Subgroups — #29835504

Order 15616.

490 conjugacy classes of subgroups.

The subgroup lattice has 220 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21456332	C₂	1	2	yes	maximal
2	#21448100	C₂	1	2	yes	maximal, complemented
3	#21448098	C₂	1	1	yes	maximal, characteristic, complemented
4	#21448095	C₂	1	2	yes	maximal, complemented
5	#21448093	C₂	1	2	yes	maximal, complemented
6	#21447602	C₂	1	2	yes	maximal
7	#21447600	C₂	1	2	yes	maximal
8	C₄×C₁₆×D₆₁	C₂	1	1	yes	maximal, characteristic
9	C₆₁×#2253	C₂	1	1	yes	maximal, characteristic, complemented
10	#16840318	—	2	2	no	—
11	C₂×C₁₆×D₆₁	C₂²	1	1	yes	characteristic
12	C₂×C₁₆×D₆₁	C₂²	1	1	yes	characteristic
13	#16840166	C₂²	1	4	yes	—
14	#16840164	C₂²	1	2	yes	—
15	#16840161	—	2	2	no	—
16	#16840159	—	2	2	no	—
17	#16839799	C₂²	1	4	yes	—
18	#16839797	—	2	2	no	—
19	C₄×C₈×D₆₁	C₂²	1	1	yes	characteristic
20	#16839590	C₂²	1	2	yes	—
21	#16839589	C₂²	1	4	yes	—
22	#16839588	C₂²	1	2	yes	—
23	#16839587	C₂²	1	4	yes	—
24	C₄×#11758185	C₂²	1	1	yes	characteristic
25	C₁₆×Dic₆₁	C₂²	1	1	yes	characteristic
26	C₁₆×Dic₆₁	C₂²	1	1	yes	characteristic
27	C₆₁×#499	C₂²	1	2	yes	—
28	C₆₁×#367	C₂²	1	2	yes	—
29	C₆₁×#366	C₂²	1	2	yes	—
30	C₄×C₁₆×C₆₁	C₂²	1	1	yes	characteristic
31	C₂×C₈×D₆₁	D₄	1	1	yes	characteristic
32	C₂×C₈×D₆₁	C₂³	1	1	yes	characteristic
33	#14519776	—	2	4	no	—
34	#14519769	—	2	2	no	—
35	#14519755	—	4	4	no	—
36	#14519742	—	4	4	no	—
37	#14519735	—	2	4	no	—
38	C₄²×D₆₁	D₄	1	1	yes	characteristic
39	C₂×#11758185	C₂³	1	1	yes	characteristic
40	C₂×#11758185	C₂³	1	1	yes	characteristic
41	C₁₆×D₆₁	Q₈	1	4	yes	complemented
42	C₁₆×D₆₁	Q₈	1	4	yes	complemented
43	#14519711	C₂³	1	4	yes	—
44	#14519710	—	2	4	no	—
45	#14519709	—	2	2	no	—
46	#14519708	—	2	4	no	—
47	C₄×#92418	D₄	1	1	yes	characteristic
48	C₈×Dic₆₁	D₄	1	1	yes	characteristic
49	C₈×Dic₆₁	C₂³	1	1	yes	characteristic
50	#125×C₆₁	—	2	2	no	—
51	C₂×C₁₆×C₆₁	C₂³	1	1	yes	characteristic
52	C₂×C₁₆×C₆₁	C₂³	1	1	yes	characteristic
53	#107×C₆₁	C₂³	1	4	yes	—
54	#106×C₆₁	—	2	2	no	—
55	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
56	C₂×C₄×D₆₁	D₈	1	2	yes	—
57	C₂×C₄×D₆₁	C₂×D₄	1	1	yes	characteristic
58	C₂×Dic₁₂₂	—	4	4	no	—
59	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
60	C₂×C₈×C₆₁	C₂⁴	1	1	yes	characteristic
61	C₄⋊C₄×C₆₁	—	2	4	no	—
62	C₄⋊C₄×C₆₁	—	4	4	no	—
63	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
64	#11758198	—	4	4	no	—
65	C₄⋊Dic₆₁	—	2	4	no	—
66	#11758196	—	4	4	no	—
67	#11758196	—	4	4	no	—
68	C₄×Dic₆₁	—	4	4	no	—
69	C₄×Dic₆₁	D₈	1	2	yes	—
70	C₄×Dic₆₁	C₂×D₄	1	1	yes	characteristic
71	C₂×#92418	C₂×D₄	1	1	yes	characteristic
72	C₂×#92418	C₂×D₄	1	1	yes	characteristic
73	C₈×D₆₁	Q₁₆	1	4	yes	—
74	C₈×D₆₁	C₂×Q₈	1	2	yes	—
75	C₈×D₆₁	C₂×Q₈	1	2	yes	—
76	C₁₆×C₆₁	C₂×Q₈	1	2	yes	—
77	C₁₆×C₆₁	C₂×Q₈	1	2	yes	—
78	#11758185	C₂×Q₈	1	2	yes	—
79	#11758185	C₂×Q₈	1	2	yes	—
80	C₂²×D₆₁	C₂×D₈	1	1	yes	characteristic
81	C₂×C₄×C₆₁	—	4	4	no	—
82	C₂×C₄×C₆₁	C₂×D₈	1	2	yes	—
83	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
84	C₂×Dic₆₁	—	4	4	no	—
85	C₂×Dic₆₁	C₂×D₈	1	1	yes	characteristic
86	C₂×Dic₆₁	C₂×D₈	1	2	yes	—
87	C₄×D₆₁	—	2	4	no	—
88	C₄×D₆₁	C₄⋊Q₈	1	2	yes	—
89	C₄×D₆₁	C₂×Q₁₆	1	2	yes	—
90	Dic₁₂₂	—	8	8	no	—
91	C₈×C₆₁	C₂×Q₁₆	1	2	yes	—
92	C₈×C₆₁	C₂²×Q₈	1	1	yes	characteristic
93	C₈×C₆₁	C₂²×Q₈	1	1	yes	characteristic
94	#92418	C₄⋊Q₈	1	2	yes	—
95	#92418	C₂×Q₁₆	1	2	yes	—
96	#33726	—	61	1	no	maximal, Sylow
97	C₂²×C₆₁	C₂²×D₈	1	1	yes	characteristic
98	C₂×D₆₁	—	2	2	no	—
99	C₂×D₆₁	#500	1	2	yes	—
100	C₄×C₆₁	—	2	2	no	—
101	C₄×C₆₁	—	8	4	no	—
102	C₄×C₆₁	C₂×C₄⋊Q₈	1	1	yes	characteristic
103	C₄×C₆₁	C₂²×Q₁₆	1	1	yes	characteristic
104	Dic₆₁	—	8	4	no	—
105	Dic₆₁	#500	1	2	yes	—
106	Dic₆₁	#508	1	4	yes	—
107	#3160	—	61	2	no	—
108	#2254	—	61	2	no	—
109	#2253	—	61	2	no	—
110	#2253	—	61	1	no	—
111	#2253	D₆₁	1	1	yes	characteristic, complemented
112	#2252	—	61	2	no	—
113	#2158	—	61	2	no	—
114	#2158	—	61	2	no	—
115	C₂×C₄×C₁₆	—	61	1	no	—
116	C₂×C₆₁	C₂×#500	1	1	yes	characteristic
117	C₂×C₆₁	C₂×#508	1	2	yes	—
118	D₆₁	—	2	2	no	—
119	#533	—	122	2	no	—
120	C₂²×C₁₆	—	61	1	no	—
121	C₂²×C₁₆	—	61	1	no	—
122	#500	—	61	4	no	—
123	#499	—	61	2	no	—
124	#499	—	122	2	no	—
125	#499	C₂×D₆₁	1	2	yes	—
126	#498	—	122	2	no	—
127	#427	—	61	4	no	—
128	#427	—	122	2	no	—
129	C₂×C₄×C₈	—	61	1	no	—
130	#367	—	61	2	no	—
131	#367	—	61	4	no	—
132	#367	C₂×D₆₁	1	2	yes	—
133	#366	—	61	2	no	—
134	#366	—	61	4	no	—
135	#366	C₂×D₆₁	1	2	yes	—
136	C₄×C₁₆	—	61	1	no	—
137	C₄×C₁₆	—	61	1	no	—
138	C₄×C₁₆	—	61	1	no	—
139	C₄×C₁₆	C₂×D₆₁	1	1	yes	characteristic
140	C₆₁	#33726	1	1	yes	characteristic, complemented, Sylow
141	C₂²×C₈	—	61	1	no	—
142	C₂²×C₈	—	61	1	no	—
143	C₄⋊Q₈	—	122	4	no	—
144	#125	—	2	2	no	—
145	#125	—	122	2	no	—
146	C₂²⋊Q₈	—	244	4	no	—
147	C₄×Q₈	—	244	4	no	—
148	#117	—	122	4	no	—
149	C₂×C₄²	—	61	1	no	—
150	C₂×C₁₆	—	61	1	no	—
151	C₂×C₁₆	—	61	1	no	—
152	C₂×C₁₆	—	61	4	no	—
153	C₂×C₁₆	—	61	4	no	—
154	C₂×C₁₆	C₂²×D₆₁	1	1	yes	characteristic
155	C₂×C₁₆	C₂²×D₆₁	1	1	yes	characteristic
156	#107	—	61	4	no	—
157	#107	—	122	4	no	—
158	#107	C₂²×D₆₁	1	4	yes	—
159	#106	—	2	2	no	—
160	#106	—	122	4	no	—
161	#106	—	122	2	no	—
162	C₄×C₈	—	61	1	no	—
163	C₄×C₈	—	61	1	no	—
164	C₄×C₈	—	61	1	no	—
165	C₄×C₈	C₂²×D₆₁	1	1	yes	characteristic
166	C₂×Q₈	—	244	4	no	—
167	C₂²×C₄	—	61	1	no	—
168	C₂²×C₄	—	61	2	no	—
169	C₂×C₈	—	61	1	no	—
170	C₂×C₈	—	61	1	no	—
171	C₂×C₈	—	61	2	no	—
172	C₂×C₈	—	61	2	no	—
173	C₂×C₈	—	61	4	no	—
174	C₂×C₈	D₄×D₆₁	1	1	yes	characteristic
175	C₂×C₈	C₂³×D₆₁	1	1	yes	characteristic
176	C₄⋊C₄	—	2	4	no	—
177	C₄⋊C₄	—	4	4	no	—
178	C₄⋊C₄	—	122	4	no	—
179	C₄⋊C₄	—	244	4	no	—
180	C₄⋊C₄	—	244	4	no	—
181	C₂²⋊C₄	—	244	4	no	—
182	C₄²	—	61	1	no	—
183	C₄²	—	61	2	no	—
184	C₄²	—	244	4	no	—
185	C₄²	D₄×D₆₁	1	1	yes	characteristic
186	C₁₆	—	61	2	no	—
187	C₁₆	—	61	2	no	—
188	C₁₆	Q₈×D₆₁	1	2	yes	—
189	C₁₆	Q₈×D₆₁	1	2	yes	—
190	C₂³	—	61	1	no	—
191	Q₈	—	488	8	no	—
192	C₂×C₄	—	4	4	no	—
193	C₂×C₄	—	61	1	no	—
194	C₂×C₄	—	61	2	no	—
195	C₂×C₄	—	61	2	no	—
196	C₂×C₄	—	61	2	no	—
197	C₂×C₄	—	122	4	no	—
198	C₂×C₄	—	244	4	no	—
199	C₂×C₄	D₈×D₆₁	1	2	yes	—
200	C₂×C₄	C₂×D₄×D₆₁	1	1	yes	characteristic
201	C₈	—	61	2	no	—
202	C₈	—	61	2	no	—
203	C₈	Q₁₆×D₆₁	1	2	yes	—
204	C₈	C₂×Q₈×D₆₁	1	1	yes	characteristic
205	C₈	C₂×Q₈×D₆₁	1	1	yes	characteristic
206	C₂²	—	61	2	no	—
207	C₂²	—	122	2	no	—
208	C₂²	C₂×D₈×D₆₁	1	1	yes	characteristic
209	C₄	—	2	2	no	—
210	C₄	—	8	4	no	—
211	C₄	—	61	2	no	—
212	C₄	—	61	4	no	—
213	C₄	—	488	4	no	—
214	C₄	C₄⋊Q₈×D₆₁	1	1	yes	characteristic
215	C₄	C₂×Q₁₆×D₆₁	1	1	yes	characteristic
216	C₂	—	122	2	no	—
217	C₂	#500×D₆₁	1	1	yes	characteristic
218	C₂	#508×D₆₁	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₈×D₆₁
commutator, cosocle	C₂×C₈×C₆₁	C₂⁴
Frattini	C₂×C₈	C₂³×D₆₁
Fitting	C₆₁×#2253	C₂
socle	C₂²×C₆₁	C₂²×D₈
prime radical	C₂²×D₆₁	C₂×D₈
hypercenter	#2253	D₆₁
hypocenter	C₆₁	#33726
LCS	G ▷ C₂×C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2253

Complements

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

N	H ≅ G/N	type
2. #21448100	C₂	semidirect
3. #21448098	C₂	semidirect
4. #21448095	C₂	semidirect
5. #21448093	C₂	semidirect
9. C₆₁×#2253	C₂	semidirect
41. C₁₆×D₆₁	Q₈	semidirect
42. C₁₆×D₆₁	Q₈	semidirect
111. #2253	D₆₁	semidirect
140. C₆₁	#33726	semidirect

Maximal subgroups

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

subgroup	index
#21456332	2
#21448100	2
#21448098	2
#21448095	2
#21448093	2
#21447602	2
#21447600	2
C₄×C₁₆×D₆₁	2
C₆₁×#2253	2
#33726	61