Subgroups — C₁₃×#12741 #15699080

Order 3328.

930 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₃×#576	C₂	1	1	yes	maximal, characteristic
2	C₁₃×#2145	C₂	1	2	yes	maximal, complemented
3	C₁₃×#2140	C₂	1	2	yes	maximal, complemented
4	C₁₃×#2115	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₃×#1812	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₁₃×H(5,𝔽₂)	—	2	1	no	—
7	C₁₃×#576	—	2	4	no	—
8	C₁₃×#576	—	2	4	no	—
9	C₂×C₁₃×AGL(1,ℤ/8)	—	2	1	no	—
10	C₂×C₁₃×AGL(1,ℤ/8)	—	2	2	no	—
11	C₂×C₁₃×#135	C₄	1	2	yes	complemented
12	C₂×C₁₃×#135	C₂²	1	1	yes	characteristic
13	C₂²×C₁₃×D₈	C₄	1	2	yes	complemented
14	C₂²×C₁₃×D₈	C₂²	1	1	yes	characteristic
15	C₂×C₁₃×#131	C₂²	1	1	yes	characteristic
16	C₂×C₁₃×M₃₂	C₂²	1	2	yes	complemented
17	C₁₃×#427	C₂²	1	2	yes	complemented
18	C₁₃×#410	—	2	1	no	—
19	C₁₃×#360	—	2	4	no	—
20	C₁₃×#357	—	2	4	no	—
21	C₁₃×#349	—	2	2	no	—
22	C₁₃×H(3,ℤ/4)	—	2	1	no	—
23	C₁₃×H(5,𝔽₂)	—	2	8	no	—
24	C₁₃×H(5,𝔽₂)	—	4	4	no	—
25	C₂×C₁₃×#41	—	2	2	no	—
26	C₂×C₁₃×#41	—	2	1	no	—
27	C₂×C₁₃×#41	—	4	1	no	—
28	C₂×C₁₃×#41	D₄	1	1	yes	characteristic
29	C₂²×D₄×C₁₃	—	2	2	no	—
30	C₂²×D₄×C₁₃	—	2	1	no	—
31	C₂²×D₄×C₁₃	—	4	2	no	—
32	C₂²×D₄×C₁₃	—	4	1	no	—
33	C₁₃×AGL(1,ℤ/8)	—	2	8	no	—
34	C₁₃×AGL(1,ℤ/8)	—	2	16	no	—
35	C₁₃×#135	—	2	4	no	—
36	C₁₃×#135	—	2	4	no	—
37	C₁₃×#135	—	2	2	no	—
38	C₁₃×#135	D₄	1	4	yes	—
39	C₂×C₁₃×Q₁₆	C₂×C₄	1	1	yes	characteristic
40	C₂×C₁₃×SD₁₆	—	2	2	no	—
41	C₂×C₁₃×SD₁₆	—	2	1	no	—
42	C₂×C₁₃×D₈	—	2	4	no	—
43	C₂×C₁₃×D₈	—	2	4	no	—
44	C₂×C₁₃×D₈	—	2	2	no	—
45	C₂×C₁₃×D₈	—	2	1	no	—
46	C₂×C₁₃×D₈	—	2	2	no	—
47	C₂×C₁₃×D₈	C₂×C₄	1	1	yes	characteristic
48	C₂×C₁₃×D₈	C₂×C₄	1	2	yes	—
49	C₂×C₁₃×D₈	D₄	1	4	yes	—
50	C₁₃×#131	—	2	2	no	—
51	C₁₃×#131	—	2	2	no	—
52	C₂×C₁₃×M₁₆	D₄	1	1	yes	characteristic
53	C₂×C₁₃×M₁₆	D₄	1	2	yes	—
54	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
55	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
56	C₁₃×#117	—	2	2	no	—
57	C₁₃×M₃₂	—	2	2	no	—
58	C₁₃×M₃₂	—	2	2	no	—
59	C₂×C₁₃×C₁₆	—	2	2	no	—
60	C₁₃×#107	—	2	2	no	—
61	C₁₃×#106	—	2	2	no	—
62	C₁₃×#99	—	4	2	no	—
63	#12741	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
64	C₂⁴×C₁₃	—	4	2	no	—
65	C₂⁴×C₁₃	—	4	1	no	—
66	C₁₃×#41	—	2	2	no	—
67	C₁₃×#41	—	2	2	no	—
68	C₁₃×#41	—	2	8	no	—
69	C₁₃×#41	—	2	4	no	—
70	C₁₃×#41	—	4	4	no	—
71	C₁₃×#41	—	4	8	no	—
72	C₁₃×#41	—	4	2	no	—
73	C₂×Q₈×C₁₃	—	2	1	no	—
74	C₂×Q₈×C₁₃	SD₁₆	1	1	yes	characteristic
75	C₂×D₄×C₁₃	—	2	8	no	—
76	C₂×D₄×C₁₃	—	2	1	no	—
77	C₂×D₄×C₁₃	—	2	2	no	—
78	C₂×D₄×C₁₃	—	2	4	no	—
79	C₂×D₄×C₁₃	—	4	4	no	—
80	C₂×D₄×C₁₃	—	4	16	no	—
81	C₂×D₄×C₁₃	—	4	8	no	—
82	C₂×D₄×C₁₃	—	4	2	no	—
83	C₂×D₄×C₁₃	—	4	1	no	—
84	C₂×D₄×C₁₃	—	4	2	no	—
85	C₂×D₄×C₁₃	—	4	1	no	—
86	C₂×D₄×C₁₃	—	4	2	no	—
87	C₂×D₄×C₁₃	D₈	1	2	yes	—
88	C₂×D₄×C₁₃	SD₁₆	1	1	yes	characteristic
89	C₂²×C₄×C₁₃	—	4	2	no	—
90	C₂²×C₄×C₁₃	—	4	1	no	—
91	C₂²×C₄×C₁₃	C₂²⋊C₄	1	2	yes	—
92	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
93	C₁₃×Q₁₆	—	2	1	no	—
94	C₁₃×Q₁₆	C₂²⋊C₄	1	2	yes	—
95	C₁₃×SD₁₆	—	2	8	no	—
96	C₁₃×SD₁₆	—	2	4	no	—
97	C₁₃×D₈	—	2	8	no	—
98	C₁₃×D₈	—	2	2	no	—
99	C₁₃×D₈	—	2	1	no	—
100	C₁₃×D₈	—	2	4	no	—
101	C₁₃×D₈	C₂²⋊C₄	1	2	yes	—
102	C₁₃×D₈	C₂²⋊C₄	1	4	yes	—
103	C₁₃×M₁₆	—	2	2	no	—
104	C₁₃×M₁₆	—	2	4	no	—
105	C₂×C₈×C₁₃	—	2	2	no	—
106	C₂×C₈×C₁₃	—	2	2	no	—
107	C₂×C₈×C₁₃	C₂²⋊C₄	1	2	yes	—
108	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
109	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
110	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
111	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
112	C₁₃×C₄⋊C₄	—	4	2	no	—
113	C₁₃×C₂²⋊C₄	—	4	2	no	—
114	C₄²×C₁₃	—	4	2	no	—
115	C₁₃×C₁₆	—	4	2	no	—
116	C₂×#576	C₂×C₁₃	1	1	yes	characteristic
117	#2145	C₂×C₁₃	1	2	yes	complemented
118	#2140	C₂×C₁₃	1	2	yes	complemented
119	#2115	C₂×C₁₃	1	1	yes	characteristic, complemented
120	#1812	C₂×C₁₃	1	1	yes	characteristic, complemented
121	C₂³×C₁₃	—	4	8	no	—
122	C₂³×C₁₃	—	4	2	no	—
123	C₂³×C₁₃	—	4	1	no	—
124	C₂³×C₁₃	—	4	4	no	—
125	C₂³×C₁₃	—	16	2	no	—
126	C₂³×C₁₃	—	16	4	no	—
127	C₂³×C₁₃	#102	1	2	yes	—
128	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
129	Q₈×C₁₃	—	2	2	no	—
130	Q₈×C₁₃	—	2	2	no	—
131	Q₈×C₁₃	—	4	1	no	—
132	D₄×C₁₃	—	2	4	no	—
133	D₄×C₁₃	—	2	2	no	—
134	D₄×C₁₃	—	2	4	no	—
135	D₄×C₁₃	—	2	2	no	—
136	D₄×C₁₃	—	4	8	no	—
137	D₄×C₁₃	—	4	8	no	—
138	D₄×C₁₃	—	4	4	no	—
139	D₄×C₁₃	—	4	1	no	—
140	D₄×C₁₃	—	4	2	no	—
141	D₄×C₁₃	—	4	4	no	—
142	C₂×C₄×C₁₃	—	2	2	no	—
143	C₂×C₄×C₁₃	—	2	2	no	—
144	C₂×C₄×C₁₃	—	4	8	no	—
145	C₂×C₄×C₁₃	—	4	1	no	—
146	C₂×C₄×C₁₃	—	4	4	no	—
147	C₂×C₄×C₁₃	—	8	2	no	—
148	C₂×C₄×C₁₃	#102	1	2	yes	—
149	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
150	C₂×C₄×C₁₃	#120	1	2	yes	—
151	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
152	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	2	yes	—
153	C₈×C₁₃	—	2	2	no	—
154	C₈×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
155	C₈×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
156	C₈×C₁₃	#120	1	2	yes	—
157	C₂×H(5,𝔽₂)	—	2	1	no	—
158	#576	—	2	4	no	—
159	#576	—	2	4	no	—
160	C₂×AGL(1,ℤ/8)	—	2	1	no	—
161	C₂×AGL(1,ℤ/8)	—	2	2	no	—
162	C₂×#135	C₄×C₁₃	1	2	yes	complemented
163	C₂×#135	C₂²×C₁₃	1	1	yes	characteristic
164	C₂²×D₈	C₄×C₁₃	1	2	yes	complemented
165	C₂²×D₈	C₂²×C₁₃	1	1	yes	characteristic
166	C₂×#131	C₂²×C₁₃	1	1	yes	characteristic
167	C₂×M₃₂	C₂²×C₁₃	1	2	yes	complemented
168	#427	C₂²×C₁₃	1	2	yes	complemented
169	#410	—	2	1	no	—
170	#360	—	2	4	no	—
171	#357	—	2	4	no	—
172	#349	—	2	2	no	—
173	H(3,ℤ/4)	—	2	1	no	—
174	C₂²×C₁₃	—	2	2	no	—
175	C₂²×C₁₃	—	4	1	no	—
176	C₂²×C₁₃	—	4	2	no	—
177	C₂²×C₁₃	—	4	4	no	—
178	C₂²×C₁₃	—	4	2	no	—
179	C₂²×C₁₃	—	8	4	no	—
180	C₂²×C₁₃	—	8	2	no	—
181	C₂²×C₁₃	—	16	4	no	—
182	C₂²×C₁₃	—	16	8	no	—
183	C₂²×C₁₃	C₂×#102	1	1	yes	characteristic
184	C₂²×C₁₃	#447	1	2	yes	—
185	C₄×C₁₃	—	2	2	no	—
186	C₄×C₁₃	—	4	2	no	—
187	C₄×C₁₃	—	16	2	no	—
188	C₄×C₁₃	#379	1	1	yes	characteristic
189	C₄×C₁₃	#418	1	1	yes	characteristic
190	C₄×C₁₃	#450	1	2	yes	—
191	H(5,𝔽₂)	—	2	8	no	—
192	H(5,𝔽₂)	—	4	4	no	—
193	C₂×#41	—	2	2	no	—
194	C₂×#41	—	2	1	no	—
195	C₂×#41	—	4	1	no	—
196	C₂×#41	D₄×C₁₃	1	1	yes	characteristic
197	C₂²×D₄	—	2	2	no	—
198	C₂²×D₄	—	2	1	no	—
199	C₂²×D₄	—	4	2	no	—
200	C₂²×D₄	—	4	1	no	—
201	AGL(1,ℤ/8)	—	2	8	no	—
202	AGL(1,ℤ/8)	—	2	16	no	—
203	#135	—	2	4	no	—
204	#135	—	2	4	no	—
205	#135	—	2	2	no	—
206	#135	D₄×C₁₃	1	4	yes	—
207	C₂×Q₁₆	C₂×C₄×C₁₃	1	1	yes	characteristic
208	C₂×SD₁₆	—	2	2	no	—
209	C₂×SD₁₆	—	2	1	no	—
210	C₂×D₈	—	2	4	no	—
211	C₂×D₈	—	2	4	no	—
212	C₂×D₈	—	2	2	no	—
213	C₂×D₈	—	2	1	no	—
214	C₂×D₈	—	2	2	no	—
215	C₂×D₈	C₂×C₄×C₁₃	1	1	yes	characteristic
216	C₂×D₈	C₂×C₄×C₁₃	1	2	yes	—
217	C₂×D₈	D₄×C₁₃	1	4	yes	—
218	#131	—	2	2	no	—
219	#131	—	2	2	no	—
220	C₂×M₁₆	D₄×C₁₃	1	1	yes	characteristic
221	C₂×M₁₆	D₄×C₁₃	1	2	yes	—
222	C₂²×C₈	C₂×C₄×C₁₃	1	2	yes	—
223	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
224	#117	—	2	2	no	—
225	M₃₂	—	2	2	no	—
226	M₃₂	—	2	2	no	—
227	C₂×C₁₆	—	2	2	no	—
228	#107	—	2	2	no	—
229	#106	—	2	2	no	—
230	#99	—	4	2	no	—
231	C₂×C₁₃	—	2	2	no	—
232	C₂×C₁₃	—	2	1	no	—
233	C₂×C₁₃	—	4	2	no	—
234	C₂×C₁₃	—	8	4	no	—
235	C₂×C₁₃	—	8	2	no	—
236	C₂×C₁₃	#1786	1	1	yes	characteristic
237	C₂⁴	—	4	2	no	—
238	C₂⁴	—	4	1	no	—
239	#41	—	2	2	no	—
240	#41	—	2	2	no	—
241	#41	—	2	8	no	—
242	#41	—	2	4	no	—
243	#41	—	4	4	no	—
244	#41	—	4	8	no	—
245	#41	—	4	2	no	—
246	C₂×Q₈	—	2	1	no	—
247	C₂×Q₈	C₁₃×SD₁₆	1	1	yes	characteristic
248	C₂×D₄	—	2	8	no	—
249	C₂×D₄	—	2	1	no	—
250	C₂×D₄	—	2	2	no	—
251	C₂×D₄	—	2	4	no	—
252	C₂×D₄	—	4	4	no	—
253	C₂×D₄	—	4	16	no	—
254	C₂×D₄	—	4	8	no	—
255	C₂×D₄	—	4	2	no	—
256	C₂×D₄	—	4	1	no	—
257	C₂×D₄	—	4	2	no	—
258	C₂×D₄	—	4	1	no	—
259	C₂×D₄	—	4	2	no	—
260	C₂×D₄	C₁₃×D₈	1	2	yes	—
261	C₂×D₄	C₁₃×SD₁₆	1	1	yes	characteristic
262	C₂²×C₄	—	4	2	no	—
263	C₂²×C₄	—	4	1	no	—
264	C₂²×C₄	C₁₃×C₂²⋊C₄	1	2	yes	—
265	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
266	Q₁₆	—	2	1	no	—
267	Q₁₆	C₁₃×C₂²⋊C₄	1	2	yes	—
268	SD₁₆	—	2	8	no	—
269	SD₁₆	—	2	4	no	—
270	D₈	—	2	8	no	—
271	D₈	—	2	2	no	—
272	D₈	—	2	1	no	—
273	D₈	—	2	4	no	—
274	D₈	C₁₃×C₂²⋊C₄	1	2	yes	—
275	D₈	C₁₃×C₂²⋊C₄	1	4	yes	—
276	M₁₆	—	2	2	no	—
277	M₁₆	—	2	4	no	—
278	C₂×C₈	—	2	2	no	—
279	C₂×C₈	—	2	2	no	—
280	C₂×C₈	C₁₃×C₂²⋊C₄	1	2	yes	—
281	C₂×C₈	C₂²×C₄×C₁₃	1	1	yes	characteristic
282	C₂×C₈	C₂×D₄×C₁₃	1	2	yes	—
283	C₂×C₈	C₂×D₄×C₁₃	1	2	yes	—
284	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
285	C₄⋊C₄	—	4	2	no	—
286	C₂²⋊C₄	—	4	2	no	—
287	C₄²	—	4	2	no	—
288	C₁₆	—	4	2	no	—
289	C₁₃	#12741	1	1	yes	characteristic, direct product factor, Sylow
290	C₂³	—	4	8	no	—
291	C₂³	—	4	2	no	—
292	C₂³	—	4	1	no	—
293	C₂³	—	4	4	no	—
294	C₂³	—	16	2	no	—
295	C₂³	—	16	4	no	—
296	C₂³	C₁₃×#102	1	2	yes	—
297	C₂³	C₂×C₁₃×D₈	1	1	yes	characteristic
298	Q₈	—	2	2	no	—
299	Q₈	—	2	2	no	—
300	Q₈	—	4	1	no	—
301	D₄	—	2	4	no	—
302	D₄	—	2	2	no	—
303	D₄	—	2	4	no	—
304	D₄	—	2	2	no	—
305	D₄	—	4	8	no	—
306	D₄	—	4	8	no	—
307	D₄	—	4	4	no	—
308	D₄	—	4	1	no	—
309	D₄	—	4	2	no	—
310	D₄	—	4	4	no	—
311	C₂×C₄	—	2	2	no	—
312	C₂×C₄	—	2	2	no	—
313	C₂×C₄	—	4	8	no	—
314	C₂×C₄	—	4	1	no	—
315	C₂×C₄	—	4	4	no	—
316	C₂×C₄	—	8	2	no	—
317	C₂×C₄	C₁₃×#102	1	2	yes	—
318	C₂×C₄	C₂×C₁₃×C₂²⋊C₄	1	1	yes	characteristic
319	C₂×C₄	C₁₃×#120	1	2	yes	—
320	C₂×C₄	C₂×C₁₃×SD₁₆	1	1	yes	characteristic
321	C₂×C₄	C₁₃×AGL(1,ℤ/8)	1	2	yes	—
322	C₈	—	2	2	no	—
323	C₈	C₂×C₁₃×C₂²⋊C₄	1	1	yes	characteristic
324	C₈	C₂×C₁₃×C₂²⋊C₄	1	1	yes	characteristic
325	C₈	C₁₃×#120	1	2	yes	—
326	C₂²	—	2	2	no	—
327	C₂²	—	4	1	no	—
328	C₂²	—	4	2	no	—
329	C₂²	—	4	4	no	—
330	C₂²	—	4	2	no	—
331	C₂²	—	8	4	no	—
332	C₂²	—	8	2	no	—
333	C₂²	—	16	4	no	—
334	C₂²	—	16	8	no	—
335	C₂²	C₂×C₁₃×#102	1	1	yes	characteristic
336	C₂²	C₁₃×#447	1	2	yes	—
337	C₄	—	2	2	no	—
338	C₄	—	4	2	no	—
339	C₄	—	16	2	no	—
340	C₄	C₁₃×#379	1	1	yes	characteristic
341	C₄	C₁₃×#418	1	1	yes	characteristic
342	C₄	C₁₃×#450	1	2	yes	—
343	C₂	—	2	2	no	—
344	C₂	—	2	1	no	—
345	C₂	—	4	2	no	—
346	C₂	—	8	4	no	—
347	C₂	—	8	2	no	—
348	C₂	C₁₃×#1786	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₁₃	#1786
commutator	C₂×C₈	C₂²×C₄×C₁₃
Frattini, cosocle	C₂²×C₈	C₂³×C₁₃
prime radical	C₂×C₁₃×#576	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₃ ◁ C₂²×C₄×C₁₃ ◁ C₂×C₁₃×#131 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₁₃×#2145	C₂	semidirect
3. C₁₃×#2140	C₂	semidirect
4. C₁₃×#2115	C₂	semidirect
5. C₁₃×#1812	C₂	semidirect
11. C₂×C₁₃×#135	C₄	semidirect
13. C₂²×C₁₃×D₈	C₄	semidirect
16. C₂×C₁₃×M₃₂	C₂²	semidirect
17. C₁₃×#427	C₂²	semidirect
63. #12741	C₁₃	direct (N × H)
117. #2145	C₂×C₁₃	semidirect
118. #2140	C₂×C₁₃	semidirect
119. #2115	C₂×C₁₃	semidirect
120. #1812	C₂×C₁₃	semidirect
162. C₂×#135	C₄×C₁₃	semidirect
164. C₂²×D₈	C₄×C₁₃	semidirect
167. C₂×M₃₂	C₂²×C₁₃	semidirect
168. #427	C₂²×C₁₃	semidirect
289. C₁₃	#12741	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×#576	2
C₁₃×#2145	2
C₁₃×#2140	2
C₁₃×#2115	2
C₁₃×#1812	2
#12741	13

Subgroups — C13×#12741 #15699080

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#12741 #15699080