Subgroups — #29698544

Order 15616.

808 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	#21454737	C₂	1	2	yes	maximal
2	C₂×#16839885	C₂	1	1	yes	maximal, characteristic
3	#21454427	C₂	1	2	yes	maximal, complemented
4	#21446609	C₂	1	2	yes	maximal, complemented
5	#21446060	C₂	1	4	yes	maximal, complemented
6	#21445994	C₂	1	1	yes	maximal, characteristic, complemented
7	#21445870	C₂	1	2	yes	maximal, complemented
8	C₆₁×#1773	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#14519799	C₂²	1	1	yes	characteristic
10	#16840528	C₂²	1	4	yes	—
11	C₂×#14519783	C₂²	1	2	yes	—
12	#16839885	—	2	4	no	—
13	#16839885	C₄	1	8	yes	complemented
14	#16839880	—	2	4	no	—
15	#16839810	C₂²	1	4	yes	—
16	#16839809	—	2	4	no	—
17	C₂×#14519713	C₂²	1	2	yes	—
18	C₂×#14519713	C₂²	1	2	yes	—
19	C₂×#14519713	C₂²	1	1	yes	characteristic
20	#108×D₆₁	—	2	4	no	—
21	#16839786	C₂²	1	2	yes	—
22	#16839780	C₂²	1	4	yes	—
23	C₂×#14519705	C₂²	1	2	yes	—
24	#16839532	C₂²	1	4	yes	—
25	C₂×#108×C₆₁	C₂²	1	1	yes	characteristic
26	C₆₁×#427	C₂²	1	1	yes	characteristic
27	C₆₁×#405	C₂²	1	2	yes	—
28	C₂×#97×C₆₁	C₂²	1	1	yes	characteristic
29	C₆₁×#339	C₂²	1	2	yes	—
30	C₂×#11758222	C₂³	1	2	yes	—
31	#14519799	C₂×C₄	1	4	yes	—
32	#14519799	D₄	1	4	yes	—
33	C₂×Dic₂₄₄	Q₈	1	1	yes	characteristic
34	C₂×#11758190	Q₈	1	2	yes	—
35	C₂×D₄₈₈	Q₈	1	1	yes	characteristic
36	#14519785	—	4	4	no	—
37	#14519783	—	2	2	no	—
38	#14519783	—	2	2	no	—
39	#14519783	—	2	8	no	—
40	#14519783	C₂×C₄	1	4	yes	—
41	#14519783	C₂×C₄	1	4	yes	—
42	C₂×#11758194	C₂³	1	4	yes	—
43	C₂×#11758194	C₂³	1	2	yes	—
44	C₂×#11758189	—	2	2	no	—
45	M₁₆×D₆₁	—	2	8	no	—
46	C₂×C₈×D₆₁	—	2	2	no	—
47	#14519713	—	2	2	no	—
48	#14519713	—	2	2	no	—
49	#14519713	—	2	2	no	—
50	#14519713	—	2	2	no	—
51	#14519713	C₂×C₄	1	4	yes	—
52	#14519712	—	2	8	no	—
53	C₄⋊#92418	—	4	2	no	—
54	#14519705	—	2	8	no	—
55	#14519705	C₂×C₄	1	8	yes	—
56	C₂×M₁₆×C₆₁	C₂³	1	2	yes	—
57	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
58	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
59	#117×C₆₁	C₂³	1	2	yes	—
60	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
61	#108×C₆₁	D₄	1	4	yes	—
62	#107×C₆₁	—	2	1	no	—
63	#106×C₆₁	—	2	1	no	—
64	#97×C₆₁	—	2	2	no	—
65	#97×C₆₁	C₂×C₄	1	4	yes	—
66	C₄×C₈×C₆₁	—	2	2	no	—
67	C₂²×C₄×C₆₁	—	2	1	no	—
68	C₂²×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
69	C₂×C₂²⋊D₆₁	—	2	2	no	—
70	#11758222	—	2	2	no	—
71	#11758222	—	2	2	no	—
72	#11758222	C₂²×C₄	1	4	yes	—
73	#11758222	C₂×D₄	1	4	yes	—
74	C₂×D₂₄₄	C₂×Q₈	1	2	yes	—
75	C₂×C₄×D₆₁	—	2	2	no	—
76	C₂×Dic₁₂₂	C₂×Q₈	1	2	yes	—
77	M₁₆×C₆₁	—	2	2	no	—
78	M₁₆×C₆₁	C₂×D₄	1	4	yes	—
79	C₂×C₈×C₆₁	—	2	2	no	—
80	C₂×C₈×C₆₁	C₂²×C₄	1	2	yes	—
81	C₂×C₈×C₆₁	C₂²×C₄	1	2	yes	—
82	C₂×C₈×C₆₁	C₂×D₄	1	2	yes	—
83	C₂×C₈×C₆₁	C₂×D₄	1	2	yes	—
84	C₂×C₈×C₆₁	C₂×Q₈	1	1	yes	characteristic
85	C₂×C₈×C₆₁	C₂×Q₈	1	1	yes	characteristic
86	C₂×C₈×C₆₁	#41	1	1	yes	characteristic
87	C₂×C₈×C₆₁	#41	1	1	yes	characteristic
88	C₄⋊C₄×C₆₁	—	2	2	no	—
89	C₂²⋊C₄×C₆₁	—	2	2	no	—
90	C₄²×C₆₁	—	2	1	no	—
91	C₄²×C₆₁	—	2	2	no	—
92	C₄²×C₆₁	C₂²×C₄	1	2	yes	—
93	#11758194	—	2	2	no	—
94	#11758194	—	2	4	no	—
95	#11758194	—	2	4	no	—
96	#11758194	C₂²×C₄	1	4	yes	—
97	C₂×#92418	—	2	2	no	—
98	C₂×#92418	—	2	4	no	—
99	Dic₂₄₄	C₄⋊C₄	1	4	yes	—
100	D₄₈₈	C₄⋊C₄	1	4	yes	—
101	#11758190	C₄⋊C₄	1	8	yes	—
102	#11758189	—	2	4	no	—
103	#11758189	—	2	4	no	—
104	#11758189	—	4	8	no	—
105	C₈×D₆₁	—	2	4	no	—
106	C₈×D₆₁	—	2	4	no	—
107	C₈×D₆₁	—	4	8	no	—
108	C₂³×C₆₁	2₋¹⁺⁴	1	1	yes	characteristic
109	C₂²×D₆₁	—	2	2	no	—
110	C₂×C₄×C₆₁	—	2	2	no	—
111	C₂×C₄×C₆₁	—	4	2	no	—
112	C₂×C₄×C₆₁	—	8	1	no	—
113	C₂×C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
114	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
115	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
116	C₂×C₄×C₆₁	C₂²×Q₈	1	1	yes	characteristic
117	C₂×C₄×C₆₁	C₂×#41	1	1	yes	characteristic
118	C₂×C₄×C₆₁	H(5,𝔽₂)	1	1	yes	characteristic
119	C₂²⋊D₆₁	—	2	4	no	—
120	C₂²⋊D₆₁	—	2	4	no	—
121	C₂×Dic₆₁	—	2	2	no	—
122	D₂₄₄	—	2	2	no	—
123	D₂₄₄	C₂×C₄⋊C₄	1	4	yes	—
124	C₄×D₆₁	—	2	4	no	—
125	C₄×D₆₁	—	2	4	no	—
126	Dic₁₂₂	—	2	2	no	—
127	Dic₁₂₂	C₂×C₄⋊C₄	1	4	yes	—
128	C₈×C₆₁	—	2	4	no	—
129	C₈×C₆₁	C₂×C₄⋊C₄	1	2	yes	—
130	C₈×C₆₁	C₂×C₄⋊C₄	1	2	yes	—
131	C₈×C₆₁	C₄×D₄	1	2	yes	—
132	C₈×C₆₁	C₄×D₄	1	2	yes	—
133	#92418	—	2	4	no	—
134	#92418	—	4	4	no	—
135	#22695	—	61	1	no	maximal, Sylow
136	C₂²×C₆₁	—	4	1	no	—
137	C₂²×C₆₁	#520	1	1	yes	characteristic
138	C₂²×C₆₁	#547	1	1	yes	characteristic
139	C₂²×C₆₁	D₄×Q₈	1	1	yes	characteristic
140	C₂×D₆₁	—	2	4	no	—
141	C₂×D₆₁	—	8	2	no	—
142	C₄×C₆₁	—	4	2	no	—
143	C₄×C₆₁	—	8	2	no	—
144	C₄×C₆₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
145	C₄×C₆₁	C₂×C₄×D₄	1	1	yes	characteristic
146	C₄×C₆₁	D₄²	1	1	yes	characteristic
147	C₄×C₆₁	#554	1	1	yes	characteristic
148	Dic₆₁	—	2	4	no	—
149	#2957	—	61	2	no	—
150	C₂×#444	—	61	1	no	—
151	#2916	—	61	2	no	—
152	#1953	—	61	2	no	—
153	#1861	—	61	4	no	—
154	#1849	—	61	1	no	—
155	#1814	—	61	2	no	—
156	#1773	D₆₁	1	1	yes	characteristic, complemented
157	C₂×C₆₁	—	2	1	no	—
158	C₂×C₆₁	—	2	1	no	—
159	C₂×C₆₁	—	2	1	no	—
160	C₂×C₆₁	D₄×C₄⋊C₄	1	1	yes	characteristic
161	D₆₁	—	4	4	no	—
162	C₂×#135	—	61	1	no	—
163	#568	—	61	4	no	—
164	C₂×#131	—	61	2	no	—
165	#444	—	61	8	no	—
166	#444	—	122	4	no	—
167	#443	—	122	4	no	—
168	#430	—	61	4	no	—
169	#430	—	122	4	no	—
170	C₂×#108	—	61	1	no	—
171	C₂×#108	—	61	2	no	—
172	C₂×#108	—	61	2	no	—
173	C₂×#108	—	122	4	no	—
174	C₂×#108	C₂×D₆₁	1	1	yes	characteristic
175	#427	C₂×D₆₁	1	1	yes	characteristic
176	#423	—	61	2	no	—
177	#421	—	61	4	no	—
178	C₂×#104	—	61	2	no	—
179	#405	C₂×D₆₁	1	2	yes	—
180	C₂×#97	C₂×D₆₁	1	1	yes	characteristic
181	#344	—	61	4	no	—
182	#339	C₂×D₆₁	1	2	yes	—
183	C₆₁	#22695	1	1	yes	characteristic, complemented, Sylow
184	C₂×#41	—	61	2	no	—
185	#135	—	61	4	no	—
186	#135	—	61	4	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	8	no	—
193	#131	—	122	2	no	—
194	#131	—	122	2	no	—
195	#131	—	244	4	no	—
196	C₂×M₁₆	—	61	2	no	—
197	C₂×M₁₆	—	61	4	no	—
198	C₂×M₁₆	—	122	2	no	—
199	C₂×M₁₆	—	122	8	no	—
200	C₂×M₁₆	C₂²×D₆₁	1	2	yes	—
201	C₂²×C₈	—	122	2	no	—
202	C₂²×C₈	C₂²×D₆₁	1	1	yes	characteristic
203	C₂²×C₈	C₂²×D₆₁	1	1	yes	characteristic
204	#117	C₂²×D₆₁	1	2	yes	—
205	C₂×C₄²	C₂²×D₆₁	1	1	yes	characteristic
206	#108	—	61	4	no	—
207	#108	—	122	8	no	—
208	#108	—	122	2	no	—
209	#108	—	122	2	no	—
210	#108	—	122	2	no	—
211	#108	—	122	2	no	—
212	#108	D₂₄₄	1	4	yes	—
213	#107	—	2	1	no	—
214	#106	—	2	1	no	—
215	C₄⋊C₈	—	244	2	no	—
216	#104	—	61	8	no	—
217	#104	—	122	8	no	—
218	#97	—	2	2	no	—
219	#97	C₄×D₆₁	1	4	yes	—
220	C₄×C₈	—	2	2	no	—
221	#41	—	61	4	no	—
222	#41	—	61	4	no	—
223	#41	—	122	2	no	—
224	#41	—	122	2	no	—
225	C₂×Q₈	—	61	2	no	—
226	C₂×D₄	—	61	2	no	—
227	C₂×D₄	—	122	2	no	—
228	C₂²×C₄	—	2	1	no	—
229	C₂²×C₄	—	122	2	no	—
230	C₂²×C₄	C₂³×D₆₁	1	1	yes	characteristic
231	Q₁₆	—	61	4	no	—
232	SD₁₆	—	61	8	no	—
233	D₈	—	61	4	no	—
234	M₁₆	—	2	2	no	—
235	M₁₆	—	61	4	no	—
236	M₁₆	—	122	2	no	—
237	M₁₆	—	122	4	no	—
238	M₁₆	—	122	4	no	—
239	M₁₆	—	122	4	no	—
240	M₁₆	—	122	4	no	—
241	M₁₆	—	244	8	no	—
242	M₁₆	C₂×D₂₄₄	1	4	yes	—
243	C₂×C₈	—	2	2	no	—
244	C₂×C₈	—	122	2	no	—
245	C₂×C₈	—	122	4	no	—
246	C₂×C₈	—	122	4	no	—
247	C₂×C₈	—	122	4	no	—
248	C₂×C₈	—	244	8	no	—
249	C₂×C₈	C₂×C₄×D₆₁	1	2	yes	—
250	C₂×C₈	C₂×C₄×D₆₁	1	2	yes	—
251	C₂×C₈	C₂×D₂₄₄	1	2	yes	—
252	C₂×C₈	#11758222	1	1	yes	characteristic
253	C₂×C₈	#11758222	1	1	yes	characteristic
254	C₂×C₈	D₄×D₆₁	1	2	yes	—
255	C₂×C₈	Q₈×D₆₁	1	1	yes	characteristic
256	C₂×C₈	Q₈×D₆₁	1	1	yes	characteristic
257	C₄⋊C₄	—	2	2	no	—
258	C₂²⋊C₄	—	2	2	no	—
259	C₄²	—	2	1	no	—
260	C₄²	—	2	2	no	—
261	C₄²	C₂×C₄×D₆₁	1	2	yes	—
262	C₂³	—	122	2	no	—
263	C₂³	#14519824	1	1	yes	characteristic
264	Q₈	—	61	4	no	—
265	Q₈	—	122	2	no	—
266	D₄	—	61	4	no	—
267	D₄	—	122	4	no	—
268	D₄	—	122	4	no	—
269	D₄	—	122	2	no	—
270	C₂×C₄	—	2	2	no	—
271	C₂×C₄	—	4	2	no	—
272	C₂×C₄	—	8	1	no	—
273	C₂×C₄	—	122	2	no	—
274	C₂×C₄	—	122	4	no	—
275	C₂×C₄	—	122	4	no	—
276	C₂×C₄	C₂²×C₄×D₆₁	1	1	yes	characteristic
277	C₂×C₄	C₂×D₄×D₆₁	1	1	yes	characteristic
278	C₂×C₄	C₂²×D₂₄₄	1	1	yes	characteristic
279	C₂×C₄	C₂×Q₈×D₆₁	1	1	yes	characteristic
280	C₂×C₄	C₂×#11758222	1	1	yes	characteristic
281	C₂×C₄	#14519823	1	1	yes	characteristic
282	C₈	—	2	4	no	—
283	C₈	—	122	4	no	—
284	C₈	—	244	4	no	—
285	C₈	C₄⋊C₄×D₆₁	1	2	yes	—
286	C₈	C₄⋊C₄×D₆₁	1	2	yes	—
287	C₈	C₄×D₂₄₄	1	2	yes	—
288	C₈	C₄×D₂₄₄	1	2	yes	—
289	C₂²	—	4	1	no	—
290	C₂²	—	122	4	no	—
291	C₂²	—	488	2	no	—
292	C₂²	#16840242	1	1	yes	characteristic
293	C₂²	#16840417	1	1	yes	characteristic
294	C₂²	Q₈×D₂₄₄	1	1	yes	characteristic
295	C₄	—	4	2	no	—
296	C₄	—	8	2	no	—
297	C₄	—	122	4	no	—
298	C₄	C₂×C₄⋊C₄×D₆₁	1	1	yes	characteristic
299	C₄	C₂×C₄×D₂₄₄	1	1	yes	characteristic
300	C₄	D₄×D₂₄₄	1	1	yes	characteristic
301	C₄	#16840472	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	C₄⋊C₄×D₂₄₄	1	1	yes	characteristic

	subgroup	G/H
center	C₄	D₄×D₂₄₄
commutator	C₂×C₄×C₆₁	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₆₁
Fitting	C₆₁×#1773	C₂
socle	C₂×C₆₁	D₄×C₄⋊C₄
prime radical	C₂×#14519799	C₂²
cosocle	C₂²×C₄×C₆₁	C₂⁴
hypercenter	#1773	D₆₁
hypocenter	C₆₁	#22695
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #1773

Complements

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

N	H ≅ G/N	type
3. #21454427	C₂	semidirect
4. #21446609	C₂	semidirect
5. #21446060	C₂	semidirect
6. #21445994	C₂	semidirect
7. #21445870	C₂	semidirect
8. C₆₁×#1773	C₂	semidirect
13. #16839885	C₄	semidirect
156. #1773	D₆₁	semidirect
183. C₆₁	#22695	semidirect

Maximal subgroups

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

subgroup	index
#21454737	2
C₂×#16839885	2
#21454427	2
#21446609	2
#21446060	2
#21445994	2
#21445870	2
C₆₁×#1773	2
#22695	61