Subgroups — C₁₃×#23669 #15710008

Order 3328.

874 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₃×#427	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₃×#424	C₂	1	1	yes	maximal, characteristic
3	C₁₃×#2316	C₂	1	2	yes	maximal
4	C₁₃×#2088	C₂	1	4	yes	maximal, complemented
5	C₁₃×#2087	C₂	1	2	yes	maximal, complemented
6	C₁₃×#2086	C₂	1	2	yes	maximal, complemented
7	C₁₃×#1736	C₂	1	2	yes	maximal
8	C₂×C₁₃×#340	C₂	1	1	yes	maximal, characteristic
9	C₂²×C₁₃×M₁₆	C₂²	1	1	yes	characteristic
10	C₂³×C₈×C₁₃	C₂²	1	1	yes	characteristic
11	C₂×C₁₃×#117	C₂²	1	2	yes	—
12	C₂×C₁₃×#117	C₂²	1	1	yes	characteristic
13	C₂²×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
14	C₁₃×#427	—	2	4	no	—
15	C₂×C₁₃×#107	C₂²	1	1	yes	characteristic
16	C₂×C₁₃×#107	C₂²	1	1	yes	characteristic
17	C₂×C₁₃×#106	C₂²	1	2	yes	—
18	C₁₃×#424	—	2	4	no	—
19	C₂×C₁₃×C₄⋊C₈	C₂²	1	2	yes	—
20	C₂×C₁₃×C₄⋊C₈	C₂²	1	2	yes	—
21	C₁₃×#389	—	2	2	no	—
22	C₁₃×#389	—	2	2	no	—
23	C₁₃×#389	C₂²	1	4	yes	—
24	C₁₃×#389	C₂²	1	4	yes	—
25	C₁₃×#380	—	2	4	no	—
26	C₁₃×#340	C₂²	1	2	yes	—
27	C₁₃×#340	C₂²	1	2	yes	—
28	C₁₃×#340	C₂²	1	4	yes	—
29	C₁₃×#340	C₂²	1	4	yes	—
30	C₂³×C₄×C₁₃	—	2	2	no	—
31	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
32	C₂×C₁₃×M₁₆	—	2	2	no	—
33	C₂×C₁₃×M₁₆	—	2	4	no	—
34	C₂²×C₈×C₁₃	—	2	4	no	—
35	C₂²×C₈×C₁₃	D₄	1	4	yes	—
36	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
37	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
38	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
39	C₁₃×#117	—	2	2	no	—
40	C₁₃×#117	—	2	2	no	—
41	C₁₃×#117	—	2	4	no	—
42	C₁₃×#117	—	2	4	no	—
43	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
44	C₂×C₁₃×C₄⋊C₄	—	2	4	no	—
45	C₂×C₁₃×C₄⋊C₄	—	2	4	no	—
46	C₂×C₁₃×C₄⋊C₄	—	2	4	no	—
47	C₂×C₁₃×C₄⋊C₄	D₄	1	1	yes	characteristic
48	C₂×C₁₃×C₄⋊C₄	D₄	1	1	yes	characteristic
49	C₂×C₁₃×C₄⋊C₄	Q₈	1	8	yes	—
50	C₂×C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
51	C₂×C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
52	C₂×C₁₃×C₄⋊C₄	C₂³	1	4	yes	—
53	C₂×C₁₃×C₂²⋊C₄	—	2	2	no	—
54	C₂×C₁₃×C₂²⋊C₄	D₄	1	2	yes	—
55	C₂×C₄²×C₁₃	—	2	2	no	—
56	C₂×C₄²×C₁₃	C₂³	1	2	yes	—
57	C₁₃×#107	—	2	2	no	—
58	C₁₃×#107	—	2	2	no	—
59	C₁₃×#106	—	2	4	no	—
60	C₁₃×C₄⋊C₈	—	2	4	no	—
61	C₁₃×C₄⋊C₈	—	2	4	no	—
62	C₁₃×#95	—	2	4	no	—
63	C₁₃×#95	—	2	4	no	—
64	#23669	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
65	C₂⁴×C₁₃	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₁₃	—	2	4	no	—
67	C₂²×C₄×C₁₃	—	2	8	no	—
68	C₂²×C₄×C₁₃	—	2	2	no	—
69	C₂²×C₄×C₁₃	—	2	2	no	—
70	C₂²×C₄×C₁₃	—	8	2	no	—
71	C₂²×C₄×C₁₃	—	8	2	no	—
72	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
73	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
74	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
75	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
76	C₂²×C₄×C₁₃	C₂×Q₈	1	4	yes	—
77	C₂²×C₄×C₁₃	#41	1	2	yes	—
78	C₂²×C₄×C₁₃	#41	1	4	yes	—
79	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
80	C₁₃×M₁₆	—	4	4	no	—
81	C₂×C₈×C₁₃	—	2	8	no	—
82	C₂×C₈×C₁₃	—	2	2	no	—
83	C₂×C₈×C₁₃	—	2	2	no	—
84	C₂×C₈×C₁₃	—	2	2	no	—
85	C₂×C₈×C₁₃	—	2	4	no	—
86	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
87	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
88	C₁₃×C₄⋊C₄	—	2	2	no	—
89	C₁₃×C₄⋊C₄	—	2	2	no	—
90	C₁₃×C₄⋊C₄	—	2	8	no	—
91	C₁₃×C₄⋊C₄	—	2	4	no	—
92	C₁₃×C₄⋊C₄	—	2	2	no	—
93	C₁₃×C₄⋊C₄	—	2	2	no	—
94	C₁₃×C₄⋊C₄	—	2	2	no	—
95	C₁₃×C₄⋊C₄	—	2	2	no	—
96	C₁₃×C₄⋊C₄	—	8	4	no	—
97	C₁₃×C₄⋊C₄	—	8	4	no	—
98	C₁₃×C₄⋊C₄	C₂×Q₈	1	8	yes	—
99	C₁₃×C₂²⋊C₄	—	2	4	no	—
100	C₁₃×C₂²⋊C₄	—	4	4	no	—
101	C₄²×C₁₃	—	2	2	no	—
102	C₄²×C₁₃	—	2	2	no	—
103	C₄²×C₁₃	—	4	4	no	—
104	C₂×#427	C₂×C₁₃	1	1	yes	characteristic
105	C₂×#424	C₂×C₁₃	1	1	yes	characteristic
106	#2316	C₂×C₁₃	1	2	yes	—
107	#2088	C₂×C₁₃	1	4	yes	complemented
108	#2087	C₂×C₁₃	1	2	yes	complemented
109	#2086	C₂×C₁₃	1	2	yes	complemented
110	#1736	C₂×C₁₃	1	2	yes	—
111	C₂×#340	C₂×C₁₃	1	1	yes	characteristic
112	C₂³×C₁₃	—	2	1	no	—
113	C₂³×C₁₃	—	2	2	no	—
114	C₂³×C₁₃	—	2	1	no	—
115	C₂³×C₁₃	#121	1	2	yes	—
116	C₂³×C₁₃	C₂²⋊Q₈	1	4	yes	—
117	C₂³×C₁₃	C₂²×D₄	1	1	yes	characteristic
118	C₂×C₄×C₁₃	—	2	2	no	—
119	C₂×C₄×C₁₃	—	2	2	no	—
120	C₂×C₄×C₁₃	—	2	8	no	—
121	C₂×C₄×C₁₃	—	2	2	no	—
122	C₂×C₄×C₁₃	—	2	4	no	—
123	C₂×C₄×C₁₃	—	2	2	no	—
124	C₂×C₄×C₁₃	—	4	4	no	—
125	C₂×C₄×C₁₃	—	4	4	no	—
126	C₂×C₄×C₁₃	—	4	2	no	—
127	C₂×C₄×C₁₃	—	4	2	no	—
128	C₂×C₄×C₁₃	—	4	2	no	—
129	C₂×C₄×C₁₃	—	4	2	no	—
130	C₂×C₄×C₁₃	—	4	2	no	—
131	C₂×C₄×C₁₃	—	4	2	no	—
132	C₂×C₄×C₁₃	—	8	8	no	—
133	C₂×C₄×C₁₃	#121	1	2	yes	—
134	C₂×C₄×C₁₃	C₂²⋊Q₈	1	4	yes	—
135	C₂×C₄×C₁₃	#123	1	4	yes	—
136	C₂×C₄×C₁₃	#125	1	8	yes	—
137	C₂×C₄×C₁₃	C₄⋊Q₈	1	4	yes	—
138	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
139	C₂×C₄×C₁₃	C₂²×Q₈	1	2	yes	—
140	C₂×C₄×C₁₃	C₂×#41	1	2	yes	—
141	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
142	C₈×C₁₃	—	2	2	no	—
143	C₈×C₁₃	—	2	2	no	—
144	C₈×C₁₃	—	4	2	no	—
145	C₂²×M₁₆	C₂²×C₁₃	1	1	yes	characteristic
146	C₂³×C₈	C₂²×C₁₃	1	1	yes	characteristic
147	C₂×#117	C₂²×C₁₃	1	2	yes	—
148	C₂×#117	C₂²×C₁₃	1	1	yes	characteristic
149	C₂²×C₄⋊C₄	C₂²×C₁₃	1	2	yes	—
150	#427	—	2	4	no	—
151	C₂×#107	C₂²×C₁₃	1	1	yes	characteristic
152	C₂×#107	C₂²×C₁₃	1	1	yes	characteristic
153	C₂×#106	C₂²×C₁₃	1	2	yes	—
154	#424	—	2	4	no	—
155	C₂×C₄⋊C₈	C₂²×C₁₃	1	2	yes	—
156	C₂×C₄⋊C₈	C₂²×C₁₃	1	2	yes	—
157	#389	—	2	2	no	—
158	#389	—	2	2	no	—
159	#389	C₂²×C₁₃	1	4	yes	—
160	#389	C₂²×C₁₃	1	4	yes	—
161	#380	—	2	4	no	—
162	#340	C₂²×C₁₃	1	2	yes	—
163	#340	C₂²×C₁₃	1	4	yes	—
164	#340	C₂²×C₁₃	1	4	yes	—
165	#340	C₂²×C₁₃	1	2	yes	—
166	C₂²×C₁₃	—	2	4	no	—
167	C₂²×C₁₃	—	2	4	no	—
168	C₂²×C₁₃	—	2	2	no	—
169	C₂²×C₁₃	—	2	2	no	—
170	C₂²×C₁₃	#398	1	4	yes	—
171	C₂²×C₁₃	C₂×#121	1	1	yes	characteristic
172	C₂²×C₁₃	C₂×C₂²⋊Q₈	1	2	yes	—
173	C₂²×C₁₃	#576	1	1	yes	characteristic
174	C₂²×C₁₃	#577	1	2	yes	—
175	C₂²×C₁₃	#578	1	1	yes	characteristic
176	C₄×C₁₃	—	4	8	no	—
177	C₄×C₁₃	—	4	2	no	—
178	C₄×C₁₃	—	4	2	no	—
179	C₄×C₁₃	—	8	2	no	—
180	C₄×C₁₃	—	8	2	no	—
181	C₄×C₁₃	#398	1	4	yes	—
182	C₄×C₁₃	C₂×#123	1	1	yes	characteristic
183	C₄×C₁₃	C₂×#125	1	2	yes	—
184	C₄×C₁₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
185	C₂³×C₄	—	2	2	no	—
186	C₂³×C₄	C₂³×C₁₃	1	1	yes	characteristic
187	C₂×M₁₆	—	2	2	no	—
188	C₂×M₁₆	—	2	4	no	—
189	C₂²×C₈	—	2	4	no	—
190	C₂²×C₈	D₄×C₁₃	1	4	yes	—
191	C₂²×C₈	C₂³×C₁₃	1	2	yes	—
192	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
193	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
194	#117	—	2	2	no	—
195	#117	—	2	2	no	—
196	#117	—	2	4	no	—
197	#117	—	2	4	no	—
198	C₂×C₄⋊C₄	—	2	2	no	—
199	C₂×C₄⋊C₄	—	2	4	no	—
200	C₂×C₄⋊C₄	—	2	4	no	—
201	C₂×C₄⋊C₄	—	2	4	no	—
202	C₂×C₄⋊C₄	D₄×C₁₃	1	1	yes	characteristic
203	C₂×C₄⋊C₄	D₄×C₁₃	1	1	yes	characteristic
204	C₂×C₄⋊C₄	Q₈×C₁₃	1	8	yes	—
205	C₂×C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
206	C₂×C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
207	C₂×C₄⋊C₄	C₂³×C₁₃	1	4	yes	—
208	C₂×C₂²⋊C₄	—	2	2	no	—
209	C₂×C₂²⋊C₄	D₄×C₁₃	1	2	yes	—
210	C₂×C₄²	—	2	2	no	—
211	C₂×C₄²	C₂³×C₁₃	1	2	yes	—
212	#107	—	2	2	no	—
213	#107	—	2	2	no	—
214	#106	—	2	4	no	—
215	C₄⋊C₈	—	2	4	no	—
216	C₄⋊C₈	—	2	4	no	—
217	#95	—	2	4	no	—
218	#95	—	2	4	no	—
219	C₂×C₁₃	—	2	4	no	—
220	C₂×C₁₃	C₂×#398	1	1	yes	characteristic
221	C₂×C₁₃	#3060	1	1	yes	characteristic
222	C₂×C₁₃	#3061	1	1	yes	characteristic
223	C₂×C₁₃	#3081	1	2	yes	—
224	C₂×C₁₃	#3082	1	2	yes	—
225	C₂⁴	C₂×D₄×C₁₃	1	1	yes	characteristic
226	C₂²×C₄	—	2	4	no	—
227	C₂²×C₄	—	2	8	no	—
228	C₂²×C₄	—	2	2	no	—
229	C₂²×C₄	—	2	2	no	—
230	C₂²×C₄	—	8	2	no	—
231	C₂²×C₄	—	8	2	no	—
232	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
233	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
234	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
235	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
236	C₂²×C₄	C₂×Q₈×C₁₃	1	4	yes	—
237	C₂²×C₄	C₁₃×#41	1	2	yes	—
238	C₂²×C₄	C₁₃×#41	1	4	yes	—
239	C₂²×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
240	M₁₆	—	4	4	no	—
241	C₂×C₈	—	2	8	no	—
242	C₂×C₈	—	2	2	no	—
243	C₂×C₈	—	2	2	no	—
244	C₂×C₈	—	2	2	no	—
245	C₂×C₈	—	2	4	no	—
246	C₂×C₈	C₂×D₄×C₁₃	1	2	yes	—
247	C₂×C₈	C₂×D₄×C₁₃	1	2	yes	—
248	C₄⋊C₄	—	2	2	no	—
249	C₄⋊C₄	—	2	2	no	—
250	C₄⋊C₄	—	2	8	no	—
251	C₄⋊C₄	—	2	4	no	—
252	C₄⋊C₄	—	2	2	no	—
253	C₄⋊C₄	—	2	2	no	—
254	C₄⋊C₄	—	2	2	no	—
255	C₄⋊C₄	—	2	2	no	—
256	C₄⋊C₄	—	8	4	no	—
257	C₄⋊C₄	—	8	4	no	—
258	C₄⋊C₄	C₂×Q₈×C₁₃	1	8	yes	—
259	C₂²⋊C₄	—	2	4	no	—
260	C₂²⋊C₄	—	4	4	no	—
261	C₄²	—	2	2	no	—
262	C₄²	—	2	2	no	—
263	C₄²	—	4	4	no	—
264	C₁₃	#23669	1	1	yes	characteristic, direct product factor, Sylow
265	C₂³	—	2	1	no	—
266	C₂³	—	2	2	no	—
267	C₂³	—	2	1	no	—
268	C₂³	C₁₃×#121	1	2	yes	—
269	C₂³	C₁₃×C₂²⋊Q₈	1	4	yes	—
270	C₂³	C₂²×D₄×C₁₃	1	1	yes	characteristic
271	C₂×C₄	—	2	2	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	2	8	no	—
274	C₂×C₄	—	2	2	no	—
275	C₂×C₄	—	2	4	no	—
276	C₂×C₄	—	2	2	no	—
277	C₂×C₄	—	4	4	no	—
278	C₂×C₄	—	4	4	no	—
279	C₂×C₄	—	4	2	no	—
280	C₂×C₄	—	4	2	no	—
281	C₂×C₄	—	4	2	no	—
282	C₂×C₄	—	4	2	no	—
283	C₂×C₄	—	4	2	no	—
284	C₂×C₄	—	4	2	no	—
285	C₂×C₄	—	8	8	no	—
286	C₂×C₄	C₁₃×#121	1	2	yes	—
287	C₂×C₄	C₁₃×C₂²⋊Q₈	1	4	yes	—
288	C₂×C₄	C₁₃×#123	1	4	yes	—
289	C₂×C₄	C₁₃×#125	1	8	yes	—
290	C₂×C₄	C₁₃×C₄⋊Q₈	1	4	yes	—
291	C₂×C₄	C₂²×D₄×C₁₃	1	1	yes	characteristic
292	C₂×C₄	C₂²×Q₈×C₁₃	1	2	yes	—
293	C₂×C₄	C₂×C₁₃×#41	1	2	yes	—
294	C₂×C₄	C₂×C₁₃×#41	1	1	yes	characteristic
295	C₈	—	2	2	no	—
296	C₈	—	2	2	no	—
297	C₈	—	4	2	no	—
298	C₂²	—	2	2	no	—
299	C₂²	—	2	2	no	—
300	C₂²	—	2	4	no	—
301	C₂²	—	2	4	no	—
302	C₂²	C₁₃×#398	1	4	yes	—
303	C₂²	C₂×C₁₃×#121	1	1	yes	characteristic
304	C₂²	C₂×C₁₃×C₂²⋊Q₈	1	2	yes	—
305	C₂²	C₁₃×#576	1	1	yes	characteristic
306	C₂²	C₁₃×#577	1	2	yes	—
307	C₂²	C₁₃×#578	1	1	yes	characteristic
308	C₄	—	4	2	no	—
309	C₄	—	4	2	no	—
310	C₄	—	4	8	no	—
311	C₄	—	8	2	no	—
312	C₄	—	8	2	no	—
313	C₄	C₁₃×#398	1	4	yes	—
314	C₄	C₂×C₁₃×#123	1	1	yes	characteristic
315	C₄	C₂×C₁₃×#125	1	2	yes	—
316	C₄	C₂×C₁₃×C₄⋊Q₈	1	1	yes	characteristic
317	C₂	—	2	4	no	—
318	C₂	C₂×C₁₃×#398	1	1	yes	characteristic
319	C₂	C₁₃×#3060	1	1	yes	characteristic
320	C₂	C₁₃×#3061	1	1	yes	characteristic
321	C₂	C₁₃×#3081	1	2	yes	—
322	C₂	C₁₃×#3082	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₁₃	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃
prime radical	C₂⁴×C₁₃	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₃ ◁ C₂×C₁₃×#117 ◁ G

Complements

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

N	H ≅ G/N	type
4. C₁₃×#2088	C₂	semidirect
5. C₁₃×#2087	C₂	semidirect
6. C₁₃×#2086	C₂	semidirect
64. #23669	C₁₃	direct (N × H)
107. #2088	C₂×C₁₃	semidirect
108. #2087	C₂×C₁₃	semidirect
109. #2086	C₂×C₁₃	semidirect
264. C₁₃	#23669	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×#427	2
C₂×C₁₃×#424	2
C₁₃×#2316	2
C₁₃×#2088	2
C₁₃×#2087	2
C₁₃×#2086	2
C₁₃×#1736	2
C₂×C₁₃×#340	2
#23669	13

Subgroups — C13×#23669 #15710008

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#23669 #15710008