Subgroups — C₁₃×#33693 #15720032

Order 3328.

512 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#3177	C₂	1	1	yes	maximal, characteristic, complemented
2	C₁₃×#2955	C₂	1	1	yes	maximal, characteristic
3	C₁₃×#2239	C₂	1	2	yes	maximal, complemented
4	C₁₃×#2231	C₂	1	2	yes	maximal, complemented
5	C₁₃×#2214	C₂	1	2	yes	maximal, complemented
6	C₁₃×#2191	C₂	1	2	yes	maximal
7	C₁₃×#2180	C₂	1	4	yes	maximal, complemented
8	C₁₃×#2166	C₂	1	1	yes	maximal, characteristic, complemented
9	C₁₃×#575	C₂²	1	1	yes	characteristic
10	C₂×C₁₃×#131	—	2	1	no	—
11	C₁₃×#541	—	2	1	no	—
12	C₁₃×#510	C₂²	1	4	yes	—
13	C₁₃×#499	C₂²	1	1	yes	characteristic, complemented
14	C₁₃×#494	C₂²	1	1	yes	characteristic, complemented
15	C₁₃×#491	—	2	1	no	—
16	C₁₃×#470	C₂²	1	2	yes	—
17	C₁₃×#467	—	2	1	no	—
18	C₁₃×#444	—	2	1	no	—
19	C₁₃×#444	C₂²	1	2	yes	—
20	C₁₃×#443	—	2	1	no	—
21	C₁₃×#443	C₂²	1	2	yes	—
22	C₁₃×#441	C₂²	1	2	yes	complemented
23	C₄×C₁₃×Q₁₆	—	2	1	no	—
24	C₁₃×#430	C₂²	1	1	yes	characteristic
25	C₁₃×#421	—	2	1	no	—
26	C₁₃×#405	C₂²	1	1	yes	characteristic
27	C₁₃×#365	C₂²	1	4	yes	complemented
28	C₁₃×#364	C₂²	1	2	yes	complemented
29	C₁₃×#362	C₂²	1	4	yes	complemented
30	C₁₃×#360	C₂²	1	4	yes	—
31	C₁₃×#349	C₂²	1	2	yes	—
32	C₁₃×#347	C₂²	1	2	yes	complemented
33	C₂×C₁₃×#41	—	2	1	no	—
34	C₁₃×#137	—	2	2	no	—
35	C₁₃×AGL(1,ℤ/8)	—	2	2	no	—
36	C₁₃×#135	—	2	1	no	—
37	C₁₃×#135	C₂³	1	2	yes	—
38	C₂×C₁₃×Q₁₆	—	2	1	no	—
39	C₂×C₁₃×Q₁₆	C₂³	1	2	yes	—
40	C₁₃×#131	—	2	2	no	—
41	C₁₃×#131	—	2	2	no	—
42	C₁₃×#131	—	4	1	no	—
43	C₁₃×#131	—	4	1	no	—
44	C₂×C₁₃×M₁₆	—	2	1	no	—
45	C₂×C₁₃×M₁₆	—	4	1	no	—
46	C₂×C₁₃×M₁₆	C₂³	1	1	yes	characteristic
47	C₂²×C₈×C₁₃	—	4	1	no	—
48	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
49	C₁₃×C₄⋊Q₈	—	2	1	no	—
50	C₁₃×#126	—	4	2	no	—
51	C₁₃×#125	—	2	1	no	—
52	C₁₃×#125	—	4	2	no	—
53	C₁₃×C₂²⋊Q₈	—	2	2	no	—
54	C₄×Q₈×C₁₃	—	2	2	no	—
55	C₁₃×#117	D₄	1	1	yes	characteristic
56	C₁₃×Q₃₂	—	2	4	no	—
57	C₁₃×SD₃₂	—	2	4	no	—
58	C₁₃×M₃₂	C₂³	1	4	yes	—
59	C₁₃×#108	—	2	1	no	—
60	C₁₃×#108	C₂³	1	2	yes	—
61	C₁₃×#107	—	2	1	no	—
62	C₁₃×#106	C₂³	1	2	yes	—
63	C₁₃×#104	—	2	2	no	—
64	C₁₃×#104	—	2	2	no	—
65	C₁₃×#103	—	2	2	no	—
66	C₁₃×#103	—	2	2	no	—
67	C₁₃×#97	D₄	1	1	yes	characteristic
68	C₁₃×#97	C₂³	1	1	yes	characteristic
69	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
70	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
71	#33693	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
72	C₁₃×#41	—	2	2	no	—
73	C₁₃×#41	—	2	2	no	—
74	C₁₃×#41	—	4	1	no	—
75	C₁₃×#41	—	4	1	no	—
76	C₂×Q₈×C₁₃	—	2	1	no	—
77	C₂×Q₈×C₁₃	—	2	2	no	—
78	C₂×D₄×C₁₃	—	2	1	no	—
79	C₂×D₄×C₁₃	—	4	1	no	—
80	C₂²×C₄×C₁₃	—	4	1	no	—
81	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
82	C₁₃×Q₁₆	—	2	1	no	—
83	C₁₃×Q₁₆	—	2	4	no	—
84	C₁₃×Q₁₆	—	4	1	no	—
85	C₁₃×Q₁₆	—	4	1	no	—
86	C₁₃×Q₁₆	#41	1	2	yes	—
87	C₁₃×SD₁₆	—	2	2	no	—
88	C₁₃×SD₁₆	—	2	2	no	—
89	C₁₃×D₈	—	2	1	no	—
90	C₁₃×D₈	#41	1	2	yes	—
91	C₁₃×M₁₆	—	2	2	no	—
92	C₁₃×M₁₆	—	2	2	no	—
93	C₁₃×M₁₆	—	4	2	no	—
94	C₁₃×M₁₆	—	4	1	no	—
95	C₁₃×M₁₆	—	4	2	no	—
96	C₂×C₈×C₁₃	—	2	1	no	—
97	C₂×C₈×C₁₃	—	2	1	no	—
98	C₂×C₈×C₁₃	—	4	2	no	—
99	C₂×C₈×C₁₃	—	4	1	no	—
100	C₂×C₈×C₁₃	—	4	2	no	—
101	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
102	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
103	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
104	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
105	C₁₃×C₄⋊C₄	—	2	1	no	—
106	C₁₃×C₄⋊C₄	—	2	2	no	—
107	C₁₃×C₄⋊C₄	—	4	2	no	—
108	C₁₃×C₄⋊C₄	—	4	2	no	—
109	C₁₃×C₄⋊C₄	—	4	2	no	—
110	C₁₃×C₄⋊C₄	—	4	2	no	—
111	C₁₃×C₂²⋊C₄	—	2	1	no	—
112	C₁₃×C₂²⋊C₄	—	4	2	no	—
113	C₄²×C₁₃	—	4	2	no	—
114	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
115	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
116	C₁₃×C₁₆	—	2	4	no	—
117	#3177	C₂×C₁₃	1	1	yes	characteristic, complemented
118	#2955	C₂×C₁₃	1	1	yes	characteristic
119	#2239	C₂×C₁₃	1	2	yes	complemented
120	#2231	C₂×C₁₃	1	2	yes	complemented
121	#2214	C₂×C₁₃	1	2	yes	complemented
122	#2191	C₂×C₁₃	1	2	yes	—
123	#2180	C₂×C₁₃	1	4	yes	complemented
124	#2166	C₂×C₁₃	1	1	yes	characteristic, complemented
125	C₂³×C₁₃	—	4	1	no	—
126	C₂³×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
127	Q₈×C₁₃	—	2	2	no	—
128	Q₈×C₁₃	—	4	1	no	—
129	Q₈×C₁₃	—	4	2	no	—
130	Q₈×C₁₃	—	4	2	no	—
131	D₄×C₁₃	—	2	2	no	—
132	D₄×C₁₃	—	4	1	no	—
133	D₄×C₁₃	—	4	2	no	—
134	D₄×C₁₃	—	4	2	no	—
135	C₂×C₄×C₁₃	—	2	1	no	—
136	C₂×C₄×C₁₃	—	2	1	no	—
137	C₂×C₄×C₁₃	—	2	1	no	—
138	C₂×C₄×C₁₃	—	2	1	no	—
139	C₂×C₄×C₁₃	—	4	2	no	—
140	C₂×C₄×C₁₃	—	4	1	no	—
141	C₂×C₄×C₁₃	—	4	2	no	—
142	C₂×C₄×C₁₃	—	4	2	no	—
143	C₂×C₄×C₁₃	—	4	2	no	—
144	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
145	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
146	C₈×C₁₃	—	2	1	no	—
147	C₈×C₁₃	—	2	1	no	—
148	C₈×C₁₃	—	4	2	no	—
149	C₈×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
150	C₈×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
151	C₈×C₁₃	C₂×#41	1	2	yes	—
152	#575	C₂²×C₁₃	1	1	yes	characteristic
153	C₂×#131	—	2	1	no	—
154	#541	—	2	1	no	—
155	#510	C₂²×C₁₃	1	4	yes	—
156	#499	C₂²×C₁₃	1	1	yes	characteristic, complemented
157	#494	C₂²×C₁₃	1	1	yes	characteristic, complemented
158	#491	—	2	1	no	—
159	#470	C₂²×C₁₃	1	2	yes	—
160	#467	—	2	1	no	—
161	#444	—	2	1	no	—
162	#444	C₂²×C₁₃	1	2	yes	—
163	#443	—	2	1	no	—
164	#443	C₂²×C₁₃	1	2	yes	—
165	#441	C₂²×C₁₃	1	2	yes	complemented
166	C₄×Q₁₆	—	2	1	no	—
167	#430	C₂²×C₁₃	1	1	yes	characteristic
168	#421	—	2	1	no	—
169	#405	C₂²×C₁₃	1	1	yes	characteristic
170	#365	C₂²×C₁₃	1	4	yes	complemented
171	#364	C₂²×C₁₃	1	2	yes	complemented
172	#362	C₂²×C₁₃	1	4	yes	complemented
173	#360	C₂²×C₁₃	1	4	yes	—
174	#349	C₂²×C₁₃	1	2	yes	—
175	#347	C₂²×C₁₃	1	2	yes	complemented
176	C₂²×C₁₃	—	2	1	no	—
177	C₂²×C₁₃	—	4	1	no	—
178	C₂²×C₁₃	—	4	2	no	—
179	C₂²×C₁₃	—	8	2	no	—
180	C₂²×C₁₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
181	C₄×C₁₃	—	2	1	no	—
182	C₄×C₁₃	—	4	1	no	—
183	C₄×C₁₃	—	4	1	no	—
184	C₄×C₁₃	—	4	2	no	—
185	C₄×C₁₃	—	8	2	no	—
186	C₄×C₁₃	—	8	2	no	—
187	C₄×C₁₃	#525	1	1	yes	characteristic
188	C₄×C₁₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
189	C₂×#41	—	2	1	no	—
190	#137	—	2	2	no	—
191	AGL(1,ℤ/8)	—	2	2	no	—
192	#135	—	2	1	no	—
193	#135	C₂³×C₁₃	1	2	yes	—
194	C₂×Q₁₆	—	2	1	no	—
195	C₂×Q₁₆	C₂³×C₁₃	1	2	yes	—
196	#131	—	2	2	no	—
197	#131	—	2	2	no	—
198	#131	—	4	1	no	—
199	#131	—	4	1	no	—
200	C₂×M₁₆	—	2	1	no	—
201	C₂×M₁₆	—	4	1	no	—
202	C₂×M₁₆	C₂³×C₁₃	1	1	yes	characteristic
203	C₂²×C₈	—	4	1	no	—
204	C₂²×C₈	D₄×C₁₃	1	1	yes	characteristic
205	C₄⋊Q₈	—	2	1	no	—
206	#126	—	4	2	no	—
207	#125	—	2	1	no	—
208	#125	—	4	2	no	—
209	C₂²⋊Q₈	—	2	2	no	—
210	C₄×Q₈	—	2	2	no	—
211	#117	D₄×C₁₃	1	1	yes	characteristic
212	Q₃₂	—	2	4	no	—
213	SD₃₂	—	2	4	no	—
214	M₃₂	C₂³×C₁₃	1	4	yes	—
215	#108	—	2	1	no	—
216	#108	C₂³×C₁₃	1	2	yes	—
217	#107	—	2	1	no	—
218	#106	C₂³×C₁₃	1	2	yes	—
219	#104	—	2	2	no	—
220	#104	—	2	2	no	—
221	#103	—	2	2	no	—
222	#103	—	2	2	no	—
223	#97	D₄×C₁₃	1	1	yes	characteristic
224	#97	C₂³×C₁₃	1	1	yes	characteristic
225	C₄×C₈	D₄×C₁₃	1	1	yes	characteristic
226	C₄×C₈	C₂³×C₁₃	1	1	yes	characteristic
227	C₂×C₁₃	—	2	1	no	—
228	C₂×C₁₃	—	4	1	no	—
229	C₂×C₁₃	—	8	2	no	—
230	C₂×C₁₃	#3109	1	1	yes	characteristic
231	#41	—	2	2	no	—
232	#41	—	2	2	no	—
233	#41	—	4	1	no	—
234	#41	—	4	1	no	—
235	C₂×Q₈	—	2	1	no	—
236	C₂×Q₈	—	2	2	no	—
237	C₂×D₄	—	2	1	no	—
238	C₂×D₄	—	4	1	no	—
239	C₂²×C₄	—	4	1	no	—
240	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
241	Q₁₆	—	2	1	no	—
242	Q₁₆	—	2	4	no	—
243	Q₁₆	—	4	1	no	—
244	Q₁₆	—	4	1	no	—
245	Q₁₆	C₁₃×#41	1	2	yes	—
246	SD₁₆	—	2	2	no	—
247	SD₁₆	—	2	2	no	—
248	D₈	—	2	1	no	—
249	D₈	C₁₃×#41	1	2	yes	—
250	M₁₆	—	2	2	no	—
251	M₁₆	—	2	2	no	—
252	M₁₆	—	4	2	no	—
253	M₁₆	—	4	1	no	—
254	M₁₆	—	4	2	no	—
255	C₂×C₈	—	2	1	no	—
256	C₂×C₈	—	2	1	no	—
257	C₂×C₈	—	4	2	no	—
258	C₂×C₈	—	4	1	no	—
259	C₂×C₈	—	4	2	no	—
260	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
261	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
262	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
263	C₂×C₈	C₂⁴×C₁₃	1	1	yes	characteristic
264	C₄⋊C₄	—	2	1	no	—
265	C₄⋊C₄	—	2	2	no	—
266	C₄⋊C₄	—	4	2	no	—
267	C₄⋊C₄	—	4	2	no	—
268	C₄⋊C₄	—	4	2	no	—
269	C₄⋊C₄	—	4	2	no	—
270	C₂²⋊C₄	—	2	1	no	—
271	C₂²⋊C₄	—	4	2	no	—
272	C₄²	—	4	2	no	—
273	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
274	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
275	C₁₆	—	2	4	no	—
276	C₁₃	#33693	1	1	yes	characteristic, direct product factor, Sylow
277	C₂³	—	4	1	no	—
278	C₂³	C₁₃×AGL(1,ℤ/8)	1	1	yes	characteristic
279	Q₈	—	2	2	no	—
280	Q₈	—	4	1	no	—
281	Q₈	—	4	2	no	—
282	Q₈	—	4	2	no	—
283	D₄	—	2	2	no	—
284	D₄	—	4	1	no	—
285	D₄	—	4	2	no	—
286	D₄	—	4	2	no	—
287	C₂×C₄	—	2	1	no	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	4	2	no	—
292	C₂×C₄	—	4	1	no	—
293	C₂×C₄	—	4	2	no	—
294	C₂×C₄	—	4	2	no	—
295	C₂×C₄	—	4	2	no	—
296	C₂×C₄	C₁₃×AGL(1,ℤ/8)	1	1	yes	characteristic
297	C₂×C₄	C₂²×D₄×C₁₃	1	1	yes	characteristic
298	C₈	—	2	1	no	—
299	C₈	—	2	1	no	—
300	C₈	—	4	2	no	—
301	C₈	C₁₃×AGL(1,ℤ/8)	1	1	yes	characteristic
302	C₈	C₁₃×AGL(1,ℤ/8)	1	1	yes	characteristic
303	C₈	C₂×C₁₃×#41	1	2	yes	—
304	C₂²	—	2	1	no	—
305	C₂²	—	4	1	no	—
306	C₂²	—	4	2	no	—
307	C₂²	—	8	2	no	—
308	C₂²	C₂×C₁₃×AGL(1,ℤ/8)	1	1	yes	characteristic
309	C₄	—	2	1	no	—
310	C₄	—	4	1	no	—
311	C₄	—	4	1	no	—
312	C₄	—	4	2	no	—
313	C₄	—	8	2	no	—
314	C₄	—	8	2	no	—
315	C₄	C₁₃×#525	1	1	yes	characteristic
316	C₄	C₂×C₁₃×AGL(1,ℤ/8)	1	1	yes	characteristic
317	C₂	—	2	1	no	—
318	C₂	—	4	1	no	—
319	C₂	—	8	2	no	—
320	C₂	C₁₃×#3109	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₁₃	#3109
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₁₃
prime radical	C₁₃×#575	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₃ ◁ C₂×C₄×C₁₃ ◁ C₁₃×#405 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₁₃×#3177	C₂	semidirect
3. C₁₃×#2239	C₂	semidirect
4. C₁₃×#2231	C₂	semidirect
5. C₁₃×#2214	C₂	semidirect
7. C₁₃×#2180	C₂	semidirect
8. C₁₃×#2166	C₂	semidirect
13. C₁₃×#499	C₂²	semidirect
14. C₁₃×#494	C₂²	semidirect
22. C₁₃×#441	C₂²	semidirect
27. C₁₃×#365	C₂²	semidirect
28. C₁₃×#364	C₂²	semidirect
29. C₁₃×#362	C₂²	semidirect
32. C₁₃×#347	C₂²	semidirect
71. #33693	C₁₃	direct (N × H)
117. #3177	C₂×C₁₃	semidirect
119. #2239	C₂×C₁₃	semidirect
120. #2231	C₂×C₁₃	semidirect
121. #2214	C₂×C₁₃	semidirect
123. #2180	C₂×C₁₃	semidirect
124. #2166	C₂×C₁₃	semidirect
156. #499	C₂²×C₁₃	semidirect
157. #494	C₂²×C₁₃	semidirect
165. #441	C₂²×C₁₃	semidirect
170. #365	C₂²×C₁₃	semidirect
171. #364	C₂²×C₁₃	semidirect
172. #362	C₂²×C₁₃	semidirect
175. #347	C₂²×C₁₃	semidirect
276. C₁₃	#33693	direct (N × H)

Maximal subgroups

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

subgroup	index
C₁₃×#3177	2
C₁₃×#2955	2
C₁₃×#2239	2
C₁₃×#2231	2
C₁₃×#2214	2
C₁₃×#2191	2
C₁₃×#2180	2
C₁₃×#2166	2
#33693	13

Subgroups — C13×#33693 #15720032

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#33693 #15720032