Subgroups — C₁₃×#13001 #15699340

Order 3328.

572 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#3050	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₁₃×#106	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₁₃×#360	C₂	1	2	yes	maximal, complemented
4	C₁₃×#2112	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₃×#1386	C₂	1	2	yes	maximal, complemented
6	C₂×C₁₃×#135	C₂²	1	1	yes	characteristic
7	C₂³×C₈×C₁₃	C₂²	1	1	yes	characteristic
8	C₁₃×#525	—	2	1	no	—
9	C₂²×C₁₃×C₄⋊C₄	—	2	1	no	—
10	C₂×C₁₃×M₃₂	C₂²	1	2	yes	—
11	C₁₃×#467	C₄	1	2	yes	complemented
12	C₁₃×#466	C₄	1	2	yes	complemented
13	C₁₃×#465	—	2	2	no	—
14	C₁₃×#427	C₂²	1	1	yes	characteristic
15	C₂×C₁₃×#106	—	2	1	no	—
16	C₂×C₁₃×#106	—	2	4	no	—
17	C₂×C₁₃×#106	C₂²	1	2	yes	—
18	C₁₃×#416	—	2	1	no	—
19	C₁₃×#360	—	2	4	no	—
20	C₁₃×C₂²⋊C₁₆	—	2	2	no	—
21	C₂×C₁₃×#41	—	2	1	no	—
22	C₂³×C₄×C₁₃	—	4	1	no	—
23	C₂³×C₄×C₁₃	D₄	1	1	yes	characteristic
24	C₁₃×#135	—	2	2	no	—
25	C₁₃×#135	—	2	2	no	—
26	C₂×C₁₃×Q₁₆	C₂×C₄	1	1	yes	characteristic, complemented
27	C₂×C₁₃×SD₁₆	—	2	1	no	—
28	C₂×C₁₃×D₈	C₂×C₄	1	1	yes	characteristic, complemented
29	C₂²×C₈×C₁₃	—	2	2	no	—
30	C₂²×C₈×C₁₃	—	2	2	no	—
31	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
32	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
33	C₂²×C₈×C₁₃	D₄	1	2	yes	—
34	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
35	C₁₃×#123	—	4	1	no	—
36	C₁₃×C₂²⋊Q₈	—	2	2	no	—
37	C₁₃×#121	—	2	2	no	—
38	C₁₃×#120	—	4	1	no	—
39	C₄×D₄×C₁₃	—	4	2	no	—
40	C₁₃×#117	—	2	1	no	—
41	C₂×C₁₃×C₄⋊C₄	—	2	4	no	—
42	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
43	C₂×C₁₃×C₄⋊C₄	—	4	2	no	—
44	C₂×C₁₃×C₄⋊C₄	—	4	1	no	—
45	C₁₃×M₃₂	—	2	2	no	—
46	C₁₃×M₃₂	—	2	2	no	—
47	C₂×C₁₃×C₁₆	—	2	2	no	—
48	C₁₃×#107	C₂×C₄	1	2	yes	complemented
49	C₁₃×#106	—	2	4	no	—
50	C₁₃×#106	—	2	2	no	—
51	C₁₃×#106	—	2	1	no	—
52	C₁₃×#106	D₄	1	4	yes	complemented
53	C₁₃×#103	—	2	2	no	—
54	C₁₃×#102	—	2	2	no	—
55	#13001	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
56	C₂⁴×C₁₃	D₈	1	1	yes	characteristic
57	C₁₃×#41	—	4	1	no	—
58	C₁₃×#41	—	4	1	no	—
59	C₁₃×#41	—	4	1	no	—
60	C₁₃×#41	—	4	1	no	—
61	C₂×Q₈×C₁₃	—	2	1	no	—
62	C₂×D₄×C₁₃	—	2	1	no	—
63	C₂×D₄×C₁₃	—	4	2	no	—
64	C₂×D₄×C₁₃	—	4	1	no	—
65	C₂²×C₄×C₁₃	—	2	1	no	—
66	C₂²×C₄×C₁₃	—	2	1	no	—
67	C₂²×C₄×C₁₃	—	2	1	no	—
68	C₂²×C₄×C₁₃	—	2	1	no	—
69	C₂²×C₄×C₁₃	—	4	4	no	—
70	C₂²×C₄×C₁₃	—	4	2	no	—
71	C₂²×C₄×C₁₃	—	4	1	no	—
72	C₂²×C₄×C₁₃	—	8	4	no	—
73	C₂²×C₄×C₁₃	C₂²⋊C₄	1	2	yes	—
74	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
75	C₂²×C₄×C₁₃	SD₁₆	1	2	yes	—
76	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
77	C₁₃×Q₁₆	—	2	2	no	—
78	C₁₃×SD₁₆	—	4	2	no	—
79	C₁₃×D₈	—	2	2	no	—
80	C₂×C₈×C₁₃	—	2	1	no	—
81	C₂×C₈×C₁₃	—	2	1	no	—
82	C₂×C₈×C₁₃	—	2	4	no	—
83	C₂×C₈×C₁₃	—	2	4	no	—
84	C₂×C₈×C₁₃	C₂²⋊C₄	1	2	yes	—
85	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
86	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
87	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
88	C₂×C₈×C₁₃	#41	1	2	yes	—
89	C₁₃×C₄⋊C₄	—	2	2	no	—
90	C₁₃×C₄⋊C₄	—	2	4	no	—
91	C₁₃×C₄⋊C₄	—	4	4	no	—
92	C₁₃×C₄⋊C₄	—	4	2	no	—
93	C₁₃×C₄⋊C₄	—	4	2	no	—
94	C₁₃×C₂²⋊C₄	—	4	2	no	—
95	C₁₃×C₂²⋊C₄	—	4	1	no	—
96	C₁₃×C₂²⋊C₄	—	4	2	no	—
97	C₄²×C₁₃	—	4	1	no	—
98	C₁₃×C₁₆	—	4	2	no	—
99	#3050	C₂×C₁₃	1	1	yes	characteristic
100	C₂²×#106	C₂×C₁₃	1	1	yes	characteristic, complemented
101	C₂×#360	C₂×C₁₃	1	2	yes	complemented
102	#2112	C₂×C₁₃	1	1	yes	characteristic, complemented
103	#1386	C₂×C₁₃	1	2	yes	complemented
104	C₂³×C₁₃	—	2	4	no	—
105	C₂³×C₁₃	—	2	1	no	—
106	C₂³×C₁₃	—	2	1	no	—
107	C₂³×C₁₃	—	4	1	no	—
108	C₂³×C₁₃	#102	1	2	yes	—
109	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
110	Q₈×C₁₃	—	4	1	no	—
111	Q₈×C₁₃	—	4	1	no	—
112	D₄×C₁₃	—	4	1	no	—
113	D₄×C₁₃	—	4	1	no	—
114	D₄×C₁₃	—	8	4	no	—
115	D₄×C₁₃	—	8	1	no	—
116	D₄×C₁₃	—	8	1	no	—
117	C₂×C₄×C₁₃	—	2	1	no	—
118	C₂×C₄×C₁₃	—	2	1	no	—
119	C₂×C₄×C₁₃	—	2	2	no	—
120	C₂×C₄×C₁₃	—	2	2	no	—
121	C₂×C₄×C₁₃	—	2	2	no	—
122	C₂×C₄×C₁₃	—	2	2	no	—
123	C₂×C₄×C₁₃	—	4	2	no	—
124	C₂×C₄×C₁₃	—	4	1	no	—
125	C₂×C₄×C₁₃	—	4	4	no	—
126	C₂×C₄×C₁₃	—	8	8	no	—
127	C₂×C₄×C₁₃	—	8	4	no	—
128	C₂×C₄×C₁₃	—	8	1	no	—
129	C₂×C₄×C₁₃	—	8	1	no	—
130	C₂×C₄×C₁₃	#102	1	2	yes	—
131	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
132	C₂×C₄×C₁₃	#120	1	1	yes	characteristic
133	C₂×C₄×C₁₃	#123	1	1	yes	characteristic
134	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
135	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
136	C₂×C₄×C₁₃	#137	1	1	yes	characteristic
137	C₈×C₁₃	—	2	2	no	—
138	C₈×C₁₃	C₄×D₄	1	2	yes	—
139	C₈×C₁₃	#121	1	2	yes	—
140	C₂×#135	C₂²×C₁₃	1	1	yes	characteristic
141	C₂³×C₈	C₂²×C₁₃	1	1	yes	characteristic
142	#525	—	2	1	no	—
143	C₂²×C₄⋊C₄	—	2	1	no	—
144	C₂×M₃₂	C₂²×C₁₃	1	2	yes	—
145	#467	C₄×C₁₃	1	2	yes	complemented
146	#466	C₄×C₁₃	1	2	yes	complemented
147	#465	—	2	2	no	—
148	#427	C₂²×C₁₃	1	1	yes	characteristic
149	C₂×#106	—	2	1	no	—
150	C₂×#106	—	2	4	no	—
151	C₂×#106	C₂²×C₁₃	1	2	yes	—
152	#416	—	2	1	no	—
153	#360	—	2	4	no	—
154	C₂²⋊C₁₆	—	2	2	no	—
155	C₂²×C₁₃	—	2	4	no	—
156	C₂²×C₁₃	—	2	4	no	—
157	C₂²×C₁₃	—	2	2	no	—
158	C₂²×C₁₃	—	2	2	no	—
159	C₂²×C₁₃	—	2	2	no	—
160	C₂²×C₁₃	—	8	1	no	—
161	C₂²×C₁₃	—	8	2	no	—
162	C₂²×C₁₃	#360	1	4	yes	complemented
163	C₂²×C₁₃	C₂×#102	1	1	yes	characteristic
164	C₂²×C₁₃	#447	1	1	yes	characteristic
165	C₂²×C₁₃	#480	1	1	yes	characteristic
166	C₄×C₁₃	—	2	2	no	—
167	C₄×C₁₃	—	8	4	no	—
168	C₄×C₁₃	—	8	1	no	—
169	C₄×C₁₃	—	8	2	no	—
170	C₄×C₁₃	#386	1	1	yes	characteristic
171	C₄×C₁₃	#417	1	1	yes	characteristic
172	C₄×C₁₃	#448	1	1	yes	characteristic
173	C₄×C₁₃	#481	1	1	yes	characteristic
174	C₂×#41	—	2	1	no	—
175	C₂³×C₄	—	4	1	no	—
176	C₂³×C₄	D₄×C₁₃	1	1	yes	characteristic
177	#135	—	2	2	no	—
178	#135	—	2	2	no	—
179	C₂×Q₁₆	C₂×C₄×C₁₃	1	1	yes	characteristic, complemented
180	C₂×SD₁₆	—	2	1	no	—
181	C₂×D₈	C₂×C₄×C₁₃	1	1	yes	characteristic, complemented
182	C₂²×C₈	—	2	2	no	—
183	C₂²×C₈	—	2	2	no	—
184	C₂²×C₈	C₂×C₄×C₁₃	1	2	yes	—
185	C₂²×C₈	D₄×C₁₃	1	1	yes	characteristic
186	C₂²×C₈	D₄×C₁₃	1	2	yes	—
187	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
188	#123	—	4	1	no	—
189	C₂²⋊Q₈	—	2	2	no	—
190	#121	—	2	2	no	—
191	#120	—	4	1	no	—
192	C₄×D₄	—	4	2	no	—
193	#117	—	2	1	no	—
194	C₂×C₄⋊C₄	—	2	4	no	—
195	C₂×C₄⋊C₄	—	2	2	no	—
196	C₂×C₄⋊C₄	—	4	2	no	—
197	C₂×C₄⋊C₄	—	4	1	no	—
198	M₃₂	—	2	2	no	—
199	M₃₂	—	2	2	no	—
200	C₂×C₁₆	—	2	2	no	—
201	#107	C₂×C₄×C₁₃	1	2	yes	complemented
202	#106	—	2	4	no	—
203	#106	—	2	2	no	—
204	#106	—	2	1	no	—
205	#106	D₄×C₁₃	1	4	yes	complemented
206	#103	—	2	2	no	—
207	#102	—	2	2	no	—
208	C₂×C₁₃	—	2	1	no	—
209	C₂×C₁₃	—	2	4	no	—
210	C₂×C₁₃	—	2	1	no	—
211	C₂×C₁₃	—	16	1	no	—
212	C₂×C₁₃	#1874	1	1	yes	characteristic
213	C₂×C₁₃	C₂×#360	1	2	yes	—
214	C₂⁴	C₁₃×D₈	1	1	yes	characteristic
215	#41	—	4	1	no	—
216	#41	—	4	1	no	—
217	#41	—	4	1	no	—
218	#41	—	4	1	no	—
219	C₂×Q₈	—	2	1	no	—
220	C₂×D₄	—	2	1	no	—
221	C₂×D₄	—	4	2	no	—
222	C₂×D₄	—	4	1	no	—
223	C₂²×C₄	—	2	1	no	—
224	C₂²×C₄	—	2	1	no	—
225	C₂²×C₄	—	2	1	no	—
226	C₂²×C₄	—	2	1	no	—
227	C₂²×C₄	—	4	4	no	—
228	C₂²×C₄	—	4	2	no	—
229	C₂²×C₄	—	4	1	no	—
230	C₂²×C₄	—	8	4	no	—
231	C₂²×C₄	C₁₃×C₂²⋊C₄	1	2	yes	—
232	C₂²×C₄	C₁₃×D₈	1	1	yes	characteristic
233	C₂²×C₄	C₁₃×SD₁₆	1	2	yes	—
234	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
235	Q₁₆	—	2	2	no	—
236	SD₁₆	—	4	2	no	—
237	D₈	—	2	2	no	—
238	C₂×C₈	—	2	1	no	—
239	C₂×C₈	—	2	1	no	—
240	C₂×C₈	—	2	4	no	—
241	C₂×C₈	—	2	4	no	—
242	C₂×C₈	C₁₃×C₂²⋊C₄	1	2	yes	—
243	C₂×C₈	C₂²×C₄×C₁₃	1	1	yes	characteristic
244	C₂×C₈	C₂×D₄×C₁₃	1	2	yes	—
245	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
246	C₂×C₈	C₁₃×#41	1	2	yes	—
247	C₄⋊C₄	—	2	2	no	—
248	C₄⋊C₄	—	2	4	no	—
249	C₄⋊C₄	—	4	4	no	—
250	C₄⋊C₄	—	4	2	no	—
251	C₄⋊C₄	—	4	2	no	—
252	C₂²⋊C₄	—	4	2	no	—
253	C₂²⋊C₄	—	4	1	no	—
254	C₂²⋊C₄	—	4	2	no	—
255	C₄²	—	4	1	no	—
256	C₁₆	—	4	2	no	—
257	C₁₃	#13001	1	1	yes	characteristic, direct product factor, Sylow
258	C₂³	—	2	4	no	—
259	C₂³	—	2	1	no	—
260	C₂³	—	2	1	no	—
261	C₂³	—	4	1	no	—
262	C₂³	C₁₃×#102	1	2	yes	—
263	C₂³	C₂×C₁₃×D₈	1	1	yes	characteristic
264	Q₈	—	4	1	no	—
265	Q₈	—	4	1	no	—
266	D₄	—	4	1	no	—
267	D₄	—	4	1	no	—
268	D₄	—	8	4	no	—
269	D₄	—	8	1	no	—
270	D₄	—	8	1	no	—
271	C₂×C₄	—	2	1	no	—
272	C₂×C₄	—	2	1	no	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	2	2	no	—
275	C₂×C₄	—	2	2	no	—
276	C₂×C₄	—	2	2	no	—
277	C₂×C₄	—	4	2	no	—
278	C₂×C₄	—	4	1	no	—
279	C₂×C₄	—	4	4	no	—
280	C₂×C₄	—	8	8	no	—
281	C₂×C₄	—	8	4	no	—
282	C₂×C₄	—	8	1	no	—
283	C₂×C₄	—	8	1	no	—
284	C₂×C₄	C₁₃×#102	1	2	yes	—
285	C₂×C₄	C₂×C₁₃×C₂²⋊C₄	1	1	yes	characteristic
286	C₂×C₄	C₁₃×#120	1	1	yes	characteristic
287	C₂×C₄	C₁₃×#123	1	1	yes	characteristic
288	C₂×C₄	C₂×C₁₃×SD₁₆	1	1	yes	characteristic
289	C₂×C₄	C₁₃×AGL(1,ℤ/8)	1	1	yes	characteristic
290	C₂×C₄	C₁₃×#137	1	1	yes	characteristic
291	C₈	—	2	2	no	—
292	C₈	C₄×D₄×C₁₃	1	2	yes	—
293	C₈	C₁₃×#121	1	2	yes	—
294	C₂²	—	2	4	no	—
295	C₂²	—	2	4	no	—
296	C₂²	—	2	2	no	—
297	C₂²	—	2	2	no	—
298	C₂²	—	2	2	no	—
299	C₂²	—	8	1	no	—
300	C₂²	—	8	2	no	—
301	C₂²	C₁₃×#360	1	4	yes	complemented
302	C₂²	C₂×C₁₃×#102	1	1	yes	characteristic
303	C₂²	C₁₃×#447	1	1	yes	characteristic
304	C₂²	C₁₃×#480	1	1	yes	characteristic
305	C₄	—	2	2	no	—
306	C₄	—	8	4	no	—
307	C₄	—	8	1	no	—
308	C₄	—	8	2	no	—
309	C₄	C₁₃×#386	1	1	yes	characteristic
310	C₄	C₁₃×#417	1	1	yes	characteristic
311	C₄	C₁₃×#448	1	1	yes	characteristic
312	C₄	C₁₃×#481	1	1	yes	characteristic
313	C₂	—	2	1	no	—
314	C₂	—	2	4	no	—
315	C₂	—	2	1	no	—
316	C₂	—	16	1	no	—
317	C₂	C₁₃×#1874	1	1	yes	characteristic
318	C₂	C₂×C₁₃×#360	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₁₃	C₂×#102
commutator	C₂×C₈	C₂²×C₄×C₁₃
Frattini, cosocle	C₂²×C₈	C₂³×C₁₃
prime radical	C₁₃×#3050	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×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. C₂²×C₁₃×#106	C₂	semidirect
3. C₂×C₁₃×#360	C₂	semidirect
4. C₁₃×#2112	C₂	semidirect
5. C₁₃×#1386	C₂	semidirect
11. C₁₃×#467	C₄	semidirect
12. C₁₃×#466	C₄	semidirect
26. C₂×C₁₃×Q₁₆	C₂×C₄	semidirect
28. C₂×C₁₃×D₈	C₂×C₄	semidirect
48. C₁₃×#107	C₂×C₄	semidirect
52. C₁₃×#106	D₄	semidirect
55. #13001	C₁₃	direct (N × H)
100. C₂²×#106	C₂×C₁₃	semidirect
101. C₂×#360	C₂×C₁₃	semidirect
102. #2112	C₂×C₁₃	semidirect
103. #1386	C₂×C₁₃	semidirect
145. #467	C₄×C₁₃	semidirect
146. #466	C₄×C₁₃	semidirect
162. C₂²×C₁₃	#360	semidirect
179. C₂×Q₁₆	C₂×C₄×C₁₃	semidirect
181. C₂×D₈	C₂×C₄×C₁₃	semidirect
201. #107	C₂×C₄×C₁₃	semidirect
205. #106	D₄×C₁₃	semidirect
257. C₁₃	#13001	direct (N × H)
301. C₂²	C₁₃×#360	semidirect

Maximal subgroups

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

subgroup	index
C₁₃×#3050	2
C₂²×C₁₃×#106	2
C₂×C₁₃×#360	2
C₁₃×#2112	2
C₁₃×#1386	2
#13001	13

Subgroups — C13×#13001 #15699340

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#13001 #15699340