Subgroups — C₁₃×#13231 #15699570

Order 3328.

468 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#3058	C₂	1	1	yes	maximal, characteristic
2	C₁₃×#2128	C₂	1	2	yes	maximal, complemented
3	C₁₃×#2122	C₂	1	2	yes	maximal, complemented
4	C₁₃×#1812	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₃×#1757	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₁₃×D₈	C₄	1	1	yes	characteristic, complemented
7	C₂×C₁₃×#131	C₂²	1	1	yes	characteristic
8	C₁₃×#535	C₂²	1	2	yes	—
9	C₁₃×#469	—	2	1	no	—
10	C₁₃×#469	—	2	1	no	—
11	C₁₃×#466	—	2	1	no	—
12	C₁₃×#466	—	2	1	no	—
13	C₁₃×#427	—	2	1	no	—
14	C₁₃×#427	C₄	1	1	yes	characteristic, complemented
15	C₁₃×#418	C₄	1	2	yes	complemented
16	C₁₃×#413	C₂²	1	1	yes	characteristic
17	C₁₃×#410	C₂²	1	2	yes	—
18	C₁₃×#410	C₂²	1	1	yes	characteristic
19	C₁₃×#405	—	2	1	no	—
20	C₁₃×#353	—	2	4	no	—
21	C₁₃×AΣL(1,𝔽₁₆)	—	2	4	no	—
22	C₁₃×#344	—	2	1	no	—
23	C₁₃×H(3,ℤ/4)	—	2	1	no	—
24	C₂×C₁₃×#41	C₂³	1	1	yes	characteristic
25	C₂²×D₄×C₁₃	C₂×C₄	1	2	yes	—
26	C₂×C₁₃×D₈	—	2	1	no	—
27	C₂×C₁₃×D₈	—	2	1	no	—
28	C₂×C₁₃×D₈	—	4	1	no	—
29	C₂×C₁₃×D₈	—	4	1	no	—
30	C₁₃×#131	—	4	1	no	—
31	C₁₃×#131	—	4	1	no	—
32	C₂×C₁₃×M₁₆	—	2	1	no	—
33	C₂×C₁₃×M₁₆	C₂×C₄	1	1	yes	characteristic
34	C₂×C₁₃×M₁₆	D₄	1	1	yes	characteristic
35	C₂×C₁₃×M₁₆	D₄	1	1	yes	characteristic
36	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
38	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
39	C₁₃×#127	—	2	2	no	—
40	C₁₃×#124	—	2	2	no	—
41	C₁₃×#121	—	2	2	no	—
42	C₁₃×#121	—	2	2	no	—
43	C₁₃×#120	—	2	2	no	—
44	C₁₃×#120	—	2	2	no	—
45	C₁₃×#117	—	2	1	no	—
46	C₁₃×#117	—	2	2	no	—
47	C₁₃×#117	C₂×C₄	1	2	yes	—
48	C₁₃×#107	—	2	1	no	—
49	C₁₃×#107	—	4	1	no	—
50	C₁₃×#106	—	2	1	no	—
51	C₁₃×#106	—	4	1	no	—
52	C₁₃×#102	—	2	2	no	—
53	C₁₃×#102	—	2	2	no	—
54	C₁₃×#101	—	2	1	no	—
55	C₁₃×#100	D₄	1	2	yes	—
56	C₁₃×#99	—	2	1	no	—
57	C₁₃×#99	—	2	2	no	—
58	C₁₃×#99	—	2	2	no	—
59	C₁₃×#99	D₄	1	2	yes	—
60	C₁₃×#97	—	4	1	no	—
61	C₄×C₈×C₁₃	—	4	1	no	—
62	#13231	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
63	C₂⁴×C₁₃	—	2	2	no	—
64	C₁₃×#41	—	4	1	no	—
65	C₁₃×#41	—	4	1	no	—
66	C₂×Q₈×C₁₃	#41	1	1	yes	characteristic
67	C₂×D₄×C₁₃	—	2	2	no	—
68	C₂×D₄×C₁₃	—	2	2	no	—
69	C₂×D₄×C₁₃	—	4	2	no	—
70	C₂×D₄×C₁₃	—	4	2	no	—
71	C₂×D₄×C₁₃	—	4	2	no	—
72	C₂×D₄×C₁₃	—	4	2	no	—
73	C₂×D₄×C₁₃	C₂×D₄	1	1	yes	characteristic
74	C₂×D₄×C₁₃	C₂×D₄	1	1	yes	characteristic
75	C₂×D₄×C₁₃	#41	1	1	yes	characteristic
76	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
77	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
79	C₁₃×D₈	—	4	2	no	—
80	C₁₃×D₈	—	4	1	no	—
81	C₁₃×D₈	—	4	1	no	—
82	C₁₃×M₁₆	—	2	2	no	—
83	C₁₃×M₁₆	—	4	1	no	—
84	C₁₃×M₁₆	—	4	1	no	—
85	C₁₃×M₁₆	—	4	1	no	—
86	C₁₃×M₁₆	—	4	1	no	—
87	C₂×C₈×C₁₃	—	2	1	no	—
88	C₂×C₈×C₁₃	—	2	1	no	—
89	C₂×C₈×C₁₃	—	2	1	no	—
90	C₂×C₈×C₁₃	—	2	1	no	—
91	C₂×C₈×C₁₃	—	2	1	no	—
92	C₂×C₈×C₁₃	—	2	1	no	—
93	C₂×C₈×C₁₃	—	4	1	no	—
94	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
95	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
96	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
97	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
98	C₁₃×C₄⋊C₄	—	2	2	no	—
99	C₁₃×C₄⋊C₄	—	4	1	no	—
100	C₁₃×C₄⋊C₄	—	4	2	no	—
101	C₁₃×C₂²⋊C₄	—	2	2	no	—
102	C₁₃×C₂²⋊C₄	—	2	2	no	—
103	C₁₃×C₂²⋊C₄	—	4	2	no	—
104	C₁₃×C₂²⋊C₄	—	4	2	no	—
105	C₁₃×C₂²⋊C₄	—	4	2	no	—
106	C₄²×C₁₃	—	2	2	no	—
107	C₄²×C₁₃	—	4	1	no	—
108	C₄²×C₁₃	—	4	2	no	—
109	#3058	C₂×C₁₃	1	1	yes	characteristic
110	#2128	C₂×C₁₃	1	2	yes	complemented
111	#2122	C₂×C₁₃	1	2	yes	complemented
112	#1812	C₂×C₁₃	1	1	yes	characteristic, complemented
113	#1757	C₂×C₁₃	1	1	yes	characteristic, complemented
114	C₂³×C₁₃	—	4	2	no	—
115	C₂³×C₁₃	—	8	2	no	—
116	C₂³×C₁₃	—	16	2	no	—
117	C₂³×C₁₃	C₄×D₄	1	1	yes	characteristic
118	C₂³×C₁₃	#121	1	1	yes	characteristic
119	C₂³×C₁₃	#127	1	1	yes	characteristic
120	Q₈×C₁₃	—	4	1	no	—
121	D₄×C₁₃	—	4	1	no	—
122	D₄×C₁₃	—	4	1	no	—
123	D₄×C₁₃	—	4	1	no	—
124	D₄×C₁₃	—	4	2	no	—
125	D₄×C₁₃	—	4	2	no	—
126	D₄×C₁₃	—	8	2	no	—
127	D₄×C₁₃	—	16	2	no	—
128	D₄×C₁₃	—	16	2	no	—
129	C₂×C₄×C₁₃	—	2	1	no	—
130	C₂×C₄×C₁₃	—	2	1	no	—
131	C₂×C₄×C₁₃	—	2	1	no	—
132	C₂×C₄×C₁₃	—	2	1	no	—
133	C₂×C₄×C₁₃	—	2	1	no	—
134	C₂×C₄×C₁₃	—	2	1	no	—
135	C₂×C₄×C₁₃	—	4	2	no	—
136	C₂×C₄×C₁₃	—	4	2	no	—
137	C₂×C₄×C₁₃	—	8	2	no	—
138	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
139	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
140	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
141	C₂×C₄×C₁₃	#124	1	1	yes	characteristic
142	C₈×C₁₃	—	2	1	no	—
143	C₈×C₁₃	—	2	1	no	—
144	C₈×C₁₃	—	2	1	no	—
145	C₈×C₁₃	—	4	2	no	—
146	C₈×C₁₃	#99	1	1	yes	characteristic
147	C₈×C₁₃	#99	1	1	yes	characteristic
148	C₂²×D₈	C₄×C₁₃	1	1	yes	characteristic, complemented
149	C₂×#131	C₂²×C₁₃	1	1	yes	characteristic
150	#535	C₂²×C₁₃	1	2	yes	—
151	#469	—	2	1	no	—
152	#469	—	2	1	no	—
153	#466	—	2	1	no	—
154	#466	—	2	1	no	—
155	#427	—	2	1	no	—
156	#427	C₄×C₁₃	1	1	yes	characteristic, complemented
157	#418	C₄×C₁₃	1	2	yes	complemented
158	#413	C₂²×C₁₃	1	1	yes	characteristic
159	#410	C₂²×C₁₃	1	2	yes	—
160	#410	C₂²×C₁₃	1	1	yes	characteristic
161	#405	—	2	1	no	—
162	#353	—	2	4	no	—
163	AΣL(1,𝔽₁₆)	—	2	4	no	—
164	#344	—	2	1	no	—
165	H(3,ℤ/4)	—	2	1	no	—
166	C₂²×C₁₃	—	2	1	no	—
167	C₂²×C₁₃	—	2	1	no	—
168	C₂²×C₁₃	—	2	1	no	—
169	C₂²×C₁₃	—	4	1	no	—
170	C₂²×C₁₃	—	4	1	no	—
171	C₂²×C₁₃	—	4	1	no	—
172	C₂²×C₁₃	—	8	2	no	—
173	C₂²×C₁₃	—	8	2	no	—
174	C₂²×C₁₃	—	8	2	no	—
175	C₂²×C₁₃	—	16	4	no	—
176	C₂²×C₁₃	#390	1	1	yes	characteristic
177	C₄×C₁₃	—	2	1	no	—
178	C₄×C₁₃	—	2	1	no	—
179	C₄×C₁₃	—	2	1	no	—
180	C₄×C₁₃	—	8	2	no	—
181	C₄×C₁₃	—	8	2	no	—
182	C₄×C₁₃	—	16	2	no	—
183	C₄×C₁₃	C₂×#99	1	1	yes	characteristic
184	C₄×C₁₃	#410	1	1	yes	characteristic
185	C₂×#41	C₂³×C₁₃	1	1	yes	characteristic
186	C₂²×D₄	C₂×C₄×C₁₃	1	2	yes	—
187	C₂×D₈	—	2	1	no	—
188	C₂×D₈	—	2	1	no	—
189	C₂×D₈	—	4	1	no	—
190	C₂×D₈	—	4	1	no	—
191	#131	—	4	1	no	—
192	#131	—	4	1	no	—
193	C₂×M₁₆	—	2	1	no	—
194	C₂×M₁₆	C₂×C₄×C₁₃	1	1	yes	characteristic
195	C₂×M₁₆	D₄×C₁₃	1	1	yes	characteristic
196	C₂×M₁₆	D₄×C₁₃	1	1	yes	characteristic
197	C₂²×C₈	C₂×C₄×C₁₃	1	1	yes	characteristic
198	C₂²×C₈	D₄×C₁₃	1	1	yes	characteristic
199	C₂²×C₈	D₄×C₁₃	1	1	yes	characteristic
200	#127	—	2	2	no	—
201	#124	—	2	2	no	—
202	#121	—	2	2	no	—
203	#121	—	2	2	no	—
204	#120	—	2	2	no	—
205	#120	—	2	2	no	—
206	#117	—	2	1	no	—
207	#117	—	2	2	no	—
208	#117	C₂×C₄×C₁₃	1	2	yes	—
209	#107	—	2	1	no	—
210	#107	—	4	1	no	—
211	#106	—	2	1	no	—
212	#106	—	4	1	no	—
213	#102	—	2	2	no	—
214	#102	—	2	2	no	—
215	#101	—	2	1	no	—
216	#100	D₄×C₁₃	1	2	yes	—
217	#99	—	2	1	no	—
218	#99	—	2	2	no	—
219	#99	—	2	2	no	—
220	#99	D₄×C₁₃	1	2	yes	—
221	#97	—	4	1	no	—
222	C₄×C₈	—	4	1	no	—
223	C₂×C₁₃	—	2	1	no	—
224	C₂×C₁₃	—	4	1	no	—
225	C₂×C₁₃	—	4	1	no	—
226	C₂×C₁₃	—	4	1	no	—
227	C₂×C₁₃	—	16	2	no	—
228	C₂×C₁₃	#1964	1	1	yes	characteristic
229	C₂⁴	—	2	2	no	—
230	#41	—	4	1	no	—
231	#41	—	4	1	no	—
232	C₂×Q₈	C₁₃×#41	1	1	yes	characteristic
233	C₂×D₄	—	2	2	no	—
234	C₂×D₄	—	2	2	no	—
235	C₂×D₄	—	4	2	no	—
236	C₂×D₄	—	4	2	no	—
237	C₂×D₄	—	4	2	no	—
238	C₂×D₄	—	4	2	no	—
239	C₂×D₄	C₂×D₄×C₁₃	1	1	yes	characteristic
240	C₂×D₄	C₂×D₄×C₁₃	1	1	yes	characteristic
241	C₂×D₄	C₁₃×#41	1	1	yes	characteristic
242	C₂²×C₄	C₂²×C₄×C₁₃	1	1	yes	characteristic
243	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
244	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
245	D₈	—	4	2	no	—
246	D₈	—	4	1	no	—
247	D₈	—	4	1	no	—
248	M₁₆	—	2	2	no	—
249	M₁₆	—	4	1	no	—
250	M₁₆	—	4	1	no	—
251	M₁₆	—	4	1	no	—
252	M₁₆	—	4	1	no	—
253	C₂×C₈	—	2	1	no	—
254	C₂×C₈	—	2	1	no	—
255	C₂×C₈	—	2	1	no	—
256	C₂×C₈	—	2	1	no	—
257	C₂×C₈	—	2	1	no	—
258	C₂×C₈	—	2	1	no	—
259	C₂×C₈	—	4	1	no	—
260	C₂×C₈	C₁₃×C₂²⋊C₄	1	1	yes	characteristic
261	C₂×C₈	C₁₃×C₂²⋊C₄	1	1	yes	characteristic
262	C₂×C₈	C₁₃×C₂²⋊C₄	1	1	yes	characteristic
263	C₂×C₈	C₁₃×C₂²⋊C₄	1	1	yes	characteristic
264	C₄⋊C₄	—	2	2	no	—
265	C₄⋊C₄	—	4	1	no	—
266	C₄⋊C₄	—	4	2	no	—
267	C₂²⋊C₄	—	2	2	no	—
268	C₂²⋊C₄	—	2	2	no	—
269	C₂²⋊C₄	—	4	2	no	—
270	C₂²⋊C₄	—	4	2	no	—
271	C₂²⋊C₄	—	4	2	no	—
272	C₄²	—	2	2	no	—
273	C₄²	—	4	1	no	—
274	C₄²	—	4	2	no	—
275	C₁₃	#13231	1	1	yes	characteristic, direct product factor, Sylow
276	C₂³	—	4	2	no	—
277	C₂³	—	8	2	no	—
278	C₂³	—	16	2	no	—
279	C₂³	C₄×D₄×C₁₃	1	1	yes	characteristic
280	C₂³	C₁₃×#121	1	1	yes	characteristic
281	C₂³	C₁₃×#127	1	1	yes	characteristic
282	Q₈	—	4	1	no	—
283	D₄	—	4	1	no	—
284	D₄	—	4	1	no	—
285	D₄	—	4	1	no	—
286	D₄	—	4	2	no	—
287	D₄	—	4	2	no	—
288	D₄	—	8	2	no	—
289	D₄	—	16	2	no	—
290	D₄	—	16	2	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	2	1	no	—
296	C₂×C₄	—	2	1	no	—
297	C₂×C₄	—	4	2	no	—
298	C₂×C₄	—	4	2	no	—
299	C₂×C₄	—	8	2	no	—
300	C₂×C₄	C₂×C₁₃×C₂²⋊C₄	1	1	yes	characteristic
301	C₂×C₄	C₄×D₄×C₁₃	1	1	yes	characteristic
302	C₂×C₄	C₁₃×#121	1	1	yes	characteristic
303	C₂×C₄	C₁₃×#124	1	1	yes	characteristic
304	C₈	—	2	1	no	—
305	C₈	—	2	1	no	—
306	C₈	—	2	1	no	—
307	C₈	—	4	2	no	—
308	C₈	C₁₃×#99	1	1	yes	characteristic
309	C₈	C₁₃×#99	1	1	yes	characteristic
310	C₂²	—	2	1	no	—
311	C₂²	—	2	1	no	—
312	C₂²	—	2	1	no	—
313	C₂²	—	4	1	no	—
314	C₂²	—	4	1	no	—
315	C₂²	—	4	1	no	—
316	C₂²	—	8	2	no	—
317	C₂²	—	8	2	no	—
318	C₂²	—	8	2	no	—
319	C₂²	—	16	4	no	—
320	C₂²	C₁₃×#390	1	1	yes	characteristic
321	C₄	—	2	1	no	—
322	C₄	—	2	1	no	—
323	C₄	—	2	1	no	—
324	C₄	—	8	2	no	—
325	C₄	—	8	2	no	—
326	C₄	—	16	2	no	—
327	C₄	C₂×C₁₃×#99	1	1	yes	characteristic
328	C₄	C₁₃×#410	1	1	yes	characteristic
329	C₂	—	2	1	no	—
330	C₂	—	4	1	no	—
331	C₂	—	4	1	no	—
332	C₂	—	4	1	no	—
333	C₂	—	16	2	no	—
334	C₂	C₁₃×#1964	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₁₃	#1964
commutator	C₂²×C₄	C₂²×C₄×C₁₃
Frattini, cosocle	C₂×#41	C₂³×C₁₃
prime radical	C₁₃×#3058	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₃ ◁ C₂×C₄×C₁₃ ◁ C₂×C₁₃×#131 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₁₃×#2128	C₂	semidirect
3. C₁₃×#2122	C₂	semidirect
4. C₁₃×#1812	C₂	semidirect
5. C₁₃×#1757	C₂	semidirect
6. C₂²×C₁₃×D₈	C₄	semidirect
14. C₁₃×#427	C₄	semidirect
15. C₁₃×#418	C₄	semidirect
62. #13231	C₁₃	direct (N × H)
110. #2128	C₂×C₁₃	semidirect
111. #2122	C₂×C₁₃	semidirect
112. #1812	C₂×C₁₃	semidirect
113. #1757	C₂×C₁₃	semidirect
148. C₂²×D₈	C₄×C₁₃	semidirect
156. #427	C₄×C₁₃	semidirect
157. #418	C₄×C₁₃	semidirect
275. C₁₃	#13231	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×#3058	2
C₁₃×#2128	2
C₁₃×#2122	2
C₁₃×#1812	2
C₁₃×#1757	2
#13231	13

Subgroups — C13×#13231 #15699570

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#13231 #15699570