Subgroups — #29598568

Order 15616.

332 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	C₂×#16839583	C₂	1	1	yes	maximal, characteristic
2	#21445698	C₂	1	1	yes	maximal, characteristic
3	#21443935	C₂	1	2	yes	maximal
4	#21443920	C₂	1	2	yes	maximal
5	C₆₁×#3160	C₂	1	1	yes	maximal, characteristic
6	C₂×#14519717	C₂²	1	2	yes	—
7	C₂×#14519713	C₂²	1	1	yes	characteristic
8	#16839800	C₂²	1	1	yes	characteristic
9	#16839717	—	2	1	no	—
10	#16839583	—	2	1	no	—
11	#16839583	—	2	1	no	—
12	#16839522	—	2	1	no	—
13	C₆₁×#533	—	2	1	no	—
14	C₆₁×#500	—	2	1	no	—
15	C₆₁×#499	—	2	1	no	—
16	C₆₁×#499	C₄	1	2	yes	complemented
17	C₆₁×#498	C₄	1	2	yes	complemented
18	C₆₁×#427	—	2	1	no	—
19	C₆₁×#427	C₂²	1	2	yes	—
20	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
21	C₂×#11758194	—	2	1	no	—
22	#14519736	—	2	1	no	—
23	#14519717	—	2	2	no	—
24	#14519717	—	2	2	no	—
25	C₂×#11758185	—	2	2	no	—
26	#14519713	—	2	1	no	—
27	#14519713	Q₈	1	2	yes	complemented
28	#14519711	—	2	1	no	—
29	#14519709	Q₈	1	2	yes	complemented
30	#14519687	—	4	1	no	—
31	C₈×Dic₆₁	—	4	1	no	—
32	C₂²×C₈×C₆₁	D₄	1	1	yes	characteristic
33	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
34	C₄⋊Q₈×C₆₁	—	2	2	no	—
35	#125×C₆₁	—	2	2	no	—
36	C₂²⋊Q₈×C₆₁	—	4	2	no	—
37	C₄×Q₈×C₆₁	—	4	2	no	—
38	#117×C₆₁	—	2	2	no	—
39	C₂×C₄²×C₆₁	D₄	1	1	yes	characteristic
40	#107×C₆₁	—	2	2	no	—
41	#107×C₆₁	—	2	2	no	—
42	#106×C₆₁	—	2	2	no	—
43	#106×C₆₁	C₂×C₄	1	4	yes	—
44	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
45	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
46	C₄×C₈×C₆₁	D₄	1	1	yes	characteristic
47	C₄×C₈×C₆₁	D₄	1	1	yes	characteristic
48	C₂×Q₈×C₆₁	—	4	2	no	—
49	C₂²×C₄×C₆₁	—	2	1	no	—
50	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
51	C₂×C₈×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
52	C₂×C₈×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
53	C₂×C₈×C₆₁	SD₁₆	1	1	yes	characteristic
54	C₂×C₈×C₆₁	SD₁₆	1	1	yes	characteristic
55	C₂×C₈×C₆₁	Q₁₆	1	1	yes	characteristic
56	C₂×C₈×C₆₁	Q₁₆	1	1	yes	characteristic
57	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
59	C₂×C₈×C₆₁	C₂×Q₈	1	1	yes	characteristic
60	C₂×C₈×C₆₁	C₂×Q₈	1	1	yes	characteristic
61	C₂×C₈×C₆₁	#41	1	1	yes	characteristic
62	C₂×C₈×C₆₁	#41	1	1	yes	characteristic
63	C₄⋊C₄×C₆₁	—	2	4	no	—
64	C₄⋊C₄×C₆₁	—	4	4	no	—
65	C₄⋊C₄×C₆₁	—	4	2	no	—
66	C₂²⋊C₄×C₆₁	—	4	2	no	—
67	C₄²×C₆₁	—	2	1	no	—
68	C₄²×C₆₁	—	4	2	no	—
69	C₄²×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
70	C₄²×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
71	C₂²⋊Dic₆₁	—	4	1	no	—
72	C₄⋊Dic₆₁	—	2	2	no	—
73	C₄×Dic₆₁	—	4	1	no	—
74	#11758194	—	2	2	no	—
75	#11758194	—	4	1	no	—
76	C₂×#92418	—	4	1	no	—
77	#11758185	—	4	2	no	—
78	C₂³×C₆₁	AGL(1,ℤ/8)	1	1	yes	characteristic
79	Q₈×C₆₁	—	8	4	no	—
80	C₂×C₄×C₆₁	—	2	1	no	—
81	C₂×C₄×C₆₁	—	2	1	no	—
82	C₂×C₄×C₆₁	—	4	2	no	—
83	C₂×C₄×C₆₁	—	4	4	no	—
84	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
85	C₂×C₄×C₆₁	C₂²⋊Q₈	1	1	yes	characteristic
86	C₂×C₄×C₆₁	C₂²⋊Q₈	1	1	yes	characteristic
87	C₂×C₄×C₆₁	C₂×SD₁₆	1	1	yes	characteristic
88	C₂×C₄×C₆₁	C₂×Q₁₆	1	1	yes	characteristic
89	C₂×C₄×C₆₁	AGL(1,ℤ/8)	1	1	yes	characteristic
90	C₂×Dic₆₁	—	4	2	no	—
91	C₈×C₆₁	#103	1	1	yes	characteristic
92	C₈×C₆₁	#103	1	1	yes	characteristic
93	C₈×C₆₁	#103	1	1	yes	characteristic
94	C₈×C₆₁	#103	1	1	yes	characteristic
95	C₈×C₆₁	C₄×Q₈	1	1	yes	characteristic
96	C₈×C₆₁	C₄×Q₈	1	1	yes	characteristic
97	C₈×C₆₁	#124	1	1	yes	characteristic
98	C₈×C₆₁	C₄⋊Q₈	1	1	yes	characteristic
99	#92418	—	4	2	no	—
100	#13271	—	61	1	no	maximal, Sylow
101	C₂²×C₆₁	—	4	1	no	—
102	C₂²×C₆₁	#418	1	1	yes	characteristic
103	C₂²×C₆₁	#476	1	1	yes	characteristic
104	C₂²×C₆₁	#477	1	1	yes	characteristic
105	C₄×C₆₁	—	2	2	no	—
106	C₄×C₆₁	—	4	1	no	—
107	C₄×C₆₁	—	8	4	no	—
108	C₄×C₆₁	#391	1	1	yes	characteristic
109	C₄×C₆₁	C₂×#103	1	1	yes	characteristic
110	C₄×C₆₁	#476	1	1	yes	characteristic
111	C₄×C₆₁	#477	1	1	yes	characteristic
112	Dic₆₁	—	8	2	no	—
113	#3160	D₆₁	1	1	yes	characteristic
114	C₂×#364	—	61	1	no	—
115	#1772	—	61	1	no	—
116	#1389	—	61	2	no	—
117	#1384	—	61	2	no	—
118	C₂×C₆₁	—	2	1	no	—
119	C₂×C₆₁	—	2	1	no	—
120	C₂×C₆₁	—	2	1	no	—
121	C₂×C₆₁	#1987	1	1	yes	characteristic
122	#533	—	2	1	no	—
123	C₂×M₃₂	—	61	2	no	—
124	#500	—	2	1	no	—
125	#499	—	2	1	no	—
126	#499	Dic₆₁	1	2	yes	complemented
127	#498	Dic₆₁	1	2	yes	complemented
128	C₂×#108	—	61	1	no	—
129	#427	—	2	1	no	—
130	#427	—	61	1	no	—
131	#427	C₂×D₆₁	1	2	yes	—
132	#405	—	122	1	no	—
133	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
134	#364	—	122	1	no	—
135	#364	—	122	1	no	—
136	#339	—	122	1	no	—
137	C₆₁	#13271	1	1	yes	characteristic, complemented, Sylow
138	C₂×M₁₆	—	122	1	no	—
139	C₂²×C₈	C₂²⋊D₆₁	1	1	yes	characteristic
140	C₂²×C₈	C₂²×D₆₁	1	1	yes	characteristic
141	C₄⋊Q₈	—	2	2	no	—
142	#125	—	2	2	no	—
143	C₂²⋊Q₈	—	4	2	no	—
144	C₄×Q₈	—	4	2	no	—
145	#117	—	2	2	no	—
146	#117	—	122	1	no	—
147	C₂×C₄²	C₂²⋊D₆₁	1	1	yes	characteristic
148	M₃₂	—	122	2	no	—
149	M₃₂	—	122	2	no	—
150	C₂×C₁₆	—	122	2	no	—
151	#108	—	61	2	no	—
152	#108	—	122	1	no	—
153	#107	—	2	2	no	—
154	#107	—	2	2	no	—
155	#107	—	122	1	no	—
156	#106	—	2	2	no	—
157	#106	—	61	2	no	—
158	#106	C₂×Dic₆₁	1	4	yes	—
159	#97	—	244	1	no	—
160	C₄×C₈	—	244	1	no	—
161	C₄×C₈	C₂×Dic₆₁	1	1	yes	characteristic
162	C₄×C₈	C₂×Dic₆₁	1	1	yes	characteristic
163	C₄×C₈	C₂²⋊D₆₁	1	1	yes	characteristic
164	C₄×C₈	C₂²⋊D₆₁	1	1	yes	characteristic
165	C₂×Q₈	—	4	2	no	—
166	C₂²×C₄	—	2	1	no	—
167	C₂²×C₄	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
168	M₁₆	—	122	2	no	—
169	M₁₆	—	244	1	no	—
170	C₂×C₈	—	244	1	no	—
171	C₂×C₈	#11758201	1	1	yes	characteristic
172	C₂×C₈	#11758201	1	1	yes	characteristic
173	C₂×C₈	#11758202	1	1	yes	characteristic
174	C₂×C₈	#11758202	1	1	yes	characteristic
175	C₂×C₈	C₂²⋊Dic₆₁	1	1	yes	characteristic
176	C₂×C₈	C₂²⋊Dic₆₁	1	1	yes	characteristic
177	C₂×C₈	Q₈×D₆₁	1	1	yes	characteristic
178	C₂×C₈	Q₈×D₆₁	1	1	yes	characteristic
179	C₂×C₈	#11758226	1	1	yes	characteristic
180	C₂×C₈	#11758226	1	1	yes	characteristic
181	C₂×C₈	C₂²×Dic₆₁	1	1	yes	characteristic
182	C₂×C₈	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
183	C₄⋊C₄	—	2	4	no	—
184	C₄⋊C₄	—	4	4	no	—
185	C₄⋊C₄	—	4	2	no	—
186	C₄⋊C₄	—	122	2	no	—
187	C₂²⋊C₄	—	4	2	no	—
188	C₂²⋊C₄	—	244	1	no	—
189	C₄²	—	2	1	no	—
190	C₄²	—	4	2	no	—
191	C₄²	—	244	1	no	—
192	C₄²	C₂²⋊Dic₆₁	1	1	yes	characteristic
193	C₄²	C₂²⋊Dic₆₁	1	1	yes	characteristic
194	C₁₆	—	244	2	no	—
195	C₂³	#14519803	1	1	yes	characteristic
196	Q₈	—	8	4	no	—
197	C₂×C₄	—	2	1	no	—
198	C₂×C₄	—	2	1	no	—
199	C₂×C₄	—	4	2	no	—
200	C₂×C₄	—	4	4	no	—
201	C₂×C₄	—	244	2	no	—
202	C₂×C₄	C₂×C₂²⋊Dic₆₁	1	1	yes	characteristic
203	C₂×C₄	#14519753	1	1	yes	characteristic
204	C₂×C₄	#14519753	1	1	yes	characteristic
205	C₂×C₄	C₂×#11758201	1	1	yes	characteristic
206	C₂×C₄	C₂×#11758202	1	1	yes	characteristic
207	C₂×C₄	#14519803	1	1	yes	characteristic
208	C₈	—	244	2	no	—
209	C₈	#14519701	1	1	yes	characteristic
210	C₈	#14519701	1	1	yes	characteristic
211	C₈	#14519701	1	1	yes	characteristic
212	C₈	#14519701	1	1	yes	characteristic
213	C₈	Q₈×Dic₆₁	1	1	yes	characteristic
214	C₈	Q₈×Dic₆₁	1	1	yes	characteristic
215	C₈	#14519766	1	1	yes	characteristic
216	C₈	#14519775	1	1	yes	characteristic
217	C₂²	—	4	1	no	—
218	C₂²	#16839767	1	1	yes	characteristic
219	C₂²	#16840049	1	1	yes	characteristic
220	C₂²	#16840054	1	1	yes	characteristic
221	C₄	—	2	2	no	—
222	C₄	—	4	1	no	—
223	C₄	—	8	4	no	—
224	C₄	—	488	2	no	—
225	C₄	#16839669	1	1	yes	characteristic
226	C₄	C₂×#14519701	1	1	yes	characteristic
227	C₄	#16840049	1	1	yes	characteristic
228	C₄	#16840054	1	1	yes	characteristic
229	C₂	—	2	1	no	—
230	C₂	—	2	1	no	—
231	C₂	—	2	1	no	—
232	C₂	#21446768	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#16840049
commutator	C₂×C₈×C₆₁	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₆₁
Fitting	C₆₁×#3160	C₂
socle	C₂×C₆₁	#1987
prime radical	C₂³×C₆₁	AGL(1,ℤ/8)
cosocle	C₂²×C₈×C₆₁	C₂³
hypercenter	#3160	D₆₁
hypocenter	C₆₁	#13271
LCS	G ▷ C₂×C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ #3160

Complements

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

N	H ≅ G/N	type
16. C₆₁×#499	C₄	semidirect
17. C₆₁×#498	C₄	semidirect
27. #14519713	Q₈	semidirect
29. #14519709	Q₈	semidirect
126. #499	Dic₆₁	semidirect
127. #498	Dic₆₁	semidirect
137. C₆₁	#13271	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16839583	2
#21445698	2
#21443935	2
#21443920	2
C₆₁×#3160	2
#13271	61