Subgroups — #29595116

Order 15616.

426 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16840178	C₂	1	1	yes	maximal, characteristic
2	C₂×#16839545	C₂	1	1	yes	maximal, characteristic
3	#21447415	C₂	1	1	yes	maximal, characteristic, complemented
4	#21445656	C₂	1	1	yes	maximal, characteristic
5	#21443907	C₂	1	1	yes	maximal, characteristic, complemented
6	#21443901	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₄×C₁₆×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#14519783	C₂²	1	1	yes	characteristic
9	#16840178	—	2	2	no	—
10	#16840178	C₄	1	4	yes	complemented
11	C₂×#14519717	C₂²	1	1	yes	characteristic
12	C₂×#14519717	C₂²	1	1	yes	characteristic
13	C₂×#14519716	—	2	1	no	—
14	C₂×C₁₆×D₆₁	—	2	1	no	—
15	C₂×#14519705	C₄	1	1	yes	characteristic
16	#16839759	C₄	1	1	yes	characteristic
17	#16839717	C₂²	1	1	yes	characteristic
18	#16839545	C₄	1	4	yes	complemented
19	#16839537	—	2	1	no	—
20	C₁₆×Dic₆₁	—	2	1	no	—
21	#16839527	C₄	1	1	yes	characteristic
22	#16839522	C₄	1	1	yes	characteristic
23	C₂²×C₁₆×C₆₁	C₂²	1	1	yes	characteristic
24	C₂²×C₁₆×C₆₁	C₂²	1	1	yes	characteristic
25	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
26	C₄×C₁₆×C₆₁	—	2	2	no	—
27	C₂×#11758222	C₂×C₄	1	1	yes	characteristic
28	#14519783	—	2	1	no	—
29	#14519783	—	2	1	no	—
30	#14519783	C₂×C₄	1	2	yes	—
31	#14519783	C₂×C₄	1	2	yes	—
32	C₂×#11758194	C₂×C₄	1	1	yes	characteristic
33	C₂×#11758194	C₂×C₄	1	1	yes	characteristic
34	C₂×#11758189	—	2	1	no	—
35	C₂×C₈×D₆₁	—	2	1	no	—
36	#14519736	C₂×C₄	1	1	yes	characteristic
37	#14519717	—	2	1	no	—
38	#14519717	—	2	1	no	—
39	#14519717	C₂×C₄	1	2	yes	—
40	#14519717	C₂×C₄	1	2	yes	—
41	#14519716	—	2	4	no	—
42	C₂×#11758185	—	2	1	no	—
43	C₂×#11758185	—	2	1	no	—
44	C₁₆×D₆₁	—	2	4	no	—
45	#14519705	C₈	1	4	yes	—
46	#14519702	C₈	1	2	yes	—
47	#14519699	C₈	1	2	yes	—
48	#14519687	—	2	1	no	—
49	C₈×Dic₆₁	—	2	1	no	—
50	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
51	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
52	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
53	C₂×C₁₆×C₆₁	—	2	1	no	—
54	C₂×C₁₆×C₆₁	—	2	1	no	—
55	C₂×C₁₆×C₆₁	C₂×C₄	1	2	yes	—
56	C₂×C₁₆×C₆₁	C₂×C₄	1	2	yes	—
57	C₂×C₁₆×C₆₁	D₄	1	1	yes	characteristic
58	C₂×C₁₆×C₆₁	D₄	1	1	yes	characteristic
59	C₂×C₁₆×C₆₁	D₄	1	2	yes	—
60	C₄×C₈×C₆₁	—	2	1	no	—
61	C₄×C₈×C₆₁	—	2	1	no	—
62	C₂²×C₄×C₆₁	—	2	1	no	—
63	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
64	C₂×C₂²⋊D₆₁	—	2	1	no	—
65	#11758222	—	2	1	no	—
66	#11758222	—	2	1	no	—
67	#11758222	C₄²	1	2	yes	—
68	#11758222	C₂×C₈	1	2	yes	—
69	C₂×D₂₄₄	C₂×C₈	1	1	yes	characteristic
70	C₂×C₄×D₆₁	—	2	1	no	—
71	C₂×Dic₁₂₂	C₂×C₈	1	1	yes	characteristic
72	C₂×C₈×C₆₁	—	2	1	no	—
73	C₂×C₈×C₆₁	—	2	1	no	—
74	C₂×C₈×C₆₁	C₂×C₈	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	1	yes	characteristic
79	C₂×C₈×C₆₁	#41	1	1	yes	characteristic
80	C₂×C₈×C₆₁	#41	1	1	yes	characteristic
81	C₄²×C₆₁	—	2	1	no	—
82	C₄²×C₆₁	C₂×C₈	1	2	yes	—
83	C₂²⋊Dic₆₁	—	2	1	no	—
84	C₄⋊Dic₆₁	C₂×C₈	1	2	yes	—
85	C₄×Dic₆₁	—	2	1	no	—
86	#11758194	—	2	1	no	—
87	#11758194	—	2	1	no	—
88	#11758194	C₄²	1	2	yes	—
89	#11758194	C₂×C₈	1	2	yes	—
90	C₂×#92418	—	2	1	no	—
91	C₂×#92418	—	2	1	no	—
92	#11758189	—	2	2	no	—
93	#11758189	—	2	2	no	—
94	C₈×D₆₁	—	2	2	no	—
95	C₈×D₆₁	—	2	2	no	—
96	C₁₆×C₆₁	—	2	2	no	—
97	C₁₆×C₆₁	C₂²⋊C₄	1	2	yes	—
98	C₁₆×C₆₁	C₂²⋊C₄	1	2	yes	—
99	#11758185	—	2	2	no	—
100	#11758185	—	2	2	no	—
101	C₂³×C₆₁	#131	1	1	yes	characteristic
102	C₂²×D₆₁	—	2	1	no	—
103	C₂×C₄×C₆₁	—	2	4	no	—
104	C₂×C₄×C₆₁	—	2	2	no	—
105	C₂×C₄×C₆₁	C₂×C₄²	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	C₂²×C₈	1	1	yes	characteristic
109	C₂×C₄×C₆₁	C₂²×C₈	1	1	yes	characteristic
110	C₂×C₄×C₆₁	#131	1	1	yes	characteristic
111	C₂²⋊D₆₁	—	2	2	no	—
112	C₂²⋊D₆₁	—	2	2	no	—
113	C₂×Dic₆₁	—	2	1	no	—
114	C₂×Dic₆₁	—	2	2	no	—
115	D₂₄₄	—	2	1	no	—
116	D₂₄₄	C₄×C₈	1	2	yes	—
117	C₄×D₆₁	—	2	2	no	—
118	C₄×D₆₁	—	2	2	no	—
119	Dic₁₂₂	—	2	1	no	—
120	Dic₁₂₂	C₄×C₈	1	2	yes	—
121	C₈×C₆₁	—	2	2	no	—
122	C₈×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
123	C₈×C₆₁	#117	1	1	yes	characteristic
124	C₈×C₆₁	C₄×D₄	1	1	yes	characteristic
125	C₈×C₆₁	C₄×D₄	1	1	yes	characteristic
126	#92418	—	2	2	no	—
127	#92418	—	2	2	no	—
128	#12614	—	61	1	no	maximal, Sylow
129	C₂²×C₆₁	—	2	2	no	—
130	C₂²×C₆₁	#405	1	1	yes	characteristic
131	C₂²×C₆₁	C₈×D₄	1	1	yes	characteristic
132	C₂²×C₆₁	C₈×D₄	1	1	yes	characteristic
133	C₂×D₆₁	—	2	2	no	—
134	C₂×D₆₁	—	4	2	no	—
135	C₄×C₆₁	—	2	4	no	—
136	C₄×C₆₁	C₄×C₂²⋊C₄	1	1	yes	characteristic
137	C₄×C₆₁	C₂×C₄×C₈	1	1	yes	characteristic
138	C₄×C₆₁	C₈×D₄	1	1	yes	characteristic
139	C₄×C₆₁	C₈×D₄	1	1	yes	characteristic
140	Dic₆₁	—	2	2	no	—
141	Dic₆₁	—	4	2	no	—
142	C₂×#504	—	61	1	no	—
143	C₂×#350	—	61	1	no	—
144	#2108	—	61	1	no	—
145	C₂×C₄×C₁₆	D₆₁	1	1	yes	characteristic, complemented
146	#1764	—	61	1	no	—
147	#1379	—	61	1	no	—
148	#1376	—	61	1	no	—
149	C₂×C₆₁	—	2	1	no	—
150	C₂×C₆₁	—	2	1	no	—
151	C₂×C₆₁	C₈×C₂²⋊C₄	1	1	yes	characteristic
152	C₂×C₆₁	#2170	1	2	yes	—
153	D₆₁	—	4	2	no	—
154	C₂×#131	—	61	1	no	—
155	#504	—	61	4	no	—
156	#504	—	122	2	no	—
157	C₂×M₃₂	—	61	1	no	—
158	C₂×M₃₂	—	61	1	no	—
159	C₂×M₃₂	—	122	1	no	—
160	C₂²×C₁₆	—	122	1	no	—
161	C₂²×C₁₆	C₂×D₆₁	1	1	yes	characteristic
162	C₂²×C₁₆	C₂×D₆₁	1	1	yes	characteristic
163	C₂×#104	—	61	1	no	—
164	#416	—	61	1	no	—
165	#405	—	61	1	no	—
166	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
167	#350	—	61	4	no	—
168	#346	—	122	1	no	—
169	C₄×C₁₆	—	2	2	no	—
170	C₄×C₁₆	—	122	1	no	—
171	#341	—	61	1	no	—
172	#339	—	61	1	no	—
173	C₆₁	#12614	1	1	yes	characteristic, complemented, Sylow
174	C₂×#41	—	61	1	no	—
175	#131	—	61	2	no	—
176	#131	—	61	2	no	—
177	#131	—	122	1	no	—
178	#131	—	122	1	no	—
179	C₂×M₁₆	—	61	1	no	—
180	C₂×M₁₆	—	61	1	no	—
181	C₂×M₁₆	—	122	1	no	—
182	C₂²×C₈	—	122	1	no	—
183	C₂²×C₈	C₄×D₆₁	1	1	yes	characteristic
184	C₂²×C₈	C₂²×D₆₁	1	1	yes	characteristic
185	#117	—	61	1	no	—
186	C₂×C₄²	C₄×D₆₁	1	1	yes	characteristic
187	M₃₂	—	61	2	no	—
188	M₃₂	—	61	2	no	—
189	M₃₂	—	122	1	no	—
190	M₃₂	—	122	1	no	—
191	M₃₂	—	122	4	no	—
192	C₂×C₁₆	—	2	1	no	—
193	C₂×C₁₆	—	2	1	no	—
194	C₂×C₁₆	—	122	1	no	—
195	C₂×C₁₆	—	122	1	no	—
196	C₂×C₁₆	—	122	4	no	—
197	C₂×C₁₆	C₄×D₆₁	1	2	yes	—
198	C₂×C₁₆	C₄×D₆₁	1	2	yes	—
199	C₂×C₁₆	D₂₄₄	1	1	yes	characteristic
200	C₂×C₁₆	D₂₄₄	1	1	yes	characteristic
201	C₂×C₁₆	C₂²⋊D₆₁	1	2	yes	—
202	#104	—	61	4	no	—
203	#103	—	61	2	no	—
204	#102	—	61	2	no	—
205	#97	—	122	1	no	—
206	C₄×C₈	—	2	1	no	—
207	C₄×C₈	—	2	1	no	—
208	C₄×C₈	—	122	1	no	—
209	#41	—	61	2	no	—
210	#41	—	61	2	no	—
211	#41	—	122	1	no	—
212	#41	—	122	1	no	—
213	C₂×Q₈	—	61	1	no	—
214	C₂×D₄	—	61	1	no	—
215	C₂×D₄	—	122	1	no	—
216	C₂²×C₄	—	2	1	no	—
217	C₂²×C₄	—	122	1	no	—
218	C₂²×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
219	M₁₆	—	61	2	no	—
220	M₁₆	—	61	2	no	—
221	M₁₆	—	122	1	no	—
222	M₁₆	—	122	1	no	—
223	M₁₆	—	122	2	no	—
224	M₁₆	—	122	2	no	—
225	C₂×C₈	—	2	1	no	—
226	C₂×C₈	—	2	1	no	—
227	C₂×C₈	—	122	1	no	—
228	C₂×C₈	—	122	1	no	—
229	C₂×C₈	—	122	2	no	—
230	C₂×C₈	—	122	2	no	—
231	C₂×C₈	C₈×D₆₁	1	2	yes	—
232	C₂×C₈	C₂×C₄×D₆₁	1	1	yes	characteristic
233	C₂×C₈	C₂×C₄×D₆₁	1	1	yes	characteristic
234	C₂×C₈	C₂×D₂₄₄	1	1	yes	characteristic
235	C₂×C₈	#11758222	1	1	yes	characteristic
236	C₂×C₈	#11758222	1	1	yes	characteristic
237	C₂×C₈	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
238	C₄⋊C₄	—	61	2	no	—
239	C₂²⋊C₄	—	122	1	no	—
240	C₄²	—	2	1	no	—
241	C₄²	—	122	1	no	—
242	C₄²	C₈×D₆₁	1	2	yes	—
243	C₁₆	—	2	2	no	—
244	C₁₆	—	122	2	no	—
245	C₁₆	—	122	2	no	—
246	C₁₆	#11758198	1	2	yes	—
247	C₁₆	#11758198	1	2	yes	—
248	C₂³	—	122	1	no	—
249	C₂³	#14519783	1	1	yes	characteristic
250	Q₈	—	61	2	no	—
251	Q₈	—	122	1	no	—
252	D₄	—	61	2	no	—
253	D₄	—	122	2	no	—
254	D₄	—	122	2	no	—
255	D₄	—	122	1	no	—
256	C₂×C₄	—	2	4	no	—
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	122	1	no	—
259	C₂×C₄	—	122	2	no	—
260	C₂×C₄	—	122	2	no	—
261	C₂×C₄	—	122	2	no	—
262	C₂×C₄	C₄²×D₆₁	1	1	yes	characteristic
263	C₂×C₄	C₄×C₂²⋊D₆₁	1	1	yes	characteristic
264	C₂×C₄	C₄×D₂₄₄	1	1	yes	characteristic
265	C₂×C₄	C₂×C₈×D₆₁	1	1	yes	characteristic
266	C₂×C₄	C₂×C₈×D₆₁	1	1	yes	characteristic
267	C₂×C₄	#14519783	1	1	yes	characteristic
268	C₈	—	2	2	no	—
269	C₈	—	122	2	no	—
270	C₈	—	122	2	no	—
271	C₈	C₂×#11758198	1	1	yes	characteristic
272	C₈	#14519734	1	1	yes	characteristic
273	C₈	C₄×C₂²⋊D₆₁	1	1	yes	characteristic
274	C₈	C₄×D₂₄₄	1	1	yes	characteristic
275	C₂²	—	2	2	no	—
276	C₂²	—	122	2	no	—
277	C₂²	—	244	2	no	—
278	C₂²	#16839716	1	1	yes	characteristic
279	C₂²	C₈×C₂²⋊D₆₁	1	1	yes	characteristic
280	C₂²	C₈×D₂₄₄	1	1	yes	characteristic
281	C₄	—	2	4	no	—
282	C₄	—	122	2	no	—
283	C₄	—	244	2	no	—
284	C₄	C₄×#11758198	1	1	yes	characteristic
285	C₄	C₄×C₈×D₆₁	1	1	yes	characteristic
286	C₄	C₈×C₂²⋊D₆₁	1	1	yes	characteristic
287	C₄	C₈×D₂₄₄	1	1	yes	characteristic
288	C₂	—	2	1	no	—
289	C₂	—	2	1	no	—
290	C₂	—	244	2	no	—
291	C₂	C₈×#11758198	1	1	yes	characteristic
292	C₂	#21447657	1	2	yes	—

	subgroup	G/H
center	C₂×C₁₆	D₂₄₄
commutator	C₄×C₆₁	C₂×C₄×C₈
Frattini	C₂²×C₈	C₂²×D₆₁
Fitting	C₂×C₄×C₁₆×C₆₁	C₂
socle	C₂²×C₆₁	C₈×D₄
prime radical	C₂×#11758222	C₂×C₄
cosocle	C₂²×C₈×C₆₁	C₂³
hypercenter	C₂×C₄×C₁₆	D₆₁
hypocenter	C₆₁	#12614
LCS	G ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₁₆ ◁ C₂²×C₁₆ ◁ C₂×C₄×C₁₆

Complements

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

N	H ≅ G/N	type
3. #21447415	C₂	semidirect
5. #21443907	C₂	semidirect
6. #21443901	C₂	semidirect
7. C₂×C₄×C₁₆×C₆₁	C₂	semidirect
10. #16840178	C₄	semidirect
18. #16839545	C₄	semidirect
145. C₂×C₄×C₁₆	D₆₁	semidirect
173. C₆₁	#12614	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16840178	2
C₂×#16839545	2
#21447415	2
#21445656	2
#21443907	2
#21443901	2
C₂×C₄×C₁₆×C₆₁	2
#12614	61