Subgroups — #30598975

Order 15616.

462 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21461366	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#16840623	C₂	1	1	yes	maximal, characteristic, complemented
3	#21461114	C₂	1	2	yes	maximal
4	#21460665	C₂	1	2	yes	maximal, complemented
5	#131×Dic₆₁	C₂	1	1	yes	maximal, characteristic
6	#16840778	C₂²	1	2	yes	—
7	C₂×C₄⋊#92420	C₂²	1	2	yes	—
8	#16840625	—	2	1	no	—
9	#16840623	—	2	1	no	—
10	#16840623	—	2	1	no	—
11	#16840623	—	2	1	no	—
12	Q₈⋊#92420	—	2	2	no	—
13	D₄⋊#92420	—	2	2	no	—
14	C₂×#14519784	C₄	1	1	yes	characteristic
15	#41×Dic₆₁	C₂²	1	1	yes	characteristic
16	#16839720	—	2	1	no	—
17	#16839720	C₄	1	1	yes	characteristic
18	#16839717	—	2	1	no	—
19	#16839717	C₄	1	1	yes	characteristic
20	M₁₆×Dic₆₁	—	2	1	no	—
21	M₁₆×Dic₆₁	C₂²	1	1	yes	characteristic
22	C₂×C₈×Dic₆₁	—	2	1	no	—
23	C₂×C₈×Dic₆₁	C₂²	1	1	yes	characteristic
24	C₂×#131×C₆₁	C₄	1	1	yes	characteristic
25	C₂²×#92420	—	2	2	no	—
26	C₄⋊#92420	—	2	2	no	—
27	C₄⋊#92420	—	2	2	no	—
28	C₄⋊#92420	—	2	2	no	—
29	C₂²⋊#92420	—	2	2	no	—
30	C₄×#92420	—	2	2	no	—
31	#14519784	—	2	2	no	—
32	#14519784	—	2	2	no	—
33	C₂×#11758194	—	2	1	no	—
34	C₂×#11758194	C₂×C₄	1	1	yes	characteristic
35	C₂²×#92418	—	2	1	no	—
36	C₂²×#92418	C₂×C₄	1	1	yes	characteristic
37	Q₈×Dic₆₁	D₄	1	1	yes	characteristic
38	Q₈×Dic₆₁	Q₈	1	1	yes	characteristic
39	D₄×Dic₆₁	—	2	1	no	—
40	D₄×Dic₆₁	—	2	1	no	—
41	D₄×Dic₆₁	D₄	1	1	yes	characteristic
42	D₄×Dic₆₁	Q₈	1	1	yes	characteristic
43	#14519736	—	2	1	no	—
44	#14519736	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄×Dic₆₁	—	2	1	no	—
46	C₂×C₄×Dic₆₁	C₂³	1	1	yes	characteristic
47	#14519687	—	2	1	no	—
48	#14519687	—	2	1	no	—
49	#14519687	—	2	1	no	—
50	C₈×Dic₆₁	—	2	1	no	—
51	C₈×Dic₆₁	—	2	1	no	—
52	C₈×Dic₆₁	—	2	1	no	—
53	C₈×Dic₆₁	D₄	1	1	yes	characteristic
54	C₈×Dic₆₁	D₄	1	1	yes	characteristic
55	C₈×Dic₆₁	D₄	1	1	yes	characteristic
56	C₈×Dic₆₁	D₄	1	1	yes	characteristic
57	C₂×#41×C₆₁	C₂×C₄	1	1	yes	characteristic
58	#131×C₆₁	—	2	2	no	—
59	#131×C₆₁	C₈	1	4	yes	complemented
60	C₂×M₁₆×C₆₁	—	2	1	no	—
61	C₂×M₁₆×C₆₁	C₂×C₄	1	1	yes	characteristic
62	C₂²×C₈×C₆₁	—	2	1	no	—
63	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
64	#41×C₆₁	—	2	1	no	—
65	#41×C₆₁	—	2	1	no	—
66	#41×C₆₁	C₂×C₈	1	2	yes	—
67	#41×C₆₁	M₁₆	1	2	yes	—
68	C₂×Q₈×C₆₁	C₄⋊C₄	1	1	yes	characteristic
69	C₂×D₄×C₆₁	—	2	1	no	—
70	C₂×D₄×C₆₁	C₄⋊C₄	1	1	yes	characteristic
71	C₂²×C₄×C₆₁	—	2	1	no	—
72	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
73	C₂²×Dic₆₁	—	2	1	no	—
74	C₂²×Dic₆₁	—	2	1	no	—
75	C₂²×Dic₆₁	#41	1	1	yes	characteristic
76	C₂²×Dic₆₁	#41	1	1	yes	characteristic
77	C₂×#92420	—	4	2	no	—
78	C₂×#92420	—	4	2	no	—
79	C₂×#92420	—	8	2	no	—
80	M₁₆×C₆₁	—	2	1	no	—
81	M₁₆×C₆₁	—	2	4	no	—
82	M₁₆×C₆₁	C₂×C₈	1	2	yes	—
83	C₂×C₈×C₆₁	—	2	1	no	—
84	C₂×C₈×C₆₁	—	2	4	no	—
85	C₂×C₈×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
86	C₂×C₈×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
87	C₂×C₈×C₆₁	C₂×C₈	1	2	yes	—
88	C₂×C₈×C₆₁	M₁₆	1	2	yes	—
89	C₂²⋊Dic₆₁	—	2	1	no	—
90	C₂²⋊Dic₆₁	—	2	1	no	—
91	C₂²⋊Dic₆₁	—	2	1	no	—
92	C₄⋊Dic₆₁	—	2	1	no	—
93	C₄⋊Dic₆₁	—	2	1	no	—
94	C₄⋊Dic₆₁	C₄⋊C₄	1	1	yes	characteristic
95	C₄⋊Dic₆₁	C₄⋊C₄	1	1	yes	characteristic
96	C₄×Dic₆₁	—	2	1	no	—
97	C₄×Dic₆₁	—	2	1	no	—
98	C₄×Dic₆₁	—	2	1	no	—
99	C₄×Dic₆₁	C₂×D₄	1	1	yes	characteristic
100	C₄×Dic₆₁	C₂×D₄	1	1	yes	characteristic
101	C₄×Dic₆₁	C₂×D₄	1	1	yes	characteristic
102	C₄×Dic₆₁	C₂×Q₈	1	1	yes	characteristic
103	#11758194	—	2	4	no	—
104	#11758194	—	2	1	no	—
105	#11758194	M₁₆	1	2	yes	—
106	C₂×#92418	—	2	4	no	—
107	C₂×#92418	—	2	1	no	—
108	C₂×#92418	—	2	1	no	—
109	C₂×#92418	—	2	1	no	—
110	C₂×#92418	C₂²⋊C₄	1	1	yes	characteristic
111	C₂×#92418	C₂²⋊C₄	1	1	yes	characteristic
112	C₂³×C₆₁	—	2	1	no	—
113	C₂³×C₆₁	#117	1	1	yes	characteristic
114	Q₈×C₆₁	—	2	1	no	—
115	Q₈×C₆₁	C₄⋊C₈	1	2	yes	—
116	D₄×C₆₁	—	2	2	no	—
117	D₄×C₆₁	—	2	2	no	—
118	D₄×C₆₁	—	2	1	no	—
119	D₄×C₆₁	C₄⋊C₈	1	2	yes	—
120	C₂×C₄×C₆₁	—	2	1	no	—
121	C₂×C₄×C₆₁	—	2	2	no	—
122	C₂×C₄×C₆₁	—	2	2	no	—
123	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
124	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
125	C₂×C₄×C₆₁	C₂²×C₈	1	1	yes	characteristic
126	C₂×C₄×C₆₁	C₂×M₁₆	1	1	yes	characteristic
127	C₂×C₄×C₆₁	C₂×M₁₆	1	1	yes	characteristic
128	C₂×C₄×C₆₁	C₂×M₁₆	1	1	yes	characteristic
129	C₂×Dic₆₁	—	2	2	no	—
130	C₂×Dic₆₁	—	2	1	no	—
131	C₂×Dic₆₁	—	2	1	no	—
132	C₂×Dic₆₁	—	4	1	no	—
133	C₂×Dic₆₁	—	4	2	no	—
134	C₂×Dic₆₁	—	4	2	no	—
135	C₂×Dic₆₁	#121	1	1	yes	characteristic
136	C₂×Dic₆₁	#121	1	1	yes	characteristic
137	C₂×Dic₆₁	C₂²⋊Q₈	1	1	yes	characteristic
138	C₂×Dic₆₁	C₂²⋊Q₈	1	1	yes	characteristic
139	#92420	—	8	2	no	—
140	#92420	—	8	2	no	—
141	C₈×C₆₁	—	2	2	no	—
142	C₈×C₆₁	C₂²⋊C₈	1	2	yes	—
143	C₈×C₆₁	C₂²⋊C₈	1	2	yes	—
144	#92418	—	2	1	no	—
145	#92418	—	2	2	no	—
146	#92418	—	2	1	no	—
147	#8604	—	61	1	no	maximal, Sylow
148	C₂²×C₆₁	—	2	1	no	—
149	C₂²×C₆₁	—	2	1	no	—
150	C₂²×C₆₁	—	2	2	no	—
151	C₂²×C₆₁	—	4	1	no	—
152	C₂²×C₆₁	—	4	1	no	—
153	C₂²×C₆₁	#380	1	1	yes	characteristic
154	C₂²×C₆₁	#431	1	1	yes	characteristic
155	C₂²×C₆₁	#432	1	1	yes	characteristic
156	C₄×C₆₁	—	2	2	no	—
157	C₄×C₆₁	C₂×C₂²⋊C₈	1	1	yes	characteristic
158	C₄×C₆₁	#407	1	1	yes	characteristic
159	C₄×C₆₁	C₂×C₄⋊C₈	1	1	yes	characteristic
160	C₄×C₆₁	#423	1	1	yes	characteristic
161	Dic₆₁	—	2	1	no	—
162	Dic₆₁	—	2	1	no	—
163	Dic₆₁	—	2	1	no	—
164	Dic₆₁	—	4	1	no	—
165	Dic₆₁	—	4	1	no	—
166	Dic₆₁	#471	1	2	yes	—
167	C₄×#131	—	61	1	no	—
168	#1564	—	61	1	no	—
169	C₂×#334	—	61	1	no	—
170	#1477	—	61	2	no	—
171	#1283	—	61	2	no	—
172	C₂×C₆₁	—	2	1	no	—
173	C₂×C₆₁	—	2	1	no	—
174	C₂×C₆₁	—	4	2	no	—
175	C₂×C₆₁	#1797	1	1	yes	characteristic
176	C₂×C₆₁	#1814	1	1	yes	characteristic
177	C₂×C₆₁	#1815	1	1	yes	characteristic
178	C₂×#131	—	61	1	no	—
179	C₂×#131	#6923	1	1	yes	characteristic
180	C₄×#41	—	61	1	no	—
181	#431	—	61	2	no	—
182	C₂×C₄⋊C₈	—	61	2	no	—
183	#405	—	61	1	no	—
184	#405	—	61	1	no	—
185	#405	—	122	1	no	—
186	#405	—	122	1	no	—
187	C₄×M₁₆	—	61	1	no	—
188	C₄×M₁₆	—	122	1	no	—
189	C₂×C₄×C₈	—	61	1	no	—
190	C₂×C₄×C₈	—	122	1	no	—
191	#335	—	122	1	no	—
192	#334	—	122	1	no	—
193	#334	—	122	1	no	—
194	#334	—	122	1	no	—
195	Q₈⋊C₈	—	122	2	no	—
196	D₄⋊C₈	—	122	2	no	—
197	C₆₁	#8604	1	1	yes	characteristic, complemented, Sylow
198	C₂×#41	C₂×#6923	1	1	yes	characteristic
199	#131	—	2	2	no	—
200	#131	—	122	2	no	—
201	#131	—	122	2	no	—
202	#131	#92420	1	4	yes	complemented
203	C₂×M₁₆	—	2	1	no	—
204	C₂×M₁₆	—	61	1	no	—
205	C₂×M₁₆	—	122	1	no	—
206	C₂×M₁₆	C₂×#6923	1	1	yes	characteristic
207	C₂²×C₈	—	2	1	no	—
208	C₂²×C₈	—	61	1	no	—
209	C₂²×C₈	—	122	1	no	—
210	C₂²×C₈	—	122	2	no	—
211	C₂²×C₈	C₂×#6923	1	1	yes	characteristic
212	C₄×Q₈	—	61	1	no	—
213	C₄×Q₈	—	61	1	no	—
214	C₄×D₄	—	61	1	no	—
215	C₄×D₄	—	61	1	no	—
216	C₄×D₄	—	122	1	no	—
217	C₄×D₄	—	122	1	no	—
218	#117	—	61	1	no	—
219	#117	—	122	1	no	—
220	C₂×C₄²	—	61	1	no	—
221	C₂×C₄²	—	122	1	no	—
222	C₄⋊C₈	—	122	2	no	—
223	C₄⋊C₈	—	122	2	no	—
224	C₄⋊C₈	—	122	2	no	—
225	C₂²⋊C₈	—	122	2	no	—
226	#97	—	122	1	no	—
227	#97	—	122	1	no	—
228	#97	—	122	1	no	—
229	C₄×C₈	—	61	1	no	—
230	C₄×C₈	—	61	1	no	—
231	C₄×C₈	—	61	1	no	—
232	C₄×C₈	—	61	1	no	—
233	C₄×C₈	—	122	1	no	—
234	C₄×C₈	—	122	2	no	—
235	C₄×C₈	—	122	1	no	—
236	C₄×C₈	—	122	1	no	—
237	#41	—	2	1	no	—
238	#41	—	2	1	no	—
239	#41	#11758213	1	2	yes	—
240	#41	C₂×#92420	1	2	yes	—
241	C₂×Q₈	C₄⋊#6923	1	1	yes	characteristic
242	C₂×D₄	—	2	1	no	—
243	C₂×D₄	C₄⋊#6923	1	1	yes	characteristic
244	C₂²×C₄	—	2	1	no	—
245	C₂²×C₄	—	61	1	no	—
246	C₂²×C₄	—	61	1	no	—
247	C₂²×C₄	—	122	1	no	—
248	C₂²×C₄	—	122	1	no	—
249	C₂²×C₄	C₂²×#6923	1	1	yes	characteristic
250	M₁₆	—	2	1	no	—
251	M₁₆	—	2	4	no	—
252	M₁₆	—	61	2	no	—
253	M₁₆	—	122	4	no	—
254	M₁₆	—	122	1	no	—
255	M₁₆	C₂×#92420	1	2	yes	—
256	C₂×C₈	—	2	1	no	—
257	C₂×C₈	—	2	4	no	—
258	C₂×C₈	—	61	1	no	—
259	C₂×C₈	—	61	1	no	—
260	C₂×C₈	—	122	4	no	—
261	C₂×C₈	—	122	1	no	—
262	C₂×C₈	—	122	1	no	—
263	C₂×C₈	—	122	1	no	—
264	C₂×C₈	—	488	2	no	—
265	C₂×C₈	—	244	2	no	—
266	C₂×C₈	—	244	2	no	—
267	C₂×C₈	C₂×#92420	1	2	yes	—
268	C₂×C₈	#11758217	1	2	yes	—
269	C₂×C₈	C₂²⋊#6923	1	1	yes	characteristic
270	C₂×C₈	C₂²⋊#6923	1	1	yes	characteristic
271	C₄⋊C₄	—	61	1	no	—
272	C₄⋊C₄	—	61	1	no	—
273	C₄⋊C₄	—	122	1	no	—
274	C₄⋊C₄	—	122	1	no	—
275	C₂²⋊C₄	—	122	1	no	—
276	C₂²⋊C₄	—	122	1	no	—
277	C₂²⋊C₄	—	122	1	no	—
278	C₄²	—	61	1	no	—
279	C₄²	—	61	1	no	—
280	C₄²	—	61	1	no	—
281	C₄²	—	61	1	no	—
282	C₄²	—	122	1	no	—
283	C₄²	—	122	1	no	—
284	C₄²	—	122	1	no	—
285	C₂³	—	2	1	no	—
286	C₂³	#14519856	1	1	yes	characteristic
287	Q₈	—	2	1	no	—
288	Q₈	C₄⋊#92420	1	2	yes	—
289	D₄	—	2	2	no	—
290	D₄	—	2	2	no	—
291	D₄	—	2	1	no	—
292	D₄	C₄⋊#92420	1	2	yes	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	2	2	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	61	1	no	—
297	C₂×C₄	—	61	1	no	—
298	C₂×C₄	—	61	1	no	—
299	C₂×C₄	—	61	1	no	—
300	C₂×C₄	—	122	2	no	—
301	C₂×C₄	—	122	1	no	—
302	C₂×C₄	—	122	1	no	—
303	C₂×C₄	—	244	2	no	—
304	C₂×C₄	—	244	1	no	—
305	C₂×C₄	—	244	2	no	—
306	C₂×C₄	C₂×C₂²⋊#6923	1	1	yes	characteristic
307	C₂×C₄	C₂×C₄⋊#6923	1	1	yes	characteristic
308	C₂×C₄	C₂²×#92420	1	1	yes	characteristic
309	C₂×C₄	C₂×#11758217	1	1	yes	characteristic
310	C₂×C₄	D₆₁⋊M₁₆	1	1	yes	characteristic
311	C₂×C₄	C₂×#11758213	1	1	yes	characteristic
312	C₈	—	2	2	no	—
313	C₈	—	122	1	no	—
314	C₈	—	122	1	no	—
315	C₈	—	122	2	no	—
316	C₈	—	488	2	no	—
317	C₈	—	488	2	no	—
318	C₈	C₂²⋊#92420	1	2	yes	—
319	C₈	C₂²⋊#92420	1	2	yes	—
320	C₂²	—	2	1	no	—
321	C₂²	—	2	1	no	—
322	C₂²	—	2	2	no	—
323	C₂²	—	4	1	no	—
324	C₂²	—	4	1	no	—
325	C₂²	#16840704	1	1	yes	characteristic
326	C₂²	#16840778	1	1	yes	characteristic
327	C₂²	#16840783	1	1	yes	characteristic
328	C₄	—	2	2	no	—
329	C₄	—	61	2	no	—
330	C₄	—	122	1	no	—
331	C₄	—	122	1	no	—
332	C₄	—	122	1	no	—
333	C₄	—	244	1	no	—
334	C₄	—	244	1	no	—
335	C₄	C₂×C₂²⋊#92420	1	1	yes	characteristic
336	C₄	#16840733	1	1	yes	characteristic
337	C₄	C₂×C₄⋊#92420	1	1	yes	characteristic
338	C₄	#16840765	1	1	yes	characteristic
339	C₂	—	2	1	no	—
340	C₂	—	2	1	no	—
341	C₂	—	4	2	no	—
342	C₂	#21461673	1	1	yes	characteristic
343	C₂	#21461705	1	1	yes	characteristic
344	C₂	#21461707	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#16840704
commutator	C₂×C₄×C₆₁	C₂²×C₈
Frattini	C₂²×C₄	C₂²×#6923
Fitting	C₂×#131×C₆₁	C₄
socle	C₂²×C₆₁	#380
prime radical	C₂×#41×C₆₁	C₂×C₄
cosocle	C₂×C₄×Dic₆₁	C₂³
hypercenter	C₂×#131	#6923
hypocenter	C₆₁	#8604
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#131

Complements

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

N	H ≅ G/N	type
1. #21461366	C₂	semidirect
2. C₂×#16840623	C₂	semidirect
4. #21460665	C₂	semidirect
59. #131×C₆₁	C₈	semidirect
197. C₆₁	#8604	semidirect
202. #131	#92420	semidirect

Maximal subgroups

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

subgroup	index
#21461366	2
C₂×#16840623	2
#21461114	2
#21460665	2
#131×Dic₆₁	2
#8604	61