Subgroups — #29688636

Order 15616.

998 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21456247	C₂	1	1	yes	maximal, characteristic
2	C₂×#16839883	C₂	1	2	yes	maximal, complemented
3	C₂×#16839878	C₂	1	2	yes	maximal, complemented
4	#21454536	C₂	1	2	yes	maximal, complemented
5	#21446710	C₂	1	2	yes	maximal, complemented
6	#21446705	C₂	1	2	yes	maximal, complemented
7	#21446650	C₂	1	2	yes	maximal, complemented
8	#21445505	C₂	1	1	yes	maximal, characteristic, complemented
9	C₆₁×#1848	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#14519799	C₂²	1	2	yes	—
11	C₂×#14519799	C₂²	1	2	yes	—
12	C₂×#14519785	C₂²	1	4	yes	—
13	#16840312	C₂²	1	2	yes	—
14	#16840146	C₄	1	2	yes	complemented
15	#16840146	C₄	1	2	yes	complemented
16	#16840142	C₄	1	1	yes	characteristic, complemented
17	#16840139	C₄	1	1	yes	characteristic, complemented
18	#16840135	C₄	1	2	yes	complemented
19	#16839883	—	2	8	no	—
20	#16839878	—	2	8	no	—
21	#16839832	—	2	8	no	—
22	C₂×#14519703	C₂²	1	4	yes	—
23	C₂×#14519703	C₂²	1	4	yes	—
24	#16839730	C₂²	1	4	yes	—
25	C₂×#14519688	C₂²	1	1	yes	characteristic
26	C₂×#14519687	C₂²	1	1	yes	characteristic
27	C₂×#14519687	C₂²	1	1	yes	characteristic
28	C₂×C₈×Dic₆₁	C₂²	1	1	yes	characteristic
29	C₂×C₈×Dic₆₁	C₂²	1	1	yes	characteristic
30	#16839608	C₂²	1	1	yes	characteristic
31	C₂×#108×C₆₁	C₂²	1	2	yes	—
32	C₂×#108×C₆₁	C₂²	1	2	yes	—
33	C₆₁×#423	C₂²	1	2	yes	—
34	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
35	C₂×#11758222	C₂³	1	4	yes	—
36	#14519799	—	2	4	no	—
37	#14519799	—	2	4	no	—
38	#14519799	—	2	4	no	—
39	#14519799	—	2	4	no	—
40	C₂×Dic₂₄₄	C₂×C₄	1	2	yes	—
41	C₂×Dic₂₄₄	C₂×C₄	1	2	yes	—
42	C₂×#11758190	C₂×C₄	1	4	yes	—
43	C₂×#11758190	C₂×C₄	1	4	yes	—
44	C₂×D₄₈₈	C₂×C₄	1	2	yes	—
45	C₂×D₄₈₈	C₂×C₄	1	2	yes	—
46	#14519785	—	2	8	no	—
47	#14519785	—	2	8	no	—
48	C₂×#11758189	—	2	4	no	—
49	C₂²×#92418	C₂³	1	1	yes	characteristic
50	C₂²×#92418	C₂³	1	1	yes	characteristic
51	C₂×C₈×D₆₁	—	2	4	no	—
52	#14519777	C₂×C₄	1	2	yes	—
53	#14519774	C₂×C₄	1	2	yes	—
54	#14519765	C₂×C₄	1	4	yes	—
55	#14519751	—	2	4	no	—
56	C₄×D₂₄₄	—	2	4	no	—
57	C₂×C₄×Dic₆₁	C₂³	1	1	yes	characteristic
58	C₂×C₄×Dic₆₁	C₂³	1	1	yes	characteristic
59	#14519703	—	2	8	no	—
60	#14519703	—	2	8	no	—
61	#14519689	—	2	8	no	—
62	#14519688	D₄	1	4	yes	—
63	#14519687	—	2	2	no	—
64	#14519687	D₄	1	4	yes	complemented
65	C₈×Dic₆₁	—	2	2	no	—
66	C₈×Dic₆₁	D₄	1	4	yes	complemented
67	#14519684	—	2	2	no	—
68	C₂×M₁₆×C₆₁	C₂³	1	4	yes	—
69	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
70	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
71	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
72	#108×C₆₁	—	2	4	no	—
73	#108×C₆₁	—	2	4	no	—
74	C₄⋊C₈×C₆₁	—	2	4	no	—
75	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
76	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
77	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
78	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
79	C₂²×C₄×C₆₁	#41	1	2	yes	—
80	C₂²×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
81	C₂×C₂²⋊D₆₁	—	2	4	no	—
82	C₂²×Dic₆₁	—	2	1	no	—
83	C₂²×Dic₆₁	—	2	1	no	—
84	#11758222	—	2	4	no	—
85	#11758222	—	2	4	no	—
86	#11758222	—	2	4	no	—
87	#11758222	—	2	4	no	—
88	C₂×D₂₄₄	—	2	4	no	—
89	C₂×D₂₄₄	C₂²×C₄	1	4	yes	—
90	C₂×C₄×D₆₁	—	2	4	no	—
91	C₂×Dic₁₂₂	C₂²×C₄	1	4	yes	—
92	M₁₆×C₆₁	—	2	4	no	—
93	M₁₆×C₆₁	—	2	4	no	—
94	C₂×C₈×C₆₁	—	2	4	no	—
95	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
96	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
97	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
98	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
99	C₂×C₈×C₆₁	C₂×D₄	1	2	yes	—
100	C₂×C₈×C₆₁	C₂×D₄	1	2	yes	—
101	C₂×C₈×C₆₁	#41	1	2	yes	—
102	C₂×C₈×C₆₁	#41	1	2	yes	—
103	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
104	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
105	C₄²×C₆₁	C₂×D₄	1	2	yes	—
106	#11758198	—	2	8	no	—
107	C₄⋊Dic₆₁	—	2	4	no	—
108	C₄×Dic₆₁	—	2	1	no	—
109	C₄×Dic₆₁	—	2	1	no	—
110	C₄×Dic₆₁	C₂×D₄	1	2	yes	—
111	C₄×Dic₆₁	C₂×D₄	1	2	yes	—
112	C₂×#92418	—	2	1	no	—
113	C₂×#92418	—	2	1	no	—
114	C₂×#92418	C₂×D₄	1	2	yes	—
115	C₂×#92418	C₂×D₄	1	2	yes	—
116	C₂×#92418	C₂×D₄	1	2	yes	—
117	C₂×#92418	C₂×D₄	1	2	yes	—
118	Dic₂₄₄	—	2	4	no	—
119	Dic₂₄₄	—	2	4	no	—
120	D₄₈₈	—	2	4	no	—
121	D₄₈₈	—	2	4	no	—
122	#11758190	—	2	8	no	—
123	#11758190	—	2	8	no	—
124	#11758189	—	4	8	no	—
125	C₈×D₆₁	—	4	8	no	—
126	C₂³×C₆₁	C₂×#41	1	1	yes	characteristic
127	C₂²×D₆₁	—	2	4	no	—
128	C₂×C₄×C₆₁	—	4	2	no	—
129	C₂×C₄×C₆₁	C₄×D₄	1	2	yes	—
130	C₂×C₄×C₆₁	C₄×D₄	1	2	yes	—
131	C₂×C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
132	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
133	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
134	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
135	C₂×C₄×C₆₁	C₂×#41	1	1	yes	characteristic
136	C₂×C₄×C₆₁	C₂×#41	1	1	yes	characteristic
137	C₂²⋊D₆₁	—	4	4	no	—
138	C₂²⋊D₆₁	—	4	4	no	—
139	C₂×Dic₆₁	—	2	4	no	—
140	C₂×Dic₆₁	—	2	2	no	—
141	C₂×Dic₆₁	—	2	2	no	—
142	C₂×Dic₆₁	—	4	2	no	—
143	C₂×Dic₆₁	—	4	2	no	—
144	D₂₄₄	—	2	4	no	—
145	D₂₄₄	—	2	4	no	—
146	D₂₄₄	—	8	4	no	—
147	C₄×D₆₁	—	4	4	no	—
148	C₄×D₆₁	—	4	4	no	—
149	Dic₁₂₂	—	2	4	no	—
150	Dic₁₂₂	—	2	4	no	—
151	C₈×C₆₁	—	4	4	no	—
152	C₈×C₆₁	C₄×D₄	1	2	yes	—
153	C₈×C₆₁	C₄×D₄	1	2	yes	—
154	C₈×C₆₁	C₄×D₄	1	2	yes	—
155	C₈×C₆₁	C₄×D₄	1	2	yes	—
156	#92418	—	2	2	no	—
157	#92418	#127	1	4	yes	—
158	#21707	—	61	1	no	maximal, Sylow
159	C₂²×C₆₁	—	2	2	no	—
160	C₂²×C₆₁	C₂×C₄×D₄	1	1	yes	characteristic
161	C₂²×C₆₁	#532	1	1	yes	characteristic
162	C₂²×C₆₁	#532	1	1	yes	characteristic
163	C₂×D₆₁	—	4	4	no	—
164	C₂×D₆₁	—	4	8	no	—
165	C₄×C₆₁	—	2	2	no	—
166	C₄×C₆₁	—	2	2	no	—
167	C₄×C₆₁	C₂×C₄×D₄	1	1	yes	characteristic
168	C₄×C₆₁	C₂×C₄×D₄	1	1	yes	characteristic
169	C₄×C₆₁	C₂×#127	1	1	yes	characteristic
170	C₄×C₆₁	#532	1	1	yes	characteristic
171	Dic₆₁	—	4	2	no	—
172	Dic₆₁	—	4	2	no	—
173	Dic₆₁	—	4	4	no	—
174	#3147	—	61	1	no	—
175	C₂×#444	—	61	2	no	—
176	C₂×#443	—	61	2	no	—
177	#2932	—	61	2	no	—
178	#1975	—	61	2	no	—
179	#1974	—	61	2	no	—
180	#1963	—	61	2	no	—
181	#1848	D₆₁	1	1	yes	characteristic, complemented
182	#1726	—	61	1	no	—
183	C₂×C₆₁	—	2	1	no	—
184	C₂×C₆₁	—	2	1	no	—
185	C₂×C₆₁	C₄×#127	1	1	yes	characteristic
186	C₂×C₆₁	#2957	1	2	yes	—
187	D₆₁	—	8	4	no	—
188	C₂×#135	—	61	2	no	—
189	C₂×#135	—	61	2	no	—
190	C₂×#131	—	61	4	no	—
191	#532	—	61	2	no	—
192	#495	—	61	2	no	—
193	#495	—	61	2	no	—
194	#494	—	61	1	no	—
195	#493	—	61	1	no	—
196	#492	—	61	2	no	—
197	#444	—	122	8	no	—
198	#443	—	122	8	no	—
199	#435	—	122	8	no	—
200	C₂×#108	C₂×D₆₁	1	2	yes	—
201	C₂×#108	C₂×D₆₁	1	2	yes	—
202	#423	C₂×D₆₁	1	2	yes	—
203	C₂×#104	—	61	4	no	—
204	C₂×#104	—	61	4	no	—
205	#408	—	61	4	no	—
206	C₂×#97	—	61	1	no	—
207	C₂×#97	—	61	1	no	—
208	C₂×#97	—	61	1	no	—
209	C₂×C₄×C₈	—	61	1	no	—
210	C₂×C₄×C₈	—	61	1	no	—
211	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
212	#376	—	61	1	no	—
213	C₆₁	#21707	1	1	yes	characteristic, complemented, Sylow
214	C₂×#41	—	61	4	no	—
215	#135	—	122	4	no	—
216	#135	—	122	4	no	—
217	#135	—	122	4	no	—
218	#135	—	122	4	no	—
219	C₂×Q₁₆	—	61	2	no	—
220	C₂×Q₁₆	—	61	2	no	—
221	C₂×SD₁₆	—	61	4	no	—
222	C₂×SD₁₆	—	61	4	no	—
223	C₂×D₈	—	61	2	no	—
224	C₂×D₈	—	61	2	no	—
225	#131	—	122	8	no	—
226	#131	—	122	8	no	—
227	C₂×M₁₆	—	122	4	no	—
228	C₂×M₁₆	C₂²×D₆₁	1	4	yes	—
229	C₂²×C₈	—	61	1	no	—
230	C₂²×C₈	—	61	1	no	—
231	C₂²×C₈	—	122	4	no	—
232	C₂²×C₈	C₂²×D₆₁	1	1	yes	characteristic
233	C₂²×C₈	C₂²×D₆₁	1	1	yes	characteristic
234	C₄⋊Q₈	—	61	2	no	—
235	#127	—	61	2	no	—
236	#124	—	61	4	no	—
237	#121	—	122	4	no	—
238	C₄×D₄	—	122	4	no	—
239	C₂×C₄²	—	61	1	no	—
240	C₂×C₄²	—	61	1	no	—
241	C₂×C₄²	C₂²×D₆₁	1	1	yes	characteristic
242	#108	—	2	4	no	—
243	#108	—	2	4	no	—
244	C₄⋊C₈	—	2	4	no	—
245	#104	—	122	8	no	—
246	#104	—	122	8	no	—
247	C₂²⋊C₈	—	122	8	no	—
248	#97	—	61	4	no	—
249	#97	—	61	4	no	—
250	#97	—	122	2	no	—
251	C₄×C₈	—	61	4	no	—
252	C₄×C₈	—	122	2	no	—
253	C₄×C₈	C₄×D₆₁	1	1	yes	characteristic
254	C₄×C₈	C₄×D₆₁	1	1	yes	characteristic
255	C₄×C₈	C₄×D₆₁	1	1	yes	characteristic
256	C₄×C₈	C₄×D₆₁	1	1	yes	characteristic
257	#95	—	122	2	no	—
258	#41	—	122	4	no	—
259	#41	—	122	4	no	—
260	#41	—	122	4	no	—
261	#41	—	122	4	no	—
262	C₂×Q₈	—	61	4	no	—
263	C₂×D₄	—	61	4	no	—
264	C₂×D₄	—	122	4	no	—
265	C₂×D₄	—	122	4	no	—
266	C₂²×C₄	—	122	1	no	—
267	C₂²×C₄	—	122	1	no	—
268	C₂²×C₄	—	122	4	no	—
269	C₂²×C₄	#11758226	1	2	yes	—
270	C₂²×C₄	C₂³×D₆₁	1	1	yes	characteristic
271	Q₁₆	—	122	4	no	—
272	Q₁₆	—	122	4	no	—
273	SD₁₆	—	122	8	no	—
274	SD₁₆	—	122	8	no	—
275	D₈	—	122	4	no	—
276	D₈	—	122	4	no	—
277	M₁₆	—	2	4	no	—
278	M₁₆	—	2	4	no	—
279	M₁₆	—	244	8	no	—
280	C₂×C₈	—	2	4	no	—
281	C₂×C₈	—	61	2	no	—
282	C₂×C₈	—	61	2	no	—
283	C₂×C₈	—	61	2	no	—
284	C₂×C₈	—	61	2	no	—
285	C₂×C₈	—	122	1	no	—
286	C₂×C₈	—	122	1	no	—
287	C₂×C₈	—	244	8	no	—
288	C₂×C₈	C₂×C₄×D₆₁	1	1	yes	characteristic
289	C₂×C₈	C₂×C₄×D₆₁	1	1	yes	characteristic
290	C₂×C₈	C₂×C₄×D₆₁	1	1	yes	characteristic
291	C₂×C₈	C₂×C₄×D₆₁	1	1	yes	characteristic
292	C₂×C₈	D₄×D₆₁	1	2	yes	—
293	C₂×C₈	D₄×D₆₁	1	2	yes	—
294	C₂×C₈	#11758226	1	2	yes	—
295	C₂×C₈	#11758226	1	2	yes	—
296	C₄⋊C₄	—	122	4	no	—
297	C₂²⋊C₄	—	122	8	no	—
298	C₄²	—	61	2	no	—
299	C₄²	—	61	2	no	—
300	C₄²	—	122	1	no	—
301	C₄²	—	122	1	no	—
302	C₄²	C₂×C₄×D₆₁	1	1	yes	characteristic
303	C₄²	C₂×C₄×D₆₁	1	1	yes	characteristic
304	C₄²	D₄×D₆₁	1	2	yes	—
305	C₂³	—	122	4	no	—
306	C₂³	C₂×#11758226	1	1	yes	characteristic
307	Q₈	—	122	4	no	—
308	Q₈	—	122	4	no	—
309	D₄	—	122	4	no	—
310	D₄	—	122	4	no	—
311	D₄	—	488	4	no	—
312	D₄	—	244	4	no	—
313	D₄	—	244	4	no	—
314	C₂×C₄	—	4	2	no	—
315	C₂×C₄	—	122	4	no	—
316	C₂×C₄	—	122	2	no	—
317	C₂×C₄	—	122	2	no	—
318	C₂×C₄	—	244	2	no	—
319	C₂×C₄	—	244	2	no	—
320	C₂×C₄	—	244	4	no	—
321	C₂×C₄	—	244	4	no	—
322	C₂×C₄	#14519739	1	2	yes	—
323	C₂×C₄	#14519739	1	2	yes	—
324	C₂×C₄	C₂²×C₄×D₆₁	1	1	yes	characteristic
325	C₂×C₄	C₂×D₄×D₆₁	1	1	yes	characteristic
326	C₂×C₄	C₂×D₄×D₆₁	1	1	yes	characteristic
327	C₂×C₄	C₂×D₄×D₆₁	1	1	yes	characteristic
328	C₂×C₄	C₂×#11758226	1	1	yes	characteristic
329	C₂×C₄	C₂×#11758226	1	1	yes	characteristic
330	C₈	—	4	4	no	—
331	C₈	—	61	4	no	—
332	C₈	—	122	2	no	—
333	C₈	#14519739	1	2	yes	—
334	C₈	#14519739	1	2	yes	—
335	C₈	#14519739	1	2	yes	—
336	C₈	#14519739	1	2	yes	—
337	C₂²	—	2	2	no	—
338	C₂²	—	244	8	no	—
339	C₂²	—	244	4	no	—
340	C₂²	C₂×#14519739	1	1	yes	characteristic
341	C₂²	#16840313	1	1	yes	characteristic
342	C₂²	#16840313	1	1	yes	characteristic
343	C₄	—	2	2	no	—
344	C₄	—	2	2	no	—
345	C₄	—	244	2	no	—
346	C₄	—	244	2	no	—
347	C₄	—	244	4	no	—
348	C₄	C₂×#14519739	1	1	yes	characteristic
349	C₄	C₂×#14519739	1	1	yes	characteristic
350	C₄	#127×D₆₁	1	1	yes	characteristic
351	C₄	#16840313	1	1	yes	characteristic
352	C₂	—	2	1	no	—
353	C₂	—	2	1	no	—
354	C₂	—	488	4	no	—
355	C₂	#21448381	1	1	yes	characteristic
356	C₂	#21454735	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₆₁
commutator	C₂×C₄×C₆₁	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₆₁
Fitting	C₆₁×#1848	C₂
socle	C₂²×C₆₁	C₂×C₄×D₄
prime radical	#21456247	C₂
cosocle	C₂²×C₄×C₆₁	C₂⁴
hypercenter	#1848	D₆₁
hypocenter	C₆₁	#21707
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1848

Complements

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

N	H ≅ G/N	type
2. C₂×#16839883	C₂	semidirect
3. C₂×#16839878	C₂	semidirect
4. #21454536	C₂	semidirect
5. #21446710	C₂	semidirect
6. #21446705	C₂	semidirect
7. #21446650	C₂	semidirect
8. #21445505	C₂	semidirect
9. C₆₁×#1848	C₂	semidirect
14. #16840146	C₄	semidirect
15. #16840146	C₄	semidirect
16. #16840142	C₄	semidirect
17. #16840139	C₄	semidirect
18. #16840135	C₄	semidirect
64. #14519687	D₄	semidirect
66. C₈×Dic₆₁	D₄	semidirect
181. #1848	D₆₁	semidirect
213. C₆₁	#21707	semidirect

Maximal subgroups

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

subgroup	index
#21456247	2
C₂×#16839883	2
C₂×#16839878	2
#21454536	2
#21446710	2
#21446705	2
#21446650	2
#21445505	2
C₆₁×#1848	2
#21707	61