Subgroups — C₁₃×#29127 #15715466

Order 3328.

842 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×#2864	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄×C₁₃×#127	C₂	1	1	yes	maximal, characteristic
3	C₁₃×#2099	C₂	1	2	yes	maximal, complemented
4	C₁₃×#2041	C₂	1	2	yes	maximal
5	C₁₃×#1936	C₂	1	4	yes	maximal, complemented
6	C₁₃×#1804	C₂	1	2	yes	maximal, complemented
7	C₁₃×#1804	C₂	1	2	yes	maximal, complemented
8	C₁₃×#1767	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₁₃×#127	C₂²	1	1	yes	characteristic
10	C₂×C₄×D₄×C₁₃	—	2	2	no	—
11	C₂×C₄×D₄×C₁₃	C₂²	1	2	yes	—
12	C₂×C₁₃×C₄⋊C₈	C₂²	1	2	yes	—
13	C₂×C₁₃×#102	C₂²	1	4	yes	—
14	C₂×C₁₃×#102	C₂²	1	4	yes	—
15	C₂×C₁₃×#97	C₂²	1	2	yes	—
16	C₁₃×#390	—	2	2	no	—
17	C₁₃×#390	C₂²	1	2	yes	—
18	C₁₃×#389	C₂²	1	2	yes	—
19	C₁₃×#389	C₂²	1	2	yes	—
20	C₁₃×#387	—	2	2	no	—
21	C₁₃×#387	—	2	2	no	—
22	C₁₃×#384	C₂²	1	1	yes	characteristic
23	C₁₃×#382	C₂²	1	1	yes	characteristic
24	C₁₃×#382	C₂²	1	1	yes	characteristic
25	C₄³×C₁₃	C₂²	1	1	yes	characteristic
26	C₁₃×#340	C₂²	1	4	yes	—
27	C₁₃×#340	C₂²	1	4	yes	—
28	C₁₃×#336	C₂²	1	2	yes	—
29	C₂²×D₄×C₁₃	—	2	2	no	—
30	C₂²×D₄×C₁₃	C₂³	1	2	yes	—
31	C₂³×C₄×C₁₃	—	2	2	no	—
32	C₂²×C₈×C₁₃	C₂³	1	4	yes	—
33	C₁₃×#127	—	2	4	no	—
34	C₄×D₄×C₁₃	—	2	4	no	—
35	C₄×D₄×C₁₃	—	2	4	no	—
36	C₄×D₄×C₁₃	—	4	8	no	—
37	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
38	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
39	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
40	C₂×C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
41	C₂×C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
42	C₂×C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
43	C₂×C₁₃×C₂²⋊C₄	—	2	2	no	—
44	C₂×C₁₃×C₂²⋊C₄	—	2	4	no	—
45	C₂×C₄²×C₁₃	D₄	1	2	yes	—
46	C₂×C₄²×C₁₃	D₄	1	2	yes	—
47	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
48	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
49	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
50	C₁₃×C₄⋊C₈	—	2	4	no	—
51	C₁₃×#102	—	2	8	no	—
52	C₁₃×#102	—	4	4	no	—
53	C₁₃×#97	—	4	2	no	—
54	C₁₃×#95	—	2	1	no	—
55	C₁₃×#95	—	2	1	no	—
56	C₁₃×#95	—	2	1	no	—
57	C₁₃×#95	—	2	1	no	—
58	C₁₃×#95	—	2	2	no	—
59	C₁₃×#95	—	2	2	no	—
60	#29127	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
61	C₂⁴×C₁₃	—	2	2	no	—
62	C₂×D₄×C₁₃	—	2	8	no	—
63	C₂×D₄×C₁₃	—	2	2	no	—
64	C₂×D₄×C₁₃	—	2	2	no	—
65	C₂×D₄×C₁₃	—	2	2	no	—
66	C₂×D₄×C₁₃	—	2	2	no	—
67	C₂×D₄×C₁₃	—	2	2	no	—
68	C₂×D₄×C₁₃	—	4	4	no	—
69	C₂×D₄×C₁₃	—	4	4	no	—
70	C₂×D₄×C₁₃	#41	1	4	yes	—
71	C₂²×C₄×C₁₃	—	2	2	no	—
72	C₂²×C₄×C₁₃	—	2	2	no	—
73	C₂²×C₄×C₁₃	—	2	2	no	—
74	C₂²×C₄×C₁₃	—	4	8	no	—
75	C₂²×C₄×C₁₃	—	4	4	no	—
76	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
77	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
79	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
80	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
81	C₂×C₈×C₁₃	—	2	4	no	—
82	C₂×C₈×C₁₃	—	2	4	no	—
83	C₂×C₈×C₁₃	—	2	4	no	—
84	C₁₃×C₄⋊C₄	—	2	4	no	—
85	C₁₃×C₄⋊C₄	—	2	2	no	—
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	—	4	4	no	—
90	C₁₃×C₄⋊C₄	—	4	4	no	—
91	C₁₃×C₄⋊C₄	—	4	4	no	—
92	C₁₃×C₂²⋊C₄	—	4	8	no	—
93	C₁₃×C₂²⋊C₄	—	4	4	no	—
94	C₄²×C₁₃	—	2	4	no	—
95	C₄²×C₁₃	—	2	2	no	—
96	C₄²×C₁₃	—	2	1	no	—
97	C₄²×C₁₃	—	2	1	no	—
98	C₄²×C₁₃	—	4	2	no	—
99	C₄²×C₁₃	#41	1	4	yes	—
100	#2864	C₂×C₁₃	1	1	yes	characteristic, complemented
101	C₄×#127	C₂×C₁₃	1	1	yes	characteristic
102	#2099	C₂×C₁₃	1	2	yes	complemented
103	#2041	C₂×C₁₃	1	2	yes	—
104	#1936	C₂×C₁₃	1	4	yes	complemented
105	#1804	C₂×C₁₃	1	2	yes	complemented
106	#1804	C₂×C₁₃	1	2	yes	complemented
107	#1767	C₂×C₁₃	1	1	yes	characteristic, complemented
108	C₂³×C₁₃	—	2	4	no	—
109	C₂³×C₁₃	—	4	2	no	—
110	C₂³×C₁₃	—	4	2	no	—
111	C₂³×C₁₃	—	4	4	no	—
112	C₂³×C₁₃	—	4	4	no	—
113	C₂³×C₁₃	C₂²×D₄	1	1	yes	characteristic
114	D₄×C₁₃	—	2	4	no	—
115	D₄×C₁₃	—	2	4	no	—
116	D₄×C₁₃	—	4	16	no	—
117	D₄×C₁₃	—	4	2	no	—
118	D₄×C₁₃	—	4	2	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	—	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	4	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	—	8	8	no	—
135	C₂×C₄×C₁₃	#124	1	2	yes	—
136	C₂×C₄×C₁₃	#124	1	2	yes	—
137	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	4	yes	—
138	C₂×C₄×C₁₃	C₂×#41	1	2	yes	—
139	C₂×C₄×C₁₃	C₂×#41	1	2	yes	—
140	C₂×C₄×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
141	C₂×C₄×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
142	C₈×C₁₃	—	4	4	no	—
143	C₂×#127	C₂²×C₁₃	1	1	yes	characteristic
144	C₂×C₄×D₄	—	2	2	no	—
145	C₂×C₄×D₄	C₂²×C₁₃	1	2	yes	—
146	C₂×C₄⋊C₈	C₂²×C₁₃	1	2	yes	—
147	C₂×#102	C₂²×C₁₃	1	4	yes	—
148	C₂×#102	C₂²×C₁₃	1	4	yes	—
149	C₂×#97	C₂²×C₁₃	1	2	yes	—
150	#390	—	2	2	no	—
151	#390	C₂²×C₁₃	1	2	yes	—
152	#389	C₂²×C₁₃	1	2	yes	—
153	#389	C₂²×C₁₃	1	2	yes	—
154	#387	—	2	2	no	—
155	#387	—	2	2	no	—
156	#384	C₂²×C₁₃	1	1	yes	characteristic
157	#382	C₂²×C₁₃	1	1	yes	characteristic
158	#382	C₂²×C₁₃	1	1	yes	characteristic
159	C₄³	C₂²×C₁₃	1	1	yes	characteristic
160	#340	C₂²×C₁₃	1	4	yes	—
161	#340	C₂²×C₁₃	1	4	yes	—
162	#336	C₂²×C₁₃	1	2	yes	—
163	C₂²×C₁₃	—	4	4	no	—
164	C₂²×C₁₃	—	4	8	no	—
165	C₂²×C₁₃	—	8	2	no	—
166	C₂²×C₁₃	—	8	2	no	—
167	C₂²×C₁₃	—	8	4	no	—
168	C₂²×C₁₃	#525	1	1	yes	characteristic
169	C₂²×C₁₃	C₂×#124	1	1	yes	characteristic
170	C₂²×C₁₃	#537	1	1	yes	characteristic
171	C₂²×C₁₃	C₂×AGL(1,ℤ/8)	1	2	yes	—
172	C₂²×C₁₃	#575	1	2	yes	—
173	C₄×C₁₃	—	2	4	no	—
174	C₄×C₁₃	—	2	2	no	—
175	C₄×C₁₃	—	2	2	no	—
176	C₄×C₁₃	—	4	2	no	—
177	C₄×C₁₃	—	8	2	no	—
178	C₄×C₁₃	—	8	2	no	—
179	C₄×C₁₃	—	8	2	no	—
180	C₄×C₁₃	#552	1	2	yes	—
181	C₄×C₁₃	#553	1	2	yes	—
182	C₂²×D₄	—	2	2	no	—
183	C₂²×D₄	C₂³×C₁₃	1	2	yes	—
184	C₂³×C₄	—	2	2	no	—
185	C₂²×C₈	C₂³×C₁₃	1	4	yes	—
186	#127	—	2	4	no	—
187	C₄×D₄	—	2	4	no	—
188	C₄×D₄	—	2	4	no	—
189	C₄×D₄	—	4	8	no	—
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	C₂³×C₁₃	1	2	yes	—
194	C₂×C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
195	C₂×C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
196	C₂×C₂²⋊C₄	—	2	4	no	—
197	C₂×C₂²⋊C₄	—	2	2	no	—
198	C₂×C₄²	D₄×C₁₃	1	2	yes	—
199	C₂×C₄²	D₄×C₁₃	1	2	yes	—
200	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
201	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
202	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
203	C₄⋊C₈	—	2	4	no	—
204	#102	—	2	8	no	—
205	#102	—	4	4	no	—
206	#97	—	4	2	no	—
207	#95	—	2	1	no	—
208	#95	—	2	1	no	—
209	#95	—	2	1	no	—
210	#95	—	2	1	no	—
211	#95	—	2	2	no	—
212	#95	—	2	2	no	—
213	C₂×C₁₃	—	8	4	no	—
214	C₂×C₁₃	#2562	1	1	yes	characteristic
215	C₂×C₁₃	#3114	1	2	yes	—
216	C₂×C₁₃	#3135	1	1	yes	characteristic
217	C₂×C₁₃	#3138	1	1	yes	characteristic
218	C₂×C₁₃	#3251	1	2	yes	—
219	C₂⁴	—	2	2	no	—
220	C₂×D₄	—	2	8	no	—
221	C₂×D₄	—	2	2	no	—
222	C₂×D₄	—	2	2	no	—
223	C₂×D₄	—	2	2	no	—
224	C₂×D₄	—	2	2	no	—
225	C₂×D₄	—	2	2	no	—
226	C₂×D₄	—	4	4	no	—
227	C₂×D₄	—	4	4	no	—
228	C₂×D₄	C₁₃×#41	1	4	yes	—
229	C₂²×C₄	—	2	2	no	—
230	C₂²×C₄	—	2	2	no	—
231	C₂²×C₄	—	2	2	no	—
232	C₂²×C₄	—	4	8	no	—
233	C₂²×C₄	—	4	4	no	—
234	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
235	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
236	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
237	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
238	C₂²×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
239	C₂×C₈	—	2	4	no	—
240	C₂×C₈	—	2	4	no	—
241	C₂×C₈	—	2	4	no	—
242	C₄⋊C₄	—	2	4	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	—	4	4	no	—
248	C₄⋊C₄	—	4	4	no	—
249	C₄⋊C₄	—	4	4	no	—
250	C₂²⋊C₄	—	4	8	no	—
251	C₂²⋊C₄	—	4	4	no	—
252	C₄²	—	2	4	no	—
253	C₄²	—	2	2	no	—
254	C₄²	—	2	1	no	—
255	C₄²	—	2	1	no	—
256	C₄²	—	4	2	no	—
257	C₄²	C₁₃×#41	1	4	yes	—
258	C₁₃	#29127	1	1	yes	characteristic, direct product factor, Sylow
259	C₂³	—	2	4	no	—
260	C₂³	—	4	2	no	—
261	C₂³	—	4	2	no	—
262	C₂³	—	4	4	no	—
263	C₂³	—	4	4	no	—
264	C₂³	C₂²×D₄×C₁₃	1	1	yes	characteristic
265	D₄	—	2	4	no	—
266	D₄	—	2	4	no	—
267	D₄	—	4	16	no	—
268	D₄	—	4	2	no	—
269	D₄	—	4	2	no	—
270	C₂×C₄	—	2	2	no	—
271	C₂×C₄	—	2	2	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	—	2	2	no	—
277	C₂×C₄	—	4	2	no	—
278	C₂×C₄	—	4	2	no	—
279	C₂×C₄	—	4	2	no	—
280	C₂×C₄	—	4	4	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	—	8	8	no	—
286	C₂×C₄	C₁₃×#124	1	2	yes	—
287	C₂×C₄	C₁₃×#124	1	2	yes	—
288	C₂×C₄	C₁₃×AGL(1,ℤ/8)	1	4	yes	—
289	C₂×C₄	C₂×C₁₃×#41	1	2	yes	—
290	C₂×C₄	C₂×C₁₃×#41	1	2	yes	—
291	C₂×C₄	C₁₃×H(5,𝔽₂)	1	1	yes	characteristic
292	C₂×C₄	C₁₃×2₋¹⁺⁴	1	1	yes	characteristic
293	C₈	—	4	4	no	—
294	C₂²	—	4	4	no	—
295	C₂²	—	4	8	no	—
296	C₂²	—	8	2	no	—
297	C₂²	—	8	2	no	—
298	C₂²	—	8	4	no	—
299	C₂²	C₁₃×#525	1	1	yes	characteristic
300	C₂²	C₂×C₁₃×#124	1	1	yes	characteristic
301	C₂²	C₁₃×#537	1	1	yes	characteristic
302	C₂²	C₂×C₁₃×AGL(1,ℤ/8)	1	2	yes	—
303	C₂²	C₁₃×#575	1	2	yes	—
304	C₄	—	2	4	no	—
305	C₄	—	2	2	no	—
306	C₄	—	2	2	no	—
307	C₄	—	4	2	no	—
308	C₄	—	8	2	no	—
309	C₄	—	8	2	no	—
310	C₄	—	8	2	no	—
311	C₄	C₁₃×#552	1	2	yes	—
312	C₄	C₁₃×#553	1	2	yes	—
313	C₂	—	8	4	no	—
314	C₂	C₁₃×#2562	1	1	yes	characteristic
315	C₂	C₁₃×#3114	1	2	yes	—
316	C₂	C₁₃×#3135	1	1	yes	characteristic
317	C₂	C₁₃×#3138	1	1	yes	characteristic
318	C₂	C₁₃×#3251	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₁₃	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃
prime radical	C₂×C₁₃×#127	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₃ ◁ C₄³×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₁₃×#2864	C₂	semidirect
3. C₁₃×#2099	C₂	semidirect
5. C₁₃×#1936	C₂	semidirect
6. C₁₃×#1804	C₂	semidirect
7. C₁₃×#1804	C₂	semidirect
8. C₁₃×#1767	C₂	semidirect
60. #29127	C₁₃	direct (N × H)
100. #2864	C₂×C₁₃	semidirect
102. #2099	C₂×C₁₃	semidirect
104. #1936	C₂×C₁₃	semidirect
105. #1804	C₂×C₁₃	semidirect
106. #1804	C₂×C₁₃	semidirect
107. #1767	C₂×C₁₃	semidirect
258. C₁₃	#29127	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×#2864	2
C₄×C₁₃×#127	2
C₁₃×#2099	2
C₁₃×#2041	2
C₁₃×#1936	2
C₁₃×#1804	2
C₁₃×#1767	2
#29127	13

Subgroups — C13×#29127 #15715466

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#29127 #15715466