Subgroups — #13127752

Order 1664.

360 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11719390	C₂	1	1	yes	maximal, characteristic, complemented
2	D₄×Dic₂₆	C₂	1	1	yes	maximal, characteristic
3	C₁₃×#478	C₂	1	1	yes	maximal, characteristic
4	#11718726	C₂	1	1	yes	maximal, characteristic, complemented
5	#11718679	C₂	1	1	yes	maximal, characteristic
6	#11718536	C₂	1	1	yes	maximal, characteristic, complemented
7	#11718535	C₂	1	1	yes	maximal, characteristic, complemented
8	#11718517	C₂	1	1	yes	maximal, characteristic, complemented
9	#11718515	C₂	1	1	yes	maximal, characteristic, complemented
10	#11718511	C₂	1	1	yes	maximal, characteristic, complemented
11	#11718489	C₂	1	1	yes	maximal, characteristic, complemented
12	#11718427	C₂	1	1	yes	maximal, characteristic
13	#11718420	C₂	1	1	yes	maximal, characteristic
14	#11718419	C₂	1	1	yes	maximal, characteristic
15	#11718418	C₂	1	1	yes	maximal, characteristic
16	C₂²×Dic₂₆	—	2	1	no	—
17	C₁₃×#125	C₂²	1	1	yes	characteristic
18	C₄×D₄×C₁₃	C₂²	1	1	yes	characteristic
19	D₄×Dic₁₃	C₂²	1	1	yes	characteristic
20	D₄×Dic₁₃	C₂²	1	1	yes	characteristic
21	C₂×#6105	C₂²	1	1	yes	characteristic
22	#88178	—	2	1	no	—
23	#88144	C₂²	1	1	yes	characteristic
24	#88143	—	2	1	no	—
25	#88143	—	2	1	no	—
26	#88142	—	2	1	no	—
27	#88142	C₂²	1	1	yes	characteristic
28	#88142	C₂²	1	1	yes	characteristic
29	#88142	C₂²	1	1	yes	characteristic
30	#88141	—	2	1	no	—
31	#88141	C₂²	1	1	yes	characteristic
32	#88141	C₂²	1	1	yes	characteristic
33	#88132	—	2	1	no	—
34	#88132	—	2	1	no	—
35	#88123	C₂²	1	1	yes	characteristic
36	C₄×Dic₂₆	C₂²	1	1	yes	characteristic
37	C₄×Dic₂₆	C₂²	1	1	yes	characteristic
38	C₁₃×#107	C₂²	1	1	yes	characteristic
39	C₁₃×#106	C₂²	1	1	yes	characteristic
40	C₁₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
41	C₁₃×#102	C₂²	1	1	yes	characteristic
42	C₁₃×#102	C₂²	1	1	yes	characteristic
43	#88072	C₂²	1	1	yes	characteristic
44	#88058	C₂²	1	1	yes	characteristic
45	#88058	C₂²	1	1	yes	characteristic
46	#88056	C₂²	1	1	yes	characteristic
47	#88056	C₂²	1	1	yes	characteristic
48	#88055	C₂²	1	1	yes	characteristic
49	#88054	C₂²	1	1	yes	characteristic
50	#88054	C₂²	1	1	yes	characteristic
51	C₈×Dic₁₃	C₂²	1	1	yes	characteristic
52	#88050	C₂²	1	1	yes	characteristic
53	#88048	C₂²	1	1	yes	characteristic
54	#88048	C₂²	1	1	yes	characteristic
55	#88047	C₂²	1	1	yes	characteristic
56	#88047	C₂²	1	1	yes	characteristic
57	#88047	C₂²	1	1	yes	characteristic
58	#88043	C₂²	1	1	yes	characteristic
59	C₂×D₄×C₁₃	C₂³	1	1	yes	characteristic
60	C₂²×C₄×C₁₃	—	2	1	no	—
61	C₂²×Dic₁₃	—	2	1	no	—
62	C₂²×Dic₁₃	—	2	1	no	—
63	C₂×Dic₂₆	—	2	1	no	—
64	C₂×Dic₂₆	—	2	1	no	—
65	C₂×Dic₂₆	—	2	1	no	—
66	C₂×Dic₂₆	—	2	1	no	—
67	C₂×Dic₂₆	—	4	4	no	—
68	C₂×Dic₂₆	D₄	1	1	yes	characteristic
69	C₂×Dic₂₆	C₂³	1	1	yes	characteristic
70	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
71	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
72	C₁₃×C₄⋊C₄	—	2	1	no	—
73	C₁₃×C₄⋊C₄	—	2	1	no	—
74	C₁₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
75	C₁₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
76	C₁₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
77	C₁₃×C₂²⋊C₄	—	2	1	no	—
78	C₄²×C₁₃	C₂³	1	1	yes	characteristic
79	C₂²⋊Dic₁₃	—	2	1	no	—
80	C₂²⋊Dic₁₃	—	2	1	no	—
81	#6105	Q₈	1	4	yes	complemented
82	C₄⋊Dic₁₃	—	2	1	no	—
83	C₄⋊Dic₁₃	—	2	1	no	—
84	C₄⋊Dic₁₃	D₄	1	1	yes	characteristic
85	C₄⋊Dic₁₃	C₂³	1	1	yes	characteristic
86	C₄⋊Dic₁₃	C₂³	1	1	yes	characteristic
87	C₄⋊Dic₁₃	C₂³	1	1	yes	characteristic
88	#6101	—	2	1	no	—
89	#6101	—	2	1	no	—
90	#6101	—	2	1	no	—
91	#6101	—	2	1	no	—
92	#6101	—	2	1	no	—
93	#6101	—	2	1	no	—
94	#6101	—	2	1	no	—
95	#6101	D₄	1	1	yes	characteristic
96	#6101	D₄	1	1	yes	characteristic
97	C₄×Dic₁₃	—	2	1	no	—
98	C₄×Dic₁₃	—	2	1	no	—
99	C₄×Dic₁₃	—	2	1	no	—
100	C₄×Dic₁₃	C₂³	1	1	yes	characteristic
101	C₄×Dic₁₃	C₂³	1	1	yes	characteristic
102	C₂×#1055	C₂³	1	1	yes	characteristic
103	C₂×#1055	C₂³	1	1	yes	characteristic
104	#3386	—	13	1	no	maximal, Sylow
105	C₂³×C₁₃	—	2	1	no	—
106	D₄×C₁₃	—	2	1	no	—
107	D₄×C₁₃	C₂×Q₈	1	2	yes	—
108	C₂×C₄×C₁₃	—	2	1	no	—
109	C₂×C₄×C₁₃	—	2	1	no	—
110	C₂×C₄×C₁₃	—	2	1	no	—
111	C₂×C₄×C₁₃	—	4	2	no	—
112	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
113	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
114	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
115	C₂×Dic₁₃	—	2	1	no	—
116	C₂×Dic₁₃	—	2	1	no	—
117	C₂×Dic₁₃	—	2	1	no	—
118	C₂×Dic₁₃	—	2	1	no	—
119	C₂×Dic₁₃	—	4	2	no	—
120	C₂×Dic₁₃	—	4	2	no	—
121	C₂×Dic₁₃	C₂×D₄	1	1	yes	characteristic
122	C₂×Dic₁₃	C₂×D₄	1	1	yes	characteristic
123	C₂×Dic₁₃	C₂×D₄	1	1	yes	characteristic
124	C₂×Dic₁₃	C₂×D₄	1	1	yes	characteristic
125	Dic₂₆	—	2	1	no	—
126	Dic₂₆	—	2	2	no	—
127	Dic₂₆	—	4	2	no	—
128	Dic₂₆	—	4	2	no	—
129	Dic₂₆	—	4	2	no	—
130	Dic₂₆	—	4	2	no	—
131	Dic₂₆	C₂×Q₈	1	2	yes	complemented
132	C₈×C₁₃	—	2	1	no	—
133	C₈×C₁₃	—	2	1	no	—
134	#1055	—	2	1	no	—
135	#1055	C₂×Q₈	1	2	yes	complemented
136	#557	—	13	1	no	—
137	D₄×Q₈	—	13	1	no	—
138	#501	—	13	1	no	—
139	#501	—	13	1	no	—
140	#500	—	13	1	no	—
141	#478	—	13	1	no	—
142	#478	D₁₃	1	1	yes	characteristic
143	#477	—	13	1	no	—
144	#477	—	13	1	no	—
145	#475	—	13	1	no	—
146	#474	—	13	1	no	—
147	#446	—	13	1	no	—
148	#440	—	13	1	no	—
149	#440	—	13	1	no	—
150	C₄×SD₁₆	—	13	1	no	—
151	C₂²×C₁₃	—	2	2	no	—
152	C₂²×C₁₃	—	4	1	no	—
153	C₂²×C₁₃	—	4	1	no	—
154	C₂²×C₁₃	C₂²×D₄	1	1	yes	characteristic
155	C₄×C₁₃	—	2	1	no	—
156	C₄×C₁₃	—	2	1	no	—
157	C₄×C₁₃	—	4	1	no	—
158	C₄×C₁₃	—	4	1	no	—
159	C₄×C₁₃	—	4	1	no	—
160	C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
161	C₄×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
162	Dic₁₃	—	2	2	no	—
163	Dic₁₃	—	2	1	no	—
164	Dic₁₃	—	2	1	no	—
165	Dic₁₃	—	2	1	no	—
166	Dic₁₃	—	4	1	no	—
167	Dic₁₃	—	4	1	no	—
168	Dic₁₃	—	4	1	no	—
169	Dic₁₃	AGL(1,ℤ/8)	1	2	yes	—
170	C₂²×Q₈	—	26	1	no	—
171	C₂×SD₁₆	—	13	1	no	—
172	C₄⋊Q₈	—	13	1	no	—
173	C₄⋊Q₈	—	13	1	no	—
174	C₄⋊Q₈	—	13	1	no	—
175	C₄⋊Q₈	—	13	1	no	—
176	C₄⋊Q₈	—	26	1	no	—
177	#125	—	13	1	no	—
178	#125	—	26	1	no	—
179	#125	—	26	1	no	—
180	#125	C₂×D₁₃	1	1	yes	characteristic
181	C₂²⋊Q₈	—	26	1	no	—
182	C₂²⋊Q₈	—	26	1	no	—
183	C₂²⋊Q₈	—	26	1	no	—
184	C₄×Q₈	—	13	1	no	—
185	C₄×Q₈	—	13	1	no	—
186	C₄×Q₈	—	13	1	no	—
187	C₄×Q₈	—	13	1	no	—
188	C₄×Q₈	—	26	1	no	—
189	C₄×D₄	—	13	1	no	—
190	C₄×D₄	—	13	1	no	—
191	C₄×D₄	C₂×D₁₃	1	1	yes	characteristic
192	#107	—	13	1	no	—
193	#107	—	13	1	no	—
194	#107	—	13	1	no	—
195	#107	—	13	1	no	—
196	#107	—	13	1	no	—
197	#107	C₂×D₁₃	1	1	yes	characteristic
198	#106	—	13	1	no	—
199	#106	—	13	1	no	—
200	#106	C₂×D₁₃	1	1	yes	characteristic
201	C₄⋊C₈	—	13	1	no	—
202	C₄⋊C₈	—	13	1	no	—
203	C₄⋊C₈	C₂×D₁₃	1	1	yes	characteristic
204	#103	—	13	1	no	—
205	#103	—	13	1	no	—
206	#103	—	13	1	no	—
207	#102	—	13	1	no	—
208	#102	C₂×D₁₃	1	1	yes	characteristic
209	#102	C₂×D₁₃	1	1	yes	characteristic
210	#97	—	13	1	no	—
211	#97	—	13	1	no	—
212	C₄×C₈	—	13	1	no	—
213	C₂×C₁₃	—	4	2	no	—
214	C₂×C₁₃	D₄×Q₈	1	1	yes	characteristic
215	C₂×C₁₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
216	C₂×C₁₃	#578	1	1	yes	characteristic
217	C₂×Q₈	—	13	1	no	—
218	C₂×Q₈	—	13	1	no	—
219	C₂×Q₈	—	26	1	no	—
220	C₂×Q₈	—	26	1	no	—
221	C₂×Q₈	—	26	1	no	—
222	C₂×Q₈	—	26	1	no	—
223	C₂×Q₈	—	52	4	no	—
224	C₂×D₄	C₂²×D₁₃	1	1	yes	characteristic
225	C₂²×C₄	—	2	1	no	—
226	C₂²×C₄	—	26	1	no	—
227	C₂²×C₄	—	26	1	no	—
228	SD₁₆	—	13	4	no	—
229	C₂×C₈	—	13	1	no	—
230	C₂×C₈	—	13	1	no	—
231	C₂×C₈	C₂²×D₁₃	1	1	yes	characteristic
232	C₂×C₈	C₂²×D₁₃	1	1	yes	characteristic
233	C₄⋊C₄	—	2	1	no	—
234	C₄⋊C₄	—	2	1	no	—
235	C₄⋊C₄	—	13	1	no	—
236	C₄⋊C₄	—	13	1	no	—
237	C₄⋊C₄	—	13	1	no	—
238	C₄⋊C₄	—	13	1	no	—
239	C₄⋊C₄	—	13	1	no	—
240	C₄⋊C₄	—	13	1	no	—
241	C₄⋊C₄	—	26	1	no	—
242	C₄⋊C₄	—	26	1	no	—
243	C₄⋊C₄	—	26	1	no	—
244	C₄⋊C₄	—	26	1	no	—
245	C₄⋊C₄	—	26	1	no	—
246	C₄⋊C₄	—	26	1	no	—
247	C₄⋊C₄	—	26	1	no	—
248	C₄⋊C₄	—	26	1	no	—
249	C₄⋊C₄	—	26	1	no	—
250	C₄⋊C₄	C₂²×D₁₃	1	1	yes	characteristic
251	C₄⋊C₄	C₂²×D₁₃	1	1	yes	characteristic
252	C₄⋊C₄	C₂²×D₁₃	1	1	yes	characteristic
253	C₂²⋊C₄	—	2	1	no	—
254	C₂²⋊C₄	—	26	1	no	—
255	C₂²⋊C₄	—	26	1	no	—
256	C₄²	—	13	1	no	—
257	C₄²	—	13	1	no	—
258	C₄²	—	26	1	no	—
259	C₄²	—	26	1	no	—
260	C₄²	—	26	1	no	—
261	C₄²	C₂²×D₁₃	1	1	yes	characteristic
262	C₁₃	#3386	1	1	yes	characteristic, complemented, Sylow
263	C₂³	—	2	1	no	—
264	Q₈	—	13	2	no	—
265	Q₈	—	26	2	no	—
266	Q₈	—	26	1	no	—
267	Q₈	—	52	2	no	—
268	Q₈	—	52	2	no	—
269	Q₈	—	52	2	no	—
270	Q₈	—	52	2	no	—
271	D₄	—	2	1	no	—
272	D₄	Q₈×D₁₃	1	2	yes	—
273	C₂×C₄	—	2	1	no	—
274	C₂×C₄	—	2	1	no	—
275	C₂×C₄	—	2	1	no	—
276	C₂×C₄	—	4	2	no	—
277	C₂×C₄	—	13	1	no	—
278	C₂×C₄	—	13	1	no	—
279	C₂×C₄	—	13	1	no	—
280	C₂×C₄	—	13	1	no	—
281	C₂×C₄	—	26	1	no	—
282	C₂×C₄	—	26	1	no	—
283	C₂×C₄	—	26	1	no	—
284	C₂×C₄	—	26	1	no	—
285	C₂×C₄	—	52	2	no	—
286	C₂×C₄	—	52	2	no	—
287	C₂×C₄	D₄×D₁₃	1	1	yes	characteristic
288	C₂×C₄	D₄×D₁₃	1	1	yes	characteristic
289	C₂×C₄	C₂³×D₁₃	1	1	yes	characteristic
290	C₈	—	2	1	no	—
291	C₈	—	2	1	no	—
292	C₈	—	13	2	no	—
293	C₈	—	26	1	no	—
294	C₂²	—	2	2	no	—
295	C₂²	—	4	1	no	—
296	C₂²	—	4	1	no	—
297	C₂²	C₂×D₄×D₁₃	1	1	yes	characteristic
298	C₄	—	2	1	no	—
299	C₄	—	2	1	no	—
300	C₄	—	4	1	no	—
301	C₄	—	4	1	no	—
302	C₄	—	4	1	no	—
303	C₄	—	13	2	no	—
304	C₄	—	26	1	no	—
305	C₄	—	26	1	no	—
306	C₄	—	26	1	no	—
307	C₄	—	26	2	no	—
308	C₄	—	52	1	no	—
309	C₄	—	52	1	no	—
310	C₄	—	52	1	no	—
311	C₄	C₂×Q₈×D₁₃	1	1	yes	characteristic
312	C₄	#88257	1	1	yes	characteristic
313	C₂	—	4	2	no	—
314	C₂	#11719337	1	1	yes	characteristic
315	C₂	#11719476	1	1	yes	characteristic
316	C₂	D₁₃×AGL(1,ℤ/8)	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₃
commutator, cosocle	C₂×C₄×C₁₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₃
Fitting	C₁₃×#478	C₂
socle	C₂²×C₁₃	C₂²×D₄
prime radical	C₂×D₄×C₁₃	C₂³
hypercenter	#478	D₁₃
hypocenter	C₁₃	#3386
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #478

Complements

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

N	H ≅ G/N	type
1. #11719390	C₂	semidirect
4. #11718726	C₂	semidirect
6. #11718536	C₂	semidirect
7. #11718535	C₂	semidirect
8. #11718517	C₂	semidirect
9. #11718515	C₂	semidirect
10. #11718511	C₂	semidirect
11. #11718489	C₂	semidirect
81. #6105	Q₈	semidirect
131. Dic₂₆	C₂×Q₈	semidirect
135. #1055	C₂×Q₈	semidirect
262. C₁₃	#3386	semidirect

Maximal subgroups

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

subgroup	index
#11719390	2
D₄×Dic₂₆	2
C₁₃×#478	2
#11718726	2
#11718679	2
#11718536	2
#11718535	2
#11718517	2
#11718515	2
#11718511	2
#11718489	2
#11718427	2
#11718420	2
#11718419	2
#11718418	2
#3386	13