Subgroups — C₁₃×#23811 #15710150

Order 3328.

1038 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₃×#440	C₂	1	2	yes	maximal, complemented
2	C₂×C₄×C₁₃×SD₁₆	C₂	1	2	yes	maximal, complemented
3	C₁₃×#2328	C₂	1	1	yes	maximal, characteristic
4	C₁₃×#2327	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₃×#2097	C₂	1	2	yes	maximal, complemented
6	C₁₃×#2092	C₂	1	2	yes	maximal, complemented
7	C₁₃×#2014	C₂	1	2	yes	maximal
8	C₁₃×#2013	C₂	1	2	yes	maximal, complemented
9	C₁₃×#1746	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×C₁₃×SD₁₆	C₂²	1	2	yes	—
11	C₂²×C₁₃×SD₁₆	C₂²	1	2	yes	—
12	C₂×C₁₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
13	C₂×C₁₃×#127	C₂²	1	1	yes	characteristic
14	C₂×C₄×Q₈×C₁₃	C₂²	1	2	yes	—
15	C₂×C₄×D₄×C₁₃	C₂²	1	2	yes	—
16	C₁₃×#440	—	2	8	no	—
17	C₄×C₁₃×SD₁₆	—	2	8	no	—
18	C₂×C₁₃×#107	C₂²	1	2	yes	—
19	C₂×C₁₃×#106	C₂²	1	2	yes	—
20	C₂×C₁₃×#103	C₂²	1	2	yes	—
21	C₂×C₁₃×#103	C₂²	1	2	yes	—
22	C₂×C₁₃×#102	C₂²	1	2	yes	—
23	C₂×C₁₃×#102	C₂²	1	2	yes	—
24	C₂×C₁₃×#97	C₂²	1	1	yes	characteristic
25	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
26	C₁₃×#391	C₂²	1	2	yes	—
27	C₁₃×#390	C₂²	1	2	yes	—
28	C₁₃×#387	—	2	4	no	—
29	C₁₃×#386	—	2	4	no	—
30	C₁₃×#382	C₂²	1	1	yes	characteristic
31	C₁₃×#382	C₂²	1	1	yes	characteristic
32	C₁₃×#376	C₂²	1	1	yes	characteristic
33	C₁₃×#336	C₂²	1	2	yes	—
34	C₁₃×#336	C₂²	1	2	yes	—
35	C₂²×Q₈×C₁₃	C₂³	1	2	yes	—
36	C₂²×D₄×C₁₃	—	2	2	no	—
37	C₂²×D₄×C₁₃	C₂³	1	2	yes	—
38	C₂³×C₄×C₁₃	—	2	2	no	—
39	C₂×C₁₃×SD₁₆	—	2	4	no	—
40	C₂×C₁₃×SD₁₆	—	2	4	no	—
41	C₂×C₁₃×SD₁₆	—	2	4	no	—
42	C₂×C₁₃×SD₁₆	—	2	4	no	—
43	C₂×C₁₃×SD₁₆	—	2	4	no	—
44	C₂×C₁₃×SD₁₆	—	2	4	no	—
45	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
46	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
47	C₁₃×C₄⋊Q₈	—	2	4	no	—
48	C₁₃×#127	—	2	4	no	—
49	C₄×Q₈×C₁₃	—	2	4	no	—
50	C₄×Q₈×C₁₃	—	2	4	no	—
51	C₄×D₄×C₁₃	—	2	4	no	—
52	C₄×D₄×C₁₃	—	2	4	no	—
53	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
54	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
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	4	no	—
59	C₂×C₄²×C₁₃	—	2	2	no	—
60	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
61	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
62	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
63	C₁₃×#107	—	2	4	no	—
64	C₁₃×#106	—	2	4	no	—
65	C₁₃×#103	—	2	4	no	—
66	C₁₃×#103	—	2	4	no	—
67	C₁₃×#102	—	2	4	no	—
68	C₁₃×#102	—	2	4	no	—
69	C₁₃×#97	D₄	1	4	yes	complemented
70	C₄×C₈×C₁₃	D₄	1	4	yes	complemented
71	C₁₃×#95	—	2	2	no	—
72	C₁₃×#95	—	2	2	no	—
73	C₁₃×#95	—	2	4	no	—
74	C₁₃×#95	D₄	1	4	yes	—
75	#23811	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
76	C₂⁴×C₁₃	—	2	2	no	—
77	C₂×Q₈×C₁₃	—	2	2	no	—
78	C₂×Q₈×C₁₃	—	2	2	no	—
79	C₂×Q₈×C₁₃	—	2	2	no	—
80	C₂×Q₈×C₁₃	—	2	2	no	—
81	C₂×Q₈×C₁₃	—	2	2	no	—
82	C₂×Q₈×C₁₃	#41	1	4	yes	—
83	C₂×D₄×C₁₃	—	2	8	no	—
84	C₂×D₄×C₁₃	—	2	2	no	—
85	C₂×D₄×C₁₃	—	2	2	no	—
86	C₂×D₄×C₁₃	—	2	2	no	—
87	C₂×D₄×C₁₃	—	2	2	no	—
88	C₂×D₄×C₁₃	—	2	2	no	—
89	C₂×D₄×C₁₃	—	8	4	no	—
90	C₂×D₄×C₁₃	#41	1	4	yes	—
91	C₂²×C₄×C₁₃	—	2	2	no	—
92	C₂²×C₄×C₁₃	—	2	2	no	—
93	C₂²×C₄×C₁₃	—	2	2	no	—
94	C₂²×C₄×C₁₃	—	4	4	no	—
95	C₂²×C₄×C₁₃	—	8	4	no	—
96	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
97	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
98	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
99	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
100	C₁₃×SD₁₆	—	2	16	no	—
101	C₁₃×SD₁₆	—	2	16	no	—
102	C₂×C₈×C₁₃	—	2	2	no	—
103	C₂×C₈×C₁₃	—	2	2	no	—
104	C₂×C₈×C₁₃	—	2	2	no	—
105	C₂×C₈×C₁₃	—	2	2	no	—
106	C₂×C₈×C₁₃	C₂×D₄	1	4	yes	—
107	C₂×C₈×C₁₃	C₂×D₄	1	4	yes	—
108	C₁₃×C₄⋊C₄	—	2	8	no	—
109	C₁₃×C₄⋊C₄	—	2	2	no	—
110	C₁₃×C₄⋊C₄	—	2	2	no	—
111	C₁₃×C₄⋊C₄	—	2	2	no	—
112	C₁₃×C₄⋊C₄	—	2	2	no	—
113	C₁₃×C₄⋊C₄	—	4	4	no	—
114	C₁₃×C₂²⋊C₄	—	4	4	no	—
115	C₁₃×C₂²⋊C₄	—	8	4	no	—
116	C₄²×C₁₃	—	2	2	no	—
117	C₄²×C₁₃	—	2	2	no	—
118	C₄²×C₁₃	—	4	4	no	—
119	C₄²×C₁₃	C₂×D₄	1	2	yes	—
120	C₄²×C₁₃	C₂×D₄	1	2	yes	—
121	C₂×#440	C₂×C₁₃	1	2	yes	complemented
122	C₂×C₄×SD₁₆	C₂×C₁₃	1	2	yes	complemented
123	#2328	C₂×C₁₃	1	1	yes	characteristic
124	#2327	C₂×C₁₃	1	1	yes	characteristic, complemented
125	#2097	C₂×C₁₃	1	2	yes	complemented
126	#2092	C₂×C₁₃	1	2	yes	complemented
127	#2014	C₂×C₁₃	1	2	yes	—
128	#2013	C₂×C₁₃	1	2	yes	complemented
129	#1746	C₂×C₁₃	1	1	yes	characteristic, complemented
130	C₂³×C₁₃	—	2	4	no	—
131	C₂³×C₁₃	—	4	2	no	—
132	C₂³×C₁₃	—	4	2	no	—
133	C₂³×C₁₃	—	4	4	no	—
134	C₂³×C₁₃	—	4	4	no	—
135	C₂³×C₁₃	C₂²×D₄	1	1	yes	characteristic
136	Q₈×C₁₃	—	2	4	no	—
137	Q₈×C₁₃	—	2	4	no	—
138	Q₈×C₁₃	—	4	2	no	—
139	Q₈×C₁₃	—	4	2	no	—
140	D₄×C₁₃	—	2	4	no	—
141	D₄×C₁₃	—	2	4	no	—
142	D₄×C₁₃	—	4	2	no	—
143	D₄×C₁₃	—	4	2	no	—
144	D₄×C₁₃	—	8	8	no	—
145	C₂×C₄×C₁₃	—	2	4	no	—
146	C₂×C₄×C₁₃	—	2	2	no	—
147	C₂×C₄×C₁₃	—	2	2	no	—
148	C₂×C₄×C₁₃	—	2	4	no	—
149	C₂×C₄×C₁₃	—	2	4	no	—
150	C₂×C₄×C₁₃	—	2	4	no	—
151	C₂×C₄×C₁₃	—	4	2	no	—
152	C₂×C₄×C₁₃	—	4	2	no	—
153	C₂×C₄×C₁₃	—	4	2	no	—
154	C₂×C₄×C₁₃	—	4	2	no	—
155	C₂×C₄×C₁₃	—	4	4	no	—
156	C₂×C₄×C₁₃	—	4	4	no	—
157	C₂×C₄×C₁₃	—	8	8	no	—
158	C₂×C₄×C₁₃	#127	1	4	yes	—
159	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
160	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
161	C₂×C₄×C₁₃	C₂×#41	1	2	yes	—
162	C₂×C₄×C₁₃	C₂×#41	1	2	yes	—
163	C₈×C₁₃	—	2	4	no	—
164	C₈×C₁₃	—	2	4	no	—
165	C₂²×SD₁₆	C₂²×C₁₃	1	2	yes	—
166	C₂²×SD₁₆	C₂²×C₁₃	1	2	yes	—
167	C₂×C₄⋊Q₈	C₂²×C₁₃	1	1	yes	characteristic
168	C₂×#127	C₂²×C₁₃	1	1	yes	characteristic
169	C₂×C₄×Q₈	C₂²×C₁₃	1	2	yes	—
170	C₂×C₄×D₄	C₂²×C₁₃	1	2	yes	—
171	#440	—	2	8	no	—
172	C₄×SD₁₆	—	2	8	no	—
173	C₂×#107	C₂²×C₁₃	1	2	yes	—
174	C₂×#106	C₂²×C₁₃	1	2	yes	—
175	C₂×#103	C₂²×C₁₃	1	2	yes	—
176	C₂×#103	C₂²×C₁₃	1	2	yes	—
177	C₂×#102	C₂²×C₁₃	1	2	yes	—
178	C₂×#102	C₂²×C₁₃	1	2	yes	—
179	C₂×#97	C₂²×C₁₃	1	1	yes	characteristic
180	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
181	#391	C₂²×C₁₃	1	2	yes	—
182	#390	C₂²×C₁₃	1	2	yes	—
183	#387	—	2	4	no	—
184	#386	—	2	4	no	—
185	#382	C₂²×C₁₃	1	1	yes	characteristic
186	#382	C₂²×C₁₃	1	1	yes	characteristic
187	#376	C₂²×C₁₃	1	1	yes	characteristic
188	#336	C₂²×C₁₃	1	2	yes	—
189	#336	C₂²×C₁₃	1	2	yes	—
190	C₂²×C₁₃	—	4	8	no	—
191	C₂²×C₁₃	—	4	4	no	—
192	C₂²×C₁₃	—	8	4	no	—
193	C₂²×C₁₃	—	8	2	no	—
194	C₂²×C₁₃	—	8	2	no	—
195	C₂²×C₁₃	#525	1	1	yes	characteristic
196	C₂²×C₁₃	#525	1	1	yes	characteristic
197	C₂²×C₁₃	C₂×#127	1	1	yes	characteristic
198	C₂²×C₁₃	#577	1	4	yes	—
199	C₄×C₁₃	—	2	4	no	—
200	C₄×C₁₃	—	4	2	no	—
201	C₄×C₁₃	—	4	2	no	—
202	C₄×C₁₃	—	4	4	no	—
203	C₄×C₁₃	—	8	2	no	—
204	C₄×C₁₃	—	8	2	no	—
205	C₄×C₁₃	#525	1	2	yes	—
206	C₄×C₁₃	#525	1	2	yes	—
207	C₂²×Q₈	C₂³×C₁₃	1	2	yes	—
208	C₂²×D₄	—	2	2	no	—
209	C₂²×D₄	C₂³×C₁₃	1	2	yes	—
210	C₂³×C₄	—	2	2	no	—
211	C₂×SD₁₆	—	2	4	no	—
212	C₂×SD₁₆	—	2	4	no	—
213	C₂×SD₁₆	—	2	4	no	—
214	C₂×SD₁₆	—	2	4	no	—
215	C₂×SD₁₆	—	2	4	no	—
216	C₂×SD₁₆	—	2	4	no	—
217	C₂²×C₈	C₂³×C₁₃	1	2	yes	—
218	C₂²×C₈	C₂³×C₁₃	1	2	yes	—
219	C₄⋊Q₈	—	2	4	no	—
220	#127	—	2	4	no	—
221	C₄×Q₈	—	2	4	no	—
222	C₄×Q₈	—	2	4	no	—
223	C₄×D₄	—	2	4	no	—
224	C₄×D₄	—	2	4	no	—
225	C₂×C₄⋊C₄	—	2	2	no	—
226	C₂×C₄⋊C₄	—	2	2	no	—
227	C₂×C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
228	C₂×C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
229	C₂×C₂²⋊C₄	—	2	2	no	—
230	C₂×C₂²⋊C₄	—	2	4	no	—
231	C₂×C₄²	—	2	2	no	—
232	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
233	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
234	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
235	#107	—	2	4	no	—
236	#106	—	2	4	no	—
237	#103	—	2	4	no	—
238	#103	—	2	4	no	—
239	#102	—	2	4	no	—
240	#102	—	2	4	no	—
241	#97	D₄×C₁₃	1	4	yes	complemented
242	C₄×C₈	D₄×C₁₃	1	4	yes	complemented
243	#95	—	2	2	no	—
244	#95	—	2	2	no	—
245	#95	—	2	4	no	—
246	#95	D₄×C₁₃	1	4	yes	—
247	C₂×C₁₃	—	8	4	no	—
248	C₂×C₁₃	#2397	1	1	yes	characteristic
249	C₂×C₁₃	#3123	1	2	yes	—
250	C₂×C₁₃	#3126	1	2	yes	—
251	C₂×C₁₃	#3154	1	2	yes	—
252	C₂⁴	—	2	2	no	—
253	C₂×Q₈	—	2	2	no	—
254	C₂×Q₈	—	2	2	no	—
255	C₂×Q₈	—	2	2	no	—
256	C₂×Q₈	—	2	2	no	—
257	C₂×Q₈	—	2	2	no	—
258	C₂×Q₈	C₁₃×#41	1	4	yes	—
259	C₂×D₄	—	2	8	no	—
260	C₂×D₄	—	2	2	no	—
261	C₂×D₄	—	2	2	no	—
262	C₂×D₄	—	2	2	no	—
263	C₂×D₄	—	2	2	no	—
264	C₂×D₄	—	2	2	no	—
265	C₂×D₄	—	8	4	no	—
266	C₂×D₄	C₁₃×#41	1	4	yes	—
267	C₂²×C₄	—	2	2	no	—
268	C₂²×C₄	—	2	2	no	—
269	C₂²×C₄	—	2	2	no	—
270	C₂²×C₄	—	4	4	no	—
271	C₂²×C₄	—	8	4	no	—
272	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
273	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
274	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
275	C₂²×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
276	SD₁₆	—	2	16	no	—
277	SD₁₆	—	2	16	no	—
278	C₂×C₈	—	2	2	no	—
279	C₂×C₈	—	2	2	no	—
280	C₂×C₈	—	2	2	no	—
281	C₂×C₈	—	2	2	no	—
282	C₂×C₈	C₂×D₄×C₁₃	1	4	yes	—
283	C₂×C₈	C₂×D₄×C₁₃	1	4	yes	—
284	C₄⋊C₄	—	2	8	no	—
285	C₄⋊C₄	—	2	2	no	—
286	C₄⋊C₄	—	2	2	no	—
287	C₄⋊C₄	—	2	2	no	—
288	C₄⋊C₄	—	2	2	no	—
289	C₄⋊C₄	—	4	4	no	—
290	C₂²⋊C₄	—	4	4	no	—
291	C₂²⋊C₄	—	8	4	no	—
292	C₄²	—	2	2	no	—
293	C₄²	—	2	2	no	—
294	C₄²	—	4	4	no	—
295	C₄²	C₂×D₄×C₁₃	1	2	yes	—
296	C₄²	C₂×D₄×C₁₃	1	2	yes	—
297	C₁₃	#23811	1	1	yes	characteristic, direct product factor, Sylow
298	C₂³	—	2	4	no	—
299	C₂³	—	4	2	no	—
300	C₂³	—	4	2	no	—
301	C₂³	—	4	4	no	—
302	C₂³	—	4	4	no	—
303	C₂³	C₂²×D₄×C₁₃	1	1	yes	characteristic
304	Q₈	—	2	4	no	—
305	Q₈	—	2	4	no	—
306	Q₈	—	4	2	no	—
307	Q₈	—	4	2	no	—
308	D₄	—	2	4	no	—
309	D₄	—	2	4	no	—
310	D₄	—	4	2	no	—
311	D₄	—	4	2	no	—
312	D₄	—	8	8	no	—
313	C₂×C₄	—	2	4	no	—
314	C₂×C₄	—	2	2	no	—
315	C₂×C₄	—	2	2	no	—
316	C₂×C₄	—	2	4	no	—
317	C₂×C₄	—	2	4	no	—
318	C₂×C₄	—	2	4	no	—
319	C₂×C₄	—	4	2	no	—
320	C₂×C₄	—	4	2	no	—
321	C₂×C₄	—	4	2	no	—
322	C₂×C₄	—	4	2	no	—
323	C₂×C₄	—	4	4	no	—
324	C₂×C₄	—	4	4	no	—
325	C₂×C₄	—	8	8	no	—
326	C₂×C₄	C₁₃×#127	1	4	yes	—
327	C₂×C₄	C₂²×D₄×C₁₃	1	1	yes	characteristic
328	C₂×C₄	C₂²×D₄×C₁₃	1	1	yes	characteristic
329	C₂×C₄	C₂×C₁₃×#41	1	2	yes	—
330	C₂×C₄	C₂×C₁₃×#41	1	2	yes	—
331	C₈	—	2	4	no	—
332	C₈	—	2	4	no	—
333	C₂²	—	4	8	no	—
334	C₂²	—	4	4	no	—
335	C₂²	—	8	4	no	—
336	C₂²	—	8	2	no	—
337	C₂²	—	8	2	no	—
338	C₂²	C₁₃×#525	1	1	yes	characteristic
339	C₂²	C₁₃×#525	1	1	yes	characteristic
340	C₂²	C₂×C₁₃×#127	1	1	yes	characteristic
341	C₂²	C₁₃×#577	1	4	yes	—
342	C₄	—	2	4	no	—
343	C₄	—	4	2	no	—
344	C₄	—	4	2	no	—
345	C₄	—	4	4	no	—
346	C₄	—	8	2	no	—
347	C₄	—	8	2	no	—
348	C₄	C₁₃×#525	1	2	yes	—
349	C₄	C₁₃×#525	1	2	yes	—
350	C₂	—	8	4	no	—
351	C₂	C₁₃×#2397	1	1	yes	characteristic
352	C₂	C₁₃×#3123	1	2	yes	—
353	C₂	C₁₃×#3126	1	2	yes	—
354	C₂	C₁₃×#3154	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₁₃	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃
prime radical	C₂×C₁₃×#127	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₃ ◁ C₁₃×#376 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₁₃×#440	C₂	semidirect
2. C₂×C₄×C₁₃×SD₁₆	C₂	semidirect
4. C₁₃×#2327	C₂	semidirect
5. C₁₃×#2097	C₂	semidirect
6. C₁₃×#2092	C₂	semidirect
8. C₁₃×#2013	C₂	semidirect
9. C₁₃×#1746	C₂	semidirect
69. C₁₃×#97	D₄	semidirect
70. C₄×C₈×C₁₃	D₄	semidirect
75. #23811	C₁₃	direct (N × H)
121. C₂×#440	C₂×C₁₃	semidirect
122. C₂×C₄×SD₁₆	C₂×C₁₃	semidirect
124. #2327	C₂×C₁₃	semidirect
125. #2097	C₂×C₁₃	semidirect
126. #2092	C₂×C₁₃	semidirect
128. #2013	C₂×C₁₃	semidirect
129. #1746	C₂×C₁₃	semidirect
241. #97	D₄×C₁₃	semidirect
242. C₄×C₈	D₄×C₁₃	semidirect
297. C₁₃	#23811	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×#440	2
C₂×C₄×C₁₃×SD₁₆	2
C₁₃×#2328	2
C₁₃×#2327	2
C₁₃×#2097	2
C₁₃×#2092	2
C₁₃×#2014	2
C₁₃×#2013	2
C₁₃×#1746	2
#23811	13

Subgroups — C13×#23811 #15710150

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#23811 #15710150