Subgroups — C₁₃×#19268 #15705607

Order 3328.

702 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₃×#494	C₂	1	1	yes	maximal, characteristic, complemented
2	C₁₃×#3079	C₂	1	2	yes	maximal
3	C₈×C₁₃×#41	C₂	1	1	yes	maximal, characteristic
4	C₁₃×#1670	C₂	1	4	yes	maximal, complemented
5	C₁₃×#1669	C₂	1	4	yes	maximal, complemented
6	C₁₃×#1567	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₃×#1549	C₂	1	2	yes	maximal, complemented
8	C₂²×C₁₃×Q₁₆	—	2	2	no	—
9	C₂×C₁₃×C₄⋊Q₈	C₂²	1	2	yes	—
10	C₄×C₁₃×#41	C₂²	1	1	yes	characteristic
11	C₁₃×#494	—	2	1	no	—
12	C₁₃×#494	—	2	1	no	—
13	C₁₃×#494	C₂²	1	4	yes	—
14	C₁₃×#477	—	2	2	no	—
15	C₁₃×#476	—	2	2	no	—
16	C₁₃×#475	C₂²	1	4	yes	—
17	C₁₃×#474	C₂²	1	4	yes	—
18	C₈×Q₈×C₁₃	C₂²	1	2	yes	—
19	C₈×D₄×C₁₃	—	2	2	no	—
20	C₈×D₄×C₁₃	C₂²	1	2	yes	—
21	C₁₃×#431	—	2	1	no	—
22	C₁₃×#431	C₂²	1	1	yes	characteristic
23	C₁₃×#428	—	2	2	no	—
24	C₁₃×#423	C₂²	1	2	yes	—
25	C₁₃×#417	—	2	2	no	—
26	C₂×C₄×C₈×C₁₃	—	2	1	no	—
27	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
28	C₁₃×#335	C₂²	1	4	yes	—
29	C₁₃×#332	C₂²	1	8	yes	—
30	C₂×C₁₃×#41	D₄	1	1	yes	characteristic
31	C₂²×Q₈×C₁₃	—	2	2	no	—
32	C₂×C₁₃×Q₁₆	—	2	4	no	—
33	C₂×C₁₃×Q₁₆	—	2	4	no	—
34	C₂×C₁₃×Q₁₆	—	4	8	no	—
35	C₂×C₁₃×M₁₆	—	2	2	no	—
36	C₂²×C₈×C₁₃	—	2	2	no	—
37	C₂²×C₈×C₁₃	D₄	1	2	yes	—
38	C₁₃×C₄⋊Q₈	—	2	2	no	—
39	C₁₃×C₄⋊Q₈	—	2	2	no	—
40	C₁₃×C₄⋊Q₈	—	2	2	no	—
41	C₁₃×C₄⋊Q₈	C₂³	1	4	yes	—
42	C₄×Q₈×C₁₃	D₄	1	1	yes	characteristic
43	C₄×Q₈×C₁₃	C₂³	1	1	yes	characteristic
44	C₄×D₄×C₁₃	—	2	1	no	—
45	C₄×D₄×C₁₃	—	2	1	no	—
46	C₄×D₄×C₁₃	D₄	1	1	yes	characteristic
47	C₄×D₄×C₁₃	C₂³	1	1	yes	characteristic
48	C₁₃×#117	—	2	1	no	—
49	C₁₃×#117	D₄	1	1	yes	characteristic
50	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
51	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
52	C₂×C₄²×C₁₃	—	2	1	no	—
53	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
54	C₁₃×#107	—	2	4	no	—
55	C₁₃×#106	—	2	4	no	—
56	C₁₃×C₄⋊C₈	—	2	2	no	—
57	C₁₃×C₄⋊C₈	—	2	2	no	—
58	C₁₃×C₄⋊C₈	C₂³	1	2	yes	—
59	C₁₃×C₄⋊C₈	C₂³	1	4	yes	—
60	C₁₃×#103	—	2	4	no	—
61	C₁₃×#102	—	2	4	no	—
62	C₁₃×C₂²⋊C₈	—	2	1	no	—
63	C₁₃×C₂²⋊C₈	—	2	2	no	—
64	C₄×C₈×C₁₃	—	2	2	no	—
65	C₄×C₈×C₁₃	—	2	1	no	—
66	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
67	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
68	C₄×C₈×C₁₃	C₂³	1	2	yes	—
69	#19268	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
70	C₁₃×#41	—	2	2	no	—
71	C₁₃×#41	Q₁₆	1	4	yes	complemented
72	C₂×Q₈×C₁₃	—	2	4	no	—
73	C₂×Q₈×C₁₃	—	4	8	no	—
74	C₂×Q₈×C₁₃	—	4	2	no	—
75	C₂×Q₈×C₁₃	C₂×D₄	1	1	yes	characteristic
76	C₂×D₄×C₁₃	—	2	1	no	—
77	C₂×D₄×C₁₃	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₄×C₁₃	—	2	1	no	—
79	C₂²×C₄×C₁₃	—	2	1	no	—
80	C₂²×C₄×C₁₃	—	2	1	no	—
81	C₂²×C₄×C₁₃	—	4	2	no	—
82	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
83	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
84	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
85	C₁₃×Q₁₆	—	4	8	no	—
86	C₁₃×Q₁₆	—	4	8	no	—
87	C₁₃×M₁₆	—	4	4	no	—
88	C₂×C₈×C₁₃	—	2	4	no	—
89	C₂×C₈×C₁₃	—	2	2	no	—
90	C₂×C₈×C₁₃	—	2	4	no	—
91	C₂×C₈×C₁₃	—	4	4	no	—
92	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
93	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
94	C₁₃×C₄⋊C₄	—	2	1	no	—
95	C₁₃×C₄⋊C₄	—	2	1	no	—
96	C₁₃×C₄⋊C₄	—	2	4	no	—
97	C₁₃×C₄⋊C₄	—	2	4	no	—
98	C₁₃×C₄⋊C₄	—	4	2	no	—
99	C₁₃×C₄⋊C₄	—	4	2	no	—
100	C₁₃×C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
101	C₁₃×C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
102	C₁₃×C₂²⋊C₄	—	2	1	no	—
103	C₁₃×C₂²⋊C₄	—	2	1	no	—
104	C₁₃×C₂²⋊C₄	—	2	1	no	—
105	C₄²×C₁₃	—	2	1	no	—
106	C₄²×C₁₃	—	2	1	no	—
107	C₄²×C₁₃	—	2	1	no	—
108	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
109	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
110	C₄²×C₁₃	#41	1	1	yes	characteristic
111	C₄²×C₁₃	C₂⁴	1	1	yes	characteristic
112	C₂×#494	C₂×C₁₃	1	1	yes	characteristic, complemented
113	#3079	C₂×C₁₃	1	2	yes	—
114	C₈×#41	C₂×C₁₃	1	1	yes	characteristic
115	#1670	C₂×C₁₃	1	4	yes	complemented
116	#1669	C₂×C₁₃	1	4	yes	complemented
117	#1567	C₂×C₁₃	1	1	yes	characteristic, complemented
118	#1549	C₂×C₁₃	1	2	yes	complemented
119	C₂³×C₁₃	—	2	1	no	—
120	C₂³×C₁₃	#121	1	1	yes	characteristic
121	Q₈×C₁₃	—	2	1	no	—
122	Q₈×C₁₃	—	4	8	no	—
123	Q₈×C₁₃	—	8	4	no	—
124	Q₈×C₁₃	C₂×Q₁₆	1	2	yes	complemented
125	D₄×C₁₃	—	2	4	no	—
126	D₄×C₁₃	—	2	1	no	—
127	D₄×C₁₃	C₂×Q₁₆	1	2	yes	complemented
128	C₂×C₄×C₁₃	—	2	1	no	—
129	C₂×C₄×C₁₃	—	2	2	no	—
130	C₂×C₄×C₁₃	—	2	4	no	—
131	C₂×C₄×C₁₃	—	2	2	no	—
132	C₂×C₄×C₁₃	—	2	1	no	—
133	C₂×C₄×C₁₃	—	2	1	no	—
134	C₂×C₄×C₁₃	—	4	2	no	—
135	C₂×C₄×C₁₃	—	4	2	no	—
136	C₂×C₄×C₁₃	—	4	4	no	—
137	C₂×C₄×C₁₃	—	8	4	no	—
138	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
139	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
140	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
141	C₂×C₄×C₁₃	C₂×Q₁₆	1	2	yes	—
142	C₂×C₄×C₁₃	#135	1	2	yes	—
143	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
144	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
145	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
146	C₈×C₁₃	—	2	2	no	—
147	C₈×C₁₃	—	4	4	no	—
148	C₈×C₁₃	—	4	2	no	—
149	C₈×C₁₃	AGL(1,ℤ/8)	1	4	yes	—
150	C₂²×Q₁₆	—	2	2	no	—
151	C₂×C₄⋊Q₈	C₂²×C₁₃	1	2	yes	—
152	C₄×#41	C₂²×C₁₃	1	1	yes	characteristic
153	#494	—	2	1	no	—
154	#494	—	2	1	no	—
155	#494	C₂²×C₁₃	1	4	yes	—
156	#477	—	2	2	no	—
157	#476	—	2	2	no	—
158	#475	C₂²×C₁₃	1	4	yes	—
159	#474	C₂²×C₁₃	1	4	yes	—
160	C₈×Q₈	C₂²×C₁₃	1	2	yes	—
161	C₈×D₄	—	2	2	no	—
162	C₈×D₄	C₂²×C₁₃	1	2	yes	—
163	#431	—	2	1	no	—
164	#431	C₂²×C₁₃	1	1	yes	characteristic
165	#428	—	2	2	no	—
166	#423	C₂²×C₁₃	1	2	yes	—
167	#417	—	2	2	no	—
168	C₂×C₄×C₈	—	2	1	no	—
169	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
170	#335	C₂²×C₁₃	1	4	yes	—
171	#332	C₂²×C₁₃	1	8	yes	—
172	C₂²×C₁₃	—	2	1	no	—
173	C₂²×C₁₃	—	2	1	no	—
174	C₂²×C₁₃	—	2	2	no	—
175	C₂²×C₁₃	—	4	1	no	—
176	C₂²×C₁₃	—	4	1	no	—
177	C₂²×C₁₃	#467	1	2	yes	—
178	C₂²×C₁₃	C₂×#121	1	1	yes	characteristic
179	C₄×C₁₃	—	2	1	no	—
180	C₄×C₁₃	—	2	2	no	—
181	C₄×C₁₃	—	2	1	no	—
182	C₄×C₁₃	—	4	1	no	—
183	C₄×C₁₃	—	4	1	no	—
184	C₄×C₁₃	—	8	4	no	—
185	C₄×C₁₃	#467	1	2	yes	—
186	C₄×C₁₃	#473	1	2	yes	—
187	C₄×C₁₃	C₂²×Q₁₆	1	1	yes	characteristic
188	C₄×C₁₃	C₂×#135	1	1	yes	characteristic
189	C₄×C₁₃	C₂×AGL(1,ℤ/8)	1	2	yes	—
190	C₂×#41	D₄×C₁₃	1	1	yes	characteristic
191	C₂²×Q₈	—	2	2	no	—
192	C₂×Q₁₆	—	2	4	no	—
193	C₂×Q₁₆	—	2	4	no	—
194	C₂×Q₁₆	—	4	8	no	—
195	C₂×M₁₆	—	2	2	no	—
196	C₂²×C₈	—	2	2	no	—
197	C₂²×C₈	D₄×C₁₃	1	2	yes	—
198	C₄⋊Q₈	—	2	2	no	—
199	C₄⋊Q₈	—	2	2	no	—
200	C₄⋊Q₈	—	2	2	no	—
201	C₄⋊Q₈	C₂³×C₁₃	1	4	yes	—
202	C₄×Q₈	D₄×C₁₃	1	1	yes	characteristic
203	C₄×Q₈	C₂³×C₁₃	1	1	yes	characteristic
204	C₄×D₄	—	2	1	no	—
205	C₄×D₄	—	2	1	no	—
206	C₄×D₄	D₄×C₁₃	1	1	yes	characteristic
207	C₄×D₄	C₂³×C₁₃	1	1	yes	characteristic
208	#117	—	2	1	no	—
209	#117	D₄×C₁₃	1	1	yes	characteristic
210	C₂×C₄⋊C₄	—	2	2	no	—
211	C₂×C₄⋊C₄	—	2	2	no	—
212	C₂×C₄²	—	2	1	no	—
213	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
214	#107	—	2	4	no	—
215	#106	—	2	4	no	—
216	C₄⋊C₈	—	2	2	no	—
217	C₄⋊C₈	—	2	2	no	—
218	C₄⋊C₈	C₂³×C₁₃	1	2	yes	—
219	C₄⋊C₈	C₂³×C₁₃	1	4	yes	—
220	#103	—	2	4	no	—
221	#102	—	2	4	no	—
222	C₂²⋊C₈	—	2	1	no	—
223	C₂²⋊C₈	—	2	2	no	—
224	C₄×C₈	—	2	2	no	—
225	C₄×C₈	—	2	1	no	—
226	C₄×C₈	D₄×C₁₃	1	1	yes	characteristic
227	C₄×C₈	D₄×C₁₃	1	1	yes	characteristic
228	C₄×C₈	C₂³×C₁₃	1	2	yes	—
229	C₂×C₁₃	—	2	2	no	—
230	C₂×C₁₃	—	4	2	no	—
231	C₂×C₁₃	#3038	1	1	yes	characteristic
232	C₂×C₁₃	C₂×#467	1	1	yes	characteristic
233	C₂×C₁₃	C₂×#473	1	1	yes	characteristic
234	#41	—	2	2	no	—
235	#41	C₁₃×Q₁₆	1	4	yes	complemented
236	C₂×Q₈	—	2	4	no	—
237	C₂×Q₈	—	4	8	no	—
238	C₂×Q₈	—	4	2	no	—
239	C₂×Q₈	C₂×D₄×C₁₃	1	1	yes	characteristic
240	C₂×D₄	—	2	1	no	—
241	C₂×D₄	C₂×D₄×C₁₃	1	1	yes	characteristic
242	C₂²×C₄	—	2	1	no	—
243	C₂²×C₄	—	2	1	no	—
244	C₂²×C₄	—	2	1	no	—
245	C₂²×C₄	—	4	2	no	—
246	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
247	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
248	C₂²×C₄	C₁₃×#41	1	1	yes	characteristic
249	Q₁₆	—	4	8	no	—
250	Q₁₆	—	4	8	no	—
251	M₁₆	—	4	4	no	—
252	C₂×C₈	—	2	4	no	—
253	C₂×C₈	—	2	2	no	—
254	C₂×C₈	—	2	4	no	—
255	C₂×C₈	—	4	4	no	—
256	C₂×C₈	C₂×D₄×C₁₃	1	2	yes	—
257	C₂×C₈	C₂×D₄×C₁₃	1	2	yes	—
258	C₄⋊C₄	—	2	1	no	—
259	C₄⋊C₄	—	2	1	no	—
260	C₄⋊C₄	—	2	4	no	—
261	C₄⋊C₄	—	2	4	no	—
262	C₄⋊C₄	—	4	2	no	—
263	C₄⋊C₄	—	4	2	no	—
264	C₄⋊C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
265	C₄⋊C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
266	C₂²⋊C₄	—	2	1	no	—
267	C₂²⋊C₄	—	2	1	no	—
268	C₂²⋊C₄	—	2	1	no	—
269	C₄²	—	2	1	no	—
270	C₄²	—	2	1	no	—
271	C₄²	—	2	1	no	—
272	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
273	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
274	C₄²	C₁₃×#41	1	1	yes	characteristic
275	C₄²	C₂⁴×C₁₃	1	1	yes	characteristic
276	C₁₃	#19268	1	1	yes	characteristic, direct product factor, Sylow
277	C₂³	—	2	1	no	—
278	C₂³	C₁₃×#121	1	1	yes	characteristic
279	Q₈	—	2	1	no	—
280	Q₈	—	4	8	no	—
281	Q₈	—	8	4	no	—
282	Q₈	C₂×C₁₃×Q₁₆	1	2	yes	complemented
283	D₄	—	2	4	no	—
284	D₄	—	2	1	no	—
285	D₄	C₂×C₁₃×Q₁₆	1	2	yes	complemented
286	C₂×C₄	—	2	1	no	—
287	C₂×C₄	—	2	4	no	—
288	C₂×C₄	—	2	2	no	—
289	C₂×C₄	—	2	2	no	—
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	4	2	no	—
293	C₂×C₄	—	4	2	no	—
294	C₂×C₄	—	4	4	no	—
295	C₂×C₄	—	8	4	no	—
296	C₂×C₄	C₁₃×#121	1	1	yes	characteristic
297	C₂×C₄	C₁₃×#121	1	1	yes	characteristic
298	C₂×C₄	C₁₃×#121	1	1	yes	characteristic
299	C₂×C₄	C₂×C₁₃×Q₁₆	1	2	yes	—
300	C₂×C₄	C₁₃×#135	1	2	yes	—
301	C₂×C₄	C₂²×D₄×C₁₃	1	1	yes	characteristic
302	C₂×C₄	C₂²×D₄×C₁₃	1	1	yes	characteristic
303	C₂×C₄	C₂×C₁₃×#41	1	1	yes	characteristic
304	C₈	—	2	2	no	—
305	C₈	—	4	4	no	—
306	C₈	—	4	2	no	—
307	C₈	C₁₃×AGL(1,ℤ/8)	1	4	yes	—
308	C₂²	—	2	1	no	—
309	C₂²	—	2	1	no	—
310	C₂²	—	2	2	no	—
311	C₂²	—	4	1	no	—
312	C₂²	—	4	1	no	—
313	C₂²	C₁₃×#467	1	2	yes	—
314	C₂²	C₂×C₁₃×#121	1	1	yes	characteristic
315	C₄	—	2	1	no	—
316	C₄	—	2	2	no	—
317	C₄	—	2	1	no	—
318	C₄	—	4	1	no	—
319	C₄	—	4	1	no	—
320	C₄	—	8	4	no	—
321	C₄	C₁₃×#467	1	2	yes	—
322	C₄	C₁₃×#473	1	2	yes	—
323	C₄	C₂²×C₁₃×Q₁₆	1	1	yes	characteristic
324	C₄	C₂×C₁₃×#135	1	1	yes	characteristic
325	C₄	C₂×C₁₃×AGL(1,ℤ/8)	1	2	yes	—
326	C₂	—	2	2	no	—
327	C₂	—	4	2	no	—
328	C₂	C₁₃×#3038	1	1	yes	characteristic
329	C₂	C₂×C₁₃×#467	1	1	yes	characteristic
330	C₂	C₂×C₁₃×#473	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₃	C₂×#121
commutator, Frattini, cosocle	C₄²	C₂⁴×C₁₃
prime radical	C₂×C₁₃×#41	D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃ ◁ C₂×C₄²×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₁₃×#494	C₂	semidirect
4. C₁₃×#1670	C₂	semidirect
5. C₁₃×#1669	C₂	semidirect
6. C₁₃×#1567	C₂	semidirect
7. C₁₃×#1549	C₂	semidirect
69. #19268	C₁₃	direct (N × H)
71. C₁₃×#41	Q₁₆	semidirect
112. C₂×#494	C₂×C₁₃	semidirect
115. #1670	C₂×C₁₃	semidirect
116. #1669	C₂×C₁₃	semidirect
117. #1567	C₂×C₁₃	semidirect
118. #1549	C₂×C₁₃	semidirect
124. Q₈×C₁₃	C₂×Q₁₆	semidirect
127. D₄×C₁₃	C₂×Q₁₆	semidirect
235. #41	C₁₃×Q₁₆	semidirect
276. C₁₃	#19268	direct (N × H)
282. Q₈	C₂×C₁₃×Q₁₆	semidirect
285. D₄	C₂×C₁₃×Q₁₆	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₁₃×#494	2
C₁₃×#3079	2
C₈×C₁₃×#41	2
C₁₃×#1670	2
C₁₃×#1669	2
C₁₃×#1567	2
C₁₃×#1549	2
#19268	13

Subgroups — C13×#19268 #15705607

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#19268 #15705607