Subgroups — C₁₃×#9106 #15695445

Order 3328.

522 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	C₁₃×#2983	C₂	1	1	yes	maximal, characteristic, complemented
2	C₁₃×#2326	C₂	1	1	yes	maximal, characteristic
3	C₁₃×#1478	C₂	1	2	yes	maximal
4	C₁₃×#1461	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₃×#1274	C₂	1	2	yes	maximal, complemented
6	C₂×C₁₃×#124	C₄	1	1	yes	characteristic
7	C₂×C₄×Q₈×C₁₃	C₂²	1	1	yes	characteristic
8	C₂×C₄×D₄×C₁₃	C₂²	1	1	yes	characteristic
9	C₁₃×#446	—	2	4	no	—
10	C₁₃×#432	—	2	2	no	—
11	C₁₃×#431	C₂²	1	2	yes	—
12	C₁₃×#431	C₂²	1	2	yes	—
13	C₁₃×#391	C₄	1	1	yes	characteristic
14	C₁₃×#390	C₄	1	1	yes	characteristic
15	C₁₃×#388	—	2	2	no	—
16	C₁₃×#385	—	2	2	no	—
17	C₁₃×#384	C₄	1	1	yes	characteristic
18	C₁₃×#376	C₂²	1	1	yes	characteristic
19	C₁₃×D₄⋊C₈	—	2	4	no	—
20	C₁₃×#324	—	2	2	no	—
21	C₂²×Q₈×C₁₃	C₂×C₄	1	1	yes	characteristic
22	C₂²×D₄×C₁₃	C₂×C₄	1	1	yes	characteristic
23	C₂³×C₄×C₁₃	—	2	1	no	—
24	C₁₃×#124	—	2	2	no	—
25	C₁₃×#124	—	2	2	no	—
26	C₄×Q₈×C₁₃	—	2	2	no	—
27	C₄×Q₈×C₁₃	—	2	2	no	—
28	C₄×D₄×C₁₃	—	2	2	no	—
29	C₄×D₄×C₁₃	—	2	2	no	—
30	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
31	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
32	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
33	C₂×C₁₃×C₄⋊C₄	D₄	1	2	yes	—
34	C₂×C₁₃×C₂²⋊C₄	—	2	2	no	—
35	C₂×C₁₃×C₂²⋊C₄	—	2	1	no	—
36	C₂×C₁₃×C₂²⋊C₄	—	2	2	no	—
37	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄²×C₁₃	D₄	1	2	yes	—
40	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
41	C₁₃×C₄⋊C₈	—	2	4	no	—
42	C₁₃×C₄⋊C₈	—	2	2	no	—
43	C₁₃×C₄⋊C₈	—	2	2	no	—
44	C₁₃×C₂²⋊C₈	—	2	2	no	—
45	C₁₃×C₂²⋊C₈	—	2	2	no	—
46	C₁₃×#97	—	2	4	no	—
47	C₄×C₈×C₁₃	—	2	2	no	—
48	C₄×C₈×C₁₃	D₄	1	4	yes	—
49	C₁₃×#95	—	2	2	no	—
50	C₁₃×#95	—	2	1	no	—
51	C₁₃×#95	—	2	1	no	—
52	#9106	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
53	C₂⁴×C₁₃	—	2	1	no	—
54	C₂×Q₈×C₁₃	—	2	2	no	—
55	C₂×Q₈×C₁₃	—	2	1	no	—
56	C₂×Q₈×C₁₃	—	2	1	no	—
57	C₂×Q₈×C₁₃	—	2	2	no	—
58	C₂×D₄×C₁₃	—	2	2	no	—
59	C₂×D₄×C₁₃	—	2	1	no	—
60	C₂×D₄×C₁₃	—	2	2	no	—
61	C₂×D₄×C₁₃	M₁₆	1	2	yes	—
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	—	4	2	no	—
66	C₂²×C₄×C₁₃	—	8	2	no	—
67	C₂²×C₄×C₁₃	C₂²⋊C₄	1	2	yes	—
68	C₂²×C₄×C₁₃	C₂²⋊C₄	1	2	yes	—
69	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
70	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
72	C₂×C₈×C₁₃	—	2	4	no	—
73	C₂×C₈×C₁₃	—	4	2	no	—
74	C₁₃×C₄⋊C₄	—	2	2	no	—
75	C₁₃×C₄⋊C₄	—	2	4	no	—
76	C₁₃×C₄⋊C₄	—	2	2	no	—
77	C₁₃×C₄⋊C₄	—	8	2	no	—
78	C₁₃×C₂²⋊C₄	—	2	8	no	—
79	C₁₃×C₂²⋊C₄	—	4	1	no	—
80	C₁₃×C₂²⋊C₄	—	4	1	no	—
81	C₁₃×C₂²⋊C₄	—	8	2	no	—
82	C₄²×C₁₃	—	2	2	no	—
83	C₄²×C₁₃	—	2	2	no	—
84	C₄²×C₁₃	—	2	1	no	—
85	C₄²×C₁₃	—	2	2	no	—
86	C₄²×C₁₃	M₁₆	1	2	yes	—
87	C₄²×C₁₃	C₂×D₄	1	2	yes	—
88	C₄²×C₁₃	#41	1	2	yes	—
89	#2983	C₂×C₁₃	1	1	yes	characteristic, complemented
90	#2326	C₂×C₁₃	1	1	yes	characteristic
91	#1478	C₂×C₁₃	1	2	yes	—
92	#1461	C₂×C₁₃	1	1	yes	characteristic, complemented
93	#1274	C₂×C₁₃	1	2	yes	complemented
94	C₂³×C₁₃	—	2	2	no	—
95	C₂³×C₁₃	—	4	2	no	—
96	C₂³×C₁₃	—	4	2	no	—
97	C₂³×C₁₃	—	4	1	no	—
98	C₂³×C₁₃	—	4	1	no	—
99	C₂³×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
100	Q₈×C₁₃	—	4	2	no	—
101	Q₈×C₁₃	—	4	2	no	—
102	D₄×C₁₃	—	2	4	no	—
103	D₄×C₁₃	—	4	2	no	—
104	C₂×C₄×C₁₃	—	2	2	no	—
105	C₂×C₄×C₁₃	—	2	2	no	—
106	C₂×C₄×C₁₃	—	2	2	no	—
107	C₂×C₄×C₁₃	—	2	2	no	—
108	C₂×C₄×C₁₃	—	2	2	no	—
109	C₂×C₄×C₁₃	—	4	1	no	—
110	C₂×C₄×C₁₃	—	4	1	no	—
111	C₂×C₄×C₁₃	—	4	1	no	—
112	C₂×C₄×C₁₃	—	4	1	no	—
113	C₂×C₄×C₁₃	—	4	2	no	—
114	C₂×C₄×C₁₃	—	4	1	no	—
115	C₂×C₄×C₁₃	—	4	2	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	—	8	4	no	—
120	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
121	C₂×C₄×C₁₃	#120	1	1	yes	characteristic
122	C₂×C₄×C₁₃	#121	1	2	yes	—
123	C₂×C₄×C₁₃	#123	1	1	yes	characteristic
124	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
125	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
126	C₂×C₄×C₁₃	#131	1	2	yes	—
127	C₈×C₁₃	—	4	4	no	—
128	C₈×C₁₃	—	8	2	no	—
129	C₂×#124	C₄×C₁₃	1	1	yes	characteristic
130	C₂×C₄×Q₈	C₂²×C₁₃	1	1	yes	characteristic
131	C₂×C₄×D₄	C₂²×C₁₃	1	1	yes	characteristic
132	#446	—	2	4	no	—
133	#432	—	2	2	no	—
134	#431	C₂²×C₁₃	1	2	yes	—
135	#431	C₂²×C₁₃	1	2	yes	—
136	#391	C₄×C₁₃	1	1	yes	characteristic
137	#390	C₄×C₁₃	1	1	yes	characteristic
138	#388	—	2	2	no	—
139	#385	—	2	2	no	—
140	#384	C₄×C₁₃	1	1	yes	characteristic
141	#376	C₂²×C₁₃	1	1	yes	characteristic
142	D₄⋊C₈	—	2	4	no	—
143	#324	—	2	2	no	—
144	C₂²×C₁₃	—	2	1	no	—
145	C₂²×C₁₃	—	2	1	no	—
146	C₂²×C₁₃	—	4	4	no	—
147	C₂²×C₁₃	—	4	2	no	—
148	C₂²×C₁₃	—	8	2	no	—
149	C₂²×C₁₃	—	8	1	no	—
150	C₂²×C₁₃	—	8	1	no	—
151	C₂²×C₁₃	#386	1	1	yes	characteristic
152	C₂²×C₁₃	#407	1	1	yes	characteristic
153	C₂²×C₁₃	#408	1	1	yes	characteristic
154	C₄×C₁₃	—	2	2	no	—
155	C₄×C₁₃	—	2	1	no	—
156	C₄×C₁₃	—	2	1	no	—
157	C₄×C₁₃	—	4	2	no	—
158	C₄×C₁₃	—	4	2	no	—
159	C₄×C₁₃	—	4	2	no	—
160	C₄×C₁₃	—	8	1	no	—
161	C₄×C₁₃	—	8	1	no	—
162	C₄×C₁₃	#435	1	2	yes	—
163	C₄×C₁₃	#436	1	2	yes	—
164	C₂²×Q₈	C₂×C₄×C₁₃	1	1	yes	characteristic
165	C₂²×D₄	C₂×C₄×C₁₃	1	1	yes	characteristic
166	C₂³×C₄	—	2	1	no	—
167	#124	—	2	2	no	—
168	#124	—	2	2	no	—
169	C₄×Q₈	—	2	2	no	—
170	C₄×Q₈	—	2	2	no	—
171	C₄×D₄	—	2	2	no	—
172	C₄×D₄	—	2	2	no	—
173	C₂×C₄⋊C₄	—	2	2	no	—
174	C₂×C₄⋊C₄	C₂×C₄×C₁₃	1	1	yes	characteristic
175	C₂×C₄⋊C₄	C₂×C₄×C₁₃	1	1	yes	characteristic
176	C₂×C₄⋊C₄	D₄×C₁₃	1	2	yes	—
177	C₂×C₂²⋊C₄	—	2	2	no	—
178	C₂×C₂²⋊C₄	—	2	1	no	—
179	C₂×C₂²⋊C₄	—	2	2	no	—
180	C₂×C₄²	C₂×C₄×C₁₃	1	1	yes	characteristic
181	C₂×C₄²	C₂×C₄×C₁₃	1	1	yes	characteristic
182	C₂×C₄²	D₄×C₁₃	1	2	yes	—
183	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
184	C₄⋊C₈	—	2	4	no	—
185	C₄⋊C₈	—	2	2	no	—
186	C₄⋊C₈	—	2	2	no	—
187	C₂²⋊C₈	—	2	2	no	—
188	C₂²⋊C₈	—	2	2	no	—
189	#97	—	2	4	no	—
190	C₄×C₈	—	2	2	no	—
191	C₄×C₈	D₄×C₁₃	1	4	yes	—
192	#95	—	2	2	no	—
193	#95	—	2	1	no	—
194	#95	—	2	1	no	—
195	C₂×C₁₃	—	2	1	no	—
196	C₂×C₁₃	—	2	1	no	—
197	C₂×C₁₃	—	8	2	no	—
198	C₂×C₁₃	#1870	1	1	yes	characteristic
199	C₂×C₁₃	#1910	1	1	yes	characteristic
200	C₂×C₁₃	#1912	1	1	yes	characteristic
201	C₂⁴	—	2	1	no	—
202	C₂×Q₈	—	2	2	no	—
203	C₂×Q₈	—	2	1	no	—
204	C₂×Q₈	—	2	1	no	—
205	C₂×Q₈	—	2	2	no	—
206	C₂×D₄	—	2	2	no	—
207	C₂×D₄	—	2	1	no	—
208	C₂×D₄	—	2	2	no	—
209	C₂×D₄	C₁₃×M₁₆	1	2	yes	—
210	C₂²×C₄	—	2	1	no	—
211	C₂²×C₄	—	2	1	no	—
212	C₂²×C₄	—	2	1	no	—
213	C₂²×C₄	—	4	2	no	—
214	C₂²×C₄	—	8	2	no	—
215	C₂²×C₄	C₁₃×C₂²⋊C₄	1	2	yes	—
216	C₂²×C₄	C₁₃×C₂²⋊C₄	1	2	yes	—
217	C₂²×C₄	C₂²×C₄×C₁₃	1	1	yes	characteristic
218	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
219	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
220	C₂×C₈	—	2	4	no	—
221	C₂×C₈	—	4	2	no	—
222	C₄⋊C₄	—	2	2	no	—
223	C₄⋊C₄	—	2	4	no	—
224	C₄⋊C₄	—	2	2	no	—
225	C₄⋊C₄	—	8	2	no	—
226	C₂²⋊C₄	—	2	8	no	—
227	C₂²⋊C₄	—	4	1	no	—
228	C₂²⋊C₄	—	4	1	no	—
229	C₂²⋊C₄	—	8	2	no	—
230	C₄²	—	2	2	no	—
231	C₄²	—	2	2	no	—
232	C₄²	—	2	1	no	—
233	C₄²	—	2	2	no	—
234	C₄²	C₁₃×M₁₆	1	2	yes	—
235	C₄²	C₂×D₄×C₁₃	1	2	yes	—
236	C₄²	C₁₃×#41	1	2	yes	—
237	C₁₃	#9106	1	1	yes	characteristic, direct product factor, Sylow
238	C₂³	—	2	2	no	—
239	C₂³	—	4	2	no	—
240	C₂³	—	4	2	no	—
241	C₂³	—	4	1	no	—
242	C₂³	—	4	1	no	—
243	C₂³	C₂×C₁₃×C₂²⋊C₄	1	1	yes	characteristic
244	Q₈	—	4	2	no	—
245	Q₈	—	4	2	no	—
246	D₄	—	2	4	no	—
247	D₄	—	4	2	no	—
248	C₂×C₄	—	2	2	no	—
249	C₂×C₄	—	2	2	no	—
250	C₂×C₄	—	2	2	no	—
251	C₂×C₄	—	2	2	no	—
252	C₂×C₄	—	2	2	no	—
253	C₂×C₄	—	4	1	no	—
254	C₂×C₄	—	4	1	no	—
255	C₂×C₄	—	4	1	no	—
256	C₂×C₄	—	4	1	no	—
257	C₂×C₄	—	4	2	no	—
258	C₂×C₄	—	4	1	no	—
259	C₂×C₄	—	4	2	no	—
260	C₂×C₄	—	4	1	no	—
261	C₂×C₄	—	4	1	no	—
262	C₂×C₄	—	4	1	no	—
263	C₂×C₄	—	8	4	no	—
264	C₂×C₄	C₄×D₄×C₁₃	1	2	yes	—
265	C₂×C₄	C₁₃×#120	1	1	yes	characteristic
266	C₂×C₄	C₁₃×#121	1	2	yes	—
267	C₂×C₄	C₁₃×#123	1	1	yes	characteristic
268	C₂×C₄	C₂×C₁₃×M₁₆	1	1	yes	characteristic
269	C₂×C₄	C₂×C₁₃×M₁₆	1	1	yes	characteristic
270	C₂×C₄	C₁₃×#131	1	2	yes	—
271	C₈	—	4	4	no	—
272	C₈	—	8	2	no	—
273	C₂²	—	2	1	no	—
274	C₂²	—	2	1	no	—
275	C₂²	—	4	4	no	—
276	C₂²	—	4	2	no	—
277	C₂²	—	8	2	no	—
278	C₂²	—	8	1	no	—
279	C₂²	—	8	1	no	—
280	C₂²	C₁₃×#386	1	1	yes	characteristic
281	C₂²	C₁₃×#407	1	1	yes	characteristic
282	C₂²	C₁₃×#408	1	1	yes	characteristic
283	C₄	—	2	2	no	—
284	C₄	—	2	1	no	—
285	C₄	—	2	1	no	—
286	C₄	—	4	2	no	—
287	C₄	—	4	2	no	—
288	C₄	—	4	2	no	—
289	C₄	—	8	1	no	—
290	C₄	—	8	1	no	—
291	C₄	C₁₃×#435	1	2	yes	—
292	C₄	C₁₃×#436	1	2	yes	—
293	C₂	—	2	1	no	—
294	C₂	—	2	1	no	—
295	C₂	—	8	2	no	—
296	C₂	C₁₃×#1870	1	1	yes	characteristic
297	C₂	C₁₃×#1910	1	1	yes	characteristic
298	C₂	C₁₃×#1912	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₃	#386
commutator	C₂²×C₄	C₂²×C₄×C₁₃
Frattini, cosocle	C₂×C₄²	C₂³×C₁₃
prime radical	C₂²×D₄×C₁₃	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃ ◁ C₂×C₄²×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₁₃×#2983	C₂	semidirect
4. C₁₃×#1461	C₂	semidirect
5. C₁₃×#1274	C₂	semidirect
52. #9106	C₁₃	direct (N × H)
89. #2983	C₂×C₁₃	semidirect
92. #1461	C₂×C₁₃	semidirect
93. #1274	C₂×C₁₃	semidirect
237. C₁₃	#9106	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×#2983	2
C₁₃×#2326	2
C₁₃×#1478	2
C₁₃×#1461	2
C₁₃×#1274	2
#9106	13

Subgroups — C13×#9106 #15695445

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#9106 #15695445