Subgroups — #30612120

Order 15616.

602 conjugacy classes of subgroups.

The subgroup lattice has 406 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21462738	C₂	1	1	yes	maximal, characteristic
2	#21462734	C₂	1	1	yes	maximal, characteristic
3	#21461395	C₂	1	2	yes	maximal, complemented
4	#21461392	C₂	1	2	yes	maximal, complemented
5	#21461390	C₂	1	2	yes	maximal, complemented
6	#21461388	C₂	1	2	yes	maximal, complemented
7	#21461372	C₂	1	1	yes	maximal, characteristic, complemented
8	#21461370	C₂	1	1	yes	maximal, characteristic, complemented
9	#21461366	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#16840626	C₂	1	1	yes	maximal, characteristic, complemented
11	#21454459	C₂	1	1	yes	maximal, characteristic
12	#16840813	—	2	1	no	—
13	Q₈×#92420	—	2	1	no	—
14	#16840797	—	2	1	no	—
15	#16840793	—	2	2	no	—
16	D₄×#92420	—	2	2	no	—
17	D₄×#92420	—	2	1	no	—
18	#16840783	C₂²	1	2	yes	—
19	#16840778	C₂²	1	1	yes	characteristic
20	#16840778	C₂²	1	1	yes	characteristic
21	#16840778	C₂²	1	2	yes	—
22	#16840765	C₂²	1	2	yes	—
23	C₂×C₄⋊#92420	C₂²	1	1	yes	characteristic
24	C₂×C₄⋊#92420	C₂²	1	1	yes	characteristic
25	C₂×C₄⋊#92420	C₂²	1	2	yes	—
26	#16840626	C₂²	1	2	yes	—
27	#16840626	C₂²	1	2	yes	—
28	#16840626	C₂²	1	2	yes	—
29	#16840625	C₂²	1	1	yes	characteristic
30	#16840625	C₂²	1	1	yes	characteristic
31	#16840625	C₂²	1	2	yes	—
32	#16840624	C₂²	1	2	yes	—
33	#16840623	C₂²	1	1	yes	characteristic
34	#16840623	C₂²	1	1	yes	characteristic
35	#16840623	C₂²	1	2	yes	—
36	#41×Dic₆₁	C₂²	1	1	yes	characteristic
37	#16839820	C₄	1	2	yes	—
38	#16839820	C₄	1	2	yes	—
39	#16839812	C₂²	1	2	yes	—
40	#16839812	C₂²	1	2	yes	—
41	#16839790	C₄	1	1	yes	characteristic
42	#16839788	C₄	1	1	yes	characteristic
43	#16839729	C₄	1	1	yes	characteristic
44	M₁₆×Dic₆₁	C₂²	1	1	yes	characteristic
45	C₂×C₈×Dic₆₁	C₂²	1	1	yes	characteristic
46	C₆₁×#408	C₄	1	1	yes	characteristic
47	C₂×#11758217	—	2	2	no	—
48	C₂²×#92420	—	2	1	no	—
49	C₂²×#92420	—	2	2	no	—
50	C₂²×#92420	—	2	1	no	—
51	#14519844	—	2	2	no	—
52	C₄⋊#92420	—	2	1	no	—
53	C₄⋊#92420	—	2	2	no	—
54	C₄⋊#92420	—	2	1	no	—
55	C₄⋊#92420	C₂³	1	2	yes	—
56	C₄⋊#92420	C₂³	1	1	yes	characteristic
57	C₄⋊#92420	C₂³	1	2	yes	—
58	C₄⋊#92420	C₂³	1	1	yes	characteristic
59	C₄⋊#92420	C₂³	1	1	yes	characteristic
60	C₄⋊#92420	C₂³	1	1	yes	characteristic
61	C₂²⋊#92420	—	2	1	no	—
62	C₂²⋊#92420	—	2	2	no	—
63	C₂²⋊#92420	—	2	1	no	—
64	C₂²⋊#92420	—	2	2	no	—
65	#14519832	—	2	2	no	—
66	C₄×#92420	—	2	1	no	—
67	C₄×#92420	—	2	2	no	—
68	C₄×#92420	—	2	1	no	—
69	C₂×#11758194	C₂×C₄	1	1	yes	characteristic
70	C₂²×#92418	C₂×C₄	1	1	yes	characteristic
71	Q₈×Dic₆₁	D₄	1	1	yes	characteristic
72	Q₈×Dic₆₁	Q₈	1	1	yes	characteristic
73	D₄×Dic₆₁	D₄	1	2	yes	—
74	D₄×Dic₆₁	D₄	1	1	yes	characteristic
75	D₄×Dic₆₁	Q₈	1	2	yes	—
76	D₄×Dic₆₁	Q₈	1	1	yes	characteristic
77	#14519736	C₂×C₄	1	1	yes	characteristic
78	#14519736	C₂×C₄	1	2	yes	—
79	C₂×C₄×Dic₆₁	C₂³	1	1	yes	characteristic
80	C₂×C₄×Dic₆₁	C₂³	1	2	yes	—
81	#14519706	C₂×C₄	1	2	yes	—
82	#14519706	C₂×C₄	1	2	yes	—
83	#14519706	C₂×C₄	1	2	yes	—
84	#14519706	C₂×C₄	1	2	yes	—
85	C₂²⋊#92418	C₂×C₄	1	2	yes	—
86	C₂²⋊#92418	C₂×C₄	1	2	yes	—
87	#14519687	C₂×C₄	1	2	yes	—
88	C₈×Dic₆₁	C₂×C₄	1	1	yes	characteristic
89	C₈×Dic₆₁	C₂×C₄	1	1	yes	characteristic
90	C₈×Dic₆₁	C₂³	1	1	yes	characteristic
91	C₈×Dic₆₁	C₂³	1	1	yes	characteristic
92	C₈×Dic₆₁	C₂³	1	2	yes	—
93	C₂×#41×C₆₁	C₂×C₄	1	1	yes	characteristic
94	C₂×M₁₆×C₆₁	C₂×C₄	1	1	yes	characteristic
95	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
96	C₂²⋊C₈×C₆₁	C₂×C₄	1	2	yes	—
97	C₂²⋊C₈×C₆₁	C₂×C₄	1	2	yes	—
98	#41×C₆₁	—	2	2	no	—
99	#41×C₆₁	—	2	2	no	—
100	C₂×Q₈×C₆₁	C₄⋊C₄	1	1	yes	characteristic
101	C₂×D₄×C₆₁	C₄⋊C₄	1	2	yes	—
102	C₂×D₄×C₆₁	C₄⋊C₄	1	1	yes	characteristic
103	C₂²×C₄×C₆₁	C₂²×C₄	1	2	yes	—
104	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
105	C₂²×Dic₆₁	C₂×D₄	1	1	yes	characteristic
106	C₂²×Dic₆₁	C₂×D₄	1	2	yes	—
107	C₂²×Dic₆₁	C₂×Q₈	1	2	yes	—
108	C₂²×Dic₆₁	C₂×Q₈	1	1	yes	characteristic
109	#11758217	—	8	2	no	—
110	C₂×#92420	—	2	1	no	—
111	C₂×#92420	—	2	1	no	—
112	C₂×#92420	—	2	1	no	—
113	C₂×#92420	—	2	2	no	—
114	C₂×#92420	—	2	1	no	—
115	C₂×#92420	—	2	2	no	—
116	C₂×#92420	—	8	1	no	—
117	C₂×#92420	—	8	2	no	—
118	C₂×#92420	—	8	1	no	—
119	M₁₆×C₆₁	—	2	2	no	—
120	C₂×C₈×C₆₁	—	2	2	no	—
121	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
122	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
123	C₂×C₈×C₆₁	C₂²×C₄	1	2	yes	—
124	C₂²⋊Dic₆₁	C₄⋊C₄	1	2	yes	—
125	C₂²⋊Dic₆₁	C₄⋊C₄	1	2	yes	—
126	C₂²⋊Dic₆₁	C₄⋊C₄	1	2	yes	—
127	C₄⋊Dic₆₁	C₄⋊C₄	1	2	yes	—
128	C₄⋊Dic₆₁	C₄⋊C₄	1	1	yes	characteristic
129	C₄⋊Dic₆₁	C₄⋊C₄	1	2	yes	—
130	C₄⋊Dic₆₁	C₄⋊C₄	1	1	yes	characteristic
131	C₄×Dic₆₁	C₂²×C₄	1	1	yes	characteristic
132	C₄×Dic₆₁	C₂²×C₄	1	2	yes	—
133	C₄×Dic₆₁	C₂×D₄	1	1	yes	characteristic
134	C₄×Dic₆₁	C₂×D₄	1	2	yes	—
135	C₄×Dic₆₁	C₂×Q₈	1	2	yes	—
136	C₄×Dic₆₁	C₂×Q₈	1	1	yes	characteristic
137	C₄×Dic₆₁	C₂⁴	1	1	yes	characteristic
138	#11758194	—	2	2	no	—
139	C₂×#92418	—	2	2	no	—
140	C₂×#92418	C₂²×C₄	1	1	yes	characteristic
141	C₂×#92418	C₂²×C₄	1	1	yes	characteristic
142	C₂×#92418	C₂²×C₄	1	2	yes	—
143	C₂³×C₆₁	C₂×C₄⋊C₄	1	2	yes	—
144	C₂³×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
145	Q₈×C₆₁	—	2	2	no	—
146	D₄×C₆₁	—	2	4	no	—
147	D₄×C₆₁	—	2	2	no	—
148	C₂×C₄×C₆₁	—	2	2	no	—
149	C₂×C₄×C₆₁	—	2	2	no	—
150	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	2	yes	—
151	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
152	C₂×C₄×C₆₁	#131	1	2	yes	—
153	C₂×C₄×C₆₁	#131	1	2	yes	—
154	C₂×C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
155	C₂×Dic₆₁	—	2	1	no	—
156	C₂×Dic₆₁	—	2	2	no	—
157	C₂×Dic₆₁	—	2	1	no	—
158	C₂×Dic₆₁	—	2	2	no	—
159	C₂×Dic₆₁	—	4	1	no	—
160	C₂×Dic₆₁	—	4	2	no	—
161	C₂×Dic₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
162	C₂×Dic₆₁	C₂×C₄⋊C₄	1	2	yes	—
163	C₂×Dic₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
164	C₂×Dic₆₁	C₂×C₄⋊C₄	1	2	yes	—
165	C₂×Dic₆₁	C₂²×D₄	1	1	yes	characteristic
166	C₂×Dic₆₁	C₂²×Q₈	1	1	yes	characteristic
167	#92420	—	4	2	no	—
168	#92420	—	4	1	no	—
169	#92420	—	4	2	no	—
170	#92420	—	4	1	no	—
171	#92420	—	4	1	no	—
172	#92420	—	4	1	no	—
173	C₈×C₆₁	—	2	1	no	—
174	C₈×C₆₁	—	2	1	no	—
175	C₈×C₆₁	—	2	2	no	—
176	#92418	—	2	1	no	—
177	#92418	—	2	1	no	—
178	#92418	—	2	2	no	—
179	#17812	—	61	1	no	maximal, Sylow
180	C₂²×C₆₁	—	2	1	no	—
181	C₂²×C₆₁	—	2	1	no	—
182	C₂²×C₆₁	—	2	2	no	—
183	C₂²×C₆₁	—	2	2	no	—
184	C₂²×C₆₁	—	2	2	no	—
185	C₂²×C₆₁	#424	1	2	yes	—
186	C₂²×C₆₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
187	C₄×C₆₁	—	2	2	no	—
188	C₄×C₆₁	#424	1	2	yes	—
189	C₄×C₆₁	C₂×#131	1	1	yes	characteristic
190	C₄×C₆₁	C₂×#131	1	1	yes	characteristic
191	Dic₆₁	—	2	1	no	—
192	Dic₆₁	—	2	2	no	—
193	Dic₆₁	—	2	1	no	—
194	Dic₆₁	—	2	1	no	—
195	Dic₆₁	—	2	2	no	—
196	Dic₆₁	#576	1	1	yes	characteristic
197	Dic₆₁	#578	1	1	yes	characteristic
198	#2989	—	61	1	no	—
199	#2988	—	61	1	no	—
200	#2922	—	61	1	no	—
201	#1574	—	61	2	no	—
202	#1573	—	61	2	no	—
203	#1572	—	61	2	no	—
204	#1571	—	61	2	no	—
205	#1565	—	61	1	no	—
206	#1565	—	61	1	no	—
207	#1564	—	61	1	no	—
208	C₂×#335	—	61	1	no	—
209	C₂×C₆₁	—	2	2	no	—
210	C₂×C₆₁	—	4	2	no	—
211	C₂×C₆₁	C₂×#424	1	1	yes	characteristic
212	C₂×C₆₁	#2913	1	1	yes	characteristic
213	C₂×C₆₁	#2916	1	1	yes	characteristic
214	C₄×#41	—	61	1	no	—
215	#446	—	122	1	no	—
216	C₈×Q₈	—	122	1	no	—
217	#436	—	122	1	no	—
218	#435	—	122	2	no	—
219	C₈×D₄	—	122	2	no	—
220	C₈×D₄	—	122	1	no	—
221	#433	—	61	2	no	—
222	#433	—	61	2	no	—
223	#432	—	61	2	no	—
224	#431	—	61	2	no	—
225	#431	—	61	2	no	—
226	#431	—	61	1	no	—
227	#431	—	61	1	no	—
228	#431	—	61	2	no	—
229	#424	—	61	1	no	—
230	#424	—	61	1	no	—
231	#423	—	61	2	no	—
232	C₂×C₄⋊C₈	—	61	1	no	—
233	C₂×C₄⋊C₈	—	61	1	no	—
234	C₂×C₄⋊C₈	—	61	2	no	—
235	#408	—	61	1	no	—
236	#408	#6923	1	1	yes	characteristic
237	C₄×M₁₆	—	61	1	no	—
238	C₂×C₄×C₈	—	61	1	no	—
239	#335	—	61	2	no	—
240	#335	—	61	2	no	—
241	#335	—	61	1	no	—
242	#335	—	61	1	no	—
243	#335	—	61	2	no	—
244	#335	—	61	2	no	—
245	#334	—	61	1	no	—
246	#334	—	61	1	no	—
247	#334	—	61	2	no	—
248	#334	—	61	2	no	—
249	C₆₁	#17812	1	1	yes	characteristic, complemented, Sylow
250	C₂×#41	C₂×#6923	1	1	yes	characteristic
251	C₂×M₁₆	—	61	1	no	—
252	C₂×M₁₆	—	122	2	no	—
253	C₂×M₁₆	C₂×#6923	1	1	yes	characteristic
254	C₂²×C₈	—	61	1	no	—
255	C₂²×C₈	—	122	1	no	—
256	C₂²×C₈	—	122	2	no	—
257	C₂²×C₈	—	122	1	no	—
258	C₂²×C₈	C₂×#6923	1	1	yes	characteristic
259	C₄×Q₈	—	61	1	no	—
260	C₄×Q₈	—	61	1	no	—
261	C₄×D₄	—	61	2	no	—
262	C₄×D₄	—	61	1	no	—
263	C₄×D₄	—	61	2	no	—
264	C₄×D₄	—	61	1	no	—
265	#117	—	61	1	no	—
266	#117	—	61	2	no	—
267	C₂×C₄²	—	61	1	no	—
268	C₂×C₄²	—	61	2	no	—
269	C₄⋊C₈	—	61	2	no	—
270	C₄⋊C₈	—	61	2	no	—
271	C₄⋊C₈	—	61	2	no	—
272	C₄⋊C₈	—	61	2	no	—
273	C₄⋊C₈	—	61	1	no	—
274	C₄⋊C₈	—	61	1	no	—
275	C₄⋊C₈	—	61	1	no	—
276	C₄⋊C₈	—	61	2	no	—
277	C₄⋊C₈	—	61	2	no	—
278	C₄⋊C₈	—	61	1	no	—
279	C₄⋊C₈	—	122	1	no	—
280	C₄⋊C₈	—	122	2	no	—
281	C₄⋊C₈	—	122	1	no	—
282	C₄⋊C₈	—	122	2	no	—
283	C₂²⋊C₈	—	61	2	no	—
284	C₂²⋊C₈	—	61	2	no	—
285	C₂²⋊C₈	—	122	1	no	—
286	C₂²⋊C₈	—	122	2	no	—
287	C₂²⋊C₈	—	122	1	no	—
288	C₂²⋊C₈	—	122	2	no	—
289	C₂²⋊C₈	C₂×#6923	1	2	yes	—
290	C₂²⋊C₈	C₂×#6923	1	2	yes	—
291	#97	—	61	2	no	—
292	#97	—	122	2	no	—
293	C₄×C₈	—	61	1	no	—
294	C₄×C₈	—	61	2	no	—
295	C₄×C₈	—	61	1	no	—
296	C₄×C₈	—	61	1	no	—
297	C₄×C₈	—	61	1	no	—
298	C₄×C₈	—	122	1	no	—
299	C₄×C₈	—	122	2	no	—
300	C₄×C₈	—	122	1	no	—
301	#41	—	2	2	no	—
302	#41	—	2	2	no	—
303	C₂×Q₈	C₄⋊#6923	1	1	yes	characteristic
304	C₂×D₄	C₄⋊#6923	1	2	yes	—
305	C₂×D₄	C₄⋊#6923	1	1	yes	characteristic
306	C₂²×C₄	—	61	1	no	—
307	C₂²×C₄	—	61	2	no	—
308	C₂²×C₄	—	61	2	no	—
309	C₂²×C₄	—	61	1	no	—
310	C₂²×C₄	C₂²×#6923	1	2	yes	—
311	C₂²×C₄	C₂²×#6923	1	1	yes	characteristic
312	M₁₆	—	2	2	no	—
313	M₁₆	—	122	2	no	—
314	M₁₆	—	488	2	no	—
315	C₂×C₈	—	2	2	no	—
316	C₂×C₈	—	61	1	no	—
317	C₂×C₈	—	61	1	no	—
318	C₂×C₈	—	61	2	no	—
319	C₂×C₈	—	122	2	no	—
320	C₂×C₈	—	122	1	no	—
321	C₂×C₈	—	122	1	no	—
322	C₂×C₈	—	122	1	no	—
323	C₂×C₈	—	122	2	no	—
324	C₂×C₈	—	122	1	no	—
325	C₂×C₈	—	122	2	no	—
326	C₂×C₈	—	488	1	no	—
327	C₂×C₈	—	488	2	no	—
328	C₂×C₈	—	488	1	no	—
329	C₂×C₈	C₂²×#6923	1	1	yes	characteristic
330	C₂×C₈	C₂²×#6923	1	1	yes	characteristic
331	C₂×C₈	C₂²×#6923	1	2	yes	—
332	C₄⋊C₄	—	61	2	no	—
333	C₄⋊C₄	—	61	2	no	—
334	C₄⋊C₄	—	61	1	no	—
335	C₄⋊C₄	—	61	1	no	—
336	C₂²⋊C₄	—	61	2	no	—
337	C₂²⋊C₄	—	61	2	no	—
338	C₂²⋊C₄	—	61	2	no	—
339	C₄²	—	61	1	no	—
340	C₄²	—	61	2	no	—
341	C₄²	—	61	2	no	—
342	C₄²	—	61	1	no	—
343	C₄²	—	61	1	no	—
344	C₄²	—	61	1	no	—
345	C₄²	—	61	2	no	—
346	C₂³	C₂×C₄⋊#6923	1	2	yes	—
347	C₂³	C₂×C₄⋊#6923	1	1	yes	characteristic
348	Q₈	—	2	2	no	—
349	D₄	—	2	4	no	—
350	D₄	—	2	2	no	—
351	C₂×C₄	—	2	2	no	—
352	C₂×C₄	—	2	2	no	—
353	C₂×C₄	—	61	1	no	—
354	C₂×C₄	—	61	1	no	—
355	C₂×C₄	—	61	2	no	—
356	C₂×C₄	—	61	1	no	—
357	C₂×C₄	—	61	2	no	—
358	C₂×C₄	—	61	1	no	—
359	C₂×C₄	—	122	1	no	—
360	C₂×C₄	—	122	2	no	—
361	C₂×C₄	—	122	1	no	—
362	C₂×C₄	—	122	2	no	—
363	C₂×C₄	—	244	1	no	—
364	C₂×C₄	—	244	2	no	—
365	C₂×C₄	C₂×C₄⋊#6923	1	2	yes	—
366	C₂×C₄	C₂×C₄⋊#6923	1	1	yes	characteristic
367	C₂×C₄	#14519864	1	2	yes	—
368	C₂×C₄	#14519864	1	2	yes	—
369	C₂×C₄	C₂³×#6923	1	1	yes	characteristic
370	C₈	—	2	1	no	—
371	C₈	—	2	1	no	—
372	C₈	—	2	2	no	—
373	C₈	—	122	1	no	—
374	C₈	—	122	1	no	—
375	C₈	—	122	2	no	—
376	C₈	—	244	1	no	—
377	C₈	—	244	1	no	—
378	C₈	—	244	2	no	—
379	C₈	—	244	1	no	—
380	C₈	—	244	2	no	—
381	C₈	—	244	1	no	—
382	C₂²	—	2	1	no	—
383	C₂²	—	2	1	no	—
384	C₂²	—	2	2	no	—
385	C₂²	—	2	2	no	—
386	C₂²	—	2	2	no	—
387	C₂²	#16840768	1	2	yes	—
388	C₂²	C₂²×C₄⋊#6923	1	1	yes	characteristic
389	C₄	—	2	2	no	—
390	C₄	—	61	1	no	—
391	C₄	—	61	1	no	—
392	C₄	—	122	1	no	—
393	C₄	—	122	2	no	—
394	C₄	—	122	1	no	—
395	C₄	—	122	2	no	—
396	C₄	—	122	1	no	—
397	C₄	#16840768	1	2	yes	—
398	C₄	C₂×#14519864	1	1	yes	characteristic
399	C₄	C₂×#14519864	1	1	yes	characteristic
400	C₂	—	2	2	no	—
401	C₂	—	4	2	no	—
402	C₂	C₂×#16840768	1	1	yes	characteristic
403	C₂	#21462474	1	1	yes	characteristic
404	C₂	#21462480	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×C₄⋊#6923
commutator	C₂×C₄×C₆₁	C₂³×C₄
Frattini	C₂×C₄	C₂³×#6923
Fitting	C₆₁×#408	C₄
socle	C₂²×C₆₁	C₂²×C₄⋊C₄
prime radical	C₂×#41×C₆₁	C₂×C₄
cosocle	C₄×Dic₆₁	C₂⁴
hypercenter	#408	#6923
hypocenter	C₆₁	#17812
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₂×#41 ◁ #408

Complements

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

N	H ≅ G/N	type
3. #21461395	C₂	semidirect
4. #21461392	C₂	semidirect
5. #21461390	C₂	semidirect
6. #21461388	C₂	semidirect
7. #21461372	C₂	semidirect
8. #21461370	C₂	semidirect
9. #21461366	C₂	semidirect
10. C₂×#16840626	C₂	semidirect
249. C₆₁	#17812	semidirect

Maximal subgroups

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

subgroup	index
#21462738	2
#21462734	2
#21461395	2
#21461392	2
#21461390	2
#21461388	2
#21461372	2
#21461370	2
#21461366	2
C₂×#16840626	2
#21454459	2
#17812	61