Subgroups — C₁₃×#32828 #15719167

Order 3328.

570 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#2647	C₂	1	1	yes	maximal, characteristic
2	C₁₃×#2642	C₂	1	1	yes	maximal, characteristic
3	C₁₃×#2097	C₂	1	2	yes	maximal
4	C₁₃×#2072	C₂	1	2	yes	maximal
5	C₁₃×#2068	C₂	1	2	yes	maximal
6	C₁₃×#2066	C₂	1	2	yes	maximal
7	C₁₃×#2059	C₂	1	2	yes	maximal
8	C₁₃×#2043	C₂	1	2	yes	maximal
9	C₁₃×#1832	C₂	1	1	yes	maximal, characteristic
10	C₂×C₁₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
11	C₂×C₁₃×#125	C₂²	1	1	yes	characteristic
12	C₂×C₁₃×#107	C₂²	1	2	yes	—
13	C₂×C₁₃×#106	C₂²	1	2	yes	—
14	C₂×C₁₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
15	C₂×C₁₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
16	C₂×C₁₃×#103	C₂²	1	2	yes	—
17	C₂×C₁₃×#103	C₂²	1	2	yes	—
18	C₂×C₁₃×#103	C₂²	1	2	yes	—
19	C₂×C₁₃×#103	C₂²	1	2	yes	—
20	C₁₃×#400	—	2	2	no	—
21	C₁₃×#400	—	2	2	no	—
22	C₁₃×#400	—	2	2	no	—
23	C₁₃×#398	—	2	2	no	—
24	C₁₃×#391	C₂²	1	2	yes	—
25	C₁₃×#391	C₂²	1	2	yes	—
26	C₁₃×#389	C₂²	1	2	yes	—
27	C₁₃×#389	C₂²	1	2	yes	—
28	C₁₃×#384	C₂²	1	1	yes	characteristic
29	C₁₃×#382	C₂²	1	1	yes	characteristic
30	C₁₃×#376	C₂²	1	1	yes	characteristic
31	C₁₃×#340	C₂²	1	2	yes	—
32	C₁₃×#340	C₂²	1	2	yes	—
33	C₁₃×#336	C₂²	1	2	yes	—
34	C₁₃×#336	C₂²	1	2	yes	—
35	C₂²×Q₈×C₁₃	C₂³	1	2	yes	—
36	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
37	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
38	C₁₃×C₄⋊Q₈	—	4	1	no	—
39	C₁₃×C₄⋊Q₈	—	4	1	no	—
40	C₁₃×#125	—	4	1	no	—
41	C₁₃×#125	—	4	1	no	—
42	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
43	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
44	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
45	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
46	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
47	C₂×C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
48	C₂×C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
49	C₂×C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
50	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
51	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
52	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
53	C₁₃×#107	—	4	2	no	—
54	C₁₃×#106	—	4	2	no	—
55	C₁₃×C₄⋊C₈	—	4	1	no	—
56	C₁₃×C₄⋊C₈	—	4	1	no	—
57	C₁₃×#103	—	4	2	no	—
58	C₁₃×#103	—	4	2	no	—
59	C₁₃×#103	—	4	2	no	—
60	C₁₃×#103	—	4	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	—	2	4	no	—
66	C₁₃×#95	—	2	2	no	—
67	C₁₃×#95	D₄	1	2	yes	—
68	C₁₃×#95	D₄	1	2	yes	—
69	#32828	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
70	C₂×Q₈×C₁₃	—	2	2	no	—
71	C₂×Q₈×C₁₃	—	2	2	no	—
72	C₂×Q₈×C₁₃	—	2	2	no	—
73	C₂×Q₈×C₁₃	—	2	2	no	—
74	C₂×Q₈×C₁₃	—	2	2	no	—
75	C₂×Q₈×C₁₃	—	2	2	no	—
76	C₂²×C₄×C₁₃	—	2	2	no	—
77	C₂²×C₄×C₁₃	—	2	2	no	—
78	C₂²×C₄×C₁₃	—	2	2	no	—
79	C₂²×C₄×C₁₃	—	2	2	no	—
80	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
81	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
82	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
83	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
84	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
85	C₂×C₈×C₁₃	—	2	2	no	—
86	C₂×C₈×C₁₃	—	2	2	no	—
87	C₂×C₈×C₁₃	—	2	2	no	—
88	C₂×C₈×C₁₃	—	2	2	no	—
89	C₂×C₈×C₁₃	—	2	2	no	—
90	C₂×C₈×C₁₃	—	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	—	2	2	no	—
95	C₁₃×C₄⋊C₄	—	2	2	no	—
96	C₁₃×C₄⋊C₄	—	2	2	no	—
97	C₁₃×C₄⋊C₄	—	4	4	no	—
98	C₁₃×C₄⋊C₄	—	4	4	no	—
99	C₁₃×C₄⋊C₄	—	8	2	no	—
100	C₁₃×C₄⋊C₄	—	8	2	no	—
101	C₁₃×C₄⋊C₄	—	8	2	no	—
102	C₄²×C₁₃	—	2	1	no	—
103	C₄²×C₁₃	—	2	1	no	—
104	C₄²×C₁₃	—	2	1	no	—
105	C₄²×C₁₃	—	2	1	no	—
106	C₄²×C₁₃	—	2	1	no	—
107	C₄²×C₁₃	—	2	1	no	—
108	#2647	C₂×C₁₃	1	1	yes	characteristic
109	#2642	C₂×C₁₃	1	1	yes	characteristic
110	#2097	C₂×C₁₃	1	2	yes	—
111	#2072	C₂×C₁₃	1	2	yes	—
112	#2068	C₂×C₁₃	1	2	yes	—
113	#2066	C₂×C₁₃	1	2	yes	—
114	#2059	C₂×C₁₃	1	2	yes	—
115	#2043	C₂×C₁₃	1	2	yes	—
116	#1832	C₂×C₁₃	1	1	yes	characteristic
117	C₂³×C₁₃	C₂²×D₄	1	1	yes	characteristic
118	Q₈×C₁₃	—	4	2	no	—
119	Q₈×C₁₃	—	4	2	no	—
120	Q₈×C₁₃	—	4	2	no	—
121	Q₈×C₁₃	—	4	2	no	—
122	C₂×C₄×C₁₃	—	2	2	no	—
123	C₂×C₄×C₁₃	—	2	2	no	—
124	C₂×C₄×C₁₃	—	2	1	no	—
125	C₂×C₄×C₁₃	—	2	1	no	—
126	C₂×C₄×C₁₃	—	2	2	no	—
127	C₂×C₄×C₁₃	—	2	2	no	—
128	C₂×C₄×C₁₃	—	2	2	no	—
129	C₂×C₄×C₁₃	—	2	2	no	—
130	C₂×C₄×C₁₃	—	2	2	no	—
131	C₂×C₄×C₁₃	—	2	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	—	4	2	no	—
138	C₂×C₄×C₁₃	—	4	2	no	—
139	C₂×C₄×C₁₃	—	4	2	no	—
140	C₂×C₄×C₁₃	—	4	2	no	—
141	C₂×C₄×C₁₃	—	4	2	no	—
142	C₂×C₄×C₁₃	—	4	2	no	—
143	C₂×C₄×C₁₃	—	4	2	no	—
144	C₂×C₄×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
145	C₂×C₄×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
146	C₂×C₄×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
147	C₂×C₄×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
148	C₂×C₄×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
149	C₂×C₄×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
150	C₈×C₁₃	—	4	2	no	—
151	C₈×C₁₃	—	4	2	no	—
152	C₂×C₄⋊Q₈	C₂²×C₁₃	1	1	yes	characteristic
153	C₂×#125	C₂²×C₁₃	1	1	yes	characteristic
154	C₂×#107	C₂²×C₁₃	1	2	yes	—
155	C₂×#106	C₂²×C₁₃	1	2	yes	—
156	C₂×C₄⋊C₈	C₂²×C₁₃	1	1	yes	characteristic
157	C₂×C₄⋊C₈	C₂²×C₁₃	1	1	yes	characteristic
158	C₂×#103	C₂²×C₁₃	1	2	yes	—
159	C₂×#103	C₂²×C₁₃	1	2	yes	—
160	C₂×#103	C₂²×C₁₃	1	2	yes	—
161	C₂×#103	C₂²×C₁₃	1	2	yes	—
162	#400	—	2	2	no	—
163	#400	—	2	2	no	—
164	#400	—	2	2	no	—
165	#398	—	2	2	no	—
166	#391	C₂²×C₁₃	1	2	yes	—
167	#391	C₂²×C₁₃	1	2	yes	—
168	#389	C₂²×C₁₃	1	2	yes	—
169	#389	C₂²×C₁₃	1	2	yes	—
170	#384	C₂²×C₁₃	1	1	yes	characteristic
171	#382	C₂²×C₁₃	1	1	yes	characteristic
172	#376	C₂²×C₁₃	1	1	yes	characteristic
173	#340	C₂²×C₁₃	1	2	yes	—
174	#340	C₂²×C₁₃	1	2	yes	—
175	#336	C₂²×C₁₃	1	2	yes	—
176	#336	C₂²×C₁₃	1	2	yes	—
177	C₂²×C₁₃	#535	1	1	yes	characteristic
178	C₂²×C₁₃	#536	1	1	yes	characteristic
179	C₂²×C₁₃	#537	1	1	yes	characteristic
180	C₂²×C₁₃	#577	1	2	yes	—
181	C₂²×C₁₃	#578	1	2	yes	—
182	C₄×C₁₃	—	4	2	no	—
183	C₄×C₁₃	—	4	2	no	—
184	C₄×C₁₃	—	4	1	no	—
185	C₄×C₁₃	—	4	1	no	—
186	C₄×C₁₃	—	8	2	no	—
187	C₄×C₁₃	—	8	2	no	—
188	C₄×C₁₃	—	8	2	no	—
189	C₄×C₁₃	—	8	2	no	—
190	C₄×C₁₃	#562	1	1	yes	characteristic
191	C₄×C₁₃	#562	1	1	yes	characteristic
192	C₄×C₁₃	#563	1	1	yes	characteristic
193	C₄×C₁₃	#563	1	1	yes	characteristic
194	C₂²×Q₈	C₂³×C₁₃	1	2	yes	—
195	C₂²×C₈	C₂³×C₁₃	1	2	yes	—
196	C₂²×C₈	C₂³×C₁₃	1	2	yes	—
197	C₄⋊Q₈	—	4	1	no	—
198	C₄⋊Q₈	—	4	1	no	—
199	#125	—	4	1	no	—
200	#125	—	4	1	no	—
201	C₂×C₄⋊C₄	—	2	2	no	—
202	C₂×C₄⋊C₄	—	2	2	no	—
203	C₂×C₄⋊C₄	—	2	2	no	—
204	C₂×C₄⋊C₄	—	2	2	no	—
205	C₂×C₄⋊C₄	—	2	2	no	—
206	C₂×C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
207	C₂×C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
208	C₂×C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
209	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
210	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
211	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
212	#107	—	4	2	no	—
213	#106	—	4	2	no	—
214	C₄⋊C₈	—	4	1	no	—
215	C₄⋊C₈	—	4	1	no	—
216	#103	—	4	2	no	—
217	#103	—	4	2	no	—
218	#103	—	4	2	no	—
219	#103	—	4	2	no	—
220	#95	—	2	2	no	—
221	#95	—	2	2	no	—
222	#95	—	2	2	no	—
223	#95	—	2	2	no	—
224	#95	—	2	4	no	—
225	#95	—	2	2	no	—
226	#95	D₄×C₁₃	1	2	yes	—
227	#95	D₄×C₁₃	1	2	yes	—
228	C₂×C₁₃	#2820	1	1	yes	characteristic
229	C₂×C₁₃	#3225	1	2	yes	—
230	C₂×C₁₃	#3242	1	2	yes	—
231	C₂×C₁₃	#3260	1	1	yes	characteristic
232	C₂×C₁₃	#3261	1	1	yes	characteristic
233	C₂×Q₈	—	2	2	no	—
234	C₂×Q₈	—	2	2	no	—
235	C₂×Q₈	—	2	2	no	—
236	C₂×Q₈	—	2	2	no	—
237	C₂×Q₈	—	2	2	no	—
238	C₂×Q₈	—	2	2	no	—
239	C₂²×C₄	—	2	2	no	—
240	C₂²×C₄	—	2	2	no	—
241	C₂²×C₄	—	2	2	no	—
242	C₂²×C₄	—	2	2	no	—
243	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
244	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
245	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
246	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
247	C₂²×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
248	C₂×C₈	—	2	2	no	—
249	C₂×C₈	—	2	2	no	—
250	C₂×C₈	—	2	2	no	—
251	C₂×C₈	—	2	2	no	—
252	C₂×C₈	—	2	2	no	—
253	C₂×C₈	—	2	2	no	—
254	C₄⋊C₄	—	2	2	no	—
255	C₄⋊C₄	—	2	2	no	—
256	C₄⋊C₄	—	2	2	no	—
257	C₄⋊C₄	—	2	2	no	—
258	C₄⋊C₄	—	2	2	no	—
259	C₄⋊C₄	—	2	2	no	—
260	C₄⋊C₄	—	4	4	no	—
261	C₄⋊C₄	—	4	4	no	—
262	C₄⋊C₄	—	8	2	no	—
263	C₄⋊C₄	—	8	2	no	—
264	C₄⋊C₄	—	8	2	no	—
265	C₄²	—	2	1	no	—
266	C₄²	—	2	1	no	—
267	C₄²	—	2	1	no	—
268	C₄²	—	2	1	no	—
269	C₄²	—	2	1	no	—
270	C₄²	—	2	1	no	—
271	C₁₃	#32828	1	1	yes	characteristic, direct product factor, Sylow
272	C₂³	C₂²×D₄×C₁₃	1	1	yes	characteristic
273	Q₈	—	4	2	no	—
274	Q₈	—	4	2	no	—
275	Q₈	—	4	2	no	—
276	Q₈	—	4	2	no	—
277	C₂×C₄	—	2	2	no	—
278	C₂×C₄	—	2	2	no	—
279	C₂×C₄	—	2	1	no	—
280	C₂×C₄	—	2	1	no	—
281	C₂×C₄	—	2	2	no	—
282	C₂×C₄	—	2	2	no	—
283	C₂×C₄	—	2	2	no	—
284	C₂×C₄	—	2	2	no	—
285	C₂×C₄	—	2	2	no	—
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	4	2	no	—
288	C₂×C₄	—	4	2	no	—
289	C₂×C₄	—	4	2	no	—
290	C₂×C₄	—	4	2	no	—
291	C₂×C₄	—	4	2	no	—
292	C₂×C₄	—	4	2	no	—
293	C₂×C₄	—	4	2	no	—
294	C₂×C₄	—	4	2	no	—
295	C₂×C₄	—	4	2	no	—
296	C₂×C₄	—	4	2	no	—
297	C₂×C₄	—	4	2	no	—
298	C₂×C₄	—	4	2	no	—
299	C₂×C₄	C₁₃×H(5,𝔽₂)	1	1	yes	characteristic
300	C₂×C₄	C₁₃×H(5,𝔽₂)	1	1	yes	characteristic
301	C₂×C₄	C₁₃×H(5,𝔽₂)	1	1	yes	characteristic
302	C₂×C₄	C₁₃×2₋¹⁺⁴	1	1	yes	characteristic
303	C₂×C₄	C₁₃×2₋¹⁺⁴	1	1	yes	characteristic
304	C₂×C₄	C₁₃×2₋¹⁺⁴	1	1	yes	characteristic
305	C₈	—	4	2	no	—
306	C₈	—	4	2	no	—
307	C₂²	C₁₃×#535	1	1	yes	characteristic
308	C₂²	C₁₃×#536	1	1	yes	characteristic
309	C₂²	C₁₃×#537	1	1	yes	characteristic
310	C₂²	C₁₃×#577	1	2	yes	—
311	C₂²	C₁₃×#578	1	2	yes	—
312	C₄	—	4	2	no	—
313	C₄	—	4	2	no	—
314	C₄	—	4	1	no	—
315	C₄	—	4	1	no	—
316	C₄	—	8	2	no	—
317	C₄	—	8	2	no	—
318	C₄	—	8	2	no	—
319	C₄	—	8	2	no	—
320	C₄	C₁₃×#562	1	1	yes	characteristic
321	C₄	C₁₃×#562	1	1	yes	characteristic
322	C₄	C₁₃×#563	1	1	yes	characteristic
323	C₄	C₁₃×#563	1	1	yes	characteristic
324	C₂	C₁₃×#2820	1	1	yes	characteristic
325	C₂	C₁₃×#3225	1	2	yes	—
326	C₂	C₁₃×#3242	1	2	yes	—
327	C₂	C₁₃×#3260	1	1	yes	characteristic
328	C₂	C₁₃×#3261	1	1	yes	characteristic

	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 ◁ C₁₃×#376 ◁ G

Complements

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

N	H ≅ G/N	type
69. #32828	C₁₃	direct (N × H)
271. C₁₃	#32828	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×#2647	2
C₁₃×#2642	2
C₁₃×#2097	2
C₁₃×#2072	2
C₁₃×#2068	2
C₁₃×#2066	2
C₁₃×#2059	2
C₁₃×#2043	2
C₁₃×#1832	2
#32828	13

Subgroups — C13×#32828 #15719167

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#32828 #15719167