Subgroups — C₁₃×#20523 #15706862

Order 3328.

1450 conjugacy classes of subgroups.

The subgroup lattice has 238 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₁₃×C₂²⋊C₈	C₂	1	2	yes	maximal
2	C₂×C₄×C₁₃×C₂²⋊C₄	C₂	1	1	yes	maximal, characteristic
3	C₈×C₁₃×C₂²⋊C₄	C₂	1	8	yes	maximal, complemented
4	C₄×C₁₃×C₂²⋊C₈	C₂	1	4	yes	maximal, complemented
5	C₂⁴×C₄×C₁₃	C₂²	1	1	yes	characteristic
6	C₂³×C₈×C₁₃	C₂²	1	4	yes	—
7	C₂²×C₁₃×C₂²⋊C₄	C₄	1	2	yes	—
8	C₂²×C₄²×C₁₃	C₂²	1	2	yes	—
9	C₂×C₁₃×C₂²⋊C₈	C₄	1	16	yes	complemented
10	C₂×C₁₃×C₂²⋊C₈	C₂²	1	4	yes	—
11	C₂×C₁₃×C₂²⋊C₈	C₂²	1	4	yes	—
12	C₂×C₄×C₈×C₁₃	C₂²	1	8	yes	complemented
13	C₄×C₁₃×C₂²⋊C₄	C₄	1	4	yes	—
14	C₄×C₁₃×C₂²⋊C₄	C₂²	1	4	yes	—
15	C₂×C₁₃×#95	C₄	1	2	yes	—
16	C₁₃×#336	C₂²	1	8	yes	complemented
17	C₂⁵×C₁₃	C₂×C₄	1	1	yes	characteristic
18	C₂³×C₄×C₁₃	—	4	4	no	—
19	C₂³×C₄×C₁₃	C₂×C₄	1	2	yes	—
20	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
21	C₂³×C₄×C₁₃	C₂×C₄	1	8	yes	—
22	C₂³×C₄×C₁₃	C₂×C₄	1	4	yes	—
23	C₂³×C₄×C₁₃	C₂³	1	2	yes	—
24	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
25	C₂²×C₈×C₁₃	—	4	8	no	—
26	C₂²×C₈×C₁₃	C₂×C₄	1	16	yes	complemented
27	C₂²×C₈×C₁₃	C₂³	1	8	yes	—
28	C₂×C₁₃×C₂²⋊C₄	C₈	1	16	yes	complemented
29	C₂×C₁₃×C₂²⋊C₄	C₂×C₄	1	8	yes	—
30	C₂×C₄²×C₁₃	—	4	4	no	—
31	C₂×C₄²×C₁₃	C₂×C₄	1	4	yes	—
32	C₂×C₄²×C₁₃	C₂³	1	4	yes	—
33	C₁₃×C₂²⋊C₈	C₂×C₄	1	16	yes	—
34	C₁₃×C₂²⋊C₈	C₂×C₄	1	16	yes	complemented
35	C₄×C₈×C₁₃	—	4	8	no	—
36	C₁₃×#95	C₂×C₄	1	8	yes	—
37	#20523	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
38	C₂⁴×C₁₃	—	4	2	no	—
39	C₂⁴×C₁₃	—	4	2	no	—
40	C₂⁴×C₁₃	C₄²	1	4	yes	—
41	C₂⁴×C₁₃	C₂×C₈	1	8	yes	—
42	C₂⁴×C₁₃	C₂²×C₄	1	2	yes	—
43	C₂⁴×C₁₃	C₂²×C₄	1	1	yes	characteristic
44	C₂²×C₄×C₁₃	—	2	4	no	—
45	C₂²×C₄×C₁₃	—	2	8	no	—
46	C₂²×C₄×C₁₃	—	4	16	no	—
47	C₂²×C₄×C₁₃	—	4	4	no	—
48	C₂²×C₄×C₁₃	—	4	4	no	—
49	C₂²×C₄×C₁₃	—	4	2	no	—
50	C₂²×C₄×C₁₃	—	4	4	no	—
51	C₂²×C₄×C₁₃	C₄²	1	8	yes	—
52	C₂²×C₄×C₁₃	C₄²	1	4	yes	—
53	C₂²×C₄×C₁₃	C₂×C₈	1	16	yes	complemented
54	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
55	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
56	C₂²×C₄×C₁₃	C₂²×C₄	1	4	yes	—
57	C₂²×C₄×C₁₃	C₂²×C₄	1	4	yes	—
58	C₂²×C₄×C₁₃	C₂²×C₄	1	4	yes	—
59	C₂²×C₄×C₁₃	C₂²×C₄	1	8	yes	—
60	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
61	C₂×C₈×C₁₃	—	2	16	no	—
62	C₂×C₈×C₁₃	—	4	16	no	—
63	C₂×C₈×C₁₃	C₂²×C₄	1	16	yes	—
64	C₁₃×C₂²⋊C₄	C₂×C₈	1	32	yes	complemented
65	C₄²×C₁₃	—	2	8	no	—
66	C₄²×C₁₃	—	4	4	no	—
67	C₂²×C₂²⋊C₈	C₂×C₁₃	1	2	yes	—
68	C₂×C₄×C₂²⋊C₄	C₂×C₁₃	1	1	yes	characteristic
69	C₈×C₂²⋊C₄	C₂×C₁₃	1	8	yes	complemented
70	C₄×C₂²⋊C₈	C₂×C₁₃	1	4	yes	complemented
71	C₂³×C₁₃	—	2	2	no	—
72	C₂³×C₁₃	—	2	4	no	—
73	C₂³×C₁₃	—	2	2	no	—
74	C₂³×C₁₃	—	2	4	no	—
75	C₂³×C₁₃	—	4	4	no	—
76	C₂³×C₁₃	—	4	4	no	—
77	C₂³×C₁₃	—	4	16	no	—
78	C₂³×C₁₃	C₄×C₈	1	16	yes	complemented
79	C₂³×C₁₃	C₂×C₄²	1	2	yes	—
80	C₂³×C₁₃	C₂×C₄²	1	4	yes	—
81	C₂³×C₁₃	C₂²×C₈	1	8	yes	—
82	C₂³×C₁₃	C₂²×C₈	1	4	yes	—
83	C₂³×C₁₃	C₂³×C₄	1	1	yes	characteristic
84	C₂×C₄×C₁₃	—	2	8	no	—
85	C₂×C₄×C₁₃	—	2	2	no	—
86	C₂×C₄×C₁₃	—	2	4	no	—
87	C₂×C₄×C₁₃	—	2	8	no	—
88	C₂×C₄×C₁₃	—	2	8	no	—
89	C₂×C₄×C₁₃	—	2	8	no	—
90	C₂×C₄×C₁₃	—	2	8	no	—
91	C₂×C₄×C₁₃	—	4	8	no	—
92	C₂×C₄×C₁₃	—	4	4	no	—
93	C₂×C₄×C₁₃	—	4	16	no	—
94	C₂×C₄×C₁₃	C₂×C₄²	1	4	yes	—
95	C₂×C₄×C₁₃	C₂×C₄²	1	2	yes	—
96	C₂×C₄×C₁₃	C₂²×C₈	1	16	yes	—
97	C₂×C₄×C₁₃	C₂³×C₄	1	2	yes	—
98	C₂×C₄×C₁₃	H(5,𝔽₂)	1	4	yes	—
99	C₈×C₁₃	—	2	16	no	—
100	C₂⁴×C₄	C₂²×C₁₃	1	1	yes	characteristic
101	C₂³×C₈	C₂²×C₁₃	1	4	yes	—
102	C₂²×C₂²⋊C₄	C₄×C₁₃	1	2	yes	—
103	C₂²×C₄²	C₂²×C₁₃	1	2	yes	—
104	C₂×C₂²⋊C₈	C₄×C₁₃	1	16	yes	complemented
105	C₂×C₂²⋊C₈	C₂²×C₁₃	1	4	yes	—
106	C₂×C₂²⋊C₈	C₂²×C₁₃	1	4	yes	—
107	C₂×C₄×C₈	C₂²×C₁₃	1	8	yes	complemented
108	C₄×C₂²⋊C₄	C₄×C₁₃	1	4	yes	—
109	C₄×C₂²⋊C₄	C₂²×C₁₃	1	4	yes	—
110	C₂×#95	C₄×C₁₃	1	2	yes	—
111	#336	C₂²×C₁₃	1	8	yes	complemented
112	C₂²×C₁₃	—	2	2	no	—
113	C₂²×C₁₃	—	2	4	no	—
114	C₂²×C₁₃	—	2	8	no	—
115	C₂²×C₁₃	—	2	16	no	—
116	C₂²×C₁₃	—	2	4	no	—
117	C₂²×C₁₃	—	2	2	no	—
118	C₂²×C₁₃	—	4	16	no	—
119	C₂²×C₁₃	C₂×C₄×C₈	1	8	yes	—
120	C₂²×C₁₃	C₂×C₄×C₈	1	4	yes	—
121	C₂²×C₁₃	C₂²×C₄²	1	1	yes	characteristic
122	C₂²×C₁₃	#518	1	2	yes	—
123	C₂²×C₁₃	C₂³×C₈	1	2	yes	—
124	C₂²×C₁₃	#568	1	2	yes	—
125	C₄×C₁₃	—	2	16	no	—
126	C₄×C₁₃	—	2	2	no	—
127	C₄×C₁₃	—	2	4	no	—
128	C₄×C₁₃	#518	1	4	yes	—
129	C₂⁵	C₂×C₄×C₁₃	1	1	yes	characteristic
130	C₂³×C₄	—	4	4	no	—
131	C₂³×C₄	C₂×C₄×C₁₃	1	1	yes	characteristic
132	C₂³×C₄	C₂×C₄×C₁₃	1	2	yes	—
133	C₂³×C₄	C₂×C₄×C₁₃	1	8	yes	—
134	C₂³×C₄	C₂×C₄×C₁₃	1	4	yes	—
135	C₂³×C₄	C₂³×C₁₃	1	2	yes	—
136	C₂³×C₄	C₂³×C₁₃	1	1	yes	characteristic
137	C₂²×C₈	—	4	8	no	—
138	C₂²×C₈	C₂×C₄×C₁₃	1	16	yes	complemented
139	C₂²×C₈	C₂³×C₁₃	1	8	yes	—
140	C₂×C₂²⋊C₄	C₈×C₁₃	1	16	yes	complemented
141	C₂×C₂²⋊C₄	C₂×C₄×C₁₃	1	8	yes	—
142	C₂×C₄²	—	4	4	no	—
143	C₂×C₄²	C₂×C₄×C₁₃	1	4	yes	—
144	C₂×C₄²	C₂³×C₁₃	1	4	yes	—
145	C₂²⋊C₈	C₂×C₄×C₁₃	1	16	yes	—
146	C₂²⋊C₈	C₂×C₄×C₁₃	1	16	yes	complemented
147	C₄×C₈	—	4	8	no	—
148	#95	C₂×C₄×C₁₃	1	8	yes	—
149	C₂×C₁₃	—	2	4	no	—
150	C₂×C₁₃	—	2	8	no	—
151	C₂×C₁₃	#2273	1	1	yes	characteristic
152	C₂×C₁₃	C₂²×C₄×C₈	1	1	yes	characteristic
153	C₂×C₁₃	#2873	1	1	yes	characteristic
154	C₂×C₁₃	#2972	1	4	yes	—
155	C₂⁴	—	4	2	no	—
156	C₂⁴	—	4	2	no	—
157	C₂⁴	C₄²×C₁₃	1	4	yes	—
158	C₂⁴	C₂×C₈×C₁₃	1	8	yes	—
159	C₂⁴	C₂²×C₄×C₁₃	1	2	yes	—
160	C₂⁴	C₂²×C₄×C₁₃	1	1	yes	characteristic
161	C₂²×C₄	—	2	4	no	—
162	C₂²×C₄	—	2	8	no	—
163	C₂²×C₄	—	4	16	no	—
164	C₂²×C₄	—	4	4	no	—
165	C₂²×C₄	—	4	4	no	—
166	C₂²×C₄	—	4	2	no	—
167	C₂²×C₄	—	4	4	no	—
168	C₂²×C₄	C₄²×C₁₃	1	4	yes	—
169	C₂²×C₄	C₄²×C₁₃	1	8	yes	—
170	C₂²×C₄	C₂×C₈×C₁₃	1	16	yes	complemented
171	C₂²×C₄	C₂²×C₄×C₁₃	1	2	yes	—
172	C₂²×C₄	C₂²×C₄×C₁₃	1	1	yes	characteristic
173	C₂²×C₄	C₂²×C₄×C₁₃	1	4	yes	—
174	C₂²×C₄	C₂²×C₄×C₁₃	1	4	yes	—
175	C₂²×C₄	C₂²×C₄×C₁₃	1	4	yes	—
176	C₂²×C₄	C₂²×C₄×C₁₃	1	8	yes	—
177	C₂²×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
178	C₂×C₈	—	2	16	no	—
179	C₂×C₈	—	4	16	no	—
180	C₂×C₈	C₂²×C₄×C₁₃	1	16	yes	—
181	C₂²⋊C₄	C₂×C₈×C₁₃	1	32	yes	complemented
182	C₄²	—	2	8	no	—
183	C₄²	—	4	4	no	—
184	C₁₃	#20523	1	1	yes	characteristic, direct product factor, Sylow
185	C₂³	—	2	2	no	—
186	C₂³	—	2	4	no	—
187	C₂³	—	2	2	no	—
188	C₂³	—	2	4	no	—
189	C₂³	—	4	4	no	—
190	C₂³	—	4	4	no	—
191	C₂³	—	4	16	no	—
192	C₂³	C₄×C₈×C₁₃	1	16	yes	complemented
193	C₂³	C₂×C₄²×C₁₃	1	2	yes	—
194	C₂³	C₂×C₄²×C₁₃	1	4	yes	—
195	C₂³	C₂²×C₈×C₁₃	1	8	yes	—
196	C₂³	C₂²×C₈×C₁₃	1	4	yes	—
197	C₂³	C₂³×C₄×C₁₃	1	1	yes	characteristic
198	C₂×C₄	—	2	8	no	—
199	C₂×C₄	—	2	2	no	—
200	C₂×C₄	—	2	4	no	—
201	C₂×C₄	—	2	8	no	—
202	C₂×C₄	—	2	8	no	—
203	C₂×C₄	—	2	8	no	—
204	C₂×C₄	—	2	8	no	—
205	C₂×C₄	—	4	8	no	—
206	C₂×C₄	—	4	4	no	—
207	C₂×C₄	—	4	16	no	—
208	C₂×C₄	C₂×C₄²×C₁₃	1	4	yes	—
209	C₂×C₄	C₂×C₄²×C₁₃	1	2	yes	—
210	C₂×C₄	C₂²×C₈×C₁₃	1	16	yes	—
211	C₂×C₄	C₂³×C₄×C₁₃	1	2	yes	—
212	C₂×C₄	C₁₃×H(5,𝔽₂)	1	4	yes	—
213	C₈	—	2	16	no	—
214	C₂²	—	2	4	no	—
215	C₂²	—	2	2	no	—
216	C₂²	—	2	2	no	—
217	C₂²	—	2	4	no	—
218	C₂²	—	2	8	no	—
219	C₂²	—	2	16	no	—
220	C₂²	—	4	16	no	—
221	C₂²	C₂×C₄×C₈×C₁₃	1	8	yes	—
222	C₂²	C₂×C₄×C₈×C₁₃	1	4	yes	—
223	C₂²	C₂²×C₄²×C₁₃	1	1	yes	characteristic
224	C₂²	C₁₃×#518	1	2	yes	—
225	C₂²	C₂³×C₈×C₁₃	1	2	yes	—
226	C₂²	C₁₃×#568	1	2	yes	—
227	C₄	—	2	2	no	—
228	C₄	—	2	4	no	—
229	C₄	—	2	16	no	—
230	C₄	C₁₃×#518	1	4	yes	—
231	C₂	—	2	4	no	—
232	C₂	—	2	8	no	—
233	C₂	C₁₃×#2273	1	1	yes	characteristic
234	C₂	C₂²×C₄×C₈×C₁₃	1	1	yes	characteristic
235	C₂	C₁₃×#2873	1	1	yes	characteristic
236	C₂	C₁₃×#2972	1	4	yes	—

	subgroup	G/H
center	C₂²×C₄×C₁₃	C₂⁴
commutator	C₂	C₂²×C₄×C₈×C₁₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃
socle	C₂³×C₁₃	C₂³×C₄
prime radical	C₂⁵×C₁₃	C₂×C₄
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₈×C₁₃×C₂²⋊C₄	C₂	semidirect
4. C₄×C₁₃×C₂²⋊C₈	C₂	semidirect
9. C₂×C₁₃×C₂²⋊C₈	C₄	semidirect
12. C₂×C₄×C₈×C₁₃	C₂²	semidirect
16. C₁₃×#336	C₂²	semidirect
26. C₂²×C₈×C₁₃	C₂×C₄	semidirect
28. C₂×C₁₃×C₂²⋊C₄	C₈	semidirect
34. C₁₃×C₂²⋊C₈	C₂×C₄	semidirect
37. #20523	C₁₃	direct (N × H)
53. C₂²×C₄×C₁₃	C₂×C₈	semidirect
64. C₁₃×C₂²⋊C₄	C₂×C₈	semidirect
69. C₈×C₂²⋊C₄	C₂×C₁₃	semidirect
70. C₄×C₂²⋊C₈	C₂×C₁₃	semidirect
78. C₂³×C₁₃	C₄×C₈	semidirect
104. C₂×C₂²⋊C₈	C₄×C₁₃	semidirect
107. C₂×C₄×C₈	C₂²×C₁₃	semidirect
111. #336	C₂²×C₁₃	semidirect
138. C₂²×C₈	C₂×C₄×C₁₃	semidirect
140. C₂×C₂²⋊C₄	C₈×C₁₃	semidirect
146. C₂²⋊C₈	C₂×C₄×C₁₃	semidirect
170. C₂²×C₄	C₂×C₈×C₁₃	semidirect
181. C₂²⋊C₄	C₂×C₈×C₁₃	semidirect
184. C₁₃	#20523	direct (N × H)
192. C₂³	C₄×C₈×C₁₃	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²×C₁₃×C₂²⋊C₈	2
C₂×C₄×C₁₃×C₂²⋊C₄	2
C₈×C₁₃×C₂²⋊C₄	2
C₄×C₁₃×C₂²⋊C₈	2
#20523	13

Subgroups — C13×#20523 #15706862

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#20523 #15706862