Subgroups — C₁₃×#32514 #15718853

Order 3328.

574 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#2507	C₂	1	2	yes	maximal
2	C₁₃×#2088	C₂	1	2	yes	maximal
3	C₁₃×#2086	C₂	1	2	yes	maximal
4	C₁₃×#2083	C₂	1	2	yes	maximal
5	C₁₃×#1947	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	2	yes	—
12	C₂×C₁₃×#107	C₂²	1	1	yes	characteristic
13	C₂×C₁₃×#106	C₂²	1	1	yes	characteristic
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×#398	—	2	2	no	—
19	C₁₃×#398	—	2	2	no	—
20	C₁₃×#389	—	2	2	no	—
21	C₁₃×#389	C₂²	1	2	yes	—
22	C₁₃×#389	C₂²	1	2	yes	—
23	C₁₃×#389	C₂²	1	2	yes	—
24	C₁₃×#389	C₂²	1	2	yes	—
25	C₁₃×#387	—	2	2	no	—
26	C₁₃×#384	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	—	2	2	no	—
43	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
44	C₂×C₁₃×C₄⋊C₄	D₄	1	2	yes	—
45	C₂×C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
46	C₂×C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
47	C₂×C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
48	C₂×C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
49	C₂×C₄²×C₁₃	—	2	2	no	—
50	C₂×C₄²×C₁₃	C₂³	1	2	yes	—
51	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
52	C₁₃×#107	—	4	1	no	—
53	C₁₃×#107	—	4	1	no	—
54	C₁₃×#107	—	4	2	no	—
55	C₁₃×#106	—	2	2	no	—
56	C₁₃×#106	Q₈	1	4	yes	—
57	C₁₃×C₄⋊C₈	—	2	4	no	—
58	C₁₃×C₄⋊C₈	—	4	1	no	—
59	C₁₃×#97	—	4	1	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	#32514	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	1	yes	characteristic
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	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	4	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	—	2	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	—	8	2	no	—
99	C₁₃×C₄⋊C₄	—	8	2	no	—
100	C₁₃×C₄⋊C₄	C₂×Q₈	1	4	yes	—
101	C₄²×C₁₃	—	2	2	no	—
102	C₄²×C₁₃	—	2	1	no	—
103	C₄²×C₁₃	—	2	2	no	—
104	C₄²×C₁₃	—	2	1	no	—
105	C₄²×C₁₃	—	8	2	no	—
106	#2507	C₂×C₁₃	1	2	yes	—
107	#2088	C₂×C₁₃	1	2	yes	—
108	#2086	C₂×C₁₃	1	2	yes	—
109	#2083	C₂×C₁₃	1	2	yes	—
110	#1947	C₂×C₁₃	1	2	yes	—
111	#1944	C₂×C₁₃	1	1	yes	characteristic
112	#1944	C₂×C₁₃	1	1	yes	characteristic
113	#1940	C₂×C₁₃	1	1	yes	characteristic
114	#1919	C₂×C₁₃	1	2	yes	—
115	C₂³×C₁₃	C₂²×D₄	1	1	yes	characteristic
116	C₂×C₄×C₁₃	—	2	2	no	—
117	C₂×C₄×C₁₃	—	2	2	no	—
118	C₂×C₄×C₁₃	—	2	1	no	—
119	C₂×C₄×C₁₃	—	2	1	no	—
120	C₂×C₄×C₁₃	—	2	2	no	—
121	C₂×C₄×C₁₃	—	2	4	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	—	2	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	—	4	2	no	—
137	C₂×C₄×C₁₃	#137	1	4	yes	—
138	C₂×C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
139	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
140	C₂×C₄×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
141	C₂×C₄×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
142	C₂×C₄×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
143	C₂×C₄×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
144	C₈×C₁₃	—	2	2	no	—
145	C₈×C₁₃	—	2	2	no	—
146	C₈×C₁₃	—	4	2	no	—
147	C₂×#107	C₂²×C₁₃	1	1	yes	characteristic
148	C₂×#107	C₂²×C₁₃	1	2	yes	—
149	C₂×#107	C₂²×C₁₃	1	1	yes	characteristic
150	C₂×#106	C₂²×C₁₃	1	1	yes	characteristic
151	C₂×#106	C₂²×C₁₃	1	1	yes	characteristic
152	C₂×C₄⋊C₈	C₂²×C₁₃	1	2	yes	—
153	C₂×C₄⋊C₈	C₂²×C₁₃	1	1	yes	characteristic
154	C₂×#97	C₂²×C₁₃	1	1	yes	characteristic
155	#398	—	2	2	no	—
156	#398	—	2	2	no	—
157	#389	—	2	2	no	—
158	#389	C₂²×C₁₃	1	2	yes	—
159	#389	C₂²×C₁₃	1	2	yes	—
160	#389	C₂²×C₁₃	1	2	yes	—
161	#389	C₂²×C₁₃	1	2	yes	—
162	#387	—	2	2	no	—
163	#384	C₂²×C₁₃	1	2	yes	—
164	C₄×C₄⋊C₄	C₂²×C₁₃	1	2	yes	—
165	C₄×C₄⋊C₄	C₂²×C₁₃	1	1	yes	characteristic
166	#340	C₂²×C₁₃	1	2	yes	—
167	#340	C₂²×C₁₃	1	2	yes	—
168	#340	C₂²×C₁₃	1	2	yes	—
169	#340	C₂²×C₁₃	1	2	yes	—
170	#340	C₂²×C₁₃	1	2	yes	—
171	#336	C₂²×C₁₃	1	2	yes	—
172	C₂²×C₁₃	#537	1	1	yes	characteristic
173	C₂²×C₁₃	#547	1	1	yes	characteristic
174	C₂²×C₁₃	D₄×Q₈	1	1	yes	characteristic
175	C₂²×C₁₃	C₂×#137	1	2	yes	—
176	C₂²×C₁₃	#576	1	2	yes	—
177	C₄×C₁₃	—	2	4	no	—
178	C₄×C₁₃	—	4	2	no	—
179	C₄×C₁₃	—	4	4	no	—
180	C₄×C₁₃	—	4	1	no	—
181	C₄×C₁₃	—	4	1	no	—
182	C₄×C₁₃	—	8	2	no	—
183	C₄×C₁₃	—	8	2	no	—
184	C₄×C₁₃	—	8	2	no	—
185	C₄×C₁₃	#539	1	1	yes	characteristic
186	C₄×C₁₃	#544	1	1	yes	characteristic
187	C₄×C₁₃	#557	1	1	yes	characteristic
188	C₄×C₁₃	#564	1	1	yes	characteristic
189	C₂²×C₈	C₂³×C₁₃	1	2	yes	—
190	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
191	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
192	C₂×C₄⋊C₄	—	2	2	no	—
193	C₂×C₄⋊C₄	—	2	2	no	—
194	C₂×C₄⋊C₄	—	2	2	no	—
195	C₂×C₄⋊C₄	—	2	2	no	—
196	C₂×C₄⋊C₄	—	2	2	no	—
197	C₂×C₄⋊C₄	—	2	2	no	—
198	C₂×C₄⋊C₄	D₄×C₁₃	1	2	yes	—
199	C₂×C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
200	C₂×C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
201	C₂×C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
202	C₂×C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
203	C₂×C₄²	—	2	2	no	—
204	C₂×C₄²	C₂³×C₁₃	1	2	yes	—
205	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
206	#107	—	4	1	no	—
207	#107	—	4	1	no	—
208	#107	—	4	2	no	—
209	#106	—	2	2	no	—
210	#106	Q₈×C₁₃	1	4	yes	—
211	C₄⋊C₈	—	2	4	no	—
212	C₄⋊C₈	—	4	1	no	—
213	#97	—	4	1	no	—
214	#95	—	2	2	no	—
215	#95	—	2	2	no	—
216	#95	—	2	2	no	—
217	#95	—	2	2	no	—
218	#95	—	2	2	no	—
219	#95	D₄×C₁₃	1	1	yes	characteristic
220	#95	D₄×C₁₃	1	1	yes	characteristic
221	C₂×C₁₃	#2726	1	1	yes	characteristic
222	C₂×C₁₃	#3255	1	1	yes	characteristic
223	C₂×C₁₃	#3259	1	1	yes	characteristic
224	C₂×C₁₃	#3346	1	2	yes	—
225	C₂×C₁₃	#3385	1	2	yes	—
226	C₂²×C₄	—	2	2	no	—
227	C₂²×C₄	—	2	2	no	—
228	C₂²×C₄	—	2	2	no	—
229	C₂²×C₄	—	2	2	no	—
230	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
231	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
232	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
233	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
234	C₂²×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
235	C₂×C₈	—	2	1	no	—
236	C₂×C₈	—	2	1	no	—
237	C₂×C₈	—	2	1	no	—
238	C₂×C₈	—	2	1	no	—
239	C₂×C₈	—	2	2	no	—
240	C₂×C₈	—	2	2	no	—
241	C₂×C₈	—	2	2	no	—
242	C₂×C₈	C₂×Q₈×C₁₃	1	2	yes	—
243	C₂×C₈	C₁₃×#41	1	2	yes	—
244	C₄⋊C₄	—	2	4	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	—	2	2	no	—
250	C₄⋊C₄	—	2	2	no	—
251	C₄⋊C₄	—	2	2	no	—
252	C₄⋊C₄	—	4	4	no	—
253	C₄⋊C₄	—	8	2	no	—
254	C₄⋊C₄	—	8	2	no	—
255	C₄⋊C₄	—	8	2	no	—
256	C₄⋊C₄	—	8	2	no	—
257	C₄⋊C₄	—	8	2	no	—
258	C₄⋊C₄	C₂×Q₈×C₁₃	1	4	yes	—
259	C₄²	—	2	2	no	—
260	C₄²	—	2	1	no	—
261	C₄²	—	2	2	no	—
262	C₄²	—	2	1	no	—
263	C₄²	—	8	2	no	—
264	C₁₃	#32514	1	1	yes	characteristic, direct product factor, Sylow
265	C₂³	C₂²×D₄×C₁₃	1	1	yes	characteristic
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	—	2	2	no	—
268	C₂×C₄	—	2	1	no	—
269	C₂×C₄	—	2	1	no	—
270	C₂×C₄	—	2	2	no	—
271	C₂×C₄	—	2	4	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	2	2	no	—
275	C₂×C₄	—	2	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	—	4	2	no	—
285	C₂×C₄	—	4	2	no	—
286	C₂×C₄	—	4	2	no	—
287	C₂×C₄	C₁₃×#137	1	4	yes	—
288	C₂×C₄	C₂²×Q₈×C₁₃	1	1	yes	characteristic
289	C₂×C₄	C₂×C₁₃×#41	1	1	yes	characteristic
290	C₂×C₄	C₁₃×H(5,𝔽₂)	1	1	yes	characteristic
291	C₂×C₄	C₁₃×2₋¹⁺⁴	1	1	yes	characteristic
292	C₂×C₄	C₁₃×2₋¹⁺⁴	1	1	yes	characteristic
293	C₂×C₄	C₁₃×2₋¹⁺⁴	1	1	yes	characteristic
294	C₈	—	2	2	no	—
295	C₈	—	2	2	no	—
296	C₈	—	4	2	no	—
297	C₂²	C₁₃×#537	1	1	yes	characteristic
298	C₂²	C₁₃×#547	1	1	yes	characteristic
299	C₂²	D₄×Q₈×C₁₃	1	1	yes	characteristic
300	C₂²	C₂×C₁₃×#137	1	2	yes	—
301	C₂²	C₁₃×#576	1	2	yes	—
302	C₄	—	2	4	no	—
303	C₄	—	4	2	no	—
304	C₄	—	4	4	no	—
305	C₄	—	4	1	no	—
306	C₄	—	4	1	no	—
307	C₄	—	8	2	no	—
308	C₄	—	8	2	no	—
309	C₄	—	8	2	no	—
310	C₄	C₁₃×#539	1	1	yes	characteristic
311	C₄	C₁₃×#544	1	1	yes	characteristic
312	C₄	C₁₃×#557	1	1	yes	characteristic
313	C₄	C₁₃×#564	1	1	yes	characteristic
314	C₂	C₁₃×#2726	1	1	yes	characteristic
315	C₂	C₁₃×#3255	1	1	yes	characteristic
316	C₂	C₁₃×#3259	1	1	yes	characteristic
317	C₂	C₁₃×#3346	1	2	yes	—
318	C₂	C₁₃×#3385	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. #32514	C₁₃	direct (N × H)
264. C₁₃	#32514	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×#2507	2
C₁₃×#2088	2
C₁₃×#2086	2
C₁₃×#2083	2
C₁₃×#1947	2
C₁₃×#1944	2
C₁₃×#1940	2
C₁₃×#1919	2
#32514	13

Subgroups — C13×#32514 #15718853

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#32514 #15718853