Subgroups — #29598524

Order 15616.

566 conjugacy classes of subgroups.

The subgroup lattice has 264 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, complemented
2	C₂×#16839575	C₂	1	2	yes	maximal, complemented
3	#21447535	C₂	1	2	yes	maximal, complemented
4	#21445698	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₆₁×#493	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	2	no	—
11	#16839575	—	2	4	no	—
12	#16839563	—	2	4	no	—
13	#16839522	—	2	1	no	—
14	C₂²×D₈×C₆₁	—	2	1	no	—
15	C₂²×D₈×C₆₁	C₂²	1	2	yes	—
16	C₂×#127×C₆₁	—	2	1	no	—
17	C₆₁×#493	—	2	2	no	—
18	C₆₁×#493	C₄	1	4	yes	complemented
19	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
20	C₂×#11758194	—	2	1	no	—
21	#14519736	—	2	1	no	—
22	#14519717	—	2	2	no	—
23	#14519717	—	2	2	no	—
24	C₂×#11758185	—	2	2	no	—
25	#14519713	—	2	1	no	—
26	#14519713	D₄	1	2	yes	complemented
27	#14519711	D₄	1	2	yes	complemented
28	#14519709	—	2	1	no	—
29	#14519687	—	4	1	no	—
30	C₈×Dic₆₁	—	4	1	no	—
31	C₂²×D₄×C₆₁	—	2	2	no	—
32	C₂²×D₄×C₆₁	—	4	2	no	—
33	C₂×D₈×C₆₁	—	2	4	no	—
34	C₂×D₈×C₆₁	—	2	2	no	—
35	C₂×D₈×C₆₁	—	2	2	no	—
36	C₂×D₈×C₆₁	—	2	2	no	—
37	C₂×D₈×C₆₁	—	2	2	no	—
38	C₂×D₈×C₆₁	—	2	2	no	—
39	C₂×D₈×C₆₁	—	4	1	no	—
40	C₂×D₈×C₆₁	—	4	1	no	—
41	C₂×D₈×C₆₁	—	4	2	no	—
42	C₂×D₈×C₆₁	C₂×C₄	1	4	yes	—
43	C₂²×C₈×C₆₁	D₄	1	1	yes	characteristic
44	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
45	#127×C₆₁	—	2	4	no	—
46	#127×C₆₁	—	4	2	no	—
47	C₂×C₄²×C₆₁	D₄	1	1	yes	characteristic
48	C₄×C₈×C₆₁	C₂×C₄	1	2	yes	—
49	C₄×C₈×C₆₁	D₄	1	2	yes	—
50	C₂⁴×C₆₁	—	4	2	no	—
51	C₂×D₄×C₆₁	—	2	4	no	—
52	C₂×D₄×C₆₁	—	4	8	no	—
53	C₂×D₄×C₆₁	—	4	2	no	—
54	C₂×D₄×C₆₁	—	4	2	no	—
55	C₂×D₄×C₆₁	—	4	2	no	—
56	C₂×D₄×C₆₁	—	4	2	no	—
57	C₂×D₄×C₆₁	—	4	2	no	—
58	C₂×D₄×C₆₁	—	8	8	no	—
59	C₂²×C₄×C₆₁	—	2	1	no	—
60	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
61	D₈×C₆₁	—	2	4	no	—
62	D₈×C₆₁	—	2	4	no	—
63	D₈×C₆₁	—	4	4	no	—
64	D₈×C₆₁	—	4	4	no	—
65	D₈×C₆₁	—	4	2	no	—
66	D₈×C₆₁	—	4	2	no	—
67	C₂×C₈×C₆₁	C₂²⋊C₄	1	2	yes	—
68	C₂×C₈×C₆₁	D₈	1	1	yes	characteristic
69	C₂×C₈×C₆₁	D₈	1	1	yes	characteristic
70	C₂×C₈×C₆₁	SD₁₆	1	2	yes	—
71	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
72	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
73	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
74	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
75	C₂×C₈×C₆₁	#41	1	1	yes	characteristic
76	C₂×C₈×C₆₁	#41	1	1	yes	characteristic
77	C₄²×C₆₁	—	2	1	no	—
78	C₄²×C₆₁	C₂²⋊C₄	1	2	yes	—
79	C₂²⋊Dic₆₁	—	4	1	no	—
80	C₄⋊Dic₆₁	—	2	2	no	—
81	C₄×Dic₆₁	—	4	1	no	—
82	#11758194	—	2	2	no	—
83	#11758194	—	4	1	no	—
84	C₂×#92418	—	4	1	no	—
85	#11758185	—	4	2	no	—
86	C₂³×C₆₁	—	4	4	no	—
87	C₂³×C₆₁	—	8	8	no	—
88	C₂³×C₆₁	—	8	2	no	—
89	C₂³×C₆₁	—	8	2	no	—
90	C₂³×C₆₁	AGL(1,ℤ/8)	1	1	yes	characteristic
91	D₄×C₆₁	—	4	4	no	—
92	D₄×C₆₁	—	4	4	no	—
93	D₄×C₆₁	—	8	16	no	—
94	D₄×C₆₁	—	8	2	no	—
95	D₄×C₆₁	—	8	2	no	—
96	C₂×C₄×C₆₁	—	2	4	no	—
97	C₂×C₄×C₆₁	—	2	2	no	—
98	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
99	C₂×C₄×C₆₁	#121	1	1	yes	characteristic
100	C₂×C₄×C₆₁	#121	1	1	yes	characteristic
101	C₂×C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
102	C₂×C₄×C₆₁	C₂×SD₁₆	1	1	yes	characteristic
103	C₂×C₄×C₆₁	AGL(1,ℤ/8)	1	1	yes	characteristic
104	C₂×Dic₆₁	—	4	2	no	—
105	C₈×C₆₁	#102	1	2	yes	—
106	C₈×C₆₁	#102	1	2	yes	—
107	C₈×C₆₁	C₄×D₄	1	1	yes	characteristic
108	C₈×C₆₁	C₄×D₄	1	1	yes	characteristic
109	C₈×C₆₁	#124	1	1	yes	characteristic
110	C₈×C₆₁	#127	1	1	yes	characteristic
111	#92418	—	4	2	no	—
112	#13261	—	61	1	no	maximal, Sylow
113	C₂²×C₆₁	—	2	2	no	—
114	C₂²×C₆₁	—	8	8	no	—
115	C₂²×C₆₁	—	8	4	no	—
116	C₂²×C₆₁	—	16	4	no	—
117	C₂²×C₆₁	—	16	4	no	—
118	C₂²×C₆₁	#418	1	1	yes	characteristic
119	C₂²×C₆₁	#459	1	1	yes	characteristic
120	C₂²×C₆₁	#460	1	1	yes	characteristic
121	C₄×C₆₁	—	2	4	no	—
122	C₄×C₆₁	#390	1	1	yes	characteristic
123	C₄×C₆₁	C₂×#102	1	1	yes	characteristic
124	C₄×C₆₁	#459	1	1	yes	characteristic
125	C₄×C₆₁	#460	1	1	yes	characteristic
126	Dic₆₁	—	8	2	no	—
127	C₂×#493	D₆₁	1	1	yes	characteristic
128	C₂×#364	—	61	1	no	—
129	C₂×#361	—	61	2	no	—
130	#2140	—	61	2	no	—
131	#1772	—	61	1	no	—
132	C₂×C₆₁	—	2	1	no	—
133	C₂×C₆₁	—	2	1	no	—
134	C₂×C₆₁	—	16	4	no	—
135	C₂×C₆₁	#1982	1	1	yes	characteristic
136	C₂×C₆₁	#2213	1	2	yes	—
137	C₂²×D₈	—	2	1	no	—
138	C₂²×D₈	C₂×D₆₁	1	2	yes	—
139	C₂×#127	—	2	1	no	—
140	C₂×M₃₂	—	61	2	no	—
141	#493	—	2	2	no	—
142	#493	Dic₆₁	1	4	yes	complemented
143	C₂×#108	—	61	1	no	—
144	#427	—	61	1	no	—
145	#405	—	122	1	no	—
146	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
147	#364	—	122	2	no	—
148	#361	—	122	4	no	—
149	#357	—	122	4	no	—
150	#339	—	122	1	no	—
151	C₆₁	#13261	1	1	yes	characteristic, complemented, Sylow
152	C₂²×D₄	—	2	2	no	—
153	C₂²×D₄	—	4	2	no	—
154	C₂×D₈	—	2	4	no	—
155	C₂×D₈	—	2	2	no	—
156	C₂×D₈	—	2	2	no	—
157	C₂×D₈	—	2	2	no	—
158	C₂×D₈	—	2	2	no	—
159	C₂×D₈	—	2	2	no	—
160	C₂×D₈	—	4	1	no	—
161	C₂×D₈	—	4	1	no	—
162	C₂×D₈	—	4	2	no	—
163	C₂×D₈	C₂×Dic₆₁	1	4	yes	—
164	C₂×M₁₆	—	122	1	no	—
165	C₂²×C₈	C₂²⋊D₆₁	1	1	yes	characteristic
166	C₂²×C₈	C₂²×D₆₁	1	1	yes	characteristic
167	#127	—	2	4	no	—
168	#127	—	4	2	no	—
169	#117	—	122	1	no	—
170	C₂×C₄²	C₂²⋊D₆₁	1	1	yes	characteristic
171	M₃₂	—	122	2	no	—
172	M₃₂	—	122	2	no	—
173	C₂×C₁₆	—	122	2	no	—
174	#108	—	61	2	no	—
175	#108	—	122	1	no	—
176	#107	—	61	2	no	—
177	#106	—	122	1	no	—
178	#97	—	244	1	no	—
179	C₄×C₈	—	244	1	no	—
180	C₄×C₈	C₂×Dic₆₁	1	2	yes	—
181	C₄×C₈	C₂²⋊D₆₁	1	2	yes	—
182	C₂⁴	—	4	2	no	—
183	C₂×D₄	—	2	4	no	—
184	C₂×D₄	—	4	8	no	—
185	C₂×D₄	—	4	2	no	—
186	C₂×D₄	—	4	2	no	—
187	C₂×D₄	—	4	2	no	—
188	C₂×D₄	—	4	2	no	—
189	C₂×D₄	—	4	2	no	—
190	C₂×D₄	—	8	8	no	—
191	C₂²×C₄	—	2	1	no	—
192	C₂²×C₄	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
193	D₈	—	2	4	no	—
194	D₈	—	2	4	no	—
195	D₈	—	4	4	no	—
196	D₈	—	4	4	no	—
197	D₈	—	4	2	no	—
198	D₈	—	4	2	no	—
199	M₁₆	—	122	2	no	—
200	M₁₆	—	244	1	no	—
201	C₂×C₈	—	244	1	no	—
202	C₂×C₈	#11758199	1	1	yes	characteristic
203	C₂×C₈	#11758199	1	1	yes	characteristic
204	C₂×C₈	#11758200	1	2	yes	—
205	C₂×C₈	C₂²⋊Dic₆₁	1	2	yes	—
206	C₂×C₈	D₄×D₆₁	1	1	yes	characteristic
207	C₂×C₈	D₄×D₆₁	1	1	yes	characteristic
208	C₂×C₈	#11758224	1	1	yes	characteristic
209	C₂×C₈	#11758224	1	1	yes	characteristic
210	C₂×C₈	C₂²×Dic₆₁	1	1	yes	characteristic
211	C₂×C₈	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
212	C₄⋊C₄	—	122	2	no	—
213	C₂²⋊C₄	—	244	1	no	—
214	C₄²	—	2	1	no	—
215	C₄²	—	244	1	no	—
216	C₄²	C₂²⋊Dic₆₁	1	2	yes	—
217	C₁₆	—	244	2	no	—
218	C₂³	—	4	4	no	—
219	C₂³	—	8	8	no	—
220	C₂³	—	8	2	no	—
221	C₂³	—	8	2	no	—
222	C₂³	#14519803	1	1	yes	characteristic
223	D₄	—	4	4	no	—
224	D₄	—	4	4	no	—
225	D₄	—	8	16	no	—
226	D₄	—	8	2	no	—
227	D₄	—	8	2	no	—
228	C₂×C₄	—	2	4	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	244	2	no	—
231	C₂×C₄	C₂×C₂²⋊Dic₆₁	1	1	yes	characteristic
232	C₂×C₄	#14519748	1	1	yes	characteristic
233	C₂×C₄	#14519748	1	1	yes	characteristic
234	C₂×C₄	C₂×#11758199	1	1	yes	characteristic
235	C₂×C₄	C₂×#11758200	1	1	yes	characteristic
236	C₂×C₄	#14519803	1	1	yes	characteristic
237	C₈	—	244	2	no	—
238	C₈	#14519698	1	2	yes	—
239	C₈	#14519698	1	2	yes	—
240	C₈	D₄×Dic₆₁	1	1	yes	characteristic
241	C₈	D₄×Dic₆₁	1	1	yes	characteristic
242	C₈	#14519764	1	1	yes	characteristic
243	C₈	#14519773	1	1	yes	characteristic
244	C₂²	—	2	2	no	—
245	C₂²	—	8	8	no	—
246	C₂²	—	8	4	no	—
247	C₂²	—	16	4	no	—
248	C₂²	—	16	4	no	—
249	C₂²	#16839767	1	1	yes	characteristic
250	C₂²	#16839954	1	1	yes	characteristic
251	C₂²	#16839959	1	1	yes	characteristic
252	C₄	—	2	4	no	—
253	C₄	—	488	2	no	—
254	C₄	#16839665	1	1	yes	characteristic
255	C₄	C₂×#14519698	1	1	yes	characteristic
256	C₄	#16839954	1	1	yes	characteristic
257	C₄	#16839959	1	1	yes	characteristic
258	C₂	—	2	1	no	—
259	C₂	—	2	1	no	—
260	C₂	—	16	4	no	—
261	C₂	#21446747	1	1	yes	characteristic
262	C₂	#21447892	1	2	yes	—

	subgroup	G/H
center	C₂²	#16839954
commutator	C₂×C₈×C₆₁	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₆₁
Fitting, prime radical	C₂×C₆₁×#493	C₂
socle	C₂²×C₆₁	#459
cosocle	C₂²×C₈×C₆₁	C₂³
hypercenter	C₂×#493	D₆₁
hypocenter	C₆₁	#13261
LCS	G ▷ C₂×C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#493

Complements

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

N	H ≅ G/N	type
1. C₂×#16839583	C₂	semidirect
2. C₂×#16839575	C₂	semidirect
3. #21447535	C₂	semidirect
4. #21445698	C₂	semidirect
18. C₆₁×#493	C₄	semidirect
26. #14519713	D₄	semidirect
27. #14519711	D₄	semidirect
142. #493	Dic₆₁	semidirect
151. C₆₁	#13261	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
C₂×#16839575	2
#21447535	2
#21445698	2
C₂×C₆₁×#493	2
#13261	61