Subgroups — C₁₃×#2591 #13111649

Order 1664.

596 conjugacy classes of subgroups.

The subgroup lattice has 212 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₃×#121	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₁₃×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₃×#400	C₂	1	2	yes	maximal, complemented
4	C₁₃×#394	C₂	1	2	yes	maximal, complemented
5	C₁₃×#388	C₂	1	4	yes	maximal, complemented
6	C₁₃×#386	C₂	1	2	yes	maximal, complemented
7	C₁₃×#386	C₂	1	2	yes	maximal, complemented
8	C₄×C₁₃×C₂²⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×D₄×C₁₃	C₂²	1	2	yes	—
10	C₂²×D₄×C₁₃	C₂²	1	1	yes	characteristic
11	C₂³×C₄×C₁₃	C₂²	1	1	yes	characteristic
12	C₂³×C₄×C₁₃	C₂²	1	1	yes	characteristic
13	C₂³×C₄×C₁₃	C₂²	1	1	yes	characteristic
14	C₁₃×#121	—	2	4	no	—
15	C₂×C₁₃×C₄⋊C₄	—	2	4	no	—
16	C₂×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
17	C₂×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
18	C₂×C₁₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
19	C₂×C₁₃×C₂²⋊C₄	C₂²	1	2	yes	—
20	C₂×C₁₃×C₂²⋊C₄	C₂²	1	4	yes	—
21	C₂×C₁₃×C₂²⋊C₄	C₂²	1	2	yes	—
22	C₂×C₁₃×C₂²⋊C₄	C₂²	1	4	yes	—
23	C₂×C₄²×C₁₃	C₂²	1	2	yes	—
24	C₁₃×#95	C₂²	1	2	yes	—
25	C₁₃×#95	C₂²	1	4	yes	—
26	C₁₃×#95	C₂²	1	4	yes	—
27	C₂⁴×C₁₃	C₂³	1	2	yes	—
28	C₂⁴×C₁₃	C₂³	1	1	yes	characteristic
29	C₂×D₄×C₁₃	—	2	4	no	—
30	C₂×D₄×C₁₃	—	2	8	no	—
31	C₂×D₄×C₁₃	—	2	2	no	—
32	C₂×D₄×C₁₃	—	4	2	no	—
33	C₂²×C₄×C₁₃	—	2	2	no	—
34	C₂²×C₄×C₁₃	—	2	4	no	—
35	C₂²×C₄×C₁₃	—	2	2	no	—
36	C₂²×C₄×C₁₃	—	4	2	no	—
37	C₂²×C₄×C₁₃	—	4	2	no	—
38	C₂²×C₄×C₁₃	D₄	1	4	yes	—
39	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
40	C₂²×C₄×C₁₃	C₂³	1	4	yes	—
41	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
42	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
43	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
44	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
45	C₁₃×C₄⋊C₄	—	2	4	no	—
46	C₁₃×C₄⋊C₄	—	2	4	no	—
47	C₁₃×C₄⋊C₄	—	2	2	no	—
48	C₁₃×C₂²⋊C₄	—	2	4	no	—
49	C₁₃×C₂²⋊C₄	—	2	4	no	—
50	C₁₃×C₂²⋊C₄	—	2	8	no	—
51	C₁₃×C₂²⋊C₄	—	4	4	no	—
52	C₄²×C₁₃	—	4	2	no	—
53	#2591	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
54	C₂³×C₁₃	—	2	2	no	—
55	C₂³×C₁₃	—	2	2	no	—
56	C₂³×C₁₃	—	2	2	no	—
57	C₂³×C₁₃	—	2	4	no	—
58	C₂³×C₁₃	—	2	4	no	—
59	C₂³×C₁₃	—	2	4	no	—
60	C₂³×C₁₃	C₂×D₄	1	2	yes	—
61	C₂³×C₁₃	#41	1	4	yes	—
62	C₂³×C₁₃	C₂⁴	1	1	yes	characteristic
63	D₄×C₁₃	—	4	4	no	—
64	D₄×C₁₃	—	4	8	no	—
65	C₂×C₄×C₁₃	—	2	8	no	—
66	C₂×C₄×C₁₃	—	2	2	no	—
67	C₂×C₄×C₁₃	—	2	2	no	—
68	C₂×C₄×C₁₃	—	2	2	no	—
69	C₂×C₄×C₁₃	—	2	2	no	—
70	C₂×C₄×C₁₃	—	2	2	no	—
71	C₂×C₄×C₁₃	—	2	2	no	—
72	C₂×C₄×C₁₃	—	2	4	no	—
73	C₂×C₄×C₁₃	—	2	4	no	—
74	C₂×C₄×C₁₃	—	2	4	no	—
75	C₂×C₄×C₁₃	—	2	4	no	—
76	C₂×C₄×C₁₃	—	2	4	no	—
77	C₂×C₄×C₁₃	—	4	4	no	—
78	C₂×C₄×C₁₃	—	4	4	no	—
79	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
80	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
81	C₂×C₄×C₁₃	#41	1	4	yes	—
82	C₂×#121	C₂×C₁₃	1	1	yes	characteristic
83	C₂²×C₄⋊C₄	C₂×C₁₃	1	1	yes	characteristic, complemented
84	#400	C₂×C₁₃	1	2	yes	complemented
85	#394	C₂×C₁₃	1	2	yes	complemented
86	#388	C₂×C₁₃	1	4	yes	complemented
87	#386	C₂×C₁₃	1	2	yes	complemented
88	#386	C₂×C₁₃	1	2	yes	complemented
89	C₄×C₂²⋊C₄	C₂×C₁₃	1	1	yes	characteristic, complemented
90	C₂²×C₁₃	—	2	2	no	—
91	C₂²×C₁₃	—	2	2	no	—
92	C₂²×C₁₃	—	2	8	no	—
93	C₂²×C₁₃	—	4	2	no	—
94	C₂²×C₁₃	—	4	2	no	—
95	C₂²×C₁₃	—	4	2	no	—
96	C₂²×C₁₃	—	4	8	no	—
97	C₂²×C₁₃	#124	1	4	yes	—
98	C₂²×C₁₃	C₂²×D₄	1	1	yes	characteristic
99	C₂²×C₁₃	C₂×#41	1	2	yes	—
100	C₂²×C₁₃	C₂×#41	1	2	yes	—
101	C₂²×C₁₃	H(5,𝔽₂)	1	2	yes	—
102	C₄×C₁₃	—	2	4	no	—
103	C₄×C₁₃	—	2	2	no	—
104	C₄×C₁₃	—	2	2	no	—
105	C₄×C₁₃	—	4	2	no	—
106	C₄×C₁₃	—	4	2	no	—
107	C₄×C₁₃	—	4	4	no	—
108	C₂²×D₄	C₂²×C₁₃	1	2	yes	—
109	C₂²×D₄	C₂²×C₁₃	1	1	yes	characteristic
110	C₂³×C₄	C₂²×C₁₃	1	1	yes	characteristic
111	C₂³×C₄	C₂²×C₁₃	1	1	yes	characteristic
112	C₂³×C₄	C₂²×C₁₃	1	1	yes	characteristic
113	#121	—	2	4	no	—
114	C₂×C₄⋊C₄	—	2	4	no	—
115	C₂×C₄⋊C₄	C₂²×C₁₃	1	2	yes	—
116	C₂×C₄⋊C₄	C₂²×C₁₃	1	2	yes	—
117	C₂×C₄⋊C₄	C₂²×C₁₃	1	1	yes	characteristic
118	C₂×C₂²⋊C₄	C₂²×C₁₃	1	2	yes	—
119	C₂×C₂²⋊C₄	C₂²×C₁₃	1	2	yes	—
120	C₂×C₂²⋊C₄	C₂²×C₁₃	1	4	yes	—
121	C₂×C₂²⋊C₄	C₂²×C₁₃	1	4	yes	—
122	C₂×C₄²	C₂²×C₁₃	1	2	yes	—
123	#95	C₂²×C₁₃	1	2	yes	—
124	#95	C₂²×C₁₃	1	4	yes	—
125	#95	C₂²×C₁₃	1	4	yes	—
126	C₂×C₁₃	—	2	4	no	—
127	C₂×C₁₃	—	8	2	no	—
128	C₂×C₁₃	#525	1	1	yes	characteristic
129	C₂×C₁₃	C₂×#124	1	1	yes	characteristic
130	C₂×C₁₃	#534	1	1	yes	characteristic
131	C₂×C₁₃	#553	1	4	yes	—
132	C₂⁴	C₂³×C₁₃	1	2	yes	—
133	C₂⁴	C₂³×C₁₃	1	1	yes	characteristic
134	C₂×D₄	—	2	4	no	—
135	C₂×D₄	—	2	8	no	—
136	C₂×D₄	—	2	2	no	—
137	C₂×D₄	—	4	2	no	—
138	C₂²×C₄	—	2	2	no	—
139	C₂²×C₄	—	2	4	no	—
140	C₂²×C₄	—	2	2	no	—
141	C₂²×C₄	—	4	2	no	—
142	C₂²×C₄	—	4	2	no	—
143	C₂²×C₄	D₄×C₁₃	1	4	yes	—
144	C₂²×C₄	C₂³×C₁₃	1	2	yes	—
145	C₂²×C₄	C₂³×C₁₃	1	4	yes	—
146	C₂²×C₄	C₂³×C₁₃	1	2	yes	—
147	C₂²×C₄	C₂³×C₁₃	1	1	yes	characteristic
148	C₂²×C₄	C₂³×C₁₃	1	1	yes	characteristic
149	C₂²×C₄	C₂³×C₁₃	1	2	yes	—
150	C₄⋊C₄	—	2	4	no	—
151	C₄⋊C₄	—	2	4	no	—
152	C₄⋊C₄	—	2	2	no	—
153	C₂²⋊C₄	—	2	4	no	—
154	C₂²⋊C₄	—	2	4	no	—
155	C₂²⋊C₄	—	2	8	no	—
156	C₂²⋊C₄	—	4	4	no	—
157	C₄²	—	4	2	no	—
158	C₁₃	#2591	1	1	yes	characteristic, direct product factor, Sylow
159	C₂³	—	2	2	no	—
160	C₂³	—	2	2	no	—
161	C₂³	—	2	2	no	—
162	C₂³	—	2	4	no	—
163	C₂³	—	2	4	no	—
164	C₂³	—	2	4	no	—
165	C₂³	C₂×D₄×C₁₃	1	2	yes	—
166	C₂³	C₁₃×#41	1	4	yes	—
167	C₂³	C₂⁴×C₁₃	1	1	yes	characteristic
168	D₄	—	4	4	no	—
169	D₄	—	4	8	no	—
170	C₂×C₄	—	2	8	no	—
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	—	2	2	no	—
173	C₂×C₄	—	2	2	no	—
174	C₂×C₄	—	2	2	no	—
175	C₂×C₄	—	2	2	no	—
176	C₂×C₄	—	2	2	no	—
177	C₂×C₄	—	2	4	no	—
178	C₂×C₄	—	2	4	no	—
179	C₂×C₄	—	2	4	no	—
180	C₂×C₄	—	2	4	no	—
181	C₂×C₄	—	2	4	no	—
182	C₂×C₄	—	4	4	no	—
183	C₂×C₄	—	4	4	no	—
184	C₂×C₄	C₂×D₄×C₁₃	1	2	yes	—
185	C₂×C₄	C₂×D₄×C₁₃	1	2	yes	—
186	C₂×C₄	C₁₃×#41	1	4	yes	—
187	C₂²	—	2	2	no	—
188	C₂²	—	2	2	no	—
189	C₂²	—	2	8	no	—
190	C₂²	—	4	2	no	—
191	C₂²	—	4	2	no	—
192	C₂²	—	4	2	no	—
193	C₂²	—	4	8	no	—
194	C₂²	C₁₃×#124	1	4	yes	—
195	C₂²	C₂²×D₄×C₁₃	1	1	yes	characteristic
196	C₂²	C₂×C₁₃×#41	1	2	yes	—
197	C₂²	C₂×C₁₃×#41	1	2	yes	—
198	C₂²	C₁₃×H(5,𝔽₂)	1	2	yes	—
199	C₄	—	2	4	no	—
200	C₄	—	2	2	no	—
201	C₄	—	2	2	no	—
202	C₄	—	4	2	no	—
203	C₄	—	4	2	no	—
204	C₄	—	4	4	no	—
205	C₂	—	2	4	no	—
206	C₂	—	8	2	no	—
207	C₂	C₁₃×#525	1	1	yes	characteristic
208	C₂	C₂×C₁₃×#124	1	1	yes	characteristic
209	C₂	C₁₃×#534	1	1	yes	characteristic
210	C₂	C₁₃×#553	1	4	yes	—

	subgroup	G/H
center, socle	C₂³×C₁₃	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₁₃
prime radical	C₂×C₁₃×#121	C₂
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. C₂²×C₁₃×C₄⋊C₄	C₂	semidirect
3. C₁₃×#400	C₂	semidirect
4. C₁₃×#394	C₂	semidirect
5. C₁₃×#388	C₂	semidirect
6. C₁₃×#386	C₂	semidirect
7. C₁₃×#386	C₂	semidirect
8. C₄×C₁₃×C₂²⋊C₄	C₂	semidirect
53. #2591	C₁₃	direct (N × H)
83. C₂²×C₄⋊C₄	C₂×C₁₃	semidirect
84. #400	C₂×C₁₃	semidirect
85. #394	C₂×C₁₃	semidirect
86. #388	C₂×C₁₃	semidirect
87. #386	C₂×C₁₃	semidirect
88. #386	C₂×C₁₃	semidirect
89. C₄×C₂²⋊C₄	C₂×C₁₃	semidirect
158. C₁₃	#2591	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×#121	2
C₂²×C₁₃×C₄⋊C₄	2
C₁₃×#400	2
C₁₃×#394	2
C₁₃×#388	2
C₁₃×#386	2
C₄×C₁₃×C₂²⋊C₄	2
#2591	13

Subgroups — C13×#2591 #13111649

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#2591 #13111649