Subgroups — #15852165

Order 3328.

770 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13125581	C₂	1	1	yes	maximal, characteristic, complemented
2	#13125550	C₂	1	2	yes	maximal
3	#13125378	C₂	1	2	yes	maximal, complemented
4	#13114511	C₂	1	2	yes	maximal, complemented
5	#13114505	C₂	1	1	yes	maximal, characteristic, complemented
6	#13114467	C₂	1	1	yes	maximal, characteristic, complemented
7	#13114456	C₂	1	2	yes	maximal, complemented
8	#13114452	C₂	1	1	yes	maximal, characteristic, complemented
9	#13114422	C₂	1	1	yes	maximal, characteristic, complemented
10	#13113139	C₂	1	1	yes	maximal, characteristic, complemented
11	C₁₃×#3152	C₂	1	1	yes	maximal, characteristic
12	C₂×C₁₃×#137	—	2	2	no	—
13	C₁₃×#532	C₂²	1	1	yes	characteristic
14	C₂×C₁₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
15	#11719175	C₂²	1	2	yes	—
16	C₁₃×#497	—	2	2	no	—
17	C₁₃×#494	C₂²	1	1	yes	characteristic
18	C₁₃×#494	C₂²	1	1	yes	characteristic
19	C₁₃×#492	C₂²	1	1	yes	characteristic
20	C₁₃×#492	C₂²	1	1	yes	characteristic
21	C₄×C₁₃×M₁₆	C₂²	1	1	yes	characteristic
22	#11718798	—	2	2	no	—
23	#11718754	C₂²	1	2	yes	—
24	#11718741	—	2	4	no	—
25	#11718740	C₂²	1	2	yes	—
26	#11718740	C₂²	1	2	yes	—
27	#11718739	C₂²	1	2	yes	—
28	#11718739	C₂²	1	2	yes	—
29	#11718739	C₂²	1	2	yes	—
30	#11718737	C₂²	1	2	yes	—
31	#11718731	C₂²	1	2	yes	—
32	#11718703	—	2	4	no	—
33	#11718679	—	2	4	no	—
34	#11718625	—	2	4	no	—
35	C₄×#6099	C₂²	1	2	yes	—
36	#11718377	—	2	2	no	—
37	#11718360	C₂²	1	1	yes	characteristic
38	#11718342	C₂²	1	2	yes	—
39	#11718341	C₂²	1	1	yes	characteristic
40	#11718047	C₂²	1	1	yes	characteristic
41	#11718046	C₂²	1	2	yes	—
42	C₈×#1055	C₂²	1	1	yes	characteristic
43	C₂×C₁₃×#41	—	2	1	no	—
44	C₂²×Q₈×C₁₃	—	2	1	no	—
45	C₁₃×#137	—	4	8	no	—
46	C₂×C₁₃×Q₁₆	—	2	2	no	—
47	C₂×C₁₃×Q₁₆	—	2	2	no	—
48	C₂×C₁₃×SD₁₆	—	2	2	no	—
49	C₂×C₁₃×SD₁₆	—	2	2	no	—
50	C₂×C₁₃×M₁₆	D₄	1	2	yes	—
51	C₁₃×C₄⋊Q₈	—	2	2	no	—
52	C₁₃×C₄⋊Q₈	—	2	1	no	—
53	C₁₃×C₄⋊Q₈	C₂³	1	1	yes	characteristic
54	C₁₃×C₄⋊Q₈	C₂³	1	1	yes	characteristic
55	C₁₃×C₄⋊Q₈	C₂³	1	1	yes	characteristic
56	C₁₃×#127	C₂³	1	1	yes	characteristic
57	C₁₃×#124	—	2	1	no	—
58	C₁₃×#121	—	2	2	no	—
59	C₄×D₄×C₁₃	—	2	2	no	—
60	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
61	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
62	C₂×#6105	—	2	4	no	—
63	#88180	—	2	2	no	—
64	#88178	—	2	2	no	—
65	#88178	—	2	2	no	—
66	C₂×#6099	D₄	1	4	yes	—
67	C₂×Dic₅₂	—	2	2	no	—
68	#88124	—	2	2	no	—
69	#88123	C₂³	1	2	yes	—
70	#88123	C₂³	1	2	yes	—
71	C₄×Dic₂₆	—	2	2	no	—
72	C₄×Dic₂₆	—	2	2	no	—
73	C₁₃×#97	D₄	1	2	yes	—
74	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
75	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
76	#88072	—	2	4	no	—
77	#88057	—	2	2	no	—
78	#88056	—	2	4	no	—
79	#88050	—	2	4	no	—
80	#88048	—	2	4	no	—
81	#88048	—	2	4	no	—
82	#88047	—	2	4	no	—
83	#88047	—	2	4	no	—
84	#88047	—	2	4	no	—
85	#88043	D₄	1	4	yes	—
86	C₄×#1055	C₂³	1	2	yes	—
87	C₄×#1055	C₂³	1	2	yes	—
88	#19746	—	13	1	no	maximal, Sylow
89	C₁₃×#41	—	4	4	no	—
90	C₂×Q₈×C₁₃	—	2	1	no	—
91	C₂×Q₈×C₁₃	—	2	1	no	—
92	C₂×Q₈×C₁₃	—	2	1	no	—
93	C₂×Q₈×C₁₃	—	4	4	no	—
94	C₂×Q₈×C₁₃	—	4	1	no	—
95	C₂×D₄×C₁₃	—	2	2	no	—
96	C₂×D₄×C₁₃	—	2	1	no	—
97	C₂×D₄×C₁₃	—	4	1	no	—
98	C₂²×C₄×C₁₃	—	4	1	no	—
99	C₂²×C₄×C₁₃	—	4	1	no	—
100	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
101	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
102	C₂×Dic₂₆	—	2	2	no	—
103	C₂×Dic₂₆	—	2	2	no	—
104	C₁₃×Q₁₆	—	4	4	no	—
105	C₁₃×Q₁₆	—	4	4	no	—
106	C₁₃×SD₁₆	—	4	4	no	—
107	C₁₃×SD₁₆	—	4	4	no	—
108	C₁₃×M₁₆	—	2	4	no	—
109	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
110	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
111	C₁₃×C₄⋊C₄	—	2	2	no	—
112	C₁₃×C₄⋊C₄	—	2	2	no	—
113	C₁₃×C₄⋊C₄	—	2	2	no	—
114	C₁₃×C₄⋊C₄	—	4	2	no	—
115	C₁₃×C₂²⋊C₄	—	4	2	no	—
116	C₄²×C₁₃	C₂×D₄	1	2	yes	—
117	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
118	C₄²×C₁₃	C₂⁴	1	1	yes	characteristic
119	C₂²⋊Dic₁₃	—	4	2	no	—
120	#6105	—	4	8	no	—
121	C₄⋊Dic₁₃	—	2	2	no	—
122	C₄⋊Dic₁₃	—	2	2	no	—
123	C₄⋊Dic₁₃	—	2	2	no	—
124	C₄⋊Dic₁₃	—	2	2	no	—
125	#6101	—	4	2	no	—
126	#6101	—	4	2	no	—
127	C₄×Dic₁₃	—	4	2	no	—
128	#6099	—	2	8	no	—
129	C₂×#1055	C₂×D₄	1	4	yes	—
130	C₂×#1055	C₂×D₄	1	4	yes	—
131	Dic₅₂	—	4	4	no	—
132	#3185	—	13	1	no	—
133	#3182	—	13	2	no	—
134	#3165	—	13	2	no	—
135	#3152	D₁₃	1	1	yes	characteristic
136	#1723	—	13	2	no	—
137	#1722	—	13	1	no	—
138	#1716	—	13	1	no	—
139	#1714	—	13	2	no	—
140	#1713	—	13	1	no	—
141	#1705	—	13	1	no	—
142	#1455	—	13	1	no	—
143	C₂³×C₁₃	—	4	1	no	—
144	C₂³×C₁₃	#127	1	1	yes	characteristic
145	Q₈×C₁₃	—	4	2	no	—
146	Q₈×C₁₃	—	4	2	no	—
147	Q₈×C₁₃	—	4	2	no	—
148	Q₈×C₁₃	—	8	2	no	—
149	D₄×C₁₃	—	4	2	no	—
150	D₄×C₁₃	—	4	4	no	—
151	D₄×C₁₃	—	8	2	no	—
152	C₂×C₄×C₁₃	—	2	2	no	—
153	C₂×C₄×C₁₃	—	2	4	no	—
154	C₂×C₄×C₁₃	—	4	1	no	—
155	C₂×C₄×C₁₃	—	4	1	no	—
156	C₂×C₄×C₁₃	—	4	1	no	—
157	C₂×C₄×C₁₃	—	8	2	no	—
158	C₂×C₄×C₁₃	—	8	2	no	—
159	C₂×C₄×C₁₃	#127	1	2	yes	—
160	C₂×C₄×C₁₃	#127	1	1	yes	characteristic
161	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
162	C₂×C₄×C₁₃	C₂²×D₄	1	2	yes	—
163	C₂×Dic₁₃	—	4	2	no	—
164	C₂×Dic₁₃	—	4	2	no	—
165	Dic₂₆	—	4	4	no	—
166	Dic₂₆	—	4	4	no	—
167	C₈×C₁₃	—	2	2	no	—
168	C₈×C₁₃	AGL(1,ℤ/8)	1	4	yes	—
169	#1055	—	2	4	no	—
170	#1055	#137	1	8	yes	complemented
171	C₂×#137	—	2	2	no	—
172	#533	—	13	2	no	—
173	#532	C₂×D₁₃	1	1	yes	characteristic
174	C₂×C₄⋊Q₈	C₂×D₁₃	1	1	yes	characteristic
175	#501	—	26	4	no	—
176	#500	—	13	2	no	—
177	#500	—	13	2	no	—
178	#500	—	13	2	no	—
179	#498	—	13	2	no	—
180	#498	—	13	2	no	—
181	#498	—	13	1	no	—
182	#497	—	2	2	no	—
183	#494	—	13	1	no	—
184	#494	C₂×D₁₃	1	1	yes	characteristic
185	#494	C₂×D₁₃	1	1	yes	characteristic
186	#492	—	13	2	no	—
187	#492	C₂×D₁₃	1	1	yes	characteristic
188	#492	C₂×D₁₃	1	1	yes	characteristic
189	#487	—	13	2	no	—
190	#487	—	13	2	no	—
191	#486	—	13	2	no	—
192	#470	—	26	2	no	—
193	#468	—	26	4	no	—
194	#441	—	26	2	no	—
195	#440	—	26	4	no	—
196	#428	—	26	4	no	—
197	C₄×M₁₆	—	13	2	no	—
198	C₄×M₁₆	C₂×D₁₃	1	1	yes	characteristic
199	#322	—	13	2	no	—
200	#322	—	13	1	no	—
201	C₈²	—	13	1	no	—
202	C₂²×C₁₃	—	2	1	no	—
203	C₂²×C₁₃	—	2	2	no	—
204	C₂²×C₁₃	—	8	1	no	—
205	C₂²×C₁₃	—	8	2	no	—
206	C₂²×C₁₃	C₂×#127	1	1	yes	characteristic
207	C₄×C₁₃	—	2	1	no	—
208	C₄×C₁₃	—	2	2	no	—
209	C₄×C₁₃	—	8	1	no	—
210	C₄×C₁₃	—	8	1	no	—
211	C₄×C₁₃	—	8	1	no	—
212	C₄×C₁₃	C₂×AGL(1,ℤ/8)	1	2	yes	—
213	C₄×C₁₃	C₂×#137	1	4	yes	—
214	Dic₁₃	—	8	2	no	—
215	Dic₁₃	—	8	2	no	—
216	C₂×#41	—	2	1	no	—
217	C₂²×Q₈	—	2	1	no	—
218	#137	—	4	8	no	—
219	C₂×Q₁₆	—	2	2	no	—
220	C₂×Q₁₆	—	2	2	no	—
221	C₂×Q₁₆	—	26	2	no	—
222	C₂×SD₁₆	—	2	2	no	—
223	C₂×SD₁₆	—	2	2	no	—
224	C₂×SD₁₆	—	26	4	no	—
225	C₂×M₁₆	—	13	4	no	—
226	C₂×M₁₆	C₂²⋊D₁₃	1	2	yes	—
227	C₄⋊Q₈	—	2	2	no	—
228	C₄⋊Q₈	—	2	1	no	—
229	C₄⋊Q₈	—	13	2	no	—
230	C₄⋊Q₈	—	13	2	no	—
231	C₄⋊Q₈	C₂²×D₁₃	1	1	yes	characteristic
232	C₄⋊Q₈	C₂²×D₁₃	1	1	yes	characteristic
233	C₄⋊Q₈	C₂²×D₁₃	1	1	yes	characteristic
234	#127	C₂²×D₁₃	1	1	yes	characteristic
235	#125	—	26	2	no	—
236	#124	—	2	1	no	—
237	C₂²⋊Q₈	—	26	2	no	—
238	C₂²⋊Q₈	—	26	2	no	—
239	#121	—	2	2	no	—
240	C₄×Q₈	—	26	2	no	—
241	C₄×Q₈	—	26	2	no	—
242	C₄×D₄	—	2	2	no	—
243	#117	—	26	2	no	—
244	C₂×C₄⋊C₄	—	2	2	no	—
245	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
246	#107	—	26	4	no	—
247	#107	—	26	4	no	—
248	#107	—	26	4	no	—
249	#106	—	26	4	no	—
250	#106	—	26	4	no	—
251	#106	—	26	2	no	—
252	#103	—	26	4	no	—
253	#103	—	26	4	no	—
254	#102	—	26	4	no	—
255	#97	—	13	4	no	—
256	#97	C₂²⋊D₁₃	1	2	yes	—
257	C₄×C₈	—	13	2	no	—
258	C₄×C₈	—	13	2	no	—
259	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
260	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
261	C₂×C₁₃	—	4	1	no	—
262	C₂×C₁₃	—	16	1	no	—
263	C₂×C₁₃	#3151	1	1	yes	characteristic
264	C₂×C₁₃	#3152	1	2	yes	—
265	#41	—	4	4	no	—
266	C₂×Q₈	—	2	1	no	—
267	C₂×Q₈	—	2	1	no	—
268	C₂×Q₈	—	2	1	no	—
269	C₂×Q₈	—	4	4	no	—
270	C₂×Q₈	—	4	1	no	—
271	C₂×Q₈	—	26	2	no	—
272	C₂×Q₈	—	26	2	no	—
273	C₂×D₄	—	2	2	no	—
274	C₂×D₄	—	2	1	no	—
275	C₂×D₄	—	4	1	no	—
276	C₂²×C₄	—	4	1	no	—
277	C₂²×C₄	—	4	1	no	—
278	C₂²×C₄	D₄×D₁₃	1	2	yes	—
279	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
280	Q₁₆	—	4	4	no	—
281	Q₁₆	—	4	4	no	—
282	Q₁₆	—	52	4	no	—
283	SD₁₆	—	4	4	no	—
284	SD₁₆	—	4	4	no	—
285	SD₁₆	—	52	8	no	—
286	M₁₆	—	2	4	no	—
287	M₁₆	—	26	8	no	—
288	C₂×C₈	—	13	4	no	—
289	C₂×C₈	—	13	4	no	—
290	C₂×C₈	C₂×C₂²⋊D₁₃	1	2	yes	—
291	C₂×C₈	C₂×C₂²⋊D₁₃	1	2	yes	—
292	C₄⋊C₄	—	2	2	no	—
293	C₄⋊C₄	—	2	2	no	—
294	C₄⋊C₄	—	2	2	no	—
295	C₄⋊C₄	—	4	2	no	—
296	C₄⋊C₄	—	26	2	no	—
297	C₄⋊C₄	—	26	2	no	—
298	C₄⋊C₄	—	26	2	no	—
299	C₄⋊C₄	—	26	2	no	—
300	C₄⋊C₄	—	52	2	no	—
301	C₄⋊C₄	—	52	2	no	—
302	C₂²⋊C₄	—	4	2	no	—
303	C₂²⋊C₄	—	52	2	no	—
304	C₄²	—	52	2	no	—
305	C₄²	D₄×D₁₃	1	2	yes	—
306	C₄²	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
307	C₄²	C₂³×D₁₃	1	1	yes	characteristic
308	C₁₃	#19746	1	1	yes	characteristic, complemented, Sylow
309	C₂³	—	4	1	no	—
310	C₂³	#88194	1	1	yes	characteristic
311	Q₈	—	4	2	no	—
312	Q₈	—	4	2	no	—
313	Q₈	—	4	2	no	—
314	Q₈	—	8	2	no	—
315	Q₈	—	52	4	no	—
316	Q₈	—	52	4	no	—
317	D₄	—	4	4	no	—
318	D₄	—	4	2	no	—
319	D₄	—	8	2	no	—
320	C₂×C₄	—	2	4	no	—
321	C₂×C₄	—	2	2	no	—
322	C₂×C₄	—	4	1	no	—
323	C₂×C₄	—	4	1	no	—
324	C₂×C₄	—	4	1	no	—
325	C₂×C₄	—	8	2	no	—
326	C₂×C₄	—	8	2	no	—
327	C₂×C₄	—	52	2	no	—
328	C₂×C₄	—	52	2	no	—
329	C₂×C₄	#88194	1	2	yes	—
330	C₂×C₄	#88194	1	1	yes	characteristic
331	C₂×C₄	C₂×D₄×D₁₃	1	2	yes	—
332	C₂×C₄	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
333	C₈	—	2	2	no	—
334	C₈	—	13	8	no	—
335	C₈	—	26	4	no	—
336	C₈	#88186	1	4	yes	—
337	C₂²	—	2	1	no	—
338	C₂²	—	2	2	no	—
339	C₂²	—	8	1	no	—
340	C₂²	—	8	2	no	—
341	C₂²	C₂×#88194	1	1	yes	characteristic
342	C₄	—	2	1	no	—
343	C₄	—	2	2	no	—
344	C₄	—	8	1	no	—
345	C₄	—	8	1	no	—
346	C₄	—	8	1	no	—
347	C₄	—	104	2	no	—
348	C₄	—	104	2	no	—
349	C₄	D₁₃×#137	1	4	yes	—
350	C₄	C₂×#88186	1	2	yes	—
351	C₂	—	4	1	no	—
352	C₂	—	16	1	no	—
353	C₂	#13125286	1	1	yes	characteristic
354	C₂	#13125291	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×#88194
commutator, cosocle	C₄²×C₁₃	C₂⁴
Frattini	C₄²	C₂³×D₁₃
Fitting	C₁₃×#3152	C₂
socle	C₂²×C₁₃	C₂×#127
prime radical	C₁₃×#532	C₂²
hypercenter	#3152	D₁₃
hypocenter	C₁₃	#19746
LCS	G ▷ C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #3152

Complements

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

N	H ≅ G/N	type
1. #13125581	C₂	semidirect
3. #13125378	C₂	semidirect
4. #13114511	C₂	semidirect
5. #13114505	C₂	semidirect
6. #13114467	C₂	semidirect
7. #13114456	C₂	semidirect
8. #13114452	C₂	semidirect
9. #13114422	C₂	semidirect
10. #13113139	C₂	semidirect
170. #1055	#137	semidirect
308. C₁₃	#19746	semidirect

Maximal subgroups

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

subgroup	index
#13125581	2
#13125550	2
#13125378	2
#13114511	2
#13114505	2
#13114467	2
#13114456	2
#13114452	2
#13114422	2
#13113139	2
C₁₃×#3152	2
#19746	13