Subgroups — C₁₃×#2664 #13111722

Order 1664.

434 conjugacy classes of subgroups.

The subgroup lattice has 248 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
2	C₂×C₁₃×#125	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
5	C₁₃×#398	C₂	1	1	yes	maximal, characteristic, complemented
6	C₁₃×#398	C₂	1	2	yes	maximal, complemented
7	C₁₃×#397	C₂	1	1	yes	maximal, characteristic
8	C₁₃×#396	C₂	1	2	yes	maximal
9	C₁₃×#389	C₂	1	2	yes	maximal, complemented
10	C₁₃×#388	C₂	1	2	yes	maximal
11	C₄×C₁₃×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
12	C₁₃×#126	—	2	4	no	—
13	C₁₃×#125	—	2	4	no	—
14	C₂×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
15	C₂×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
16	C₂×C₁₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
17	C₂×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
18	C₂×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
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	1	yes	characteristic
23	C₂×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
24	C₂×C₁₃×C₂²⋊C₄	C₂²	1	1	yes	characteristic
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	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
31	C₁₃×#95	C₂²	1	1	yes	characteristic
32	C₁₃×#95	C₂²	1	1	yes	characteristic
33	C₁₃×#95	C₂²	1	2	yes	—
34	C₁₃×#95	C₂²	1	2	yes	—
35	C₁₃×#95	C₂²	1	2	yes	—
36	C₁₃×#95	C₂²	1	2	yes	—
37	C₂⁴×C₁₃	C₂³	1	1	yes	characteristic
38	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
39	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
40	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
41	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
42	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
43	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
44	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
45	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
46	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
47	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
48	C₁₃×C₄⋊C₄	—	2	4	no	—
49	C₁₃×C₄⋊C₄	—	2	4	no	—
50	C₁₃×C₄⋊C₄	—	2	2	no	—
51	C₁₃×C₄⋊C₄	—	2	2	no	—
52	C₁₃×C₄⋊C₄	—	2	2	no	—
53	C₁₃×C₄⋊C₄	—	2	2	no	—
54	C₁₃×C₄⋊C₄	—	2	2	no	—
55	C₁₃×C₄⋊C₄	—	2	4	no	—
56	C₁₃×C₄⋊C₄	—	2	4	no	—
57	C₁₃×C₄⋊C₄	—	4	2	no	—
58	C₁₃×C₂²⋊C₄	—	2	2	no	—
59	C₁₃×C₂²⋊C₄	—	2	2	no	—
60	C₁₃×C₂²⋊C₄	—	2	2	no	—
61	C₁₃×C₂²⋊C₄	—	2	4	no	—
62	C₁₃×C₂²⋊C₄	—	4	2	no	—
63	C₄²×C₁₃	—	4	2	no	—
64	C₄²×C₁₃	D₄	1	4	yes	—
65	#2664	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
66	C₂³×C₁₃	—	2	1	no	—
67	C₂³×C₁₃	—	2	2	no	—
68	C₂³×C₁₃	—	2	1	no	—
69	C₂³×C₁₃	—	2	2	no	—
70	C₂³×C₁₃	—	2	1	no	—
71	C₂³×C₁₃	C₂⁴	1	1	yes	characteristic
72	C₂×C₄×C₁₃	—	2	2	no	—
73	C₂×C₄×C₁₃	—	2	1	no	—
74	C₂×C₄×C₁₃	—	2	2	no	—
75	C₂×C₄×C₁₃	—	2	2	no	—
76	C₂×C₄×C₁₃	—	2	1	no	—
77	C₂×C₄×C₁₃	—	2	2	no	—
78	C₂×C₄×C₁₃	—	2	1	no	—
79	C₂×C₄×C₁₃	—	2	1	no	—
80	C₂×C₄×C₁₃	—	2	1	no	—
81	C₂×C₄×C₁₃	—	2	2	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	2	no	—
89	C₂×C₄×C₁₃	—	2	1	no	—
90	C₂×C₄×C₁₃	—	2	2	no	—
91	C₂×C₄×C₁₃	—	2	1	no	—
92	C₂×C₄×C₁₃	—	2	2	no	—
93	C₂×C₄×C₁₃	—	2	2	no	—
94	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
95	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
96	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
97	C₂×C₄×C₁₃	#41	1	4	yes	—
98	C₂×#126	C₂×C₁₃	1	1	yes	characteristic
99	C₂×#125	C₂×C₁₃	1	1	yes	characteristic, complemented
100	#400	C₂×C₁₃	1	1	yes	characteristic, complemented
101	#399	C₂×C₁₃	1	1	yes	characteristic
102	#398	C₂×C₁₃	1	1	yes	characteristic, complemented
103	#398	C₂×C₁₃	1	2	yes	complemented
104	#397	C₂×C₁₃	1	1	yes	characteristic
105	#396	C₂×C₁₃	1	2	yes	—
106	#389	C₂×C₁₃	1	2	yes	complemented
107	#388	C₂×C₁₃	1	2	yes	—
108	C₄×C₄⋊C₄	C₂×C₁₃	1	1	yes	characteristic, complemented
109	C₂²×C₁₃	—	4	1	no	—
110	C₂²×C₁₃	—	4	1	no	—
111	C₂²×C₁₃	—	4	1	no	—
112	C₂²×C₁₃	—	4	2	no	—
113	C₂²×C₁₃	—	4	2	no	—
114	C₂²×C₁₃	C₂²×D₄	1	1	yes	characteristic
115	C₂²×C₁₃	C₂×#41	1	2	yes	—
116	C₂²×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
117	C₂²×C₁₃	2₋¹⁺⁴	1	2	yes	—
118	C₂²×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
119	C₄×C₁₃	—	2	2	no	—
120	C₄×C₁₃	—	2	2	no	—
121	C₄×C₁₃	—	2	2	no	—
122	C₄×C₁₃	—	2	4	no	—
123	C₄×C₁₃	—	4	1	no	—
124	C₄×C₁₃	—	4	2	no	—
125	C₄×C₁₃	—	4	1	no	—
126	C₄×C₁₃	—	4	1	no	—
127	C₄×C₁₃	—	4	2	no	—
128	C₄×C₁₃	—	4	2	no	—
129	#126	—	2	4	no	—
130	#125	—	2	4	no	—
131	C₂×C₄⋊C₄	C₂²×C₁₃	1	2	yes	—
132	C₂×C₄⋊C₄	C₂²×C₁₃	1	2	yes	—
133	C₂×C₄⋊C₄	C₂²×C₁₃	1	1	yes	characteristic
134	C₂×C₄⋊C₄	C₂²×C₁₃	1	1	yes	characteristic
135	C₂×C₄⋊C₄	C₂²×C₁₃	1	1	yes	characteristic
136	C₂×C₄⋊C₄	C₂²×C₁₃	1	1	yes	characteristic
137	C₂×C₄⋊C₄	C₂²×C₁₃	1	1	yes	characteristic
138	C₂×C₄⋊C₄	C₂²×C₁₃	1	2	yes	—
139	C₂×C₄⋊C₄	C₂²×C₁₃	1	2	yes	—
140	C₂×C₄⋊C₄	C₂²×C₁₃	1	2	yes	—
141	C₂×C₂²⋊C₄	C₂²×C₁₃	1	1	yes	characteristic
142	C₂×C₂²⋊C₄	C₂²×C₁₃	1	1	yes	characteristic
143	C₂×C₂²⋊C₄	C₂²×C₁₃	1	1	yes	characteristic
144	C₂×C₂²⋊C₄	C₂²×C₁₃	1	2	yes	—
145	C₂×C₂²⋊C₄	C₂²×C₁₃	1	2	yes	—
146	C₂×C₄²	C₂²×C₁₃	1	2	yes	—
147	C₂×C₄²	C₂²×C₁₃	1	1	yes	characteristic
148	#95	C₂²×C₁₃	1	1	yes	characteristic
149	#95	C₂²×C₁₃	1	1	yes	characteristic
150	#95	C₂²×C₁₃	1	2	yes	—
151	#95	C₂²×C₁₃	1	2	yes	—
152	#95	C₂²×C₁₃	1	2	yes	—
153	#95	C₂²×C₁₃	1	2	yes	—
154	C₂×C₁₃	—	8	1	no	—
155	C₂×C₁₃	#532	1	1	yes	characteristic
156	C₂×C₁₃	#536	1	1	yes	characteristic
157	C₂×C₁₃	#537	1	1	yes	characteristic
158	C₂×C₁₃	#539	1	2	yes	—
159	C₂×C₁₃	#541	1	2	yes	—
160	C₂⁴	C₂³×C₁₃	1	1	yes	characteristic
161	C₂²×C₄	C₂³×C₁₃	1	2	yes	—
162	C₂²×C₄	C₂³×C₁₃	1	1	yes	characteristic
163	C₂²×C₄	C₂³×C₁₃	1	1	yes	characteristic
164	C₂²×C₄	C₂³×C₁₃	1	1	yes	characteristic
165	C₂²×C₄	C₂³×C₁₃	1	2	yes	—
166	C₂²×C₄	C₂³×C₁₃	1	2	yes	—
167	C₂²×C₄	C₂³×C₁₃	1	1	yes	characteristic
168	C₂²×C₄	C₂³×C₁₃	1	1	yes	characteristic
169	C₂²×C₄	C₂³×C₁₃	1	1	yes	characteristic
170	C₂²×C₄	C₂³×C₁₃	1	2	yes	—
171	C₄⋊C₄	—	2	4	no	—
172	C₄⋊C₄	—	2	4	no	—
173	C₄⋊C₄	—	2	2	no	—
174	C₄⋊C₄	—	2	2	no	—
175	C₄⋊C₄	—	2	2	no	—
176	C₄⋊C₄	—	2	2	no	—
177	C₄⋊C₄	—	2	2	no	—
178	C₄⋊C₄	—	2	4	no	—
179	C₄⋊C₄	—	2	4	no	—
180	C₄⋊C₄	—	4	2	no	—
181	C₂²⋊C₄	—	2	2	no	—
182	C₂²⋊C₄	—	2	2	no	—
183	C₂²⋊C₄	—	2	2	no	—
184	C₂²⋊C₄	—	2	4	no	—
185	C₂²⋊C₄	—	4	2	no	—
186	C₄²	—	4	2	no	—
187	C₄²	D₄×C₁₃	1	4	yes	—
188	C₁₃	#2664	1	1	yes	characteristic, direct product factor, Sylow
189	C₂³	—	2	1	no	—
190	C₂³	—	2	2	no	—
191	C₂³	—	2	1	no	—
192	C₂³	—	2	2	no	—
193	C₂³	—	2	1	no	—
194	C₂³	C₂⁴×C₁₃	1	1	yes	characteristic
195	C₂×C₄	—	2	2	no	—
196	C₂×C₄	—	2	1	no	—
197	C₂×C₄	—	2	2	no	—
198	C₂×C₄	—	2	2	no	—
199	C₂×C₄	—	2	1	no	—
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	—	2	1	no	—
202	C₂×C₄	—	2	1	no	—
203	C₂×C₄	—	2	1	no	—
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	2	2	no	—
211	C₂×C₄	—	2	2	no	—
212	C₂×C₄	—	2	1	no	—
213	C₂×C₄	—	2	2	no	—
214	C₂×C₄	—	2	1	no	—
215	C₂×C₄	—	2	2	no	—
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	C₂×D₄×C₁₃	1	2	yes	—
218	C₂×C₄	C₂×D₄×C₁₃	1	2	yes	—
219	C₂×C₄	C₂×D₄×C₁₃	1	2	yes	—
220	C₂×C₄	C₁₃×#41	1	4	yes	—
221	C₂²	—	4	1	no	—
222	C₂²	—	4	1	no	—
223	C₂²	—	4	1	no	—
224	C₂²	—	4	2	no	—
225	C₂²	—	4	2	no	—
226	C₂²	C₂²×D₄×C₁₃	1	1	yes	characteristic
227	C₂²	C₂×C₁₃×#41	1	2	yes	—
228	C₂²	C₁₃×H(5,𝔽₂)	1	1	yes	characteristic
229	C₂²	C₁₃×2₋¹⁺⁴	1	2	yes	—
230	C₂²	C₁₃×2₋¹⁺⁴	1	1	yes	characteristic
231	C₄	—	2	2	no	—
232	C₄	—	2	2	no	—
233	C₄	—	2	2	no	—
234	C₄	—	2	4	no	—
235	C₄	—	4	1	no	—
236	C₄	—	4	2	no	—
237	C₄	—	4	1	no	—
238	C₄	—	4	1	no	—
239	C₄	—	4	2	no	—
240	C₄	—	4	2	no	—
241	C₂	—	8	1	no	—
242	C₂	C₁₃×#532	1	1	yes	characteristic
243	C₂	C₁₃×#536	1	1	yes	characteristic
244	C₂	C₁₃×#537	1	1	yes	characteristic
245	C₂	C₁₃×#539	1	2	yes	—
246	C₂	C₁₃×#541	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₁₃	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₁₃
prime radical	C₂⁴×C₁₃	C₂³
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×#125	C₂	semidirect
3. C₁₃×#400	C₂	semidirect
5. C₁₃×#398	C₂	semidirect
6. C₁₃×#398	C₂	semidirect
9. C₁₃×#389	C₂	semidirect
11. C₄×C₁₃×C₄⋊C₄	C₂	semidirect
65. #2664	C₁₃	direct (N × H)
99. C₂×#125	C₂×C₁₃	semidirect
100. #400	C₂×C₁₃	semidirect
102. #398	C₂×C₁₃	semidirect
103. #398	C₂×C₁₃	semidirect
106. #389	C₂×C₁₃	semidirect
108. C₄×C₄⋊C₄	C₂×C₁₃	semidirect
188. C₁₃	#2664	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×C₁₃×#125	2
C₁₃×#400	2
C₁₃×#399	2
C₁₃×#398	2
C₁₃×#397	2
C₁₃×#396	2
C₁₃×#389	2
C₁₃×#388	2
C₄×C₁₃×C₄⋊C₄	2
#2664	13

Subgroups — C13×#2664 #13111722

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#2664 #13111722