Subgroups — #29595315

Order 15616.

412 conjugacy classes of subgroups.

The subgroup lattice has 290 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16839885	C₂	1	1	yes	maximal, characteristic
2	#21447566	C₂	1	1	yes	maximal, characteristic
3	#21447546	C₂	1	1	yes	maximal, characteristic
4	C₂×#16839540	C₂	1	1	yes	maximal, characteristic, complemented
5	#21443930	C₂	1	1	yes	maximal, characteristic, complemented
6	#21443922	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₁×#1778	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#14519799	C₂²	1	1	yes	characteristic
9	C₂×#14519783	—	2	1	no	—
10	C₂×#14519717	C₂²	1	1	yes	characteristic
11	C₂×#14519717	C₂²	1	1	yes	characteristic
12	#16839885	—	2	2	no	—
13	#16839885	C₄	1	4	yes	—
14	C₂×#14519713	C₂²	1	1	yes	characteristic
15	C₂×#14519705	—	2	1	no	—
16	#16839577	C₄	1	2	yes	complemented
17	#16839574	C₄	1	2	yes	complemented
18	#16839568	C₄	1	4	yes	complemented
19	#16839540	—	2	2	no	—
20	C₂×#108×C₆₁	C₂²	1	1	yes	characteristic
21	C₂×#108×C₆₁	C₂²	1	1	yes	characteristic
22	C₄×M₁₆×C₆₁	—	2	1	no	—
23	C₂×#97×C₆₁	C₂²	1	1	yes	characteristic
24	C₆₁×#341	—	2	1	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×#11758189	—	4	1	no	—
37	C₂×C₈×D₆₁	—	4	1	no	—
38	#14519717	—	2	1	no	—
39	#14519717	—	2	1	no	—
40	#14519717	—	2	1	no	—
41	#14519717	C₂×C₄	1	2	yes	—
42	C₂×#11758185	—	2	1	no	—
43	C₂×#11758185	C₂×C₄	1	2	yes	—
44	#14519713	—	2	1	no	—
45	#14519713	C₂×C₄	1	2	yes	—
46	#14519705	—	2	4	no	—
47	C₂×M₁₆×C₆₁	—	2	1	no	—
48	C₂×M₁₆×C₆₁	—	2	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	#108×C₆₁	—	2	1	no	—
53	#108×C₆₁	—	2	1	no	—
54	#108×C₆₁	C₂×C₄	1	2	yes	—
55	#108×C₆₁	C₂×C₄	1	2	yes	—
56	#97×C₆₁	—	4	1	no	—
57	#97×C₆₁	C₂×C₄	1	2	yes	—
58	#97×C₆₁	D₄	1	2	yes	—
59	C₄×C₈×C₆₁	—	4	1	no	—
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	M₁₆×C₆₁	—	2	2	no	—
71	M₁₆×C₆₁	—	2	2	no	—
72	M₁₆×C₆₁	—	4	1	no	—
73	M₁₆×C₆₁	—	4	1	no	—
74	C₂×C₈×C₆₁	—	2	1	no	—
75	C₂×C₈×C₆₁	—	2	1	no	—
76	C₂×C₈×C₆₁	—	4	1	no	—
77	C₂×C₈×C₆₁	—	4	1	no	—
78	C₂×C₈×C₆₁	C₂²⋊C₄	1	2	yes	—
79	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
81	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
82	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
83	C₂×C₈×C₆₁	#41	1	1	yes	characteristic
84	C₂×C₈×C₆₁	#41	1	1	yes	characteristic
85	C₄²×C₆₁	—	2	1	no	—
86	C₄²×C₆₁	C₂²⋊C₄	1	2	yes	—
87	#11758194	—	2	2	no	—
88	#11758194	—	4	1	no	—
89	C₂×#92418	—	4	1	no	—
90	Dic₂₄₄	—	2	1	no	—
91	Dic₂₄₄	C₄²	1	2	yes	—
92	D₄₈₈	—	2	1	no	—
93	D₄₈₈	C₄²	1	2	yes	—
94	#11758190	—	2	2	no	—
95	#11758190	—	2	2	no	—
96	#11758189	—	4	4	no	—
97	C₈×D₆₁	—	4	4	no	—
98	#11758185	—	2	2	no	—
99	#11758185	—	2	2	no	—
100	C₂³×C₆₁	#137	1	1	yes	characteristic
101	C₂²×D₆₁	—	4	1	no	—
102	C₂×C₄×C₆₁	—	2	2	no	—
103	C₂×C₄×C₆₁	—	8	1	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	AGL(1,ℤ/8)	1	1	yes	characteristic
110	C₂²⋊D₆₁	—	4	2	no	—
111	C₂²⋊D₆₁	—	4	2	no	—
112	C₂×Dic₆₁	—	4	1	no	—
113	D₂₄₄	—	2	2	no	—
114	D₂₄₄	—	4	1	no	—
115	C₄×D₆₁	—	4	2	no	—
116	C₄×D₆₁	—	4	2	no	—
117	Dic₁₂₂	—	2	2	no	—
118	Dic₁₂₂	—	4	1	no	—
119	C₈×C₆₁	—	2	2	no	—
120	C₈×C₆₁	—	4	2	no	—
121	C₈×C₆₁	—	4	2	no	—
122	C₈×C₆₁	C₂×C₄²	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	—	4	2	no	—
127	#12651	—	61	1	no	maximal, Sylow
128	C₂²×C₆₁	—	4	1	no	—
129	C₂²×C₆₁	#417	1	1	yes	characteristic
130	C₂²×C₆₁	#440	1	1	yes	characteristic
131	C₂²×C₆₁	#440	1	1	yes	characteristic
132	C₂×D₆₁	—	4	2	no	—
133	C₂×D₆₁	—	8	2	no	—
134	C₄×C₆₁	—	4	2	no	—
135	C₄×C₆₁	C₄×C₂²⋊C₄	1	1	yes	characteristic
136	C₄×C₆₁	#418	1	1	yes	characteristic
137	C₄×C₆₁	#442	1	1	yes	characteristic
138	C₄×C₆₁	#442	1	1	yes	characteristic
139	Dic₆₁	—	4	2	no	—
140	C₂×#444	—	61	1	no	—
141	#2148	—	61	1	no	—
142	#2143	—	61	1	no	—
143	C₂×#347	—	61	1	no	—
144	#1778	D₆₁	1	1	yes	characteristic, complemented
145	#1388	—	61	1	no	—
146	#1385	—	61	1	no	—
147	C₂×C₆₁	—	2	1	no	—
148	C₂×C₆₁	—	2	1	no	—
149	C₂×C₆₁	—	2	1	no	—
150	C₂×C₆₁	#1762	1	1	yes	characteristic
151	D₆₁	—	8	2	no	—
152	C₂×#135	—	61	1	no	—
153	C₂×#131	—	122	1	no	—
154	C₂×M₃₂	—	61	1	no	—
155	C₂×M₃₂	—	61	1	no	—
156	#444	—	61	4	no	—
157	#444	—	122	2	no	—
158	C₂×#108	—	61	1	no	—
159	C₂×#108	C₂×D₆₁	1	1	yes	characteristic
160	C₂×#108	C₂×D₆₁	1	1	yes	characteristic
161	C₂×#104	—	122	1	no	—
162	C₄×M₁₆	—	2	1	no	—
163	C₂×#97	C₂×D₆₁	1	1	yes	characteristic
164	#362	—	61	2	no	—
165	#361	—	61	2	no	—
166	#359	—	61	4	no	—
167	#347	—	122	2	no	—
168	#341	—	2	1	no	—
169	C₆₁	#12651	1	1	yes	characteristic, complemented, Sylow
170	C₂×#41	—	122	1	no	—
171	#135	—	61	2	no	—
172	#135	—	61	2	no	—
173	#135	—	122	1	no	—
174	#135	—	122	1	no	—
175	C₂×Q₁₆	—	61	1	no	—
176	C₂×SD₁₆	—	122	1	no	—
177	C₂×D₈	—	61	1	no	—
178	#131	—	122	4	no	—
179	#131	—	244	2	no	—
180	C₂×M₁₆	—	2	1	no	—
181	C₂×M₁₆	—	2	1	no	—
182	C₂×M₁₆	—	122	1	no	—
183	C₂×M₁₆	—	244	1	no	—
184	C₂²×C₈	—	244	1	no	—
185	C₂²×C₈	C₂²⋊D₆₁	1	1	yes	characteristic
186	C₂²×C₈	C₂²×D₆₁	1	1	yes	characteristic
187	C₂×C₄²	C₂²⋊D₆₁	1	1	yes	characteristic
188	M₃₂	—	61	2	no	—
189	M₃₂	—	122	1	no	—
190	M₃₂	—	122	1	no	—
191	M₃₂	—	122	1	no	—
192	C₂×C₁₆	—	61	2	no	—
193	C₂×C₁₆	—	122	1	no	—
194	#108	—	2	1	no	—
195	#108	—	2	1	no	—
196	#108	—	61	2	no	—
197	#108	—	122	1	no	—
198	#108	C₄×D₆₁	1	2	yes	—
199	#108	C₄×D₆₁	1	2	yes	—
200	#104	—	122	4	no	—
201	#97	—	4	1	no	—
202	#97	C₄×D₆₁	1	2	yes	—
203	#97	D₂₄₄	1	2	yes	—
204	C₄×C₈	—	4	1	no	—
205	#41	—	122	2	no	—
206	#41	—	122	2	no	—
207	#41	—	244	1	no	—
208	#41	—	244	1	no	—
209	C₂×Q₈	—	122	1	no	—
210	C₂×D₄	—	122	1	no	—
211	C₂×D₄	—	244	1	no	—
212	C₂²×C₄	—	2	1	no	—
213	C₂²×C₄	—	244	1	no	—
214	C₂²×C₄	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
215	Q₁₆	—	61	2	no	—
216	Q₁₆	—	122	1	no	—
217	SD₁₆	—	122	2	no	—
218	SD₁₆	—	122	2	no	—
219	D₈	—	61	2	no	—
220	D₈	—	122	1	no	—
221	M₁₆	—	2	2	no	—
222	M₁₆	—	2	2	no	—
223	M₁₆	—	4	1	no	—
224	M₁₆	—	4	1	no	—
225	M₁₆	—	122	2	no	—
226	M₁₆	—	244	4	no	—
227	M₁₆	—	244	1	no	—
228	C₂×C₈	—	2	1	no	—
229	C₂×C₈	—	2	1	no	—
230	C₂×C₈	—	4	1	no	—
231	C₂×C₈	—	4	1	no	—
232	C₂×C₈	—	244	4	no	—
233	C₂×C₈	—	244	1	no	—
234	C₂×C₈	#11758198	1	2	yes	—
235	C₂×C₈	C₂×C₄×D₆₁	1	1	yes	characteristic
236	C₂×C₈	C₂×C₄×D₆₁	1	1	yes	characteristic
237	C₂×C₈	C₂×C₄×D₆₁	1	1	yes	characteristic
238	C₂×C₈	C₂×D₂₄₄	1	1	yes	characteristic
239	C₂×C₈	#11758222	1	1	yes	characteristic
240	C₂×C₈	#11758222	1	1	yes	characteristic
241	C₄²	—	2	1	no	—
242	C₄²	#11758198	1	2	yes	—
243	C₁₆	—	122	2	no	—
244	C₁₆	—	122	2	no	—
245	C₂³	—	244	1	no	—
246	C₂³	#14519808	1	1	yes	characteristic
247	Q₈	—	122	2	no	—
248	Q₈	—	244	1	no	—
249	D₄	—	122	2	no	—
250	D₄	—	244	2	no	—
251	D₄	—	244	2	no	—
252	D₄	—	244	1	no	—
253	C₂×C₄	—	2	2	no	—
254	C₂×C₄	—	8	1	no	—
255	C₂×C₄	—	244	1	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×C₂²⋊D₆₁	1	1	yes	characteristic
260	C₂×C₄	C₄×C₂²⋊D₆₁	1	1	yes	characteristic
261	C₂×C₄	#14519802	1	1	yes	characteristic
262	C₂×C₄	#14519802	1	1	yes	characteristic
263	C₂×C₄	#14519803	1	1	yes	characteristic
264	C₈	—	2	2	no	—
265	C₈	—	4	2	no	—
266	C₈	—	4	2	no	—
267	C₈	—	244	2	no	—
268	C₈	C₄²×D₆₁	1	1	yes	characteristic
269	C₈	#14519734	1	1	yes	characteristic
270	C₈	C₄×D₂₄₄	1	1	yes	characteristic
271	C₈	C₄×D₂₄₄	1	1	yes	characteristic
272	C₂²	—	4	1	no	—
273	C₂²	—	488	2	no	—
274	C₂²	—	244	2	no	—
275	C₂²	#16839761	1	1	yes	characteristic
276	C₂²	#16839860	1	1	yes	characteristic
277	C₂²	#16839860	1	1	yes	characteristic
278	C₄	—	4	2	no	—
279	C₄	—	244	2	no	—
280	C₄	C₄×#11758198	1	1	yes	characteristic
281	C₄	#16839766	1	1	yes	characteristic
282	C₄	#16839872	1	1	yes	characteristic
283	C₄	#16839872	1	1	yes	characteristic
284	C₂	—	2	1	no	—
285	C₂	—	2	1	no	—
286	C₂	—	2	1	no	—
287	C₂	—	488	2	no	—
288	C₂	#21445647	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#16839872
commutator	C₈×C₆₁	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₆₁
Fitting	C₆₁×#1778	C₂
socle	C₂×C₆₁	#1762
prime radical	C₂×#14519799	C₂²
cosocle	C₂²×C₈×C₆₁	C₂³
hypercenter	#1778	D₆₁
hypocenter	C₆₁	#12651
LCS	G ▷ C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#97 ◁ #1778

Complements

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

N	H ≅ G/N	type
4. C₂×#16839540	C₂	semidirect
5. #21443930	C₂	semidirect
6. #21443922	C₂	semidirect
7. C₆₁×#1778	C₂	semidirect
16. #16839577	C₄	semidirect
17. #16839574	C₄	semidirect
18. #16839568	C₄	semidirect
144. #1778	D₆₁	semidirect
169. C₆₁	#12651	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16839885	2
#21447566	2
#21447546	2
C₂×#16839540	2
#21443930	2
#21443922	2
C₆₁×#1778	2
#12651	61