Subgroups — #30602143

Order 15616.

566 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21461752	C₂	1	1	yes	maximal, characteristic
2	C₄²×#92420	C₂	1	1	yes	maximal, characteristic
3	#21460655	C₂	1	2	yes	maximal
4	#21460655	C₂	1	2	yes	maximal
5	#21448390	C₂	1	1	yes	maximal, characteristic
6	C₂×#14519844	—	2	1	no	—
7	C₂×#14519844	—	2	1	no	—
8	C₂×C₄⋊#92420	C₂²	1	2	yes	—
9	C₂×C₄×#92420	—	2	1	no	—
10	C₂×C₄×#92420	—	2	1	no	—
11	C₂×C₄×#92420	C₂²	1	2	yes	—
12	C₂×#14519776	C₄	1	1	yes	characteristic
13	C₂×#14519742	—	2	1	no	—
14	#16839666	—	2	1	no	—
15	#16839660	C₄	1	1	yes	characteristic
16	#16839657	—	2	1	no	—
17	#16839631	C₄	1	1	yes	characteristic
18	C₄⋊C₄×Dic₆₁	C₂²	1	1	yes	characteristic
19	C₄⋊C₄×Dic₆₁	C₂²	1	1	yes	characteristic
20	#16839613	—	2	1	no	—
21	C₄²×Dic₆₁	C₂²	1	1	yes	characteristic
22	C₆₁×#389	C₄	1	1	yes	characteristic
23	C₂²×#92420	—	2	2	no	—
24	C₂²×#92420	—	2	2	no	—
25	#14519844	—	4	2	no	—
26	#14519844	—	4	2	no	—
27	C₄⋊#92420	—	2	4	no	—
28	C₄×#92420	—	2	4	no	—
29	C₄×#92420	—	2	4	no	—
30	C₄×#92420	—	2	2	no	—
31	C₄×#92420	Q₈	1	4	yes	complemented
32	C₂²×Dic₁₂₂	—	2	1	no	—
33	#14519776	—	2	2	no	—
34	#14519776	C₈	1	4	yes	complemented
35	#14519742	—	4	2	no	—
36	#14519742	—	4	2	no	—
37	C₂×C₄⋊Dic₆₁	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄⋊Dic₆₁	C₂×C₄	1	1	yes	characteristic
39	C₂×#11758196	—	2	1	no	—
40	C₂×#11758196	—	2	1	no	—
41	C₂×#11758196	—	2	1	no	—
42	C₂×#11758196	—	2	1	no	—
43	C₂×C₄×Dic₆₁	—	2	1	no	—
44	C₂×C₄×Dic₆₁	—	2	1	no	—
45	C₂×C₄×Dic₆₁	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄×Dic₆₁	D₄	1	1	yes	characteristic
47	C₂×C₄×Dic₆₁	D₄	1	1	yes	characteristic
48	C₂×C₄×Dic₆₁	D₄	1	1	yes	characteristic
49	C₂×C₄×Dic₆₁	D₄	1	1	yes	characteristic
50	C₂×C₄×Dic₆₁	C₂³	1	1	yes	characteristic
51	#14519684	—	2	1	no	—
52	#14519684	—	2	1	no	—
53	#14519684	—	2	1	no	—
54	C₂×C₄⋊C₄×C₆₁	—	2	1	no	—
55	C₂×C₄⋊C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
56	C₂×C₄⋊C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
57	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
58	#95×C₆₁	—	2	1	no	—
59	C₂²×C₄×C₆₁	—	2	1	no	—
60	C₂²×C₄×C₆₁	—	2	1	no	—
61	C₂²×C₄×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
62	C₂²×C₄×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
63	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
64	C₂²×Dic₆₁	—	2	1	no	—
65	C₂²×Dic₆₁	—	2	1	no	—
66	C₂²×Dic₆₁	C₂²⋊C₄	1	1	yes	characteristic
67	C₂²×Dic₆₁	C₂²⋊C₄	1	1	yes	characteristic
68	C₂²×Dic₆₁	C₂×D₄	1	1	yes	characteristic
69	C₂²×Dic₆₁	C₂×D₄	1	1	yes	characteristic
70	C₂×Dic₁₂₂	—	2	4	no	—
71	C₂×Dic₁₂₂	—	4	2	no	—
72	C₂×Dic₁₂₂	—	4	2	no	—
73	C₂×#92420	—	2	4	no	—
74	C₂×#92420	—	2	4	no	—
75	C₂×#92420	—	2	4	no	—
76	C₂×#92420	—	4	2	no	—
77	C₂×#92420	—	4	2	no	—
78	C₂×#92420	—	4	2	no	—
79	C₄⋊C₄×C₆₁	—	2	1	no	—
80	C₄⋊C₄×C₆₁	—	2	2	no	—
81	C₄⋊C₄×C₆₁	—	4	2	no	—
82	C₄⋊C₄×C₆₁	C₂×C₈	1	2	yes	—
83	C₄²×C₆₁	—	2	2	no	—
84	C₄⋊Dic₆₁	—	2	1	no	—
85	C₄⋊Dic₆₁	—	2	2	no	—
86	C₄⋊Dic₆₁	C₂×C₈	1	2	yes	—
87	#11758196	—	2	4	no	—
88	#11758196	—	4	2	no	—
89	#11758196	—	4	2	no	—
90	#11758196	—	4	2	no	—
91	C₄×Dic₆₁	—	2	2	no	—
92	C₄×Dic₆₁	—	2	2	no	—
93	C₄×Dic₆₁	—	4	1	no	—
94	C₄×Dic₆₁	—	4	1	no	—
95	C₄×Dic₆₁	—	4	1	no	—
96	C₄×Dic₆₁	—	4	1	no	—
97	C₄×Dic₆₁	C₂×C₈	1	2	yes	—
98	C₄×Dic₆₁	M₁₆	1	2	yes	—
99	C₄×Dic₆₁	D₈	1	2	yes	—
100	C₄×Dic₆₁	SD₁₆	1	2	yes	—
101	C₄×Dic₆₁	SD₁₆	1	2	yes	—
102	C₄×Dic₆₁	Q₁₆	1	2	yes	—
103	C₄×Dic₆₁	C₂×Q₈	1	2	yes	—
104	C₄×Dic₆₁	#41	1	2	yes	—
105	C₂³×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
106	C₂×C₄×C₆₁	—	2	1	no	—
107	C₂×C₄×C₆₁	—	2	1	no	—
108	C₂×C₄×C₆₁	—	2	1	no	—
109	C₂×C₄×C₆₁	—	2	1	no	—
110	C₂×C₄×C₆₁	—	2	2	no	—
111	C₂×C₄×C₆₁	—	4	1	no	—
112	C₂×C₄×C₆₁	—	4	1	no	—
113	C₂×C₄×C₆₁	—	4	1	no	—
114	C₂×C₄×C₆₁	—	4	1	no	—
115	C₂×C₄×C₆₁	—	4	1	no	—
116	C₂×C₄×C₆₁	#102	1	2	yes	—
117	C₂×C₄×C₆₁	#103	1	2	yes	—
118	C₂×C₄×C₆₁	C₄×Q₈	1	2	yes	—
119	C₂×C₄×C₆₁	C₂²×C₈	1	1	yes	characteristic
120	C₂×C₄×C₆₁	C₂×M₁₆	1	1	yes	characteristic
121	C₂×C₄×C₆₁	#131	1	2	yes	—
122	C₂×Dic₆₁	—	2	1	no	—
123	C₂×Dic₆₁	—	2	1	no	—
124	C₂×Dic₆₁	—	2	2	no	—
125	C₂×Dic₆₁	—	4	1	no	—
126	C₂×Dic₆₁	—	4	1	no	—
127	C₂×Dic₆₁	—	4	1	no	—
128	C₂×Dic₆₁	—	4	1	no	—
129	C₂×Dic₆₁	—	4	1	no	—
130	C₂×Dic₆₁	C₂²⋊C₈	1	2	yes	—
131	C₂×Dic₆₁	C₂²⋊C₈	1	2	yes	—
132	C₂×Dic₆₁	#102	1	2	yes	—
133	C₂×Dic₆₁	#103	1	2	yes	—
134	C₂×Dic₆₁	#104	1	2	yes	—
135	C₂×Dic₆₁	#104	1	2	yes	—
136	C₂×Dic₆₁	C₂²⋊Q₈	1	2	yes	—
137	C₂×Dic₆₁	#124	1	1	yes	characteristic
138	C₂×Dic₆₁	C₄⋊Q₈	1	1	yes	characteristic
139	C₂×Dic₆₁	C₂×D₈	1	1	yes	characteristic
140	C₂×Dic₆₁	C₂×SD₁₆	1	1	yes	characteristic
141	C₂×Dic₆₁	C₂×SD₁₆	1	1	yes	characteristic
142	C₂×Dic₆₁	C₂×Q₁₆	1	1	yes	characteristic
143	Dic₁₂₂	—	4	8	no	—
144	#92420	—	2	8	no	—
145	#92420	—	8	2	no	—
146	#9870	—	61	1	no	maximal, Sylow
147	C₂²×C₆₁	#391	1	1	yes	characteristic
148	C₂²×C₆₁	C₂×C₂²⋊C₈	1	1	yes	characteristic
149	C₂²×C₆₁	#408	1	1	yes	characteristic
150	C₂²×C₆₁	C₂×#102	1	1	yes	characteristic
151	C₂²×C₆₁	C₂×#103	1	1	yes	characteristic
152	C₂²×C₆₁	C₂×#104	1	1	yes	characteristic
153	C₂²×C₆₁	C₂×#104	1	1	yes	characteristic
154	C₄×C₆₁	—	2	2	no	—
155	C₄×C₆₁	—	2	2	no	—
156	C₄×C₆₁	—	4	2	no	—
157	C₄×C₆₁	—	8	1	no	—
158	C₄×C₆₁	C₈×Q₈	1	2	yes	—
159	C₄×C₆₁	#446	1	2	yes	—
160	Dic₆₁	—	2	2	no	—
161	Dic₆₁	—	4	2	no	—
162	Dic₆₁	—	8	1	no	—
163	Dic₆₁	D₄⋊C₈	1	4	yes	—
164	Dic₆₁	Q₈⋊C₈	1	4	yes	—
165	Dic₆₁	#486	1	1	yes	characteristic
166	Dic₆₁	#487	1	1	yes	characteristic
167	Dic₆₁	#498	1	1	yes	characteristic
168	Dic₆₁	#500	1	1	yes	characteristic
169	C₄×C₄⋊Q₈	—	61	1	no	—
170	#1842	—	61	1	no	—
171	C₄²×C₈	—	61	1	no	—
172	#1279	—	61	2	no	—
173	#1279	—	61	2	no	—
174	C₂×C₆₁	C₂×D₄⋊C₈	1	1	yes	characteristic
175	C₂×C₆₁	C₂×Q₈⋊C₈	1	1	yes	characteristic
176	C₂×C₆₁	#1492	1	1	yes	characteristic
177	C₂×C₆₁	#1493	1	1	yes	characteristic
178	C₂×C₆₁	#1984	1	1	yes	characteristic
179	C₂×C₆₁	#1990	1	1	yes	characteristic
180	C₂×C₆₁	#1994	1	1	yes	characteristic
181	C₂×C₄⋊Q₈	—	61	1	no	—
182	C₂×C₄×Q₈	—	122	1	no	—
183	C₂×C₄⋊C₈	—	61	2	no	—
184	C₂×C₄⋊C₈	—	122	1	no	—
185	C₂×C₄⋊C₈	—	122	1	no	—
186	C₂×C₄×C₈	—	61	2	no	—
187	C₂×C₄×C₈	—	122	1	no	—
188	C₂×C₄×C₈	—	122	1	no	—
189	#391	—	122	1	no	—
190	#389	—	61	1	no	—
191	#389	—	122	1	no	—
192	#389	#6923	1	1	yes	characteristic
193	#384	—	61	1	no	—
194	C₄×C₄⋊C₄	—	61	1	no	—
195	C₄×C₄⋊C₄	—	61	1	no	—
196	C₄×C₄⋊C₄	—	122	1	no	—
197	C₄³	—	61	1	no	—
198	C₆₁	#9870	1	1	yes	characteristic, complemented, Sylow
199	C₂²×Q₈	—	122	1	no	—
200	C₂²×C₈	—	122	2	no	—
201	C₂²×C₈	—	122	2	no	—
202	C₄⋊Q₈	—	61	4	no	—
203	C₄⋊Q₈	—	122	2	no	—
204	C₄×Q₈	—	244	2	no	—
205	C₄×Q₈	—	244	2	no	—
206	C₂×C₄⋊C₄	—	2	1	no	—
207	C₂×C₄⋊C₄	—	61	1	no	—
208	C₂×C₄⋊C₄	—	61	1	no	—
209	C₂×C₄⋊C₄	—	122	1	no	—
210	C₂×C₄⋊C₄	—	122	1	no	—
211	C₂×C₄⋊C₄	—	122	1	no	—
212	C₂×C₄⋊C₄	—	122	1	no	—
213	C₂×C₄⋊C₄	C₂×#6923	1	1	yes	characteristic
214	C₂×C₄⋊C₄	C₂×#6923	1	1	yes	characteristic
215	C₂×C₄²	—	61	1	no	—
216	C₂×C₄²	—	61	1	no	—
217	C₂×C₄²	—	61	1	no	—
218	C₂×C₄²	—	61	1	no	—
219	C₂×C₄²	—	61	1	no	—
220	C₂×C₄²	—	61	1	no	—
221	C₂×C₄²	—	122	1	no	—
222	C₂×C₄²	—	122	1	no	—
223	C₂×C₄²	C₂×#6923	1	1	yes	characteristic
224	C₄⋊C₈	—	122	4	no	—
225	C₄⋊C₈	—	244	2	no	—
226	C₄⋊C₈	—	244	2	no	—
227	C₄×C₈	—	61	4	no	—
228	C₄×C₈	—	122	2	no	—
229	C₄×C₈	—	122	4	no	—
230	C₄×C₈	—	122	4	no	—
231	#95	—	2	1	no	—
232	#95	—	122	1	no	—
233	#95	—	122	1	no	—
234	#95	—	122	1	no	—
235	C₂×Q₈	—	122	4	no	—
236	C₂×Q₈	—	244	2	no	—
237	C₂×Q₈	—	244	2	no	—
238	C₂²×C₄	—	2	1	no	—
239	C₂²×C₄	—	2	1	no	—
240	C₂²×C₄	—	61	1	no	—
241	C₂²×C₄	—	61	1	no	—
242	C₂²×C₄	—	61	1	no	—
243	C₂²×C₄	—	61	1	no	—
244	C₂²×C₄	—	122	1	no	—
245	C₂²×C₄	—	122	1	no	—
246	C₂²×C₄	C₂²⋊#6923	1	1	yes	characteristic
247	C₂²×C₄	C₂²⋊#6923	1	1	yes	characteristic
248	C₂²×C₄	C₂²×#6923	1	1	yes	characteristic
249	C₂×C₈	—	122	4	no	—
250	C₂×C₈	—	122	4	no	—
251	C₂×C₈	—	122	4	no	—
252	C₂×C₈	—	244	2	no	—
253	C₂×C₈	—	244	2	no	—
254	C₂×C₈	—	244	2	no	—
255	C₄⋊C₄	—	2	1	no	—
256	C₄⋊C₄	—	2	2	no	—
257	C₄⋊C₄	—	4	2	no	—
258	C₄⋊C₄	—	61	2	no	—
259	C₄⋊C₄	—	122	4	no	—
260	C₄⋊C₄	—	122	1	no	—
261	C₄⋊C₄	—	122	2	no	—
262	C₄⋊C₄	—	244	2	no	—
263	C₄⋊C₄	—	244	2	no	—
264	C₄⋊C₄	—	244	2	no	—
265	C₄⋊C₄	D₆₁⋊C₈	1	2	yes	—
266	C₄²	—	2	2	no	—
267	C₄²	—	61	2	no	—
268	C₄²	—	61	2	no	—
269	C₄²	—	61	2	no	—
270	C₄²	—	61	2	no	—
271	C₄²	—	61	2	no	—
272	C₄²	—	61	2	no	—
273	C₄²	—	61	2	no	—
274	C₄²	—	61	2	no	—
275	C₄²	—	122	2	no	—
276	C₄²	—	122	2	no	—
277	C₄²	—	244	1	no	—
278	C₄²	—	244	1	no	—
279	C₄²	—	244	1	no	—
280	C₄²	—	244	1	no	—
281	C₂³	C₂×C₂²⋊#6923	1	1	yes	characteristic
282	Q₈	—	244	8	no	—
283	C₂×C₄	—	2	1	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	2	1	no	—
286	C₂×C₄	—	2	1	no	—
287	C₂×C₄	—	2	2	no	—
288	C₂×C₄	—	4	1	no	—
289	C₂×C₄	—	4	1	no	—
290	C₂×C₄	—	4	1	no	—
291	C₂×C₄	—	4	1	no	—
292	C₂×C₄	—	4	1	no	—
293	C₂×C₄	—	61	1	no	—
294	C₂×C₄	—	61	1	no	—
295	C₂×C₄	—	61	2	no	—
296	C₂×C₄	—	61	2	no	—
297	C₂×C₄	—	61	2	no	—
298	C₂×C₄	—	61	1	no	—
299	C₂×C₄	—	61	1	no	—
300	C₂×C₄	—	61	2	no	—
301	C₂×C₄	—	61	1	no	—
302	C₂×C₄	—	61	1	no	—
303	C₂×C₄	—	61	2	no	—
304	C₂×C₄	—	61	2	no	—
305	C₂×C₄	—	61	2	no	—
306	C₂×C₄	—	122	1	no	—
307	C₂×C₄	—	122	1	no	—
308	C₂×C₄	—	122	2	no	—
309	C₂×C₄	—	244	1	no	—
310	C₂×C₄	—	244	1	no	—
311	C₂×C₄	—	244	1	no	—
312	C₂×C₄	—	244	1	no	—
313	C₂×C₄	—	244	1	no	—
314	C₂×C₄	#14519839	1	2	yes	—
315	C₂×C₄	#14519840	1	2	yes	—
316	C₂×C₄	Q₈×#6923	1	2	yes	—
317	C₂×C₄	C₂×D₆₁⋊C₈	1	1	yes	characteristic
318	C₂×C₄	D₆₁⋊M₁₆	1	1	yes	characteristic
319	C₂×C₄	#14519864	1	2	yes	—
320	C₈	—	122	8	no	—
321	C₈	—	488	2	no	—
322	C₂²	(C₂×#6923)⋊Q₈	1	1	yes	characteristic
323	C₂²	D₆₁⋊(C₂²⋊C₈)	1	1	yes	characteristic
324	C₂²	#16840735	1	1	yes	characteristic
325	C₂²	C₂×#14519839	1	1	yes	characteristic
326	C₂²	C₂×#14519840	1	1	yes	characteristic
327	C₂²	D₆₁⋊#104	1	1	yes	characteristic
328	C₂²	D₆₁⋊#104	1	1	yes	characteristic
329	C₄	—	2	2	no	—
330	C₄	—	2	2	no	—
331	C₄	—	4	2	no	—
332	C₄	—	8	1	no	—
333	C₄	—	61	4	no	—
334	C₄	—	61	4	no	—
335	C₄	—	61	1	no	—
336	C₄	—	61	1	no	—
337	C₄	—	61	1	no	—
338	C₄	—	61	1	no	—
339	C₄	—	122	2	no	—
340	C₄	—	488	1	no	—
341	C₄	—	244	2	no	—
342	C₄	#16840811	1	2	yes	—
343	C₄	#16840814	1	2	yes	—
344	C₂	D₆₁⋊(D₄⋊C₈)	1	1	yes	characteristic
345	C₂	D₆₁⋊(Q₈⋊C₈)	1	1	yes	characteristic
346	C₂	#21461164	1	1	yes	characteristic
347	C₂	#21461167	1	1	yes	characteristic
348	C₂	#21461990	1	1	yes	characteristic
349	C₂	#21462000	1	1	yes	characteristic
350	C₂	#21462005	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊#6923
commutator	C₂×C₄×C₆₁	C₂²×C₈
Frattini	C₂²×C₄	C₂²×#6923
Fitting	C₆₁×#389	C₄
socle, prime radical	C₂³×C₆₁	C₂×C₂²⋊C₄
cosocle	C₂×C₄×Dic₆₁	C₂³
hypercenter	#389	#6923
hypocenter	C₆₁	#9870
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #389

Complements

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

N	H ≅ G/N	type
31. C₄×#92420	Q₈	semidirect
34. #14519776	C₈	semidirect
198. C₆₁	#9870	semidirect

Maximal subgroups

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

subgroup	index
#21461752	2
C₄²×#92420	2
#21460655	2
#21448390	2
#9870	61