Subgroups — C₁₃×#23744 #15710083

Order 3328.

810 conjugacy classes of subgroups.

The subgroup lattice has 314 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#2296	C₂	1	1	yes	maximal, characteristic
2	C₁₃×#2285	C₂	1	1	yes	maximal, characteristic
3	C₁₃×#2101	C₂	1	4	yes	maximal, complemented
4	C₁₃×#2100	C₂	1	2	yes	maximal, complemented
5	C₁₃×#2099	C₂	1	2	yes	maximal, complemented
6	C₁₃×#1735	C₂	1	2	yes	maximal
7	C₁₃×#1735	C₂	1	2	yes	maximal
8	C₂×C₁₃×#336	C₂	1	1	yes	maximal, characteristic
9	C₂³×C₈×C₁₃	C₂²	1	2	yes	—
10	C₂×C₁₃×#117	C₂²	1	2	yes	—
11	C₂×C₁₃×#117	C₂²	1	1	yes	characteristic
12	C₂×C₁₃×#117	C₂²	1	1	yes	characteristic
13	C₂²×C₄²×C₁₃	C₂²	1	1	yes	characteristic
14	C₁₃×#389	C₂²	1	2	yes	—
15	C₁₃×#389	C₂²	1	2	yes	—
16	C₁₃×#385	—	2	2	no	—
17	C₁₃×#384	C₂²	1	1	yes	characteristic
18	C₁₃×#384	C₂²	1	1	yes	characteristic
19	C₁₃×#382	C₂²	1	2	yes	—
20	C₁₃×#380	—	2	2	no	—
21	C₄×C₁₃×C₄⋊C₄	—	2	2	no	—
22	C₄×C₁₃×C₄⋊C₄	—	2	2	no	—
23	C₁₃×#340	C₂²	1	4	yes	—
24	C₁₃×#340	C₂²	1	4	yes	—
25	C₁₃×#340	C₂²	1	4	yes	—
26	C₁₃×#340	C₂²	1	4	yes	—
27	C₁₃×#336	C₂²	1	2	yes	—
28	C₁₃×#336	C₂²	1	2	yes	—
29	C₂³×C₄×C₁₃	D₄	1	2	yes	—
30	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
31	C₂²×C₈×C₁₃	—	2	4	no	—
32	C₂²×C₈×C₁₃	—	2	4	no	—
33	C₂²×C₈×C₁₃	—	2	4	no	—
34	C₂²×C₈×C₁₃	C₂³	1	4	yes	—
35	C₁₃×#117	—	2	2	no	—
36	C₁₃×#117	—	2	4	no	—
37	C₁₃×#117	—	2	8	no	—
38	C₁₃×#117	Q₈	1	4	yes	—
39	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
40	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
41	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
42	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
43	C₂×C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
44	C₂×C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
45	C₂×C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
46	C₂×C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
47	C₂×C₁₃×C₂²⋊C₄	—	2	1	no	—
48	C₂×C₁₃×C₂²⋊C₄	—	2	2	no	—
49	C₂×C₁₃×C₂²⋊C₄	—	2	1	no	—
50	C₂×C₄²×C₁₃	—	2	1	no	—
51	C₂×C₄²×C₁₃	—	2	4	no	—
52	C₂×C₄²×C₁₃	—	2	1	no	—
53	C₂×C₄²×C₁₃	—	2	2	no	—
54	C₂×C₄²×C₁₃	D₄	1	2	yes	—
55	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
56	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
57	C₁₃×#95	—	2	4	no	—
58	C₁₃×#95	—	2	2	no	—
59	C₁₃×#95	—	2	2	no	—
60	#23744	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
61	C₂⁴×C₁₃	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₄×C₁₃	—	2	4	no	—
63	C₂²×C₄×C₁₃	—	2	2	no	—
64	C₂²×C₄×C₁₃	—	2	4	no	—
65	C₂²×C₄×C₁₃	—	2	2	no	—
66	C₂²×C₄×C₁₃	—	2	2	no	—
67	C₂²×C₄×C₁₃	—	2	2	no	—
68	C₂²×C₄×C₁₃	—	4	2	no	—
69	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
70	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
71	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₁₃	C₂×Q₈	1	2	yes	—
73	C₂²×C₄×C₁₃	#41	1	2	yes	—
74	C₂²×C₄×C₁₃	#41	1	4	yes	—
75	C₂²×C₄×C₁₃	#41	1	4	yes	—
76	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
77	C₂×C₈×C₁₃	—	2	4	no	—
78	C₂×C₈×C₁₃	—	2	4	no	—
79	C₂×C₈×C₁₃	—	2	4	no	—
80	C₂×C₈×C₁₃	—	4	8	no	—
81	C₁₃×C₄⋊C₄	—	2	2	no	—
82	C₁₃×C₄⋊C₄	—	2	4	no	—
83	C₁₃×C₄⋊C₄	—	2	4	no	—
84	C₁₃×C₄⋊C₄	—	2	4	no	—
85	C₁₃×C₄⋊C₄	—	4	4	no	—
86	C₁₃×C₄⋊C₄	—	4	4	no	—
87	C₁₃×C₄⋊C₄	—	4	4	no	—
88	C₁₃×C₄⋊C₄	—	4	4	no	—
89	C₁₃×C₄⋊C₄	C₂×Q₈	1	4	yes	—
90	C₁₃×C₂²⋊C₄	—	2	4	no	—
91	C₁₃×C₂²⋊C₄	—	4	2	no	—
92	C₁₃×C₂²⋊C₄	—	4	4	no	—
93	C₄²×C₁₃	—	2	4	no	—
94	C₄²×C₁₃	—	2	4	no	—
95	C₄²×C₁₃	—	2	2	no	—
96	C₄²×C₁₃	—	2	2	no	—
97	C₄²×C₁₃	—	4	2	no	—
98	C₄²×C₁₃	—	4	4	no	—
99	#2296	C₂×C₁₃	1	1	yes	characteristic
100	#2285	C₂×C₁₃	1	1	yes	characteristic
101	#2101	C₂×C₁₃	1	4	yes	complemented
102	#2100	C₂×C₁₃	1	2	yes	complemented
103	#2099	C₂×C₁₃	1	2	yes	complemented
104	#1735	C₂×C₁₃	1	2	yes	—
105	#1735	C₂×C₁₃	1	2	yes	—
106	C₂×#336	C₂×C₁₃	1	1	yes	characteristic
107	C₂³×C₁₃	—	2	2	no	—
108	C₂³×C₁₃	—	2	2	no	—
109	C₂³×C₁₃	C₂²⋊Q₈	1	2	yes	—
110	C₂³×C₁₃	#123	1	2	yes	—
111	C₂³×C₁₃	#124	1	2	yes	—
112	C₂³×C₁₃	C₂²×D₄	1	1	yes	characteristic
113	C₂×C₄×C₁₃	—	2	2	no	—
114	C₂×C₄×C₁₃	—	2	2	no	—
115	C₂×C₄×C₁₃	—	2	4	no	—
116	C₂×C₄×C₁₃	—	2	2	no	—
117	C₂×C₄×C₁₃	—	2	2	no	—
118	C₂×C₄×C₁₃	—	2	2	no	—
119	C₂×C₄×C₁₃	—	4	8	no	—
120	C₂×C₄×C₁₃	—	4	2	no	—
121	C₂×C₄×C₁₃	—	4	2	no	—
122	C₂×C₄×C₁₃	—	4	2	no	—
123	C₂×C₄×C₁₃	—	4	2	no	—
124	C₂×C₄×C₁₃	—	4	2	no	—
125	C₂×C₄×C₁₃	—	4	2	no	—
126	C₂×C₄×C₁₃	—	4	4	no	—
127	C₂×C₄×C₁₃	—	4	2	no	—
128	C₂×C₄×C₁₃	—	4	4	no	—
129	C₂×C₄×C₁₃	C₂²⋊Q₈	1	2	yes	—
130	C₂×C₄×C₁₃	#123	1	2	yes	—
131	C₂×C₄×C₁₃	#124	1	2	yes	—
132	C₂×C₄×C₁₃	#125	1	8	yes	—
133	C₂×C₄×C₁₃	#126	1	8	yes	—
134	C₂×C₄×C₁₃	#135	1	4	yes	—
135	C₂×C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
136	C₂×C₄×C₁₃	C₂×#41	1	2	yes	—
137	C₂×C₄×C₁₃	C₂×#41	1	2	yes	—
138	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
139	C₈×C₁₃	—	4	4	no	—
140	C₂³×C₈	C₂²×C₁₃	1	2	yes	—
141	C₂×#117	C₂²×C₁₃	1	2	yes	—
142	C₂×#117	C₂²×C₁₃	1	1	yes	characteristic
143	C₂×#117	C₂²×C₁₃	1	1	yes	characteristic
144	C₂²×C₄²	C₂²×C₁₃	1	1	yes	characteristic
145	#389	C₂²×C₁₃	1	2	yes	—
146	#389	C₂²×C₁₃	1	2	yes	—
147	#385	—	2	2	no	—
148	#384	C₂²×C₁₃	1	1	yes	characteristic
149	#384	C₂²×C₁₃	1	1	yes	characteristic
150	#382	C₂²×C₁₃	1	2	yes	—
151	#380	—	2	2	no	—
152	C₄×C₄⋊C₄	—	2	2	no	—
153	C₄×C₄⋊C₄	—	2	2	no	—
154	#340	C₂²×C₁₃	1	4	yes	—
155	#340	C₂²×C₁₃	1	4	yes	—
156	#340	C₂²×C₁₃	1	4	yes	—
157	#340	C₂²×C₁₃	1	4	yes	—
158	#336	C₂²×C₁₃	1	2	yes	—
159	#336	C₂²×C₁₃	1	2	yes	—
160	C₂²×C₁₃	—	2	4	no	—
161	C₂²×C₁₃	—	2	4	no	—
162	C₂²×C₁₃	—	2	2	no	—
163	C₂²×C₁₃	—	2	2	no	—
164	C₂²×C₁₃	#400	1	4	yes	—
165	C₂²×C₁₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
166	C₂²×C₁₃	C₂×#123	1	1	yes	characteristic
167	C₂²×C₁₃	C₂×#124	1	1	yes	characteristic
168	C₂²×C₁₃	C₂×#135	1	2	yes	—
169	C₂²×C₁₃	#575	1	2	yes	—
170	C₄×C₁₃	—	2	4	no	—
171	C₄×C₁₃	—	4	4	no	—
172	C₄×C₁₃	—	4	2	no	—
173	C₄×C₁₃	—	8	2	no	—
174	C₄×C₁₃	—	8	2	no	—
175	C₄×C₁₃	—	8	2	no	—
176	C₄×C₁₃	#400	1	4	yes	—
177	C₄×C₁₃	C₂×#125	1	2	yes	—
178	C₄×C₁₃	C₂×#126	1	2	yes	—
179	C₂³×C₄	D₄×C₁₃	1	2	yes	—
180	C₂³×C₄	C₂³×C₁₃	1	1	yes	characteristic
181	C₂²×C₈	—	2	4	no	—
182	C₂²×C₈	—	2	4	no	—
183	C₂²×C₈	—	2	4	no	—
184	C₂²×C₈	C₂³×C₁₃	1	4	yes	—
185	#117	—	2	2	no	—
186	#117	—	2	4	no	—
187	#117	—	2	8	no	—
188	#117	Q₈×C₁₃	1	4	yes	—
189	C₂×C₄⋊C₄	—	2	2	no	—
190	C₂×C₄⋊C₄	—	2	2	no	—
191	C₂×C₄⋊C₄	—	2	2	no	—
192	C₂×C₄⋊C₄	—	2	2	no	—
193	C₂×C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
194	C₂×C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
195	C₂×C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
196	C₂×C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
197	C₂×C₂²⋊C₄	—	2	1	no	—
198	C₂×C₂²⋊C₄	—	2	2	no	—
199	C₂×C₂²⋊C₄	—	2	1	no	—
200	C₂×C₄²	—	2	1	no	—
201	C₂×C₄²	—	2	4	no	—
202	C₂×C₄²	—	2	1	no	—
203	C₂×C₄²	—	2	2	no	—
204	C₂×C₄²	D₄×C₁₃	1	2	yes	—
205	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
206	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
207	#95	—	2	4	no	—
208	#95	—	2	2	no	—
209	#95	—	2	2	no	—
210	C₂×C₁₃	—	2	4	no	—
211	C₂×C₁₃	C₂×#400	1	1	yes	characteristic
212	C₂×C₁₃	#3076	1	2	yes	—
213	C₂×C₁₃	#3091	1	2	yes	—
214	C₂×C₁₃	#3131	1	1	yes	characteristic
215	C₂×C₁₃	#3135	1	1	yes	characteristic
216	C₂⁴	C₂×D₄×C₁₃	1	1	yes	characteristic
217	C₂²×C₄	—	2	4	no	—
218	C₂²×C₄	—	2	2	no	—
219	C₂²×C₄	—	2	4	no	—
220	C₂²×C₄	—	2	2	no	—
221	C₂²×C₄	—	2	2	no	—
222	C₂²×C₄	—	2	2	no	—
223	C₂²×C₄	—	4	2	no	—
224	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
225	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
226	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
227	C₂²×C₄	C₂×Q₈×C₁₃	1	2	yes	—
228	C₂²×C₄	C₁₃×#41	1	2	yes	—
229	C₂²×C₄	C₁₃×#41	1	4	yes	—
230	C₂²×C₄	C₁₃×#41	1	4	yes	—
231	C₂²×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
232	C₂×C₈	—	2	4	no	—
233	C₂×C₈	—	2	4	no	—
234	C₂×C₈	—	2	4	no	—
235	C₂×C₈	—	4	8	no	—
236	C₄⋊C₄	—	2	2	no	—
237	C₄⋊C₄	—	2	4	no	—
238	C₄⋊C₄	—	2	4	no	—
239	C₄⋊C₄	—	2	4	no	—
240	C₄⋊C₄	—	4	4	no	—
241	C₄⋊C₄	—	4	4	no	—
242	C₄⋊C₄	—	4	4	no	—
243	C₄⋊C₄	—	4	4	no	—
244	C₄⋊C₄	C₂×Q₈×C₁₃	1	4	yes	—
245	C₂²⋊C₄	—	2	4	no	—
246	C₂²⋊C₄	—	4	2	no	—
247	C₂²⋊C₄	—	4	4	no	—
248	C₄²	—	2	4	no	—
249	C₄²	—	2	4	no	—
250	C₄²	—	2	2	no	—
251	C₄²	—	2	2	no	—
252	C₄²	—	4	2	no	—
253	C₄²	—	4	4	no	—
254	C₁₃	#23744	1	1	yes	characteristic, direct product factor, Sylow
255	C₂³	—	2	2	no	—
256	C₂³	—	2	2	no	—
257	C₂³	C₁₃×C₂²⋊Q₈	1	2	yes	—
258	C₂³	C₁₃×#123	1	2	yes	—
259	C₂³	C₁₃×#124	1	2	yes	—
260	C₂³	C₂²×D₄×C₁₃	1	1	yes	characteristic
261	C₂×C₄	—	2	2	no	—
262	C₂×C₄	—	2	2	no	—
263	C₂×C₄	—	2	4	no	—
264	C₂×C₄	—	2	2	no	—
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	—	4	8	no	—
268	C₂×C₄	—	4	2	no	—
269	C₂×C₄	—	4	2	no	—
270	C₂×C₄	—	4	2	no	—
271	C₂×C₄	—	4	2	no	—
272	C₂×C₄	—	4	2	no	—
273	C₂×C₄	—	4	2	no	—
274	C₂×C₄	—	4	4	no	—
275	C₂×C₄	—	4	2	no	—
276	C₂×C₄	—	4	4	no	—
277	C₂×C₄	C₁₃×C₂²⋊Q₈	1	2	yes	—
278	C₂×C₄	C₁₃×#123	1	2	yes	—
279	C₂×C₄	C₁₃×#124	1	2	yes	—
280	C₂×C₄	C₁₃×#125	1	8	yes	—
281	C₂×C₄	C₁₃×#126	1	8	yes	—
282	C₂×C₄	C₁₃×#135	1	4	yes	—
283	C₂×C₄	C₂²×Q₈×C₁₃	1	1	yes	characteristic
284	C₂×C₄	C₂×C₁₃×#41	1	2	yes	—
285	C₂×C₄	C₂×C₁₃×#41	1	2	yes	—
286	C₂×C₄	C₂×C₁₃×#41	1	1	yes	characteristic
287	C₈	—	4	4	no	—
288	C₂²	—	2	4	no	—
289	C₂²	—	2	4	no	—
290	C₂²	—	2	2	no	—
291	C₂²	—	2	2	no	—
292	C₂²	C₁₃×#400	1	4	yes	—
293	C₂²	C₂×C₁₃×C₂²⋊Q₈	1	1	yes	characteristic
294	C₂²	C₂×C₁₃×#123	1	1	yes	characteristic
295	C₂²	C₂×C₁₃×#124	1	1	yes	characteristic
296	C₂²	C₂×C₁₃×#135	1	2	yes	—
297	C₂²	C₁₃×#575	1	2	yes	—
298	C₄	—	2	4	no	—
299	C₄	—	4	4	no	—
300	C₄	—	4	2	no	—
301	C₄	—	8	2	no	—
302	C₄	—	8	2	no	—
303	C₄	—	8	2	no	—
304	C₄	C₁₃×#400	1	4	yes	—
305	C₄	C₂×C₁₃×#125	1	2	yes	—
306	C₄	C₂×C₁₃×#126	1	2	yes	—
307	C₂	—	2	4	no	—
308	C₂	C₂×C₁₃×#400	1	1	yes	characteristic
309	C₂	C₁₃×#3076	1	2	yes	—
310	C₂	C₁₃×#3091	1	2	yes	—
311	C₂	C₁₃×#3131	1	1	yes	characteristic
312	C₂	C₁₃×#3135	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
3. C₁₃×#2101	C₂	semidirect
4. C₁₃×#2100	C₂	semidirect
5. C₁₃×#2099	C₂	semidirect
60. #23744	C₁₃	direct (N × H)
101. #2101	C₂×C₁₃	semidirect
102. #2100	C₂×C₁₃	semidirect
103. #2099	C₂×C₁₃	semidirect
254. C₁₃	#23744	direct (N × H)

Maximal subgroups

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

subgroup	index
C₁₃×#2296	2
C₁₃×#2285	2
C₁₃×#2101	2
C₁₃×#2100	2
C₁₃×#2099	2
C₁₃×#1735	2
C₂×C₁₃×#336	2
#23744	13

Subgroups — C13×#23744 #15710083

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#23744 #15710083