Subgroups — #29595285

Order 15616.

486 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16839875	C₂	1	1	yes	maximal, characteristic
2	C₂×#16839576	C₂	1	1	yes	maximal, characteristic
3	#21447536	C₂	1	1	yes	maximal, characteristic
4	#21445703	C₂	1	1	yes	maximal, characteristic, complemented
5	#21443925	C₂	1	1	yes	maximal, characteristic, complemented
6	#21443918	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₆₁×#347	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×D₄₈₈	C₂²	1	1	yes	characteristic
9	C₂×C₄×D₂₄₄	—	2	1	no	—
10	#16839875	—	2	2	no	—
11	#16839875	C₄	1	4	yes	complemented
12	C₂×#14519713	C₂²	1	1	yes	characteristic
13	#16839800	C₂²	1	1	yes	characteristic
14	C₂×#14519709	C₂²	1	1	yes	characteristic
15	C₂×#14519699	—	2	1	no	—
16	#16839717	—	2	1	no	—
17	#16839576	C₄	1	4	yes	complemented
18	#16839564	C₄	1	4	yes	complemented
19	#16839522	—	2	1	no	—
20	C₂×M₃₂×C₆₁	C₂²	1	1	yes	characteristic
21	C₂×M₃₂×C₆₁	C₂²	1	1	yes	characteristic
22	C₂×#97×C₆₁	C₂²	1	1	yes	characteristic
23	C₆₁×#347	—	2	2	no	—
24	C₂²×D₂₄₄	—	2	1	no	—
25	C₂²×C₄×D₆₁	—	4	1	no	—
26	C₂×D₄₈₈	—	2	1	no	—
27	C₂×D₄₈₈	—	2	1	no	—
28	C₂×D₄₈₈	—	2	1	no	—
29	C₂×D₄₈₈	C₂×C₄	1	2	yes	—
30	C₂×D₄₈₈	C₂×C₄	1	2	yes	—
31	C₂×D₄₈₈	C₂×C₄	1	2	yes	—
32	C₂×#11758194	—	2	1	no	—
33	C₄×D₂₄₄	—	2	4	no	—
34	C₄×D₂₄₄	—	4	2	no	—
35	#14519736	—	2	1	no	—
36	C₂×C₄⋊Dic₆₁	—	2	1	no	—
37	C₂×#11758198	—	4	1	no	—
38	#14519713	—	2	1	no	—
39	#14519713	C₂×C₄	1	2	yes	complemented
40	#14519711	C₂×C₄	1	2	yes	complemented
41	#14519709	—	2	1	no	—
42	#14519709	—	2	1	no	—
43	#14519709	C₂×C₄	1	2	yes	—
44	#14519699	—	2	4	no	—
45	#14519687	—	4	1	no	—
46	C₈×Dic₆₁	—	4	1	no	—
47	C₂²×C₈×C₆₁	D₄	1	1	yes	characteristic
48	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
49	C₂×C₄²×C₆₁	D₄	1	1	yes	characteristic
50	M₃₂×C₆₁	—	2	1	no	—
51	M₃₂×C₆₁	—	2	1	no	—
52	M₃₂×C₆₁	—	2	1	no	—
53	M₃₂×C₆₁	C₂×C₄	1	2	yes	complemented
54	C₂×C₁₆×C₆₁	—	2	1	no	—
55	C₂×C₁₆×C₆₁	C₂×C₄	1	2	yes	complemented
56	#97×C₆₁	C₂×C₄	1	2	yes	—
57	#97×C₆₁	D₄	1	2	yes	—
58	C₂³×D₆₁	—	4	1	no	—
59	C₂²×C₄×C₆₁	—	2	1	no	—
60	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
61	C₂²×Dic₆₁	—	4	1	no	—
62	C₂×D₂₄₄	—	2	2	no	—
63	C₂×D₂₄₄	—	2	2	no	—
64	C₂×D₂₄₄	—	2	2	no	—
65	C₂×D₂₄₄	—	4	1	no	—
66	C₂×D₂₄₄	—	4	1	no	—
67	C₂×D₂₄₄	—	4	1	no	—
68	C₂×C₄×D₆₁	—	4	4	no	—
69	C₂×C₄×D₆₁	—	8	4	no	—
70	C₂×C₈×C₆₁	—	2	1	no	—
71	C₂×C₈×C₆₁	—	2	1	no	—
72	C₂×C₈×C₆₁	C₂²⋊C₄	1	2	yes	—
73	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
74	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
75	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
76	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
77	C₂×C₈×C₆₁	#41	1	1	yes	characteristic
78	C₂×C₈×C₆₁	#41	1	1	yes	characteristic
79	C₄²×C₆₁	—	2	1	no	—
80	C₄²×C₆₁	C₂²⋊C₄	1	2	yes	—
81	C₂²⋊Dic₆₁	—	4	1	no	—
82	#11758198	—	4	4	no	—
83	C₄⋊Dic₆₁	—	2	2	no	—
84	C₄⋊Dic₆₁	—	2	2	no	—
85	C₄⋊Dic₆₁	—	4	1	no	—
86	C₄×Dic₆₁	—	4	1	no	—
87	#11758194	—	2	2	no	—
88	#11758194	—	4	1	no	—
89	C₂×#92418	—	4	1	no	—
90	D₄₈₈	—	2	2	no	—
91	D₄₈₈	—	2	2	no	—
92	D₄₈₈	—	2	2	no	—
93	D₄₈₈	C₄²	1	4	yes	—
94	C₁₆×C₆₁	—	2	2	no	—
95	C₁₆×C₆₁	—	2	2	no	—
96	C₂³×C₆₁	AGL(1,ℤ/8)	1	1	yes	characteristic
97	C₂²×D₆₁	—	4	2	no	—
98	C₂²×D₆₁	—	4	2	no	—
99	C₂²×D₆₁	—	4	2	no	—
100	C₂²×D₆₁	—	8	2	no	—
101	C₂²×D₆₁	—	8	2	no	—
102	C₂×C₄×C₆₁	—	2	2	no	—
103	C₂×C₄×C₆₁	—	4	2	no	—
104	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
105	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
106	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
107	C₂×C₄×C₆₁	AGL(1,ℤ/8)	1	1	yes	characteristic
108	C₂×C₄×C₆₁	AGL(1,ℤ/8)	1	1	yes	characteristic
109	C₂×C₄×C₆₁	#137	1	1	yes	characteristic
110	C₂×Dic₆₁	—	4	2	no	—
111	C₂×Dic₆₁	—	4	2	no	—
112	C₂×Dic₆₁	—	8	2	no	—
113	D₂₄₄	—	2	4	no	—
114	D₂₄₄	—	4	2	no	—
115	D₂₄₄	—	4	2	no	—
116	D₂₄₄	—	4	2	no	—
117	C₄×D₆₁	—	8	8	no	—
118	C₈×C₆₁	—	2	2	no	—
119	C₈×C₆₁	C₂×C₄²	1	1	yes	characteristic
120	C₈×C₆₁	#117	1	1	yes	characteristic
121	C₈×C₆₁	C₄×D₄	1	1	yes	characteristic
122	C₈×C₆₁	C₄×D₄	1	1	yes	characteristic
123	#92418	—	4	2	no	—
124	#12647	—	61	1	no	maximal, Sylow
125	C₂²×C₆₁	—	2	2	no	—
126	C₂²×C₆₁	#418	1	1	yes	characteristic
127	C₂²×C₆₁	#440	1	1	yes	characteristic
128	C₂²×C₆₁	#442	1	1	yes	characteristic
129	C₂×D₆₁	—	4	4	no	—
130	C₂×D₆₁	—	8	2	no	—
131	C₂×D₆₁	—	8	2	no	—
132	C₂×D₆₁	—	8	4	no	—
133	C₂×D₆₁	—	8	4	no	—
134	C₄×C₆₁	—	4	2	no	—
135	C₄×C₆₁	C₄×C₂²⋊C₄	1	1	yes	characteristic
136	C₄×C₆₁	#417	1	1	yes	characteristic
137	C₄×C₆₁	#440	1	1	yes	characteristic
138	C₄×C₆₁	#442	1	1	yes	characteristic
139	Dic₆₁	—	8	2	no	—
140	Dic₆₁	—	8	2	no	—
141	C₂×#442	—	61	1	no	—
142	C₂×#361	—	61	1	no	—
143	#2140	—	61	1	no	—
144	C₂×#347	D₆₁	1	1	yes	characteristic, complemented
145	#1773	—	61	1	no	—
146	#1386	—	61	1	no	—
147	#1383	—	61	1	no	—
148	C₂×C₆₁	—	2	1	no	—
149	C₂×C₆₁	—	2	1	no	—
150	C₂×C₆₁	#1762	1	1	yes	characteristic
151	C₂×C₆₁	AGL(1,ℤ/16)	1	2	yes	—
152	D₆₁	—	8	4	no	—
153	C₂²×D₈	—	61	1	no	—
154	C₂×C₄×D₄	—	122	1	no	—
155	C₂×M₃₂	C₂×D₆₁	1	1	yes	characteristic
156	C₂×M₃₂	C₂×D₆₁	1	1	yes	characteristic
157	#442	—	61	4	no	—
158	#442	—	122	2	no	—
159	C₂×#108	—	61	1	no	—
160	#427	—	61	1	no	—
161	C₂×#106	—	61	1	no	—
162	C₂×#102	—	122	1	no	—
163	#405	—	122	1	no	—
164	C₂×#97	C₂×D₆₁	1	1	yes	characteristic
165	#361	—	61	4	no	—
166	#357	—	61	4	no	—
167	#347	—	2	2	no	—
168	#339	—	122	1	no	—
169	C₆₁	#12647	1	1	yes	characteristic, complemented, Sylow
170	C₂²×D₄	—	122	1	no	—
171	C₂³×C₄	—	244	1	no	—
172	C₂×D₈	—	61	2	no	—
173	C₂×D₈	—	61	2	no	—
174	C₂×D₈	—	61	2	no	—
175	C₂×D₈	—	122	1	no	—
176	C₂×D₈	—	122	1	no	—
177	C₂×D₈	—	122	1	no	—
178	C₂×M₁₆	—	122	1	no	—
179	C₂²×C₈	D₂₄₄	1	1	yes	characteristic
180	C₂²×C₈	C₂²×D₆₁	1	1	yes	characteristic
181	C₄×D₄	—	122	4	no	—
182	C₄×D₄	—	244	2	no	—
183	#117	—	122	1	no	—
184	C₂×C₄⋊C₄	—	122	1	no	—
185	C₂×C₂²⋊C₄	—	244	1	no	—
186	C₂×C₄²	D₂₄₄	1	1	yes	characteristic
187	M₃₂	—	2	1	no	—
188	M₃₂	—	2	1	no	—
189	M₃₂	—	2	1	no	—
190	M₃₂	C₄×D₆₁	1	2	yes	complemented
191	C₂×C₁₆	—	2	1	no	—
192	C₂×C₁₆	C₄×D₆₁	1	2	yes	complemented
193	#108	—	61	2	no	—
194	#108	—	122	1	no	—
195	#107	—	61	2	no	—
196	#106	—	61	2	no	—
197	#106	—	122	1	no	—
198	#106	—	122	1	no	—
199	#102	—	122	4	no	—
200	#97	—	244	1	no	—
201	#97	C₄×D₆₁	1	2	yes	—
202	#97	C₂²⋊D₆₁	1	2	yes	—
203	C₄×C₈	—	244	1	no	—
204	C₂⁴	—	244	1	no	—
205	C₂×D₄	—	122	2	no	—
206	C₂×D₄	—	122	2	no	—
207	C₂×D₄	—	122	2	no	—
208	C₂×D₄	—	244	1	no	—
209	C₂×D₄	—	244	1	no	—
210	C₂×D₄	—	244	1	no	—
211	C₂²×C₄	—	2	1	no	—
212	C₂²×C₄	—	488	4	no	—
213	C₂²×C₄	—	244	4	no	—
214	C₂²×C₄	—	244	1	no	—
215	C₂²×C₄	C₂×D₂₄₄	1	1	yes	characteristic
216	D₈	—	61	4	no	—
217	D₈	—	122	2	no	—
218	D₈	—	122	2	no	—
219	D₈	—	122	2	no	—
220	M₁₆	—	122	2	no	—
221	M₁₆	—	244	1	no	—
222	C₂×C₈	—	2	1	no	—
223	C₂×C₈	—	2	1	no	—
224	C₂×C₈	—	244	1	no	—
225	C₂×C₈	#11758198	1	2	yes	—
226	C₂×C₈	C₂×C₄×D₆₁	1	1	yes	characteristic
227	C₂×C₈	C₂×C₄×D₆₁	1	1	yes	characteristic
228	C₂×C₈	C₂×C₄×D₆₁	1	1	yes	characteristic
229	C₂×C₈	#11758222	1	1	yes	characteristic
230	C₂×C₈	#11758222	1	1	yes	characteristic
231	C₂×C₈	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
232	C₄⋊C₄	—	122	2	no	—
233	C₄⋊C₄	—	122	2	no	—
234	C₄⋊C₄	—	244	1	no	—
235	C₂²⋊C₄	—	244	4	no	—
236	C₂²⋊C₄	—	244	1	no	—
237	C₄²	—	2	1	no	—
238	C₄²	—	244	1	no	—
239	C₄²	#11758198	1	2	yes	—
240	C₁₆	—	2	2	no	—
241	C₁₆	—	2	2	no	—
242	C₂³	—	488	2	no	—
243	C₂³	—	488	2	no	—
244	C₂³	—	244	2	no	—
245	C₂³	—	244	2	no	—
246	C₂³	—	244	2	no	—
247	C₂³	#14519805	1	1	yes	characteristic
248	D₄	—	122	4	no	—
249	D₄	—	244	2	no	—
250	D₄	—	244	2	no	—
251	D₄	—	244	2	no	—
252	C₂×C₄	—	2	2	no	—
253	C₂×C₄	—	4	2	no	—
254	C₂×C₄	—	488	8	no	—
255	C₂×C₄	—	488	2	no	—
256	C₂×C₄	—	244	2	no	—
257	C₂×C₄	—	244	2	no	—
258	C₂×C₄	C₂×#11758198	1	1	yes	characteristic
259	C₂×C₄	C₄×D₂₄₄	1	1	yes	characteristic
260	C₂×C₄	C₄×D₂₄₄	1	1	yes	characteristic
261	C₂×C₄	#14519805	1	1	yes	characteristic
262	C₂×C₄	#14519805	1	1	yes	characteristic
263	C₂×C₄	#14519810	1	1	yes	characteristic
264	C₈	—	2	2	no	—
265	C₈	—	244	2	no	—
266	C₈	C₄²×D₆₁	1	1	yes	characteristic
267	C₈	#14519734	1	1	yes	characteristic
268	C₈	C₄×C₂²⋊D₆₁	1	1	yes	characteristic
269	C₈	C₄×C₂²⋊D₆₁	1	1	yes	characteristic
270	C₂²	—	2	2	no	—
271	C₂²	—	488	2	no	—
272	C₂²	—	488	2	no	—
273	C₂²	—	488	4	no	—
274	C₂²	—	488	4	no	—
275	C₂²	—	244	4	no	—
276	C₂²	#16839768	1	1	yes	characteristic
277	C₂²	#16839864	1	1	yes	characteristic
278	C₂²	#16839875	1	1	yes	characteristic
279	C₄	—	4	2	no	—
280	C₄	—	488	2	no	—
281	C₄	—	488	2	no	—
282	C₄	C₄×#11758198	1	1	yes	characteristic
283	C₄	#16839764	1	1	yes	characteristic
284	C₄	#16839864	1	1	yes	characteristic
285	C₄	#16839875	1	1	yes	characteristic
286	C₂	—	2	1	no	—
287	C₂	—	2	1	no	—
288	C₂	—	488	4	no	—
289	C₂	#21445645	1	1	yes	characteristic
290	C₂	#21447719	1	2	yes	—

	subgroup	G/H
center	C₂²	#16839875
commutator	C₈×C₆₁	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₆₁
Fitting	C₂×C₆₁×#347	C₂
socle	C₂²×C₆₁	#442
prime radical	C₂²×D₄₈₈	C₂²
cosocle	C₂²×C₈×C₆₁	C₂³
hypercenter	C₂×#347	D₆₁
hypocenter	C₆₁	#12647
LCS	G ▷ C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#97 ◁ C₂×#347

Complements

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

N	H ≅ G/N	type
4. #21445703	C₂	semidirect
5. #21443925	C₂	semidirect
6. #21443918	C₂	semidirect
7. C₂×C₆₁×#347	C₂	semidirect
11. #16839875	C₄	semidirect
17. #16839576	C₄	semidirect
18. #16839564	C₄	semidirect
39. #14519713	C₂×C₄	semidirect
40. #14519711	C₂×C₄	semidirect
53. M₃₂×C₆₁	C₂×C₄	semidirect
55. C₂×C₁₆×C₆₁	C₂×C₄	semidirect
144. C₂×#347	D₆₁	semidirect
169. C₆₁	#12647	semidirect
190. M₃₂	C₄×D₆₁	semidirect
192. C₂×C₁₆	C₄×D₆₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16839875	2
C₂×#16839576	2
#21447536	2
#21445703	2
#21443925	2
#21443918	2
C₂×C₆₁×#347	2
#12647	61