Subgroups — C₁₃×#2498 #13111556

Order 1664.

546 conjugacy classes of subgroups.

The subgroup lattice has 274 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₃×#123	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₃×#117	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₃×#400	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₃×#399	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₃×#398	C₂	1	1	yes	maximal, characteristic, complemented
6	C₁₃×#397	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₃×#388	C₂	1	2	yes	maximal, complemented
8	C₁₃×#387	C₂	1	2	yes	maximal, complemented
9	C₁₃×#386	C₂	1	2	yes	maximal
10	C₁₃×#380	C₂	1	2	yes	maximal, complemented
11	C₂×C₁₃×#95	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×D₄×C₁₃	C₂²	1	1	yes	characteristic
13	C₂³×C₄×C₁₃	C₂²	1	2	yes	—
14	C₂³×C₄×C₁₃	C₂²	1	1	yes	characteristic
15	C₁₃×#123	—	2	4	no	—
16	C₁₃×#117	—	2	4	no	—
17	C₂×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
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	1	yes	characteristic
22	C₂×C₁₃×C₄⋊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×C₄²×C₁₃	C₂²	1	2	yes	—
30	C₁₃×#95	C₂²	1	2	yes	—
31	C₁₃×#95	C₂²	1	2	yes	—
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	1	yes	characteristic
39	C₂×D₄×C₁₃	—	2	2	no	—
40	C₂×D₄×C₁₃	—	2	4	no	—
41	C₂²×C₄×C₁₃	—	2	4	no	—
42	C₂²×C₄×C₁₃	—	2	4	no	—
43	C₂²×C₄×C₁₃	—	2	4	no	—
44	C₂²×C₄×C₁₃	—	4	2	no	—
45	C₂²×C₄×C₁₃	D₄	1	4	yes	—
46	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
47	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
48	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
49	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
50	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
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	2	yes	—
55	C₁₃×C₄⋊C₄	—	2	2	no	—
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	4	no	—
60	C₁₃×C₄⋊C₄	—	4	2	no	—
61	C₁₃×C₂²⋊C₄	—	2	4	no	—
62	C₁₃×C₂²⋊C₄	—	2	2	no	—
63	C₁₃×C₂²⋊C₄	—	2	2	no	—
64	C₁₃×C₂²⋊C₄	—	2	2	no	—
65	C₁₃×C₂²⋊C₄	—	2	4	no	—
66	C₁₃×C₂²⋊C₄	—	4	2	no	—
67	C₄²×C₁₃	—	2	4	no	—
68	#2498	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
69	C₂³×C₁₃	—	2	2	no	—
70	C₂³×C₁₃	—	2	1	no	—
71	C₂³×C₁₃	—	2	1	no	—
72	C₂³×C₁₃	—	2	1	no	—
73	C₂³×C₁₃	—	2	2	no	—
74	C₂³×C₁₃	—	2	2	no	—
75	C₂³×C₁₃	—	2	1	no	—
76	C₂³×C₁₃	—	2	1	no	—
77	C₂³×C₁₃	C₂×D₄	1	2	yes	—
78	C₂³×C₁₃	#41	1	4	yes	—
79	C₂³×C₁₃	C₂⁴	1	1	yes	characteristic
80	D₄×C₁₃	—	4	4	no	—
81	C₂×C₄×C₁₃	—	2	8	no	—
82	C₂×C₄×C₁₃	—	2	2	no	—
83	C₂×C₄×C₁₃	—	2	2	no	—
84	C₂×C₄×C₁₃	—	2	2	no	—
85	C₂×C₄×C₁₃	—	2	2	no	—
86	C₂×C₄×C₁₃	—	2	2	no	—
87	C₂×C₄×C₁₃	—	2	2	no	—
88	C₂×C₄×C₁₃	—	2	1	no	—
89	C₂×C₄×C₁₃	—	2	1	no	—
90	C₂×C₄×C₁₃	—	2	1	no	—
91	C₂×C₄×C₁₃	—	2	2	no	—
92	C₂×C₄×C₁₃	—	2	2	no	—
93	C₂×C₄×C₁₃	—	2	2	no	—
94	C₂×C₄×C₁₃	—	2	2	no	—
95	C₂×C₄×C₁₃	—	2	2	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	1	no	—
100	C₂×C₄×C₁₃	—	2	2	no	—
101	C₂×C₄×C₁₃	—	2	1	no	—
102	C₂×C₄×C₁₃	—	4	8	no	—
103	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
104	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
105	C₂×C₄×C₁₃	#41	1	4	yes	—
106	C₂×#123	C₂×C₁₃	1	1	yes	characteristic
107	C₂×#117	C₂×C₁₃	1	1	yes	characteristic, complemented
108	#400	C₂×C₁₃	1	1	yes	characteristic, complemented
109	#399	C₂×C₁₃	1	1	yes	characteristic, complemented
110	#398	C₂×C₁₃	1	1	yes	characteristic, complemented
111	#397	C₂×C₁₃	1	1	yes	characteristic, complemented
112	#388	C₂×C₁₃	1	2	yes	complemented
113	#387	C₂×C₁₃	1	2	yes	complemented
114	#386	C₂×C₁₃	1	2	yes	—
115	#380	C₂×C₁₃	1	2	yes	complemented
116	C₂×#95	C₂×C₁₃	1	1	yes	characteristic, complemented
117	C₂²×C₁₃	—	2	2	no	—
118	C₂²×C₁₃	—	2	2	no	—
119	C₂²×C₁₃	—	2	4	no	—
120	C₂²×C₁₃	—	2	4	no	—
121	C₂²×C₁₃	—	4	1	no	—
122	C₂²×C₁₃	—	4	1	no	—
123	C₂²×C₁₃	—	4	1	no	—
124	C₂²×C₁₃	—	4	2	no	—
125	C₂²×C₁₃	—	4	2	no	—
126	C₂²×C₁₃	#123	1	4	yes	—
127	C₂²×C₁₃	C₂²×D₄	1	1	yes	characteristic
128	C₂²×C₁₃	C₂×#41	1	2	yes	—
129	C₂²×C₁₃	C₂×#41	1	2	yes	—
130	C₂²×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
131	C₂²×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
132	C₄×C₁₃	—	2	2	no	—
133	C₄×C₁₃	—	2	2	no	—
134	C₄×C₁₃	—	2	4	no	—
135	C₄×C₁₃	—	4	1	no	—
136	C₄×C₁₃	—	4	2	no	—
137	C₄×C₁₃	—	4	1	no	—
138	C₄×C₁₃	—	4	2	no	—
139	C₄×C₁₃	—	4	1	no	—
140	C₄×C₁₃	—	4	2	no	—
141	C₂²×D₄	C₂²×C₁₃	1	1	yes	characteristic
142	C₂³×C₄	C₂²×C₁₃	1	2	yes	—
143	C₂³×C₄	C₂²×C₁₃	1	1	yes	characteristic
144	#123	—	2	4	no	—
145	#117	—	2	4	no	—
146	C₂×C₄⋊C₄	C₂²×C₁₃	1	1	yes	characteristic
147	C₂×C₄⋊C₄	C₂²×C₁₃	1	1	yes	characteristic
148	C₂×C₄⋊C₄	C₂²×C₁₃	1	1	yes	characteristic
149	C₂×C₄⋊C₄	C₂²×C₁₃	1	1	yes	characteristic
150	C₂×C₄⋊C₄	C₂²×C₁₃	1	2	yes	—
151	C₂×C₄⋊C₄	C₂²×C₁₃	1	2	yes	—
152	C₂×C₂²⋊C₄	C₂²×C₁₃	1	2	yes	—
153	C₂×C₂²⋊C₄	C₂²×C₁₃	1	1	yes	characteristic
154	C₂×C₂²⋊C₄	C₂²×C₁₃	1	1	yes	characteristic
155	C₂×C₂²⋊C₄	C₂²×C₁₃	1	1	yes	characteristic
156	C₂×C₂²⋊C₄	C₂²×C₁₃	1	2	yes	—
157	C₂×C₂²⋊C₄	C₂²×C₁₃	1	2	yes	—
158	C₂×C₄²	C₂²×C₁₃	1	2	yes	—
159	#95	C₂²×C₁₃	1	2	yes	—
160	#95	C₂²×C₁₃	1	1	yes	characteristic
161	#95	C₂²×C₁₃	1	1	yes	characteristic
162	#95	C₂²×C₁₃	1	2	yes	—
163	#95	C₂²×C₁₃	1	2	yes	—
164	#95	C₂²×C₁₃	1	2	yes	—
165	#95	C₂²×C₁₃	1	2	yes	—
166	C₂×C₁₃	—	2	4	no	—
167	C₂×C₁₃	—	8	1	no	—
168	C₂×C₁₃	C₂×#123	1	1	yes	characteristic
169	C₂×C₁₃	#532	1	1	yes	characteristic
170	C₂×C₁₃	#537	1	1	yes	characteristic
171	C₂×C₁₃	#551	1	2	yes	—
172	C₂×C₁₃	#552	1	2	yes	—
173	C₂⁴	C₂³×C₁₃	1	1	yes	characteristic
174	C₂⁴	C₂³×C₁₃	1	1	yes	characteristic
175	C₂×D₄	—	2	2	no	—
176	C₂×D₄	—	2	4	no	—
177	C₂²×C₄	—	2	4	no	—
178	C₂²×C₄	—	2	4	no	—
179	C₂²×C₄	—	2	4	no	—
180	C₂²×C₄	—	4	2	no	—
181	C₂²×C₄	D₄×C₁₃	1	4	yes	—
182	C₂²×C₄	C₂³×C₁₃	1	2	yes	—
183	C₂²×C₄	C₂³×C₁₃	1	1	yes	characteristic
184	C₂²×C₄	C₂³×C₁₃	1	2	yes	—
185	C₂²×C₄	C₂³×C₁₃	1	2	yes	—
186	C₂²×C₄	C₂³×C₁₃	1	1	yes	characteristic
187	C₂²×C₄	C₂³×C₁₃	1	1	yes	characteristic
188	C₂²×C₄	C₂³×C₁₃	1	1	yes	characteristic
189	C₂²×C₄	C₂³×C₁₃	1	1	yes	characteristic
190	C₂²×C₄	C₂³×C₁₃	1	2	yes	—
191	C₄⋊C₄	—	2	2	no	—
192	C₄⋊C₄	—	2	2	no	—
193	C₄⋊C₄	—	2	2	no	—
194	C₄⋊C₄	—	2	2	no	—
195	C₄⋊C₄	—	2	4	no	—
196	C₄⋊C₄	—	4	2	no	—
197	C₂²⋊C₄	—	2	4	no	—
198	C₂²⋊C₄	—	2	2	no	—
199	C₂²⋊C₄	—	2	2	no	—
200	C₂²⋊C₄	—	2	2	no	—
201	C₂²⋊C₄	—	2	4	no	—
202	C₂²⋊C₄	—	4	2	no	—
203	C₄²	—	2	4	no	—
204	C₁₃	#2498	1	1	yes	characteristic, direct product factor, Sylow
205	C₂³	—	2	2	no	—
206	C₂³	—	2	1	no	—
207	C₂³	—	2	1	no	—
208	C₂³	—	2	1	no	—
209	C₂³	—	2	2	no	—
210	C₂³	—	2	2	no	—
211	C₂³	—	2	1	no	—
212	C₂³	—	2	1	no	—
213	C₂³	C₂×D₄×C₁₃	1	2	yes	—
214	C₂³	C₁₃×#41	1	4	yes	—
215	C₂³	C₂⁴×C₁₃	1	1	yes	characteristic
216	D₄	—	4	4	no	—
217	C₂×C₄	—	2	8	no	—
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	—	2	2	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	2	1	no	—
226	C₂×C₄	—	2	1	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	2	2	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	2	no	—
235	C₂×C₄	—	2	1	no	—
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	2	1	no	—
238	C₂×C₄	—	4	8	no	—
239	C₂×C₄	C₂×D₄×C₁₃	1	2	yes	—
240	C₂×C₄	C₂×D₄×C₁₃	1	2	yes	—
241	C₂×C₄	C₁₃×#41	1	4	yes	—
242	C₂²	—	2	2	no	—
243	C₂²	—	2	2	no	—
244	C₂²	—	2	4	no	—
245	C₂²	—	2	4	no	—
246	C₂²	—	4	1	no	—
247	C₂²	—	4	1	no	—
248	C₂²	—	4	1	no	—
249	C₂²	—	4	2	no	—
250	C₂²	—	4	2	no	—
251	C₂²	C₁₃×#123	1	4	yes	—
252	C₂²	C₂²×D₄×C₁₃	1	1	yes	characteristic
253	C₂²	C₂×C₁₃×#41	1	2	yes	—
254	C₂²	C₂×C₁₃×#41	1	2	yes	—
255	C₂²	C₁₃×H(5,𝔽₂)	1	1	yes	characteristic
256	C₂²	C₁₃×2₋¹⁺⁴	1	1	yes	characteristic
257	C₄	—	2	2	no	—
258	C₄	—	2	2	no	—
259	C₄	—	2	4	no	—
260	C₄	—	4	1	no	—
261	C₄	—	4	2	no	—
262	C₄	—	4	1	no	—
263	C₄	—	4	2	no	—
264	C₄	—	4	1	no	—
265	C₄	—	4	2	no	—
266	C₂	—	2	4	no	—
267	C₂	—	8	1	no	—
268	C₂	C₂×C₁₃×#123	1	1	yes	characteristic
269	C₂	C₁₃×#532	1	1	yes	characteristic
270	C₂	C₁₃×#537	1	1	yes	characteristic
271	C₂	C₁₃×#551	1	2	yes	—
272	C₂	C₁₃×#552	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₁₃	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₁₃
prime radical	C₂²×D₄×C₁₃	C₂²
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
2. C₂×C₁₃×#117	C₂	semidirect
3. C₁₃×#400	C₂	semidirect
4. C₁₃×#399	C₂	semidirect
5. C₁₃×#398	C₂	semidirect
6. C₁₃×#397	C₂	semidirect
7. C₁₃×#388	C₂	semidirect
8. C₁₃×#387	C₂	semidirect
10. C₁₃×#380	C₂	semidirect
11. C₂×C₁₃×#95	C₂	semidirect
68. #2498	C₁₃	direct (N × H)
107. C₂×#117	C₂×C₁₃	semidirect
108. #400	C₂×C₁₃	semidirect
109. #399	C₂×C₁₃	semidirect
110. #398	C₂×C₁₃	semidirect
111. #397	C₂×C₁₃	semidirect
112. #388	C₂×C₁₃	semidirect
113. #387	C₂×C₁₃	semidirect
115. #380	C₂×C₁₃	semidirect
116. C₂×#95	C₂×C₁₃	semidirect
204. C₁₃	#2498	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×#123	2
C₂×C₁₃×#117	2
C₁₃×#400	2
C₁₃×#399	2
C₁₃×#398	2
C₁₃×#397	2
C₁₃×#388	2
C₁₃×#387	2
C₁₃×#386	2
C₁₃×#380	2
C₂×C₁₃×#95	2
#2498	13

Subgroups — C13×#2498 #13111556

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#2498 #13111556