Subgroups — C₁₃×#25910 #15712249

Order 3328.

1066 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₃×#533	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₃×#491	C₂	1	2	yes	maximal, complemented
3	C₁₃×#2857	C₂	1	1	yes	maximal, characteristic
4	C₁₃×#2024	C₂	1	4	yes	maximal, complemented
5	C₁₃×#1992	C₂	1	2	yes	maximal, complemented
6	C₁₃×#1873	C₂	1	4	yes	maximal
7	C₁₃×#1749	C₂	1	1	yes	maximal, characteristic
8	C₁₃×#533	—	2	8	no	—
9	C₂×C₁₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
10	C₂×C₁₃×C₄⋊Q₈	C₂²	1	2	yes	—
11	C₂×C₁₃×#125	C₂²	1	2	yes	—
12	C₂×C₁₃×C₂²⋊Q₈	—	2	2	no	—
13	C₂×C₁₃×C₂²⋊Q₈	C₂²	1	2	yes	—
14	C₂×C₄×Q₈×C₁₃	—	2	2	no	—
15	C₂×C₁₃×#117	C₂²	1	2	yes	—
16	C₂²×C₄²×C₁₃	C₂²	1	1	yes	characteristic
17	C₁₃×#491	—	2	8	no	—
18	C₂×C₁₃×#103	C₂²	1	8	yes	—
19	C₂×C₁₃×C₂²⋊C₈	C₂²	1	2	yes	—
20	C₂×C₁₃×#97	C₂²	1	2	yes	—
21	C₁₃×#391	C₂²	1	2	yes	—
22	C₁₃×#389	—	2	2	no	—
23	C₁₃×#384	C₂²	1	1	yes	characteristic
24	C₁₃×#380	—	2	2	no	—
25	C₁₃×#340	C₂²	1	8	yes	—
26	C₁₃×#336	C₂²	1	2	yes	—
27	C₂²×Q₈×C₁₃	—	2	2	no	—
28	C₂²×Q₈×C₁₃	C₂³	1	2	yes	—
29	C₂³×C₄×C₁₃	D₄	1	2	yes	—
30	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
31	C₂²×C₈×C₁₃	C₂³	1	4	yes	—
32	C₁₃×C₄⋊Q₈	—	2	8	no	—
33	C₁₃×C₄⋊Q₈	—	2	4	no	—
34	C₁₃×#125	—	2	4	no	—
35	C₁₃×#125	D₄	1	8	yes	—
36	C₁₃×C₂²⋊Q₈	—	2	16	no	—
37	C₁₃×C₂²⋊Q₈	—	2	8	no	—
38	C₄×Q₈×C₁₃	—	2	16	no	—
39	C₁₃×#117	—	2	8	no	—
40	C₂×C₁₃×C₄⋊C₄	—	2	4	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	—	2	2	no	—
45	C₂×C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
46	C₂×C₁₃×C₄⋊C₄	C₂³	1	4	yes	—
47	C₂×C₁₃×C₂²⋊C₄	—	2	2	no	—
48	C₂×C₁₃×C₂²⋊C₄	—	2	2	no	—
49	C₂×C₄²×C₁₃	—	2	2	no	—
50	C₂×C₄²×C₁₃	—	2	4	no	—
51	C₂×C₄²×C₁₃	D₄	1	2	yes	—
52	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
53	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
54	C₁₃×#103	—	2	16	no	—
55	C₁₃×C₂²⋊C₈	—	2	4	no	—
56	C₁₃×#97	—	2	4	no	—
57	C₁₃×#95	—	2	4	no	—
58	#25910	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
59	C₂⁴×C₁₃	C₂×D₄	1	1	yes	characteristic
60	C₂×Q₈×C₁₃	—	2	8	no	—
61	C₂×Q₈×C₁₃	—	2	4	no	—
62	C₂×Q₈×C₁₃	—	2	4	no	—
63	C₂×Q₈×C₁₃	—	2	4	no	—
64	C₂×Q₈×C₁₃	—	4	4	no	—
65	C₂×Q₈×C₁₃	—	4	4	no	—
66	C₂²×C₄×C₁₃	—	2	2	no	—
67	C₂²×C₄×C₁₃	—	2	2	no	—
68	C₂²×C₄×C₁₃	—	2	4	no	—
69	C₂²×C₄×C₁₃	—	2	4	no	—
70	C₂²×C₄×C₁₃	—	2	4	no	—
71	C₂²×C₄×C₁₃	—	2	2	no	—
72	C₂²×C₄×C₁₃	—	2	2	no	—
73	C₂²×C₄×C₁₃	—	4	2	no	—
74	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
75	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
77	C₂²×C₄×C₁₃	#41	1	4	yes	—
78	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
79	C₂×C₈×C₁₃	—	2	4	no	—
80	C₂×C₈×C₁₃	—	2	4	no	—
81	C₂×C₈×C₁₃	—	2	4	no	—
82	C₁₃×C₄⋊C₄	—	2	8	no	—
83	C₁₃×C₄⋊C₄	—	2	16	no	—
84	C₁₃×C₄⋊C₄	—	2	4	no	—
85	C₁₃×C₄⋊C₄	—	2	4	no	—
86	C₁₃×C₄⋊C₄	—	4	4	no	—
87	C₁₃×C₄⋊C₄	—	4	4	no	—
88	C₁₃×C₄⋊C₄	—	4	4	no	—
89	C₁₃×C₄⋊C₄	C₂×D₄	1	8	yes	—
90	C₁₃×C₂²⋊C₄	—	2	8	no	—
91	C₁₃×C₂²⋊C₄	—	4	4	no	—
92	C₄²×C₁₃	—	2	8	no	—
93	C₄²×C₁₃	—	2	4	no	—
94	C₄²×C₁₃	—	2	2	no	—
95	C₄²×C₁₃	C₂×D₄	1	4	yes	—
96	C₂×#533	C₂×C₁₃	1	1	yes	characteristic
97	C₂×#491	C₂×C₁₃	1	2	yes	complemented
98	#2857	C₂×C₁₃	1	1	yes	characteristic
99	#2024	C₂×C₁₃	1	4	yes	complemented
100	#1992	C₂×C₁₃	1	2	yes	complemented
101	#1873	C₂×C₁₃	1	4	yes	—
102	#1749	C₂×C₁₃	1	1	yes	characteristic
103	C₂³×C₁₃	—	2	1	no	—
104	C₂³×C₁₃	—	2	1	no	—
105	C₂³×C₁₃	—	2	2	no	—
106	C₂³×C₁₃	—	2	2	no	—
107	C₂³×C₁₃	#124	1	2	yes	—
108	C₂³×C₁₃	C₂²×D₄	1	1	yes	characteristic
109	Q₈×C₁₃	—	4	16	no	—
110	Q₈×C₁₃	—	4	8	no	—
111	C₂×C₄×C₁₃	—	2	1	no	—
112	C₂×C₄×C₁₃	—	2	2	no	—
113	C₂×C₄×C₁₃	—	2	1	no	—
114	C₂×C₄×C₁₃	—	2	2	no	—
115	C₂×C₄×C₁₃	—	2	2	no	—
116	C₂×C₄×C₁₃	—	2	2	no	—
117	C₂×C₄×C₁₃	—	2	8	no	—
118	C₂×C₄×C₁₃	—	2	8	no	—
119	C₂×C₄×C₁₃	—	4	8	no	—
120	C₂×C₄×C₁₃	—	4	4	no	—
121	C₂×C₄×C₁₃	—	4	4	no	—
122	C₂×C₄×C₁₃	—	4	2	no	—
123	C₂×C₄×C₁₃	—	4	2	no	—
124	C₂×C₄×C₁₃	—	4	2	no	—
125	C₂×C₄×C₁₃	—	4	2	no	—
126	C₂×C₄×C₁₃	—	4	4	no	—
127	C₂×C₄×C₁₃	#120	1	4	yes	—
128	C₂×C₄×C₁₃	#124	1	2	yes	—
129	C₂×C₄×C₁₃	#137	1	4	yes	—
130	C₂×C₄×C₁₃	C₂²×D₄	1	2	yes	—
131	C₂×C₄×C₁₃	C₂×#41	1	2	yes	—
132	C₂×C₄×C₁₃	2₋¹⁺⁴	1	2	yes	—
133	C₈×C₁₃	—	4	4	no	—
134	#533	—	2	8	no	—
135	C₂×C₄⋊Q₈	C₂²×C₁₃	1	1	yes	characteristic
136	C₂×C₄⋊Q₈	C₂²×C₁₃	1	2	yes	—
137	C₂×#125	C₂²×C₁₃	1	2	yes	—
138	C₂×C₂²⋊Q₈	—	2	2	no	—
139	C₂×C₂²⋊Q₈	C₂²×C₁₃	1	2	yes	—
140	C₂×C₄×Q₈	—	2	2	no	—
141	C₂×#117	C₂²×C₁₃	1	2	yes	—
142	C₂²×C₄²	C₂²×C₁₃	1	1	yes	characteristic
143	#491	—	2	8	no	—
144	C₂×#103	C₂²×C₁₃	1	8	yes	—
145	C₂×C₂²⋊C₈	C₂²×C₁₃	1	2	yes	—
146	C₂×#97	C₂²×C₁₃	1	2	yes	—
147	#391	C₂²×C₁₃	1	2	yes	—
148	#389	—	2	2	no	—
149	#384	C₂²×C₁₃	1	1	yes	characteristic
150	#380	—	2	2	no	—
151	#340	C₂²×C₁₃	1	8	yes	—
152	#336	C₂²×C₁₃	1	2	yes	—
153	C₂²×C₁₃	—	2	2	no	—
154	C₂²×C₁₃	—	2	2	no	—
155	C₂²×C₁₃	—	2	2	no	—
156	C₂²×C₁₃	—	4	2	no	—
157	C₂²×C₁₃	—	4	2	no	—
158	C₂²×C₁₃	C₂×#120	1	1	yes	characteristic
159	C₂²×C₁₃	C₂×#124	1	1	yes	characteristic
160	C₂²×C₁₃	#536	1	1	yes	characteristic
161	C₂²×C₁₃	C₂×#137	1	2	yes	—
162	C₂²×C₁₃	#575	1	2	yes	—
163	C₄×C₁₃	—	2	2	no	—
164	C₄×C₁₃	—	2	4	no	—
165	C₄×C₁₃	—	2	4	no	—
166	C₄×C₁₃	—	4	8	no	—
167	C₄×C₁₃	—	8	2	no	—
168	C₄×C₁₃	—	8	2	no	—
169	C₄×C₁₃	#548	1	4	yes	—
170	C₂²×Q₈	—	2	2	no	—
171	C₂²×Q₈	C₂³×C₁₃	1	2	yes	—
172	C₂³×C₄	D₄×C₁₃	1	2	yes	—
173	C₂³×C₄	C₂³×C₁₃	1	1	yes	characteristic
174	C₂²×C₈	C₂³×C₁₃	1	4	yes	—
175	C₄⋊Q₈	—	2	8	no	—
176	C₄⋊Q₈	—	2	4	no	—
177	#125	—	2	4	no	—
178	#125	D₄×C₁₃	1	8	yes	—
179	C₂²⋊Q₈	—	2	16	no	—
180	C₂²⋊Q₈	—	2	8	no	—
181	C₄×Q₈	—	2	16	no	—
182	#117	—	2	8	no	—
183	C₂×C₄⋊C₄	—	2	4	no	—
184	C₂×C₄⋊C₄	—	2	2	no	—
185	C₂×C₄⋊C₄	—	2	2	no	—
186	C₂×C₄⋊C₄	—	2	2	no	—
187	C₂×C₄⋊C₄	—	2	2	no	—
188	C₂×C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
189	C₂×C₄⋊C₄	C₂³×C₁₃	1	4	yes	—
190	C₂×C₂²⋊C₄	—	2	2	no	—
191	C₂×C₂²⋊C₄	—	2	2	no	—
192	C₂×C₄²	—	2	2	no	—
193	C₂×C₄²	—	2	4	no	—
194	C₂×C₄²	D₄×C₁₃	1	2	yes	—
195	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
196	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
197	#103	—	2	16	no	—
198	C₂²⋊C₈	—	2	4	no	—
199	#97	—	2	4	no	—
200	#95	—	2	4	no	—
201	C₂×C₁₃	—	4	2	no	—
202	C₂×C₁₃	#2436	1	1	yes	characteristic
203	C₂×C₁₃	#3007	1	2	yes	—
204	C₂×C₁₃	#3135	1	1	yes	characteristic
205	C₂×C₁₃	#3139	1	1	yes	characteristic
206	C₂×C₁₃	#3237	1	2	yes	—
207	C₂⁴	C₂×D₄×C₁₃	1	1	yes	characteristic
208	C₂×Q₈	—	2	8	no	—
209	C₂×Q₈	—	2	4	no	—
210	C₂×Q₈	—	2	4	no	—
211	C₂×Q₈	—	2	4	no	—
212	C₂×Q₈	—	4	4	no	—
213	C₂×Q₈	—	4	4	no	—
214	C₂²×C₄	—	2	2	no	—
215	C₂²×C₄	—	2	2	no	—
216	C₂²×C₄	—	2	4	no	—
217	C₂²×C₄	—	2	4	no	—
218	C₂²×C₄	—	2	4	no	—
219	C₂²×C₄	—	2	2	no	—
220	C₂²×C₄	—	2	2	no	—
221	C₂²×C₄	—	4	2	no	—
222	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
223	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
224	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
225	C₂²×C₄	C₁₃×#41	1	4	yes	—
226	C₂²×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
227	C₂×C₈	—	2	4	no	—
228	C₂×C₈	—	2	4	no	—
229	C₂×C₈	—	2	4	no	—
230	C₄⋊C₄	—	2	8	no	—
231	C₄⋊C₄	—	2	16	no	—
232	C₄⋊C₄	—	2	4	no	—
233	C₄⋊C₄	—	2	4	no	—
234	C₄⋊C₄	—	4	4	no	—
235	C₄⋊C₄	—	4	4	no	—
236	C₄⋊C₄	—	4	4	no	—
237	C₄⋊C₄	C₂×D₄×C₁₃	1	8	yes	—
238	C₂²⋊C₄	—	2	8	no	—
239	C₂²⋊C₄	—	4	4	no	—
240	C₄²	—	2	8	no	—
241	C₄²	—	2	4	no	—
242	C₄²	—	2	2	no	—
243	C₄²	C₂×D₄×C₁₃	1	4	yes	—
244	C₁₃	#25910	1	1	yes	characteristic, direct product factor, Sylow
245	C₂³	—	2	1	no	—
246	C₂³	—	2	1	no	—
247	C₂³	—	2	2	no	—
248	C₂³	—	2	2	no	—
249	C₂³	C₁₃×#124	1	2	yes	—
250	C₂³	C₂²×D₄×C₁₃	1	1	yes	characteristic
251	Q₈	—	4	16	no	—
252	Q₈	—	4	8	no	—
253	C₂×C₄	—	2	1	no	—
254	C₂×C₄	—	2	2	no	—
255	C₂×C₄	—	2	1	no	—
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	2	2	no	—
259	C₂×C₄	—	2	8	no	—
260	C₂×C₄	—	2	8	no	—
261	C₂×C₄	—	4	8	no	—
262	C₂×C₄	—	4	4	no	—
263	C₂×C₄	—	4	4	no	—
264	C₂×C₄	—	4	2	no	—
265	C₂×C₄	—	4	2	no	—
266	C₂×C₄	—	4	2	no	—
267	C₂×C₄	—	4	2	no	—
268	C₂×C₄	—	4	4	no	—
269	C₂×C₄	C₁₃×#120	1	4	yes	—
270	C₂×C₄	C₁₃×#124	1	2	yes	—
271	C₂×C₄	C₁₃×#137	1	4	yes	—
272	C₂×C₄	C₂²×D₄×C₁₃	1	2	yes	—
273	C₂×C₄	C₂×C₁₃×#41	1	2	yes	—
274	C₂×C₄	C₁₃×2₋¹⁺⁴	1	2	yes	—
275	C₈	—	4	4	no	—
276	C₂²	—	2	2	no	—
277	C₂²	—	2	2	no	—
278	C₂²	—	2	2	no	—
279	C₂²	—	4	2	no	—
280	C₂²	—	4	2	no	—
281	C₂²	C₂×C₁₃×#120	1	1	yes	characteristic
282	C₂²	C₂×C₁₃×#124	1	1	yes	characteristic
283	C₂²	C₁₃×#536	1	1	yes	characteristic
284	C₂²	C₂×C₁₃×#137	1	2	yes	—
285	C₂²	C₁₃×#575	1	2	yes	—
286	C₄	—	2	4	no	—
287	C₄	—	2	4	no	—
288	C₄	—	2	2	no	—
289	C₄	—	4	8	no	—
290	C₄	—	8	2	no	—
291	C₄	—	8	2	no	—
292	C₄	C₁₃×#548	1	4	yes	—
293	C₂	—	4	2	no	—
294	C₂	C₁₃×#2436	1	1	yes	characteristic
295	C₂	C₁₃×#3007	1	2	yes	—
296	C₂	C₁₃×#3135	1	1	yes	characteristic
297	C₂	C₁₃×#3139	1	1	yes	characteristic
298	C₂	C₁₃×#3237	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₁₃	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃
prime radical	C₂⁴×C₁₃	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₃ ◁ C₂²×C₄²×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₁₃×#491	C₂	semidirect
4. C₁₃×#2024	C₂	semidirect
5. C₁₃×#1992	C₂	semidirect
58. #25910	C₁₃	direct (N × H)
97. C₂×#491	C₂×C₁₃	semidirect
99. #2024	C₂×C₁₃	semidirect
100. #1992	C₂×C₁₃	semidirect
244. C₁₃	#25910	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₁₃×#533	2
C₂×C₁₃×#491	2
C₁₃×#2857	2
C₁₃×#2024	2
C₁₃×#1992	2
C₁₃×#1873	2
C₁₃×#1749	2
#25910	13

Subgroups — C13×#25910 #15712249

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#25910 #15712249