Subgroups — #15850607

Order 3328.

750 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	#13125415	C₂	1	1	yes	maximal, characteristic
2	#13125396	C₂	1	2	yes	maximal, complemented
3	#13125161	C₂	1	2	yes	maximal, complemented
4	#13114449	C₂	1	2	yes	maximal, complemented
5	#13114445	C₂	1	1	yes	maximal, characteristic, complemented
6	#13114443	C₂	1	1	yes	maximal, characteristic, complemented
7	#13114439	C₂	1	2	yes	maximal, complemented
8	#13114430	C₂	1	1	yes	maximal, characteristic, complemented
9	#13114429	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₈×#1055	C₂	1	1	yes	maximal, characteristic, complemented
11	C₁₃×#3170	C₂	1	1	yes	maximal, characteristic, complemented
12	C₁₃×#533	C₂²	1	2	yes	—
13	#11719182	C₂²	1	2	yes	—
14	C₁₃×#500	C₂²	1	1	yes	characteristic
15	C₁₃×#494	C₂²	1	1	yes	characteristic
16	C₁₃×#487	C₂²	1	2	yes	—
17	C₁₃×#467	—	2	2	no	—
18	C₄×C₁₃×Q₁₆	—	2	2	no	—
19	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
20	#11718776	—	2	2	no	—
21	#11718754	C₂²	1	2	yes	—
22	#11718754	C₂²	1	2	yes	—
23	#11718749	C₂²	1	2	yes	—
24	#11718749	C₂²	1	2	yes	—
25	#11718749	C₂²	1	2	yes	—
26	#11718746	C₂²	1	2	yes	—
27	#11718742	C₂²	1	2	yes	—
28	#11718740	C₂²	1	2	yes	—
29	#11718729	—	2	4	no	—
30	#11718709	—	2	4	no	—
31	C₄×#6106	—	2	4	no	—
32	#11718663	—	2	4	no	—
33	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
34	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
35	#11718356	C₂²	1	1	yes	characteristic
36	#11718355	C₂²	1	2	yes	—
37	C₄×D₁₀₄	—	2	2	no	—
38	#11718343	C₂²	1	1	yes	characteristic
39	C₈×#1055	C₂²	1	1	yes	characteristic
40	C₈×#1055	C₂²	1	1	yes	characteristic
41	C₈×#1055	C₂²	1	1	yes	characteristic
42	C₈×#1055	C₂²	1	1	yes	characteristic
43	C₂×#6127	—	2	2	no	—
44	C₂×C₁₃×Q₁₆	—	2	2	no	—
45	C₂²×C₈×C₁₃	D₄	1	2	yes	—
46	C₁₃×C₄⋊Q₈	C₂³	1	2	yes	—
47	C₁₃×C₄⋊Q₈	C₂³	1	2	yes	—
48	C₁₃×#125	—	2	2	no	—
49	C₁₃×C₂²⋊Q₈	—	2	2	no	—
50	C₁₃×C₂²⋊Q₈	—	2	2	no	—
51	C₄×Q₈×C₁₃	—	2	2	no	—
52	C₄×Q₈×C₁₃	—	2	2	no	—
53	C₁₃×#117	—	2	2	no	—
54	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
55	C₂×#6106	—	2	4	no	—
56	#88184	—	2	2	no	—
57	#88184	—	2	2	no	—
58	C₂²×#1055	D₄	1	2	yes	—
59	C₂²×#1055	D₄	1	2	yes	—
60	C₂×D₁₀₄	—	2	2	no	—
61	#88129	—	2	2	no	—
62	#88128	C₂³	1	2	yes	—
63	C₄×D₅₂	—	2	2	no	—
64	C₄×D₅₂	—	2	2	no	—
65	#88123	C₂³	1	2	yes	—
66	C₁₃×#107	—	2	2	no	—
67	C₁₃×#103	—	2	4	no	—
68	C₄×C₈×C₁₃	D₄	1	2	yes	—
69	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
70	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
71	#88075	—	2	4	no	—
72	#88061	—	2	4	no	—
73	#88058	—	2	2	no	—
74	#88050	—	2	4	no	—
75	#88050	—	2	4	no	—
76	#88049	—	2	4	no	—
77	#88049	—	2	4	no	—
78	#88049	—	2	4	no	—
79	#88048	—	2	4	no	—
80	C₄×#1055	D₄	1	2	yes	—
81	C₄×#1055	D₄	1	2	yes	—
82	C₄×#1055	C₂³	1	1	yes	characteristic
83	C₄×#1055	C₂³	1	1	yes	characteristic
84	C₄×#1055	C₂³	1	1	yes	characteristic
85	C₄×#1055	C₂³	1	1	yes	characteristic
86	#19600	—	13	1	no	maximal, Sylow
87	C₂×Q₈×C₁₃	—	2	2	no	—
88	C₂×Q₈×C₁₃	—	2	2	no	—
89	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
90	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
91	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
92	C₂×C₂²⋊D₁₃	—	4	2	no	—
93	#6127	—	4	8	no	—
94	C₂×D₅₂	—	2	2	no	—
95	C₂×D₅₂	—	2	2	no	—
96	C₂×D₅₂	—	2	2	no	—
97	C₂×C₄×D₁₃	—	4	2	no	—
98	C₂×Dic₂₆	—	2	2	no	—
99	C₁₃×Q₁₆	—	4	4	no	—
100	C₂×C₈×C₁₃	—	2	4	no	—
101	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
102	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
103	C₁₃×C₄⋊C₄	—	2	2	no	—
104	C₁₃×C₄⋊C₄	—	2	2	no	—
105	C₁₃×C₄⋊C₄	—	2	2	no	—
106	C₁₃×C₄⋊C₄	—	2	2	no	—
107	C₁₃×C₄⋊C₄	—	4	2	no	—
108	C₁₃×C₄⋊C₄	—	4	2	no	—
109	C₁₃×C₂²⋊C₄	—	4	2	no	—
110	C₄²×C₁₃	—	4	2	no	—
111	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
112	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
113	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
114	C₄²×C₁₃	C₂⁴	1	1	yes	characteristic
115	#6106	—	4	8	no	—
116	#6103	—	4	2	no	—
117	#6103	—	4	2	no	—
118	C₄⋊Dic₁₃	—	2	2	no	—
119	C₄⋊Dic₁₃	—	2	2	no	—
120	C₂×#1055	—	2	4	no	—
121	C₂×#1055	—	2	4	no	—
122	C₂×#1055	C₂×D₄	1	2	yes	—
123	C₂×#1055	C₂×D₄	1	2	yes	—
124	C₂×#1055	C₂×D₄	1	2	yes	—
125	C₂×#1055	C₂×D₄	1	2	yes	—
126	D₁₀₄	—	4	4	no	—
127	#3170	D₁₃	1	1	yes	characteristic, complemented
128	#3169	—	13	1	no	—
129	#3167	—	13	2	no	—
130	#3131	—	13	2	no	—
131	#1711	—	13	2	no	—
132	#1710	—	13	1	no	—
133	#1709	—	13	1	no	—
134	#1708	—	13	2	no	—
135	#1707	—	13	1	no	—
136	#1706	—	13	1	no	—
137	C₂×C₈²	—	13	1	no	—
138	C₂³×C₁₃	#127	1	1	yes	characteristic
139	C₂²×D₁₃	—	4	2	no	—
140	Q₈×C₁₃	—	4	4	no	—
141	Q₈×C₁₃	—	4	4	no	—
142	C₂×C₄×C₁₃	—	2	2	no	—
143	C₂×C₄×C₁₃	—	2	2	no	—
144	C₂×C₄×C₁₃	—	4	2	no	—
145	C₂×C₄×C₁₃	—	4	2	no	—
146	C₂×C₄×C₁₃	#127	1	1	yes	characteristic
147	C₂×C₄×C₁₃	#127	1	1	yes	characteristic
148	C₂×C₄×C₁₃	#127	1	1	yes	characteristic
149	C₂×C₄×C₁₃	#135	1	4	yes	—
150	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
151	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
152	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
153	C₂²⋊D₁₃	—	8	4	no	—
154	C₂×Dic₁₃	—	4	2	no	—
155	D₅₂	—	4	4	no	—
156	D₅₂	—	4	4	no	—
157	D₅₂	—	4	4	no	—
158	C₄×D₁₃	—	8	4	no	—
159	Dic₂₆	—	4	4	no	—
160	C₈×C₁₃	—	2	2	no	—
161	C₈×C₁₃	#135	1	4	yes	—
162	#1055	—	2	2	no	—
163	#1055	—	2	2	no	—
164	#1055	—	2	2	no	—
165	#1055	#135	1	4	yes	—
166	#533	C₂×D₁₃	1	2	yes	—
167	#532	—	13	2	no	—
168	#500	—	13	1	no	—
169	#500	C₂×D₁₃	1	1	yes	characteristic
170	#498	—	13	2	no	—
171	#494	C₂×D₁₃	1	1	yes	characteristic
172	#493	—	13	1	no	—
173	#492	—	13	2	no	—
174	#488	—	26	4	no	—
175	#487	—	13	2	no	—
176	#487	—	13	2	no	—
177	#487	—	13	2	no	—
178	#487	C₂×D₁₃	1	2	yes	—
179	#486	—	13	2	no	—
180	#486	—	13	2	no	—
181	#486	—	13	2	no	—
182	#486	—	13	2	no	—
183	#467	—	2	2	no	—
184	#466	—	26	2	no	—
185	#465	—	26	4	no	—
186	C₄×Q₁₆	—	2	2	no	—
187	C₄×SD₁₆	—	26	4	no	—
188	C₄×D₈	—	26	2	no	—
189	#416	—	26	4	no	—
190	C₂×C₄×C₈	—	13	1	no	—
191	C₂×C₄×C₈	—	13	1	no	—
192	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
193	C₈²	—	13	1	no	—
194	C₈²	—	13	1	no	—
195	C₈²	—	13	1	no	—
196	C₈²	—	13	1	no	—
197	C₂²×C₁₃	—	2	1	no	—
198	C₂²×C₁₃	—	2	1	no	—
199	C₂²×C₁₃	#495	1	2	yes	—
200	C₂²×C₁₃	C₂×#127	1	1	yes	characteristic
201	C₂×D₁₃	—	8	2	no	—
202	C₂×D₁₃	—	8	2	no	—
203	C₂×D₁₃	—	8	2	no	—
204	C₄×C₁₃	—	2	1	no	—
205	C₄×C₁₃	—	2	1	no	—
206	C₄×C₁₃	—	8	2	no	—
207	C₄×C₁₃	—	8	2	no	—
208	C₄×C₁₃	#495	1	2	yes	—
209	C₄×C₁₃	C₂×#135	1	2	yes	—
210	C₄×C₁₃	C₂×#135	1	2	yes	—
211	C₄×C₁₃	C₂×#135	1	2	yes	—
212	Dic₁₃	—	8	2	no	—
213	C₂×#41	—	26	2	no	—
214	C₂×Q₁₆	—	2	2	no	—
215	C₂×SD₁₆	—	26	4	no	—
216	C₂×D₈	—	26	2	no	—
217	C₂²×C₈	—	13	2	no	—
218	C₂²×C₈	—	13	2	no	—
219	C₂²×C₈	C₂²⋊D₁₃	1	2	yes	—
220	C₄⋊Q₈	—	13	2	no	—
221	C₄⋊Q₈	C₂²×D₁₃	1	2	yes	—
222	C₄⋊Q₈	C₂²×D₁₃	1	2	yes	—
223	#127	—	13	2	no	—
224	#125	—	2	2	no	—
225	#124	—	26	2	no	—
226	C₂²⋊Q₈	—	2	2	no	—
227	C₂²⋊Q₈	—	2	2	no	—
228	#121	—	26	2	no	—
229	#121	—	26	2	no	—
230	C₄×Q₈	—	2	2	no	—
231	C₄×Q₈	—	2	2	no	—
232	C₄×D₄	—	26	2	no	—
233	C₄×D₄	—	26	2	no	—
234	#117	—	2	2	no	—
235	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
236	#107	—	2	2	no	—
237	#107	—	26	2	no	—
238	#106	—	26	4	no	—
239	#103	—	2	4	no	—
240	#103	—	26	4	no	—
241	#103	—	26	4	no	—
242	#103	—	26	4	no	—
243	#102	—	26	4	no	—
244	#102	—	26	4	no	—
245	#102	—	26	4	no	—
246	#102	—	26	4	no	—
247	C₄×C₈	—	13	2	no	—
248	C₄×C₈	—	13	2	no	—
249	C₄×C₈	—	13	1	no	—
250	C₄×C₈	—	13	1	no	—
251	C₄×C₈	—	13	1	no	—
252	C₄×C₈	—	13	1	no	—
253	C₄×C₈	C₂²⋊D₁₃	1	2	yes	—
254	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
255	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
256	C₂×C₁₃	—	2	2	no	—
257	C₂×C₁₃	C₂×#495	1	1	yes	characteristic
258	C₂×C₁₃	#3147	1	1	yes	characteristic
259	C₂×C₁₃	#3147	1	1	yes	characteristic
260	D₁₃	—	16	2	no	—
261	#41	—	52	8	no	—
262	C₂×Q₈	—	2	2	no	—
263	C₂×Q₈	—	2	2	no	—
264	C₂×Q₈	—	26	2	no	—
265	C₂×D₄	—	26	2	no	—
266	C₂×D₄	—	26	2	no	—
267	C₂×D₄	—	26	2	no	—
268	C₂×D₄	—	52	2	no	—
269	C₂²×C₄	—	52	2	no	—
270	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
271	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
272	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
273	Q₁₆	—	4	4	no	—
274	SD₁₆	—	52	8	no	—
275	D₈	—	52	4	no	—
276	C₂×C₈	—	2	4	no	—
277	C₂×C₈	—	13	2	no	—
278	C₂×C₈	—	13	2	no	—
279	C₂×C₈	—	13	2	no	—
280	C₂×C₈	—	13	2	no	—
281	C₂×C₈	—	26	4	no	—
282	C₂×C₈	—	26	4	no	—
283	C₂×C₈	C₂×C₂²⋊D₁₃	1	2	yes	—
284	C₂×C₈	C₂×C₂²⋊D₁₃	1	2	yes	—
285	C₄⋊C₄	—	2	2	no	—
286	C₄⋊C₄	—	2	2	no	—
287	C₄⋊C₄	—	2	2	no	—
288	C₄⋊C₄	—	2	2	no	—
289	C₄⋊C₄	—	4	2	no	—
290	C₄⋊C₄	—	4	2	no	—
291	C₄⋊C₄	—	26	2	no	—
292	C₄⋊C₄	—	26	2	no	—
293	C₂²⋊C₄	—	4	2	no	—
294	C₂²⋊C₄	—	52	2	no	—
295	C₂²⋊C₄	—	52	2	no	—
296	C₄²	—	4	2	no	—
297	C₄²	D₄×D₁₃	1	1	yes	characteristic
298	C₄²	D₄×D₁₃	1	1	yes	characteristic
299	C₄²	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
300	C₄²	C₂³×D₁₃	1	1	yes	characteristic
301	C₁₃	#19600	1	1	yes	characteristic, complemented, Sylow
302	C₂³	—	52	2	no	—
303	C₂³	#88194	1	1	yes	characteristic
304	Q₈	—	4	4	no	—
305	Q₈	—	4	4	no	—
306	Q₈	—	52	4	no	—
307	D₄	—	52	4	no	—
308	D₄	—	52	4	no	—
309	D₄	—	52	4	no	—
310	D₄	—	104	4	no	—
311	C₂×C₄	—	2	2	no	—
312	C₂×C₄	—	2	2	no	—
313	C₂×C₄	—	4	2	no	—
314	C₂×C₄	—	4	2	no	—
315	C₂×C₄	—	52	2	no	—
316	C₂×C₄	—	104	4	no	—
317	C₂×C₄	#88173	1	4	yes	—
318	C₂×C₄	#88194	1	1	yes	characteristic
319	C₂×C₄	#88194	1	1	yes	characteristic
320	C₂×C₄	#88194	1	1	yes	characteristic
321	C₂×C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
322	C₂×C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
323	C₂×C₄	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
324	C₈	—	2	2	no	—
325	C₈	—	13	4	no	—
326	C₈	—	26	2	no	—
327	C₈	—	26	2	no	—
328	C₈	—	26	2	no	—
329	C₈	#88204	1	4	yes	—
330	C₂²	—	2	1	no	—
331	C₂²	—	2	1	no	—
332	C₂²	—	104	2	no	—
333	C₂²	—	104	2	no	—
334	C₂²	—	104	2	no	—
335	C₂²	#11718852	1	2	yes	—
336	C₂²	C₂×#88194	1	1	yes	characteristic
337	C₄	—	2	1	no	—
338	C₄	—	2	1	no	—
339	C₄	—	8	2	no	—
340	C₄	—	8	2	no	—
341	C₄	—	104	2	no	—
342	C₄	#11718852	1	2	yes	—
343	C₄	C₂×#88173	1	2	yes	—
344	C₄	D₁₃×#135	1	2	yes	—
345	C₄	C₂×#88204	1	2	yes	—
346	C₂	—	2	2	no	—
347	C₂	—	208	2	no	—
348	C₂	C₂×#11718852	1	1	yes	characteristic
349	C₂	#13125260	1	1	yes	characteristic
350	C₂	#13125262	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#88194
commutator, cosocle	C₄²×C₁₃	C₂⁴
Frattini	C₄²	C₂³×D₁₃
Fitting	C₁₃×#3170	C₂
socle	C₂²×C₁₃	C₂×#127
prime radical	#13125415	C₂
hypercenter	#3170	D₁₃
hypocenter	C₁₃	#19600
LCS	G ▷ C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3170

Complements

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

N	H ≅ G/N	type
2. #13125396	C₂	semidirect
3. #13125161	C₂	semidirect
4. #13114449	C₂	semidirect
5. #13114445	C₂	semidirect
6. #13114443	C₂	semidirect
7. #13114439	C₂	semidirect
8. #13114430	C₂	semidirect
9. #13114429	C₂	semidirect
10. C₂×C₈×#1055	C₂	semidirect
11. C₁₃×#3170	C₂	semidirect
127. #3170	D₁₃	semidirect
301. C₁₃	#19600	semidirect

Maximal subgroups

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

subgroup	index
#13125415	2
#13125396	2
#13125161	2
#13114449	2
#13114445	2
#13114443	2
#13114439	2
#13114430	2
#13114429	2
C₂×C₈×#1055	2
C₁₃×#3170	2
#19600	13