Subgroups — #13131611

Order 1664.

458 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11719629	C₂	1	1	yes	maximal, characteristic, complemented
2	#11719625	C₂	1	1	yes	maximal, characteristic, complemented
3	#11719482	C₂	1	1	yes	maximal, characteristic
4	D₁₃⋊#104	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#88118	C₂	1	1	yes	maximal, characteristic, complemented
6	#11719112	C₂	1	2	yes	maximal, complemented
7	#11719111	C₂	1	2	yes	maximal, complemented
8	#11719108	C₂	1	2	yes	maximal, complemented
9	#11719107	C₂	1	2	yes	maximal, complemented
10	#11719100	C₂	1	1	yes	maximal, characteristic, complemented
11	M₁₆×#261	C₂	1	1	yes	maximal, characteristic, complemented
12	#41×D₁₃	C₂²	1	1	yes	characteristic
13	C₂×#6131	C₂²	1	1	yes	characteristic
14	#88240	—	2	1	no	—
15	#88240	—	2	1	no	—
16	#88240	C₂²	1	2	yes	complemented
17	#88240	C₂²	1	2	yes	complemented
18	#88240	C₂²	1	2	yes	complemented
19	#88238	—	2	2	no	—
20	#88238	C₂²	1	2	yes	complemented
21	C₂×C₄×#261	C₂²	1	1	yes	characteristic
22	D₁₃⋊M₁₆	—	2	2	no	—
23	D₁₃⋊M₁₆	—	2	1	no	—
24	D₁₃⋊M₁₆	C₂²	1	1	yes	characteristic
25	D₁₃⋊M₁₆	C₂²	1	1	yes	characteristic
26	C₂×#6118	—	2	1	no	—
27	C₂×#6118	C₂²	1	1	yes	characteristic
28	C₁₃×#137	C₄	1	1	yes	characteristic, complemented
29	#88203	C₄	1	1	yes	characteristic, complemented
30	#88196	C₄	1	1	yes	characteristic, complemented
31	#88173	C₂²	1	2	yes	—
32	#88172	C₄	1	2	yes	complemented
33	#88170	C₂²	1	2	yes	—
34	#88168	C₄	1	2	yes	complemented
35	#88162	C₄	1	1	yes	characteristic, complemented
36	M₁₆×D₁₃	C₂²	1	1	yes	characteristic
37	#88118	C₂²	1	2	yes	complemented
38	#88118	C₂²	1	2	yes	complemented
39	#88118	C₂²	1	1	yes	characteristic, complemented
40	#88118	C₂²	1	1	yes	characteristic, complemented
41	#88116	C₂²	1	1	yes	characteristic, complemented
42	#88116	C₂²	1	1	yes	characteristic, complemented
43	#88104	C₂²	1	2	yes	complemented
44	#88103	C₂²	1	2	yes	complemented
45	#88100	C₂²	1	2	yes	complemented
46	C₈×#261	C₂²	1	2	yes	complemented
47	C₂²×#261	—	2	1	no	—
48	C₁₃×#41	C₂×C₄	1	1	yes	characteristic
49	C₂×Q₈×C₁₃	C₂×C₄	1	1	yes	characteristic
50	#6131	—	2	1	no	—
51	#6131	—	2	1	no	—
52	#6131	C₂×C₄	1	2	yes	—
53	#6131	C₂³	1	2	yes	—
54	#6131	C₂³	1	1	yes	characteristic
55	Q₈×D₁₃	C₂×C₄	1	1	yes	characteristic
56	#6129	—	2	1	no	—
57	#6129	C₂³	1	1	yes	characteristic
58	D₄×D₁₃	—	2	1	no	—
59	D₄×D₁₃	C₂×C₄	1	1	yes	characteristic
60	#6127	—	2	1	no	—
61	#6127	C₂×C₄	1	1	yes	characteristic
62	C₂×D₅₂	—	2	1	no	—
63	C₂×D₅₂	C₂×C₄	1	1	yes	characteristic
64	C₂×C₄×D₁₃	—	2	1	no	—
65	C₂×C₄×D₁₃	—	2	1	no	—
66	C₂×C₄×D₁₃	C₂³	1	1	yes	characteristic
67	#6122	—	2	1	no	—
68	#6122	—	2	2	no	—
69	#6122	D₄	1	2	yes	—
70	C₂×#1057	—	2	1	no	—
71	C₂×#1057	—	2	2	no	—
72	C₄×#261	—	2	1	no	—
73	C₄×#261	C₂³	1	1	yes	characteristic
74	C₄×#261	C₂³	1	1	yes	characteristic
75	#6118	—	2	1	no	—
76	#6118	—	2	2	no	—
77	#6118	—	2	2	no	—
78	#6118	—	2	2	no	—
79	#6118	D₄	1	1	yes	characteristic
80	#6118	D₄	1	1	yes	characteristic
81	#6118	C₂³	1	1	yes	characteristic
82	#6118	C₂³	1	2	yes	—
83	#6118	C₂³	1	1	yes	characteristic
84	D₁₃⋊C₈	—	2	1	no	—
85	D₁₃⋊C₈	—	2	2	no	—
86	D₁₃⋊C₈	—	2	2	no	—
87	D₁₃⋊C₈	—	2	2	no	—
88	D₁₃⋊C₈	C₂³	1	2	yes	—
89	C₁₃×Q₁₆	C₂×C₄	1	2	yes	complemented
90	C₁₃×SD₁₆	C₂×C₄	1	2	yes	complemented
91	C₁₃×M₁₆	C₂×C₄	1	1	yes	characteristic
92	#6107	C₂×C₄	1	2	yes	complemented
93	#6106	C₂×C₄	1	2	yes	complemented
94	#6106	C₂×C₄	1	2	yes	complemented
95	#6104	C₂×C₄	1	2	yes	complemented
96	#6099	C₂×C₄	1	1	yes	characteristic
97	D₁₀₄	C₂×C₄	1	2	yes	complemented
98	#6095	C₂×C₄	1	2	yes	complemented
99	#6094	C₂×C₄	1	2	yes	—
100	C₈×D₁₃	C₂³	1	2	yes	—
101	#2956	—	13	1	no	maximal, Sylow
102	C₂²×D₁₃	—	2	1	no	—
103	C₂²×D₁₃	—	2	1	no	—
104	C₂²×D₁₃	#41	1	1	yes	characteristic
105	C₂×#261	—	2	1	no	—
106	C₂×#261	—	2	1	no	—
107	C₂×#261	—	4	2	no	—
108	Q₈×C₁₃	—	2	1	no	—
109	Q₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
110	Q₈×C₁₃	C₂²×C₄	1	2	yes	—
111	D₄×C₁₃	—	2	1	no	—
112	D₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
113	C₂×C₄×C₁₃	—	2	1	no	—
114	C₂×C₄×C₁₃	—	2	1	no	—
115	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
116	C₂²⋊D₁₃	—	2	1	no	—
117	C₂²⋊D₁₃	—	2	1	no	—
118	C₂²⋊D₁₃	—	2	1	no	—
119	C₂×Dic₁₃	—	2	1	no	—
120	C₂×Dic₁₃	C₂×D₄	1	1	yes	characteristic
121	D₅₂	—	2	2	no	—
122	D₅₂	—	2	2	no	—
123	D₅₂	—	2	1	no	—
124	D₅₂	—	2	1	no	—
125	D₅₂	C₂²×C₄	1	2	yes	—
126	D₅₂	C₂²×C₄	1	1	yes	characteristic
127	C₄×D₁₃	—	2	1	no	—
128	C₄×D₁₃	—	2	1	no	—
129	C₄×D₁₃	—	2	2	no	—
130	C₄×D₁₃	—	2	1	no	—
131	C₄×D₁₃	—	2	2	no	—
132	C₄×D₁₃	C₂²×C₄	1	1	yes	characteristic
133	C₄×D₁₃	C₂×D₄	1	1	yes	characteristic
134	C₄×D₁₃	#41	1	1	yes	characteristic
135	C₄×D₁₃	C₂⁴	1	1	yes	characteristic
136	Dic₂₆	—	2	1	no	—
137	Dic₂₆	C₂²×C₄	1	1	yes	characteristic
138	#1057	—	2	1	no	—
139	#1057	—	2	1	no	—
140	#1057	—	2	2	no	—
141	#1057	C₂×D₄	1	2	yes	—
142	#1057	C₂×D₄	1	2	yes	—
143	C₈×C₁₃	C₂²×C₄	1	2	yes	—
144	#1055	C₂²×C₄	1	2	yes	—
145	#575	—	13	1	no	—
146	#568	—	13	1	no	—
147	#568	—	13	1	no	—
148	#444	—	13	2	no	—
149	#444	—	13	2	no	—
150	#443	—	13	2	no	—
151	#443	—	13	2	no	—
152	#430	—	13	1	no	—
153	C₂×#104	—	13	1	no	—
154	C₂×#104	—	13	1	no	—
155	C₄×M₁₆	—	13	1	no	—
156	C₂²×C₁₃	—	2	1	no	—
157	C₂²×C₁₃	C₄×D₄	1	1	yes	characteristic
158	C₂×D₁₃	—	2	2	no	—
159	C₂×D₁₃	—	2	1	no	—
160	C₂×D₁₃	—	4	1	no	—
161	C₂×D₁₃	—	8	1	no	—
162	C₂×D₁₃	—	8	1	no	—
163	C₂×D₁₃	C₄×D₄	1	1	yes	characteristic
164	C₂×D₁₃	C₂×#41	1	1	yes	characteristic
165	#261	—	2	2	no	—
166	#261	—	4	1	no	—
167	C₄×C₁₃	—	2	1	no	—
168	C₄×C₁₃	—	2	2	no	—
169	C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
170	C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
171	Dic₁₃	—	2	1	no	—
172	Dic₁₃	C₄×D₄	1	1	yes	characteristic
173	Dic₁₃	C₂²×D₄	1	1	yes	characteristic
174	C₂×#41	—	13	1	no	—
175	C₂×#41	—	13	1	no	—
176	#137	—	13	1	no	—
177	#137	—	13	2	no	—
178	#137	#261	1	1	yes	characteristic, complemented
179	AGL(1,ℤ/8)	—	13	1	no	—
180	AGL(1,ℤ/8)	—	13	2	no	—
181	AGL(1,ℤ/8)	—	13	1	no	—
182	#135	—	13	2	no	—
183	#135	—	13	2	no	—
184	#131	—	13	2	no	—
185	#131	—	13	2	no	—
186	#131	—	13	2	no	—
187	#131	—	13	2	no	—
188	#131	—	26	2	no	—
189	#131	—	26	1	no	—
190	#131	—	26	1	no	—
191	C₂×M₁₆	—	13	1	no	—
192	C₂×M₁₆	—	13	1	no	—
193	C₂×M₁₆	—	13	1	no	—
194	C₂×M₁₆	—	13	1	no	—
195	C₂×M₁₆	—	26	1	no	—
196	C₂×M₁₆	—	26	1	no	—
197	C₂×M₁₆	—	26	2	no	—
198	C₂×C₄²	—	13	1	no	—
199	#108	—	13	2	no	—
200	#108	—	13	2	no	—
201	#104	—	13	1	no	—
202	#104	—	13	1	no	—
203	#104	—	13	2	no	—
204	#104	—	13	1	no	—
205	#104	—	13	2	no	—
206	#104	—	13	1	no	—
207	#97	—	13	2	no	—
208	C₄×C₈	—	13	2	no	—
209	C₂×C₁₃	—	2	1	no	—
210	C₂×C₁₃	—	4	1	no	—
211	C₂×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
212	D₁₃	—	2	1	no	—
213	D₁₃	—	2	1	no	—
214	D₁₃	—	4	2	no	—
215	D₁₃	—	4	1	no	—
216	#41	—	13	1	no	—
217	#41	—	13	2	no	—
218	#41	—	13	2	no	—
219	#41	—	13	1	no	—
220	#41	—	13	1	no	—
221	#41	—	26	1	no	—
222	#41	—	26	1	no	—
223	#41	—	26	1	no	—
224	#41	—	26	1	no	—
225	#41	C₂×#261	1	1	yes	characteristic
226	C₂×Q₈	—	13	1	no	—
227	C₂×Q₈	C₂×#261	1	1	yes	characteristic
228	C₂×D₄	—	13	1	no	—
229	C₂×D₄	—	13	1	no	—
230	C₂×D₄	—	26	1	no	—
231	C₂×D₄	—	26	1	no	—
232	C₂²×C₄	—	13	1	no	—
233	C₂²×C₄	—	26	1	no	—
234	C₂²×C₄	—	26	1	no	—
235	C₂²×C₄	—	26	1	no	—
236	Q₁₆	—	13	2	no	—
237	Q₁₆	C₂×#261	1	2	yes	complemented
238	SD₁₆	—	13	2	no	—
239	SD₁₆	—	13	2	no	—
240	SD₁₆	—	13	2	no	—
241	SD₁₆	C₂×#261	1	2	yes	complemented
242	D₈	—	13	2	no	—
243	D₈	—	13	2	no	—
244	M₁₆	—	13	1	no	—
245	M₁₆	—	13	1	no	—
246	M₁₆	—	13	2	no	—
247	M₁₆	—	13	1	no	—
248	M₁₆	—	13	2	no	—
249	M₁₆	—	13	1	no	—
250	M₁₆	—	13	1	no	—
251	M₁₆	—	13	2	no	—
252	M₁₆	—	26	2	no	—
253	M₁₆	—	26	2	no	—
254	M₁₆	—	26	2	no	—
255	M₁₆	—	26	2	no	—
256	M₁₆	—	26	1	no	—
257	M₁₆	—	26	1	no	—
258	M₁₆	C₂×#261	1	1	yes	characteristic
259	C₂×C₈	—	13	2	no	—
260	C₂×C₈	—	13	2	no	—
261	C₂×C₈	—	26	2	no	—
262	C₂×C₈	—	26	2	no	—
263	C₂×C₈	—	26	2	no	—
264	C₂×C₈	—	26	2	no	—
265	C₂×C₈	—	26	1	no	—
266	C₂×C₈	—	26	1	no	—
267	C₄²	—	13	1	no	—
268	C₄²	—	13	1	no	—
269	C₄²	—	26	1	no	—
270	C₁₃	#2956	1	1	yes	characteristic, complemented, Sylow
271	C₂³	—	13	1	no	—
272	C₂³	—	26	1	no	—
273	C₂³	—	26	1	no	—
274	Q₈	—	2	1	no	—
275	Q₈	—	13	1	no	—
276	Q₈	—	26	1	no	—
277	Q₈	C₂²×#261	1	1	yes	characteristic
278	Q₈	C₂²×#261	1	2	yes	—
279	D₄	—	2	1	no	—
280	D₄	—	13	2	no	—
281	D₄	—	13	1	no	—
282	D₄	—	26	1	no	—
283	D₄	—	26	2	no	—
284	D₄	—	26	2	no	—
285	D₄	—	26	1	no	—
286	D₄	—	26	1	no	—
287	D₄	—	26	1	no	—
288	D₄	—	26	1	no	—
289	D₄	C₂²×#261	1	1	yes	characteristic
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	13	1	no	—
293	C₂×C₄	—	13	1	no	—
294	C₂×C₄	—	13	1	no	—
295	C₂×C₄	—	13	1	no	—
296	C₂×C₄	—	13	1	no	—
297	C₂×C₄	—	26	1	no	—
298	C₂×C₄	—	26	2	no	—
299	C₂×C₄	—	26	2	no	—
300	C₂×C₄	—	26	1	no	—
301	C₂×C₄	—	26	1	no	—
302	C₂×C₄	—	26	1	no	—
303	C₂×C₄	—	26	1	no	—
304	C₂×C₄	—	26	1	no	—
305	C₂×C₄	—	52	2	no	—
306	C₂×C₄	C₂²×#261	1	1	yes	characteristic
307	C₈	—	13	2	no	—
308	C₈	—	13	2	no	—
309	C₈	—	13	2	no	—
310	C₈	—	26	1	no	—
311	C₈	—	26	1	no	—
312	C₈	—	26	2	no	—
313	C₈	C₂²×#261	1	2	yes	—
314	C₂²	—	2	1	no	—
315	C₂²	—	13	1	no	—
316	C₂²	—	13	1	no	—
317	C₂²	—	26	2	no	—
318	C₂²	—	26	1	no	—
319	C₂²	—	52	1	no	—
320	C₂²	—	104	1	no	—
321	C₂²	—	104	1	no	—
322	C₂²	D₄×#261	1	1	yes	characteristic
323	C₄	—	2	1	no	—
324	C₄	—	2	2	no	—
325	C₄	—	13	1	no	—
326	C₄	—	13	1	no	—
327	C₄	—	26	2	no	—
328	C₄	—	26	1	no	—
329	C₄	—	52	1	no	—
330	C₄	D₄×#261	1	1	yes	characteristic
331	C₄	C₂³×#261	1	1	yes	characteristic
332	C₂	—	2	1	no	—
333	C₂	—	4	1	no	—
334	C₂	—	26	1	no	—
335	C₂	—	26	1	no	—
336	C₂	—	52	2	no	—
337	C₂	—	52	1	no	—
338	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×#137	C₄
socle	C₂×C₁₃	C₂×C₄×D₄
prime radical	#11719482	C₂
cosocle	C₄×D₁₃	C₂⁴
hypercenter	#137	#261
hypocenter	C₁₃	#2956
LCS	G ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #137

Complements

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

N	H ≅ G/N	type
1. #11719629	C₂	semidirect
2. #11719625	C₂	semidirect
4. D₁₃⋊#104	C₂	semidirect
5. C₂×#88118	C₂	semidirect
6. #11719112	C₂	semidirect
7. #11719111	C₂	semidirect
8. #11719108	C₂	semidirect
9. #11719107	C₂	semidirect
10. #11719100	C₂	semidirect
11. M₁₆×#261	C₂	semidirect
16. #88240	C₂²	semidirect
17. #88240	C₂²	semidirect
18. #88240	C₂²	semidirect
20. #88238	C₂²	semidirect
28. C₁₃×#137	C₄	semidirect
29. #88203	C₄	semidirect
30. #88196	C₄	semidirect
32. #88172	C₄	semidirect
34. #88168	C₄	semidirect
35. #88162	C₄	semidirect
37. #88118	C₂²	semidirect
38. #88118	C₂²	semidirect
39. #88118	C₂²	semidirect
40. #88118	C₂²	semidirect
41. #88116	C₂²	semidirect
42. #88116	C₂²	semidirect
43. #88104	C₂²	semidirect
44. #88103	C₂²	semidirect
45. #88100	C₂²	semidirect
46. C₈×#261	C₂²	semidirect
89. C₁₃×Q₁₆	C₂×C₄	semidirect
90. C₁₃×SD₁₆	C₂×C₄	semidirect
92. #6107	C₂×C₄	semidirect
93. #6106	C₂×C₄	semidirect
94. #6106	C₂×C₄	semidirect
95. #6104	C₂×C₄	semidirect
97. D₁₀₄	C₂×C₄	semidirect
98. #6095	C₂×C₄	semidirect
178. #137	#261	semidirect
237. Q₁₆	C₂×#261	semidirect
241. SD₁₆	C₂×#261	semidirect
270. C₁₃	#2956	semidirect

Maximal subgroups

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

subgroup	index
#11719629	2
#11719625	2
#11719482	2
D₁₃⋊#104	2
C₂×#88118	2
#11719112	2
#11719111	2
#11719108	2
#11719107	2
#11719100	2
M₁₆×#261	2
#2956	13