Subgroups — #30612063

Order 15616.

666 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21462738	C₂	1	2	yes	maximal
2	#21461383	C₂	1	4	yes	maximal, complemented
3	#21461381	C₂	1	2	yes	maximal, complemented
4	#21461376	C₂	1	2	yes	maximal, complemented
5	#21461372	C₂	1	2	yes	maximal, complemented
6	#21461366	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#16840625	C₂	1	1	yes	maximal, characteristic, complemented
8	#131×Dic₆₁	C₂	1	1	yes	maximal, characteristic
9	#16840813	—	2	2	no	—
10	#16840797	—	2	2	no	—
11	#16840793	—	2	4	no	—
12	#16840783	C₂²	1	4	yes	—
13	#16840778	C₂²	1	2	yes	—
14	#16840765	C₂²	1	4	yes	—
15	C₂×C₄⋊#92420	C₂²	1	2	yes	—
16	#16840626	C₂²	1	4	yes	—
17	#16840625	C₂²	1	2	yes	—
18	#16840625	C₂²	1	2	yes	—
19	#16840625	C₂²	1	1	yes	characteristic
20	#16840625	C₂²	1	1	yes	characteristic
21	#16840624	C₂²	1	4	yes	—
22	#16840623	C₂²	1	2	yes	—
23	C₂×#14519784	C₄	1	1	yes	characteristic
24	#41×Dic₆₁	C₂²	1	1	yes	characteristic
25	#16839720	C₄	1	1	yes	characteristic
26	#16839720	C₄	1	2	yes	—
27	#16839717	C₄	1	2	yes	—
28	#16839717	C₄	1	1	yes	characteristic
29	M₁₆×Dic₆₁	C₂²	1	1	yes	characteristic
30	M₁₆×Dic₆₁	C₂²	1	2	yes	—
31	C₂×C₈×Dic₆₁	C₂²	1	1	yes	characteristic
32	C₂×C₈×Dic₆₁	C₂²	1	2	yes	—
33	C₂×#131×C₆₁	C₄	1	1	yes	characteristic
34	C₂×#11758217	—	2	4	no	—
35	C₂²×#92420	—	2	2	no	—
36	#14519844	—	2	4	no	—
37	C₄⋊#92420	—	2	2	no	—
38	C₄⋊#92420	C₂³	1	4	yes	—
39	C₄⋊#92420	C₂³	1	2	yes	—
40	C₄⋊#92420	C₂³	1	2	yes	—
41	C₂²⋊#92420	—	2	2	no	—
42	C₂²⋊#92420	—	2	4	no	—
43	#14519832	—	2	4	no	—
44	C₄×#92420	—	2	2	no	—
45	#14519784	—	2	4	no	—
46	C₂×#11758194	C₂×C₄	1	2	yes	—
47	C₂×#11758194	C₂×C₄	1	1	yes	characteristic
48	C₂²×#92418	C₂×C₄	1	1	yes	characteristic
49	C₂²×#92418	C₂×C₄	1	2	yes	—
50	Q₈×Dic₆₁	D₄	1	1	yes	characteristic
51	Q₈×Dic₆₁	Q₈	1	1	yes	characteristic
52	D₄×Dic₆₁	D₄	1	2	yes	—
53	D₄×Dic₆₁	D₄	1	1	yes	characteristic
54	D₄×Dic₆₁	Q₈	1	2	yes	—
55	D₄×Dic₆₁	Q₈	1	1	yes	characteristic
56	#14519736	C₂×C₄	1	2	yes	—
57	#14519736	C₂×C₄	1	1	yes	characteristic
58	C₂×C₄×Dic₆₁	C₂³	1	1	yes	characteristic
59	C₂×C₄×Dic₆₁	C₂³	1	2	yes	—
60	#14519687	C₂×C₄	1	2	yes	—
61	#14519687	C₂×C₄	1	2	yes	—
62	#14519687	C₂×C₄	1	2	yes	—
63	C₈×Dic₆₁	C₂×C₄	1	1	yes	characteristic
64	C₈×Dic₆₁	C₂×C₄	1	2	yes	—
65	C₈×Dic₆₁	C₂×C₄	1	2	yes	—
66	C₈×Dic₆₁	C₂×C₄	1	1	yes	characteristic
67	C₈×Dic₆₁	C₂³	1	1	yes	characteristic
68	C₈×Dic₆₁	C₂³	1	1	yes	characteristic
69	C₈×Dic₆₁	C₂³	1	2	yes	—
70	C₂×#41×C₆₁	C₂×C₄	1	1	yes	characteristic
71	#131×C₆₁	—	2	4	no	—
72	C₂×M₁₆×C₆₁	C₂×C₄	1	1	yes	characteristic
73	C₂×M₁₆×C₆₁	C₂×C₄	1	2	yes	—
74	C₂²×C₈×C₆₁	C₂×C₄	1	2	yes	—
75	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
76	#41×C₆₁	—	2	2	no	—
77	#41×C₆₁	—	2	2	no	—
78	C₂×Q₈×C₆₁	C₄⋊C₄	1	1	yes	characteristic
79	C₂×D₄×C₆₁	C₄⋊C₄	1	2	yes	—
80	C₂×D₄×C₆₁	C₄⋊C₄	1	1	yes	characteristic
81	C₂²×C₄×C₆₁	C₂²×C₄	1	2	yes	—
82	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
83	C₂²×Dic₆₁	C₂×D₄	1	1	yes	characteristic
84	C₂²×Dic₆₁	C₂×D₄	1	2	yes	—
85	C₂²×Dic₆₁	C₂×Q₈	1	2	yes	—
86	C₂²×Dic₆₁	C₂×Q₈	1	1	yes	characteristic
87	#11758217	—	8	4	no	—
88	C₂×#92420	—	2	2	no	—
89	C₂×#92420	—	2	2	no	—
90	C₂×#92420	—	2	4	no	—
91	C₂×#92420	—	8	2	no	—
92	M₁₆×C₆₁	—	2	2	no	—
93	M₁₆×C₆₁	—	2	4	no	—
94	C₂×C₈×C₆₁	—	2	4	no	—
95	C₂×C₈×C₆₁	M₁₆	1	4	yes	complemented
96	C₂×C₈×C₆₁	C₂²×C₄	1	2	yes	—
97	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
98	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
99	C₂²⋊Dic₆₁	C₄⋊C₄	1	2	yes	—
100	C₂²⋊Dic₆₁	C₄⋊C₄	1	2	yes	—
101	C₂²⋊Dic₆₁	C₄⋊C₄	1	2	yes	—
102	C₄⋊Dic₆₁	C₄⋊C₄	1	2	yes	—
103	C₄⋊Dic₆₁	C₄⋊C₄	1	1	yes	characteristic
104	C₄⋊Dic₆₁	C₄⋊C₄	1	2	yes	—
105	C₄⋊Dic₆₁	C₄⋊C₄	1	1	yes	characteristic
106	C₄×Dic₆₁	C₂²×C₄	1	1	yes	characteristic
107	C₄×Dic₆₁	C₂²×C₄	1	2	yes	—
108	C₄×Dic₆₁	C₂×D₄	1	1	yes	characteristic
109	C₄×Dic₆₁	C₂×D₄	1	2	yes	—
110	C₄×Dic₆₁	C₂×Q₈	1	2	yes	—
111	C₄×Dic₆₁	C₂×Q₈	1	1	yes	characteristic
112	C₄×Dic₆₁	C₂⁴	1	1	yes	characteristic
113	#11758194	—	2	4	no	—
114	#11758194	M₁₆	1	4	yes	complemented
115	C₂×#92418	—	2	2	no	—
116	C₂×#92418	—	2	4	no	—
117	C₂×#92418	C₂²×C₄	1	1	yes	characteristic
118	C₂×#92418	C₂²×C₄	1	1	yes	characteristic
119	C₂×#92418	C₂²×C₄	1	2	yes	—
120	C₂³×C₆₁	C₂×C₄⋊C₄	1	2	yes	—
121	C₂³×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
122	Q₈×C₆₁	—	2	2	no	—
123	D₄×C₆₁	—	2	4	no	—
124	D₄×C₆₁	—	2	2	no	—
125	C₂×C₄×C₆₁	—	2	2	no	—
126	C₂×C₄×C₆₁	—	2	2	no	—
127	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	2	yes	—
128	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
129	C₂×C₄×C₆₁	C₂×M₁₆	1	2	yes	—
130	C₂×C₄×C₆₁	C₂×M₁₆	1	2	yes	—
131	C₂×C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
132	C₂×Dic₆₁	—	2	1	no	—
133	C₂×Dic₆₁	—	2	2	no	—
134	C₂×Dic₆₁	—	2	1	no	—
135	C₂×Dic₆₁	—	2	2	no	—
136	C₂×Dic₆₁	—	4	1	no	—
137	C₂×Dic₆₁	—	4	2	no	—
138	C₂×Dic₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
139	C₂×Dic₆₁	C₂×C₄⋊C₄	1	2	yes	—
140	C₂×Dic₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
141	C₂×Dic₆₁	C₂×C₄⋊C₄	1	2	yes	—
142	C₂×Dic₆₁	C₂²×D₄	1	1	yes	characteristic
143	C₂×Dic₆₁	C₂²×Q₈	1	1	yes	characteristic
144	#92420	—	4	4	no	—
145	#92420	—	4	2	no	—
146	#92420	—	4	2	no	—
147	C₈×C₆₁	—	2	2	no	—
148	C₈×C₆₁	C₂×M₁₆	1	2	yes	—
149	C₈×C₆₁	C₂×M₁₆	1	2	yes	—
150	#92418	—	2	1	no	—
151	#92418	—	2	1	no	—
152	#92418	C₂×M₁₆	1	4	yes	—
153	#17799	—	61	1	no	maximal, Sylow
154	C₂²×C₆₁	—	2	1	no	—
155	C₂²×C₆₁	—	2	1	no	—
156	C₂²×C₆₁	—	2	2	no	—
157	C₂²×C₆₁	—	2	2	no	—
158	C₂²×C₆₁	—	2	2	no	—
159	C₂²×C₆₁	#423	1	2	yes	—
160	C₂²×C₆₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
161	C₄×C₆₁	—	2	2	no	—
162	C₄×C₆₁	#423	1	2	yes	—
163	C₄×C₆₁	C₂²×M₁₆	1	1	yes	characteristic
164	C₄×C₆₁	C₂²×M₁₆	1	1	yes	characteristic
165	Dic₆₁	—	2	1	no	—
166	Dic₆₁	—	2	2	no	—
167	Dic₆₁	—	2	1	no	—
168	Dic₆₁	—	2	1	no	—
169	Dic₆₁	—	2	2	no	—
170	Dic₆₁	#576	1	1	yes	characteristic
171	Dic₆₁	#578	1	1	yes	characteristic
172	#2989	—	61	2	no	—
173	C₄×#131	—	61	1	no	—
174	#1569	—	61	4	no	—
175	#1568	—	61	2	no	—
176	#1567	—	61	2	no	—
177	#1565	—	61	2	no	—
178	#1564	—	61	1	no	—
179	C₂×#335	—	61	1	no	—
180	C₂×C₆₁	—	2	2	no	—
181	C₂×C₆₁	—	4	2	no	—
182	C₂×C₆₁	C₂×#423	1	1	yes	characteristic
183	C₂×C₆₁	#2913	1	1	yes	characteristic
184	C₂×C₆₁	#2916	1	1	yes	characteristic
185	C₂×#131	—	61	1	no	—
186	C₂×#131	#6923	1	1	yes	characteristic
187	C₄×#41	—	61	1	no	—
188	#446	—	122	2	no	—
189	#436	—	122	2	no	—
190	#435	—	122	4	no	—
191	#432	—	61	4	no	—
192	#431	—	61	2	no	—
193	#423	—	61	4	no	—
194	C₂×C₄⋊C₈	—	61	2	no	—
195	#405	—	61	1	no	—
196	#405	—	61	2	no	—
197	#405	—	61	2	no	—
198	#405	—	61	1	no	—
199	C₄×M₁₆	—	61	1	no	—
200	C₄×M₁₆	—	61	2	no	—
201	C₂×C₄×C₈	—	61	1	no	—
202	C₂×C₄×C₈	—	61	2	no	—
203	#335	—	61	2	no	—
204	#335	—	61	2	no	—
205	#335	—	61	1	no	—
206	#335	—	61	1	no	—
207	#335	—	61	4	no	—
208	#334	—	61	2	no	—
209	#334	—	61	4	no	—
210	C₆₁	#17799	1	1	yes	characteristic, complemented, Sylow
211	C₂×#41	C₂×#6923	1	1	yes	characteristic
212	#131	—	2	4	no	—
213	#131	—	122	4	no	—
214	C₂×M₁₆	—	61	1	no	—
215	C₂×M₁₆	—	61	2	no	—
216	C₂×M₁₆	—	122	4	no	—
217	C₂×M₁₆	C₂×#6923	1	1	yes	characteristic
218	C₂×M₁₆	C₂×#6923	1	2	yes	—
219	C₂²×C₈	—	61	2	no	—
220	C₂²×C₈	—	61	1	no	—
221	C₂²×C₈	—	122	2	no	—
222	C₂²×C₈	C₂×#6923	1	2	yes	—
223	C₂²×C₈	C₂×#6923	1	1	yes	characteristic
224	C₄×Q₈	—	61	1	no	—
225	C₄×Q₈	—	61	1	no	—
226	C₄×D₄	—	61	2	no	—
227	C₄×D₄	—	61	1	no	—
228	C₄×D₄	—	61	2	no	—
229	C₄×D₄	—	61	1	no	—
230	#117	—	61	2	no	—
231	#117	—	61	1	no	—
232	C₂×C₄²	—	61	1	no	—
233	C₂×C₄²	—	61	2	no	—
234	C₄⋊C₈	—	61	2	no	—
235	C₄⋊C₈	—	61	4	no	—
236	C₄⋊C₈	—	61	2	no	—
237	C₄⋊C₈	—	122	2	no	—
238	C₄⋊C₈	—	122	4	no	—
239	C₂²⋊C₈	—	122	2	no	—
240	C₂²⋊C₈	—	122	4	no	—
241	#97	—	61	2	no	—
242	#97	—	61	2	no	—
243	#97	—	61	2	no	—
244	#97	—	122	4	no	—
245	C₄×C₈	—	61	2	no	—
246	C₄×C₈	—	61	1	no	—
247	C₄×C₈	—	61	1	no	—
248	C₄×C₈	—	61	2	no	—
249	C₄×C₈	—	61	2	no	—
250	C₄×C₈	—	61	1	no	—
251	C₄×C₈	—	61	1	no	—
252	C₄×C₈	—	122	2	no	—
253	#41	—	2	2	no	—
254	#41	—	2	2	no	—
255	C₂×Q₈	C₄⋊#6923	1	1	yes	characteristic
256	C₂×D₄	C₄⋊#6923	1	2	yes	—
257	C₂×D₄	C₄⋊#6923	1	1	yes	characteristic
258	C₂²×C₄	—	61	1	no	—
259	C₂²×C₄	—	61	2	no	—
260	C₂²×C₄	—	61	2	no	—
261	C₂²×C₄	—	61	1	no	—
262	C₂²×C₄	C₂²×#6923	1	2	yes	—
263	C₂²×C₄	C₂²×#6923	1	1	yes	characteristic
264	M₁₆	—	2	2	no	—
265	M₁₆	—	2	4	no	—
266	M₁₆	—	61	4	no	—
267	M₁₆	—	122	4	no	—
268	M₁₆	—	488	4	no	—
269	C₂×C₈	—	2	4	no	—
270	C₂×C₈	—	61	2	no	—
271	C₂×C₈	—	61	1	no	—
272	C₂×C₈	—	61	1	no	—
273	C₂×C₈	—	122	4	no	—
274	C₂×C₈	—	122	2	no	—
275	C₂×C₈	—	122	2	no	—
276	C₂×C₈	—	122	2	no	—
277	C₂×C₈	—	122	4	no	—
278	C₂×C₈	—	488	2	no	—
279	C₂×C₈	#11758217	1	4	yes	complemented
280	C₂×C₈	C₂²×#6923	1	2	yes	—
281	C₂×C₈	C₂²×#6923	1	1	yes	characteristic
282	C₂×C₈	C₂²×#6923	1	1	yes	characteristic
283	C₄⋊C₄	—	61	2	no	—
284	C₄⋊C₄	—	61	2	no	—
285	C₄⋊C₄	—	61	1	no	—
286	C₄⋊C₄	—	61	1	no	—
287	C₂²⋊C₄	—	61	2	no	—
288	C₂²⋊C₄	—	61	2	no	—
289	C₂²⋊C₄	—	61	2	no	—
290	C₄²	—	61	1	no	—
291	C₄²	—	61	2	no	—
292	C₄²	—	61	2	no	—
293	C₄²	—	61	1	no	—
294	C₄²	—	61	1	no	—
295	C₄²	—	61	1	no	—
296	C₄²	—	61	2	no	—
297	C₂³	C₂×C₄⋊#6923	1	2	yes	—
298	C₂³	C₂×C₄⋊#6923	1	1	yes	characteristic
299	Q₈	—	2	2	no	—
300	D₄	—	2	4	no	—
301	D₄	—	2	2	no	—
302	C₂×C₄	—	2	2	no	—
303	C₂×C₄	—	2	2	no	—
304	C₂×C₄	—	61	1	no	—
305	C₂×C₄	—	61	1	no	—
306	C₂×C₄	—	61	2	no	—
307	C₂×C₄	—	61	1	no	—
308	C₂×C₄	—	61	2	no	—
309	C₂×C₄	—	61	1	no	—
310	C₂×C₄	—	122	1	no	—
311	C₂×C₄	—	122	2	no	—
312	C₂×C₄	—	122	1	no	—
313	C₂×C₄	—	122	2	no	—
314	C₂×C₄	—	244	1	no	—
315	C₂×C₄	—	244	2	no	—
316	C₂×C₄	C₂×C₄⋊#6923	1	2	yes	—
317	C₂×C₄	C₂×C₄⋊#6923	1	1	yes	characteristic
318	C₂×C₄	C₂×#11758217	1	2	yes	—
319	C₂×C₄	D₆₁⋊M₁₆	1	2	yes	—
320	C₂×C₄	C₂³×#6923	1	1	yes	characteristic
321	C₈	—	2	2	no	—
322	C₈	—	61	4	no	—
323	C₈	—	122	1	no	—
324	C₈	—	122	1	no	—
325	C₈	—	244	2	no	—
326	C₈	—	244	4	no	—
327	C₈	—	244	2	no	—
328	C₈	C₂×#11758217	1	2	yes	—
329	C₈	C₂×#11758217	1	2	yes	—
330	C₂²	—	2	1	no	—
331	C₂²	—	2	1	no	—
332	C₂²	—	2	2	no	—
333	C₂²	—	2	2	no	—
334	C₂²	—	2	2	no	—
335	C₂²	#16840765	1	2	yes	—
336	C₂²	C₂²×C₄⋊#6923	1	1	yes	characteristic
337	C₄	—	2	2	no	—
338	C₄	—	61	1	no	—
339	C₄	—	61	1	no	—
340	C₄	—	122	1	no	—
341	C₄	—	122	2	no	—
342	C₄	—	122	1	no	—
343	C₄	—	122	2	no	—
344	C₄	—	122	1	no	—
345	C₄	#16840765	1	2	yes	—
346	C₄	C₂²×#11758217	1	1	yes	characteristic
347	C₄	C₂×D₆₁⋊M₁₆	1	1	yes	characteristic
348	C₂	—	2	2	no	—
349	C₂	—	4	2	no	—
350	C₂	C₂×#16840765	1	1	yes	characteristic
351	C₂	#21462474	1	1	yes	characteristic
352	C₂	#21462481	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×C₄⋊#6923
commutator	C₂×C₄×C₆₁	C₂³×C₄
Frattini	C₂×C₄	C₂³×#6923
Fitting	C₂×#131×C₆₁	C₄
socle	C₂²×C₆₁	C₂²×C₄⋊C₄
prime radical	C₂×#41×C₆₁	C₂×C₄
cosocle	C₄×Dic₆₁	C₂⁴
hypercenter	C₂×#131	#6923
hypocenter	C₆₁	#17799
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₂×#41 ◁ C₂×#131

Complements

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

N	H ≅ G/N	type
2. #21461383	C₂	semidirect
3. #21461381	C₂	semidirect
4. #21461376	C₂	semidirect
5. #21461372	C₂	semidirect
6. #21461366	C₂	semidirect
7. C₂×#16840625	C₂	semidirect
95. C₂×C₈×C₆₁	M₁₆	semidirect
114. #11758194	M₁₆	semidirect
210. C₆₁	#17799	semidirect
279. C₂×C₈	#11758217	semidirect

Maximal subgroups

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

subgroup	index
#21462738	2
#21461383	2
#21461381	2
#21461376	2
#21461372	2
#21461366	2
C₂×#16840625	2
#131×Dic₆₁	2
#17799	61