Subgroups — C₁₃×#2509 #13111567

Order 1664.

380 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#400	C₂	1	2	yes	maximal
2	C₁₃×#400	C₂	1	2	yes	maximal
3	C₁₃×#389	C₂	1	2	yes	maximal
4	C₁₃×#389	C₂	1	1	yes	maximal, characteristic
5	C₁₃×#389	C₂	1	1	yes	maximal, characteristic
6	C₁₃×#389	C₂	1	1	yes	maximal, characteristic
7	C₁₃×#389	C₂	1	1	yes	maximal, characteristic
8	C₁₃×#387	C₂	1	2	yes	maximal
9	C₁₃×#382	C₂	1	1	yes	maximal, characteristic
10	C₄×C₁₃×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
11	C₄×C₁₃×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
12	C₂×C₁₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
13	C₂×C₁₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
14	C₂×C₁₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
15	C₂×C₁₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
16	C₂×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
17	C₂×C₁₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
18	C₂×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
19	C₂×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
20	C₂×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
21	C₂×C₁₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
22	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
23	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
24	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
25	C₂×C₄²×C₁₃	C₂²	1	2	yes	—
26	C₁₃×#95	C₂²	1	1	yes	characteristic
27	C₁₃×#95	C₂²	1	1	yes	characteristic
28	C₁₃×#95	C₂²	1	2	yes	—
29	C₁₃×#95	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	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
36	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
37	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
38	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
39	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
40	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
41	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
42	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
43	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
44	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
45	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
46	C₁₃×C₄⋊C₄	—	2	2	no	—
47	C₁₃×C₄⋊C₄	—	2	4	no	—
48	C₁₃×C₄⋊C₄	—	2	2	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	—	4	2	no	—
54	C₁₃×C₄⋊C₄	—	4	2	no	—
55	C₁₃×C₄⋊C₄	Q₈	1	4	yes	—
56	C₄²×C₁₃	—	2	2	no	—
57	C₄²×C₁₃	—	2	2	no	—
58	C₄²×C₁₃	—	2	2	no	—
59	C₄²×C₁₃	—	4	2	no	—
60	#2509	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
61	C₂³×C₁₃	C₂⁴	1	1	yes	characteristic
62	C₂×C₄×C₁₃	—	2	1	no	—
63	C₂×C₄×C₁₃	—	2	1	no	—
64	C₂×C₄×C₁₃	—	2	1	no	—
65	C₂×C₄×C₁₃	—	2	2	no	—
66	C₂×C₄×C₁₃	—	2	1	no	—
67	C₂×C₄×C₁₃	—	2	2	no	—
68	C₂×C₄×C₁₃	—	2	1	no	—
69	C₂×C₄×C₁₃	—	2	1	no	—
70	C₂×C₄×C₁₃	—	2	1	no	—
71	C₂×C₄×C₁₃	—	2	1	no	—
72	C₂×C₄×C₁₃	—	2	2	no	—
73	C₂×C₄×C₁₃	—	2	2	no	—
74	C₂×C₄×C₁₃	—	2	2	no	—
75	C₂×C₄×C₁₃	—	2	2	no	—
76	C₂×C₄×C₁₃	—	2	2	no	—
77	C₂×C₄×C₁₃	—	2	1	no	—
78	C₂×C₄×C₁₃	—	2	2	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	1	no	—
83	C₂×C₄×C₁₃	—	2	1	no	—
84	C₂×C₄×C₁₃	—	2	2	no	—
85	C₂×C₄×C₁₃	—	2	1	no	—
86	C₂×C₄×C₁₃	—	2	2	no	—
87	C₂×C₄×C₁₃	—	2	1	no	—
88	C₂×C₄×C₁₃	—	2	1	no	—
89	C₂×C₄×C₁₃	C₂×Q₈	1	4	yes	—
90	C₂×C₄×C₁₃	C₂×Q₈	1	2	yes	—
91	C₂×C₄×C₁₃	#41	1	2	yes	—
92	C₂×C₄×C₁₃	#41	1	2	yes	—
93	C₂×C₄×C₁₃	#41	1	2	yes	—
94	C₂×C₄×C₁₃	#41	1	2	yes	—
95	#400	C₂×C₁₃	1	2	yes	—
96	#400	C₂×C₁₃	1	2	yes	—
97	#389	C₂×C₁₃	1	2	yes	—
98	#389	C₂×C₁₃	1	1	yes	characteristic
99	#389	C₂×C₁₃	1	1	yes	characteristic
100	#389	C₂×C₁₃	1	1	yes	characteristic
101	#389	C₂×C₁₃	1	1	yes	characteristic
102	#387	C₂×C₁₃	1	2	yes	—
103	#382	C₂×C₁₃	1	1	yes	characteristic
104	C₄×C₄⋊C₄	C₂×C₁₃	1	1	yes	characteristic
105	C₄×C₄⋊C₄	C₂×C₁₃	1	1	yes	characteristic
106	C₂²×C₁₃	C₂²×Q₈	1	1	yes	characteristic
107	C₂²×C₁₃	C₂×#41	1	1	yes	characteristic
108	C₂²×C₁₃	C₂×#41	1	1	yes	characteristic
109	C₂²×C₁₃	C₂×#41	1	1	yes	characteristic
110	C₂²×C₁₃	C₂×#41	1	1	yes	characteristic
111	C₂²×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
112	C₂²×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
113	C₄×C₁₃	—	2	2	no	—
114	C₄×C₁₃	—	2	2	no	—
115	C₄×C₁₃	—	2	4	no	—
116	C₄×C₁₃	—	4	2	no	—
117	C₄×C₁₃	—	4	1	no	—
118	C₄×C₁₃	—	4	2	no	—
119	C₄×C₁₃	—	4	1	no	—
120	C₄×C₁₃	—	4	2	no	—
121	C₄×C₁₃	—	4	1	no	—
122	C₄×C₁₃	—	4	1	no	—
123	C₄×C₁₃	#125	1	4	yes	—
124	C₂×C₄⋊C₄	C₂²×C₁₃	1	1	yes	characteristic
125	C₂×C₄⋊C₄	C₂²×C₁₃	1	1	yes	characteristic
126	C₂×C₄⋊C₄	C₂²×C₁₃	1	1	yes	characteristic
127	C₂×C₄⋊C₄	C₂²×C₁₃	1	1	yes	characteristic
128	C₂×C₄⋊C₄	C₂²×C₁₃	1	2	yes	—
129	C₂×C₄⋊C₄	C₂²×C₁₃	1	1	yes	characteristic
130	C₂×C₄⋊C₄	C₂²×C₁₃	1	2	yes	—
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×C₄²	C₂²×C₁₃	1	1	yes	characteristic
135	C₂×C₄²	C₂²×C₁₃	1	1	yes	characteristic
136	C₂×C₄²	C₂²×C₁₃	1	1	yes	characteristic
137	C₂×C₄²	C₂²×C₁₃	1	2	yes	—
138	#95	C₂²×C₁₃	1	1	yes	characteristic
139	#95	C₂²×C₁₃	1	1	yes	characteristic
140	#95	C₂²×C₁₃	1	2	yes	—
141	#95	C₂²×C₁₃	1	2	yes	—
142	#95	C₂²×C₁₃	1	2	yes	—
143	#95	C₂²×C₁₃	1	2	yes	—
144	#95	C₂²×C₁₃	1	2	yes	—
145	#95	C₂²×C₁₃	1	2	yes	—
146	#95	C₂²×C₁₃	1	2	yes	—
147	C₂×C₁₃	C₂×#125	1	1	yes	characteristic
148	C₂×C₁₃	#529	1	1	yes	characteristic
149	C₂×C₁₃	#542	1	1	yes	characteristic
150	C₂×C₁₃	#553	1	1	yes	characteristic
151	C₂×C₁₃	#554	1	1	yes	characteristic
152	C₂×C₁₃	#556	1	1	yes	characteristic
153	C₂×C₁₃	#557	1	1	yes	characteristic
154	C₂²×C₄	C₂³×C₁₃	1	1	yes	characteristic
155	C₂²×C₄	C₂³×C₁₃	1	1	yes	characteristic
156	C₂²×C₄	C₂³×C₁₃	1	1	yes	characteristic
157	C₂²×C₄	C₂³×C₁₃	1	2	yes	—
158	C₂²×C₄	C₂³×C₁₃	1	2	yes	—
159	C₂²×C₄	C₂³×C₁₃	1	1	yes	characteristic
160	C₂²×C₄	C₂³×C₁₃	1	2	yes	—
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	C₄⋊C₄	—	2	2	no	—
166	C₄⋊C₄	—	2	4	no	—
167	C₄⋊C₄	—	2	2	no	—
168	C₄⋊C₄	—	2	4	no	—
169	C₄⋊C₄	—	2	2	no	—
170	C₄⋊C₄	—	2	2	no	—
171	C₄⋊C₄	—	2	2	no	—
172	C₄⋊C₄	—	4	2	no	—
173	C₄⋊C₄	—	4	2	no	—
174	C₄⋊C₄	Q₈×C₁₃	1	4	yes	—
175	C₄²	—	2	2	no	—
176	C₄²	—	2	2	no	—
177	C₄²	—	2	2	no	—
178	C₄²	—	4	2	no	—
179	C₁₃	#2509	1	1	yes	characteristic, direct product factor, Sylow
180	C₂³	C₂⁴×C₁₃	1	1	yes	characteristic
181	C₂×C₄	—	2	1	no	—
182	C₂×C₄	—	2	1	no	—
183	C₂×C₄	—	2	1	no	—
184	C₂×C₄	—	2	2	no	—
185	C₂×C₄	—	2	1	no	—
186	C₂×C₄	—	2	2	no	—
187	C₂×C₄	—	2	1	no	—
188	C₂×C₄	—	2	1	no	—
189	C₂×C₄	—	2	1	no	—
190	C₂×C₄	—	2	1	no	—
191	C₂×C₄	—	2	2	no	—
192	C₂×C₄	—	2	2	no	—
193	C₂×C₄	—	2	2	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	2	2	no	—
196	C₂×C₄	—	2	1	no	—
197	C₂×C₄	—	2	2	no	—
198	C₂×C₄	—	2	1	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	2	no	—
204	C₂×C₄	—	2	1	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	2	1	no	—
207	C₂×C₄	—	2	1	no	—
208	C₂×C₄	C₂×Q₈×C₁₃	1	4	yes	—
209	C₂×C₄	C₂×Q₈×C₁₃	1	2	yes	—
210	C₂×C₄	C₁₃×#41	1	2	yes	—
211	C₂×C₄	C₁₃×#41	1	2	yes	—
212	C₂×C₄	C₁₃×#41	1	2	yes	—
213	C₂×C₄	C₁₃×#41	1	2	yes	—
214	C₂²	C₂²×Q₈×C₁₃	1	1	yes	characteristic
215	C₂²	C₂×C₁₃×#41	1	1	yes	characteristic
216	C₂²	C₂×C₁₃×#41	1	1	yes	characteristic
217	C₂²	C₂×C₁₃×#41	1	1	yes	characteristic
218	C₂²	C₂×C₁₃×#41	1	1	yes	characteristic
219	C₂²	C₁₃×H(5,𝔽₂)	1	1	yes	characteristic
220	C₂²	C₁₃×2₋¹⁺⁴	1	1	yes	characteristic
221	C₄	—	2	2	no	—
222	C₄	—	2	2	no	—
223	C₄	—	2	4	no	—
224	C₄	—	4	2	no	—
225	C₄	—	4	1	no	—
226	C₄	—	4	2	no	—
227	C₄	—	4	1	no	—
228	C₄	—	4	2	no	—
229	C₄	—	4	1	no	—
230	C₄	—	4	1	no	—
231	C₄	C₁₃×#125	1	4	yes	—
232	C₂	C₂×C₁₃×#125	1	1	yes	characteristic
233	C₂	C₁₃×#529	1	1	yes	characteristic
234	C₂	C₁₃×#542	1	1	yes	characteristic
235	C₂	C₁₃×#553	1	1	yes	characteristic
236	C₂	C₁₃×#554	1	1	yes	characteristic
237	C₂	C₁₃×#556	1	1	yes	characteristic
238	C₂	C₁₃×#557	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₁₃	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₁₃
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
60. #2509	C₁₃	direct (N × H)
179. C₁₃	#2509	direct (N × H)

Maximal subgroups

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

subgroup	index
C₁₃×#400	2
C₁₃×#389	2
C₁₃×#387	2
C₁₃×#382	2
C₄×C₁₃×C₄⋊C₄	2
#2509	13

Subgroups — C13×#2509 #13111567

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#2509 #13111567