋊C₄×C₆₁	—	4	2	no	—
57	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
58	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
59	C₄²×C₆₁	#41	1	1	yes	characteristic
60	C₄²×C₆₁	#41	1	1	yes	characteristic
61	C₄⋊Dic₆₁	—	4	4	no	—
62	C₄⋊Dic₆₁	—	4	4	no	—
63	C₄⋊Dic₆₁	—	4	4	no	—
64	C₂×#92418	—	2	2	no	—
65	C₂×#92418	—	2	2	no	—
66	C₂×#92418	—	2	2	no	—
67	C₂×#92418	—	2	2	no	—
68	C₂×#92418	D₈	1	4	yes	—
69	C₂×#92418	Q₁₆	1	4	yes	—
70	C₂³×C₆₁	#127	1	1	yes	characteristic
71	Q₈×C₆₁	—	4	8	no	—
72	Q₈×C₆₁	—	8	4	no	—
73	C₂×C₄×C₆₁	—	4	4	no	—
74	C₂×C₄×C₆₁	—	8	2	no	—
75	C₂×C₄×C₆₁	—	8	2	no	—
76	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
77	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
78	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
79	C₂×C₄×C₆₁	#121	1	1	yes	characteristic
80	C₂×C₄×C₆₁	#121	1	1	yes	characteristic
81	C₂×C₄×C₆₁	#124	1	1	yes	characteristic
82	C₂×C₄×C₆₁	C₂×D₈	1	2	yes	—
83	C₂×C₄×C₆₁	C₂×SD₁₆	1	2	yes	—
84	C₂×C₄×C₆₁	C₂×Q₁₆	1	2	yes	—
85	C₂×C₄×C₆₁	C₂×Q₁₆	1	2	yes	—
86	C₂×C₄×C₆₁	#135	1	2	yes	—
87	C₂×C₄×C₆₁	#135	1	2	yes	—
88	C₂×Dic₆₁	—	8	2	no	—
89	C₂×Dic₆₁	—	8	2	no	—
90	C₂×Dic₆₁	—	8	2	no	—
91	C₈×C₆₁	#103	1	8	yes	complemented
92	#92418	—	2	4	no	—
93	#92418	—	2	4	no	—
94	#9823	—	61	1	no	maximal, Sylow
95	C₂²×C₆₁	#390	1	1	yes	characteristic
96	C₂²×C₆₁	#492	1	1	yes	characteristic
97	C₂²×C₆₁	#493	1	1	yes	characteristic
98	C₂²×C₆₁	#494	1	1	yes	characteristic
99	C₂²×C₆₁	#494	1	1	yes	characteristic
100	C₂²×C₆₁	#495	1	1	yes	characteristic
101	C₂²×C₆₁	#495	1	1	yes	characteristic
102	C₄×C₆₁	—	8	4	no	—
103	C₄×C₆₁	C₂×#103	1	2	yes	—
104	C₄×C₆₁	C₄×D₈	1	2	yes	—
105	C₄×C₆₁	C₄×Q₁₆	1	2	yes	—
106	C₄×C₆₁	#466	1	2	yes	—
107	C₄×C₆₁	#467	1	2	yes	—
108	C₄×C₆₁	#487	1	2	yes	—
109	Dic₆₁	—	16	2	no	—
110	C₂×#500	D₆₁	1	1	yes	characteristic
111	#1990	—	61	2	no	—
112	#1956	—	61	2	no	—
113	#1845	—	61	1	no	—
114	C₂×C₈²	—	61	1	no	—
115	C₂×C₆₁	#1706	1	1	yes	characteristic
116	C₂×C₆₁	#1707	1	1	yes	characteristic
117	C₂×C₆₁	#1710	1	1	yes	characteristic
118	C₂×C₆₁	#1713	1	1	yes	characteristic
119	C₂×C₆₁	#1956	1	1	yes	characteristic
120	C₂×C₆₁	#1969	1	1	yes	characteristic
121	C₂×C₆₁	#1970	1	1	yes	characteristic
122	C₂×C₄⋊Q₈	C₂×D₆₁	1	2	yes	—
123	#500	—	2	2	no	—
124	#500	Dic₆₁	1	4	yes	complemented
125	C₂×#107	—	2	2	no	—
126	C₂×#107	—	122	2	no	—
127	C₂×#103	—	122	4	no	—
128	C₂×C₄×C₈	—	61	1	no	—
129	C₂×C₄×C₈	—	61	1	no	—
130	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
131	#384	—	61	2	no	—
132	#340	—	122	4	no	—
133	C₈²	—	122	2	no	—
134	C₆₁	#9823	1	1	yes	characteristic, complemented, Sylow
135	C₂²×Q₈	—	2	2	no	—
136	C₂²×C₈	—	61	2	no	—
137	C₂²×C₈	—	61	2	no	—
138	C₂²×C₈	C₂²⋊D₆₁	1	2	yes	—
139	C₄⋊Q₈	—	2	2	no	—
140	C₄⋊Q₈	—	2	2	no	—
141	C₄⋊Q₈	—	2	2	no	—
142	C₄⋊Q₈	C₂×Dic₆₁	1	4	yes	—
143	C₂×C₄⋊C₄	—	2	2	no	—
144	C₂×C₄⋊C₄	—	2	2	no	—
145	C₂×C₄⋊C₄	—	122	2	no	—
146	C₂×C₄⋊C₄	—	122	2	no	—
147	C₂×C₄⋊C₄	—	122	2	no	—
148	C₂×C₄²	C₂²×D₆₁	1	1	yes	characteristic
149	#107	—	2	8	no	—
150	#107	—	244	4	no	—
151	#103	—	244	8	no	—
152	C₄×C₈	—	122	1	no	—
153	C₄×C₈	—	122	1	no	—
154	C₄×C₈	—	122	1	no	—
155	C₄×C₈	—	122	1	no	—
156	C₄×C₈	C₂×Dic₆₁	1	2	yes	—
157	C₄×C₈	C₂²⋊D₆₁	1	2	yes	—
158	C₂×Q₈	—	2	4	no	—
159	C₂×Q₈	—	4	2	no	—
160	C₂×Q₈	—	4	4	no	—
161	C₂×Q₈	—	4	4	no	—
162	C₂²×C₄	—	4	2	no	—
163	C₂²×C₄	—	244	2	no	—
164	C₂²×C₄	D₄×D₆₁	1	1	yes	characteristic
165	C₂²×C₄	D₄×D₆₁	1	1	yes	characteristic
166	C₂²×C₄	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
167	C₂×C₈	—	61	4	no	—
168	C₂×C₈	—	61	4	no	—
169	C₂×C₈	—	122	2	no	—
170	C₂×C₈	—	122	2	no	—
171	C₂×C₈	—	122	2	no	—
172	C₂×C₈	—	122	2	no	—
173	C₂×C₈	#11758201	1	4	yes	—
174	C₂×C₈	#11758202	1	4	yes	—
175	C₂×C₈	C₂²⋊Dic₆₁	1	4	yes	—
176	C₄⋊C₄	—	2	4	no	—
177	C₄⋊C₄	—	2	4	no	—
178	C₄⋊C₄	—	4	2	no	—
179	C₄⋊C₄	—	4	2	no	—
180	C₄⋊C₄	—	244	4	no	—
181	C₄⋊C₄	—	244	4	no	—
182	C₄⋊C₄	—	244	4	no	—
183	C₄²	#11758224	1	1	yes	characteristic
184	C₄²	#11758224	1	1	yes	characteristic
185	C₄²	C₂²×Dic₆₁	1	1	yes	characteristic
186	C₄²	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
187	C₂³	#14519773	1	1	yes	characteristic
188	Q₈	—	4	8	no	—
189	Q₈	—	8	4	no	—
190	C₂×C₄	—	4	4	no	—
191	C₂×C₄	—	8	2	no	—
192	C₂×C₄	—	8	2	no	—
193	C₂×C₄	—	488	2	no	—
194	C₂×C₄	—	488	2	no	—
195	C₂×C₄	—	488	2	no	—
196	C₂×C₄	C₂×C₂²⋊Dic₆₁	1	1	yes	characteristic
197	C₂×C₄	D₄×Dic₆₁	1	1	yes	characteristic
198	C₂×C₄	D₄×Dic₆₁	1	1	yes	characteristic
199	C₂×C₄	#14519748	1	1	yes	characteristic
200	C₂×C₄	#14519748	1	1	yes	characteristic
201	C₂×C₄	#14519764	1	1	yes	characteristic
202	C₂×C₄	D₈×D₆₁	1	2	yes	—
203	C₂×C₄	C₂×#11758201	1	2	yes	—
204	C₂×C₄	Q₁₆×D₆₁	1	2	yes	—
205	C₂×C₄	C₂×#11758202	1	2	yes	—
206	C₂×C₄	#14519796	1	2	yes	—
207	C₂×C₄	#14519800	1	2	yes	—
208	C₈	—	122	4	no	—
209	C₈	—	122	4	no	—
210	C₈	#14519701	1	8	yes	complemented
211	C₂²	#16839665	1	1	yes	characteristic
212	C₂²	#16840136	1	1	yes	characteristic
213	C₂²	#16840137	1	1	yes	characteristic
214	C₂²	#16840140	1	1	yes	characteristic
215	C₂²	#16840141	1	1	yes	characteristic
216	C₂²	#16840145	1	1	yes	characteristic
217	C₂²	#16840147	1	1	yes	characteristic
218	C₄	—	8	4	no	—
219	C₄	—	976	2	no	—
220	C₄	C₂×#14519701	1	2	yes	—
221	C₄	D₈×Dic₆₁	1	2	yes	—
222	C₄	Q₁₆×Dic₆₁	1	2	yes	—
223	C₄	#16839995	1	2	yes	—
224	C₄	#16840000	1	2	yes	—
225	C₄	#16840112	1	2	yes	—
226	C₂	#21445412	1	1	yes	characteristic
227	C₂	#21445417	1	1	yes	characteristic
228	C₂	#21445428	1	1	yes	characteristic
229	C₂	#21445436	1	1	yes	characteristic
230	C₂	#21446622	1	1	yes	characteristic
231	C₂	#21446681	1	1	yes	characteristic
232	C₂	#21446686	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#14519773
commutator	C₄²×C₆₁	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₁
Fitting	C₂×C₆₁×#500	C₂
socle, prime radical	C₂³×C₆₁	#127
cosocle	C₂×C₄²×C₆₁	C₂³
hypercenter	C₂×#500	D₆₁
hypocenter	C₆₁	#9823
LCS	G ▷ C₄²×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#500

Complements

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

N	H ≅ G/N	type
15. C₆₁×#500	C₄	semidirect
91. C₈×C₆₁	#103	semidirect
124. #500	Dic₆₁	semidirect
134. C₆₁	#9823	semidirect
210. C₈	#14519701	semidirect

Maximal subgroups

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

subgroup	index
#21446781	2
#21446622	2
#21445979	2
C₂×C₈×#92418	2
C₂×C₆₁×#500	2
#9823	61