Subgroups — C₁₃×#2541 #13111599

Order 1664.

948 conjugacy classes of subgroups.

The subgroup lattice has 220 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₃×#127	C₂	1	2	yes	maximal, complemented
2	C₂×C₁₃×#127	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₁₃×#121	C₂	1	4	yes	maximal, complemented
4	C₂×C₄×D₄×C₁₃	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₃×#392	C₂	1	4	yes	maximal, complemented
6	C₁₃×#390	C₂	1	2	yes	maximal, complemented
7	C₄×C₁₃×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×D₄×C₁₃	C₂²	1	2	yes	—
9	C₂²×D₄×C₁₃	C₂²	1	2	yes	—
10	C₂²×D₄×C₁₃	C₂²	1	8	yes	—
11	C₂²×D₄×C₁₃	C₂²	1	1	yes	characteristic
12	C₂²×D₄×C₁₃	C₂²	1	2	yes	—
13	C₂³×C₄×C₁₃	C₂²	1	2	yes	—
14	C₁₃×#127	—	2	8	no	—
15	C₁₃×#127	—	2	4	no	—
16	C₁₃×#121	—	2	16	no	—
17	C₄×D₄×C₁₃	—	2	4	no	—
18	C₂×C₁₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
19	C₂×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
20	C₂×C₁₃×C₂²⋊C₄	C₂²	1	8	yes	—
21	C₂×C₁₃×C₂²⋊C₄	C₂²	1	2	yes	—
22	C₂×C₄²×C₁₃	C₂²	1	2	yes	—
23	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
24	C₁₃×#95	C₂²	1	2	yes	—
25	C₂⁴×C₁₃	C₂³	1	4	yes	—
26	C₂⁴×C₁₃	C₂³	1	2	yes	—
27	C₂×D₄×C₁₃	—	2	16	no	—
28	C₂×D₄×C₁₃	—	2	2	no	—
29	C₂×D₄×C₁₃	—	2	16	no	—
30	C₂×D₄×C₁₃	—	2	8	no	—
31	C₂×D₄×C₁₃	—	2	4	no	—
32	C₂×D₄×C₁₃	—	2	4	no	—
33	C₂×D₄×C₁₃	—	2	4	no	—
34	C₂×D₄×C₁₃	—	2	4	no	—
35	C₂×D₄×C₁₃	—	2	2	no	—
36	C₂×D₄×C₁₃	—	2	4	no	—
37	C₂×D₄×C₁₃	—	4	8	no	—
38	C₂×D₄×C₁₃	—	4	4	no	—
39	C₂×D₄×C₁₃	D₄	1	4	yes	complemented
40	C₂²×C₄×C₁₃	—	2	4	no	—
41	C₂²×C₄×C₁₃	—	2	8	no	—
42	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
43	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
44	C₂²×C₄×C₁₃	C₂³	1	4	yes	—
45	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
46	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
47	C₁₃×C₄⋊C₄	—	2	2	no	—
48	C₁₃×C₄⋊C₄	D₄	1	8	yes	complemented
49	C₁₃×C₂²⋊C₄	—	2	4	no	—
50	C₁₃×C₂²⋊C₄	—	2	16	no	—
51	C₄²×C₁₃	—	2	4	no	—
52	C₄²×C₁₃	D₄	1	4	yes	complemented
53	#2541	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
54	C₂³×C₁₃	—	2	2	no	—
55	C₂³×C₁₃	—	2	4	no	—
56	C₂³×C₁₃	—	2	2	no	—
57	C₂³×C₁₃	—	2	4	no	—
58	C₂³×C₁₃	—	2	8	no	—
59	C₂³×C₁₃	—	2	4	no	—
60	C₂³×C₁₃	—	2	4	no	—
61	C₂³×C₁₃	—	2	4	no	—
62	C₂³×C₁₃	—	2	8	no	—
63	C₂³×C₁₃	C₂×D₄	1	4	yes	—
64	C₂³×C₁₃	C₂⁴	1	1	yes	characteristic
65	D₄×C₁₃	—	2	8	no	—
66	D₄×C₁₃	—	4	32	no	—
67	D₄×C₁₃	—	4	8	no	—
68	D₄×C₁₃	—	4	8	no	—
69	D₄×C₁₃	—	4	8	no	—
70	C₂×C₄×C₁₃	—	2	4	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	2	no	—
75	C₂×C₄×C₁₃	—	4	8	no	—
76	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
77	C₂×C₄×C₁₃	C₂×D₄	1	8	yes	—
78	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
79	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
80	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
81	C₂×C₄×C₁₃	C₂×D₄	1	4	yes	—
82	C₂×C₄×C₁₃	#41	1	2	yes	—
83	C₂×#127	C₂×C₁₃	1	2	yes	complemented
84	C₂×#127	C₂×C₁₃	1	1	yes	characteristic, complemented
85	C₂×#121	C₂×C₁₃	1	4	yes	complemented
86	C₂×C₄×D₄	C₂×C₁₃	1	1	yes	characteristic, complemented
87	#392	C₂×C₁₃	1	4	yes	complemented
88	#390	C₂×C₁₃	1	2	yes	complemented
89	C₄×C₄⋊C₄	C₂×C₁₃	1	1	yes	characteristic, complemented
90	C₂²×C₁₃	—	2	4	no	—
91	C₂²×C₁₃	—	2	8	no	—
92	C₂²×C₁₃	—	4	2	no	—
93	C₂²×C₁₃	—	4	2	no	—
94	C₂²×C₁₃	—	4	4	no	—
95	C₂²×C₁₃	—	4	4	no	—
96	C₂²×C₁₃	—	4	4	no	—
97	C₂²×C₁₃	—	4	4	no	—
98	C₂²×C₁₃	—	4	8	no	—
99	C₂²×C₁₃	—	4	8	no	—
100	C₂²×C₁₃	C₂²×D₄	1	1	yes	characteristic
101	C₂²×C₁₃	C₂²×D₄	1	1	yes	characteristic
102	C₂²×C₁₃	C₂²×D₄	1	2	yes	—
103	C₂²×C₁₃	C₂×#41	1	1	yes	characteristic
104	C₂²×C₁₃	H(5,𝔽₂)	1	2	yes	—
105	C₄×C₁₃	—	2	2	no	—
106	C₄×C₁₃	—	2	8	no	—
107	C₄×C₁₃	—	4	2	no	—
108	C₄×C₁₃	#121	1	4	yes	complemented
109	C₄×C₁₃	#127	1	4	yes	complemented
110	C₂²×D₄	C₂²×C₁₃	1	2	yes	—
111	C₂²×D₄	C₂²×C₁₃	1	2	yes	—
112	C₂²×D₄	C₂²×C₁₃	1	8	yes	—
113	C₂²×D₄	C₂²×C₁₃	1	1	yes	characteristic
114	C₂²×D₄	C₂²×C₁₃	1	2	yes	—
115	C₂³×C₄	C₂²×C₁₃	1	2	yes	—
116	#127	—	2	8	no	—
117	#127	—	2	4	no	—
118	#121	—	2	16	no	—
119	C₄×D₄	—	2	4	no	—
120	C₂×C₄⋊C₄	C₂²×C₁₃	1	1	yes	characteristic
121	C₂×C₄⋊C₄	C₂²×C₁₃	1	2	yes	—
122	C₂×C₂²⋊C₄	C₂²×C₁₃	1	2	yes	—
123	C₂×C₂²⋊C₄	C₂²×C₁₃	1	8	yes	—
124	C₂×C₄²	C₂²×C₁₃	1	2	yes	—
125	C₂×C₄²	C₂²×C₁₃	1	1	yes	characteristic
126	#95	C₂²×C₁₃	1	2	yes	—
127	C₂×C₁₃	—	4	4	no	—
128	C₂×C₁₃	—	8	4	no	—
129	C₂×C₁₃	C₂×#121	1	1	yes	characteristic
130	C₂×C₁₃	C₂×#127	1	1	yes	characteristic
131	C₂×C₁₃	#535	1	1	yes	characteristic
132	C₂×C₁₃	D₄²	1	2	yes	—
133	C₂×C₁₃	#550	1	2	yes	—
134	C₂⁴	C₂³×C₁₃	1	4	yes	—
135	C₂⁴	C₂³×C₁₃	1	2	yes	—
136	C₂×D₄	—	2	16	no	—
137	C₂×D₄	—	2	2	no	—
138	C₂×D₄	—	2	16	no	—
139	C₂×D₄	—	2	8	no	—
140	C₂×D₄	—	2	4	no	—
141	C₂×D₄	—	2	4	no	—
142	C₂×D₄	—	2	4	no	—
143	C₂×D₄	—	2	4	no	—
144	C₂×D₄	—	2	2	no	—
145	C₂×D₄	—	2	4	no	—
146	C₂×D₄	—	4	8	no	—
147	C₂×D₄	—	4	4	no	—
148	C₂×D₄	D₄×C₁₃	1	4	yes	complemented
149	C₂²×C₄	—	2	4	no	—
150	C₂²×C₄	—	2	8	no	—
151	C₂²×C₄	C₂³×C₁₃	1	2	yes	—
152	C₂²×C₄	C₂³×C₁₃	1	1	yes	characteristic
153	C₂²×C₄	C₂³×C₁₃	1	4	yes	—
154	C₂²×C₄	C₂³×C₁₃	1	1	yes	characteristic
155	C₂²×C₄	C₂³×C₁₃	1	1	yes	characteristic
156	C₄⋊C₄	—	2	2	no	—
157	C₄⋊C₄	D₄×C₁₃	1	8	yes	complemented
158	C₂²⋊C₄	—	2	4	no	—
159	C₂²⋊C₄	—	2	16	no	—
160	C₄²	—	2	4	no	—
161	C₄²	D₄×C₁₃	1	4	yes	complemented
162	C₁₃	#2541	1	1	yes	characteristic, direct product factor, Sylow
163	C₂³	—	2	2	no	—
164	C₂³	—	2	4	no	—
165	C₂³	—	2	2	no	—
166	C₂³	—	2	4	no	—
167	C₂³	—	2	8	no	—
168	C₂³	—	2	4	no	—
169	C₂³	—	2	4	no	—
170	C₂³	—	2	4	no	—
171	C₂³	—	2	8	no	—
172	C₂³	C₂×D₄×C₁₃	1	4	yes	—
173	C₂³	C₂⁴×C₁₃	1	1	yes	characteristic
174	D₄	—	2	8	no	—
175	D₄	—	4	32	no	—
176	D₄	—	4	8	no	—
177	D₄	—	4	8	no	—
178	D₄	—	4	8	no	—
179	C₂×C₄	—	2	4	no	—
180	C₂×C₄	—	2	2	no	—
181	C₂×C₄	—	2	4	no	—
182	C₂×C₄	—	2	4	no	—
183	C₂×C₄	—	2	2	no	—
184	C₂×C₄	—	4	8	no	—
185	C₂×C₄	C₂×D₄×C₁₃	1	2	yes	—
186	C₂×C₄	C₂×D₄×C₁₃	1	8	yes	—
187	C₂×C₄	C₂×D₄×C₁₃	1	2	yes	—
188	C₂×C₄	C₂×D₄×C₁₃	1	2	yes	—
189	C₂×C₄	C₂×D₄×C₁₃	1	2	yes	—
190	C₂×C₄	C₂×D₄×C₁₃	1	4	yes	—
191	C₂×C₄	C₁₃×#41	1	2	yes	—
192	C₂²	—	2	4	no	—
193	C₂²	—	2	8	no	—
194	C₂²	—	4	2	no	—
195	C₂²	—	4	2	no	—
196	C₂²	—	4	4	no	—
197	C₂²	—	4	4	no	—
198	C₂²	—	4	4	no	—
199	C₂²	—	4	4	no	—
200	C₂²	—	4	8	no	—
201	C₂²	—	4	8	no	—
202	C₂²	C₂²×D₄×C₁₃	1	1	yes	characteristic
203	C₂²	C₂²×D₄×C₁₃	1	1	yes	characteristic
204	C₂²	C₂²×D₄×C₁₃	1	2	yes	—
205	C₂²	C₂×C₁₃×#41	1	1	yes	characteristic
206	C₂²	C₁₃×H(5,𝔽₂)	1	2	yes	—
207	C₄	—	2	2	no	—
208	C₄	—	2	8	no	—
209	C₄	—	4	2	no	—
210	C₄	C₁₃×#121	1	4	yes	complemented
211	C₄	C₁₃×#127	1	4	yes	complemented
212	C₂	—	4	4	no	—
213	C₂	—	8	4	no	—
214	C₂	C₂×C₁₃×#121	1	1	yes	characteristic
215	C₂	C₂×C₁₃×#127	1	1	yes	characteristic
216	C₂	C₁₃×#535	1	1	yes	characteristic
217	C₂	D₄²×C₁₃	1	2	yes	—
218	C₂	C₁₃×#550	1	2	yes	—

	subgroup	G/H
center, socle	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
1. C₂×C₁₃×#127	C₂	semidirect
2. C₂×C₁₃×#127	C₂	semidirect
3. C₂×C₁₃×#121	C₂	semidirect
4. C₂×C₄×D₄×C₁₃	C₂	semidirect
5. C₁₃×#392	C₂	semidirect
6. C₁₃×#390	C₂	semidirect
7. C₄×C₁₃×C₄⋊C₄	C₂	semidirect
39. C₂×D₄×C₁₃	D₄	semidirect
48. C₁₃×C₄⋊C₄	D₄	semidirect
52. C₄²×C₁₃	D₄	semidirect
53. #2541	C₁₃	direct (N × H)
83. C₂×#127	C₂×C₁₃	semidirect
84. C₂×#127	C₂×C₁₃	semidirect
85. C₂×#121	C₂×C₁₃	semidirect
86. C₂×C₄×D₄	C₂×C₁₃	semidirect
87. #392	C₂×C₁₃	semidirect
88. #390	C₂×C₁₃	semidirect
89. C₄×C₄⋊C₄	C₂×C₁₃	semidirect
108. C₄×C₁₃	#121	semidirect
109. C₄×C₁₃	#127	semidirect
148. C₂×D₄	D₄×C₁₃	semidirect
157. C₄⋊C₄	D₄×C₁₃	semidirect
161. C₄²	D₄×C₁₃	semidirect
162. C₁₃	#2541	direct (N × H)
210. C₄	C₁₃×#121	semidirect
211. C₄	C₁₃×#127	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₁₃×#127	2
C₂×C₁₃×#121	2
C₂×C₄×D₄×C₁₃	2
C₁₃×#392	2
C₁₃×#390	2
C₄×C₁₃×C₄⋊C₄	2
#2541	13

Subgroups — C13×#2541 #13111599

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#2541 #13111599