Subgroups — C₁₃×#32518 #15718857

Order 3328.

558 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#2508	C₂	1	2	yes	maximal
2	C₁₃×#2088	C₂	1	2	yes	maximal
3	C₁₃×#2088	C₂	1	2	yes	maximal
4	C₁₃×#2085	C₂	1	2	yes	maximal
5	C₁₃×#1949	C₂	1	2	yes	maximal
6	C₁₃×#1944	C₂	1	1	yes	maximal, characteristic
7	C₁₃×#1944	C₂	1	1	yes	maximal, characteristic
8	C₁₃×#1940	C₂	1	1	yes	maximal, characteristic
9	C₁₃×#1919	C₂	1	2	yes	maximal
10	C₂×C₁₃×#107	C₂²	1	1	yes	characteristic
11	C₂×C₁₃×#107	C₂²	1	1	yes	characteristic
12	C₂×C₁₃×#106	C₂²	1	1	yes	characteristic
13	C₂×C₁₃×#106	C₂²	1	2	yes	—
14	C₂×C₁₃×#106	C₂²	1	1	yes	characteristic
15	C₂×C₁₃×C₄⋊C₈	C₂²	1	2	yes	—
16	C₂×C₁₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
17	C₂×C₁₃×#97	C₂²	1	1	yes	characteristic
18	C₁₃×#400	—	2	2	no	—
19	C₁₃×#398	—	2	2	no	—
20	C₁₃×#389	C₂²	1	2	yes	—
21	C₁₃×#389	C₂²	1	2	yes	—
22	C₁₃×#389	C₂²	1	2	yes	—
23	C₁₃×#389	C₂²	1	2	yes	—
24	C₁₃×#387	—	2	2	no	—
25	C₁₃×#382	—	2	2	no	—
26	C₁₃×#382	C₂²	1	2	yes	—
27	C₄×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
28	C₄×C₁₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
29	C₁₃×#340	C₂²	1	2	yes	—
30	C₁₃×#340	C₂²	1	2	yes	—
31	C₁₃×#340	C₂²	1	2	yes	—
32	C₁₃×#340	C₂²	1	2	yes	—
33	C₁₃×#340	C₂²	1	2	yes	—
34	C₁₃×#336	C₂²	1	2	yes	—
35	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
36	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
37	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
38	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
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	D₄	1	2	yes	—
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×C₄²×C₁₃	—	2	2	no	—
48	C₂×C₄²×C₁₃	C₂³	1	2	yes	—
49	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
50	C₁₃×#107	—	4	1	no	—
51	C₁₃×#107	—	4	1	no	—
52	C₁₃×#106	—	2	2	no	—
53	C₁₃×#106	—	4	2	no	—
54	C₁₃×#106	Q₈	1	4	yes	—
55	C₁₃×C₄⋊C₈	—	2	4	no	—
56	C₁₃×C₄⋊C₈	—	4	1	no	—
57	C₁₃×#97	—	4	1	no	—
58	C₁₃×#95	—	2	2	no	—
59	C₁₃×#95	—	2	2	no	—
60	C₁₃×#95	—	2	2	no	—
61	C₁₃×#95	—	2	2	no	—
62	C₁₃×#95	—	2	2	no	—
63	C₁₃×#95	—	2	2	no	—
64	C₁₃×#95	—	2	2	no	—
65	C₁₃×#95	D₄	1	1	yes	characteristic
66	C₁₃×#95	D₄	1	1	yes	characteristic
67	#32518	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
68	C₂²×C₄×C₁₃	—	2	2	no	—
69	C₂²×C₄×C₁₃	—	2	2	no	—
70	C₂²×C₄×C₁₃	—	2	2	no	—
71	C₂²×C₄×C₁₃	—	2	2	no	—
72	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
73	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
74	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
75	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
77	C₂×C₈×C₁₃	—	2	1	no	—
78	C₂×C₈×C₁₃	—	2	1	no	—
79	C₂×C₈×C₁₃	—	2	1	no	—
80	C₂×C₈×C₁₃	—	2	1	no	—
81	C₂×C₈×C₁₃	—	2	2	no	—
82	C₂×C₈×C₁₃	—	2	2	no	—
83	C₂×C₈×C₁₃	—	2	2	no	—
84	C₂×C₈×C₁₃	C₂×Q₈	1	2	yes	—
85	C₂×C₈×C₁₃	#41	1	2	yes	—
86	C₁₃×C₄⋊C₄	—	2	2	no	—
87	C₁₃×C₄⋊C₄	—	2	2	no	—
88	C₁₃×C₄⋊C₄	—	2	2	no	—
89	C₁₃×C₄⋊C₄	—	2	2	no	—
90	C₁₃×C₄⋊C₄	—	2	2	no	—
91	C₁₃×C₄⋊C₄	—	2	2	no	—
92	C₁₃×C₄⋊C₄	—	2	2	no	—
93	C₁₃×C₄⋊C₄	—	4	2	no	—
94	C₁₃×C₄⋊C₄	—	4	4	no	—
95	C₁₃×C₄⋊C₄	—	8	2	no	—
96	C₁₃×C₄⋊C₄	—	8	2	no	—
97	C₁₃×C₄⋊C₄	—	8	2	no	—
98	C₁₃×C₄⋊C₄	C₂×Q₈	1	4	yes	—
99	C₄²×C₁₃	—	2	2	no	—
100	C₄²×C₁₃	—	2	1	no	—
101	C₄²×C₁₃	—	2	1	no	—
102	C₄²×C₁₃	—	2	2	no	—
103	C₄²×C₁₃	—	4	4	no	—
104	#2508	C₂×C₁₃	1	2	yes	—
105	#2088	C₂×C₁₃	1	2	yes	—
106	#2088	C₂×C₁₃	1	2	yes	—
107	#2085	C₂×C₁₃	1	2	yes	—
108	#1949	C₂×C₁₃	1	2	yes	—
109	#1944	C₂×C₁₃	1	1	yes	characteristic
110	#1944	C₂×C₁₃	1	1	yes	characteristic
111	#1940	C₂×C₁₃	1	1	yes	characteristic
112	#1919	C₂×C₁₃	1	2	yes	—
113	C₂³×C₁₃	C₂²×D₄	1	1	yes	characteristic
114	C₂×C₄×C₁₃	—	2	1	no	—
115	C₂×C₄×C₁₃	—	2	1	no	—
116	C₂×C₄×C₁₃	—	2	2	no	—
117	C₂×C₄×C₁₃	—	2	2	no	—
118	C₂×C₄×C₁₃	—	2	4	no	—
119	C₂×C₄×C₁₃	—	2	2	no	—
120	C₂×C₄×C₁₃	—	2	2	no	—
121	C₂×C₄×C₁₃	—	2	2	no	—
122	C₂×C₄×C₁₃	—	2	2	no	—
123	C₂×C₄×C₁₃	—	2	2	no	—
124	C₂×C₄×C₁₃	—	2	2	no	—
125	C₂×C₄×C₁₃	—	4	2	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	—	4	2	no	—
133	C₂×C₄×C₁₃	—	4	2	no	—
134	C₂×C₄×C₁₃	—	4	2	no	—
135	C₂×C₄×C₁₃	—	4	2	no	—
136	C₂×C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
137	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
138	C₂×C₄×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
139	C₂×C₄×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
140	C₂×C₄×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
141	C₂×C₄×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
142	C₈×C₁₃	—	2	2	no	—
143	C₈×C₁₃	—	2	2	no	—
144	C₈×C₁₃	—	4	2	no	—
145	C₂×#107	C₂²×C₁₃	1	1	yes	characteristic
146	C₂×#107	C₂²×C₁₃	1	1	yes	characteristic
147	C₂×#106	C₂²×C₁₃	1	1	yes	characteristic
148	C₂×#106	C₂²×C₁₃	1	2	yes	—
149	C₂×#106	C₂²×C₁₃	1	1	yes	characteristic
150	C₂×C₄⋊C₈	C₂²×C₁₃	1	2	yes	—
151	C₂×C₄⋊C₈	C₂²×C₁₃	1	1	yes	characteristic
152	C₂×#97	C₂²×C₁₃	1	1	yes	characteristic
153	#400	—	2	2	no	—
154	#398	—	2	2	no	—
155	#389	C₂²×C₁₃	1	2	yes	—
156	#389	C₂²×C₁₃	1	2	yes	—
157	#389	C₂²×C₁₃	1	2	yes	—
158	#389	C₂²×C₁₃	1	2	yes	—
159	#387	—	2	2	no	—
160	#382	—	2	2	no	—
161	#382	C₂²×C₁₃	1	2	yes	—
162	C₄×C₄⋊C₄	C₂²×C₁₃	1	2	yes	—
163	C₄×C₄⋊C₄	C₂²×C₁₃	1	1	yes	characteristic
164	#340	C₂²×C₁₃	1	2	yes	—
165	#340	C₂²×C₁₃	1	2	yes	—
166	#340	C₂²×C₁₃	1	2	yes	—
167	#340	C₂²×C₁₃	1	2	yes	—
168	#340	C₂²×C₁₃	1	2	yes	—
169	#336	C₂²×C₁₃	1	2	yes	—
170	C₂²×C₁₃	#537	1	1	yes	characteristic
171	C₂²×C₁₃	#547	1	1	yes	characteristic
172	C₂²×C₁₃	D₄×Q₈	1	1	yes	characteristic
173	C₂²×C₁₃	#575	1	2	yes	—
174	C₂²×C₁₃	#577	1	2	yes	—
175	C₄×C₁₃	—	4	2	no	—
176	C₄×C₁₃	—	4	2	no	—
177	C₄×C₁₃	—	4	4	no	—
178	C₄×C₁₃	—	4	1	no	—
179	C₄×C₁₃	—	4	1	no	—
180	C₄×C₁₃	—	8	2	no	—
181	C₄×C₁₃	—	8	2	no	—
182	C₄×C₁₃	—	8	2	no	—
183	C₄×C₁₃	#539	1	1	yes	characteristic
184	C₄×C₁₃	#544	1	1	yes	characteristic
185	C₄×C₁₃	#557	1	1	yes	characteristic
186	C₄×C₁₃	#564	1	1	yes	characteristic
187	C₂²×C₈	C₂³×C₁₃	1	2	yes	—
188	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
189	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
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	—	2	2	no	—
194	C₂×C₄⋊C₄	D₄×C₁₃	1	2	yes	—
195	C₂×C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
196	C₂×C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
197	C₂×C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
198	C₂×C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
199	C₂×C₄²	—	2	2	no	—
200	C₂×C₄²	C₂³×C₁₃	1	2	yes	—
201	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
202	#107	—	4	1	no	—
203	#107	—	4	1	no	—
204	#106	—	2	2	no	—
205	#106	—	4	2	no	—
206	#106	Q₈×C₁₃	1	4	yes	—
207	C₄⋊C₈	—	2	4	no	—
208	C₄⋊C₈	—	4	1	no	—
209	#97	—	4	1	no	—
210	#95	—	2	2	no	—
211	#95	—	2	2	no	—
212	#95	—	2	2	no	—
213	#95	—	2	2	no	—
214	#95	—	2	2	no	—
215	#95	—	2	2	no	—
216	#95	—	2	2	no	—
217	#95	D₄×C₁₃	1	1	yes	characteristic
218	#95	D₄×C₁₃	1	1	yes	characteristic
219	C₂×C₁₃	#2726	1	1	yes	characteristic
220	C₂×C₁₃	#3256	1	1	yes	characteristic
221	C₂×C₁₃	#3263	1	1	yes	characteristic
222	C₂×C₁₃	#3348	1	2	yes	—
223	C₂×C₁₃	#3388	1	2	yes	—
224	C₂²×C₄	—	2	2	no	—
225	C₂²×C₄	—	2	2	no	—
226	C₂²×C₄	—	2	2	no	—
227	C₂²×C₄	—	2	2	no	—
228	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
229	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
230	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
231	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
232	C₂²×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
233	C₂×C₈	—	2	1	no	—
234	C₂×C₈	—	2	1	no	—
235	C₂×C₈	—	2	1	no	—
236	C₂×C₈	—	2	1	no	—
237	C₂×C₈	—	2	2	no	—
238	C₂×C₈	—	2	2	no	—
239	C₂×C₈	—	2	2	no	—
240	C₂×C₈	C₂×Q₈×C₁₃	1	2	yes	—
241	C₂×C₈	C₁₃×#41	1	2	yes	—
242	C₄⋊C₄	—	2	2	no	—
243	C₄⋊C₄	—	2	2	no	—
244	C₄⋊C₄	—	2	2	no	—
245	C₄⋊C₄	—	2	2	no	—
246	C₄⋊C₄	—	2	2	no	—
247	C₄⋊C₄	—	2	2	no	—
248	C₄⋊C₄	—	2	2	no	—
249	C₄⋊C₄	—	4	2	no	—
250	C₄⋊C₄	—	4	4	no	—
251	C₄⋊C₄	—	8	2	no	—
252	C₄⋊C₄	—	8	2	no	—
253	C₄⋊C₄	—	8	2	no	—
254	C₄⋊C₄	C₂×Q₈×C₁₃	1	4	yes	—
255	C₄²	—	2	2	no	—
256	C₄²	—	2	1	no	—
257	C₄²	—	2	1	no	—
258	C₄²	—	2	2	no	—
259	C₄²	—	4	4	no	—
260	C₁₃	#32518	1	1	yes	characteristic, direct product factor, Sylow
261	C₂³	C₂²×D₄×C₁₃	1	1	yes	characteristic
262	C₂×C₄	—	2	1	no	—
263	C₂×C₄	—	2	1	no	—
264	C₂×C₄	—	2	2	no	—
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	2	4	no	—
267	C₂×C₄	—	2	2	no	—
268	C₂×C₄	—	2	2	no	—
269	C₂×C₄	—	2	2	no	—
270	C₂×C₄	—	2	2	no	—
271	C₂×C₄	—	2	2	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	4	2	no	—
274	C₂×C₄	—	4	2	no	—
275	C₂×C₄	—	4	2	no	—
276	C₂×C₄	—	4	2	no	—
277	C₂×C₄	—	4	2	no	—
278	C₂×C₄	—	4	2	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	C₂²×Q₈×C₁₃	1	1	yes	characteristic
285	C₂×C₄	C₂×C₁₃×#41	1	1	yes	characteristic
286	C₂×C₄	C₁₃×H(5,𝔽₂)	1	1	yes	characteristic
287	C₂×C₄	C₁₃×2₋¹⁺⁴	1	1	yes	characteristic
288	C₂×C₄	C₁₃×2₋¹⁺⁴	1	1	yes	characteristic
289	C₂×C₄	C₁₃×2₋¹⁺⁴	1	1	yes	characteristic
290	C₈	—	2	2	no	—
291	C₈	—	2	2	no	—
292	C₈	—	4	2	no	—
293	C₂²	C₁₃×#537	1	1	yes	characteristic
294	C₂²	C₁₃×#547	1	1	yes	characteristic
295	C₂²	D₄×Q₈×C₁₃	1	1	yes	characteristic
296	C₂²	C₁₃×#575	1	2	yes	—
297	C₂²	C₁₃×#577	1	2	yes	—
298	C₄	—	4	1	no	—
299	C₄	—	4	2	no	—
300	C₄	—	4	2	no	—
301	C₄	—	4	4	no	—
302	C₄	—	4	1	no	—
303	C₄	—	8	2	no	—
304	C₄	—	8	2	no	—
305	C₄	—	8	2	no	—
306	C₄	C₁₃×#539	1	1	yes	characteristic
307	C₄	C₁₃×#544	1	1	yes	characteristic
308	C₄	C₁₃×#557	1	1	yes	characteristic
309	C₄	C₁₃×#564	1	1	yes	characteristic
310	C₂	C₁₃×#2726	1	1	yes	characteristic
311	C₂	C₁₃×#3256	1	1	yes	characteristic
312	C₂	C₁₃×#3263	1	1	yes	characteristic
313	C₂	C₁₃×#3348	1	2	yes	—
314	C₂	C₁₃×#3388	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
67. #32518	C₁₃	direct (N × H)
260. C₁₃	#32518	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×#2508	2
C₁₃×#2088	2
C₁₃×#2085	2
C₁₃×#1949	2
C₁₃×#1944	2
C₁₃×#1940	2
C₁₃×#1919	2
#32518	13

Subgroups — C13×#32518 #15718857

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#32518 #15718857