Subgroups — #15761444

Order 3328.

318 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13115781	C₂	1	1	yes	maximal, characteristic
2	#13115612	C₂	1	1	yes	maximal, characteristic
3	#13113291	C₂	1	1	yes	maximal, characteristic
4	#13113126	C₂	1	1	yes	maximal, characteristic
5	#13112738	C₂	1	1	yes	maximal, characteristic
6	#13112675	C₂	1	1	yes	maximal, characteristic
7	C₁₃×#1542	C₂	1	1	yes	maximal, characteristic
8	C₁₃×#533	C₂²	1	1	yes	characteristic
9	C₁₃×#431	C₂²	1	1	yes	characteristic
10	C₂×C₁₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
11	C₂×#88059	—	2	1	no	—
12	C₄×C₄⋊Dic₁₃	C₂²	1	1	yes	characteristic
13	#11718585	C₂²	1	1	yes	characteristic
14	C₄×#6099	C₂²	1	1	yes	characteristic
15	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
16	C₁₃×#333	C₄	1	1	yes	characteristic, complemented
17	C₁₃×#333	C₄	1	1	yes	characteristic, complemented
18	C₁₃×#332	C₄	1	1	yes	characteristic, complemented
19	C₁₃×#332	C₄	1	1	yes	characteristic, complemented
20	#11718144	—	2	1	no	—
21	Q₈⋊#1055	—	2	1	no	—
22	Q₈⋊#1055	—	2	1	no	—
23	#11718124	—	2	1	no	—
24	#11718124	—	2	1	no	—
25	#11718047	—	2	1	no	—
26	#11718046	—	2	1	no	—
27	C₂²×C₈×C₁₃	—	2	1	no	—
28	C₁₃×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
29	C₁₃×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
30	C₁₃×#125	—	2	1	no	—
31	C₁₃×C₂²⋊Q₈	—	2	1	no	—
32	C₁₃×C₂²⋊Q₈	—	2	1	no	—
33	C₄×Q₈×C₁₃	—	2	1	no	—
34	C₄×Q₈×C₁₃	—	2	1	no	—
35	C₁₃×#117	—	2	1	no	—
36	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
37	C₂×C₄⋊Dic₁₃	—	2	1	no	—
38	C₂×C₄×Dic₁₃	—	2	1	no	—
39	C₂×#6099	—	2	1	no	—
40	C₂×#6099	D₄	1	1	yes	characteristic
41	C₂×#6099	D₄	1	1	yes	characteristic
42	C₂²×#1055	—	2	1	no	—
43	C₁₃×C₄⋊C₈	—	2	1	no	—
44	C₁₃×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
45	C₁₃×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
46	C₁₃×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
47	C₁₃×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
48	C₁₃×C₂²⋊C₈	—	2	1	no	—
49	C₄×C₈×C₁₃	—	2	1	no	—
50	#88071	—	2	1	no	—
51	#88059	—	4	2	no	—
52	C₄⋊#1055	—	2	1	no	—
53	C₄⋊#1055	—	2	1	no	—
54	#88043	—	2	1	no	—
55	C₄×#1055	—	2	1	no	—
56	C₄×#1055	—	2	1	no	—
57	C₄×#1055	D₄	1	1	yes	characteristic
58	C₄×#1055	D₄	1	1	yes	characteristic
59	#9591	—	13	1	no	maximal, Sylow
60	C₂×Q₈×C₁₃	—	2	1	no	—
61	C₂×Q₈×C₁₃	—	2	1	no	—
62	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
64	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
65	C₂²×Dic₁₃	—	4	1	no	—
66	C₂×C₈×C₁₃	—	2	1	no	—
67	C₂×C₈×C₁₃	—	2	1	no	—
68	C₂×C₈×C₁₃	—	2	1	no	—
69	C₂×C₈×C₁₃	—	2	1	no	—
70	C₂×C₈×C₁₃	—	4	1	no	—
71	C₂×C₈×C₁₃	—	4	1	no	—
72	C₁₃×C₄⋊C₄	—	2	1	no	—
73	C₁₃×C₄⋊C₄	—	2	1	no	—
74	C₁₃×C₄⋊C₄	—	2	1	no	—
75	C₁₃×C₄⋊C₄	—	2	1	no	—
76	C₁₃×C₄⋊C₄	—	4	1	no	—
77	C₁₃×C₄⋊C₄	—	4	1	no	—
78	C₁₃×C₂²⋊C₄	—	4	1	no	—
79	C₄²×C₁₃	—	4	1	no	—
80	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
81	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
82	C₄²×C₁₃	#41	1	1	yes	characteristic
83	C₄²×C₁₃	#41	1	1	yes	characteristic
84	C₄⋊Dic₁₃	—	2	2	no	—
85	C₄⋊Dic₁₃	—	4	1	no	—
86	C₄×Dic₁₃	—	4	1	no	—
87	C₄×Dic₁₃	—	4	1	no	—
88	#6099	—	2	1	no	—
89	#6099	—	2	1	no	—
90	#6099	—	2	1	no	—
91	#6099	—	2	1	no	—
92	#6099	—	4	1	no	—
93	#6099	—	4	1	no	—
94	C₂×#1055	—	2	1	no	—
95	C₂×#1055	—	2	1	no	—
96	C₂×#1055	—	2	1	no	—
97	C₂×#1055	—	2	1	no	—
98	C₂×#1055	—	4	2	no	—
99	C₂×#1055	—	4	1	no	—
100	#1986	—	13	1	no	—
101	#1950	—	13	1	no	—
102	#1542	D₁₃	1	1	yes	characteristic
103	#1493	—	13	1	no	—
104	#1451	—	13	1	no	—
105	#1303	—	13	1	no	—
106	#1279	—	13	1	no	—
107	C₂³×C₁₃	#124	1	1	yes	characteristic
108	Q₈×C₁₃	—	4	1	no	—
109	Q₈×C₁₃	—	4	1	no	—
110	Q₈×C₁₃	—	8	1	no	—
111	C₂×C₄×C₁₃	—	2	1	no	—
112	C₂×C₄×C₁₃	—	2	1	no	—
113	C₂×C₄×C₁₃	—	4	1	no	—
114	C₂×C₄×C₁₃	—	4	1	no	—
115	C₂×C₄×C₁₃	—	4	1	no	—
116	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
117	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
118	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
119	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
120	C₂×C₄×C₁₃	#123	1	1	yes	characteristic
121	C₂×C₄×C₁₃	#126	1	1	yes	characteristic
122	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
123	C₂×C₄×C₁₃	#137	1	1	yes	characteristic
124	C₂×C₄×C₁₃	#137	1	1	yes	characteristic
125	C₂×C₄×C₁₃	#137	1	1	yes	characteristic
126	C₂×Dic₁₃	—	4	2	no	—
127	C₂×Dic₁₃	—	8	1	no	—
128	C₂×Dic₁₃	—	8	1	no	—
129	C₈×C₁₃	—	4	1	no	—
130	C₈×C₁₃	—	4	1	no	—
131	C₈×C₁₃	—	4	1	no	—
132	C₈×C₁₃	—	4	1	no	—
133	#1055	—	2	2	no	—
134	#1055	—	4	1	no	—
135	#1055	—	4	1	no	—
136	#1055	—	4	1	no	—
137	#1055	—	8	1	no	—
138	#533	C₂×D₁₃	1	1	yes	characteristic
139	#431	C₂×D₁₃	1	1	yes	characteristic
140	C₂×#108	—	26	1	no	—
141	#423	—	13	1	no	—
142	C₂×C₄⋊C₈	C₂×D₁₃	1	1	yes	characteristic
143	C₄×M₁₆	—	13	1	no	—
144	C₂×C₄×C₈	—	13	1	no	—
145	C₄×C₄⋊C₄	—	13	1	no	—
146	#341	—	26	1	no	—
147	#339	—	26	1	no	—
148	#339	—	26	1	no	—
149	#333	Dic₁₃	1	1	yes	characteristic, complemented
150	#333	Dic₁₃	1	1	yes	characteristic, complemented
151	#332	Dic₁₃	1	1	yes	characteristic, complemented
152	#332	Dic₁₃	1	1	yes	characteristic, complemented
153	Q₈⋊C₈	—	26	1	no	—
154	Q₈⋊C₈	—	26	1	no	—
155	#322	—	26	1	no	—
156	#322	—	26	1	no	—
157	C₂²×C₁₃	—	2	1	no	—
158	C₂²×C₁₃	—	2	1	no	—
159	C₂²×C₁₃	#388	1	1	yes	characteristic
160	C₂²×C₁₃	#489	1	1	yes	characteristic
161	C₂²×C₁₃	#491	1	1	yes	characteristic
162	C₄×C₁₃	—	2	1	no	—
163	C₄×C₁₃	—	2	1	no	—
164	C₄×C₁₃	—	2	1	no	—
165	C₄×C₁₃	—	8	1	no	—
166	C₄×C₁₃	—	8	1	no	—
167	C₄×C₁₃	#440	1	1	yes	characteristic
168	C₄×C₁₃	#441	1	1	yes	characteristic
169	C₄×C₁₃	#443	1	1	yes	characteristic
170	C₄×C₁₃	#444	1	1	yes	characteristic
171	C₄×C₁₃	#468	1	1	yes	characteristic
172	C₄×C₁₃	#470	1	1	yes	characteristic
173	Dic₁₃	—	8	2	no	—
174	C₂×M₁₆	—	13	1	no	—
175	C₂×M₁₆	—	13	1	no	—
176	C₂×M₁₆	—	26	1	no	—
177	C₂²×C₈	—	2	1	no	—
178	C₂²×C₈	—	26	1	no	—
179	C₄⋊Q₈	C₂×Dic₁₃	1	1	yes	characteristic
180	C₄⋊Q₈	C₂×Dic₁₃	1	1	yes	characteristic
181	#125	—	2	1	no	—
182	C₂²⋊Q₈	—	2	1	no	—
183	C₂²⋊Q₈	—	2	1	no	—
184	C₄×Q₈	—	2	1	no	—
185	C₄×Q₈	—	2	1	no	—
186	#117	—	2	1	no	—
187	C₂×C₄⋊C₄	—	26	1	no	—
188	C₂×C₄²	—	26	1	no	—
189	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
190	#108	—	52	2	no	—
191	C₄⋊C₈	—	2	1	no	—
192	C₄⋊C₈	—	26	1	no	—
193	C₄⋊C₈	—	26	1	no	—
194	C₄⋊C₈	C₂×Dic₁₃	1	1	yes	characteristic
195	C₄⋊C₈	C₂×Dic₁₃	1	1	yes	characteristic
196	C₄⋊C₈	C₂×Dic₁₃	1	1	yes	characteristic
197	C₄⋊C₈	C₂×Dic₁₃	1	1	yes	characteristic
198	C₂²⋊C₈	—	2	1	no	—
199	#97	—	26	1	no	—
200	C₄×C₈	—	2	1	no	—
201	C₄×C₈	—	13	1	no	—
202	C₄×C₈	—	13	1	no	—
203	C₄×C₈	—	26	1	no	—
204	C₄×C₈	—	26	1	no	—
205	#95	—	26	1	no	—
206	C₂×C₁₃	—	2	1	no	—
207	C₂×C₁₃	—	2	1	no	—
208	C₂×C₁₃	#1929	1	1	yes	characteristic
209	C₂×C₁₃	#1932	1	1	yes	characteristic
210	C₂×C₁₃	#1935	1	1	yes	characteristic
211	C₂×Q₈	—	2	1	no	—
212	C₂×Q₈	—	2	1	no	—
213	C₂²×C₄	—	52	1	no	—
214	C₂²×C₄	#6127	1	1	yes	characteristic
215	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
216	C₂²×C₄	#6129	1	1	yes	characteristic
217	M₁₆	—	26	1	no	—
218	M₁₆	—	26	1	no	—
219	M₁₆	—	26	1	no	—
220	M₁₆	—	26	1	no	—
221	M₁₆	—	52	1	no	—
222	M₁₆	—	52	1	no	—
223	C₂×C₈	—	2	1	no	—
224	C₂×C₈	—	2	1	no	—
225	C₂×C₈	—	2	1	no	—
226	C₂×C₈	—	2	1	no	—
227	C₂×C₈	—	4	1	no	—
228	C₂×C₈	—	4	1	no	—
229	C₂×C₈	—	26	1	no	—
230	C₂×C₈	—	26	1	no	—
231	C₂×C₈	—	26	1	no	—
232	C₂×C₈	—	26	1	no	—
233	C₂×C₈	—	52	2	no	—
234	C₂×C₈	—	52	1	no	—
235	C₄⋊C₄	—	2	1	no	—
236	C₄⋊C₄	—	2	1	no	—
237	C₄⋊C₄	—	2	1	no	—
238	C₄⋊C₄	—	2	1	no	—
239	C₄⋊C₄	—	4	1	no	—
240	C₄⋊C₄	—	4	1	no	—
241	C₄⋊C₄	—	26	2	no	—
242	C₄⋊C₄	—	52	1	no	—
243	C₂²⋊C₄	—	4	1	no	—
244	C₄²	—	4	1	no	—
245	C₄²	—	52	1	no	—
246	C₄²	—	52	1	no	—
247	C₄²	#6127	1	1	yes	characteristic
248	C₄²	D₄×D₁₃	1	1	yes	characteristic
249	C₄²	#6129	1	1	yes	characteristic
250	C₄²	C₂²×Dic₁₃	1	1	yes	characteristic
251	C₁₃	#9591	1	1	yes	characteristic, complemented, Sylow
252	C₂³	#88139	1	1	yes	characteristic
253	Q₈	—	4	1	no	—
254	Q₈	—	4	1	no	—
255	Q₈	—	8	1	no	—
256	C₂×C₄	—	2	1	no	—
257	C₂×C₄	—	2	1	no	—
258	C₂×C₄	—	4	1	no	—
259	C₂×C₄	—	4	1	no	—
260	C₂×C₄	—	4	1	no	—
261	C₂×C₄	—	52	2	no	—
262	C₂×C₄	—	104	1	no	—
263	C₂×C₄	—	104	1	no	—
264	C₂×C₄	#88133	1	1	yes	characteristic
265	C₂×C₄	#88137	1	1	yes	characteristic
266	C₂×C₄	#88138	1	1	yes	characteristic
267	C₂×C₄	#88168	1	1	yes	characteristic
268	C₂×C₄	#88169	1	1	yes	characteristic
269	C₂×C₄	#88172	1	1	yes	characteristic
270	C₂×C₄	#88172	1	1	yes	characteristic
271	C₂×C₄	#88180	1	1	yes	characteristic
272	C₂×C₄	D₄×Dic₁₃	1	1	yes	characteristic
273	C₂×C₄	D₄×Dic₁₃	1	1	yes	characteristic
274	C₈	—	4	1	no	—
275	C₈	—	4	1	no	—
276	C₈	—	4	1	no	—
277	C₈	—	4	1	no	—
278	C₈	—	26	2	no	—
279	C₈	—	52	1	no	—
280	C₈	—	52	1	no	—
281	C₈	—	52	1	no	—
282	C₈	—	104	1	no	—
283	C₂²	—	2	1	no	—
284	C₂²	—	2	1	no	—
285	C₂²	#11718456	1	1	yes	characteristic
286	C₂²	#11718458	1	1	yes	characteristic
287	C₂²	#11718612	1	1	yes	characteristic
288	C₄	—	2	1	no	—
289	C₄	—	2	1	no	—
290	C₄	—	2	1	no	—
291	C₄	—	8	1	no	—
292	C₄	—	8	1	no	—
293	C₄	—	104	2	no	—
294	C₄	#11718473	1	1	yes	characteristic
295	C₄	#11718477	1	1	yes	characteristic
296	C₄	#11718825	1	1	yes	characteristic
297	C₄	#11718845	1	1	yes	characteristic
298	C₄	#11718857	1	1	yes	characteristic
299	C₄	#11718858	1	1	yes	characteristic
300	C₂	—	2	1	no	—
301	C₂	—	2	1	no	—
302	C₂	#13115486	1	1	yes	characteristic
303	C₂	#13115507	1	1	yes	characteristic
304	C₂	#13115528	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#88138
commutator	C₄²×C₁₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₃
Fitting	C₁₃×#1542	C₂
socle	C₂²×C₁₃	#388
prime radical	C₂³×C₁₃	#124
cosocle	C₂×C₄²×C₁₃	C₂³
hypercenter	#1542	D₁₃
hypocenter	C₁₃	#9591
LCS	G ▷ C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1542

Complements

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

N	H ≅ G/N	type
16. C₁₃×#333	C₄	semidirect
17. C₁₃×#333	C₄	semidirect
18. C₁₃×#332	C₄	semidirect
19. C₁₃×#332	C₄	semidirect
149. #333	Dic₁₃	semidirect
150. #333	Dic₁₃	semidirect
151. #332	Dic₁₃	semidirect
152. #332	Dic₁₃	semidirect
251. C₁₃	#9591	semidirect

Maximal subgroups

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

subgroup	index
#13115781	2
#13115612	2
#13113291	2
#13113126	2
#13112738	2
#13112675	2
C₁₃×#1542	2
#9591	13