Subgroups — C₁₃×#21842 #15708181

Order 3328.

1010 conjugacy classes of subgroups.

The subgroup lattice has 344 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₃×#533	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₃×#432	C₂	1	1	yes	maximal, characteristic
3	C₁₃×#2283	C₂	1	1	yes	maximal, characteristic
4	C₁₃×#1993	C₂	1	2	yes	maximal, complemented
5	C₁₃×#1993	C₂	1	2	yes	maximal, complemented
6	C₁₃×#1989	C₂	1	4	yes	maximal, complemented
7	C₁₃×#1825	C₂	1	4	yes	maximal
8	C₁₃×#533	—	2	4	no	—
9	C₁₃×#533	C₄	1	8	yes	complemented
10	C₂×C₁₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
11	C₂×C₁₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
12	C₂×C₁₃×#125	C₂²	1	2	yes	—
13	C₂×C₁₃×C₂²⋊Q₈	—	2	1	no	—
14	C₂×C₁₃×C₂²⋊Q₈	—	2	1	no	—
15	C₂×C₄×Q₈×C₁₃	—	2	1	no	—
16	C₂×C₄×Q₈×C₁₃	—	2	1	no	—
17	C₂×C₁₃×#117	C₂²	1	2	yes	—
18	C₂²×C₄²×C₁₃	C₂²	1	1	yes	characteristic
19	C₁₃×#432	—	2	4	no	—
20	C₂×C₁₃×C₄⋊C₈	C₂²	1	2	yes	—
21	C₂×C₁₃×C₂²⋊C₈	C₂²	1	2	yes	—
22	C₂×C₁₃×#97	C₂²	1	2	yes	—
23	C₁₃×#383	—	2	1	no	—
24	C₁₃×#382	C₂²	1	2	yes	—
25	C₁₃×#381	—	2	1	no	—
26	C₁₃×#380	C₂²	1	2	yes	—
27	C₄×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
28	C₄×C₁₃×C₂²⋊C₄	—	2	1	no	—
29	C₁₃×#376	—	2	1	no	—
30	C₁₃×#340	C₂²	1	8	yes	—
31	C₁₃×#340	C₂²	1	8	yes	—
32	C₂²×Q₈×C₁₃	—	2	1	no	—
33	C₂²×Q₈×C₁₃	—	2	1	no	—
34	C₂³×C₄×C₁₃	D₄	1	1	yes	characteristic
35	C₂³×C₄×C₁₃	D₄	1	1	yes	characteristic
36	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
37	C₂²×C₈×C₁₃	C₂³	1	4	yes	—
38	C₁₃×C₄⋊Q₈	—	2	2	no	—
39	C₁₃×C₄⋊Q₈	—	2	2	no	—
40	C₁₃×C₄⋊Q₈	C₂×C₄	1	4	yes	—
41	C₁₃×C₄⋊Q₈	C₂×C₄	1	4	yes	—
42	C₁₃×#125	—	2	4	no	—
43	C₁₃×#125	C₂×C₄	1	8	yes	—
44	C₁₃×C₂²⋊Q₈	—	2	8	no	—
45	C₁₃×C₂²⋊Q₈	—	2	8	no	—
46	C₄×Q₈×C₁₃	—	2	8	no	—
47	C₄×Q₈×C₁₃	—	2	8	no	—
48	C₁₃×#117	—	2	4	no	—
49	C₁₃×#117	C₂×C₄	1	8	yes	—
50	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
51	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
52	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
53	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
54	C₂×C₁₃×C₄⋊C₄	C₂³	1	4	yes	—
55	C₂×C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
56	C₂×C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
57	C₂×C₁₃×C₂²⋊C₄	—	2	2	no	—
58	C₂×C₁₃×C₂²⋊C₄	—	2	2	no	—
59	C₂×C₄²×C₁₃	—	2	2	no	—
60	C₂×C₄²×C₁₃	—	2	2	no	—
61	C₂×C₄²×C₁₃	C₂×C₄	1	4	yes	—
62	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
63	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
64	C₂×C₄²×C₁₃	D₄	1	4	yes	—
65	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
66	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
67	C₁₃×C₄⋊C₈	—	2	4	no	—
68	C₁₃×C₂²⋊C₈	—	2	4	no	—
69	C₁₃×#97	—	2	4	no	—
70	C₁₃×#95	—	2	2	no	—
71	C₁₃×#95	—	2	2	no	—
72	C₁₃×#95	—	2	2	no	—
73	C₁₃×#95	—	2	2	no	—
74	#21842	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
75	C₂⁴×C₁₃	C₂×D₄	1	1	yes	characteristic
76	C₂×Q₈×C₁₃	—	2	4	no	—
77	C₂×Q₈×C₁₃	—	2	4	no	—
78	C₂×Q₈×C₁₃	—	4	4	no	—
79	C₂×Q₈×C₁₃	—	4	4	no	—
80	C₂²×C₄×C₁₃	—	2	4	no	—
81	C₂²×C₄×C₁₃	—	2	2	no	—
82	C₂²×C₄×C₁₃	—	2	2	no	—
83	C₂²×C₄×C₁₃	—	2	4	no	—
84	C₂²×C₄×C₁₃	—	4	2	no	—
85	C₂²×C₄×C₁₃	—	4	2	no	—
86	C₂²×C₄×C₁₃	C₂²⋊C₄	1	4	yes	—
87	C₂²×C₄×C₁₃	C₂²⋊C₄	1	4	yes	—
88	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
89	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
90	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
91	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
92	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
93	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
94	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
95	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
96	C₂×C₈×C₁₃	—	2	4	no	—
97	C₂×C₈×C₁₃	—	2	8	no	—
98	C₁₃×C₄⋊C₄	—	2	8	no	—
99	C₁₃×C₄⋊C₄	—	2	8	no	—
100	C₁₃×C₄⋊C₄	—	2	4	no	—
101	C₁₃×C₄⋊C₄	—	2	4	no	—
102	C₁₃×C₄⋊C₄	—	2	2	no	—
103	C₁₃×C₄⋊C₄	—	2	2	no	—
104	C₁₃×C₄⋊C₄	—	2	8	no	—
105	C₁₃×C₄⋊C₄	C₂²×C₄	1	4	yes	—
106	C₁₃×C₄⋊C₄	C₂²×C₄	1	4	yes	—
107	C₁₃×C₂²⋊C₄	—	2	8	no	—
108	C₁₃×C₂²⋊C₄	—	4	4	no	—
109	C₄²×C₁₃	—	2	8	no	—
110	C₄²×C₁₃	—	4	4	no	—
111	C₄²×C₁₃	C₂²⋊C₄	1	4	yes	—
112	C₄²×C₁₃	C₂²⋊C₄	1	4	yes	—
113	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
114	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
115	C₄²×C₁₃	C₂×D₄	1	2	yes	—
116	C₄²×C₁₃	C₂×D₄	1	2	yes	—
117	C₂×#533	C₂×C₁₃	1	1	yes	characteristic
118	C₂×#432	C₂×C₁₃	1	1	yes	characteristic
119	#2283	C₂×C₁₃	1	1	yes	characteristic
120	#1993	C₂×C₁₃	1	2	yes	complemented
121	#1993	C₂×C₁₃	1	2	yes	complemented
122	#1989	C₂×C₁₃	1	4	yes	complemented
123	#1825	C₂×C₁₃	1	4	yes	—
124	C₂³×C₁₃	—	2	2	no	—
125	C₂³×C₁₃	—	2	2	no	—
126	C₂³×C₁₃	—	2	2	no	—
127	C₂³×C₁₃	C₂×C₂²⋊C₄	1	2	yes	—
128	C₂³×C₁₃	C₂²×D₄	1	1	yes	characteristic
129	Q₈×C₁₃	—	4	8	no	—
130	Q₈×C₁₃	—	4	8	no	—
131	C₂×C₄×C₁₃	—	2	2	no	—
132	C₂×C₄×C₁₃	—	2	2	no	—
133	C₂×C₄×C₁₃	—	2	2	no	—
134	C₂×C₄×C₁₃	—	2	2	no	—
135	C₂×C₄×C₁₃	—	2	2	no	—
136	C₂×C₄×C₁₃	—	2	4	no	—
137	C₂×C₄×C₁₃	—	2	4	no	—
138	C₂×C₄×C₁₃	—	2	4	no	—
139	C₂×C₄×C₁₃	—	2	4	no	—
140	C₂×C₄×C₁₃	—	4	4	no	—
141	C₂×C₄×C₁₃	—	4	4	no	—
142	C₂×C₄×C₁₃	—	4	4	no	—
143	C₂×C₄×C₁₃	—	4	4	no	—
144	C₂×C₄×C₁₃	—	4	4	no	—
145	C₂×C₄×C₁₃	—	4	4	no	—
146	C₂×C₄×C₁₃	—	4	4	no	—
147	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	2	yes	—
148	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	2	yes	—
149	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	2	yes	—
150	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	2	yes	—
151	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	2	yes	—
152	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
153	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
154	C₂×C₄×C₁₃	2₋¹⁺⁴	1	4	yes	—
155	C₈×C₁₃	—	4	4	no	—
156	#533	—	2	4	no	—
157	#533	C₄×C₁₃	1	8	yes	complemented
158	C₂×C₄⋊Q₈	C₂²×C₁₃	1	1	yes	characteristic
159	C₂×C₄⋊Q₈	C₂²×C₁₃	1	1	yes	characteristic
160	C₂×#125	C₂²×C₁₃	1	2	yes	—
161	C₂×C₂²⋊Q₈	—	2	1	no	—
162	C₂×C₂²⋊Q₈	—	2	1	no	—
163	C₂×C₄×Q₈	—	2	1	no	—
164	C₂×C₄×Q₈	—	2	1	no	—
165	C₂×#117	C₂²×C₁₃	1	2	yes	—
166	C₂²×C₄²	C₂²×C₁₃	1	1	yes	characteristic
167	#432	—	2	4	no	—
168	C₂×C₄⋊C₈	C₂²×C₁₃	1	2	yes	—
169	C₂×C₂²⋊C₈	C₂²×C₁₃	1	2	yes	—
170	C₂×#97	C₂²×C₁₃	1	2	yes	—
171	#383	—	2	1	no	—
172	#382	C₂²×C₁₃	1	2	yes	—
173	#381	—	2	1	no	—
174	#380	C₂²×C₁₃	1	2	yes	—
175	C₄×C₄⋊C₄	C₂²×C₁₃	1	2	yes	—
176	C₄×C₂²⋊C₄	—	2	1	no	—
177	#376	—	2	1	no	—
178	#340	C₂²×C₁₃	1	8	yes	—
179	#340	C₂²×C₁₃	1	8	yes	—
180	C₂²×C₁₃	—	2	2	no	—
181	C₂²×C₁₃	—	2	2	no	—
182	C₂²×C₁₃	—	2	2	no	—
183	C₂²×C₁₃	—	4	4	no	—
184	C₂²×C₁₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
185	C₂²×C₁₃	#536	1	2	yes	—
186	C₂²×C₁₃	#575	1	2	yes	—
187	C₂²×C₁₃	#575	1	2	yes	—
188	C₄×C₁₃	—	2	2	no	—
189	C₄×C₁₃	—	2	2	no	—
190	C₄×C₁₃	—	2	2	no	—
191	C₄×C₁₃	—	2	2	no	—
192	C₄×C₁₃	—	2	2	no	—
193	C₄×C₁₃	—	4	4	no	—
194	C₄×C₁₃	—	4	4	no	—
195	C₄×C₁₃	—	8	4	no	—
196	C₄×C₁₃	#519	1	2	yes	—
197	C₄×C₁₃	#536	1	2	yes	—
198	C₂²×Q₈	—	2	1	no	—
199	C₂²×Q₈	—	2	1	no	—
200	C₂³×C₄	D₄×C₁₃	1	1	yes	characteristic
201	C₂³×C₄	D₄×C₁₃	1	1	yes	characteristic
202	C₂³×C₄	C₂³×C₁₃	1	1	yes	characteristic
203	C₂²×C₈	C₂³×C₁₃	1	4	yes	—
204	C₄⋊Q₈	—	2	2	no	—
205	C₄⋊Q₈	—	2	2	no	—
206	C₄⋊Q₈	C₂×C₄×C₁₃	1	4	yes	—
207	C₄⋊Q₈	C₂×C₄×C₁₃	1	4	yes	—
208	#125	—	2	4	no	—
209	#125	C₂×C₄×C₁₃	1	8	yes	—
210	C₂²⋊Q₈	—	2	8	no	—
211	C₂²⋊Q₈	—	2	8	no	—
212	C₄×Q₈	—	2	8	no	—
213	C₄×Q₈	—	2	8	no	—
214	#117	—	2	4	no	—
215	#117	C₂×C₄×C₁₃	1	8	yes	—
216	C₂×C₄⋊C₄	—	2	2	no	—
217	C₂×C₄⋊C₄	—	2	2	no	—
218	C₂×C₄⋊C₄	—	2	1	no	—
219	C₂×C₄⋊C₄	—	2	1	no	—
220	C₂×C₄⋊C₄	C₂³×C₁₃	1	4	yes	—
221	C₂×C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
222	C₂×C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
223	C₂×C₂²⋊C₄	—	2	2	no	—
224	C₂×C₂²⋊C₄	—	2	2	no	—
225	C₂×C₄²	—	2	2	no	—
226	C₂×C₄²	—	2	2	no	—
227	C₂×C₄²	C₂×C₄×C₁₃	1	4	yes	—
228	C₂×C₄²	D₄×C₁₃	1	1	yes	characteristic
229	C₂×C₄²	D₄×C₁₃	1	1	yes	characteristic
230	C₂×C₄²	D₄×C₁₃	1	4	yes	—
231	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
232	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
233	C₄⋊C₈	—	2	4	no	—
234	C₂²⋊C₈	—	2	4	no	—
235	#97	—	2	4	no	—
236	#95	—	2	2	no	—
237	#95	—	2	2	no	—
238	#95	—	2	2	no	—
239	#95	—	2	2	no	—
240	C₂×C₁₃	—	4	2	no	—
241	C₂×C₁₃	#2310	1	1	yes	characteristic
242	C₂×C₁₃	#2896	1	1	yes	characteristic
243	C₂×C₁₃	#2896	1	1	yes	characteristic
244	C₂×C₁₃	#3240	1	4	yes	—
245	C₂⁴	C₂×D₄×C₁₃	1	1	yes	characteristic
246	C₂×Q₈	—	2	4	no	—
247	C₂×Q₈	—	2	4	no	—
248	C₂×Q₈	—	4	4	no	—
249	C₂×Q₈	—	4	4	no	—
250	C₂²×C₄	—	2	4	no	—
251	C₂²×C₄	—	2	2	no	—
252	C₂²×C₄	—	2	2	no	—
253	C₂²×C₄	—	2	4	no	—
254	C₂²×C₄	—	4	2	no	—
255	C₂²×C₄	—	4	2	no	—
256	C₂²×C₄	C₁₃×C₂²⋊C₄	1	4	yes	—
257	C₂²×C₄	C₁₃×C₂²⋊C₄	1	4	yes	—
258	C₂²×C₄	C₂²×C₄×C₁₃	1	2	yes	—
259	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
260	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
261	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
262	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
263	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
264	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
265	C₂²×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
266	C₂×C₈	—	2	4	no	—
267	C₂×C₈	—	2	8	no	—
268	C₄⋊C₄	—	2	8	no	—
269	C₄⋊C₄	—	2	8	no	—
270	C₄⋊C₄	—	2	4	no	—
271	C₄⋊C₄	—	2	4	no	—
272	C₄⋊C₄	—	2	2	no	—
273	C₄⋊C₄	—	2	2	no	—
274	C₄⋊C₄	—	2	8	no	—
275	C₄⋊C₄	C₂²×C₄×C₁₃	1	4	yes	—
276	C₄⋊C₄	C₂²×C₄×C₁₃	1	4	yes	—
277	C₂²⋊C₄	—	2	8	no	—
278	C₂²⋊C₄	—	4	4	no	—
279	C₄²	—	2	8	no	—
280	C₄²	—	4	4	no	—
281	C₄²	C₁₃×C₂²⋊C₄	1	4	yes	—
282	C₄²	C₁₃×C₂²⋊C₄	1	4	yes	—
283	C₄²	C₂²×C₄×C₁₃	1	2	yes	—
284	C₄²	C₂²×C₄×C₁₃	1	2	yes	—
285	C₄²	C₂×D₄×C₁₃	1	2	yes	—
286	C₄²	C₂×D₄×C₁₃	1	2	yes	—
287	C₁₃	#21842	1	1	yes	characteristic, direct product factor, Sylow
288	C₂³	—	2	2	no	—
289	C₂³	—	2	2	no	—
290	C₂³	—	2	2	no	—
291	C₂³	C₂×C₁₃×C₂²⋊C₄	1	2	yes	—
292	C₂³	C₂²×D₄×C₁₃	1	1	yes	characteristic
293	Q₈	—	4	8	no	—
294	Q₈	—	4	8	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	2	2	no	—
297	C₂×C₄	—	2	2	no	—
298	C₂×C₄	—	2	2	no	—
299	C₂×C₄	—	2	2	no	—
300	C₂×C₄	—	2	4	no	—
301	C₂×C₄	—	2	4	no	—
302	C₂×C₄	—	2	4	no	—
303	C₂×C₄	—	2	4	no	—
304	C₂×C₄	—	4	4	no	—
305	C₂×C₄	—	4	4	no	—
306	C₂×C₄	—	4	4	no	—
307	C₂×C₄	—	4	4	no	—
308	C₂×C₄	—	4	4	no	—
309	C₂×C₄	—	4	4	no	—
310	C₂×C₄	—	4	4	no	—
311	C₂×C₄	C₂×C₁₃×C₂²⋊C₄	1	2	yes	—
312	C₂×C₄	C₂×C₁₃×C₂²⋊C₄	1	2	yes	—
313	C₂×C₄	C₂×C₁₃×C₂²⋊C₄	1	2	yes	—
314	C₂×C₄	C₂×C₁₃×C₂²⋊C₄	1	2	yes	—
315	C₂×C₄	C₂×C₁₃×C₂²⋊C₄	1	2	yes	—
316	C₂×C₄	C₂³×C₄×C₁₃	1	1	yes	characteristic
317	C₂×C₄	C₂²×D₄×C₁₃	1	1	yes	characteristic
318	C₂×C₄	C₁₃×2₋¹⁺⁴	1	4	yes	—
319	C₈	—	4	4	no	—
320	C₂²	—	2	2	no	—
321	C₂²	—	2	2	no	—
322	C₂²	—	2	2	no	—
323	C₂²	—	4	4	no	—
324	C₂²	C₂²×C₁₃×C₂²⋊C₄	1	1	yes	characteristic
325	C₂²	C₁₃×#536	1	2	yes	—
326	C₂²	C₁₃×#575	1	2	yes	—
327	C₂²	C₁₃×#575	1	2	yes	—
328	C₄	—	2	2	no	—
329	C₄	—	2	2	no	—
330	C₄	—	2	2	no	—
331	C₄	—	2	2	no	—
332	C₄	—	2	2	no	—
333	C₄	—	4	4	no	—
334	C₄	—	4	4	no	—
335	C₄	—	8	4	no	—
336	C₄	C₁₃×#519	1	2	yes	—
337	C₄	C₁₃×#536	1	2	yes	—
338	C₂	—	4	2	no	—
339	C₂	C₁₃×#2310	1	1	yes	characteristic
340	C₂	C₁₃×#2896	1	1	yes	characteristic
341	C₂	C₁₃×#2896	1	1	yes	characteristic
342	C₂	C₁₃×#3240	1	4	yes	—

	subgroup	G/H
center, socle	C₂³×C₁₃	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₁₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃
prime radical	C₂⁴×C₁₃	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₃ ◁ C₂²×C₄²×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₁₃×#1993	C₂	semidirect
5. C₁₃×#1993	C₂	semidirect
6. C₁₃×#1989	C₂	semidirect
9. C₁₃×#533	C₄	semidirect
74. #21842	C₁₃	direct (N × H)
120. #1993	C₂×C₁₃	semidirect
121. #1993	C₂×C₁₃	semidirect
122. #1989	C₂×C₁₃	semidirect
157. #533	C₄×C₁₃	semidirect
287. C₁₃	#21842	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×#533	2
C₂×C₁₃×#432	2
C₁₃×#2283	2
C₁₃×#1993	2
C₁₃×#1989	2
C₁₃×#1825	2
#21842	13

Subgroups — C13×#21842 #15708181

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#21842 #15708181