Subgroups — C₁₃×#23380 #15709719

Order 3328.

612 conjugacy classes of subgroups.

The subgroup lattice has 232 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#2401	C₂	1	1	yes	maximal, characteristic
2	C₁₃×#1940	C₂	1	3	yes	maximal, complemented
3	C₁₃×#1924	C₂	1	3	yes	maximal
4	C₁₃×#1917	C₂	1	3	yes	maximal, complemented
5	C₁₃×#1850	C₂	1	3	yes	maximal
6	C₁₃×#1843	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₃×#1840	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂³×C₈×C₁₃	C₂²	1	1	yes	characteristic
9	C₂×C₁₃×C₄⋊C₈	C₂²	1	3	yes	—
10	C₂×C₁₃×C₄⋊C₈	C₂²	1	3	yes	—
11	C₂×C₁₃×C₄⋊C₈	C₂²	1	3	yes	—
12	C₂×C₁₃×C₂²⋊C₈	C₂²	1	3	yes	—
13	C₂×C₁₃×#97	C₂²	1	3	yes	—
14	C₁₃×#401	C₄	1	1	yes	characteristic
15	C₁₃×#400	C₄	1	3	yes	—
16	C₁₃×#397	C₄	1	3	yes	—
17	C₁₃×#396	C₄	1	1	yes	characteristic
18	C₄×C₁₃×C₄⋊C₄	C₂²	1	3	yes	—
19	C₄×C₁₃×C₂²⋊C₄	C₂²	1	3	yes	—
20	C₁₃×#376	C₂²	1	1	yes	characteristic
21	C₁₃×#336	C₂²	1	3	yes	—
22	C₁₃×#336	C₂²	1	3	yes	—
23	C₁₃×#336	C₂²	1	3	yes	—
24	C₁₃×#336	C₂²	1	3	yes	—
25	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
26	C₂²×C₈×C₁₃	—	4	3	no	—
27	C₂²×C₈×C₁₃	C₂³	1	3	yes	—
28	C₂²×C₈×C₁₃	C₂³	1	3	yes	—
29	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
30	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
31	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	3	yes	—
32	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	3	yes	—
33	C₂×C₁₃×C₂²⋊C₄	C₂×C₄	1	3	yes	—
34	C₂×C₁₃×C₂²⋊C₄	C₂×C₄	1	3	yes	—
35	C₂×C₄²×C₁₃	C₂³	1	3	yes	—
36	C₂×C₄²×C₁₃	C₂³	1	3	yes	—
37	C₁₃×C₄⋊C₈	—	4	3	no	—
38	C₁₃×C₄⋊C₈	—	4	3	no	—
39	C₁₃×C₄⋊C₈	—	4	3	no	—
40	C₁₃×C₂²⋊C₈	—	4	3	no	—
41	C₁₃×#97	—	4	3	no	—
42	C₁₃×#95	C₂×C₄	1	3	yes	—
43	C₁₃×#95	C₂×C₄	1	3	yes	—
44	C₁₃×#95	C₂×C₄	1	3	yes	—
45	C₁₃×#95	C₂×C₄	1	3	yes	—
46	C₁₃×#95	C₂×C₄	1	1	yes	characteristic
47	C₁₃×#95	C₂×C₄	1	3	yes	—
48	#23380	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
49	C₂⁴×C₁₃	C₂²×C₄	1	1	yes	characteristic
50	C₂²×C₄×C₁₃	—	2	3	no	—
51	C₂²×C₄×C₁₃	—	2	3	no	—
52	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
53	C₂²×C₄×C₁₃	C₂²×C₄	1	3	yes	—
54	C₂²×C₄×C₁₃	C₂²×C₄	1	3	yes	—
55	C₂²×C₄×C₁₃	C₂²×C₄	1	3	yes	—
56	C₂²×C₄×C₁₃	C₂²×C₄	1	3	yes	—
57	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
58	C₂×C₈×C₁₃	—	2	3	no	—
59	C₂×C₈×C₁₃	—	2	3	no	—
60	C₂×C₈×C₁₃	—	2	3	no	—
61	C₂×C₈×C₁₃	—	2	3	no	—
62	C₂×C₈×C₁₃	—	2	3	no	—
63	C₂×C₈×C₁₃	—	2	3	no	—
64	C₂×C₈×C₁₃	—	2	6	no	—
65	C₂×C₈×C₁₃	—	4	4	no	—
66	C₁₃×C₄⋊C₄	—	4	3	no	—
67	C₁₃×C₄⋊C₄	—	4	3	no	—
68	C₁₃×C₂²⋊C₄	—	2	6	no	—
69	C₁₃×C₂²⋊C₄	—	4	3	no	—
70	C₄²×C₁₃	—	2	3	no	—
71	C₄²×C₁₃	—	2	3	no	—
72	C₄²×C₁₃	—	2	3	no	—
73	C₄²×C₁₃	—	2	3	no	—
74	#2401	C₂×C₁₃	1	1	yes	characteristic
75	#1940	C₂×C₁₃	1	3	yes	complemented
76	#1924	C₂×C₁₃	1	3	yes	—
77	#1917	C₂×C₁₃	1	3	yes	complemented
78	#1850	C₂×C₁₃	1	3	yes	—
79	#1843	C₂×C₁₃	1	1	yes	characteristic, complemented
80	#1840	C₂×C₁₃	1	1	yes	characteristic, complemented
81	C₂³×C₁₃	—	2	3	no	—
82	C₂³×C₁₃	—	2	3	no	—
83	C₂³×C₁₃	#131	1	2	yes	—
84	C₂³×C₁₃	C₂³×C₄	1	1	yes	characteristic
85	C₂×C₄×C₁₃	—	2	3	no	—
86	C₂×C₄×C₁₃	—	2	3	no	—
87	C₂×C₄×C₁₃	—	2	3	no	—
88	C₂×C₄×C₁₃	—	2	3	no	—
89	C₂×C₄×C₁₃	—	2	3	no	—
90	C₂×C₄×C₁₃	—	2	3	no	—
91	C₂×C₄×C₁₃	—	2	3	no	—
92	C₂×C₄×C₁₃	—	2	3	no	—
93	C₂×C₄×C₁₃	—	2	3	no	—
94	C₂×C₄×C₁₃	—	2	3	no	—
95	C₂×C₄×C₁₃	—	2	3	no	—
96	C₂×C₄×C₁₃	—	2	3	no	—
97	C₂×C₄×C₁₃	—	2	3	no	—
98	C₂×C₄×C₁₃	—	4	1	no	—
99	C₂×C₄×C₁₃	—	4	3	no	—
100	C₂×C₄×C₁₃	H(5,𝔽₂)	1	3	yes	—
101	C₂×C₄×C₁₃	2₋¹⁺⁴	1	3	yes	—
102	C₈×C₁₃	—	4	3	no	—
103	C₈×C₁₃	—	4	1	no	—
104	C₈×C₁₃	—	4	1	no	—
105	C₈×C₁₃	—	4	3	no	—
106	C₂³×C₈	C₂²×C₁₃	1	1	yes	characteristic
107	C₂×C₄⋊C₈	C₂²×C₁₃	1	3	yes	—
108	C₂×C₄⋊C₈	C₂²×C₁₃	1	3	yes	—
109	C₂×C₄⋊C₈	C₂²×C₁₃	1	3	yes	—
110	C₂×C₂²⋊C₈	C₂²×C₁₃	1	3	yes	—
111	C₂×#97	C₂²×C₁₃	1	3	yes	—
112	#401	C₄×C₁₃	1	1	yes	characteristic
113	#400	C₄×C₁₃	1	3	yes	—
114	#397	C₄×C₁₃	1	3	yes	—
115	#396	C₄×C₁₃	1	1	yes	characteristic
116	C₄×C₄⋊C₄	C₂²×C₁₃	1	3	yes	—
117	C₄×C₂²⋊C₄	C₂²×C₁₃	1	3	yes	—
118	#376	C₂²×C₁₃	1	1	yes	characteristic
119	#336	C₂²×C₁₃	1	3	yes	—
120	#336	C₂²×C₁₃	1	3	yes	—
121	#336	C₂²×C₁₃	1	3	yes	—
122	#336	C₂²×C₁₃	1	3	yes	—
123	C₂²×C₁₃	—	2	6	no	—
124	C₂²×C₁₃	—	4	1	no	—
125	C₂²×C₁₃	—	4	3	no	—
126	C₂²×C₁₃	#520	1	3	yes	—
127	C₂²×C₁₃	C₂×#131	1	1	yes	characteristic
128	C₂²×C₁₃	#568	1	3	yes	—
129	C₄×C₁₃	—	4	1	no	—
130	C₄×C₁₃	—	4	3	no	—
131	C₄×C₁₃	—	4	3	no	—
132	C₄×C₁₃	—	4	3	no	—
133	C₄×C₁₃	—	4	3	no	—
134	C₄×C₁₃	#560	1	1	yes	characteristic
135	C₄×C₁₃	#563	1	3	yes	—
136	C₂³×C₄	C₂³×C₁₃	1	1	yes	characteristic
137	C₂²×C₈	—	4	3	no	—
138	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
139	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
140	C₂²×C₈	C₂³×C₁₃	1	3	yes	—
141	C₂²×C₈	C₂³×C₁₃	1	3	yes	—
142	C₂×C₄⋊C₄	C₂×C₄×C₁₃	1	3	yes	—
143	C₂×C₄⋊C₄	C₂×C₄×C₁₃	1	3	yes	—
144	C₂×C₂²⋊C₄	C₂×C₄×C₁₃	1	3	yes	—
145	C₂×C₂²⋊C₄	C₂×C₄×C₁₃	1	3	yes	—
146	C₂×C₄²	C₂³×C₁₃	1	3	yes	—
147	C₂×C₄²	C₂³×C₁₃	1	3	yes	—
148	C₄⋊C₈	—	4	3	no	—
149	C₄⋊C₈	—	4	3	no	—
150	C₄⋊C₈	—	4	3	no	—
151	C₂²⋊C₈	—	4	3	no	—
152	#97	—	4	3	no	—
153	#95	C₂×C₄×C₁₃	1	1	yes	characteristic
154	#95	C₂×C₄×C₁₃	1	3	yes	—
155	#95	C₂×C₄×C₁₃	1	3	yes	—
156	#95	C₂×C₄×C₁₃	1	3	yes	—
157	#95	C₂×C₄×C₁₃	1	3	yes	—
158	#95	C₂×C₄×C₁₃	1	3	yes	—
159	C₂×C₁₃	—	4	2	no	—
160	C₂×C₁₃	#2383	1	1	yes	characteristic
161	C₂×C₁₃	#2992	1	3	yes	—
162	C₂×C₁₃	#2995	1	3	yes	—
163	C₂⁴	C₂²×C₄×C₁₃	1	1	yes	characteristic
164	C₂²×C₄	—	2	3	no	—
165	C₂²×C₄	—	2	3	no	—
166	C₂²×C₄	C₂²×C₄×C₁₃	1	1	yes	characteristic
167	C₂²×C₄	C₂²×C₄×C₁₃	1	3	yes	—
168	C₂²×C₄	C₂²×C₄×C₁₃	1	3	yes	—
169	C₂²×C₄	C₂²×C₄×C₁₃	1	3	yes	—
170	C₂²×C₄	C₂²×C₄×C₁₃	1	3	yes	—
171	C₂²×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
172	C₂×C₈	—	2	3	no	—
173	C₂×C₈	—	2	3	no	—
174	C₂×C₈	—	2	3	no	—
175	C₂×C₈	—	2	3	no	—
176	C₂×C₈	—	2	3	no	—
177	C₂×C₈	—	2	3	no	—
178	C₂×C₈	—	2	6	no	—
179	C₂×C₈	—	4	4	no	—
180	C₄⋊C₄	—	4	3	no	—
181	C₄⋊C₄	—	4	3	no	—
182	C₂²⋊C₄	—	2	6	no	—
183	C₂²⋊C₄	—	4	3	no	—
184	C₄²	—	2	3	no	—
185	C₄²	—	2	3	no	—
186	C₄²	—	2	3	no	—
187	C₄²	—	2	3	no	—
188	C₁₃	#23380	1	1	yes	characteristic, direct product factor, Sylow
189	C₂³	—	2	3	no	—
190	C₂³	—	2	3	no	—
191	C₂³	C₁₃×#131	1	2	yes	—
192	C₂³	C₂³×C₄×C₁₃	1	1	yes	characteristic
193	C₂×C₄	—	2	3	no	—
194	C₂×C₄	—	2	3	no	—
195	C₂×C₄	—	2	3	no	—
196	C₂×C₄	—	2	3	no	—
197	C₂×C₄	—	2	3	no	—
198	C₂×C₄	—	2	3	no	—
199	C₂×C₄	—	2	3	no	—
200	C₂×C₄	—	2	3	no	—
201	C₂×C₄	—	2	3	no	—
202	C₂×C₄	—	2	3	no	—
203	C₂×C₄	—	2	3	no	—
204	C₂×C₄	—	2	3	no	—
205	C₂×C₄	—	2	3	no	—
206	C₂×C₄	—	4	1	no	—
207	C₂×C₄	—	4	3	no	—
208	C₂×C₄	C₁₃×H(5,𝔽₂)	1	3	yes	—
209	C₂×C₄	C₁₃×2₋¹⁺⁴	1	3	yes	—
210	C₈	—	4	3	no	—
211	C₈	—	4	1	no	—
212	C₈	—	4	1	no	—
213	C₈	—	4	3	no	—
214	C₂²	—	2	6	no	—
215	C₂²	—	4	1	no	—
216	C₂²	—	4	3	no	—
217	C₂²	C₁₃×#520	1	3	yes	—
218	C₂²	C₂×C₁₃×#131	1	1	yes	characteristic
219	C₂²	C₁₃×#568	1	3	yes	—
220	C₄	—	4	1	no	—
221	C₄	—	4	3	no	—
222	C₄	—	4	3	no	—
223	C₄	—	4	3	no	—
224	C₄	—	4	3	no	—
225	C₄	C₁₃×#560	1	1	yes	characteristic
226	C₄	C₁₃×#563	1	3	yes	—
227	C₂	—	4	2	no	—
228	C₂	C₁₃×#2383	1	1	yes	characteristic
229	C₂	C₁₃×#2992	1	3	yes	—
230	C₂	C₁₃×#2995	1	3	yes	—

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

Complements

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

N	H ≅ G/N	type
2. C₁₃×#1940	C₂	semidirect
4. C₁₃×#1917	C₂	semidirect
6. C₁₃×#1843	C₂	semidirect
7. C₁₃×#1840	C₂	semidirect
48. #23380	C₁₃	direct (N × H)
75. #1940	C₂×C₁₃	semidirect
77. #1917	C₂×C₁₃	semidirect
79. #1843	C₂×C₁₃	semidirect
80. #1840	C₂×C₁₃	semidirect
188. C₁₃	#23380	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×#2401	2
C₁₃×#1940	2
C₁₃×#1924	2
C₁₃×#1917	2
C₁₃×#1850	2
C₁₃×#1843	2
C₁₃×#1840	2
#23380	13

Subgroups — C13×#23380 #15709719

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#23380 #15709719