Subgroups — #29595249

Order 15616.

438 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16839880	C₂	1	1	yes	maximal, characteristic
2	C₂×#16839570	C₂	1	1	yes	maximal, characteristic
3	#21447527	C₂	1	1	yes	maximal, characteristic
4	#21445698	C₂	1	1	yes	maximal, characteristic, complemented
5	#21443915	C₂	1	1	yes	maximal, characteristic, complemented
6	#21443913	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₄×C₁₆×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#14519799	C₂²	1	1	yes	characteristic
9	C₂×#14519783	—	2	1	no	—
10	#16839880	—	2	2	no	—
11	#16839880	C₄	1	4	yes	complemented
12	C₂×#14519713	C₂²	1	1	yes	characteristic
13	C₂×#14519713	C₂²	1	1	yes	characteristic
14	#16839800	C₂²	1	1	yes	characteristic
15	C₂×#14519705	—	2	1	no	—
16	#16839717	—	2	1	no	—
17	#16839570	C₄	1	4	yes	complemented
18	#16839567	C₄	1	2	yes	complemented
19	#16839564	C₄	1	2	yes	complemented
20	#16839522	—	2	1	no	—
21	C₂²×C₁₆×C₆₁	C₂²	1	1	yes	characteristic
22	C₂²×C₁₆×C₆₁	C₂²	1	1	yes	characteristic
23	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
24	C₄×C₁₆×C₆₁	—	2	2	no	—
25	C₂×#11758222	—	2	1	no	—
26	#14519799	—	2	1	no	—
27	#14519799	—	2	1	no	—
28	#14519799	C₂×C₄	1	2	yes	—
29	#14519799	C₂×C₄	1	2	yes	—
30	C₂×Dic₂₄₄	C₂×C₄	1	1	yes	characteristic
31	C₂×#11758190	—	2	1	no	—
32	C₂×D₄₈₈	C₂×C₄	1	1	yes	characteristic
33	#14519783	—	2	4	no	—
34	#14519783	—	4	2	no	—
35	C₂×#11758194	—	2	1	no	—
36	C₂×#11758194	—	2	1	no	—
37	C₂×#11758189	—	4	1	no	—
38	C₂×C₈×D₆₁	—	4	1	no	—
39	#14519736	—	2	1	no	—
40	#14519713	—	2	1	no	—
41	#14519713	—	2	1	no	—
42	#14519713	C₂×C₄	1	2	yes	—
43	#14519713	C₂×C₄	1	2	yes	—
44	#14519711	C₂×C₄	1	2	yes	—
45	#14519709	—	2	1	no	—
46	#14519705	—	2	4	no	—
47	#14519687	—	4	1	no	—
48	C₈×Dic₆₁	—	4	1	no	—
49	C₂²×C₈×C₆₁	D₄	1	1	yes	characteristic
50	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
51	C₂×C₄²×C₆₁	D₄	1	1	yes	characteristic
52	C₂×C₁₆×C₆₁	—	2	1	no	—
53	C₂×C₁₆×C₆₁	—	2	1	no	—
54	C₂×C₁₆×C₆₁	—	2	1	no	—
55	C₂×C₁₆×C₆₁	—	2	1	no	—
56	C₂×C₁₆×C₆₁	C₂×C₄	1	2	yes	—
57	C₂×C₁₆×C₆₁	C₂×C₄	1	2	yes	—
58	C₄×C₈×C₆₁	C₂×C₄	1	2	yes	—
59	C₄×C₈×C₆₁	D₄	1	2	yes	—
60	C₂²×C₄×C₆₁	—	2	1	no	—
61	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
62	C₂×C₂²⋊D₆₁	—	4	1	no	—
63	#11758222	—	2	2	no	—
64	#11758222	—	2	2	no	—
65	#11758222	—	4	1	no	—
66	#11758222	—	4	1	no	—
67	C₂×D₂₄₄	—	2	1	no	—
68	C₂×C₄×D₆₁	—	4	1	no	—
69	C₂×Dic₁₂₂	—	2	1	no	—
70	C₂×C₈×C₆₁	C₂²⋊C₄	1	2	yes	—
71	C₂×C₈×C₆₁	D₈	1	1	yes	characteristic
72	C₂×C₈×C₆₁	D₈	1	1	yes	characteristic
73	C₂×C₈×C₆₁	SD₁₆	1	2	yes	—
74	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
75	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
76	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
78	C₂×C₈×C₆₁	#41	1	1	yes	characteristic
79	C₂×C₈×C₆₁	#41	1	1	yes	characteristic
80	C₄²×C₆₁	—	2	1	no	—
81	C₄²×C₆₁	C₂²⋊C₄	1	2	yes	—
82	C₂²⋊Dic₆₁	—	4	1	no	—
83	C₄⋊Dic₆₁	—	2	2	no	—
84	C₄×Dic₆₁	—	4	1	no	—
85	#11758194	—	2	2	no	—
86	#11758194	—	2	2	no	—
87	#11758194	—	4	1	no	—
88	#11758194	—	4	1	no	—
89	C₂×#92418	—	4	1	no	—
90	C₂×#92418	—	4	1	no	—
91	Dic₂₄₄	—	2	1	no	—
92	Dic₂₄₄	C₄²	1	2	yes	—
93	D₄₈₈	—	2	1	no	—
94	D₄₈₈	C₄²	1	2	yes	—
95	#11758190	—	2	2	no	—
96	#11758190	—	2	2	no	—
97	#11758189	—	4	4	no	—
98	C₈×D₆₁	—	4	4	no	—
99	C₁₆×C₆₁	—	2	2	no	—
100	C₁₆×C₆₁	—	2	2	no	—
101	C₂³×C₆₁	#135	1	1	yes	characteristic
102	C₂²×D₆₁	—	4	1	no	—
103	C₂×C₄×C₆₁	—	2	4	no	—
104	C₂×C₄×C₆₁	—	2	2	no	—
105	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
106	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
107	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
108	C₂×C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
109	C₂×C₄×C₆₁	C₂×SD₁₆	1	1	yes	characteristic
110	C₂×C₄×C₆₁	#135	1	1	yes	characteristic
111	C₂²⋊D₆₁	—	4	2	no	—
112	C₂²⋊D₆₁	—	4	2	no	—
113	C₂×Dic₆₁	—	4	1	no	—
114	C₂×Dic₆₁	—	4	2	no	—
115	D₂₄₄	—	2	2	no	—
116	D₂₄₄	—	4	1	no	—
117	C₄×D₆₁	—	4	2	no	—
118	C₄×D₆₁	—	4	2	no	—
119	Dic₁₂₂	—	2	2	no	—
120	Dic₁₂₂	—	4	1	no	—
121	C₈×C₆₁	#102	1	2	yes	—
122	C₈×C₆₁	#102	1	2	yes	—
123	C₈×C₆₁	C₂×C₄²	1	1	yes	characteristic
124	C₈×C₆₁	#117	1	1	yes	characteristic
125	C₈×C₆₁	C₄×D₄	1	1	yes	characteristic
126	C₈×C₆₁	C₄×D₄	1	1	yes	characteristic
127	#92418	—	4	2	no	—
128	#92418	—	4	2	no	—
129	#12641	—	61	1	no	maximal, Sylow
130	C₂²×C₆₁	—	2	2	no	—
131	C₂²×C₆₁	#416	1	1	yes	characteristic
132	C₂²×C₆₁	C₄×D₈	1	1	yes	characteristic
133	C₂²×C₆₁	C₄×SD₁₆	1	1	yes	characteristic
134	C₂×D₆₁	—	4	2	no	—
135	C₂×D₆₁	—	8	2	no	—
136	C₄×C₆₁	—	2	4	no	—
137	C₄×C₆₁	C₄×C₂²⋊C₄	1	1	yes	characteristic
138	C₄×C₆₁	C₂×#102	1	1	yes	characteristic
139	C₄×C₆₁	C₄×D₈	1	1	yes	characteristic
140	C₄×C₆₁	C₄×SD₁₆	1	1	yes	characteristic
141	Dic₆₁	—	4	2	no	—
142	Dic₆₁	—	8	2	no	—
143	C₂×#443	—	61	1	no	—
144	C₂×#359	—	61	1	no	—
145	#2138	—	61	1	no	—
146	C₂×C₄×C₁₆	D₆₁	1	1	yes	characteristic, complemented
147	#1772	—	61	1	no	—
148	#1382	—	61	1	no	—
149	#1381	—	61	1	no	—
150	C₂×C₆₁	—	2	1	no	—
151	C₂×C₆₁	—	2	1	no	—
152	C₂×C₆₁	C₄×#102	1	1	yes	characteristic
153	C₂×C₆₁	#2178	1	2	yes	—
154	D₆₁	—	8	2	no	—
155	C₂×#135	—	61	1	no	—
156	C₂×#131	—	122	1	no	—
157	C₂²×C₁₆	C₂×D₆₁	1	1	yes	characteristic
158	C₂²×C₁₆	C₂×D₆₁	1	1	yes	characteristic
159	#443	—	61	4	no	—
160	#443	—	122	2	no	—
161	C₂×#108	—	61	1	no	—
162	C₂×#108	—	61	1	no	—
163	#427	—	61	1	no	—
164	C₂×#104	—	122	1	no	—
165	#405	—	122	1	no	—
166	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
167	#359	—	61	4	no	—
168	#358	—	61	2	no	—
169	#357	—	61	2	no	—
170	C₄×C₁₆	—	2	2	no	—
171	#339	—	122	1	no	—
172	C₆₁	#12641	1	1	yes	characteristic, complemented, Sylow
173	C₂×#41	—	122	1	no	—
174	#135	—	61	2	no	—
175	#135	—	61	2	no	—
176	#135	—	122	1	no	—
177	#135	—	122	1	no	—
178	C₂×Q₁₆	—	61	1	no	—
179	C₂×SD₁₆	—	122	1	no	—
180	C₂×D₈	—	61	1	no	—
181	#131	—	122	4	no	—
182	#131	—	244	2	no	—
183	C₂×M₁₆	—	122	1	no	—
184	C₂×M₁₆	—	122	1	no	—
185	C₂×M₁₆	—	244	1	no	—
186	C₂²×C₈	—	244	1	no	—
187	C₂²×C₈	D₂₄₄	1	1	yes	characteristic
188	C₂²×C₈	C₂²×D₆₁	1	1	yes	characteristic
189	#117	—	122	1	no	—
190	C₂×C₄²	D₂₄₄	1	1	yes	characteristic
191	C₂×C₁₆	—	2	1	no	—
192	C₂×C₁₆	—	2	1	no	—
193	C₂×C₁₆	—	2	1	no	—
194	C₂×C₁₆	—	2	1	no	—
195	C₂×C₁₆	C₄×D₆₁	1	2	yes	—
196	C₂×C₁₆	C₄×D₆₁	1	2	yes	—
197	#108	—	61	2	no	—
198	#108	—	61	2	no	—
199	#108	—	122	1	no	—
200	#108	—	122	1	no	—
201	#107	—	61	2	no	—
202	#106	—	122	1	no	—
203	#104	—	122	4	no	—
204	#97	—	244	1	no	—
205	C₄×C₈	—	244	1	no	—
206	C₄×C₈	C₄×D₆₁	1	2	yes	—
207	C₄×C₈	C₂²⋊D₆₁	1	2	yes	—
208	#41	—	122	2	no	—
209	#41	—	122	2	no	—
210	#41	—	244	1	no	—
211	#41	—	244	1	no	—
212	C₂×Q₈	—	122	1	no	—
213	C₂×D₄	—	122	1	no	—
214	C₂×D₄	—	244	1	no	—
215	C₂²×C₄	—	2	1	no	—
216	C₂²×C₄	—	244	1	no	—
217	C₂²×C₄	C₂×D₂₄₄	1	1	yes	characteristic
218	Q₁₆	—	61	2	no	—
219	Q₁₆	—	122	1	no	—
220	SD₁₆	—	122	2	no	—
221	SD₁₆	—	122	2	no	—
222	D₈	—	61	2	no	—
223	D₈	—	122	1	no	—
224	M₁₆	—	122	2	no	—
225	M₁₆	—	122	2	no	—
226	M₁₆	—	244	1	no	—
227	M₁₆	—	244	1	no	—
228	M₁₆	—	244	4	no	—
229	C₂×C₈	—	244	1	no	—
230	C₂×C₈	—	244	1	no	—
231	C₂×C₈	—	244	4	no	—
232	C₂×C₈	#11758190	1	2	yes	—
233	C₂×C₈	D₄₈₈	1	1	yes	characteristic
234	C₂×C₈	D₄₈₈	1	1	yes	characteristic
235	C₂×C₈	#11758198	1	2	yes	—
236	C₂×C₈	C₂×C₄×D₆₁	1	1	yes	characteristic
237	C₂×C₈	C₂×C₄×D₆₁	1	1	yes	characteristic
238	C₂×C₈	C₂×C₄×D₆₁	1	1	yes	characteristic
239	C₂×C₈	#11758222	1	1	yes	characteristic
240	C₂×C₈	#11758222	1	1	yes	characteristic
241	C₂×C₈	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
242	C₄⋊C₄	—	122	2	no	—
243	C₂²⋊C₄	—	244	1	no	—
244	C₄²	—	2	1	no	—
245	C₄²	—	244	1	no	—
246	C₄²	#11758198	1	2	yes	—
247	C₁₆	—	2	2	no	—
248	C₁₆	—	2	2	no	—
249	C₂³	—	244	1	no	—
250	C₂³	#14519799	1	1	yes	characteristic
251	Q₈	—	122	2	no	—
252	Q₈	—	244	1	no	—
253	D₄	—	122	2	no	—
254	D₄	—	244	2	no	—
255	D₄	—	244	2	no	—
256	D₄	—	244	1	no	—
257	C₂×C₄	—	2	4	no	—
258	C₂×C₄	—	2	2	no	—
259	C₂×C₄	—	244	2	no	—
260	C₂×C₄	—	244	1	no	—
261	C₂×C₄	—	244	2	no	—
262	C₂×C₄	—	244	2	no	—
263	C₂×C₄	C₂×#11758198	1	1	yes	characteristic
264	C₂×C₄	C₄×D₂₄₄	1	1	yes	characteristic
265	C₂×C₄	C₄×D₂₄₄	1	1	yes	characteristic
266	C₂×C₄	C₂×D₄₈₈	1	1	yes	characteristic
267	C₂×C₄	C₂×#11758190	1	1	yes	characteristic
268	C₂×C₄	#14519799	1	1	yes	characteristic
269	C₈	—	244	2	no	—
270	C₈	—	244	2	no	—
271	C₈	#14519699	1	2	yes	—
272	C₈	#14519699	1	2	yes	—
273	C₈	C₄²×D₆₁	1	1	yes	characteristic
274	C₈	#14519734	1	1	yes	characteristic
275	C₈	C₄×C₂²⋊D₆₁	1	1	yes	characteristic
276	C₈	C₄×C₂²⋊D₆₁	1	1	yes	characteristic
277	C₂²	—	2	2	no	—
278	C₂²	—	488	2	no	—
279	C₂²	—	244	2	no	—
280	C₂²	#16839759	1	1	yes	characteristic
281	C₂²	C₄×D₄₈₈	1	1	yes	characteristic
282	C₂²	C₄×#11758190	1	1	yes	characteristic
283	C₄	—	2	4	no	—
284	C₄	—	488	2	no	—
285	C₄	—	244	2	no	—
286	C₄	C₄×#11758198	1	1	yes	characteristic
287	C₄	C₂×#14519699	1	1	yes	characteristic
288	C₄	C₄×D₄₈₈	1	1	yes	characteristic
289	C₄	C₄×#11758190	1	1	yes	characteristic
290	C₂	—	2	1	no	—
291	C₂	—	2	1	no	—
292	C₂	—	488	2	no	—
293	C₂	C₄×#14519699	1	1	yes	characteristic
294	C₂	#21447701	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	D₄₈₈
commutator	C₈×C₆₁	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₆₁
Fitting	C₂×C₄×C₁₆×C₆₁	C₂
socle	C₂²×C₆₁	C₄×D₈
prime radical	C₂×#14519799	C₂²
cosocle	C₂²×C₈×C₆₁	C₂³
hypercenter	C₂×C₄×C₁₆	D₆₁
hypocenter	C₆₁	#12641
LCS	G ▷ C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆

Complements

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

N	H ≅ G/N	type
4. #21445698	C₂	semidirect
5. #21443915	C₂	semidirect
6. #21443913	C₂	semidirect
7. C₂×C₄×C₁₆×C₆₁	C₂	semidirect
11. #16839880	C₄	semidirect
17. #16839570	C₄	semidirect
18. #16839567	C₄	semidirect
19. #16839564	C₄	semidirect
146. C₂×C₄×C₁₆	D₆₁	semidirect
172. C₆₁	#12641	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16839880	2
C₂×#16839570	2
#21447527	2
#21445698	2
#21443915	2
#21443913	2
C₂×C₄×C₁₆×C₆₁	2
#12641	61