Subgroups — #30622659

Order 15616.

750 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21462352	C₂	1	2	yes	maximal
2	#21461997	C₂	1	2	yes	maximal
3	#21461923	C₂	1	2	yes	maximal
4	#21461883	C₂	1	4	yes	maximal
5	#107×#6923	C₂	1	2	yes	maximal
6	C₄×#14519846	C₂	1	1	yes	maximal, characteristic
7	C₈×C₄⋊#6923	C₂	1	1	yes	maximal, characteristic
8	#500×D₆₁	C₂	1	1	yes	maximal, characteristic
9	C₂×#14519846	C₂²	1	1	yes	characteristic
10	C₂×#14519846	C₂²	1	1	yes	characteristic
11	D₆₁⋊(C₄⋊C₈)	C₂²	1	1	yes	characteristic
12	C₂×C₈×#6923	C₂²	1	2	yes	—
13	(C₂×#6923)⋊Q₈	—	2	4	no	—
14	#16840713	—	2	2	no	—
15	#16840713	C₂²	1	4	yes	—
16	C₄⋊C₄×#6923	—	2	2	no	—
17	C₄⋊C₄×#6923	C₂²	1	4	yes	—
18	C₄×C₄⋊#6923	C₂²	1	2	yes	—
19	C₄×C₄⋊#6923	C₂²	1	1	yes	characteristic
20	#16840636	C₂²	1	2	yes	—
21	#16840635	C₂²	1	4	yes	—
22	#16840635	C₂²	1	4	yes	—
23	(C₂×#6923)⋊C₈	C₂²	1	2	yes	—
24	C₄⋊Q₈×D₆₁	C₂²	1	2	yes	—
25	#16840167	C₄	1	1	yes	characteristic, complemented
26	#16840166	C₄	1	2	yes	complemented
27	#16840166	C₄	1	2	yes	complemented
28	#16840165	C₄	1	2	yes	complemented
29	#107×D₆₁	C₂²	1	2	yes	—
30	#107×D₆₁	C₂²	1	2	yes	—
31	C₄×C₈×D₆₁	C₂²	1	1	yes	characteristic
32	C₆₁×#500	C₄	1	1	yes	characteristic, complemented
33	C₂×D₆₁⋊C₈	C₂³	1	2	yes	—
34	C₂×C₄⋊#6923	—	2	4	no	—
35	C₂×C₄⋊#6923	D₄	1	1	yes	characteristic
36	C₂×C₄⋊#6923	D₄	1	1	yes	characteristic
37	C₂×C₄⋊#6923	C₂³	1	2	yes	—
38	C₂×C₄⋊#6923	C₂³	1	2	yes	—
39	C₂×C₄×#6923	—	2	2	no	—
40	C₂×C₄×#6923	—	2	4	no	—
41	C₂×C₄×#6923	C₂³	1	2	yes	—
42	#14519846	—	2	2	no	—
43	#14519846	Q₈	1	4	yes	complemented
44	#14519844	—	2	1	no	—
45	C₄⋊#92420	—	2	1	no	—
46	C₈×#6923	—	2	4	no	—
47	(C₂×#6923)⋊C₄	—	2	4	no	—
48	(C₂×#6923)⋊C₄	—	2	4	no	—
49	(C₂×#6923)⋊C₄	—	2	2	no	—
50	(C₂×#6923)⋊C₄	D₄	1	2	yes	—
51	C₂×Q₈×D₆₁	—	2	2	no	—
52	C₂×C₈×D₆₁	C₂³	1	1	yes	characteristic
53	C₂×C₈×D₆₁	C₂³	1	1	yes	characteristic
54	#14519777	C₂×C₄	1	2	yes	—
55	#14519776	—	2	2	no	—
56	#14519776	C₂×C₄	1	4	yes	—
57	#14519775	—	2	2	no	—
58	C₄⋊C₄×D₆₁	—	2	2	no	—
59	C₄⋊C₄×D₆₁	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	#14519710	C₂×C₄	1	4	yes	—
64	#14519710	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	C₂²×#6923	—	2	2	no	—
74	C₂²×#6923	C₂×D₄	1	2	yes	—
75	C₂²×#6923	C₂×D₄	1	2	yes	—
76	C₂×Q₈×C₆₁	—	2	2	no	—
77	Q₈×D₆₁	—	8	2	no	—
78	Q₈×D₆₁	—	8	2	no	—
79	C₂×C₄×D₆₁	—	2	4	no	—
80	C₂×C₄×D₆₁	C₂×D₄	1	1	yes	characteristic
81	C₂×C₄×D₆₁	C₂×D₄	1	1	yes	characteristic
82	C₂×C₄×D₆₁	C₂⁴	1	1	yes	characteristic
83	C₂×Dic₁₂₂	—	2	2	no	—
84	C₂×Dic₁₂₂	—	2	4	no	—
85	C₂×#92420	—	2	2	no	—
86	C₄⋊#6923	—	2	2	no	—
87	C₄⋊#6923	—	2	2	no	—
88	C₄⋊#6923	—	2	2	no	—
89	C₄⋊#6923	—	2	2	no	—
90	C₄⋊#6923	—	8	4	no	—
91	C₄⋊#6923	Q₁₆	1	4	yes	—
92	C₄⋊#6923	C₂×Q₈	1	4	yes	—
93	C₄×#6923	—	2	2	no	—
94	C₄×#6923	—	2	2	no	—
95	C₄×#6923	—	4	4	no	—
96	C₄×#6923	—	8	4	no	—
97	D₆₁⋊C₈	—	2	2	no	—
98	D₆₁⋊C₈	—	2	2	no	—
99	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
100	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
101	C₄⋊C₄×C₆₁	—	2	2	no	—
102	C₄⋊C₄×C₆₁	C₂²×C₄	1	4	yes	—
103	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
104	C₄⋊Dic₆₁	—	2	2	no	—
105	C₄⋊Dic₆₁	C₂²×C₄	1	4	yes	—
106	#11758196	—	2	4	no	—
107	#11758196	—	2	4	no	—
108	C₄×Dic₆₁	C₂²×C₄	1	1	yes	characteristic
109	C₄×Dic₆₁	#41	1	1	yes	characteristic
110	C₄×Dic₆₁	#41	1	1	yes	characteristic
111	C₂×#92418	C₂²×C₄	1	1	yes	characteristic
112	C₂×#92418	C₂²×C₄	1	1	yes	characteristic
113	C₈×D₆₁	—	2	1	no	—
114	C₈×D₆₁	—	2	1	no	—
115	C₈×D₆₁	C₂×Q₈	1	2	yes	—
116	C₈×D₆₁	#41	1	2	yes	—
117	C₂²×D₆₁	C₂²×D₄	1	1	yes	characteristic
118	C₂×#6923	—	2	2	no	—
119	C₂×#6923	—	2	2	no	—
120	C₂×#6923	—	2	2	no	—
121	C₂×#6923	—	2	4	no	—
122	C₂×#6923	—	2	2	no	—
123	C₂×#6923	—	2	2	no	—
124	C₂×#6923	—	4	2	no	—
125	C₂×#6923	—	4	2	no	—
126	C₂×#6923	—	4	2	no	—
127	C₂×#6923	—	4	2	no	—
128	C₂×#6923	—	4	2	no	—
129	C₂×#6923	C₂×Q₁₆	1	4	yes	—
130	Q₈×C₆₁	—	8	2	no	—
131	C₂×C₄×C₆₁	—	2	4	no	—
132	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
133	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
134	C₂×C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
135	C₂×Dic₆₁	—	2	4	no	—
136	C₂×Dic₆₁	C₄×D₄	1	1	yes	characteristic
137	C₂×Dic₆₁	C₄×D₄	1	1	yes	characteristic
138	C₂×Dic₆₁	C₂×#41	1	1	yes	characteristic
139	C₄×D₆₁	—	2	1	no	—
140	C₄×D₆₁	—	2	1	no	—
141	C₄×D₆₁	—	4	4	no	—
142	C₄×D₆₁	—	4	4	no	—
143	C₄×D₆₁	C₂×Q₁₆	1	2	yes	—
144	C₄×D₆₁	#135	1	2	yes	—
145	C₄×D₆₁	C₂²×Q₈	1	1	yes	characteristic
146	C₄×D₆₁	C₂×#41	1	1	yes	characteristic
147	C₄×D₆₁	2₋¹⁺⁴	1	1	yes	characteristic
148	C₄×D₆₁	2₋¹⁺⁴	1	1	yes	characteristic
149	Dic₁₂₂	—	8	4	no	—
150	Dic₁₂₂	—	8	2	no	—
151	#92420	—	4	2	no	—
152	C₈×C₆₁	—	2	1	no	—
153	C₈×C₆₁	C₄×Q₈	1	2	yes	—
154	#92418	—	2	1	no	—
155	#92418	C₄×Q₈	1	2	yes	—
156	#23158	—	61	1	no	maximal, Sylow
157	C₂²×C₆₁	C₂×C₄×D₄	1	1	yes	characteristic
158	C₂×D₆₁	#547	1	1	yes	characteristic
159	C₂×D₆₁	D₄×Q₈	1	1	yes	characteristic
160	C₂×D₆₁	C₂²×Q₁₆	1	1	yes	characteristic
161	C₂×D₆₁	C₂×#135	1	1	yes	characteristic
162	C₂×D₆₁	#576	1	1	yes	characteristic
163	C₂×D₆₁	#576	1	1	yes	characteristic
164	#6923	—	2	4	no	—
165	#6923	—	4	2	no	—
166	#6923	—	4	4	no	—
167	#6923	—	8	2	no	—
168	C₄×C₆₁	—	2	1	no	—
169	C₄×C₆₁	—	4	4	no	—
170	C₄×C₆₁	C₄×Q₁₆	1	2	yes	—
171	C₄×C₆₁	C₂×C₄×Q₈	1	1	yes	characteristic
172	C₄×C₆₁	#519	1	1	yes	characteristic
173	Dic₆₁	—	2	1	no	—
174	Dic₆₁	—	4	4	no	—
175	Dic₆₁	C₄×Q₁₆	1	2	yes	—
176	Dic₆₁	#556	1	1	yes	characteristic
177	Dic₆₁	#557	1	1	yes	characteristic
178	C₂×#500	—	61	1	no	—
179	#2369	—	61	2	no	—
180	#1987	—	61	2	no	—
181	#1942	—	61	2	no	—
182	#1921	—	61	4	no	—
183	C₄×#107	—	61	1	no	—
184	C₄×#107	—	61	2	no	—
185	C₈×C₄⋊C₄	—	61	1	no	—
186	C₂×C₆₁	#2365	1	1	yes	characteristic
187	C₂×C₆₁	C₂×C₄×Q₁₆	1	1	yes	characteristic
188	C₂×C₆₁	#2948	1	1	yes	characteristic
189	D₆₁	#3338	1	1	yes	characteristic
190	D₆₁	#3341	1	1	yes	characteristic
191	D₆₁	#3380	1	1	yes	characteristic
192	D₆₁	Q₈×Q₁₆	1	1	yes	characteristic
193	C₂×C₄⋊Q₈	—	61	2	no	—
194	#500	—	61	2	no	—
195	#500	—	61	2	no	—
196	#500	—	61	2	no	—
197	#500	—	61	1	no	—
198	#500	#6923	1	1	yes	characteristic, complemented
199	C₂×#107	—	61	1	no	—
200	C₂×#107	—	61	2	no	—
201	C₂×#107	—	61	2	no	—
202	C₂×#107	—	61	1	no	—
203	C₂×C₄⋊C₈	—	61	1	no	—
204	C₂×C₄×C₈	—	61	1	no	—
205	C₂×C₄×C₈	—	61	2	no	—
206	#391	—	122	4	no	—
207	#389	—	61	4	no	—
208	#389	—	122	2	no	—
209	C₄×C₄⋊C₄	—	61	1	no	—
210	C₄×C₄⋊C₄	—	61	4	no	—
211	C₄×C₄⋊C₄	—	61	2	no	—
212	C₄×C₄⋊C₄	—	122	2	no	—
213	#340	—	61	2	no	—
214	#340	—	61	4	no	—
215	#340	—	61	4	no	—
216	#336	—	61	2	no	—
217	C₆₁	#23158	1	1	yes	characteristic, complemented, Sylow
218	C₂²×Q₈	—	122	2	no	—
219	C₂²×C₈	—	61	1	no	—
220	C₂²×C₈	—	61	1	no	—
221	C₂²×C₈	—	61	2	no	—
222	C₄⋊Q₈	—	61	4	no	—
223	C₄⋊Q₈	—	61	2	no	—
224	C₄⋊Q₈	—	122	2	no	—
225	C₄⋊Q₈	—	122	2	no	—
226	C₄⋊Q₈	C₂×#6923	1	2	yes	—
227	C₂×C₄⋊C₄	—	61	1	no	—
228	C₂×C₄⋊C₄	—	61	2	no	—
229	C₂×C₄⋊C₄	—	61	4	no	—
230	C₂×C₄⋊C₄	—	61	1	no	—
231	C₂×C₄⋊C₄	—	61	2	no	—
232	C₂×C₄⋊C₄	—	122	2	no	—
233	C₂×C₄⋊C₄	—	122	4	no	—
234	C₂×C₄²	—	61	1	no	—
235	C₂×C₄²	—	61	2	no	—
236	C₂×C₄²	—	122	4	no	—
237	C₂×C₄²	—	122	2	no	—
238	#107	—	61	4	no	—
239	#107	—	61	2	no	—
240	#107	—	61	2	no	—
241	#107	—	61	4	no	—
242	#107	—	61	4	no	—
243	#107	—	122	2	no	—
244	#107	C₂×#6923	1	2	yes	—
245	#107	C₂×#6923	1	2	yes	—
246	C₄⋊C₈	—	122	1	no	—
247	C₄⋊C₈	—	122	1	no	—
248	C₄×C₈	—	61	1	no	—
249	C₄×C₈	—	61	1	no	—
250	C₄×C₈	—	61	1	no	—
251	C₄×C₈	—	122	4	no	—
252	C₄×C₈	C₂×#6923	1	1	yes	characteristic
253	#95	—	61	2	no	—
254	#95	—	122	4	no	—
255	#95	—	122	2	no	—
256	#95	—	122	4	no	—
257	C₂×Q₈	—	2	2	no	—
258	C₂×Q₈	—	122	4	no	—
259	C₂×Q₈	—	122	2	no	—
260	C₂×Q₈	—	488	2	no	—
261	C₂×Q₈	—	488	2	no	—
262	C₂²×C₄	—	61	1	no	—
263	C₂²×C₄	—	61	1	no	—
264	C₂²×C₄	—	61	2	no	—
265	C₂²×C₄	—	61	1	no	—
266	C₂²×C₄	—	61	2	no	—
267	C₂²×C₄	—	122	4	no	—
268	C₂²×C₄	—	122	2	no	—
269	C₂²×C₄	—	122	2	no	—
270	C₂×C₈	—	61	1	no	—
271	C₂×C₈	—	61	2	no	—
272	C₂×C₈	—	61	2	no	—
273	C₂×C₈	—	61	1	no	—
274	C₂×C₈	—	122	1	no	—
275	C₂×C₈	—	122	1	no	—
276	C₂×C₈	—	122	2	no	—
277	C₂×C₈	—	122	2	no	—
278	C₂×C₈	—	122	2	no	—
279	C₂×C₈	C₂²×#6923	1	1	yes	characteristic
280	C₂×C₈	C₂²×#6923	1	1	yes	characteristic
281	C₄⋊C₄	—	2	2	no	—
282	C₄⋊C₄	—	61	4	no	—
283	C₄⋊C₄	—	61	4	no	—
284	C₄⋊C₄	—	61	4	no	—
285	C₄⋊C₄	—	122	4	no	—
286	C₄⋊C₄	—	122	4	no	—
287	C₄⋊C₄	—	122	2	no	—
288	C₄⋊C₄	—	122	2	no	—
289	C₄⋊C₄	—	122	2	no	—
290	C₄⋊C₄	—	122	2	no	—
291	C₄⋊C₄	—	122	2	no	—
292	C₄⋊C₄	—	488	4	no	—
293	C₄⋊C₄	C₂²×#6923	1	4	yes	—
294	C₄²	—	61	1	no	—
295	C₄²	—	61	1	no	—
296	C₄²	—	61	1	no	—
297	C₄²	—	122	2	no	—
298	C₄²	—	122	2	no	—
299	C₄²	—	488	4	no	—
300	C₄²	—	244	4	no	—
301	C₄²	C₂²×#6923	1	1	yes	characteristic
302	C₂³	—	61	1	no	—
303	Q₈	—	8	2	no	—
304	Q₈	—	488	2	no	—
305	Q₈	—	488	4	no	—
306	C₂×C₄	—	2	4	no	—
307	C₂×C₄	—	61	1	no	—
308	C₂×C₄	—	61	1	no	—
309	C₂×C₄	—	61	2	no	—
310	C₂×C₄	—	61	2	no	—
311	C₂×C₄	—	61	1	no	—
312	C₂×C₄	—	61	1	no	—
313	C₂×C₄	—	61	1	no	—
314	C₂×C₄	—	61	1	no	—
315	C₂×C₄	—	61	4	no	—
316	C₂×C₄	—	61	1	no	—
317	C₂×C₄	—	122	1	no	—
318	C₂×C₄	—	122	1	no	—
319	C₂×C₄	—	122	2	no	—
320	C₂×C₄	—	122	2	no	—
321	C₂×C₄	—	122	2	no	—
322	C₂×C₄	—	122	4	no	—
323	C₂×C₄	—	122	2	no	—
324	C₂×C₄	—	122	2	no	—
325	C₂×C₄	—	122	4	no	—
326	C₂×C₄	—	244	4	no	—
327	C₂×C₄	—	244	4	no	—
328	C₂×C₄	—	244	2	no	—
329	C₂×C₄	—	244	2	no	—
330	C₂×C₄	—	244	2	no	—
331	C₂×C₄	—	244	2	no	—
332	C₂×C₄	—	244	2	no	—
333	C₂×C₄	D₄×#6923	1	1	yes	characteristic
334	C₂×C₄	D₄×#6923	1	1	yes	characteristic
335	C₂×C₄	C₂³×#6923	1	1	yes	characteristic
336	C₈	—	2	1	no	—
337	C₈	—	61	2	no	—
338	C₈	—	122	1	no	—
339	C₈	—	244	2	no	—
340	C₈	Q₈×#6923	1	2	yes	—
341	C₂²	—	61	1	no	—
342	C₂²	—	61	1	no	—
343	C₂²	—	61	1	no	—
344	C₂²	—	61	1	no	—
345	C₂²	—	61	1	no	—
346	C₂²	—	61	1	no	—
347	C₂²	C₂×D₄×#6923	1	1	yes	characteristic
348	C₄	—	2	1	no	—
349	C₄	—	4	4	no	—
350	C₄	—	61	2	no	—
351	C₄	—	61	1	no	—
352	C₄	—	61	1	no	—
353	C₄	—	122	4	no	—
354	C₄	—	122	1	no	—
355	C₄	—	488	2	no	—
356	C₄	—	244	2	no	—
357	C₄	—	244	4	no	—
358	C₄	—	244	4	no	—
359	C₄	Q₁₆×#6923	1	2	yes	—
360	C₄	C₂×Q₈×#6923	1	1	yes	characteristic
361	C₄	#16840830	1	1	yes	characteristic
362	C₂	—	61	1	no	—
363	C₂	—	61	1	no	—
364	C₂	—	61	1	no	—
365	C₂	—	61	1	no	—
366	C₂	#21462348	1	1	yes	characteristic
367	C₂	C₂×Q₁₆×#6923	1	1	yes	characteristic
368	C₂	#21462578	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	C₂²×D₄
cosocle	C₂×C₄×D₆₁	C₂⁴
hypercenter	#500	#6923
hypocenter	C₆₁	#23158
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. #16840166	C₄	semidirect
27. #16840166	C₄	semidirect
28. #16840165	C₄	semidirect
32. C₆₁×#500	C₄	semidirect
43. #14519846	Q₈	semidirect
198. #500	#6923	semidirect
217. C₆₁	#23158	semidirect

Maximal subgroups

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

subgroup	index
#21462352	2
#21461997	2
#21461923	2
#21461883	2
#107×#6923	2
C₄×#14519846	2
C₈×C₄⋊#6923	2
#500×D₆₁	2
#23158	61