Subgroups — #29833267

Order 15616.

648 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21458921	C₂	1	2	yes	maximal, complemented
2	C₂×#16839880	C₂	1	1	yes	maximal, characteristic
3	#21447660	C₂	1	4	yes	maximal, complemented
4	#21447655	C₂	1	4	yes	maximal, complemented
5	#21447448	C₂	1	2	yes	maximal, complemented
6	C₄×#14519717	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₆₁×#364	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#14519799	C₄	1	1	yes	characteristic
9	C₂×#14519783	C₂²	1	2	yes	—
10	#16840180	C₂²	1	8	yes	complemented
11	#16840178	—	2	4	no	—
12	C₂×#14519717	C₂²	1	1	yes	characteristic
13	C₂×#14519717	C₂²	1	1	yes	characteristic
14	M₃₂×D₆₁	—	2	4	no	—
15	#16839880	C₄	1	2	yes	—
16	#16839880	C₄	1	2	yes	—
17	#16839880	C₂²	1	2	yes	—
18	#16839880	C₂²	1	2	yes	—
19	C₂×#14519713	C₄	1	1	yes	characteristic
20	C₂×#14519705	C₄	1	2	yes	—
21	#16839547	C₂²	1	8	yes	complemented
22	#16839538	C₂²	1	2	yes	complemented
23	C₄×#11758185	C₂²	1	2	yes	complemented
24	C₂×M₃₂×C₆₁	C₂²	1	2	yes	—
25	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
26	C₆₁×#364	C₂²	1	2	yes	complemented
27	C₆₁×#364	C₂²	1	2	yes	complemented
28	C₂×#11758222	C₂×C₄	1	2	yes	—
29	#14519799	C₈	1	2	yes	—
30	#14519799	C₈	1	2	yes	—
31	#14519799	C₂×C₄	1	2	yes	—
32	#14519799	C₂×C₄	1	2	yes	—
33	C₂×Dic₂₄₄	C₈	1	1	yes	characteristic
34	C₂×#11758190	C₈	1	2	yes	—
35	C₂×D₄₈₈	C₈	1	1	yes	characteristic
36	#14519783	—	2	2	no	—
37	#14519783	—	2	2	no	—
38	#14519783	C₂×C₄	1	4	yes	—
39	#14519783	C₂³	1	4	yes	—
40	C₂×#11758194	C₂×C₄	1	2	yes	—
41	C₂×#11758189	—	2	2	no	—
42	C₂×C₈×D₆₁	—	2	2	no	—
43	#14519717	—	2	1	no	—
44	#14519717	—	2	1	no	—
45	#14519717	D₄	1	2	yes	—
46	#14519717	D₄	1	2	yes	—
47	#14519716	—	2	8	no	—
48	C₂×#11758185	C₂³	1	2	yes	—
49	C₂×#11758185	C₂³	1	2	yes	—
50	C₁₆×D₆₁	—	2	8	no	—
51	#14519713	C₂×C₄	1	2	yes	—
52	#14519713	C₂×C₄	1	2	yes	—
53	#14519705	C₂×C₄	1	4	yes	—
54	#14519705	C₂×C₄	1	4	yes	—
55	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
56	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
57	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
58	M₃₂×C₆₁	—	2	2	no	—
59	M₃₂×C₆₁	C₂³	1	4	yes	—
60	C₂×C₁₆×C₆₁	—	2	2	no	—
61	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
62	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
63	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
64	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
65	C₂²×C₄×C₆₁	—	2	1	no	—
66	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
67	C₂×C₂²⋊D₆₁	—	2	2	no	—
68	#11758222	—	2	2	no	—
69	#11758222	—	2	2	no	—
70	#11758222	C₂×C₈	1	4	yes	—
71	#11758222	C₂²×C₄	1	4	yes	—
72	C₂×D₂₄₄	C₂×C₈	1	2	yes	—
73	C₂×C₄×D₆₁	—	2	2	no	—
74	C₂×Dic₁₂₂	C₂×C₈	1	2	yes	—
75	C₂×C₈×C₆₁	C₂×C₈	1	1	yes	characteristic
76	C₂×C₈×C₆₁	C₂×C₈	1	1	yes	characteristic
77	C₂×C₈×C₆₁	C₂×C₈	1	1	yes	characteristic
78	C₂×C₈×C₆₁	C₂×C₈	1	1	yes	characteristic
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	C₂×D₄	1	1	yes	characteristic
84	C₂×C₈×C₆₁	#41	1	1	yes	characteristic
85	C₂×C₈×C₆₁	#41	1	1	yes	characteristic
86	C₂×C₈×C₆₁	C₂⁴	1	1	yes	characteristic
87	C₄²×C₆₁	—	2	1	no	—
88	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
89	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
90	#11758194	—	2	2	no	—
91	#11758194	C₂²×C₄	1	4	yes	—
92	C₂×#92418	—	2	2	no	—
93	Dic₂₄₄	C₂×C₈	1	2	yes	—
94	Dic₂₄₄	C₂×C₈	1	2	yes	—
95	D₄₈₈	C₂×C₈	1	2	yes	—
96	D₄₈₈	C₂×C₈	1	2	yes	—
97	#11758190	C₂×C₈	1	4	yes	—
98	#11758190	C₂×C₈	1	4	yes	—
99	#11758189	—	2	4	no	—
100	#11758189	—	2	4	no	—
101	C₈×D₆₁	—	2	4	no	—
102	C₈×D₆₁	—	2	4	no	—
103	C₁₆×C₆₁	—	2	4	no	—
104	#11758185	—	2	2	no	—
105	#11758185	C₂×D₄	1	4	yes	—
106	C₂³×C₆₁	#131	1	1	yes	characteristic
107	C₂²×D₆₁	—	2	2	no	—
108	C₂×C₄×C₆₁	—	2	1	no	—
109	C₂×C₄×C₆₁	—	2	1	no	—
110	C₂×C₄×C₆₁	—	4	2	no	—
111	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
112	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
113	C₂×C₄×C₆₁	C₂²×C₈	1	1	yes	characteristic
114	C₂×C₄×C₆₁	C₂²×C₈	1	1	yes	characteristic
115	C₂×C₄×C₆₁	#131	1	1	yes	characteristic
116	C₂×C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
117	C₂²⋊D₆₁	—	2	4	no	—
118	C₂²⋊D₆₁	—	2	4	no	—
119	C₂×Dic₆₁	—	2	2	no	—
120	D₂₄₄	—	2	2	no	—
121	D₂₄₄	C₂²×C₈	1	4	yes	—
122	C₄×D₆₁	—	2	4	no	—
123	C₄×D₆₁	—	2	4	no	—
124	Dic₁₂₂	—	2	2	no	—
125	Dic₁₂₂	C₂²×C₈	1	4	yes	—
126	C₈×C₆₁	C₄×D₄	1	1	yes	characteristic
127	C₈×C₆₁	C₄×D₄	1	1	yes	characteristic
128	C₈×C₆₁	C₂²×C₈	1	1	yes	characteristic
129	C₈×C₆₁	C₂²×C₈	1	1	yes	characteristic
130	C₈×C₆₁	C₂²×C₈	1	1	yes	characteristic
131	C₈×C₆₁	C₂²×C₈	1	1	yes	characteristic
132	C₈×C₆₁	C₂²×D₄	1	1	yes	characteristic
133	C₈×C₆₁	C₂×#41	1	1	yes	characteristic
134	#92418	—	2	4	no	—
135	#33443	—	61	1	no	maximal, Sylow
136	C₂²×C₆₁	—	4	1	no	—
137	C₂²×C₆₁	C₈×D₄	1	1	yes	characteristic
138	C₂²×C₆₁	C₈×D₄	1	1	yes	characteristic
139	C₂²×C₆₁	C₂×#131	1	1	yes	characteristic
140	C₂×D₆₁	—	2	4	no	—
141	C₂×D₆₁	—	8	2	no	—
142	C₄×C₆₁	—	2	2	no	—
143	C₄×C₆₁	—	4	1	no	—
144	C₄×C₆₁	C₈×D₄	1	1	yes	characteristic
145	C₄×C₆₁	C₈×D₄	1	1	yes	characteristic
146	C₄×C₆₁	C₂×C₄×D₄	1	1	yes	characteristic
147	C₄×C₆₁	C₂³×C₈	1	1	yes	characteristic
148	Dic₆₁	—	2	4	no	—
149	#3407	—	61	2	no	—
150	C₂×#443	—	61	1	no	—
151	#2171	—	61	4	no	—
152	#2170	—	61	4	no	—
153	C₂×#364	D₆₁	1	1	yes	characteristic, complemented
154	#2117	—	61	2	no	—
155	C₄×M₃₂	—	61	1	no	—
156	C₂×C₆₁	—	2	1	no	—
157	C₂×C₆₁	—	2	1	no	—
158	C₂×C₆₁	—	2	1	no	—
159	C₂×C₆₁	C₂×C₈×D₄	1	1	yes	characteristic
160	D₆₁	—	4	4	no	—
161	C₂×#135	—	61	1	no	—
162	C₂×#131	—	61	2	no	—
163	#504	—	61	8	no	—
164	#504	—	122	4	no	—
165	C₂×M₃₂	—	61	1	no	—
166	C₂×M₃₂	—	61	1	no	—
167	C₂×M₃₂	—	122	4	no	—
168	C₂×M₃₂	C₂×D₆₁	1	2	yes	—
169	#443	—	61	2	no	—
170	#443	—	61	2	no	—
171	#443	—	61	2	no	—
172	#443	—	61	2	no	—
173	C₂×#108	—	61	1	no	—
174	C₂×#104	—	61	2	no	—
175	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
176	#364	C₂×D₆₁	1	2	yes	complemented
177	#364	C₂×D₆₁	1	2	yes	complemented
178	#350	—	61	8	no	—
179	#346	—	61	2	no	—
180	C₄×C₁₆	—	61	2	no	—
181	C₆₁	#33443	1	1	yes	characteristic, complemented, Sylow
182	C₂×#41	—	61	2	no	—
183	#135	—	61	2	no	—
184	#135	—	61	2	no	—
185	#135	—	61	2	no	—
186	#135	—	61	2	no	—
187	C₂×Q₁₆	—	61	1	no	—
188	C₂×SD₁₆	—	61	2	no	—
189	C₂×D₈	—	61	1	no	—
190	#131	—	61	4	no	—
191	#131	—	61	4	no	—
192	#131	—	122	2	no	—
193	#131	—	122	2	no	—
194	C₂×M₁₆	—	61	2	no	—
195	C₂×M₁₆	—	122	2	no	—
196	C₂²×C₈	—	122	2	no	—
197	C₂²×C₈	C₄×D₆₁	1	1	yes	characteristic
198	C₂²×C₈	C₂²×D₆₁	1	1	yes	characteristic
199	C₂×C₄²	C₄×D₆₁	1	1	yes	characteristic
200	M₃₂	—	2	2	no	—
201	M₃₂	—	61	2	no	—
202	M₃₂	—	61	2	no	—
203	M₃₂	—	122	8	no	—
204	M₃₂	—	122	1	no	—
205	M₃₂	—	122	1	no	—
206	M₃₂	C₂²×D₆₁	1	4	yes	—
207	C₂×C₁₆	—	2	2	no	—
208	C₂×C₁₆	—	61	2	no	—
209	C₂×C₁₆	—	61	2	no	—
210	C₂×C₁₆	—	122	8	no	—
211	#108	—	61	2	no	—
212	#108	—	61	2	no	—
213	#104	—	61	4	no	—
214	#104	—	61	4	no	—
215	C₄×C₈	C₄×D₆₁	1	1	yes	characteristic
216	C₄×C₈	C₄×D₆₁	1	1	yes	characteristic
217	C₄×C₈	C₂²×D₆₁	1	1	yes	characteristic
218	C₄×C₈	C₂²×D₆₁	1	1	yes	characteristic
219	#41	—	61	4	no	—
220	#41	—	61	4	no	—
221	#41	—	122	2	no	—
222	#41	—	122	2	no	—
223	C₂×Q₈	—	61	2	no	—
224	C₂×D₄	—	61	2	no	—
225	C₂×D₄	—	122	2	no	—
226	C₂²×C₄	—	2	1	no	—
227	C₂²×C₄	—	122	2	no	—
228	C₂²×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
229	Q₁₆	—	61	2	no	—
230	Q₁₆	—	61	2	no	—
231	SD₁₆	—	61	4	no	—
232	SD₁₆	—	61	4	no	—
233	D₈	—	61	2	no	—
234	D₈	—	61	2	no	—
235	M₁₆	—	61	4	no	—
236	M₁₆	—	122	2	no	—
237	M₁₆	—	122	4	no	—
238	M₁₆	—	122	4	no	—
239	C₂×C₈	—	122	2	no	—
240	C₂×C₈	—	122	4	no	—
241	C₂×C₈	—	122	4	no	—
242	C₂×C₈	C₈×D₆₁	1	1	yes	characteristic
243	C₂×C₈	C₈×D₆₁	1	1	yes	characteristic
244	C₂×C₈	C₈×D₆₁	1	1	yes	characteristic
245	C₂×C₈	C₈×D₆₁	1	1	yes	characteristic
246	C₂×C₈	C₂×C₄×D₆₁	1	1	yes	characteristic
247	C₂×C₈	C₂×C₄×D₆₁	1	1	yes	characteristic
248	C₂×C₈	C₂×C₄×D₆₁	1	1	yes	characteristic
249	C₂×C₈	D₄×D₆₁	1	1	yes	characteristic
250	C₂×C₈	D₄×D₆₁	1	1	yes	characteristic
251	C₂×C₈	#11758226	1	1	yes	characteristic
252	C₂×C₈	#11758226	1	1	yes	characteristic
253	C₂×C₈	C₂³×D₆₁	1	1	yes	characteristic
254	C₄²	—	2	1	no	—
255	C₄²	C₂×C₄×D₆₁	1	1	yes	characteristic
256	C₄²	C₂×C₄×D₆₁	1	1	yes	characteristic
257	C₁₆	—	2	4	no	—
258	C₁₆	—	61	4	no	—
259	C₁₆	—	122	2	no	—
260	C₂³	—	122	2	no	—
261	C₂³	#14519785	1	1	yes	characteristic
262	Q₈	—	61	4	no	—
263	Q₈	—	122	2	no	—
264	D₄	—	61	4	no	—
265	D₄	—	122	4	no	—
266	D₄	—	122	4	no	—
267	D₄	—	122	2	no	—
268	C₂×C₄	—	2	1	no	—
269	C₂×C₄	—	2	1	no	—
270	C₂×C₄	—	4	2	no	—
271	C₂×C₄	—	122	2	no	—
272	C₂×C₄	—	122	4	no	—
273	C₂×C₄	—	122	4	no	—
274	C₂×C₄	#14519739	1	1	yes	characteristic
275	C₂×C₄	#14519739	1	1	yes	characteristic
276	C₂×C₄	C₂×C₈×D₆₁	1	1	yes	characteristic
277	C₂×C₄	C₂×C₈×D₆₁	1	1	yes	characteristic
278	C₂×C₄	#14519785	1	1	yes	characteristic
279	C₂×C₄	C₂²×C₄×D₆₁	1	1	yes	characteristic
280	C₈	—	122	4	no	—
281	C₈	#14519739	1	1	yes	characteristic
282	C₈	#14519739	1	1	yes	characteristic
283	C₈	C₂×C₈×D₆₁	1	1	yes	characteristic
284	C₈	C₂×C₈×D₆₁	1	1	yes	characteristic
285	C₈	C₂×C₈×D₆₁	1	1	yes	characteristic
286	C₈	C₂×C₈×D₆₁	1	1	yes	characteristic
287	C₈	C₂×D₄×D₆₁	1	1	yes	characteristic
288	C₈	C₂×#11758226	1	1	yes	characteristic
289	C₂²	—	4	1	no	—
290	C₂²	—	122	4	no	—
291	C₂²	—	488	2	no	—
292	C₂²	#16839827	1	1	yes	characteristic
293	C₂²	#16839827	1	1	yes	characteristic
294	C₂²	C₂×#14519785	1	1	yes	characteristic
295	C₄	—	2	2	no	—
296	C₄	—	4	1	no	—
297	C₄	—	122	4	no	—
298	C₄	#16839827	1	1	yes	characteristic
299	C₄	#16839827	1	1	yes	characteristic
300	C₄	C₂×#14519739	1	1	yes	characteristic
301	C₄	C₂²×C₈×D₆₁	1	1	yes	characteristic
302	C₂	—	2	1	no	—
303	C₂	—	2	1	no	—
304	C₂	—	2	1	no	—
305	C₂	—	244	4	no	—
306	C₂	C₂×#16839827	1	1	yes	characteristic

	subgroup	G/H
center	C₈	C₂×D₄×D₆₁
commutator	C₄×C₆₁	C₂³×C₈
Frattini	C₂×C₈	C₂³×D₆₁
Fitting	C₂×C₆₁×#364	C₂
socle	C₂×C₆₁	C₂×C₈×D₄
prime radical	C₂×#14519799	C₄
cosocle	C₂×C₈×C₆₁	C₂⁴
hypercenter	C₂×#364	D₆₁
hypocenter	C₆₁	#33443
LCS	G ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ C₂×#364

Complements

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

N	H ≅ G/N	type
1. #21458921	C₂	semidirect
3. #21447660	C₂	semidirect
4. #21447655	C₂	semidirect
5. #21447448	C₂	semidirect
6. C₄×#14519717	C₂	semidirect
7. C₂×C₆₁×#364	C₂	semidirect
10. #16840180	C₂²	semidirect
21. #16839547	C₂²	semidirect
22. #16839538	C₂²	semidirect
23. C₄×#11758185	C₂²	semidirect
26. C₆₁×#364	C₂²	semidirect
27. C₆₁×#364	C₂²	semidirect
153. C₂×#364	D₆₁	semidirect
176. #364	C₂×D₆₁	semidirect
177. #364	C₂×D₆₁	semidirect
181. C₆₁	#33443	semidirect

Maximal subgroups

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

subgroup	index
#21458921	2
C₂×#16839880	2
#21447660	2
#21447655	2
#21447448	2
C₄×#14519717	2
C₂×C₆₁×#364	2
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