Subgroups — C₁₃×#13128 #15699467

Order 3328.

328 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×#364	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₁₃×#349	C₂	1	1	yes	maximal, characteristic
3	C₂×C₁₃×#349	C₂	1	1	yes	maximal, characteristic
4	C₂×C₁₃×#347	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₃×#1753	C₂	1	1	yes	maximal, characteristic
6	C₁₃×#1376	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₃×#1375	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×C₁₃×M₁₆	C₂²	1	1	yes	characteristic
9	C₂×C₁₃×M₃₂	C₂²	1	1	yes	characteristic
10	C₂×C₁₃×M₃₂	C₂²	1	1	yes	characteristic
11	C₂×C₁₃×M₃₂	C₂²	1	1	yes	characteristic
12	C₂×C₁₃×M₃₂	C₂²	1	1	yes	characteristic
13	C₂×C₁₃×C₂²⋊C₈	C₄	1	1	yes	characteristic
14	C₂×C₁₃×#97	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
16	C₄×C₁₃×C₂²⋊C₄	C₄	1	1	yes	characteristic
17	C₁₃×#364	—	2	2	no	—
18	C₁₃×#349	—	2	2	no	—
19	C₁₃×#349	—	2	2	no	—
20	C₁₃×#347	—	2	2	no	—
21	C₁₃×#336	C₄	1	1	yes	characteristic
22	C₁₃×#336	C₄	1	1	yes	characteristic
23	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
24	C₂×C₁₃×M₁₆	—	2	1	no	—
25	C₂×C₁₃×M₁₆	—	2	1	no	—
26	C₂×C₁₃×M₁₆	—	2	1	no	—
27	C₂×C₁₃×M₁₆	—	2	1	no	—
28	C₂×C₁₃×M₁₆	—	2	1	no	—
29	C₂×C₁₃×M₁₆	—	2	1	no	—
30	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
34	C₂×C₁₃×C₂²⋊C₄	—	2	1	no	—
35	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
37	C₁₃×M₃₂	—	2	1	no	—
38	C₁₃×M₃₂	—	2	1	no	—
39	C₁₃×M₃₂	—	2	1	no	—
40	C₁₃×M₃₂	—	2	1	no	—
41	C₁₃×M₃₂	—	2	1	no	—
42	C₁₃×M₃₂	—	2	1	no	—
43	C₁₃×M₃₂	D₄	1	2	yes	—
44	C₁₃×M₃₂	D₄	1	2	yes	—
45	C₂×C₁₃×C₁₆	—	2	1	no	—
46	C₂×C₁₃×C₁₆	—	2	1	no	—
47	C₂×C₁₃×C₁₆	—	2	1	no	—
48	C₂×C₁₃×C₁₆	—	2	1	no	—
49	C₁₃×C₂²⋊C₈	—	2	2	no	—
50	C₁₃×#97	—	2	1	no	—
51	C₁₃×#97	—	2	1	no	—
52	C₄×C₈×C₁₃	—	2	1	no	—
53	C₄×C₈×C₁₃	—	2	1	no	—
54	C₁₃×#95	—	2	1	no	—
55	#13128	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
56	C₂⁴×C₁₃	M₁₆	1	1	yes	characteristic
57	C₂²×C₄×C₁₃	—	2	1	no	—
58	C₂²×C₄×C₁₃	—	2	1	no	—
59	C₂²×C₄×C₁₃	—	2	1	no	—
60	C₂²×C₄×C₁₃	—	2	1	no	—
61	C₂²×C₄×C₁₃	—	2	1	no	—
62	C₂²×C₄×C₁₃	—	2	1	no	—
63	C₂²×C₄×C₁₃	—	2	1	no	—
64	C₂²×C₄×C₁₃	—	2	1	no	—
65	C₂²×C₄×C₁₃	M₁₆	1	1	yes	characteristic
66	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
67	C₁₃×M₁₆	—	4	1	no	—
68	C₁₃×M₁₆	—	4	1	no	—
69	C₁₃×M₁₆	—	4	1	no	—
70	C₁₃×M₁₆	—	4	1	no	—
71	C₂×C₈×C₁₃	—	2	1	no	—
72	C₂×C₈×C₁₃	—	2	1	no	—
73	C₂×C₈×C₁₃	—	2	1	no	—
74	C₂×C₈×C₁₃	—	2	1	no	—
75	C₂×C₈×C₁₃	—	2	1	no	—
76	C₂×C₈×C₁₃	—	2	1	no	—
77	C₂×C₈×C₁₃	—	2	1	no	—
78	C₂×C₈×C₁₃	—	2	1	no	—
79	C₂×C₈×C₁₃	M₁₆	1	2	yes	—
80	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
81	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
82	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
83	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
84	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
85	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
86	C₁₃×C₂²⋊C₄	—	4	2	no	—
87	C₄²×C₁₃	—	2	1	no	—
88	C₄²×C₁₃	—	2	1	no	—
89	C₄²×C₁₃	—	2	1	no	—
90	C₄²×C₁₃	—	2	1	no	—
91	C₁₃×C₁₆	—	2	2	no	—
92	C₁₃×C₁₆	—	2	2	no	—
93	C₁₃×C₁₆	—	4	1	no	—
94	C₁₃×C₁₆	—	4	1	no	—
95	C₂×#364	C₂×C₁₃	1	1	yes	characteristic, complemented
96	C₂×#349	C₂×C₁₃	1	1	yes	characteristic
97	C₂×#349	C₂×C₁₃	1	1	yes	characteristic
98	C₂×#347	C₂×C₁₃	1	1	yes	characteristic, complemented
99	#1753	C₂×C₁₃	1	1	yes	characteristic
100	#1376	C₂×C₁₃	1	1	yes	characteristic, complemented
101	#1375	C₂×C₁₃	1	1	yes	characteristic, complemented
102	C₂³×C₁₃	—	2	1	no	—
103	C₂³×C₁₃	—	2	1	no	—
104	C₂³×C₁₃	—	2	1	no	—
105	C₂³×C₁₃	—	4	1	no	—
106	C₂³×C₁₃	—	4	1	no	—
107	C₂³×C₁₃	C₂×M₁₆	1	1	yes	characteristic
108	C₂×C₄×C₁₃	—	2	2	no	—
109	C₂×C₄×C₁₃	—	2	1	no	—
110	C₂×C₄×C₁₃	—	2	1	no	—
111	C₂×C₄×C₁₃	—	2	1	no	—
112	C₂×C₄×C₁₃	—	4	1	no	—
113	C₂×C₄×C₁₃	—	4	1	no	—
114	C₂×C₄×C₁₃	—	4	1	no	—
115	C₂×C₄×C₁₃	—	4	1	no	—
116	C₂×C₄×C₁₃	—	4	1	no	—
117	C₂×C₄×C₁₃	—	4	1	no	—
118	C₂×C₄×C₁₃	—	4	1	no	—
119	C₂×C₄×C₁₃	—	4	1	no	—
120	C₂×C₄×C₁₃	—	4	1	no	—
121	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
122	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
123	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
124	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
125	C₂×C₄×C₁₃	#131	1	1	yes	characteristic
126	C₂×C₄×C₁₃	#131	1	1	yes	characteristic
127	C₈×C₁₃	—	2	2	no	—
128	C₈×C₁₃	—	4	1	no	—
129	C₈×C₁₃	—	4	1	no	—
130	C₈×C₁₃	#121	1	1	yes	characteristic
131	C₈×C₁₃	#123	1	1	yes	characteristic
132	C₈×C₁₃	#124	1	1	yes	characteristic
133	C₈×C₁₃	#126	1	1	yes	characteristic
134	C₂²×M₁₆	C₂²×C₁₃	1	1	yes	characteristic
135	C₂×M₃₂	C₂²×C₁₃	1	1	yes	characteristic
136	C₂×M₃₂	C₂²×C₁₃	1	1	yes	characteristic
137	C₂×M₃₂	C₂²×C₁₃	1	1	yes	characteristic
138	C₂×M₃₂	C₂²×C₁₃	1	1	yes	characteristic
139	C₂×C₂²⋊C₈	C₄×C₁₃	1	1	yes	characteristic
140	C₂×#97	C₂²×C₁₃	1	1	yes	characteristic
141	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
142	C₄×C₂²⋊C₄	C₄×C₁₃	1	1	yes	characteristic
143	#364	—	2	2	no	—
144	#349	—	2	2	no	—
145	#349	—	2	2	no	—
146	#347	—	2	2	no	—
147	#336	C₄×C₁₃	1	1	yes	characteristic
148	#336	C₄×C₁₃	1	1	yes	characteristic
149	C₂²×C₁₃	—	2	1	no	—
150	C₂²×C₁₃	—	2	1	no	—
151	C₂²×C₁₃	—	4	1	no	—
152	C₂²×C₁₃	—	4	1	no	—
153	C₂²×C₁₃	—	4	1	no	—
154	C₂²×C₁₃	—	8	1	no	—
155	C₂²×C₁₃	—	8	1	no	—
156	C₂²×C₁₃	#432	1	1	yes	characteristic
157	C₂²×C₁₃	#435	1	1	yes	characteristic
158	C₂²×C₁₃	#435	1	1	yes	characteristic
159	C₄×C₁₃	—	4	2	no	—
160	C₄×C₁₃	—	4	1	no	—
161	C₄×C₁₃	—	8	1	no	—
162	C₄×C₁₃	#388	1	1	yes	characteristic
163	C₄×C₁₃	#433	1	1	yes	characteristic
164	C₄×C₁₃	#435	1	1	yes	characteristic
165	C₄×C₁₃	#436	1	1	yes	characteristic
166	C₂³×C₄	C₂×C₄×C₁₃	1	1	yes	characteristic
167	C₂×M₁₆	—	2	1	no	—
168	C₂×M₁₆	—	2	1	no	—
169	C₂×M₁₆	—	2	1	no	—
170	C₂×M₁₆	—	2	1	no	—
171	C₂×M₁₆	—	2	1	no	—
172	C₂×M₁₆	—	2	1	no	—
173	C₂²×C₈	C₂×C₄×C₁₃	1	1	yes	characteristic
174	C₂²×C₈	C₂×C₄×C₁₃	1	1	yes	characteristic
175	C₂²×C₈	C₂×C₄×C₁₃	1	1	yes	characteristic
176	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
177	C₂×C₂²⋊C₄	—	2	1	no	—
178	C₂×C₄²	C₂×C₄×C₁₃	1	1	yes	characteristic
179	C₂×C₄²	C₂×C₄×C₁₃	1	1	yes	characteristic
180	M₃₂	—	2	1	no	—
181	M₃₂	—	2	1	no	—
182	M₃₂	—	2	1	no	—
183	M₃₂	—	2	1	no	—
184	M₃₂	—	2	1	no	—
185	M₃₂	—	2	1	no	—
186	M₃₂	D₄×C₁₃	1	2	yes	—
187	M₃₂	D₄×C₁₃	1	2	yes	—
188	C₂×C₁₆	—	2	1	no	—
189	C₂×C₁₆	—	2	1	no	—
190	C₂×C₁₆	—	2	1	no	—
191	C₂×C₁₆	—	2	1	no	—
192	C₂²⋊C₈	—	2	2	no	—
193	#97	—	2	1	no	—
194	#97	—	2	1	no	—
195	C₄×C₈	—	2	1	no	—
196	C₄×C₈	—	2	1	no	—
197	#95	—	2	1	no	—
198	C₂×C₁₃	—	2	1	no	—
199	C₂×C₁₃	—	2	1	no	—
200	C₂×C₁₃	—	8	1	no	—
201	C₂×C₁₃	#1924	1	1	yes	characteristic
202	C₂×C₁₃	#2165	1	1	yes	characteristic
203	C₂×C₁₃	#2166	1	1	yes	characteristic
204	C₂⁴	C₁₃×M₁₆	1	1	yes	characteristic
205	C₂²×C₄	—	2	1	no	—
206	C₂²×C₄	—	2	1	no	—
207	C₂²×C₄	—	2	1	no	—
208	C₂²×C₄	—	2	1	no	—
209	C₂²×C₄	—	2	1	no	—
210	C₂²×C₄	—	2	1	no	—
211	C₂²×C₄	—	2	1	no	—
212	C₂²×C₄	—	2	1	no	—
213	C₂²×C₄	C₁₃×M₁₆	1	1	yes	characteristic
214	C₂²×C₄	C₂²×C₄×C₁₃	1	1	yes	characteristic
215	M₁₆	—	4	1	no	—
216	M₁₆	—	4	1	no	—
217	M₁₆	—	4	1	no	—
218	M₁₆	—	4	1	no	—
219	C₂×C₈	—	2	1	no	—
220	C₂×C₈	—	2	1	no	—
221	C₂×C₈	—	2	1	no	—
222	C₂×C₈	—	2	1	no	—
223	C₂×C₈	—	2	1	no	—
224	C₂×C₈	—	2	1	no	—
225	C₂×C₈	—	2	1	no	—
226	C₂×C₈	—	2	1	no	—
227	C₂×C₈	C₁₃×M₁₆	1	2	yes	—
228	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
229	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
230	C₂×C₈	C₁₃×#41	1	1	yes	characteristic
231	C₂×C₈	C₁₃×#41	1	1	yes	characteristic
232	C₂×C₈	C₁₃×#41	1	1	yes	characteristic
233	C₂×C₈	C₁₃×#41	1	1	yes	characteristic
234	C₂²⋊C₄	—	4	2	no	—
235	C₄²	—	2	1	no	—
236	C₄²	—	2	1	no	—
237	C₄²	—	2	1	no	—
238	C₄²	—	2	1	no	—
239	C₁₆	—	2	2	no	—
240	C₁₆	—	2	2	no	—
241	C₁₆	—	4	1	no	—
242	C₁₆	—	4	1	no	—
243	C₁₃	#13128	1	1	yes	characteristic, direct product factor, Sylow
244	C₂³	—	2	1	no	—
245	C₂³	—	2	1	no	—
246	C₂³	—	2	1	no	—
247	C₂³	—	4	1	no	—
248	C₂³	—	4	1	no	—
249	C₂³	C₂×C₁₃×M₁₆	1	1	yes	characteristic
250	C₂×C₄	—	2	2	no	—
251	C₂×C₄	—	2	1	no	—
252	C₂×C₄	—	2	1	no	—
253	C₂×C₄	—	2	1	no	—
254	C₂×C₄	—	4	1	no	—
255	C₂×C₄	—	4	1	no	—
256	C₂×C₄	—	4	1	no	—
257	C₂×C₄	—	4	1	no	—
258	C₂×C₄	—	4	1	no	—
259	C₂×C₄	—	4	1	no	—
260	C₂×C₄	—	4	1	no	—
261	C₂×C₄	—	4	1	no	—
262	C₂×C₄	—	4	1	no	—
263	C₂×C₄	C₁₃×#117	1	1	yes	characteristic
264	C₂×C₄	C₄×D₄×C₁₃	1	1	yes	characteristic
265	C₂×C₄	C₄×D₄×C₁₃	1	1	yes	characteristic
266	C₂×C₄	C₂×C₁₃×M₁₆	1	1	yes	characteristic
267	C₂×C₄	C₁₃×#131	1	1	yes	characteristic
268	C₂×C₄	C₁₃×#131	1	1	yes	characteristic
269	C₈	—	2	2	no	—
270	C₈	—	4	1	no	—
271	C₈	—	4	1	no	—
272	C₈	C₁₃×#121	1	1	yes	characteristic
273	C₈	C₁₃×#123	1	1	yes	characteristic
274	C₈	C₁₃×#124	1	1	yes	characteristic
275	C₈	C₁₃×#126	1	1	yes	characteristic
276	C₂²	—	2	1	no	—
277	C₂²	—	2	1	no	—
278	C₂²	—	4	1	no	—
279	C₂²	—	4	1	no	—
280	C₂²	—	4	1	no	—
281	C₂²	—	8	1	no	—
282	C₂²	—	8	1	no	—
283	C₂²	C₁₃×#432	1	1	yes	characteristic
284	C₂²	C₁₃×#435	1	1	yes	characteristic
285	C₂²	C₁₃×#435	1	1	yes	characteristic
286	C₄	—	4	2	no	—
287	C₄	—	4	1	no	—
288	C₄	—	8	1	no	—
289	C₄	C₁₃×#388	1	1	yes	characteristic
290	C₄	C₁₃×#433	1	1	yes	characteristic
291	C₄	C₁₃×#435	1	1	yes	characteristic
292	C₄	C₁₃×#436	1	1	yes	characteristic
293	C₂	—	2	1	no	—
294	C₂	—	2	1	no	—
295	C₂	—	8	1	no	—
296	C₂	C₁₃×#1924	1	1	yes	characteristic
297	C₂	C₁₃×#2165	1	1	yes	characteristic
298	C₂	C₁₃×#2166	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₁₃	#117
commutator	C₂²×C₄	C₂²×C₄×C₁₃
Frattini, cosocle	C₂²×C₈	C₂³×C₁₃
socle	C₂²×C₁₃	#432
prime radical	C₂⁴×C₁₃	M₁₆
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₃ ◁ C₂×C₁₃×#97 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₁₃×#364	C₂	semidirect
4. C₂×C₁₃×#347	C₂	semidirect
6. C₁₃×#1376	C₂	semidirect
7. C₁₃×#1375	C₂	semidirect
55. #13128	C₁₃	direct (N × H)
95. C₂×#364	C₂×C₁₃	semidirect
98. C₂×#347	C₂×C₁₃	semidirect
100. #1376	C₂×C₁₃	semidirect
101. #1375	C₂×C₁₃	semidirect
243. C₁₃	#13128	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×#364	2
C₂×C₁₃×#349	2
C₂×C₁₃×#347	2
C₁₃×#1753	2
C₁₃×#1376	2
C₁₃×#1375	2
#13128	13

Subgroups — C13×#13128 #15699467

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#13128 #15699467