Subgroups — #30622671

Order 15616.

638 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21462613	C₂	1	2	yes	maximal
2	#21461995	C₂	1	2	yes	maximal
3	#21461927	C₂	1	2	yes	maximal
4	#21461884	C₂	1	4	yes	maximal
5	C₄×#14519848	C₂	1	1	yes	maximal, characteristic
6	#21461623	C₂	1	2	yes	maximal
7	C₈×C₄⋊#6923	C₂	1	1	yes	maximal, characteristic
8	#21456330	C₂	1	1	yes	maximal, characteristic
9	#16840814	—	2	4	no	—
10	#16840813	—	2	2	no	—
11	Q₈×#92420	—	2	2	no	—
12	C₂×#14519848	C₂²	1	1	yes	characteristic
13	C₂×#14519848	C₂²	1	1	yes	characteristic
14	#16840769	C₂²	1	4	yes	—
15	D₆₁⋊(C₄⋊C₈)	C₂²	1	1	yes	characteristic
16	#16840727	C₂²	1	4	yes	—
17	C₄×#11758213	C₂²	1	2	yes	—
18	C₂×C₈×#6923	C₂²	1	2	yes	—
19	C₄×C₄⋊#6923	C₂²	1	1	yes	characteristic
20	#16840638	C₂²	1	2	yes	—
21	#16840632	C₂²	1	4	yes	—
22	#16840632	C₂²	1	4	yes	—
23	(C₂×#6923)⋊C₈	C₂²	1	2	yes	—
24	#16840317	C₂²	1	2	yes	—
25	#16840167	C₄	1	1	yes	characteristic, complemented
26	#16840163	C₄	1	2	yes	complemented
27	#16840163	C₄	1	2	yes	complemented
28	#16840158	C₄	1	2	yes	complemented
29	#16839799	C₂²	1	2	yes	—
30	#16839799	C₂²	1	2	yes	—
31	C₄×C₈×D₆₁	C₂²	1	1	yes	characteristic
32	C₆₁×#500	C₄	1	1	yes	characteristic, complemented
33	C₂×#11758213	C₂³	1	2	yes	—
34	C₂×#11758213	C₂³	1	2	yes	—
35	C₂×D₆₁⋊C₈	C₂³	1	2	yes	—
36	C₂×C₄⋊#6923	—	2	1	no	—
37	C₂×C₄×#6923	C₂³	1	2	yes	—
38	#14519848	—	2	2	no	—
39	#14519848	Q₈	1	4	yes	complemented
40	#14519844	—	2	4	no	—
41	#14519844	D₄	1	2	yes	—
42	C₄⋊#92420	—	2	4	no	—
43	C₄⋊#92420	D₄	1	1	yes	characteristic
44	C₄⋊#92420	D₄	1	1	yes	characteristic
45	#14519832	—	2	4	no	—
46	#14519832	—	2	2	no	—
47	C₄×#92420	—	2	4	no	—
48	C₄×#92420	—	2	2	no	—
49	C₈×#6923	—	2	4	no	—
50	(C₂×#6923)⋊C₄	—	2	1	no	—
51	C₂×C₈×D₆₁	C₂³	1	1	yes	characteristic
52	C₂×C₈×D₆₁	C₂³	1	1	yes	characteristic
53	#14519777	C₂×C₄	1	2	yes	—
54	#14519768	C₂×C₄	1	4	yes	—
55	#14519757	—	2	2	no	—
56	#14519753	—	2	2	no	—
57	Q₈×Dic₆₁	—	2	2	no	—
58	#14519742	—	2	2	no	—
59	#14519735	C₂³	1	4	yes	—
60	C₄²×D₆₁	C₂³	1	1	yes	characteristic
61	#14519711	C₂×C₄	1	2	yes	—
62	#14519711	C₂×C₄	1	2	yes	—
63	#14519708	C₂×C₄	1	4	yes	—
64	#14519708	C₂×C₄	1	4	yes	—
65	C₄×#92418	C₂×C₄	1	1	yes	characteristic
66	C₈×Dic₆₁	C₂×C₄	1	1	yes	characteristic
67	C₈×Dic₆₁	C₂×C₄	1	1	yes	characteristic
68	C₄⋊Q₈×C₆₁	C₂×C₄	1	2	yes	—
69	#107×C₆₁	C₂×C₄	1	2	yes	—
70	#107×C₆₁	C₂×C₄	1	2	yes	—
71	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
72	C₂²×#6923	—	2	2	no	—
73	C₂×Q₈×C₆₁	—	2	2	no	—
74	C₂×C₄×D₆₁	#41	1	1	yes	characteristic
75	C₂×C₄×D₆₁	#41	1	1	yes	characteristic
76	C₂×C₄×D₆₁	C₂⁴	1	1	yes	characteristic
77	C₂×Dic₁₂₂	—	2	2	no	—
78	C₂×#92420	—	2	2	no	—
79	C₂×#92420	—	2	2	no	—
80	C₂×#92420	C₂×D₄	1	2	yes	—
81	C₂×#92420	C₂×D₄	1	2	yes	—
82	C₄⋊#6923	—	2	4	no	—
83	C₄×#6923	—	2	2	no	—
84	C₄×#6923	—	2	2	no	—
85	#11758213	—	2	2	no	—
86	#11758213	—	2	2	no	—
87	#11758213	—	2	2	no	—
88	#11758213	C₂×Q₈	1	4	yes	—
89	D₆₁⋊C₈	—	2	2	no	—
90	D₆₁⋊C₈	—	2	2	no	—
91	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
92	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
93	C₄⋊C₄×C₆₁	—	2	2	no	—
94	C₄⋊C₄×C₆₁	C₂²×C₄	1	4	yes	—
95	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
96	#11758198	—	2	4	no	—
97	C₄⋊Dic₆₁	—	2	2	no	—
98	C₄⋊Dic₆₁	—	2	4	no	—
99	C₄⋊Dic₆₁	C₂²×C₄	1	4	yes	—
100	#11758196	—	2	4	no	—
101	C₄×Dic₆₁	—	2	4	no	—
102	C₄×Dic₆₁	C₂²×C₄	1	1	yes	characteristic
103	C₄×Dic₆₁	C₂×D₄	1	1	yes	characteristic
104	C₄×Dic₆₁	C₂×D₄	1	1	yes	characteristic
105	C₂×#92418	C₂²×C₄	1	1	yes	characteristic
106	C₂×#92418	C₂²×C₄	1	1	yes	characteristic
107	C₈×D₆₁	—	2	1	no	—
108	C₈×D₆₁	—	2	1	no	—
109	C₈×D₆₁	C₂×Q₈	1	2	yes	—
110	C₈×D₆₁	#41	1	2	yes	—
111	C₂²×D₆₁	C₂×#41	1	1	yes	characteristic
112	C₂×#6923	—	2	4	no	—
113	C₂×#6923	—	4	2	no	—
114	C₂×#6923	—	4	2	no	—
115	Q₈×C₆₁	—	4	4	no	—
116	C₂×C₄×C₆₁	—	2	4	no	—
117	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
118	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
119	C₂×C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
120	C₂×Dic₆₁	—	2	4	no	—
121	C₂×Dic₆₁	C₄×D₄	1	1	yes	characteristic
122	C₂×Dic₆₁	C₄×D₄	1	1	yes	characteristic
123	C₂×Dic₆₁	C₂²×D₄	1	1	yes	characteristic
124	C₄×D₆₁	—	2	2	no	—
125	C₄×D₆₁	—	4	1	no	—
126	C₄×D₆₁	C₂²×Q₈	1	1	yes	characteristic
127	C₄×D₆₁	C₂×#41	1	1	yes	characteristic
128	C₄×D₆₁	2₋¹⁺⁴	1	1	yes	characteristic
129	C₄×D₆₁	2₋¹⁺⁴	1	1	yes	characteristic
130	Dic₁₂₂	—	8	2	no	—
131	#92420	—	2	2	no	—
132	#92420	—	2	2	no	—
133	#92420	—	2	4	no	—
134	#92420	—	4	2	no	—
135	C₈×C₆₁	—	2	1	no	—
136	C₈×C₆₁	C₄×Q₈	1	2	yes	—
137	#92418	—	2	1	no	—
138	#92418	C₄×Q₈	1	2	yes	—
139	#23170	—	61	1	no	maximal, Sylow
140	C₂²×C₆₁	C₂×C₄×D₄	1	1	yes	characteristic
141	C₂×D₆₁	—	2	1	no	—
142	C₂×D₆₁	—	2	1	no	—
143	C₂×D₆₁	#556	1	1	yes	characteristic
144	C₂×D₆₁	#557	1	1	yes	characteristic
145	#6923	—	4	4	no	—
146	C₄×C₆₁	—	2	1	no	—
147	C₄×C₆₁	—	4	4	no	—
148	C₄×C₆₁	#443	1	2	yes	—
149	C₄×C₆₁	C₂×C₄×Q₈	1	1	yes	characteristic
150	C₄×C₆₁	#519	1	1	yes	characteristic
151	Dic₆₁	—	2	1	no	—
152	Dic₆₁	—	2	1	no	—
153	Dic₆₁	—	4	4	no	—
154	Dic₆₁	#547	1	1	yes	characteristic
155	Dic₆₁	D₄×Q₈	1	1	yes	characteristic
156	#3160	—	61	1	no	—
157	#2962	—	61	2	no	—
158	#1986	—	61	2	no	—
159	#1945	—	61	2	no	—
160	#1922	—	61	4	no	—
161	C₄×#108	—	61	1	no	—
162	#1772	—	61	2	no	—
163	C₈×C₄⋊C₄	—	61	1	no	—
164	C₂×C₆₁	#2365	1	1	yes	characteristic
165	C₂×C₆₁	C₂×#443	1	1	yes	characteristic
166	C₂×C₆₁	#2957	1	1	yes	characteristic
167	D₆₁	—	2	1	no	—
168	D₆₁	—	2	1	no	—
169	#533	—	61	2	no	—
170	#500	—	61	1	no	—
171	#500	#6923	1	1	yes	characteristic, complemented
172	#499	—	61	2	no	—
173	#499	—	61	2	no	—
174	#498	—	61	2	no	—
175	#446	—	122	2	no	—
176	#446	—	122	4	no	—
177	C₈×Q₈	—	122	2	no	—
178	C₂×#108	—	61	1	no	—
179	C₂×#108	—	61	1	no	—
180	#427	—	61	2	no	—
181	#427	—	61	2	no	—
182	#424	—	61	4	no	—
183	C₂×C₄⋊C₈	—	61	1	no	—
184	#405	—	61	4	no	—
185	C₄×M₁₆	—	61	2	no	—
186	C₂×C₄×C₈	—	61	1	no	—
187	C₂×C₄×C₈	—	61	2	no	—
188	C₄×C₄⋊C₄	—	61	1	no	—
189	#341	—	61	2	no	—
190	#339	—	61	4	no	—
191	#339	—	61	4	no	—
192	#336	—	61	2	no	—
193	C₆₁	#23170	1	1	yes	characteristic, complemented, Sylow
194	C₂×M₁₆	—	61	2	no	—
195	C₂×M₁₆	—	61	2	no	—
196	C₂²×C₈	—	61	1	no	—
197	C₂²×C₈	—	61	2	no	—
198	C₂²×C₈	—	61	1	no	—
199	C₄⋊Q₈	—	61	2	no	—
200	C₄⋊Q₈	C₂×#6923	1	2	yes	—
201	#125	—	61	4	no	—
202	C₂²⋊Q₈	—	122	2	no	—
203	C₂²⋊Q₈	—	122	2	no	—
204	C₄×Q₈	—	122	2	no	—
205	C₄×Q₈	—	122	2	no	—
206	#117	—	61	4	no	—
207	C₂×C₄⋊C₄	—	122	1	no	—
208	C₂×C₄²	—	61	1	no	—
209	C₂×C₄²	—	61	2	no	—
210	#108	—	61	4	no	—
211	#108	—	122	2	no	—
212	#107	—	61	2	no	—
213	#107	—	61	2	no	—
214	#107	C₂×#6923	1	2	yes	—
215	#107	C₂×#6923	1	2	yes	—
216	#106	—	61	4	no	—
217	#106	—	61	4	no	—
218	C₄⋊C₈	—	61	1	no	—
219	C₄⋊C₈	—	61	1	no	—
220	C₄⋊C₈	—	61	2	no	—
221	C₄⋊C₈	—	122	4	no	—
222	C₄⋊C₈	—	122	4	no	—
223	#97	—	122	2	no	—
224	#97	—	122	4	no	—
225	C₄×C₈	—	61	1	no	—
226	C₄×C₈	—	61	1	no	—
227	C₄×C₈	—	61	1	no	—
228	C₄×C₈	—	122	2	no	—
229	C₄×C₈	—	122	4	no	—
230	C₄×C₈	—	122	4	no	—
231	C₄×C₈	C₂×#6923	1	1	yes	characteristic
232	#95	—	122	1	no	—
233	C₂×Q₈	—	2	2	no	—
234	C₂×Q₈	—	122	2	no	—
235	C₂²×C₄	—	61	1	no	—
236	C₂²×C₄	—	61	1	no	—
237	C₂²×C₄	—	61	1	no	—
238	C₂²×C₄	—	122	2	no	—
239	M₁₆	—	61	4	no	—
240	M₁₆	—	122	2	no	—
241	M₁₆	—	122	2	no	—
242	M₁₆	—	122	2	no	—
243	C₂×C₈	—	61	1	no	—
244	C₂×C₈	—	61	2	no	—
245	C₂×C₈	—	61	2	no	—
246	C₂×C₈	—	61	2	no	—
247	C₂×C₈	—	61	2	no	—
248	C₂×C₈	—	61	1	no	—
249	C₂×C₈	—	122	1	no	—
250	C₂×C₈	—	122	1	no	—
251	C₂×C₈	—	122	2	no	—
252	C₂×C₈	—	122	2	no	—
253	C₂×C₈	—	122	2	no	—
254	C₂×C₈	—	122	2	no	—
255	C₂×C₈	C₂²×#6923	1	1	yes	characteristic
256	C₂×C₈	C₂²×#6923	1	1	yes	characteristic
257	C₄⋊C₄	—	2	2	no	—
258	C₄⋊C₄	—	61	4	no	—
259	C₄⋊C₄	—	122	4	no	—
260	C₄⋊C₄	—	122	4	no	—
261	C₄⋊C₄	—	122	4	no	—
262	C₄⋊C₄	—	122	2	no	—
263	C₄⋊C₄	C₂²×#6923	1	4	yes	—
264	C₂²⋊C₄	—	122	4	no	—
265	C₄²	—	61	1	no	—
266	C₄²	—	61	1	no	—
267	C₄²	—	61	1	no	—
268	C₄²	—	122	4	no	—
269	C₄²	—	122	2	no	—
270	C₄²	—	122	2	no	—
271	C₄²	C₂²×#6923	1	1	yes	characteristic
272	C₂³	—	61	1	no	—
273	Q₈	—	4	4	no	—
274	Q₈	—	488	2	no	—
275	C₂×C₄	—	2	4	no	—
276	C₂×C₄	—	61	1	no	—
277	C₂×C₄	—	61	1	no	—
278	C₂×C₄	—	61	1	no	—
279	C₂×C₄	—	61	1	no	—
280	C₂×C₄	—	61	1	no	—
281	C₂×C₄	—	61	1	no	—
282	C₂×C₄	—	61	1	no	—
283	C₂×C₄	—	122	2	no	—
284	C₂×C₄	—	122	4	no	—
285	C₂×C₄	—	122	4	no	—
286	C₂×C₄	—	244	1	no	—
287	C₂×C₄	—	244	2	no	—
288	C₂×C₄	—	244	2	no	—
289	C₂×C₄	D₄×#6923	1	1	yes	characteristic
290	C₂×C₄	D₄×#6923	1	1	yes	characteristic
291	C₂×C₄	C₂³×#6923	1	1	yes	characteristic
292	C₈	—	2	1	no	—
293	C₈	—	61	2	no	—
294	C₈	—	122	2	no	—
295	C₈	—	122	2	no	—
296	C₈	—	122	4	no	—
297	C₈	—	122	1	no	—
298	C₈	—	244	2	no	—
299	C₈	Q₈×#6923	1	2	yes	—
300	C₂²	—	61	1	no	—
301	C₂²	—	61	1	no	—
302	C₂²	—	122	1	no	—
303	C₂²	—	122	1	no	—
304	C₂²	C₂×D₄×#6923	1	1	yes	characteristic
305	C₄	—	2	1	no	—
306	C₄	—	4	4	no	—
307	C₄	—	61	1	no	—
308	C₄	—	61	1	no	—
309	C₄	—	122	1	no	—
310	C₄	—	122	1	no	—
311	C₄	—	244	4	no	—
312	C₄	—	244	4	no	—
313	C₄	#16840804	1	2	yes	—
314	C₄	C₂×Q₈×#6923	1	1	yes	characteristic
315	C₄	#16840830	1	1	yes	characteristic
316	C₂	—	122	1	no	—
317	C₂	—	122	1	no	—
318	C₂	#21462348	1	1	yes	characteristic
319	C₂	C₂×#16840804	1	1	yes	characteristic
320	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	#23170
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
25. #16840167	C₄	semidirect
26. #16840163	C₄	semidirect
27. #16840163	C₄	semidirect
28. #16840158	C₄	semidirect
32. C₆₁×#500	C₄	semidirect
39. #14519848	Q₈	semidirect
171. #500	#6923	semidirect
193. C₆₁	#23170	semidirect

Maximal subgroups

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

subgroup	index
#21462613	2
#21461995	2
#21461927	2
#21461884	2
C₄×#14519848	2
#21461623	2
C₈×C₄⋊#6923	2
#21456330	2
#23170	61