Subgroups — C₁₃×#32866 #15719205

Order 3328.

562 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#2645	C₂	1	2	yes	maximal
2	C₁₃×#2088	C₂	1	2	yes	maximal
3	C₁₃×#2087	C₂	1	1	yes	maximal, characteristic
4	C₁₃×#2068	C₂	1	2	yes	maximal
5	C₁₃×#2066	C₂	1	2	yes	maximal
6	C₁₃×#2066	C₂	1	2	yes	maximal
7	C₁₃×#2056	C₂	1	1	yes	maximal, characteristic
8	C₁₃×#2056	C₂	1	2	yes	maximal
9	C₁₃×#1843	C₂	1	1	yes	maximal, characteristic
10	C₂²×C₁₃×Q₁₆	C₂²	1	1	yes	characteristic
11	C₂×C₁₃×#107	C₂²	1	1	yes	characteristic
12	C₂×C₁₃×#106	C₂²	1	2	yes	—
13	C₂×C₁₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
14	C₂×C₁₃×C₄⋊C₈	C₂²	1	2	yes	—
15	C₂×C₁₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
16	C₂×C₁₃×C₄⋊C₈	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×#401	—	2	2	no	—
21	C₁₃×#400	—	2	2	no	—
22	C₁₃×#400	—	2	2	no	—
23	C₁₃×#400	—	2	2	no	—
24	C₁₃×#391	C₂²	1	2	yes	—
25	C₁₃×#391	C₂²	1	2	yes	—
26	C₁₃×#391	C₂²	1	2	yes	—
27	C₁₃×#389	C₂²	1	2	yes	—
28	C₁₃×#389	C₂²	1	2	yes	—
29	C₁₃×#389	C₂²	1	2	yes	—
30	C₁₃×#376	C₂²	1	1	yes	characteristic
31	C₁₃×#340	C₂²	1	2	yes	—
32	C₁₃×#340	C₂²	1	1	yes	characteristic
33	C₁₃×#340	C₂²	1	1	yes	characteristic
34	C₁₃×#340	C₂²	1	2	yes	—
35	C₂²×Q₈×C₁₃	C₂³	1	2	yes	—
36	C₂×C₁₃×Q₁₆	—	4	2	no	—
37	C₂×C₁₃×Q₁₆	—	4	1	no	—
38	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
39	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
40	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
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	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	C₂³	1	1	yes	characteristic
48	C₂×C₄²×C₁₃	C₂³	1	2	yes	—
49	C₁₃×#107	—	4	1	no	—
50	C₁₃×#106	—	4	2	no	—
51	C₁₃×C₄⋊C₈	—	4	1	no	—
52	C₁₃×C₄⋊C₈	—	4	2	no	—
53	C₁₃×C₄⋊C₈	—	4	2	no	—
54	C₁₃×C₄⋊C₈	—	4	1	no	—
55	C₁₃×#103	—	4	2	no	—
56	C₁₃×#103	—	4	2	no	—
57	C₁₃×#103	—	4	2	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	—	2	2	no	—
66	C₁₃×#95	—	2	2	no	—
67	C₁₃×#95	D₄	1	1	yes	characteristic
68	C₁₃×#95	D₄	1	1	yes	characteristic
69	C₁₃×#95	D₄	1	2	yes	—
70	#32866	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
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	C₂×Q₈×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	—	2	2	no	—
81	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
82	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
83	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
84	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
85	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
86	C₁₃×Q₁₆	—	4	4	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	C₂×C₈×C₁₃	—	2	1	no	—
92	C₂×C₈×C₁₃	—	2	2	no	—
93	C₂×C₈×C₁₃	—	2	1	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	—	2	2	no	—
98	C₁₃×C₄⋊C₄	—	2	2	no	—
99	C₁₃×C₄⋊C₄	—	2	2	no	—
100	C₁₃×C₄⋊C₄	—	8	2	no	—
101	C₁₃×C₄⋊C₄	—	8	2	no	—
102	C₁₃×C₄⋊C₄	—	8	2	no	—
103	C₄²×C₁₃	—	2	2	no	—
104	C₄²×C₁₃	—	2	1	no	—
105	C₄²×C₁₃	—	2	2	no	—
106	C₄²×C₁₃	—	2	1	no	—
107	#2645	C₂×C₁₃	1	2	yes	—
108	#2088	C₂×C₁₃	1	2	yes	—
109	#2087	C₂×C₁₃	1	1	yes	characteristic
110	#2068	C₂×C₁₃	1	2	yes	—
111	#2066	C₂×C₁₃	1	2	yes	—
112	#2066	C₂×C₁₃	1	2	yes	—
113	#2056	C₂×C₁₃	1	2	yes	—
114	#2056	C₂×C₁₃	1	1	yes	characteristic
115	#1843	C₂×C₁₃	1	1	yes	characteristic
116	C₂³×C₁₃	C₂²×D₄	1	1	yes	characteristic
117	Q₈×C₁₃	—	4	2	no	—
118	Q₈×C₁₃	—	4	2	no	—
119	Q₈×C₁₃	—	4	2	no	—
120	Q₈×C₁₃	—	4	2	no	—
121	C₂×C₄×C₁₃	—	2	2	no	—
122	C₂×C₄×C₁₃	—	2	2	no	—
123	C₂×C₄×C₁₃	—	2	1	no	—
124	C₂×C₄×C₁₃	—	2	1	no	—
125	C₂×C₄×C₁₃	—	2	2	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	1	no	—
131	C₂×C₄×C₁₃	—	2	1	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	2	yes	—
145	C₂×C₄×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
146	C₂×C₄×C₁₃	2₋¹⁺⁴	1	2	yes	—
147	C₂×C₄×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
148	C₈×C₁₃	—	4	2	no	—
149	C₈×C₁₃	—	4	1	no	—
150	C₈×C₁₃	—	4	1	no	—
151	C₂²×Q₁₆	C₂²×C₁₃	1	1	yes	characteristic
152	C₂×#107	C₂²×C₁₃	1	1	yes	characteristic
153	C₂×#106	C₂²×C₁₃	1	2	yes	—
154	C₂×C₄⋊C₈	C₂²×C₁₃	1	1	yes	characteristic
155	C₂×C₄⋊C₈	C₂²×C₁₃	1	2	yes	—
156	C₂×C₄⋊C₈	C₂²×C₁₃	1	1	yes	characteristic
157	C₂×C₄⋊C₈	C₂²×C₁₃	1	2	yes	—
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	#401	—	2	2	no	—
162	#400	—	2	2	no	—
163	#400	—	2	2	no	—
164	#400	—	2	2	no	—
165	#391	C₂²×C₁₃	1	2	yes	—
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	#389	C₂²×C₁₃	1	2	yes	—
171	#376	C₂²×C₁₃	1	1	yes	characteristic
172	#340	C₂²×C₁₃	1	2	yes	—
173	#340	C₂²×C₁₃	1	1	yes	characteristic
174	#340	C₂²×C₁₃	1	1	yes	characteristic
175	#340	C₂²×C₁₃	1	2	yes	—
176	C₂²×C₁₃	#537	1	2	yes	—
177	C₂²×C₁₃	#537	1	1	yes	characteristic
178	C₂²×C₁₃	#577	1	2	yes	—
179	C₂²×C₁₃	#578	1	1	yes	characteristic
180	C₂²×C₁₃	#578	1	1	yes	characteristic
181	C₄×C₁₃	—	4	2	no	—
182	C₄×C₁₃	—	4	2	no	—
183	C₄×C₁₃	—	4	1	no	—
184	C₄×C₁₃	—	4	1	no	—
185	C₄×C₁₃	—	8	2	no	—
186	C₄×C₁₃	—	8	2	no	—
187	C₄×C₁₃	—	8	2	no	—
188	C₄×C₁₃	—	8	2	no	—
189	C₄×C₁₃	#560	1	1	yes	characteristic
190	C₄×C₁₃	#563	1	2	yes	—
191	C₄×C₁₃	#563	1	1	yes	characteristic
192	C₂²×Q₈	C₂³×C₁₃	1	2	yes	—
193	C₂×Q₁₆	—	4	1	no	—
194	C₂×Q₁₆	—	4	2	no	—
195	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
196	C₂²×C₈	C₂³×C₁₃	1	2	yes	—
197	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
198	C₂×C₄⋊C₄	—	2	2	no	—
199	C₂×C₄⋊C₄	—	2	2	no	—
200	C₂×C₄⋊C₄	—	2	2	no	—
201	C₂×C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
202	C₂×C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
203	C₂×C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
204	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
205	C₂×C₄²	C₂³×C₁₃	1	2	yes	—
206	#107	—	4	1	no	—
207	#106	—	4	2	no	—
208	C₄⋊C₈	—	4	1	no	—
209	C₄⋊C₈	—	4	2	no	—
210	C₄⋊C₈	—	4	2	no	—
211	C₄⋊C₈	—	4	1	no	—
212	#103	—	4	2	no	—
213	#103	—	4	2	no	—
214	#103	—	4	2	no	—
215	#95	—	2	2	no	—
216	#95	—	2	2	no	—
217	#95	—	2	2	no	—
218	#95	—	2	2	no	—
219	#95	—	2	2	no	—
220	#95	—	2	2	no	—
221	#95	—	2	2	no	—
222	#95	—	2	2	no	—
223	#95	—	2	2	no	—
224	#95	D₄×C₁₃	1	1	yes	characteristic
225	#95	D₄×C₁₃	1	1	yes	characteristic
226	#95	D₄×C₁₃	1	2	yes	—
227	C₂×C₁₃	#2823	1	1	yes	characteristic
228	C₂×C₁₃	#3258	1	2	yes	—
229	C₂×C₁₃	#3258	1	2	yes	—
230	C₂×C₁₃	#3261	1	1	yes	characteristic
231	C₂×C₁₃	#3263	1	1	yes	characteristic
232	C₂×Q₈	—	2	2	no	—
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	C₂²×C₄	—	2	2	no	—
239	C₂²×C₄	—	2	2	no	—
240	C₂²×C₄	—	2	2	no	—
241	C₂²×C₄	—	2	2	no	—
242	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
243	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
244	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
245	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
246	C₂²×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
247	Q₁₆	—	4	4	no	—
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	1	no	—
253	C₂×C₈	—	2	1	no	—
254	C₂×C₈	—	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	—	2	2	no	—
261	C₄⋊C₄	—	8	2	no	—
262	C₄⋊C₄	—	8	2	no	—
263	C₄⋊C₄	—	8	2	no	—
264	C₄²	—	2	1	no	—
265	C₄²	—	2	2	no	—
266	C₄²	—	2	1	no	—
267	C₄²	—	2	2	no	—
268	C₁₃	#32866	1	1	yes	characteristic, direct product factor, Sylow
269	C₂³	C₂²×D₄×C₁₃	1	1	yes	characteristic
270	Q₈	—	4	2	no	—
271	Q₈	—	4	2	no	—
272	Q₈	—	4	2	no	—
273	Q₈	—	4	2	no	—
274	C₂×C₄	—	2	1	no	—
275	C₂×C₄	—	2	2	no	—
276	C₂×C₄	—	2	2	no	—
277	C₂×C₄	—	2	1	no	—
278	C₂×C₄	—	2	2	no	—
279	C₂×C₄	—	2	2	no	—
280	C₂×C₄	—	2	2	no	—
281	C₂×C₄	—	2	2	no	—
282	C₂×C₄	—	2	2	no	—
283	C₂×C₄	—	2	1	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	4	2	no	—
286	C₂×C₄	—	4	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	C₁₃×H(5,𝔽₂)	1	1	yes	characteristic
298	C₂×C₄	C₁₃×H(5,𝔽₂)	1	2	yes	—
299	C₂×C₄	C₁₃×2₋¹⁺⁴	1	1	yes	characteristic
300	C₂×C₄	C₁₃×2₋¹⁺⁴	1	2	yes	—
301	C₈	—	4	2	no	—
302	C₈	—	4	1	no	—
303	C₈	—	4	1	no	—
304	C₂²	C₁₃×#537	1	1	yes	characteristic
305	C₂²	C₁₃×#537	1	2	yes	—
306	C₂²	C₁₃×#577	1	2	yes	—
307	C₂²	C₁₃×#578	1	1	yes	characteristic
308	C₂²	C₁₃×#578	1	1	yes	characteristic
309	C₄	—	4	1	no	—
310	C₄	—	4	2	no	—
311	C₄	—	4	2	no	—
312	C₄	—	4	1	no	—
313	C₄	—	8	2	no	—
314	C₄	—	8	2	no	—
315	C₄	—	8	2	no	—
316	C₄	—	8	2	no	—
317	C₄	C₁₃×#560	1	1	yes	characteristic
318	C₄	C₁₃×#563	1	1	yes	characteristic
319	C₄	C₁₃×#563	1	2	yes	—
320	C₂	C₁₃×#2823	1	1	yes	characteristic
321	C₂	C₁₃×#3258	1	2	yes	—
322	C₂	C₁₃×#3258	1	2	yes	—
323	C₂	C₁₃×#3261	1	1	yes	characteristic
324	C₂	C₁₃×#3263	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
70. #32866	C₁₃	direct (N × H)
268. C₁₃	#32866	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×#2645	2
C₁₃×#2088	2
C₁₃×#2087	2
C₁₃×#2068	2
C₁₃×#2066	2
C₁₃×#2056	2
C₁₃×#1843	2
#32866	13

Subgroups — C13×#32866 #15719205

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#32866 #15719205