Subgroups — C₁₃×#29135 #15715474

Order 3328.

642 conjugacy classes of subgroups.

The subgroup lattice has 322 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×#2327	C₂	1	1	yes	maximal, characteristic
3	C₁₃×#2101	C₂	1	2	yes	maximal
4	C₁₃×#2043	C₂	1	2	yes	maximal
5	C₁₃×#1935	C₂	1	2	yes	maximal
6	C₁₃×#1933	C₂	1	2	yes	maximal
7	C₁₃×#1805	C₂	1	2	yes	maximal
8	C₁₃×#1803	C₂	1	2	yes	maximal
9	C₁₃×#1768	C₂	1	1	yes	maximal, characteristic
10	C₂×C₁₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
11	C₂×C₄×Q₈×C₁₃	C₂²	1	2	yes	—
12	C₂×C₁₃×C₄⋊C₈	C₂²	1	2	yes	—
13	C₂×C₁₃×#103	C₂²	1	2	yes	—
14	C₂×C₁₃×#103	C₂²	1	2	yes	—
15	C₂×C₁₃×#103	C₂²	1	2	yes	—
16	C₂×C₁₃×#103	C₂²	1	2	yes	—
17	C₂×C₁₃×#97	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
19	C₁₃×#400	—	2	2	no	—
20	C₁₃×#398	—	2	2	no	—
21	C₁₃×#391	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×#387	—	2	2	no	—
26	C₁₃×#384	C₂²	1	1	yes	characteristic
27	C₁₃×#382	C₂²	1	1	yes	characteristic
28	C₁₃×#382	C₂²	1	1	yes	characteristic
29	C₁₃×#376	C₂²	1	1	yes	characteristic
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×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	—	2	4	no	—
39	C₄×Q₈×C₁₃	—	2	4	no	—
40	C₄×Q₈×C₁₃	—	2	4	no	—
41	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
42	C₂×C₁₃×C₄⋊C₄	—	2	1	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	—	2	2	no	—
51	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
52	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
53	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
54	C₁₃×C₄⋊C₈	—	2	4	no	—
55	C₁₃×#103	—	2	4	no	—
56	C₁₃×#103	—	2	4	no	—
57	C₁₃×#103	—	4	2	no	—
58	C₁₃×#103	—	4	2	no	—
59	C₁₃×#97	—	4	1	no	—
60	C₄×C₈×C₁₃	—	4	1	no	—
61	C₁₃×#95	—	2	1	no	—
62	C₁₃×#95	—	2	1	no	—
63	C₁₃×#95	—	2	2	no	—
64	C₁₃×#95	—	2	4	no	—
65	C₁₃×#95	—	2	2	no	—
66	C₁₃×#95	—	2	2	no	—
67	C₁₃×#95	D₄	1	2	yes	—
68	C₁₃×#95	D₄	1	2	yes	—
69	#29135	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	#41	1	4	yes	—
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	4	no	—
92	C₁₃×C₄⋊C₄	—	2	4	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	—	2	2	no	—
98	C₁₃×C₄⋊C₄	—	2	2	no	—
99	C₁₃×C₄⋊C₄	—	4	4	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	1	no	—
106	C₄²×C₁₃	—	4	4	no	—
107	C₄²×C₁₃	#41	1	4	yes	—
108	#2647	C₂×C₁₃	1	1	yes	characteristic
109	#2327	C₂×C₁₃	1	1	yes	characteristic
110	#2101	C₂×C₁₃	1	2	yes	—
111	#2043	C₂×C₁₃	1	2	yes	—
112	#1935	C₂×C₁₃	1	2	yes	—
113	#1933	C₂×C₁₃	1	2	yes	—
114	#1805	C₂×C₁₃	1	2	yes	—
115	#1803	C₂×C₁₃	1	2	yes	—
116	#1768	C₂×C₁₃	1	1	yes	characteristic
117	C₂³×C₁₃	C₂²×D₄	1	1	yes	characteristic
118	Q₈×C₁₃	—	2	4	no	—
119	Q₈×C₁₃	—	2	4	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	4	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	#124	1	4	yes	—
144	C₂×C₄×C₁₃	C₂×#41	1	2	yes	—
145	C₂×C₄×C₁₃	C₂×#41	1	2	yes	—
146	C₂×C₄×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
147	C₂×C₄×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
148	C₈×C₁₃	—	4	2	no	—
149	C₈×C₁₃	—	4	2	no	—
150	C₂×C₄⋊Q₈	C₂²×C₁₃	1	1	yes	characteristic
151	C₂×C₄×Q₈	C₂²×C₁₃	1	2	yes	—
152	C₂×C₄⋊C₈	C₂²×C₁₃	1	2	yes	—
153	C₂×#103	C₂²×C₁₃	1	2	yes	—
154	C₂×#103	C₂²×C₁₃	1	2	yes	—
155	C₂×#103	C₂²×C₁₃	1	2	yes	—
156	C₂×#103	C₂²×C₁₃	1	2	yes	—
157	C₂×#97	C₂²×C₁₃	1	1	yes	characteristic
158	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
159	#400	—	2	2	no	—
160	#398	—	2	2	no	—
161	#391	C₂²×C₁₃	1	2	yes	—
162	#389	C₂²×C₁₃	1	2	yes	—
163	#389	C₂²×C₁₃	1	2	yes	—
164	#387	—	2	2	no	—
165	#387	—	2	2	no	—
166	#384	C₂²×C₁₃	1	1	yes	characteristic
167	#382	C₂²×C₁₃	1	1	yes	characteristic
168	#382	C₂²×C₁₃	1	1	yes	characteristic
169	#376	C₂²×C₁₃	1	1	yes	characteristic
170	#340	C₂²×C₁₃	1	2	yes	—
171	#340	C₂²×C₁₃	1	2	yes	—
172	#340	C₂²×C₁₃	1	2	yes	—
173	#340	C₂²×C₁₃	1	2	yes	—
174	#336	C₂²×C₁₃	1	2	yes	—
175	C₂²×C₁₃	#525	1	1	yes	characteristic
176	C₂²×C₁₃	C₂×#124	1	1	yes	characteristic
177	C₂²×C₁₃	#537	1	1	yes	characteristic
178	C₂²×C₁₃	#577	1	2	yes	—
179	C₂²×C₁₃	#578	1	2	yes	—
180	C₄×C₁₃	—	2	4	no	—
181	C₄×C₁₃	—	4	4	no	—
182	C₄×C₁₃	—	4	1	no	—
183	C₄×C₁₃	—	4	1	no	—
184	C₄×C₁₃	—	4	2	no	—
185	C₄×C₁₃	—	8	2	no	—
186	C₄×C₁₃	—	8	2	no	—
187	C₄×C₁₃	—	8	2	no	—
188	C₄×C₁₃	#552	1	2	yes	—
189	C₄×C₁₃	#553	1	2	yes	—
190	C₂²×Q₈	C₂³×C₁₃	1	2	yes	—
191	C₂²×C₈	C₂³×C₁₃	1	2	yes	—
192	C₂²×C₈	C₂³×C₁₃	1	2	yes	—
193	C₄⋊Q₈	—	2	4	no	—
194	C₄×Q₈	—	2	4	no	—
195	C₄×Q₈	—	2	4	no	—
196	C₂×C₄⋊C₄	—	2	1	no	—
197	C₂×C₄⋊C₄	—	2	1	no	—
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	—	2	2	no	—
202	C₂×C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
203	C₂×C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
204	C₂×C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
205	C₂×C₄²	—	2	2	no	—
206	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
207	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
208	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
209	C₄⋊C₈	—	2	4	no	—
210	#103	—	2	4	no	—
211	#103	—	2	4	no	—
212	#103	—	4	2	no	—
213	#103	—	4	2	no	—
214	#97	—	4	1	no	—
215	C₄×C₈	—	4	1	no	—
216	#95	—	2	1	no	—
217	#95	—	2	1	no	—
218	#95	—	2	2	no	—
219	#95	—	2	4	no	—
220	#95	—	2	2	no	—
221	#95	—	2	2	no	—
222	#95	D₄×C₁₃	1	2	yes	—
223	#95	D₄×C₁₃	1	2	yes	—
224	C₂×C₁₃	#2562	1	1	yes	characteristic
225	C₂×C₁₃	#3126	1	1	yes	characteristic
226	C₂×C₁₃	#3127	1	1	yes	characteristic
227	C₂×C₁₃	#3137	1	2	yes	—
228	C₂×C₁₃	#3258	1	2	yes	—
229	C₂×Q₈	—	2	2	no	—
230	C₂×Q₈	—	2	2	no	—
231	C₂×Q₈	—	2	2	no	—
232	C₂×Q₈	—	2	2	no	—
233	C₂×Q₈	—	2	2	no	—
234	C₂×Q₈	C₁₃×#41	1	4	yes	—
235	C₂²×C₄	—	2	2	no	—
236	C₂²×C₄	—	2	2	no	—
237	C₂²×C₄	—	2	2	no	—
238	C₂²×C₄	—	2	2	no	—
239	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
240	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
241	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
242	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
243	C₂²×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
244	C₂×C₈	—	2	2	no	—
245	C₂×C₈	—	2	2	no	—
246	C₂×C₈	—	2	2	no	—
247	C₂×C₈	—	2	2	no	—
248	C₂×C₈	—	2	2	no	—
249	C₂×C₈	—	2	2	no	—
250	C₄⋊C₄	—	2	4	no	—
251	C₄⋊C₄	—	2	4	no	—
252	C₄⋊C₄	—	2	2	no	—
253	C₄⋊C₄	—	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	—	4	4	no	—
259	C₄⋊C₄	—	8	2	no	—
260	C₄⋊C₄	—	8	2	no	—
261	C₄⋊C₄	—	8	2	no	—
262	C₄²	—	2	2	no	—
263	C₄²	—	2	1	no	—
264	C₄²	—	2	1	no	—
265	C₄²	—	4	4	no	—
266	C₄²	C₁₃×#41	1	4	yes	—
267	C₁₃	#29135	1	1	yes	characteristic, direct product factor, Sylow
268	C₂³	C₂²×D₄×C₁₃	1	1	yes	characteristic
269	Q₈	—	2	4	no	—
270	Q₈	—	2	4	no	—
271	Q₈	—	4	2	no	—
272	Q₈	—	4	2	no	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	2	2	no	—
275	C₂×C₄	—	2	1	no	—
276	C₂×C₄	—	2	1	no	—
277	C₂×C₄	—	2	4	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	—	4	2	no	—
284	C₂×C₄	—	4	2	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	C₁₃×#124	1	4	yes	—
295	C₂×C₄	C₂×C₁₃×#41	1	2	yes	—
296	C₂×C₄	C₂×C₁₃×#41	1	2	yes	—
297	C₂×C₄	C₁₃×H(5,𝔽₂)	1	1	yes	characteristic
298	C₂×C₄	C₁₃×2₋¹⁺⁴	1	1	yes	characteristic
299	C₈	—	4	2	no	—
300	C₈	—	4	2	no	—
301	C₂²	C₁₃×#525	1	1	yes	characteristic
302	C₂²	C₂×C₁₃×#124	1	1	yes	characteristic
303	C₂²	C₁₃×#537	1	1	yes	characteristic
304	C₂²	C₁₃×#577	1	2	yes	—
305	C₂²	C₁₃×#578	1	2	yes	—
306	C₄	—	2	4	no	—
307	C₄	—	4	4	no	—
308	C₄	—	4	1	no	—
309	C₄	—	4	1	no	—
310	C₄	—	4	2	no	—
311	C₄	—	8	2	no	—
312	C₄	—	8	2	no	—
313	C₄	—	8	2	no	—
314	C₄	C₁₃×#552	1	2	yes	—
315	C₄	C₁₃×#553	1	2	yes	—
316	C₂	C₁₃×#2562	1	1	yes	characteristic
317	C₂	C₁₃×#3126	1	1	yes	characteristic
318	C₂	C₁₃×#3127	1	1	yes	characteristic
319	C₂	C₁₃×#3137	1	2	yes	—
320	C₂	C₁₃×#3258	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 ◁ C₁₃×#376 ◁ G

Complements

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

N	H ≅ G/N	type
69. #29135	C₁₃	direct (N × H)
267. C₁₃	#29135	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×#2327	2
C₁₃×#2101	2
C₁₃×#2043	2
C₁₃×#1935	2
C₁₃×#1933	2
C₁₃×#1805	2
C₁₃×#1803	2
C₁₃×#1768	2
#29135	13

Subgroups — C13×#29135 #15715474

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#29135 #15715474