Subgroups — #13131608

Order 1664.

456 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	D₁₃⋊#131	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#88238	C₂	1	1	yes	maximal, characteristic, complemented
3	#11719478	C₂	1	1	yes	maximal, characteristic
4	#11719166	C₂	1	1	yes	maximal, characteristic, complemented
5	#11719140	C₂	1	1	yes	maximal, characteristic, complemented
6	#11719108	C₂	1	2	yes	maximal, complemented
7	#11719107	C₂	1	2	yes	maximal, complemented
8	#11719104	C₂	1	2	yes	maximal, complemented
9	#11719103	C₂	1	2	yes	maximal, complemented
10	#11719100	C₂	1	1	yes	maximal, characteristic, complemented
11	#11719099	C₂	1	1	yes	maximal, characteristic, complemented
12	#41×D₁₃	C₂²	1	1	yes	characteristic
13	C₂×#6129	C₂²	1	1	yes	characteristic
14	C₂×#6122	—	2	1	no	—
15	C₂²×#1057	—	2	1	no	—
16	#88240	C₂²	1	1	yes	characteristic, complemented
17	#88240	C₂²	1	1	yes	characteristic, complemented
18	#88238	—	2	1	no	—
19	#88238	—	2	1	no	—
20	#88238	—	2	1	no	—
21	#88238	—	2	1	no	—
22	#88238	C₂²	1	1	yes	characteristic, complemented
23	#88238	C₂²	1	2	yes	complemented
24	#88238	C₂²	1	1	yes	characteristic, complemented
25	#88238	C₂²	1	2	yes	complemented
26	#88237	C₂²	1	1	yes	characteristic
27	D₁₃⋊M₁₆	—	2	1	no	—
28	D₁₃⋊M₁₆	C₂²	1	1	yes	characteristic
29	C₂×#6118	C₂²	1	1	yes	characteristic
30	C₂×D₁₃⋊C₈	—	2	1	no	—
31	C₂×D₁₃⋊C₈	C₂²	1	1	yes	characteristic
32	C₁₃×AGL(1,ℤ/8)	C₄	1	1	yes	characteristic, complemented
33	#88205	C₄	1	1	yes	characteristic, complemented
34	#88186	C₄	1	1	yes	characteristic, complemented
35	#88170	C₂²	1	2	yes	—
36	#88169	C₄	1	2	yes	complemented
37	#88166	C₂²	1	2	yes	—
38	#88165	C₄	1	2	yes	complemented
39	#88163	C₄	1	1	yes	characteristic, complemented
40	M₁₆×D₁₃	C₂²	1	1	yes	characteristic
41	#88118	C₂²	1	2	yes	complemented
42	#88116	C₂²	1	2	yes	complemented
43	#88116	C₂²	1	2	yes	complemented
44	#88116	C₂²	1	2	yes	complemented
45	#88104	C₂²	1	2	yes	complemented
46	#88103	C₂²	1	2	yes	complemented
47	#88100	C₂²	1	2	yes	complemented
48	C₈×#261	C₂²	1	2	yes	complemented
49	C₁₃×#41	C₂×C₄	1	1	yes	characteristic
50	C₂×D₄×C₁₃	C₂×C₄	1	1	yes	characteristic
51	C₂×C₂²⋊D₁₃	—	2	1	no	—
52	C₂²×Dic₁₃	—	2	1	no	—
53	#6131	C₂³	1	1	yes	characteristic
54	Q₈×D₁₃	C₂×C₄	1	1	yes	characteristic
55	#6129	—	2	1	no	—
56	#6129	—	2	1	no	—
57	#6129	—	2	1	no	—
58	#6129	C₂×C₄	1	2	yes	—
59	#6129	C₂³	1	1	yes	characteristic
60	#6129	C₂³	1	2	yes	—
61	D₄×D₁₃	—	2	1	no	—
62	D₄×D₁₃	C₂×C₄	1	1	yes	characteristic
63	#6127	—	2	1	no	—
64	#6127	C₂×C₄	1	1	yes	characteristic
65	C₂×C₄×D₁₃	—	2	1	no	—
66	C₂×C₄×D₁₃	C₂³	1	1	yes	characteristic
67	C₂×Dic₂₆	C₂×C₄	1	1	yes	characteristic
68	C₂²⋊#261	—	2	1	no	—
69	#6122	—	2	1	no	—
70	#6122	—	2	1	no	—
71	#6122	—	2	1	no	—
72	#6122	—	2	2	no	—
73	#6122	—	2	2	no	—
74	C₂×#1057	—	2	1	no	—
75	C₂×#1057	—	2	1	no	—
76	C₂×#1057	—	2	1	no	—
77	C₂×#1057	—	2	2	no	—
78	C₂×#1057	—	2	2	no	—
79	C₂×#1057	D₄	1	1	yes	characteristic
80	C₂×#1057	D₄	1	1	yes	characteristic
81	C₄⋊#261	—	2	1	no	—
82	C₄×#261	C₂³	1	2	yes	—
83	#6118	—	2	1	no	—
84	#6118	—	2	1	no	—
85	#6118	—	2	1	no	—
86	#6118	C₂³	1	1	yes	characteristic
87	#6118	C₂³	1	2	yes	—
88	#6118	C₂³	1	1	yes	characteristic
89	D₁₃⋊C₈	—	2	1	no	—
90	D₁₃⋊C₈	—	2	1	no	—
91	D₁₃⋊C₈	—	2	1	no	—
92	D₁₃⋊C₈	D₄	1	1	yes	characteristic
93	D₁₃⋊C₈	D₄	1	1	yes	characteristic
94	D₁₃⋊C₈	C₂³	1	1	yes	characteristic
95	D₁₃⋊C₈	C₂³	1	1	yes	characteristic
96	C₁₃×SD₁₆	C₂×C₄	1	2	yes	—
97	C₁₃×D₈	C₂×C₄	1	2	yes	—
98	C₁₃×M₁₆	C₂×C₄	1	1	yes	characteristic
99	#6107	C₂×C₄	1	2	yes	—
100	#6105	C₂×C₄	1	2	yes	—
101	#6105	C₂×C₄	1	2	yes	—
102	#6104	C₂×C₄	1	2	yes	—
103	#6099	C₂×C₄	1	1	yes	characteristic
104	Dic₅₂	C₂×C₄	1	2	yes	—
105	#6095	C₂×C₄	1	2	yes	—
106	#6094	C₂×C₄	1	2	yes	—
107	C₈×D₁₃	C₂³	1	2	yes	—
108	#2955	—	13	1	no	maximal, Sylow
109	C₂³×C₁₃	—	2	1	no	—
110	C₂²×D₁₃	—	2	1	no	—
111	C₂²×D₁₃	#41	1	1	yes	characteristic
112	C₂×#261	—	2	2	no	—
113	Q₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
114	D₄×C₁₃	—	2	1	no	—
115	D₄×C₁₃	—	2	1	no	—
116	D₄×C₁₃	C₂²×C₄	1	2	yes	—
117	D₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
118	C₂×C₄×C₁₃	—	2	1	no	—
119	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
120	C₂²⋊D₁₃	—	2	1	no	—
121	C₂²⋊D₁₃	—	2	2	no	—
122	C₂²⋊D₁₃	—	2	1	no	—
123	C₂²⋊D₁₃	—	2	2	no	—
124	C₂²⋊D₁₃	—	2	1	no	—
125	C₂×Dic₁₃	—	2	1	no	—
126	C₂×Dic₁₃	—	2	2	no	—
127	C₂×Dic₁₃	—	2	1	no	—
128	C₂×Dic₁₃	—	2	2	no	—
129	C₂×Dic₁₃	C₂×D₄	1	1	yes	characteristic
130	D₅₂	—	2	1	no	—
131	D₅₂	C₂²×C₄	1	1	yes	characteristic
132	C₄×D₁₃	—	2	1	no	—
133	C₄×D₁₃	—	2	1	no	—
134	C₄×D₁₃	—	2	1	no	—
135	C₄×D₁₃	C₂²×C₄	1	1	yes	characteristic
136	C₄×D₁₃	C₂×D₄	1	1	yes	characteristic
137	C₄×D₁₃	#41	1	1	yes	characteristic
138	C₄×D₁₃	C₂⁴	1	1	yes	characteristic
139	Dic₂₆	—	2	1	no	—
140	Dic₂₆	—	2	1	no	—
141	Dic₂₆	C₂²×C₄	1	2	yes	—
142	Dic₂₆	C₂²×C₄	1	1	yes	characteristic
143	#1057	—	2	1	no	—
144	#1057	—	2	1	no	—
145	#1057	—	2	2	no	—
146	#1057	C₂×D₄	1	1	yes	characteristic
147	#1057	C₂×D₄	1	1	yes	characteristic
148	#1057	C₂×D₄	1	1	yes	characteristic
149	#1057	C₂×D₄	1	1	yes	characteristic
150	C₈×C₁₃	C₂²×C₄	1	2	yes	—
151	#1055	C₂²×C₄	1	2	yes	—
152	#575	—	13	1	no	—
153	C₂×#131	—	13	1	no	—
154	C₂×#131	—	13	1	no	—
155	#444	—	13	2	no	—
156	#444	—	13	2	no	—
157	#443	—	13	2	no	—
158	#443	—	13	2	no	—
159	#430	—	13	1	no	—
160	#421	—	13	1	no	—
161	#421	—	13	1	no	—
162	#405	—	13	1	no	—
163	C₂²×C₁₃	—	2	2	no	—
164	C₂²×C₁₃	—	2	1	no	—
165	C₂²×C₁₃	—	4	2	no	—
166	C₂²×C₁₃	C₄×D₄	1	1	yes	characteristic
167	C₂×D₁₃	—	2	1	no	—
168	C₂×D₁₃	—	4	1	no	—
169	C₂×D₁₃	—	4	2	no	—
170	C₂×D₁₃	C₄×D₄	1	1	yes	characteristic
171	C₂×D₁₃	C₂×#41	1	1	yes	characteristic
172	#261	—	4	2	no	—
173	C₄×C₁₃	—	2	1	no	—
174	C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
175	C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
176	Dic₁₃	—	2	1	no	—
177	Dic₁₃	—	2	2	no	—
178	Dic₁₃	C₄×D₄	1	1	yes	characteristic
179	Dic₁₃	C₂²×D₄	1	1	yes	characteristic
180	C₂×#41	—	13	1	no	—
181	C₂×#41	—	13	1	no	—
182	#137	—	13	1	no	—
183	#137	—	13	2	no	—
184	#137	—	13	1	no	—
185	AGL(1,ℤ/8)	—	13	1	no	—
186	AGL(1,ℤ/8)	—	13	2	no	—
187	AGL(1,ℤ/8)	#261	1	1	yes	characteristic, complemented
188	#135	—	13	2	no	—
189	#135	—	13	2	no	—
190	#131	—	13	2	no	—
191	#131	—	13	1	no	—
192	#131	—	13	2	no	—
193	#131	—	13	1	no	—
194	#131	—	13	1	no	—
195	#131	—	13	1	no	—
196	#131	—	26	1	no	—
197	#131	—	26	1	no	—
198	#131	—	26	1	no	—
199	#131	—	26	1	no	—
200	C₂×M₁₆	—	13	1	no	—
201	C₂×M₁₆	—	13	1	no	—
202	C₂×M₁₆	—	13	1	no	—
203	C₂×M₁₆	—	26	1	no	—
204	C₂×M₁₆	—	26	1	no	—
205	C₂²×C₈	—	13	1	no	—
206	C₂²×C₈	—	26	1	no	—
207	C₂²×C₈	—	26	1	no	—
208	#117	—	13	1	no	—
209	#108	—	13	2	no	—
210	#108	—	13	2	no	—
211	#104	—	13	2	no	—
212	#104	—	13	2	no	—
213	#104	—	13	2	no	—
214	#104	—	13	2	no	—
215	#97	—	13	2	no	—
216	C₄×C₈	—	13	2	no	—
217	C₂×C₁₃	—	2	1	no	—
218	C₂×C₁₃	—	4	1	no	—
219	C₂×C₁₃	—	4	2	no	—
220	C₂×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
221	D₁₃	—	2	1	no	—
222	D₁₃	—	2	1	no	—
223	D₁₃	—	4	1	no	—
224	#41	—	13	1	no	—
225	#41	—	13	2	no	—
226	#41	—	13	1	no	—
227	#41	—	13	2	no	—
228	#41	—	13	1	no	—
229	#41	—	26	1	no	—
230	#41	—	26	1	no	—
231	#41	—	26	1	no	—
232	#41	—	26	1	no	—
233	#41	C₂×#261	1	1	yes	characteristic
234	C₂×Q₈	—	13	1	no	—
235	C₂×Q₈	—	13	1	no	—
236	C₂×D₄	—	13	1	no	—
237	C₂×D₄	—	26	1	no	—
238	C₂×D₄	—	26	1	no	—
239	C₂×D₄	C₂×#261	1	1	yes	characteristic
240	C₂²×C₄	—	13	1	no	—
241	C₂²×C₄	—	26	1	no	—
242	C₂²×C₄	—	26	1	no	—
243	Q₁₆	—	13	2	no	—
244	Q₁₆	—	13	2	no	—
245	SD₁₆	—	13	2	no	—
246	SD₁₆	—	13	2	no	—
247	SD₁₆	—	13	2	no	—
248	SD₁₆	C₂×#261	1	2	yes	—
249	D₈	—	13	2	no	—
250	D₈	C₂×#261	1	2	yes	—
251	M₁₆	—	13	1	no	—
252	M₁₆	—	13	2	no	—
253	M₁₆	—	13	1	no	—
254	M₁₆	—	13	1	no	—
255	M₁₆	—	13	2	no	—
256	M₁₆	—	26	1	no	—
257	M₁₆	—	26	1	no	—
258	M₁₆	—	26	1	no	—
259	M₁₆	—	26	2	no	—
260	M₁₆	—	26	1	no	—
261	M₁₆	—	26	2	no	—
262	M₁₆	—	26	1	no	—
263	M₁₆	—	26	1	no	—
264	M₁₆	C₂×#261	1	1	yes	characteristic
265	C₂×C₈	—	13	1	no	—
266	C₂×C₈	—	13	1	no	—
267	C₂×C₈	—	13	1	no	—
268	C₂×C₈	—	13	1	no	—
269	C₂×C₈	—	13	1	no	—
270	C₂×C₈	—	13	1	no	—
271	C₂×C₈	—	13	2	no	—
272	C₂×C₈	—	26	1	no	—
273	C₂×C₈	—	26	1	no	—
274	C₂×C₈	—	26	1	no	—
275	C₂×C₈	—	26	2	no	—
276	C₂×C₈	—	26	1	no	—
277	C₂×C₈	—	26	2	no	—
278	C₂×C₈	—	26	1	no	—
279	C₂×C₈	—	26	1	no	—
280	C₄⋊C₄	—	26	1	no	—
281	C₂²⋊C₄	—	26	1	no	—
282	C₄²	—	13	2	no	—
283	C₁₃	#2955	1	1	yes	characteristic, complemented, Sylow
284	C₂³	—	2	1	no	—
285	C₂³	—	13	1	no	—
286	C₂³	—	26	1	no	—
287	Q₈	—	13	2	no	—
288	Q₈	—	13	1	no	—
289	Q₈	—	26	1	no	—
290	Q₈	—	26	1	no	—
291	Q₈	C₂²×#261	1	1	yes	characteristic
292	D₄	—	2	1	no	—
293	D₄	—	2	1	no	—
294	D₄	—	13	1	no	—
295	D₄	—	26	1	no	—
296	D₄	—	26	1	no	—
297	D₄	—	26	1	no	—
298	D₄	—	26	2	no	—
299	D₄	—	26	2	no	—
300	D₄	—	26	1	no	—
301	D₄	C₂²×#261	1	2	yes	—
302	D₄	C₂²×#261	1	1	yes	characteristic
303	C₂×C₄	—	2	1	no	—
304	C₂×C₄	—	13	1	no	—
305	C₂×C₄	—	13	1	no	—
306	C₂×C₄	—	13	1	no	—
307	C₂×C₄	—	13	1	no	—
308	C₂×C₄	—	13	1	no	—
309	C₂×C₄	—	26	1	no	—
310	C₂×C₄	—	26	1	no	—
311	C₂×C₄	—	26	1	no	—
312	C₂×C₄	—	26	2	no	—
313	C₂×C₄	—	26	2	no	—
314	C₂×C₄	—	26	1	no	—
315	C₂×C₄	—	26	1	no	—
316	C₂×C₄	—	26	2	no	—
317	C₂×C₄	C₂²×#261	1	1	yes	characteristic
318	C₈	—	13	1	no	—
319	C₈	—	13	1	no	—
320	C₈	—	13	1	no	—
321	C₈	—	13	1	no	—
322	C₈	—	13	2	no	—
323	C₈	—	26	1	no	—
324	C₈	—	26	1	no	—
325	C₈	—	26	2	no	—
326	C₈	C₂²×#261	1	2	yes	—
327	C₂²	—	2	1	no	—
328	C₂²	—	2	2	no	—
329	C₂²	—	4	2	no	—
330	C₂²	—	13	1	no	—
331	C₂²	—	13	1	no	—
332	C₂²	—	26	1	no	—
333	C₂²	—	52	1	no	—
334	C₂²	—	52	2	no	—
335	C₂²	D₄×#261	1	1	yes	characteristic
336	C₄	—	2	1	no	—
337	C₄	—	13	1	no	—
338	C₄	—	13	1	no	—
339	C₄	—	26	1	no	—
340	C₄	—	26	2	no	—
341	C₄	—	52	2	no	—
342	C₄	D₄×#261	1	1	yes	characteristic
343	C₄	C₂³×#261	1	1	yes	characteristic
344	C₂	—	2	1	no	—
345	C₂	—	4	2	no	—
346	C₂	—	4	1	no	—
347	C₂	—	26	1	no	—
348	C₂	—	26	1	no	—
349	C₂	—	52	1	no	—
350	C₂	C₂×D₄×#261	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×D₄×#261
commutator	C₄×C₁₃	C₂³×C₄
Frattini	C₄	C₂³×#261
Fitting	C₁₃×AGL(1,ℤ/8)	C₄
socle	C₂×C₁₃	C₂×C₄×D₄
prime radical	#11719478	C₂
cosocle	C₄×D₁₃	C₂⁴
hypercenter	AGL(1,ℤ/8)	#261
hypocenter	C₁₃	#2955
LCS	G ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ AGL(1,ℤ/8)

Complements

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

N	H ≅ G/N	type
1. D₁₃⋊#131	C₂	semidirect
2. C₂×#88238	C₂	semidirect
4. #11719166	C₂	semidirect
5. #11719140	C₂	semidirect
6. #11719108	C₂	semidirect
7. #11719107	C₂	semidirect
8. #11719104	C₂	semidirect
9. #11719103	C₂	semidirect
10. #11719100	C₂	semidirect
11. #11719099	C₂	semidirect
16. #88240	C₂²	semidirect
17. #88240	C₂²	semidirect
22. #88238	C₂²	semidirect
23. #88238	C₂²	semidirect
24. #88238	C₂²	semidirect
25. #88238	C₂²	semidirect
32. C₁₃×AGL(1,ℤ/8)	C₄	semidirect
33. #88205	C₄	semidirect
34. #88186	C₄	semidirect
36. #88169	C₄	semidirect
38. #88165	C₄	semidirect
39. #88163	C₄	semidirect
41. #88118	C₂²	semidirect
42. #88116	C₂²	semidirect
43. #88116	C₂²	semidirect
44. #88116	C₂²	semidirect
45. #88104	C₂²	semidirect
46. #88103	C₂²	semidirect
47. #88100	C₂²	semidirect
48. C₈×#261	C₂²	semidirect
187. AGL(1,ℤ/8)	#261	semidirect
283. C₁₃	#2955	semidirect

Maximal subgroups

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

subgroup	index
D₁₃⋊#131	2
C₂×#88238	2
#11719478	2
#11719166	2
#11719140	2
#11719108	2
#11719107	2
#11719104	2
#11719103	2
#11719100	2
#11719099	2
#2955	13