Subgroups — #30613649

Order 15616.

600 conjugacy classes of subgroups.

The subgroup lattice has 394 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21462650	C₂	1	2	yes	maximal, complemented
2	#21461458	C₂	1	1	yes	maximal, characteristic, complemented
3	#21461452	C₂	1	1	yes	maximal, characteristic, complemented
4	#21461429	C₂	1	2	yes	maximal, complemented
5	#21461422	C₂	1	2	yes	maximal, complemented
6	#21461407	C₂	1	1	yes	maximal, characteristic, complemented
7	D₈×#92420	C₂	1	1	yes	maximal, characteristic, complemented
8	#21461366	C₂	1	1	yes	maximal, characteristic, complemented
9	#21461130	C₂	1	2	yes	maximal, complemented
10	C₈×#11758217	C₂	1	1	yes	maximal, characteristic, complemented
11	#21456384	C₂	1	1	yes	maximal, characteristic
12	#16840812	C₂²	1	2	yes	—
13	#16840811	C₂²	1	2	yes	—
14	#16840797	—	2	2	no	—
15	#16840797	C₂²	1	2	yes	—
16	#16840793	—	2	2	no	—
17	D₄×#92420	C₂²	1	2	yes	—
18	#16840786	—	2	2	no	—
19	#16840779	C₂²	1	1	yes	characteristic
20	#16840778	C₂²	1	2	yes	—
21	#16840726	—	2	2	no	—
22	C₄×#11758217	C₂²	1	1	yes	characteristic
23	#16840625	C₂²	1	1	yes	characteristic
24	#16840625	C₂²	1	1	yes	characteristic
25	#16840623	C₂²	1	2	yes	—
26	#16840612	C₂²	1	2	yes	—
27	#16840612	C₂²	1	2	yes	—
28	D₄⋊#92420	C₂²	1	2	yes	—
29	D₄⋊#92420	C₂²	1	2	yes	—
30	#92420⋊C₈	C₂²	1	1	yes	characteristic
31	#92420⋊C₈	C₂²	1	1	yes	characteristic
32	C₈×#92420	C₂²	1	1	yes	characteristic
33	C₈×#92420	C₂²	1	1	yes	characteristic
34	#16840293	C₂²	1	2	yes	—
35	#16840163	C₄	1	1	yes	characteristic
36	#16840145	C₄	1	1	yes	characteristic
37	#16840115	C₄	1	2	yes	—
38	#16840003	C₄	1	1	yes	characteristic
39	#16839989	C₄	1	2	yes	—
40	#16839856	C₂²	1	1	yes	characteristic
41	#16839853	C₂²	1	2	yes	—
42	D₈×Dic₆₁	C₂²	1	1	yes	characteristic
43	C₂×C₈×Dic₆₁	C₂²	1	1	yes	characteristic
44	C₆₁×#466	C₄	1	1	yes	characteristic
45	C₂×#11758217	—	2	2	no	—
46	C₂×#11758217	D₄	1	1	yes	characteristic
47	C₂×#11758217	D₄	1	1	yes	characteristic
48	C₂²×#92420	—	2	2	no	—
49	#14519844	—	2	2	no	—
50	#14519844	—	2	1	no	—
51	C₄⋊#92420	—	2	2	no	—
52	C₄⋊#92420	C₂³	1	2	yes	—
53	C₄⋊#92420	C₂³	1	2	yes	—
54	C₂²⋊#92420	—	2	2	no	—
55	C₂²⋊#92420	—	2	1	no	—
56	C₂²⋊#92420	—	2	2	no	—
57	C₂²⋊#92420	—	2	2	no	—
58	#14519832	—	2	2	no	—
59	#14519832	D₄	1	1	yes	characteristic
60	#14519832	D₄	1	1	yes	characteristic
61	C₄×#92420	—	2	2	no	—
62	C₄×#92420	—	2	2	no	—
63	C₄×#92420	C₂³	1	1	yes	characteristic
64	C₄×#92420	C₂³	1	1	yes	characteristic
65	C₄×#92420	C₂³	1	1	yes	characteristic
66	C₄×#92420	C₂³	1	1	yes	characteristic
67	C₂×Dic₂₄₄	C₂×C₄	1	1	yes	characteristic
68	C₂×#11758200	C₂×C₄	1	2	yes	—
69	C₂²×#92418	C₂×C₄	1	1	yes	characteristic
70	#14519770	—	2	2	no	—
71	#14519768	C₂×C₄	1	2	yes	—
72	#14519764	C₂×C₄	1	2	yes	—
73	#14519758	—	2	2	no	—
74	#14519756	C₂×C₄	1	2	yes	—
75	#14519742	C₂³	1	2	yes	—
76	D₄×Dic₆₁	—	2	2	no	—
77	D₄×Dic₆₁	C₂³	1	2	yes	—
78	#14519733	—	2	2	no	—
79	C₂×C₄×Dic₆₁	C₂³	1	1	yes	characteristic
80	#14519711	C₂×C₄	1	1	yes	characteristic
81	#14519708	C₂×C₄	1	2	yes	—
82	#14519702	C₂×C₄	1	2	yes	—
83	#14519700	C₂×C₄	1	2	yes	—
84	#14519698	C₂×C₄	1	2	yes	—
85	C₈×Dic₆₁	C₂×C₄	1	1	yes	characteristic
86	C₈×Dic₆₁	C₂×C₄	1	1	yes	characteristic
87	C₈×Dic₆₁	C₂³	1	1	yes	characteristic
88	C₈×Dic₆₁	C₂³	1	1	yes	characteristic
89	C₂×D₈×C₆₁	C₂×C₄	1	1	yes	characteristic
90	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
91	#121×C₆₁	C₂×C₄	1	2	yes	—
92	#107×C₆₁	C₂×C₄	1	1	yes	characteristic
93	#102×C₆₁	C₂×C₄	1	2	yes	—
94	C₂×D₄×C₆₁	—	2	2	no	—
95	C₂×D₄×C₆₁	C₂²×C₄	1	2	yes	—
96	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
97	C₂²×Dic₆₁	—	2	2	no	—
98	C₂²×Dic₆₁	C₂×D₄	1	1	yes	characteristic
99	C₂²×Dic₆₁	#41	1	1	yes	characteristic
100	C₂×Dic₁₂₂	C₂²×C₄	1	2	yes	—
101	#11758217	—	2	2	no	—
102	#11758217	—	2	2	no	—
103	#11758217	—	4	4	no	—
104	C₂×#92420	—	2	2	no	—
105	C₂×#92420	—	2	2	no	—
106	C₂×#92420	—	2	1	no	—
107	C₂×#92420	—	2	1	no	—
108	C₂×#92420	—	4	4	no	—
109	C₂×#92420	C₂×D₄	1	1	yes	characteristic
110	C₂×#92420	C₂×D₄	1	1	yes	characteristic
111	C₂×#92420	C₂×D₄	1	1	yes	characteristic
112	C₂×#92420	C₂×D₄	1	1	yes	characteristic
113	D₈×C₆₁	—	2	2	no	—
114	C₂×C₈×C₆₁	—	2	2	no	—
115	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
116	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
117	C₄⋊C₄×C₆₁	C₂²×C₄	1	2	yes	—
118	C₂²⋊C₄×C₆₁	—	2	2	no	—
119	C₂²⋊Dic₆₁	—	2	2	no	—
120	C₂²⋊Dic₆₁	—	2	2	no	—
121	C₂²⋊Dic₆₁	—	2	2	no	—
122	C₂²⋊Dic₆₁	—	2	2	no	—
123	#11758200	—	4	2	no	—
124	C₄⋊Dic₆₁	—	2	2	no	—
125	C₄⋊Dic₆₁	C₂²×C₄	1	2	yes	—
126	#11758196	—	2	2	no	—
127	#11758196	—	2	2	no	—
128	#11758196	—	2	2	no	—
129	C₄×Dic₆₁	—	2	2	no	—
130	C₄×Dic₆₁	C₂²×C₄	1	1	yes	characteristic
131	C₄×Dic₆₁	C₂×D₄	1	1	yes	characteristic
132	C₄×Dic₆₁	#41	1	1	yes	characteristic
133	C₄×Dic₆₁	C₂⁴	1	1	yes	characteristic
134	C₂×#92418	—	4	1	no	—
135	C₂×#92418	C₂²×C₄	1	1	yes	characteristic
136	C₂×#92418	C₂²×C₄	1	1	yes	characteristic
137	Dic₂₄₄	—	2	2	no	—
138	C₂³×C₆₁	—	2	2	no	—
139	C₂³×C₆₁	C₄×D₄	1	1	yes	characteristic
140	D₄×C₆₁	—	2	2	no	—
141	D₄×C₆₁	—	2	2	no	—
142	D₄×C₆₁	—	8	2	no	—
143	C₂×C₄×C₆₁	—	2	1	no	—
144	C₂×C₄×C₆₁	—	2	1	no	—
145	C₂×C₄×C₆₁	—	2	2	no	—
146	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
147	C₂×C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
148	C₂×Dic₆₁	—	2	1	no	—
149	C₂×Dic₆₁	—	2	1	no	—
150	C₂×Dic₆₁	—	2	2	no	—
151	C₂×Dic₆₁	—	2	2	no	—
152	C₂×Dic₆₁	—	4	1	no	—
153	C₂×Dic₆₁	—	4	2	no	—
154	C₂×Dic₆₁	—	4	2	no	—
155	C₂×Dic₆₁	C₄×D₄	1	1	yes	characteristic
156	C₂×Dic₆₁	C₄×D₄	1	1	yes	characteristic
157	C₂×Dic₆₁	C₂²×D₄	1	1	yes	characteristic
158	C₂×Dic₆₁	C₂×#41	1	1	yes	characteristic
159	Dic₁₂₂	—	2	2	no	—
160	Dic₁₂₂	—	2	2	no	—
161	#92420	—	2	1	no	—
162	#92420	—	2	1	no	—
163	#92420	—	4	1	no	—
164	#92420	—	4	1	no	—
165	#92420	—	4	2	no	—
166	#92420	—	4	2	no	—
167	#92420	#135	1	2	yes	—
168	#92420	#135	1	2	yes	—
169	C₈×C₆₁	—	2	1	no	—
170	C₈×C₆₁	#131	1	2	yes	—
171	#92418	—	2	1	no	—
172	#92418	—	2	1	no	—
173	#18069	—	61	1	no	maximal, Sylow
174	C₂²×C₆₁	—	2	1	no	—
175	C₂²×C₆₁	—	2	1	no	—
176	C₂²×C₆₁	—	2	1	no	—
177	C₂²×C₆₁	—	4	2	no	—
178	C₂²×C₆₁	—	4	2	no	—
179	C₂²×C₆₁	—	4	2	no	—
180	C₂²×C₆₁	C₂×C₄×D₄	1	1	yes	characteristic
181	C₄×C₆₁	—	2	1	no	—
182	C₄×C₆₁	—	4	2	no	—
183	C₄×C₆₁	C₂×#131	1	1	yes	characteristic
184	C₄×C₆₁	#568	1	1	yes	characteristic
185	Dic₆₁	—	2	1	no	—
186	Dic₆₁	—	2	1	no	—
187	Dic₆₁	—	2	1	no	—
188	Dic₆₁	—	4	2	no	—
189	Dic₆₁	—	4	2	no	—
190	Dic₆₁	C₂×#135	1	1	yes	characteristic
191	Dic₆₁	C₂×#135	1	1	yes	characteristic
192	#3168	—	61	1	no	—
193	#2969	—	61	2	no	—
194	#1591	—	61	1	no	—
195	#1589	—	61	1	no	—
196	#1583	—	61	2	no	—
197	#1582	—	61	2	no	—
198	C₈×Q₁₆	—	61	1	no	—
199	C₈×D₈	—	61	1	no	—
200	#1564	—	61	1	no	—
201	#1482	—	61	2	no	—
202	C₈×M₁₆	—	61	1	no	—
203	C₂×C₆₁	—	4	1	no	—
204	C₂×C₆₁	—	8	2	no	—
205	C₂×C₆₁	#2935	1	1	yes	characteristic
206	C₂×C₆₁	C₄×#135	1	1	yes	characteristic
207	C₂×C₆₁	#2957	1	1	yes	characteristic
208	#529	—	61	2	no	—
209	#499	—	61	1	no	—
210	#495	—	61	1	no	—
211	#488	—	61	2	no	—
212	#467	—	61	1	no	—
213	#466	#6923	1	1	yes	characteristic
214	#465	—	61	2	no	—
215	#446	—	61	2	no	—
216	C₈×Q₈	—	61	2	no	—
217	C₄×Q₁₆	—	61	1	no	—
218	C₄×SD₁₆	—	61	2	no	—
219	C₄×D₈	—	61	1	no	—
220	#436	—	61	2	no	—
221	#436	—	122	2	no	—
222	#435	—	122	2	no	—
223	C₈×D₄	—	61	2	no	—
224	#433	—	122	2	no	—
225	#431	—	61	1	no	—
226	#431	—	61	2	no	—
227	#405	—	122	2	no	—
228	C₄×M₁₆	—	61	1	no	—
229	C₂×C₄×C₈	—	61	1	no	—
230	#335	—	61	1	no	—
231	#335	—	61	1	no	—
232	#334	—	61	2	no	—
233	Q₈⋊C₈	—	61	2	no	—
234	Q₈⋊C₈	—	61	2	no	—
235	D₄⋊C₈	—	61	2	no	—
236	D₄⋊C₈	—	61	2	no	—
237	#322	—	61	1	no	—
238	#322	—	61	1	no	—
239	C₈²	—	61	1	no	—
240	C₈²	—	61	1	no	—
241	C₆₁	#18069	1	1	yes	characteristic, complemented, Sylow
242	C₂×Q₁₆	—	61	1	no	—
243	C₂×SD₁₆	—	61	2	no	—
244	C₂×D₈	C₂×#6923	1	1	yes	characteristic
245	C₂×M₁₆	—	61	1	no	—
246	C₂×M₁₆	—	61	1	no	—
247	C₂×M₁₆	—	122	2	no	—
248	C₂²×C₈	—	61	1	no	—
249	C₂²×C₈	—	122	2	no	—
250	C₂²×C₈	C₂×#6923	1	1	yes	characteristic
251	#126	—	122	2	no	—
252	#125	—	61	2	no	—
253	#124	—	61	2	no	—
254	#123	—	122	2	no	—
255	C₂²⋊Q₈	—	61	2	no	—
256	#121	C₂×#6923	1	2	yes	—
257	C₄×Q₈	—	61	2	no	—
258	C₄×D₄	—	61	2	no	—
259	C₄×D₄	—	122	2	no	—
260	#117	—	122	2	no	—
261	C₂×C₄²	—	61	1	no	—
262	#107	—	61	1	no	—
263	#107	C₂×#6923	1	1	yes	characteristic
264	#106	—	61	2	no	—
265	C₄⋊C₈	—	61	2	no	—
266	C₄⋊C₈	—	61	2	no	—
267	C₄⋊C₈	—	122	1	no	—
268	C₄⋊C₈	—	122	2	no	—
269	C₄⋊C₈	—	122	2	no	—
270	#103	—	61	2	no	—
271	#103	—	61	2	no	—
272	#102	—	61	2	no	—
273	#102	C₂×#6923	1	2	yes	—
274	C₂²⋊C₈	—	122	2	no	—
275	C₂²⋊C₈	—	122	2	no	—
276	C₂²⋊C₈	—	122	1	no	—
277	C₂²⋊C₈	—	122	2	no	—
278	#97	—	61	1	no	—
279	#97	—	61	1	no	—
280	#97	—	122	2	no	—
281	C₄×C₈	—	61	1	no	—
282	C₄×C₈	—	61	1	no	—
283	C₄×C₈	—	61	1	no	—
284	C₄×C₈	—	61	1	no	—
285	C₄×C₈	—	61	1	no	—
286	C₄×C₈	—	61	1	no	—
287	C₄×C₈	—	61	1	no	—
288	C₄×C₈	—	61	1	no	—
289	C₄×C₈	—	122	2	no	—
290	C₄×C₈	—	122	2	no	—
291	C₂×Q₈	—	61	2	no	—
292	C₂×D₄	—	2	2	no	—
293	C₂×D₄	C₂²×#6923	1	2	yes	—
294	C₂²×C₄	—	61	1	no	—
295	C₂²×C₄	—	61	1	no	—
296	C₂²×C₄	—	122	2	no	—
297	C₂²×C₄	C₂²×#6923	1	1	yes	characteristic
298	Q₁₆	—	122	2	no	—
299	SD₁₆	—	244	2	no	—
300	D₈	—	2	2	no	—
301	M₁₆	—	122	2	no	—
302	M₁₆	—	122	2	no	—
303	M₁₆	—	244	4	no	—
304	C₂×C₈	—	2	2	no	—
305	C₂×C₈	—	61	1	no	—
306	C₂×C₈	—	61	1	no	—
307	C₂×C₈	—	61	1	no	—
308	C₂×C₈	—	61	1	no	—
309	C₂×C₈	—	61	1	no	—
310	C₂×C₈	—	61	1	no	—
311	C₂×C₈	—	122	1	no	—
312	C₂×C₈	—	122	1	no	—
313	C₂×C₈	—	122	2	no	—
314	C₂×C₈	—	122	2	no	—
315	C₂×C₈	—	244	4	no	—
316	C₂×C₈	—	244	1	no	—
317	C₂×C₈	C₂²×#6923	1	1	yes	characteristic
318	C₂×C₈	C₂²×#6923	1	1	yes	characteristic
319	C₄⋊C₄	—	61	2	no	—
320	C₄⋊C₄	—	122	2	no	—
321	C₄⋊C₄	—	122	2	no	—
322	C₄⋊C₄	—	122	2	no	—
323	C₄⋊C₄	—	122	2	no	—
324	C₄⋊C₄	C₂²×#6923	1	2	yes	—
325	C₂²⋊C₄	—	2	2	no	—
326	C₂²⋊C₄	—	122	2	no	—
327	C₂²⋊C₄	—	122	2	no	—
328	C₂²⋊C₄	—	122	2	no	—
329	C₂²⋊C₄	—	122	2	no	—
330	C₄²	—	61	1	no	—
331	C₄²	—	61	1	no	—
332	C₄²	—	61	1	no	—
333	C₄²	—	61	1	no	—
334	C₄²	—	122	2	no	—
335	C₂³	—	2	2	no	—
336	C₂³	D₄×#6923	1	1	yes	characteristic
337	Q₈	—	122	2	no	—
338	Q₈	—	122	2	no	—
339	D₄	—	2	2	no	—
340	D₄	—	2	2	no	—
341	D₄	—	8	2	no	—
342	C₂×C₄	—	2	1	no	—
343	C₂×C₄	—	2	1	no	—
344	C₂×C₄	—	2	2	no	—
345	C₂×C₄	—	61	1	no	—
346	C₂×C₄	—	61	1	no	—
347	C₂×C₄	—	61	1	no	—
348	C₂×C₄	—	61	1	no	—
349	C₂×C₄	—	122	1	no	—
350	C₂×C₄	—	122	1	no	—
351	C₂×C₄	—	122	2	no	—
352	C₂×C₄	—	122	2	no	—
353	C₂×C₄	—	244	2	no	—
354	C₂×C₄	—	244	2	no	—
355	C₂×C₄	—	244	1	no	—
356	C₂×C₄	D₄×#6923	1	1	yes	characteristic
357	C₂×C₄	C₂³×#6923	1	1	yes	characteristic
358	C₈	—	2	1	no	—
359	C₈	—	61	2	no	—
360	C₈	—	61	2	no	—
361	C₈	—	122	1	no	—
362	C₈	—	122	1	no	—
363	C₈	—	122	1	no	—
364	C₈	—	122	1	no	—
365	C₈	—	244	2	no	—
366	C₈	—	244	2	no	—
367	C₈	—	244	1	no	—
368	C₈	—	244	1	no	—
369	C₈	#14519864	1	2	yes	—
370	C₂²	—	2	1	no	—
371	C₂²	—	2	1	no	—
372	C₂²	—	2	1	no	—
373	C₂²	—	4	2	no	—
374	C₂²	—	4	2	no	—
375	C₂²	—	4	2	no	—
376	C₂²	C₂×D₄×#6923	1	1	yes	characteristic
377	C₄	—	2	1	no	—
378	C₄	—	4	2	no	—
379	C₄	—	61	1	no	—
380	C₄	—	61	1	no	—
381	C₄	—	122	1	no	—
382	C₄	—	122	1	no	—
383	C₄	—	122	1	no	—
384	C₄	—	244	2	no	—
385	C₄	—	244	2	no	—
386	C₄	C₂×#14519864	1	1	yes	characteristic
387	C₄	#16840841	1	1	yes	characteristic
388	C₂	—	4	1	no	—
389	C₂	—	8	2	no	—
390	C₂	#21462561	1	1	yes	characteristic
391	C₂	#135×#6923	1	1	yes	characteristic
392	C₂	#21462595	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×#6923
commutator	C₂×C₄×C₆₁	C₂³×C₄
Frattini	C₂×C₄	C₂³×#6923
Fitting, prime radical	C₆₁×#466	C₄
socle	C₂²×C₆₁	C₂×C₄×D₄
cosocle	C₄×Dic₆₁	C₂⁴
hypercenter	#466	#6923
hypocenter	C₆₁	#18069
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #466

Complements

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

N	H ≅ G/N	type
1. #21462650	C₂	semidirect
2. #21461458	C₂	semidirect
3. #21461452	C₂	semidirect
4. #21461429	C₂	semidirect
5. #21461422	C₂	semidirect
6. #21461407	C₂	semidirect
7. D₈×#92420	C₂	semidirect
8. #21461366	C₂	semidirect
9. #21461130	C₂	semidirect
10. C₈×#11758217	C₂	semidirect
241. C₆₁	#18069	semidirect

Maximal subgroups

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

subgroup	index
#21462650	2
#21461458	2
#21461452	2
#21461429	2
#21461422	2
#21461407	2
D₈×#92420	2
#21461366	2
#21461130	2
C₈×#11758217	2
#21456384	2
#18069	61