Subgroups — C₁₃×#20465 #15706804

Order 3328.

930 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₃×#433	C₂	1	2	yes	maximal
2	C₄×C₁₃×#117	C₂	1	1	yes	maximal, characteristic
3	C₁₃×#1768	C₂	1	2	yes	maximal, complemented
4	C₁₃×#1767	C₂	1	2	yes	maximal, complemented
5	C₁₃×#1750	C₂	1	2	yes	maximal
6	C₁₃×#1749	C₂	1	2	yes	maximal
7	C₁₃×#1746	C₂	1	2	yes	maximal, complemented
8	C₄×C₁₃×#97	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄×C₁₃×#97	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₁₃×#117	C₂²	1	1	yes	characteristic
11	C₂×C₁₃×#117	C₂²	1	1	yes	characteristic
12	C₂²×C₄²×C₁₃	C₂²	1	1	yes	characteristic
13	C₁₃×#433	C₄	1	16	yes	complemented
14	C₂×C₁₃×C₄⋊C₈	C₂²	1	4	yes	—
15	C₂×C₁₃×C₂²⋊C₈	C₂²	1	4	yes	—
16	C₂×C₁₃×#97	C₂²	1	2	yes	—
17	C₂×C₁₃×#97	C₂²	1	2	yes	—
18	C₂×C₁₃×#97	C₂²	1	2	yes	—
19	C₂×C₁₃×#97	C₂²	1	2	yes	—
20	C₂×C₄×C₈×C₁₃	C₂²	1	2	yes	—
21	C₂×C₄×C₈×C₁₃	C₂²	1	2	yes	—
22	C₄×C₁₃×C₄⋊C₄	C₄	1	4	yes	—
23	C₄×C₁₃×C₂²⋊C₄	C₄	1	4	yes	—
24	C₁₃×#376	C₂²	1	1	yes	characteristic
25	C₁₃×#376	C₂²	1	1	yes	characteristic
26	C₄³×C₁₃	C₂²	1	1	yes	characteristic
27	C₄³×C₁₃	C₂²	1	1	yes	characteristic
28	C₁₃×#336	C₂²	1	4	yes	—
29	C₁₃×#336	C₂²	1	4	yes	—
30	C₂³×C₄×C₁₃	C₂×C₄	1	2	yes	—
31	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
32	C₂²×C₈×C₁₃	C₂³	1	4	yes	—
33	C₂²×C₈×C₁₃	C₂³	1	4	yes	—
34	C₁₃×#117	—	2	4	no	—
35	C₁₃×#117	C₂×C₄	1	8	yes	—
36	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
37	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
38	C₂×C₁₃×C₂²⋊C₄	C₂×C₄	1	2	yes	—
39	C₂×C₁₃×C₂²⋊C₄	C₂×C₄	1	2	yes	—
40	C₂×C₄²×C₁₃	—	2	4	no	—
41	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
42	C₂×C₄²×C₁₃	C₂×C₄	1	4	yes	—
43	C₂×C₄²×C₁₃	C₂×C₄	1	4	yes	—
44	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
45	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
46	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
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	C₂×C₄	1	16	yes	complemented
51	C₁₃×C₂²⋊C₈	C₂×C₄	1	16	yes	—
52	C₁₃×#97	—	2	4	no	—
53	C₁₃×#97	—	2	4	no	—
54	C₁₃×#97	—	2	4	no	—
55	C₁₃×#97	C₂×C₄	1	8	yes	complemented
56	C₄×C₈×C₁₃	—	2	4	no	—
57	C₄×C₈×C₁₃	C₂×C₄	1	8	yes	complemented
58	C₁₃×#95	C₂×C₄	1	4	yes	—
59	C₁₃×#95	C₂×C₄	1	4	yes	—
60	#20465	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
61	C₂⁴×C₁₃	C₂²×C₄	1	1	yes	characteristic
62	C₂²×C₄×C₁₃	—	2	2	no	—
63	C₂²×C₄×C₁₃	—	2	2	no	—
64	C₂²×C₄×C₁₃	—	2	4	no	—
65	C₂²×C₄×C₁₃	—	4	4	no	—
66	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
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	2	yes	—
70	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
71	C₂²×C₄×C₁₃	C₂²×C₄	1	4	yes	—
72	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
73	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
74	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
75	C₂×C₈×C₁₃	—	2	4	no	—
76	C₂×C₈×C₁₃	—	2	4	no	—
77	C₂×C₈×C₁₃	—	2	4	no	—
78	C₂×C₈×C₁₃	—	2	4	no	—
79	C₂×C₈×C₁₃	C₂²×C₄	1	8	yes	—
80	C₂×C₈×C₁₃	C₂²×C₄	1	8	yes	—
81	C₁₃×C₄⋊C₄	—	2	4	no	—
82	C₁₃×C₄⋊C₄	C₄²	1	8	yes	—
83	C₁₃×C₂²⋊C₄	—	2	4	no	—
84	C₁₃×C₂²⋊C₄	C₄²	1	8	yes	—
85	C₄²×C₁₃	—	2	4	no	—
86	C₄²×C₁₃	—	2	2	no	—
87	C₄²×C₁₃	—	2	2	no	—
88	C₄²×C₁₃	—	4	4	no	—
89	C₄²×C₁₃	—	4	4	no	—
90	C₄²×C₁₃	C₂²×C₄	1	4	yes	—
91	C₄²×C₁₃	C₂²×C₄	1	4	yes	—
92	C₄²×C₁₃	#41	1	4	yes	—
93	C₄²×C₁₃	#41	1	4	yes	—
94	C₂×#433	C₂×C₁₃	1	2	yes	—
95	C₄×#117	C₂×C₁₃	1	1	yes	characteristic
96	#1768	C₂×C₁₃	1	2	yes	complemented
97	#1767	C₂×C₁₃	1	2	yes	complemented
98	#1750	C₂×C₁₃	1	2	yes	—
99	#1749	C₂×C₁₃	1	2	yes	—
100	#1746	C₂×C₁₃	1	2	yes	complemented
101	C₄×#97	C₂×C₁₃	1	1	yes	characteristic, complemented
102	C₄×#97	C₂×C₁₃	1	1	yes	characteristic, complemented
103	C₂³×C₁₃	—	2	1	no	—
104	C₂³×C₁₃	—	2	1	no	—
105	C₂³×C₁₃	—	2	4	no	—
106	C₂³×C₁₃	C₂×C₄²	1	2	yes	—
107	C₂³×C₁₃	C₂³×C₄	1	1	yes	characteristic
108	C₂×C₄×C₁₃	—	2	1	no	—
109	C₂×C₄×C₁₃	—	2	1	no	—
110	C₂×C₄×C₁₃	—	2	4	no	—
111	C₂×C₄×C₁₃	—	2	4	no	—
112	C₂×C₄×C₁₃	—	2	4	no	—
113	C₂×C₄×C₁₃	—	2	4	no	—
114	C₂×C₄×C₁₃	—	2	8	no	—
115	C₂×C₄×C₁₃	—	2	2	no	—
116	C₂×C₄×C₁₃	—	2	2	no	—
117	C₂×C₄×C₁₃	—	2	4	no	—
118	C₂×C₄×C₁₃	—	2	4	no	—
119	C₂×C₄×C₁₃	—	4	8	no	—
120	C₂×C₄×C₁₃	C₂×C₄²	1	4	yes	—
121	C₂×C₄×C₁₃	C₂×C₄²	1	4	yes	—
122	C₂×C₄×C₁₃	C₂×C₄²	1	2	yes	—
123	C₂×C₄×C₁₃	#117	1	4	yes	—
124	C₂×C₄×C₁₃	#117	1	4	yes	—
125	C₂×C₄×C₁₃	C₂³×C₄	1	2	yes	—
126	C₂×C₄×C₁₃	C₂×#41	1	2	yes	—
127	C₂×C₄×C₁₃	C₂×#41	1	2	yes	—
128	C₈×C₁₃	—	2	8	no	—
129	C₈×C₁₃	—	2	8	no	—
130	C₂×#117	C₂²×C₁₃	1	1	yes	characteristic
131	C₂×#117	C₂²×C₁₃	1	1	yes	characteristic
132	C₂²×C₄²	C₂²×C₁₃	1	1	yes	characteristic
133	#433	C₄×C₁₃	1	16	yes	complemented
134	C₂×C₄⋊C₈	C₂²×C₁₃	1	4	yes	—
135	C₂×C₂²⋊C₈	C₂²×C₁₃	1	4	yes	—
136	C₂×#97	C₂²×C₁₃	1	2	yes	—
137	C₂×#97	C₂²×C₁₃	1	2	yes	—
138	C₂×#97	C₂²×C₁₃	1	2	yes	—
139	C₂×#97	C₂²×C₁₃	1	2	yes	—
140	C₂×C₄×C₈	C₂²×C₁₃	1	2	yes	—
141	C₂×C₄×C₈	C₂²×C₁₃	1	2	yes	—
142	C₄×C₄⋊C₄	C₄×C₁₃	1	4	yes	—
143	C₄×C₂²⋊C₄	C₄×C₁₃	1	4	yes	—
144	#376	C₂²×C₁₃	1	1	yes	characteristic
145	#376	C₂²×C₁₃	1	1	yes	characteristic
146	C₄³	C₂²×C₁₃	1	1	yes	characteristic
147	C₄³	C₂²×C₁₃	1	1	yes	characteristic
148	#336	C₂²×C₁₃	1	4	yes	—
149	#336	C₂²×C₁₃	1	4	yes	—
150	C₂²×C₁₃	—	2	4	no	—
151	C₂²×C₁₃	—	2	2	no	—
152	C₂²×C₁₃	—	4	2	no	—
153	C₂²×C₁₃	—	4	2	no	—
154	C₂²×C₁₃	C₂²×C₄²	1	1	yes	characteristic
155	C₂²×C₁₃	C₂×#117	1	1	yes	characteristic
156	C₂²×C₁₃	C₂×#117	1	1	yes	characteristic
157	C₂²×C₁₃	#568	1	4	yes	—
158	C₄×C₁₃	—	2	4	no	—
159	C₄×C₁₃	—	2	4	no	—
160	C₄×C₁₃	—	2	4	no	—
161	C₄×C₁₃	—	2	4	no	—
162	C₄×C₁₃	—	2	2	no	—
163	C₄×C₁₃	—	4	2	no	—
164	C₄×C₁₃	—	4	2	no	—
165	C₄×C₁₃	C₄×#41	1	4	yes	—
166	C₂³×C₄	C₂×C₄×C₁₃	1	2	yes	—
167	C₂³×C₄	C₂³×C₁₃	1	1	yes	characteristic
168	C₂²×C₈	C₂³×C₁₃	1	4	yes	—
169	C₂²×C₈	C₂³×C₁₃	1	4	yes	—
170	#117	—	2	4	no	—
171	#117	C₂×C₄×C₁₃	1	8	yes	—
172	C₂×C₄⋊C₄	C₂×C₄×C₁₃	1	2	yes	—
173	C₂×C₄⋊C₄	C₂×C₄×C₁₃	1	2	yes	—
174	C₂×C₂²⋊C₄	C₂×C₄×C₁₃	1	2	yes	—
175	C₂×C₂²⋊C₄	C₂×C₄×C₁₃	1	2	yes	—
176	C₂×C₄²	—	2	4	no	—
177	C₂×C₄²	C₂×C₄×C₁₃	1	2	yes	—
178	C₂×C₄²	C₂×C₄×C₁₃	1	4	yes	—
179	C₂×C₄²	C₂×C₄×C₁₃	1	4	yes	—
180	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
181	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
182	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
183	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
184	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
185	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
186	C₄⋊C₈	C₂×C₄×C₁₃	1	16	yes	complemented
187	C₂²⋊C₈	C₂×C₄×C₁₃	1	16	yes	—
188	#97	—	2	4	no	—
189	#97	—	2	4	no	—
190	#97	—	2	4	no	—
191	#97	C₂×C₄×C₁₃	1	8	yes	complemented
192	C₄×C₈	—	2	4	no	—
193	C₄×C₈	C₂×C₄×C₁₃	1	8	yes	complemented
194	#95	C₂×C₄×C₁₃	1	4	yes	—
195	#95	C₂×C₄×C₁₃	1	4	yes	—
196	C₂×C₁₃	—	4	2	no	—
197	C₂×C₁₃	C₄×#117	1	1	yes	characteristic
198	C₂×C₁₃	#2875	1	2	yes	—
199	C₂×C₁₃	#2921	1	2	yes	—
200	C₂×C₁₃	#2923	1	2	yes	—
201	C₂⁴	C₂²×C₄×C₁₃	1	1	yes	characteristic
202	C₂²×C₄	—	2	2	no	—
203	C₂²×C₄	—	2	2	no	—
204	C₂²×C₄	—	2	4	no	—
205	C₂²×C₄	—	4	4	no	—
206	C₂²×C₄	C₂²×C₄×C₁₃	1	2	yes	—
207	C₂²×C₄	C₂²×C₄×C₁₃	1	2	yes	—
208	C₂²×C₄	C₂²×C₄×C₁₃	1	2	yes	—
209	C₂²×C₄	C₂²×C₄×C₁₃	1	2	yes	—
210	C₂²×C₄	C₂²×C₄×C₁₃	1	1	yes	characteristic
211	C₂²×C₄	C₂²×C₄×C₁₃	1	4	yes	—
212	C₂²×C₄	C₂²×C₄×C₁₃	1	2	yes	—
213	C₂²×C₄	C₂²×C₄×C₁₃	1	2	yes	—
214	C₂²×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
215	C₂×C₈	—	2	4	no	—
216	C₂×C₈	—	2	4	no	—
217	C₂×C₈	—	2	4	no	—
218	C₂×C₈	—	2	4	no	—
219	C₂×C₈	C₂²×C₄×C₁₃	1	8	yes	—
220	C₂×C₈	C₂²×C₄×C₁₃	1	8	yes	—
221	C₄⋊C₄	—	2	4	no	—
222	C₄⋊C₄	C₄²×C₁₃	1	8	yes	—
223	C₂²⋊C₄	—	2	4	no	—
224	C₂²⋊C₄	C₄²×C₁₃	1	8	yes	—
225	C₄²	—	2	4	no	—
226	C₄²	—	2	2	no	—
227	C₄²	—	2	2	no	—
228	C₄²	—	4	4	no	—
229	C₄²	—	4	4	no	—
230	C₄²	C₂²×C₄×C₁₃	1	4	yes	—
231	C₄²	C₂²×C₄×C₁₃	1	4	yes	—
232	C₄²	C₁₃×#41	1	4	yes	—
233	C₄²	C₁₃×#41	1	4	yes	—
234	C₁₃	#20465	1	1	yes	characteristic, direct product factor, Sylow
235	C₂³	—	2	1	no	—
236	C₂³	—	2	1	no	—
237	C₂³	—	2	4	no	—
238	C₂³	C₂×C₄²×C₁₃	1	2	yes	—
239	C₂³	C₂³×C₄×C₁₃	1	1	yes	characteristic
240	C₂×C₄	—	2	1	no	—
241	C₂×C₄	—	2	1	no	—
242	C₂×C₄	—	2	4	no	—
243	C₂×C₄	—	2	4	no	—
244	C₂×C₄	—	2	4	no	—
245	C₂×C₄	—	2	4	no	—
246	C₂×C₄	—	2	8	no	—
247	C₂×C₄	—	2	2	no	—
248	C₂×C₄	—	2	2	no	—
249	C₂×C₄	—	2	4	no	—
250	C₂×C₄	—	2	4	no	—
251	C₂×C₄	—	4	8	no	—
252	C₂×C₄	C₂×C₄²×C₁₃	1	4	yes	—
253	C₂×C₄	C₂×C₄²×C₁₃	1	4	yes	—
254	C₂×C₄	C₂×C₄²×C₁₃	1	2	yes	—
255	C₂×C₄	C₁₃×#117	1	4	yes	—
256	C₂×C₄	C₁₃×#117	1	4	yes	—
257	C₂×C₄	C₂³×C₄×C₁₃	1	2	yes	—
258	C₂×C₄	C₂×C₁₃×#41	1	2	yes	—
259	C₂×C₄	C₂×C₁₃×#41	1	2	yes	—
260	C₈	—	2	8	no	—
261	C₈	—	2	8	no	—
262	C₂²	—	2	2	no	—
263	C₂²	—	2	4	no	—
264	C₂²	—	4	2	no	—
265	C₂²	—	4	2	no	—
266	C₂²	C₂²×C₄²×C₁₃	1	1	yes	characteristic
267	C₂²	C₂×C₁₃×#117	1	1	yes	characteristic
268	C₂²	C₂×C₁₃×#117	1	1	yes	characteristic
269	C₂²	C₁₃×#568	1	4	yes	—
270	C₄	—	2	2	no	—
271	C₄	—	2	4	no	—
272	C₄	—	2	4	no	—
273	C₄	—	2	4	no	—
274	C₄	—	2	4	no	—
275	C₄	—	4	2	no	—
276	C₄	—	4	2	no	—
277	C₄	C₄×C₁₃×#41	1	4	yes	—
278	C₂	—	4	2	no	—
279	C₂	C₄×C₁₃×#117	1	1	yes	characteristic
280	C₂	C₁₃×#2875	1	2	yes	—
281	C₂	C₁₃×#2921	1	2	yes	—
282	C₂	C₁₃×#2923	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
3. C₁₃×#1768	C₂	semidirect
4. C₁₃×#1767	C₂	semidirect
7. C₁₃×#1746	C₂	semidirect
8. C₄×C₁₃×#97	C₂	semidirect
9. C₄×C₁₃×#97	C₂	semidirect
13. C₁₃×#433	C₄	semidirect
50. C₁₃×C₄⋊C₈	C₂×C₄	semidirect
55. C₁₃×#97	C₂×C₄	semidirect
57. C₄×C₈×C₁₃	C₂×C₄	semidirect
60. #20465	C₁₃	direct (N × H)
96. #1768	C₂×C₁₃	semidirect
97. #1767	C₂×C₁₃	semidirect
100. #1746	C₂×C₁₃	semidirect
101. C₄×#97	C₂×C₁₃	semidirect
102. C₄×#97	C₂×C₁₃	semidirect
133. #433	C₄×C₁₃	semidirect
186. C₄⋊C₈	C₂×C₄×C₁₃	semidirect
191. #97	C₂×C₄×C₁₃	semidirect
193. C₄×C₈	C₂×C₄×C₁₃	semidirect
234. C₁₃	#20465	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×#433	2
C₄×C₁₃×#117	2
C₁₃×#1768	2
C₁₃×#1767	2
C₁₃×#1750	2
C₁₃×#1749	2
C₁₃×#1746	2
C₄×C₁₃×#97	2
#20465	13

Subgroups — C13×#20465 #15706804

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#20465 #15706804