Subgroups — C₁₃×#2603 #13111661

Order 1664.

582 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₃×#126	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₄×D₄×C₁₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₃×#398	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₃×#397	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₃×#394	C₂	1	1	yes	maximal, characteristic, complemented
6	C₁₃×#392	C₂	1	1	yes	maximal, characteristic
7	C₁₃×#390	C₂	1	2	yes	maximal, complemented
8	C₁₃×#388	C₂	1	2	yes	maximal, complemented
9	C₁₃×#386	C₂	1	2	yes	maximal, complemented
10	C₁₃×#385	C₂	1	2	yes	maximal, complemented
11	C₁₃×#376	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×D₄×C₁₃	C₂²	1	1	yes	characteristic
13	C₂²×D₄×C₁₃	C₂²	1	2	yes	—
14	C₂³×C₄×C₁₃	C₂²	1	2	yes	—
15	C₁₃×#126	—	2	4	no	—
16	C₄×D₄×C₁₃	—	2	4	no	—
17	C₂×C₁₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
18	C₂×C₁₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
19	C₂×C₁₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
20	C₂×C₁₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
21	C₂×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
22	C₂×C₁₃×C₂²⋊C₄	C₂²	1	2	yes	—
23	C₂×C₁₃×C₂²⋊C₄	C₂²	1	1	yes	characteristic
24	C₂×C₁₃×C₂²⋊C₄	C₂²	1	2	yes	—
25	C₂×C₁₃×C₂²⋊C₄	C₂²	1	2	yes	—
26	C₂×C₁₃×C₂²⋊C₄	C₂²	1	1	yes	characteristic
27	C₂×C₁₃×C₂²⋊C₄	C₂²	1	1	yes	characteristic
28	C₂×C₁₃×C₂²⋊C₄	C₂²	1	2	yes	—
29	C₂×C₁₃×C₂²⋊C₄	C₂²	1	2	yes	—
30	C₂×C₄²×C₁₃	C₂²	1	2	yes	—
31	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
32	C₁₃×#95	C₂²	1	2	yes	—
33	C₁₃×#95	C₂²	1	2	yes	—
34	C₁₃×#95	C₂²	1	2	yes	—
35	C₁₃×#95	C₂²	1	1	yes	characteristic
36	C₁₃×#95	C₂²	1	1	yes	characteristic
37	C₂⁴×C₁₃	C₂³	1	1	yes	characteristic
38	C₂⁴×C₁₃	C₂³	1	2	yes	—
39	C₂×D₄×C₁₃	—	2	4	no	—
40	C₂×D₄×C₁₃	—	2	4	no	—
41	C₂×D₄×C₁₃	—	2	2	no	—
42	C₂×D₄×C₁₃	—	2	4	no	—
43	C₂×D₄×C₁₃	—	4	2	no	—
44	C₂²×C₄×C₁₃	—	2	4	no	—
45	C₂²×C₄×C₁₃	—	4	2	no	—
46	C₂²×C₄×C₁₃	—	4	2	no	—
47	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
48	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
49	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
50	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
51	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
52	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
53	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
54	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
55	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
56	C₁₃×C₄⋊C₄	—	2	2	no	—
57	C₁₃×C₄⋊C₄	—	2	2	no	—
58	C₁₃×C₄⋊C₄	—	2	2	no	—
59	C₁₃×C₄⋊C₄	—	2	2	no	—
60	C₁₃×C₄⋊C₄	—	4	2	no	—
61	C₁₃×C₂²⋊C₄	—	2	4	no	—
62	C₁₃×C₂²⋊C₄	—	2	4	no	—
63	C₁₃×C₂²⋊C₄	—	2	2	no	—
64	C₁₃×C₂²⋊C₄	—	2	2	no	—
65	C₁₃×C₂²⋊C₄	—	2	2	no	—
66	C₁₃×C₂²⋊C₄	—	2	4	no	—
67	C₁₃×C₂²⋊C₄	—	4	2	no	—
68	C₁₃×C₂²⋊C₄	—	4	2	no	—
69	C₄²×C₁₃	—	2	4	no	—
70	C₄²×C₁₃	D₄	1	4	yes	complemented
71	#2603	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
72	C₂³×C₁₃	—	2	2	no	—
73	C₂³×C₁₃	—	2	2	no	—
74	C₂³×C₁₃	—	2	2	no	—
75	C₂³×C₁₃	—	2	2	no	—
76	C₂³×C₁₃	—	2	2	no	—
77	C₂³×C₁₃	—	2	2	no	—
78	C₂³×C₁₃	—	2	1	no	—
79	C₂³×C₁₃	—	2	2	no	—
80	C₂³×C₁₃	—	2	2	no	—
81	C₂³×C₁₃	—	2	1	no	—
82	C₂³×C₁₃	—	2	1	no	—
83	C₂³×C₁₃	#41	1	4	yes	—
84	C₂³×C₁₃	C₂⁴	1	1	yes	characteristic
85	D₄×C₁₃	—	2	8	no	—
86	D₄×C₁₃	—	4	8	no	—
87	C₂×C₄×C₁₃	—	2	2	no	—
88	C₂×C₄×C₁₃	—	2	2	no	—
89	C₂×C₄×C₁₃	—	2	2	no	—
90	C₂×C₄×C₁₃	—	2	2	no	—
91	C₂×C₄×C₁₃	—	2	1	no	—
92	C₂×C₄×C₁₃	—	2	1	no	—
93	C₂×C₄×C₁₃	—	2	1	no	—
94	C₂×C₄×C₁₃	—	2	1	no	—
95	C₂×C₄×C₁₃	—	2	1	no	—
96	C₂×C₄×C₁₃	—	2	2	no	—
97	C₂×C₄×C₁₃	—	2	2	no	—
98	C₂×C₄×C₁₃	—	2	2	no	—
99	C₂×C₄×C₁₃	—	2	2	no	—
100	C₂×C₄×C₁₃	—	2	2	no	—
101	C₂×C₄×C₁₃	—	2	1	no	—
102	C₂×C₄×C₁₃	—	2	2	no	—
103	C₂×C₄×C₁₃	—	2	2	no	—
104	C₂×C₄×C₁₃	—	2	1	no	—
105	C₂×C₄×C₁₃	—	2	2	no	—
106	C₂×C₄×C₁₃	—	4	8	no	—
107	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
108	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
109	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
110	C₂×C₄×C₁₃	#41	1	4	yes	—
111	C₂×#126	C₂×C₁₃	1	1	yes	characteristic, complemented
112	C₂×C₄×D₄	C₂×C₁₃	1	1	yes	characteristic, complemented
113	#398	C₂×C₁₃	1	1	yes	characteristic, complemented
114	#397	C₂×C₁₃	1	1	yes	characteristic, complemented
115	#394	C₂×C₁₃	1	1	yes	characteristic, complemented
116	#392	C₂×C₁₃	1	1	yes	characteristic
117	#390	C₂×C₁₃	1	2	yes	complemented
118	#388	C₂×C₁₃	1	2	yes	complemented
119	#386	C₂×C₁₃	1	2	yes	complemented
120	#385	C₂×C₁₃	1	2	yes	complemented
121	#376	C₂×C₁₃	1	1	yes	characteristic, complemented
122	C₂²×C₁₃	—	2	4	no	—
123	C₂²×C₁₃	—	2	4	no	—
124	C₂²×C₁₃	—	2	4	no	—
125	C₂²×C₁₃	—	4	2	no	—
126	C₂²×C₁₃	—	4	2	no	—
127	C₂²×C₁₃	—	4	2	no	—
128	C₂²×C₁₃	—	4	2	no	—
129	C₂²×C₁₃	—	4	1	no	—
130	C₂²×C₁₃	—	4	1	no	—
131	C₂²×C₁₃	—	4	1	no	—
132	C₂²×C₁₃	—	4	2	no	—
133	C₂²×C₁₃	—	4	2	no	—
134	C₂²×C₁₃	C₂²×D₄	1	1	yes	characteristic
135	C₂²×C₁₃	C₂×#41	1	2	yes	—
136	C₂²×C₁₃	C₂×#41	1	2	yes	—
137	C₂²×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
138	C₂²×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
139	C₄×C₁₃	—	2	2	no	—
140	C₄×C₁₃	—	2	2	no	—
141	C₄×C₁₃	—	2	2	no	—
142	C₄×C₁₃	—	2	4	no	—
143	C₄×C₁₃	—	4	1	no	—
144	C₄×C₁₃	—	4	1	no	—
145	C₄×C₁₃	—	4	1	no	—
146	C₄×C₁₃	—	4	2	no	—
147	C₄×C₁₃	—	4	2	no	—
148	C₂²×D₄	C₂²×C₁₃	1	1	yes	characteristic
149	C₂²×D₄	C₂²×C₁₃	1	2	yes	—
150	C₂³×C₄	C₂²×C₁₃	1	2	yes	—
151	#126	—	2	4	no	—
152	C₄×D₄	—	2	4	no	—
153	C₂×C₄⋊C₄	C₂²×C₁₃	1	1	yes	characteristic
154	C₂×C₄⋊C₄	C₂²×C₁₃	1	1	yes	characteristic
155	C₂×C₄⋊C₄	C₂²×C₁₃	1	1	yes	characteristic
156	C₂×C₄⋊C₄	C₂²×C₁₃	1	1	yes	characteristic
157	C₂×C₄⋊C₄	C₂²×C₁₃	1	2	yes	—
158	C₂×C₂²⋊C₄	C₂²×C₁₃	1	2	yes	—
159	C₂×C₂²⋊C₄	C₂²×C₁₃	1	1	yes	characteristic
160	C₂×C₂²⋊C₄	C₂²×C₁₃	1	1	yes	characteristic
161	C₂×C₂²⋊C₄	C₂²×C₁₃	1	1	yes	characteristic
162	C₂×C₂²⋊C₄	C₂²×C₁₃	1	2	yes	—
163	C₂×C₂²⋊C₄	C₂²×C₁₃	1	2	yes	—
164	C₂×C₂²⋊C₄	C₂²×C₁₃	1	2	yes	—
165	C₂×C₂²⋊C₄	C₂²×C₁₃	1	2	yes	—
166	C₂×C₄²	C₂²×C₁₃	1	2	yes	—
167	C₂×C₄²	C₂²×C₁₃	1	1	yes	characteristic
168	#95	C₂²×C₁₃	1	2	yes	—
169	#95	C₂²×C₁₃	1	1	yes	characteristic
170	#95	C₂²×C₁₃	1	1	yes	characteristic
171	#95	C₂²×C₁₃	1	2	yes	—
172	#95	C₂²×C₁₃	1	2	yes	—
173	C₂×C₁₃	—	4	4	no	—
174	C₂×C₁₃	—	8	1	no	—
175	C₂×C₁₃	#525	1	1	yes	characteristic
176	C₂×C₁₃	#532	1	1	yes	characteristic
177	C₂×C₁₃	#535	1	1	yes	characteristic
178	C₂×C₁₃	#551	1	2	yes	—
179	C₂×C₁₃	#553	1	2	yes	—
180	C₂⁴	C₂³×C₁₃	1	1	yes	characteristic
181	C₂⁴	C₂³×C₁₃	1	2	yes	—
182	C₂×D₄	—	2	4	no	—
183	C₂×D₄	—	2	4	no	—
184	C₂×D₄	—	2	2	no	—
185	C₂×D₄	—	2	4	no	—
186	C₂×D₄	—	4	2	no	—
187	C₂²×C₄	—	2	4	no	—
188	C₂²×C₄	—	4	2	no	—
189	C₂²×C₄	—	4	2	no	—
190	C₂²×C₄	C₂³×C₁₃	1	2	yes	—
191	C₂²×C₄	C₂³×C₁₃	1	1	yes	characteristic
192	C₂²×C₄	C₂³×C₁₃	1	1	yes	characteristic
193	C₂²×C₄	C₂³×C₁₃	1	2	yes	—
194	C₂²×C₄	C₂³×C₁₃	1	1	yes	characteristic
195	C₂²×C₄	C₂³×C₁₃	1	1	yes	characteristic
196	C₂²×C₄	C₂³×C₁₃	1	1	yes	characteristic
197	C₂²×C₄	C₂³×C₁₃	1	1	yes	characteristic
198	C₂²×C₄	C₂³×C₁₃	1	2	yes	—
199	C₄⋊C₄	—	2	2	no	—
200	C₄⋊C₄	—	2	2	no	—
201	C₄⋊C₄	—	2	2	no	—
202	C₄⋊C₄	—	2	2	no	—
203	C₄⋊C₄	—	4	2	no	—
204	C₂²⋊C₄	—	2	4	no	—
205	C₂²⋊C₄	—	2	4	no	—
206	C₂²⋊C₄	—	2	2	no	—
207	C₂²⋊C₄	—	2	2	no	—
208	C₂²⋊C₄	—	2	2	no	—
209	C₂²⋊C₄	—	2	4	no	—
210	C₂²⋊C₄	—	4	2	no	—
211	C₂²⋊C₄	—	4	2	no	—
212	C₄²	—	2	4	no	—
213	C₄²	D₄×C₁₃	1	4	yes	complemented
214	C₁₃	#2603	1	1	yes	characteristic, direct product factor, Sylow
215	C₂³	—	2	2	no	—
216	C₂³	—	2	2	no	—
217	C₂³	—	2	2	no	—
218	C₂³	—	2	2	no	—
219	C₂³	—	2	2	no	—
220	C₂³	—	2	2	no	—
221	C₂³	—	2	1	no	—
222	C₂³	—	2	2	no	—
223	C₂³	—	2	2	no	—
224	C₂³	—	2	1	no	—
225	C₂³	—	2	1	no	—
226	C₂³	C₁₃×#41	1	4	yes	—
227	C₂³	C₂⁴×C₁₃	1	1	yes	characteristic
228	D₄	—	2	8	no	—
229	D₄	—	4	8	no	—
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	2	1	no	—
235	C₂×C₄	—	2	1	no	—
236	C₂×C₄	—	2	1	no	—
237	C₂×C₄	—	2	1	no	—
238	C₂×C₄	—	2	1	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	2	2	no	—
243	C₂×C₄	—	2	2	no	—
244	C₂×C₄	—	2	1	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	2	1	no	—
248	C₂×C₄	—	2	2	no	—
249	C₂×C₄	—	4	8	no	—
250	C₂×C₄	C₂×D₄×C₁₃	1	2	yes	—
251	C₂×C₄	C₂×D₄×C₁₃	1	2	yes	—
252	C₂×C₄	C₂×D₄×C₁₃	1	2	yes	—
253	C₂×C₄	C₁₃×#41	1	4	yes	—
254	C₂²	—	2	4	no	—
255	C₂²	—	2	4	no	—
256	C₂²	—	2	4	no	—
257	C₂²	—	4	2	no	—
258	C₂²	—	4	2	no	—
259	C₂²	—	4	2	no	—
260	C₂²	—	4	2	no	—
261	C₂²	—	4	1	no	—
262	C₂²	—	4	1	no	—
263	C₂²	—	4	1	no	—
264	C₂²	—	4	2	no	—
265	C₂²	—	4	2	no	—
266	C₂²	C₂²×D₄×C₁₃	1	1	yes	characteristic
267	C₂²	C₂×C₁₃×#41	1	2	yes	—
268	C₂²	C₂×C₁₃×#41	1	2	yes	—
269	C₂²	C₁₃×H(5,𝔽₂)	1	1	yes	characteristic
270	C₂²	C₁₃×H(5,𝔽₂)	1	1	yes	characteristic
271	C₄	—	2	2	no	—
272	C₄	—	2	2	no	—
273	C₄	—	2	2	no	—
274	C₄	—	2	4	no	—
275	C₄	—	4	1	no	—
276	C₄	—	4	1	no	—
277	C₄	—	4	1	no	—
278	C₄	—	4	2	no	—
279	C₄	—	4	2	no	—
280	C₂	—	4	4	no	—
281	C₂	—	8	1	no	—
282	C₂	C₁₃×#525	1	1	yes	characteristic
283	C₂	C₁₃×#532	1	1	yes	characteristic
284	C₂	C₁₃×#535	1	1	yes	characteristic
285	C₂	C₁₃×#551	1	2	yes	—
286	C₂	C₁₃×#553	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₁₃	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₁₃
prime radical	C₁₃×#392	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₁₃×#126	C₂	semidirect
2. C₂×C₄×D₄×C₁₃	C₂	semidirect
3. C₁₃×#398	C₂	semidirect
4. C₁₃×#397	C₂	semidirect
5. C₁₃×#394	C₂	semidirect
7. C₁₃×#390	C₂	semidirect
8. C₁₃×#388	C₂	semidirect
9. C₁₃×#386	C₂	semidirect
10. C₁₃×#385	C₂	semidirect
11. C₁₃×#376	C₂	semidirect
70. C₄²×C₁₃	D₄	semidirect
71. #2603	C₁₃	direct (N × H)
111. C₂×#126	C₂×C₁₃	semidirect
112. C₂×C₄×D₄	C₂×C₁₃	semidirect
113. #398	C₂×C₁₃	semidirect
114. #397	C₂×C₁₃	semidirect
115. #394	C₂×C₁₃	semidirect
117. #390	C₂×C₁₃	semidirect
118. #388	C₂×C₁₃	semidirect
119. #386	C₂×C₁₃	semidirect
120. #385	C₂×C₁₃	semidirect
121. #376	C₂×C₁₃	semidirect
213. C₄²	D₄×C₁₃	semidirect
214. C₁₃	#2603	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×#126	2
C₂×C₄×D₄×C₁₃	2
C₁₃×#398	2
C₁₃×#397	2
C₁₃×#394	2
C₁₃×#392	2
C₁₃×#390	2
C₁₃×#388	2
C₁₃×#386	2
C₁₃×#385	2
C₁₃×#376	2
#2603	13

Subgroups — C13×#2603 #13111661

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#2603 #13111661