Subgroups — C₁₃×#12992 #15699331

Order 3328.

582 conjugacy classes of subgroups.

The subgroup lattice has 314 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₃×#466	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₃×#427	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₁₃×#357	C₂	1	2	yes	maximal, complemented
4	C₂×C₁₃×C₂²⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₃×#1380	C₂	1	2	yes	maximal, complemented
6	C₂²×C₁₃×D₈	C₂²	1	1	yes	characteristic
7	C₂³×C₈×C₁₃	C₂²	1	1	yes	characteristic
8	C₂×C₁₃×#121	—	2	1	no	—
9	C₂×C₁₃×#117	—	2	1	no	—
10	C₂²×C₁₃×C₁₆	C₂²	1	2	yes	—
11	C₁₃×#466	—	2	2	no	—
12	C₁₃×#466	C₄	1	4	yes	complemented
13	C₁₃×#427	—	2	4	no	—
14	C₂×C₁₃×#107	C₂²	1	1	yes	characteristic
15	C₂×C₁₃×#107	C₂²	1	2	yes	—
16	C₂×C₁₃×#106	—	2	1	no	—
17	C₂×C₁₃×#102	—	2	1	no	—
18	C₁₃×#357	—	2	4	no	—
19	C₁₃×C₂²⋊C₁₆	—	2	2	no	—
20	C₂²×D₄×C₁₃	—	2	1	no	—
21	C₂²×D₄×C₁₃	—	4	1	no	—
22	C₂³×C₄×C₁₃	D₄	1	1	yes	characteristic
23	C₂×C₁₃×D₈	—	2	2	no	—
24	C₂×C₁₃×D₈	—	2	1	no	—
25	C₂×C₁₃×D₈	—	2	2	no	—
26	C₂×C₁₃×D₈	C₂×C₄	1	2	yes	—
27	C₂²×C₈×C₁₃	—	2	2	no	—
28	C₂²×C₈×C₁₃	—	2	2	no	—
29	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
30	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
31	C₂²×C₈×C₁₃	D₄	1	2	yes	—
32	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
33	C₁₃×#121	—	2	4	no	—
34	C₁₃×#121	—	4	2	no	—
35	C₁₃×#117	—	2	4	no	—
36	C₁₃×#117	—	4	2	no	—
37	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
38	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
39	C₂×C₁₃×C₂²⋊C₄	—	4	1	no	—
40	C₂×C₁₃×C₂²⋊C₄	—	4	1	no	—
41	C₂×C₄²×C₁₃	—	4	1	no	—
42	C₂×C₁₃×C₁₆	—	2	2	no	—
43	C₂×C₁₃×C₁₆	—	2	2	no	—
44	C₂×C₁₃×C₁₆	—	2	2	no	—
45	C₁₃×#107	—	2	2	no	—
46	C₁₃×#107	—	2	1	no	—
47	C₁₃×#107	C₂×C₄	1	2	yes	—
48	C₁₃×#107	D₄	1	4	yes	complemented
49	C₁₃×#106	—	2	4	no	—
50	C₁₃×#102	—	2	4	no	—
51	#12992	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
52	C₂⁴×C₁₃	—	4	1	no	—
53	C₂⁴×C₁₃	D₈	1	1	yes	characteristic
54	C₂×D₄×C₁₃	—	2	2	no	—
55	C₂×D₄×C₁₃	—	4	4	no	—
56	C₂×D₄×C₁₃	—	4	1	no	—
57	C₂×D₄×C₁₃	—	4	1	no	—
58	C₂×D₄×C₁₃	—	4	1	no	—
59	C₂×D₄×C₁₃	—	4	1	no	—
60	C₂×D₄×C₁₃	—	4	1	no	—
61	C₂×D₄×C₁₃	—	8	4	no	—
62	C₂²×C₄×C₁₃	—	2	1	no	—
63	C₂²×C₄×C₁₃	—	2	1	no	—
64	C₂²×C₄×C₁₃	—	2	1	no	—
65	C₂²×C₄×C₁₃	—	2	1	no	—
66	C₂²×C₄×C₁₃	—	4	2	no	—
67	C₂²×C₄×C₁₃	—	4	1	no	—
68	C₂²×C₄×C₁₃	C₂²⋊C₄	1	2	yes	—
69	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
70	C₂²×C₄×C₁₃	SD₁₆	1	2	yes	—
71	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
72	C₁₃×D₈	—	2	4	no	—
73	C₁₃×D₈	—	4	2	no	—
74	C₂×C₈×C₁₃	—	2	1	no	—
75	C₂×C₈×C₁₃	—	2	1	no	—
76	C₂×C₈×C₁₃	—	2	4	no	—
77	C₂×C₈×C₁₃	—	2	4	no	—
78	C₂×C₈×C₁₃	C₂²⋊C₄	1	2	yes	—
79	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
81	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
82	C₂×C₈×C₁₃	#41	1	2	yes	—
83	C₁₃×C₄⋊C₄	—	2	2	no	—
84	C₁₃×C₄⋊C₄	—	2	4	no	—
85	C₁₃×C₄⋊C₄	—	4	1	no	—
86	C₁₃×C₄⋊C₄	—	4	2	no	—
87	C₁₃×C₂²⋊C₄	—	4	4	no	—
88	C₁₃×C₂²⋊C₄	—	4	4	no	—
89	C₄²×C₁₃	—	4	4	no	—
90	C₁₃×C₁₆	—	4	2	no	—
91	C₂×#466	C₂×C₁₃	1	1	yes	characteristic
92	C₂×#427	C₂×C₁₃	1	1	yes	characteristic, complemented
93	C₂×#357	C₂×C₁₃	1	2	yes	complemented
94	C₂×C₂²⋊C₁₆	C₂×C₁₃	1	1	yes	characteristic, complemented
95	#1380	C₂×C₁₃	1	2	yes	complemented
96	C₂³×C₁₃	—	2	1	no	—
97	C₂³×C₁₃	—	2	1	no	—
98	C₂³×C₁₃	—	2	2	no	—
99	C₂³×C₁₃	—	2	2	no	—
100	C₂³×C₁₃	—	4	2	no	—
101	C₂³×C₁₃	—	8	1	no	—
102	C₂³×C₁₃	—	8	1	no	—
103	C₂³×C₁₃	—	8	2	no	—
104	C₂³×C₁₃	—	8	2	no	—
105	C₂³×C₁₃	#102	1	2	yes	—
106	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
107	D₄×C₁₃	—	4	2	no	—
108	D₄×C₁₃	—	4	2	no	—
109	D₄×C₁₃	—	8	8	no	—
110	D₄×C₁₃	—	8	1	no	—
111	D₄×C₁₃	—	8	1	no	—
112	C₂×C₄×C₁₃	—	2	1	no	—
113	C₂×C₄×C₁₃	—	2	1	no	—
114	C₂×C₄×C₁₃	—	2	2	no	—
115	C₂×C₄×C₁₃	—	2	2	no	—
116	C₂×C₄×C₁₃	—	2	2	no	—
117	C₂×C₄×C₁₃	—	2	2	no	—
118	C₂×C₄×C₁₃	—	4	2	no	—
119	C₂×C₄×C₁₃	—	4	4	no	—
120	C₂×C₄×C₁₃	—	8	2	no	—
121	C₂×C₄×C₁₃	—	8	2	no	—
122	C₂×C₄×C₁₃	—	8	2	no	—
123	C₂×C₄×C₁₃	#102	1	2	yes	—
124	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
125	C₂×C₄×C₁₃	#120	1	1	yes	characteristic
126	C₂×C₄×C₁₃	#123	1	1	yes	characteristic
127	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
128	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
129	C₂×C₄×C₁₃	#137	1	1	yes	characteristic
130	C₈×C₁₃	—	2	2	no	—
131	C₈×C₁₃	C₄×D₄	1	2	yes	—
132	C₈×C₁₃	#121	1	2	yes	—
133	C₂²×D₈	C₂²×C₁₃	1	1	yes	characteristic
134	C₂³×C₈	C₂²×C₁₃	1	1	yes	characteristic
135	C₂×#121	—	2	1	no	—
136	C₂×#117	—	2	1	no	—
137	C₂²×C₁₆	C₂²×C₁₃	1	2	yes	—
138	#466	—	2	2	no	—
139	#466	C₄×C₁₃	1	4	yes	complemented
140	#427	—	2	4	no	—
141	C₂×#107	C₂²×C₁₃	1	1	yes	characteristic
142	C₂×#107	C₂²×C₁₃	1	2	yes	—
143	C₂×#106	—	2	1	no	—
144	C₂×#102	—	2	1	no	—
145	#357	—	2	4	no	—
146	C₂²⋊C₁₆	—	2	2	no	—
147	C₂²×C₁₃	—	2	2	no	—
148	C₂²×C₁₃	—	2	2	no	—
149	C₂²×C₁₃	—	2	2	no	—
150	C₂²×C₁₃	—	4	2	no	—
151	C₂²×C₁₃	—	4	2	no	—
152	C₂²×C₁₃	—	8	2	no	—
153	C₂²×C₁₃	—	8	2	no	—
154	C₂²×C₁₃	—	8	2	no	—
155	C₂²×C₁₃	—	16	2	no	—
156	C₂²×C₁₃	—	16	2	no	—
157	C₂²×C₁₃	C₂×#102	1	1	yes	characteristic
158	C₂²×C₁₃	#447	1	1	yes	characteristic
159	C₂²×C₁₃	#480	1	1	yes	characteristic
160	C₂²×C₁₃	#508	1	2	yes	—
161	C₂²×C₁₃	#509	1	2	yes	—
162	C₄×C₁₃	—	2	2	no	—
163	C₄×C₁₃	—	8	2	no	—
164	C₄×C₁₃	—	8	4	no	—
165	C₄×C₁₃	#386	1	1	yes	characteristic
166	C₄×C₁₃	#417	1	1	yes	characteristic
167	C₄×C₁₃	#448	1	1	yes	characteristic
168	C₄×C₁₃	#481	1	1	yes	characteristic
169	C₂²×D₄	—	2	1	no	—
170	C₂²×D₄	—	4	1	no	—
171	C₂³×C₄	D₄×C₁₃	1	1	yes	characteristic
172	C₂×D₈	—	2	2	no	—
173	C₂×D₈	—	2	1	no	—
174	C₂×D₈	—	2	2	no	—
175	C₂×D₈	C₂×C₄×C₁₃	1	2	yes	—
176	C₂²×C₈	—	2	2	no	—
177	C₂²×C₈	—	2	2	no	—
178	C₂²×C₈	C₂×C₄×C₁₃	1	2	yes	—
179	C₂²×C₈	D₄×C₁₃	1	1	yes	characteristic
180	C₂²×C₈	D₄×C₁₃	1	2	yes	—
181	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
182	#121	—	2	4	no	—
183	#121	—	4	2	no	—
184	#117	—	2	4	no	—
185	#117	—	4	2	no	—
186	C₂×C₄⋊C₄	—	2	2	no	—
187	C₂×C₄⋊C₄	—	2	1	no	—
188	C₂×C₂²⋊C₄	—	4	1	no	—
189	C₂×C₂²⋊C₄	—	4	1	no	—
190	C₂×C₄²	—	4	1	no	—
191	C₂×C₁₆	—	2	2	no	—
192	C₂×C₁₆	—	2	2	no	—
193	C₂×C₁₆	—	2	2	no	—
194	#107	—	2	2	no	—
195	#107	—	2	1	no	—
196	#107	C₂×C₄×C₁₃	1	2	yes	—
197	#107	D₄×C₁₃	1	4	yes	complemented
198	#106	—	2	4	no	—
199	#102	—	2	4	no	—
200	C₂×C₁₃	—	4	2	no	—
201	C₂×C₁₃	—	16	2	no	—
202	C₂×C₁₃	#1874	1	1	yes	characteristic
203	C₂×C₁₃	#2138	1	1	yes	characteristic
204	C₂×C₁₃	#2140	1	1	yes	characteristic
205	C₂×C₁₃	#2186	1	2	yes	—
206	C₂×C₁₃	#2234	1	2	yes	—
207	C₂⁴	—	4	1	no	—
208	C₂⁴	C₁₃×D₈	1	1	yes	characteristic
209	C₂×D₄	—	2	2	no	—
210	C₂×D₄	—	4	4	no	—
211	C₂×D₄	—	4	1	no	—
212	C₂×D₄	—	4	1	no	—
213	C₂×D₄	—	4	1	no	—
214	C₂×D₄	—	4	1	no	—
215	C₂×D₄	—	4	1	no	—
216	C₂×D₄	—	8	4	no	—
217	C₂²×C₄	—	2	1	no	—
218	C₂²×C₄	—	2	1	no	—
219	C₂²×C₄	—	2	1	no	—
220	C₂²×C₄	—	2	1	no	—
221	C₂²×C₄	—	4	2	no	—
222	C₂²×C₄	—	4	1	no	—
223	C₂²×C₄	C₁₃×C₂²⋊C₄	1	2	yes	—
224	C₂²×C₄	C₁₃×D₈	1	1	yes	characteristic
225	C₂²×C₄	C₁₃×SD₁₆	1	2	yes	—
226	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
227	D₈	—	2	4	no	—
228	D₈	—	4	2	no	—
229	C₂×C₈	—	2	1	no	—
230	C₂×C₈	—	2	1	no	—
231	C₂×C₈	—	2	4	no	—
232	C₂×C₈	—	2	4	no	—
233	C₂×C₈	C₁₃×C₂²⋊C₄	1	2	yes	—
234	C₂×C₈	C₂²×C₄×C₁₃	1	1	yes	characteristic
235	C₂×C₈	C₂×D₄×C₁₃	1	2	yes	—
236	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
237	C₂×C₈	C₁₃×#41	1	2	yes	—
238	C₄⋊C₄	—	2	2	no	—
239	C₄⋊C₄	—	2	4	no	—
240	C₄⋊C₄	—	4	1	no	—
241	C₄⋊C₄	—	4	2	no	—
242	C₂²⋊C₄	—	4	4	no	—
243	C₂²⋊C₄	—	4	4	no	—
244	C₄²	—	4	4	no	—
245	C₁₆	—	4	2	no	—
246	C₁₃	#12992	1	1	yes	characteristic, direct product factor, Sylow
247	C₂³	—	2	1	no	—
248	C₂³	—	2	1	no	—
249	C₂³	—	2	2	no	—
250	C₂³	—	2	2	no	—
251	C₂³	—	4	2	no	—
252	C₂³	—	8	1	no	—
253	C₂³	—	8	1	no	—
254	C₂³	—	8	2	no	—
255	C₂³	—	8	2	no	—
256	C₂³	C₁₃×#102	1	2	yes	—
257	C₂³	C₂×C₁₃×D₈	1	1	yes	characteristic
258	D₄	—	4	2	no	—
259	D₄	—	4	2	no	—
260	D₄	—	8	8	no	—
261	D₄	—	8	1	no	—
262	D₄	—	8	1	no	—
263	C₂×C₄	—	2	1	no	—
264	C₂×C₄	—	2	1	no	—
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	—	2	2	no	—
268	C₂×C₄	—	2	2	no	—
269	C₂×C₄	—	4	2	no	—
270	C₂×C₄	—	4	4	no	—
271	C₂×C₄	—	8	2	no	—
272	C₂×C₄	—	8	2	no	—
273	C₂×C₄	—	8	2	no	—
274	C₂×C₄	C₁₃×#102	1	2	yes	—
275	C₂×C₄	C₂×C₁₃×C₂²⋊C₄	1	1	yes	characteristic
276	C₂×C₄	C₁₃×#120	1	1	yes	characteristic
277	C₂×C₄	C₁₃×#123	1	1	yes	characteristic
278	C₂×C₄	C₂×C₁₃×SD₁₆	1	1	yes	characteristic
279	C₂×C₄	C₁₃×AGL(1,ℤ/8)	1	1	yes	characteristic
280	C₂×C₄	C₁₃×#137	1	1	yes	characteristic
281	C₈	—	2	2	no	—
282	C₈	C₄×D₄×C₁₃	1	2	yes	—
283	C₈	C₁₃×#121	1	2	yes	—
284	C₂²	—	2	2	no	—
285	C₂²	—	2	2	no	—
286	C₂²	—	2	2	no	—
287	C₂²	—	4	2	no	—
288	C₂²	—	4	2	no	—
289	C₂²	—	8	2	no	—
290	C₂²	—	8	2	no	—
291	C₂²	—	8	2	no	—
292	C₂²	—	16	2	no	—
293	C₂²	—	16	2	no	—
294	C₂²	C₂×C₁₃×#102	1	1	yes	characteristic
295	C₂²	C₁₃×#447	1	1	yes	characteristic
296	C₂²	C₁₃×#480	1	1	yes	characteristic
297	C₂²	C₁₃×#508	1	2	yes	—
298	C₂²	C₁₃×#509	1	2	yes	—
299	C₄	—	2	2	no	—
300	C₄	—	8	2	no	—
301	C₄	—	8	4	no	—
302	C₄	C₁₃×#386	1	1	yes	characteristic
303	C₄	C₁₃×#417	1	1	yes	characteristic
304	C₄	C₁₃×#448	1	1	yes	characteristic
305	C₄	C₁₃×#481	1	1	yes	characteristic
306	C₂	—	4	2	no	—
307	C₂	—	16	2	no	—
308	C₂	C₁₃×#1874	1	1	yes	characteristic
309	C₂	C₁₃×#2138	1	1	yes	characteristic
310	C₂	C₁₃×#2140	1	1	yes	characteristic
311	C₂	C₁₃×#2186	1	2	yes	—
312	C₂	C₁₃×#2234	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₁₃	C₂×D₈
commutator	C₂×C₈	C₂²×C₄×C₁₃
Frattini, cosocle	C₂²×C₈	C₂³×C₁₃
prime radical	C₂×C₁₃×#466	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₃ ◁ C₂³×C₄×C₁₃ ◁ C₂³×C₈×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₁₃×#427	C₂	semidirect
3. C₂×C₁₃×#357	C₂	semidirect
4. C₂×C₁₃×C₂²⋊C₁₆	C₂	semidirect
5. C₁₃×#1380	C₂	semidirect
12. C₁₃×#466	C₄	semidirect
48. C₁₃×#107	D₄	semidirect
51. #12992	C₁₃	direct (N × H)
92. C₂×#427	C₂×C₁₃	semidirect
93. C₂×#357	C₂×C₁₃	semidirect
94. C₂×C₂²⋊C₁₆	C₂×C₁₃	semidirect
95. #1380	C₂×C₁₃	semidirect
139. #466	C₄×C₁₃	semidirect
197. #107	D₄×C₁₃	semidirect
246. C₁₃	#12992	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₁₃×#466	2
C₂×C₁₃×#427	2
C₂×C₁₃×#357	2
C₂×C₁₃×C₂²⋊C₁₆	2
C₁₃×#1380	2
#12992	13

Subgroups — C13×#12992 #15699331

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#12992 #15699331