Subgroups — #30619282

Order 15616.

578 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21462612	C₂	1	2	yes	maximal
2	#21462007	C₂	1	2	yes	maximal
3	#21461887	C₂	1	4	yes	maximal
4	#21461884	C₂	1	4	yes	maximal
5	#21461744	C₂	1	1	yes	maximal, characteristic
6	#21461537	C₂	1	1	yes	maximal, characteristic
7	#21456330	C₂	1	1	yes	maximal, characteristic
8	#16840814	—	2	4	no	—
9	#16840813	—	2	4	no	—
10	#16840769	C₂²	1	4	yes	—
11	#16840767	C₂²	1	2	yes	—
12	#16840727	C₂²	1	4	yes	—
13	#16840721	C₂²	1	1	yes	characteristic
14	#16840698	C₂²	1	1	yes	characteristic
15	#16840638	C₂²	1	2	yes	—
16	#16840638	C₂²	1	2	yes	—
17	#16840632	C₂²	1	8	yes	—
18	(C₂×#6923)⋊C₈	C₂²	1	2	yes	—
19	(C₂×#6923)⋊C₈	C₂²	1	2	yes	—
20	#16840317	C₂²	1	2	yes	—
21	#16840167	C₄	1	1	yes	characteristic, complemented
22	#16840163	C₄	1	2	yes	complemented
23	#16840163	C₄	1	2	yes	complemented
24	#16840158	C₄	1	2	yes	complemented
25	#16839799	C₂²	1	2	yes	—
26	#16839799	C₂²	1	2	yes	—
27	C₄×C₈×D₆₁	C₂²	1	1	yes	characteristic
28	C₆₁×#500	C₄	1	1	yes	characteristic, complemented
29	C₂×#11758213	C₂³	1	4	yes	—
30	C₂×D₆₁⋊C₈	C₂³	1	2	yes	—
31	C₂×C₄×#6923	C₂³	1	2	yes	—
32	#14519844	—	2	4	no	—
33	#14519844	—	2	2	no	—
34	#14519844	—	2	2	no	—
35	C₄⋊#92420	—	2	4	no	—
36	#14519832	—	2	4	no	—
37	#14519832	D₄	1	2	yes	—
38	#14519832	D₄	1	2	yes	—
39	C₄×#92420	—	2	4	no	—
40	(C₂×#6923)⋊C₄	—	2	1	no	—
41	(C₂×#6923)⋊C₄	—	2	1	no	—
42	C₂×C₈×D₆₁	C₂³	1	1	yes	characteristic
43	C₂×C₈×D₆₁	C₂³	1	1	yes	characteristic
44	#14519777	C₂×C₄	1	2	yes	—
45	#14519768	C₂×C₄	1	4	yes	—
46	#14519757	—	2	2	no	—
47	#14519753	—	2	2	no	—
48	Q₈×Dic₆₁	—	2	2	no	—
49	#14519742	—	2	2	no	—
50	#14519735	C₂³	1	4	yes	—
51	C₄²×D₆₁	C₂³	1	1	yes	characteristic
52	#14519711	C₂×C₄	1	2	yes	—
53	#14519711	C₂×C₄	1	2	yes	—
54	#14519708	C₂×C₄	1	4	yes	—
55	#14519708	C₂×C₄	1	4	yes	—
56	C₄×#92418	C₂×C₄	1	1	yes	characteristic
57	C₈×Dic₆₁	C₂×C₄	1	1	yes	characteristic
58	C₈×Dic₆₁	C₂×C₄	1	1	yes	characteristic
59	C₄⋊Q₈×C₆₁	C₂×C₄	1	2	yes	—
60	#107×C₆₁	C₂×C₄	1	2	yes	—
61	#107×C₆₁	C₂×C₄	1	2	yes	—
62	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
63	C₂²×#6923	—	2	2	no	—
64	C₂×Q₈×C₆₁	—	2	2	no	—
65	C₂×C₄×D₆₁	#41	1	1	yes	characteristic
66	C₂×C₄×D₆₁	#41	1	1	yes	characteristic
67	C₂×C₄×D₆₁	C₂⁴	1	1	yes	characteristic
68	C₂×Dic₁₂₂	—	2	2	no	—
69	C₂×#92420	—	2	4	no	—
70	C₂×#92420	C₂×D₄	1	4	yes	—
71	C₄×#6923	—	2	2	no	—
72	C₄×#6923	—	2	2	no	—
73	#11758213	—	2	4	no	—
74	#11758213	—	2	4	no	—
75	D₆₁⋊C₈	—	2	2	no	—
76	D₆₁⋊C₈	—	2	2	no	—
77	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
79	C₄⋊C₄×C₆₁	—	2	2	no	—
80	C₄⋊C₄×C₆₁	C₂²×C₄	1	4	yes	—
81	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
82	#11758198	—	2	4	no	—
83	C₄⋊Dic₆₁	—	2	2	no	—
84	C₄⋊Dic₆₁	—	2	4	no	—
85	C₄⋊Dic₆₁	C₂²×C₄	1	4	yes	—
86	#11758196	—	2	4	no	—
87	C₄×Dic₆₁	—	2	4	no	—
88	C₄×Dic₆₁	C₂²×C₄	1	1	yes	characteristic
89	C₄×Dic₆₁	C₂×D₄	1	1	yes	characteristic
90	C₄×Dic₆₁	C₂×D₄	1	1	yes	characteristic
91	C₂×#92418	C₂²×C₄	1	1	yes	characteristic
92	C₂×#92418	C₂²×C₄	1	1	yes	characteristic
93	C₈×D₆₁	—	2	2	no	—
94	C₈×D₆₁	—	2	2	no	—
95	C₂²×D₆₁	C₂×#41	1	1	yes	characteristic
96	C₂×#6923	—	4	2	no	—
97	C₂×#6923	—	4	2	no	—
98	C₂×#6923	—	4	2	no	—
99	Q₈×C₆₁	—	8	2	no	—
100	C₂×C₄×C₆₁	—	2	4	no	—
101	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
102	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
103	C₂×C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
104	C₂×Dic₆₁	—	2	4	no	—
105	C₂×Dic₆₁	C₄×D₄	1	1	yes	characteristic
106	C₂×Dic₆₁	C₄×D₄	1	1	yes	characteristic
107	C₂×Dic₆₁	C₂²×D₄	1	1	yes	characteristic
108	C₄×D₆₁	—	4	1	no	—
109	C₄×D₆₁	—	4	1	no	—
110	C₄×D₆₁	2₋¹⁺⁴	1	2	yes	—
111	C₄×D₆₁	2₋¹⁺⁴	1	2	yes	—
112	Dic₁₂₂	—	8	2	no	—
113	#92420	—	2	4	no	—
114	#92420	—	4	4	no	—
115	C₈×C₆₁	—	2	1	no	—
116	C₈×C₆₁	—	2	1	no	—
117	#92418	—	2	1	no	—
118	#92418	—	2	1	no	—
119	#22258	—	61	1	no	maximal, Sylow
120	C₂²×C₆₁	C₂×C₄×D₄	1	1	yes	characteristic
121	C₂×D₆₁	—	2	1	no	—
122	C₂×D₆₁	—	2	1	no	—
123	C₂×D₆₁	#541	1	2	yes	—
124	#6923	—	8	2	no	—
125	C₄×C₆₁	—	2	1	no	—
126	C₄×C₆₁	—	2	1	no	—
127	C₄×C₆₁	—	4	4	no	—
128	C₄×C₆₁	#519	1	1	yes	characteristic
129	C₄×C₆₁	#519	1	1	yes	characteristic
130	Dic₆₁	—	2	1	no	—
131	Dic₆₁	—	2	1	no	—
132	Dic₆₁	—	4	4	no	—
133	Dic₆₁	#536	1	2	yes	—
134	#3160	—	61	1	no	—
135	#2961	—	61	2	no	—
136	#1995	—	61	2	no	—
137	#1922	—	61	4	no	—
138	#1922	—	61	4	no	—
139	#1837	—	61	1	no	—
140	#1746	—	61	1	no	—
141	C₂×C₆₁	#2327	1	1	yes	characteristic
142	C₂×C₆₁	#2957	1	1	yes	characteristic
143	C₂×C₆₁	#2957	1	1	yes	characteristic
144	D₆₁	—	2	2	no	—
145	#533	—	61	2	no	—
146	#500	—	61	1	no	—
147	#500	#6923	1	1	yes	characteristic, complemented
148	#499	—	61	2	no	—
149	#499	—	61	2	no	—
150	#498	—	61	2	no	—
151	#446	—	122	4	no	—
152	#446	—	122	4	no	—
153	#427	—	61	2	no	—
154	#427	—	61	2	no	—
155	#424	—	61	4	no	—
156	#423	—	61	2	no	—
157	#405	—	61	4	no	—
158	C₂×#97	—	61	1	no	—
159	C₂×C₄×C₈	—	61	1	no	—
160	#376	—	61	1	no	—
161	#341	—	61	2	no	—
162	#341	—	61	2	no	—
163	#339	—	61	8	no	—
164	#336	—	61	2	no	—
165	#336	—	61	2	no	—
166	C₆₁	#22258	1	1	yes	characteristic, complemented, Sylow
167	C₂×M₁₆	—	61	4	no	—
168	C₂²×C₈	—	61	2	no	—
169	C₂²×C₈	—	61	1	no	—
170	C₂²×C₈	—	61	1	no	—
171	C₄⋊Q₈	—	61	2	no	—
172	C₄⋊Q₈	C₂×#6923	1	2	yes	—
173	#125	—	61	4	no	—
174	C₂²⋊Q₈	—	122	2	no	—
175	C₂²⋊Q₈	—	122	2	no	—
176	C₄×Q₈	—	122	2	no	—
177	C₄×Q₈	—	122	2	no	—
178	#117	—	61	4	no	—
179	C₂×C₄²	—	61	1	no	—
180	C₂×C₄²	—	61	2	no	—
181	#107	—	61	2	no	—
182	#107	—	61	2	no	—
183	#107	C₂×#6923	1	2	yes	—
184	#107	C₂×#6923	1	2	yes	—
185	#106	—	61	4	no	—
186	#106	—	61	4	no	—
187	C₄⋊C₈	—	122	2	no	—
188	C₄⋊C₈	—	122	2	no	—
189	C₄⋊C₈	—	122	4	no	—
190	C₄⋊C₈	—	122	4	no	—
191	#97	—	61	2	no	—
192	#97	—	61	2	no	—
193	#97	—	122	4	no	—
194	C₄×C₈	—	61	1	no	—
195	C₄×C₈	—	61	1	no	—
196	C₄×C₈	—	61	1	no	—
197	C₄×C₈	—	122	4	no	—
198	C₄×C₈	C₂×#6923	1	1	yes	characteristic
199	#95	—	122	1	no	—
200	#95	—	122	1	no	—
201	C₂×Q₈	—	2	2	no	—
202	C₂×Q₈	—	122	2	no	—
203	C₂²×C₄	—	61	1	no	—
204	C₂²×C₄	—	61	1	no	—
205	C₂²×C₄	—	61	1	no	—
206	C₂²×C₄	—	122	2	no	—
207	M₁₆	—	122	4	no	—
208	M₁₆	—	122	4	no	—
209	C₂×C₈	—	61	4	no	—
210	C₂×C₈	—	61	1	no	—
211	C₂×C₈	—	61	1	no	—
212	C₂×C₈	—	122	2	no	—
213	C₂×C₈	—	122	2	no	—
214	C₂×C₈	—	122	2	no	—
215	C₂×C₈	—	122	2	no	—
216	C₂×C₈	—	122	4	no	—
217	C₂×C₈	C₂²×#6923	1	1	yes	characteristic
218	C₂×C₈	C₂²×#6923	1	1	yes	characteristic
219	C₄⋊C₄	—	2	2	no	—
220	C₄⋊C₄	—	61	4	no	—
221	C₄⋊C₄	—	122	4	no	—
222	C₄⋊C₄	—	122	4	no	—
223	C₄⋊C₄	—	122	2	no	—
224	C₄⋊C₄	C₂²×#6923	1	4	yes	—
225	C₂²⋊C₄	—	122	4	no	—
226	C₄²	—	61	1	no	—
227	C₄²	—	61	1	no	—
228	C₄²	—	61	1	no	—
229	C₄²	—	122	4	no	—
230	C₄²	—	122	2	no	—
231	C₄²	—	122	2	no	—
232	C₄²	C₂²×#6923	1	1	yes	characteristic
233	C₂³	—	61	1	no	—
234	Q₈	—	8	2	no	—
235	Q₈	—	488	2	no	—
236	C₂×C₄	—	2	4	no	—
237	C₂×C₄	—	61	1	no	—
238	C₂×C₄	—	61	2	no	—
239	C₂×C₄	—	61	2	no	—
240	C₂×C₄	—	61	1	no	—
241	C₂×C₄	—	61	1	no	—
242	C₂×C₄	—	122	4	no	—
243	C₂×C₄	—	244	1	no	—
244	C₂×C₄	—	244	1	no	—
245	C₂×C₄	—	244	2	no	—
246	C₂×C₄	—	244	2	no	—
247	C₂×C₄	—	244	2	no	—
248	C₂×C₄	D₄×#6923	1	1	yes	characteristic
249	C₂×C₄	D₄×#6923	1	1	yes	characteristic
250	C₂×C₄	C₂³×#6923	1	1	yes	characteristic
251	C₈	—	2	1	no	—
252	C₈	—	2	1	no	—
253	C₈	—	122	4	no	—
254	C₈	—	122	1	no	—
255	C₈	—	122	1	no	—
256	C₈	—	244	4	no	—
257	C₂²	—	61	2	no	—
258	C₂²	—	122	1	no	—
259	C₂²	—	122	1	no	—
260	C₂²	C₂×D₄×#6923	1	1	yes	characteristic
261	C₄	—	2	1	no	—
262	C₄	—	2	1	no	—
263	C₄	—	4	4	no	—
264	C₄	—	61	2	no	—
265	C₄	—	122	1	no	—
266	C₄	—	122	1	no	—
267	C₄	—	488	2	no	—
268	C₄	—	244	4	no	—
269	C₄	#16840830	1	1	yes	characteristic
270	C₄	#16840830	1	1	yes	characteristic
271	C₂	—	122	2	no	—
272	C₂	#21462310	1	1	yes	characteristic
273	C₂	#21462595	1	1	yes	characteristic
274	C₂	#21462599	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×#6923
commutator	C₂×C₄×C₆₁	C₂³×C₄
Frattini	C₂×C₄	C₂³×#6923
Fitting	C₆₁×#500	C₄
socle	C₂²×C₆₁	C₂×C₄×D₄
prime radical	C₂²×D₆₁	C₂×#41
cosocle	C₂×C₄×D₆₁	C₂⁴
hypercenter	#500	#6923
hypocenter	C₆₁	#22258
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #500

Complements

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

N	H ≅ G/N	type
21. #16840167	C₄	semidirect
22. #16840163	C₄	semidirect
23. #16840163	C₄	semidirect
24. #16840158	C₄	semidirect
28. C₆₁×#500	C₄	semidirect
147. #500	#6923	semidirect
166. C₆₁	#22258	semidirect

Maximal subgroups

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

subgroup	index
#21462612	2
#21462007	2
#21461887	2
#21461884	2
#21461744	2
#21461537	2
#21456330	2
#22258	61