Subgroups — #29833441

Order 15616.

642 conjugacy classes of subgroups.

The subgroup lattice has 286 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16840194	C₂	1	1	yes	maximal, characteristic
2	#21454737	C₂	1	2	yes	maximal, complemented
3	#21447706	C₂	1	4	yes	maximal, complemented
4	#21447701	C₂	1	4	yes	maximal, complemented
5	C₂×#16839592	C₂	1	1	yes	maximal, characteristic, complemented
6	#21447548	C₂	1	2	yes	maximal, complemented
7	C₆₁×#2108	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#14519799	C₂²	1	2	yes	—
9	#16840528	—	2	2	no	—
10	#16840194	C₄	1	2	yes	complemented
11	#16840194	C₄	1	2	yes	complemented
12	#16840194	C₂²	1	2	yes	—
13	#16840194	C₂²	1	2	yes	—
14	C₂×Dic₄₈₈	C₄	1	1	yes	characteristic, complemented
15	C₂×#14519722	C₄	1	2	yes	complemented
16	C₂×D₉₇₆	C₄	1	1	yes	characteristic, complemented
17	#16839885	—	2	4	no	—
18	#16839880	C₂²	1	8	yes	complemented
19	C₂×#14519713	C₂²	1	2	yes	—
20	#16839780	—	2	2	no	—
21	#16839592	C₂²	1	2	yes	complemented
22	#16839592	C₂²	1	2	yes	complemented
23	#16839570	C₂²	1	8	yes	complemented
24	C₂×M₃₂×C₆₁	C₂²	1	1	yes	characteristic
25	C₂²×C₁₆×C₆₁	C₂²	1	1	yes	characteristic
26	C₆₁×#405	C₂²	1	1	yes	characteristic
27	C₆₁×#346	C₂²	1	2	yes	complemented
28	C₄×C₁₆×C₆₁	C₂²	1	2	yes	complemented
29	C₂×#11758222	—	2	2	no	—
30	#14519799	—	2	2	no	—
31	#14519799	—	2	2	no	—
32	#14519799	C₂×C₄	1	4	yes	—
33	#14519799	C₂³	1	4	yes	—
34	C₂×Dic₂₄₄	C₂×C₄	1	2	yes	—
35	C₂×#11758190	—	2	2	no	—
36	C₂×D₄₈₈	C₂×C₄	1	2	yes	—
37	#14519785	—	4	4	no	—
38	#14519783	—	2	8	no	—
39	C₂×#11758194	—	2	2	no	—
40	M₁₆×D₆₁	—	4	4	no	—
41	Dic₄₈₈	C₂×C₄	1	2	yes	—
42	Dic₄₈₈	C₂×C₄	1	2	yes	—
43	#14519722	C₂×C₄	1	4	yes	—
44	#14519722	C₂×C₄	1	4	yes	—
45	D₉₇₆	C₂×C₄	1	2	yes	—
46	D₉₇₆	C₂×C₄	1	2	yes	—
47	#14519713	—	2	2	no	—
48	#14519713	C₂³	1	4	yes	—
49	#14519705	—	2	8	no	—
50	C₂×M₁₆×C₆₁	D₄	1	1	yes	characteristic
51	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
52	#117×C₆₁	D₄	1	1	yes	characteristic
53	M₃₂×C₆₁	—	2	1	no	—
54	M₃₂×C₆₁	—	2	1	no	—
55	C₂×C₁₆×C₆₁	C₂×C₄	1	1	yes	characteristic
56	C₂×C₁₆×C₆₁	C₂×C₄	1	1	yes	characteristic
57	C₂×C₁₆×C₆₁	C₂×C₄	1	1	yes	characteristic
58	C₂×C₁₆×C₆₁	C₂×C₄	1	1	yes	characteristic
59	C₂×C₁₆×C₆₁	C₂³	1	2	yes	—
60	C₂×C₁₆×C₆₁	C₂³	1	1	yes	characteristic
61	C₂×C₁₆×C₆₁	C₂³	1	1	yes	characteristic
62	#97×C₆₁	D₄	1	2	yes	—
63	C₄×C₈×C₆₁	C₂³	1	2	yes	—
64	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
65	C₂×C₂²⋊D₆₁	—	4	2	no	—
66	#11758222	—	2	4	no	—
67	#11758222	—	2	4	no	—
68	#11758222	—	4	2	no	—
69	#11758222	—	4	2	no	—
70	C₂×D₂₄₄	—	2	2	no	—
71	C₂×C₄×D₆₁	—	4	2	no	—
72	C₂×Dic₁₂₂	—	2	2	no	—
73	M₁₆×C₆₁	D₈	1	2	yes	—
74	M₁₆×C₆₁	D₈	1	2	yes	—
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	C₂×D₄	1	2	yes	—
79	C₂×C₈×C₆₁	#41	1	1	yes	characteristic
80	C₂×C₈×C₆₁	#41	1	1	yes	characteristic
81	C₂×C₈×C₆₁	C₂⁴	1	1	yes	characteristic
82	C₄⋊C₄×C₆₁	—	2	1	no	—
83	C₂²⋊C₄×C₆₁	—	2	1	no	—
84	C₄²×C₆₁	C₂×D₄	1	2	yes	—
85	#11758194	—	2	4	no	—
86	#11758194	—	4	2	no	—
87	C₂×#92418	—	4	2	no	—
88	Dic₂₄₄	—	2	2	no	—
89	Dic₂₄₄	C₂²×C₄	1	4	yes	—
90	D₄₈₈	—	2	2	no	—
91	D₄₈₈	C₂²×C₄	1	4	yes	—
92	#11758190	—	2	4	no	—
93	#11758190	—	2	4	no	—
94	#11758189	—	4	8	no	—
95	C₈×D₆₁	—	4	8	no	—
96	C₁₆×C₆₁	—	2	2	no	—
97	C₁₆×C₆₁	C₂²×C₄	1	1	yes	characteristic
98	C₁₆×C₆₁	C₂²×C₄	1	1	yes	characteristic
99	C₁₆×C₆₁	C₂²×C₄	1	1	yes	characteristic
100	C₁₆×C₆₁	C₂²×C₄	1	1	yes	characteristic
101	C₂³×C₆₁	#135	1	1	yes	characteristic
102	C₂²×D₆₁	—	4	2	no	—
103	C₂×C₄×C₆₁	—	2	2	no	—
104	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
105	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
106	C₂×C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
107	C₂×C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
108	C₂×C₄×C₆₁	#135	1	1	yes	characteristic
109	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
110	C₂²⋊D₆₁	—	4	4	no	—
111	C₂²⋊D₆₁	—	4	4	no	—
112	C₂×Dic₆₁	—	4	2	no	—
113	D₂₄₄	—	2	4	no	—
114	D₂₄₄	—	4	2	no	—
115	C₄×D₆₁	—	4	4	no	—
116	C₄×D₆₁	—	4	4	no	—
117	Dic₁₂₂	—	2	4	no	—
118	Dic₁₂₂	—	4	2	no	—
119	C₈×C₆₁	C₄×D₄	1	1	yes	characteristic
120	C₈×C₆₁	C₄×D₄	1	1	yes	characteristic
121	C₈×C₆₁	C₂×D₈	1	4	yes	—
122	C₈×C₆₁	C₂³×C₄	1	1	yes	characteristic
123	C₈×C₆₁	C₂×#41	1	1	yes	characteristic
124	#92418	—	4	4	no	—
125	#33465	—	61	1	no	maximal, Sylow
126	C₂²×C₆₁	—	4	1	no	—
127	C₂²×C₆₁	C₄×D₈	1	1	yes	characteristic
128	C₂²×C₆₁	C₄×D₈	1	1	yes	characteristic
129	C₂²×C₆₁	C₂×#135	1	1	yes	characteristic
130	C₂×D₆₁	—	4	4	no	—
131	C₂×D₆₁	—	16	2	no	—
132	C₄×C₆₁	—	4	2	no	—
133	C₄×C₆₁	C₄×D₈	1	1	yes	characteristic
134	C₄×C₆₁	C₄×D₈	1	1	yes	characteristic
135	C₄×C₆₁	C₂×C₄×D₄	1	1	yes	characteristic
136	C₄×C₆₁	C₂²×D₈	1	1	yes	characteristic
137	Dic₆₁	—	4	4	no	—
138	C₂×#508	—	61	1	no	—
139	#2957	—	61	2	no	—
140	#2179	—	61	4	no	—
141	#2178	—	61	4	no	—
142	C₂×#368	—	61	1	no	—
143	#2143	—	61	2	no	—
144	#2108	D₆₁	1	1	yes	characteristic, complemented
145	C₂×C₆₁	—	2	1	no	—
146	C₂×C₆₁	—	2	1	no	—
147	C₂×C₆₁	—	2	1	no	—
148	C₂×C₆₁	C₂×C₄×D₈	1	1	yes	characteristic
149	D₆₁	—	8	4	no	—
150	C₂×#135	—	61	2	no	—
151	#568	—	122	2	no	—
152	#508	—	61	2	no	—
153	#508	—	61	2	no	—
154	#508	—	61	2	no	—
155	#508	—	61	2	no	—
156	C₂×Q₃₂	—	61	1	no	—
157	C₂×SD₃₂	—	61	2	no	—
158	C₂×D₁₆	—	61	1	no	—
159	C₂×M₃₂	C₂×D₆₁	1	1	yes	characteristic
160	C₂²×C₁₆	C₂×D₆₁	1	1	yes	characteristic
161	#444	—	122	4	no	—
162	#443	—	61	8	no	—
163	C₂×#108	—	61	2	no	—
164	#421	—	122	2	no	—
165	#405	C₂×D₆₁	1	1	yes	characteristic
166	#368	—	61	2	no	—
167	#368	—	61	2	no	—
168	#359	—	61	8	no	—
169	#346	C₂×D₆₁	1	2	yes	complemented
170	C₄×C₁₆	C₂×D₆₁	1	2	yes	complemented
171	C₆₁	#33465	1	1	yes	characteristic, complemented, Sylow
172	C₂×#41	—	122	2	no	—
173	#135	—	61	4	no	—
174	#135	—	61	4	no	—
175	#135	—	122	2	no	—
176	#135	—	122	2	no	—
177	C₂×Q₁₆	—	61	2	no	—
178	C₂×SD₁₆	—	122	2	no	—
179	C₂×D₈	—	61	2	no	—
180	#131	—	122	8	no	—
181	#131	—	244	4	no	—
182	C₂×M₁₆	—	122	2	no	—
183	C₂×M₁₆	—	244	4	no	—
184	C₂×M₁₆	D₂₄₄	1	1	yes	characteristic
185	C₂²×C₈	C₂²×D₆₁	1	1	yes	characteristic
186	#117	D₂₄₄	1	1	yes	characteristic
187	Q₃₂	—	61	2	no	—
188	Q₃₂	—	61	2	no	—
189	SD₃₂	—	61	4	no	—
190	SD₃₂	—	61	4	no	—
191	D₁₆	—	61	2	no	—
192	D₁₆	—	61	2	no	—
193	M₃₂	—	2	1	no	—
194	M₃₂	—	2	1	no	—
195	C₂×C₁₆	C₄×D₆₁	1	1	yes	characteristic
196	C₂×C₁₆	C₄×D₆₁	1	1	yes	characteristic
197	C₂×C₁₆	C₄×D₆₁	1	1	yes	characteristic
198	C₂×C₁₆	C₄×D₆₁	1	1	yes	characteristic
199	C₂×C₁₆	C₂²×D₆₁	1	2	yes	—
200	C₂×C₁₆	C₂²×D₆₁	1	1	yes	characteristic
201	C₂×C₁₆	C₂²×D₆₁	1	1	yes	characteristic
202	#108	—	61	4	no	—
203	#108	—	122	2	no	—
204	#104	—	122	8	no	—
205	#97	D₂₄₄	1	2	yes	—
206	C₄×C₈	C₂²×D₆₁	1	2	yes	—
207	#41	—	122	4	no	—
208	#41	—	122	4	no	—
209	#41	—	244	2	no	—
210	#41	—	244	2	no	—
211	C₂×Q₈	—	122	2	no	—
212	C₂×D₄	—	122	2	no	—
213	C₂×D₄	—	244	2	no	—
214	C₂²×C₄	—	244	2	no	—
215	C₂²×C₄	C₂×D₂₄₄	1	1	yes	characteristic
216	Q₁₆	—	61	4	no	—
217	Q₁₆	—	122	2	no	—
218	SD₁₆	—	122	4	no	—
219	SD₁₆	—	122	4	no	—
220	D₈	—	61	4	no	—
221	D₈	—	122	2	no	—
222	M₁₆	—	122	4	no	—
223	M₁₆	—	244	2	no	—
224	M₁₆	—	244	8	no	—
225	M₁₆	D₄₈₈	1	2	yes	—
226	M₁₆	D₄₈₈	1	2	yes	—
227	C₂×C₈	—	244	2	no	—
228	C₂×C₈	—	244	8	no	—
229	C₂×C₈	C₂×C₄×D₆₁	1	1	yes	characteristic
230	C₂×C₈	C₂×C₄×D₆₁	1	1	yes	characteristic
231	C₂×C₈	C₂×D₂₄₄	1	1	yes	characteristic
232	C₂×C₈	C₂×D₂₄₄	1	2	yes	—
233	C₂×C₈	#11758222	1	1	yes	characteristic
234	C₂×C₈	#11758222	1	1	yes	characteristic
235	C₂×C₈	C₂³×D₆₁	1	1	yes	characteristic
236	C₄⋊C₄	—	2	1	no	—
237	C₂²⋊C₄	—	2	1	no	—
238	C₄²	C₂×D₂₄₄	1	2	yes	—
239	C₁₆	—	2	2	no	—
240	C₁₆	C₂×C₄×D₆₁	1	1	yes	characteristic
241	C₁₆	C₂×C₄×D₆₁	1	1	yes	characteristic
242	C₁₆	C₂×C₄×D₆₁	1	1	yes	characteristic
243	C₁₆	C₂×C₄×D₆₁	1	1	yes	characteristic
244	C₂³	—	244	2	no	—
245	C₂³	#14519799	1	1	yes	characteristic
246	Q₈	—	122	4	no	—
247	Q₈	—	244	2	no	—
248	D₄	—	122	4	no	—
249	D₄	—	244	4	no	—
250	D₄	—	244	4	no	—
251	D₄	—	244	2	no	—
252	C₂×C₄	—	2	2	no	—
253	C₂×C₄	—	244	2	no	—
254	C₂×C₄	—	244	4	no	—
255	C₂×C₄	—	244	4	no	—
256	C₂×C₄	C₄×D₂₄₄	1	1	yes	characteristic
257	C₂×C₄	C₄×D₂₄₄	1	1	yes	characteristic
258	C₂×C₄	C₂×D₄₈₈	1	1	yes	characteristic
259	C₂×C₄	C₂×D₄₈₈	1	1	yes	characteristic
260	C₂×C₄	#14519799	1	1	yes	characteristic
261	C₂×C₄	C₂²×D₂₄₄	1	1	yes	characteristic
262	C₈	—	244	4	no	—
263	C₈	C₄×D₂₄₄	1	1	yes	characteristic
264	C₈	C₄×D₂₄₄	1	1	yes	characteristic
265	C₈	C₂×D₄₈₈	1	4	yes	—
266	C₈	C₂²×C₄×D₆₁	1	1	yes	characteristic
267	C₈	C₂×#11758222	1	1	yes	characteristic
268	C₂²	—	4	1	no	—
269	C₂²	—	976	2	no	—
270	C₂²	—	244	4	no	—
271	C₂²	C₄×D₄₈₈	1	1	yes	characteristic
272	C₂²	C₄×D₄₈₈	1	1	yes	characteristic
273	C₂²	C₂×#14519799	1	1	yes	characteristic
274	C₄	—	4	2	no	—
275	C₄	—	244	4	no	—
276	C₄	C₄×D₄₈₈	1	1	yes	characteristic
277	C₄	C₄×D₄₈₈	1	1	yes	characteristic
278	C₄	C₂×C₄×D₂₄₄	1	1	yes	characteristic
279	C₄	C₂²×D₄₈₈	1	1	yes	characteristic
280	C₂	—	2	1	no	—
281	C₂	—	2	1	no	—
282	C₂	—	2	1	no	—
283	C₂	—	488	4	no	—
284	C₂	C₂×C₄×D₄₈₈	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂²×D₄₈₈
commutator	C₈×C₆₁	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₆₁
Fitting	C₆₁×#2108	C₂
socle	C₂×C₆₁	C₂×C₄×D₈
prime radical	C₂×#16840194	C₂
cosocle	C₂×C₈×C₆₁	C₂⁴
hypercenter	#2108	D₆₁
hypocenter	C₆₁	#33465
LCS	G ▷ C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₄ ◁ C₂×M₁₆ ◁ #405 ◁ #2108

Complements

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

N	H ≅ G/N	type
2. #21454737	C₂	semidirect
3. #21447706	C₂	semidirect
4. #21447701	C₂	semidirect
5. C₂×#16839592	C₂	semidirect
6. #21447548	C₂	semidirect
7. C₆₁×#2108	C₂	semidirect
10. #16840194	C₄	semidirect
11. #16840194	C₄	semidirect
14. C₂×Dic₄₈₈	C₄	semidirect
15. C₂×#14519722	C₄	semidirect
16. C₂×D₉₇₆	C₄	semidirect
18. #16839880	C₂²	semidirect
21. #16839592	C₂²	semidirect
22. #16839592	C₂²	semidirect
23. #16839570	C₂²	semidirect
27. C₆₁×#346	C₂²	semidirect
28. C₄×C₁₆×C₆₁	C₂²	semidirect
144. #2108	D₆₁	semidirect
169. #346	C₂×D₆₁	semidirect
170. C₄×C₁₆	C₂×D₆₁	semidirect
171. C₆₁	#33465	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16840194	2
#21454737	2
#21447706	2
#21447701	2
C₂×#16839592	2
#21447548	2
C₆₁×#2108	2
#33465	61