Subgroups — C₁₃×#2722 #13111780

Order 1664.

448 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×#126	C₂	1	2	yes	maximal
2	C₁₃×#398	C₂	1	2	yes	maximal, complemented
3	C₁₃×#398	C₂	1	2	yes	maximal, complemented
4	C₁₃×#397	C₂	1	1	yes	maximal, characteristic
5	C₁₃×#397	C₂	1	1	yes	maximal, characteristic
6	C₁₃×#389	C₂	1	2	yes	maximal, complemented
7	C₁₃×#387	C₂	1	2	yes	maximal, complemented
8	C₁₃×#385	C₂	1	1	yes	maximal, characteristic
9	C₁₃×#380	C₂	1	2	yes	maximal
10	C₂³×C₄×C₁₃	C₂²	1	1	yes	characteristic
11	C₁₃×#126	—	2	8	no	—
12	C₂×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
13	C₂×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
14	C₂×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
15	C₂×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
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×C₄⋊C₄	C₂²	1	1	yes	characteristic
20	C₂×C₁₃×C₂²⋊C₄	C₂²	1	2	yes	—
21	C₂×C₁₃×C₂²⋊C₄	C₂²	1	2	yes	—
22	C₂×C₁₃×C₂²⋊C₄	C₂²	1	1	yes	characteristic
23	C₂×C₁₃×C₂²⋊C₄	C₂²	1	1	yes	characteristic
24	C₂×C₄²×C₁₃	C₂²	1	2	yes	—
25	C₁₃×#95	C₂²	1	2	yes	—
26	C₁₃×#95	C₂²	1	2	yes	—
27	C₁₃×#95	C₂²	1	2	yes	—
28	C₁₃×#95	C₂²	1	1	yes	characteristic
29	C₁₃×#95	C₂²	1	2	yes	—
30	C₁₃×#95	C₂²	1	2	yes	—
31	C₁₃×#95	C₂²	1	1	yes	characteristic
32	C₂⁴×C₁₃	C₂³	1	1	yes	characteristic
33	C₂²×C₄×C₁₃	—	2	4	no	—
34	C₂²×C₄×C₁₃	—	4	1	no	—
35	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
36	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
37	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
38	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
39	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
40	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
41	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
42	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
43	C₁₃×C₄⋊C₄	—	2	4	no	—
44	C₁₃×C₄⋊C₄	—	2	4	no	—
45	C₁₃×C₄⋊C₄	—	2	4	no	—
46	C₁₃×C₄⋊C₄	—	2	4	no	—
47	C₁₃×C₄⋊C₄	—	2	4	no	—
48	C₁₃×C₄⋊C₄	—	4	2	no	—
49	C₁₃×C₄⋊C₄	—	4	1	no	—
50	C₁₃×C₄⋊C₄	—	4	1	no	—
51	C₁₃×C₂²⋊C₄	—	2	4	no	—
52	C₁₃×C₂²⋊C₄	—	2	4	no	—
53	C₁₃×C₂²⋊C₄	—	4	1	no	—
54	C₁₃×C₂²⋊C₄	D₄	1	4	yes	—
55	C₄²×C₁₃	—	2	4	no	—
56	#2722	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
57	C₂³×C₁₃	—	2	2	no	—
58	C₂³×C₁₃	—	2	1	no	—
59	C₂³×C₁₃	—	2	1	no	—
60	C₂³×C₁₃	—	2	2	no	—
61	C₂³×C₁₃	C₂×D₄	1	2	yes	—
62	C₂³×C₁₃	C₂⁴	1	1	yes	characteristic
63	C₂×C₄×C₁₃	—	2	2	no	—
64	C₂×C₄×C₁₃	—	2	2	no	—
65	C₂×C₄×C₁₃	—	2	2	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	2	no	—
73	C₂×C₄×C₁₃	—	2	2	no	—
74	C₂×C₄×C₁₃	—	2	1	no	—
75	C₂×C₄×C₁₃	—	2	1	no	—
76	C₂×C₄×C₁₃	—	2	2	no	—
77	C₂×C₄×C₁₃	—	2	2	no	—
78	C₂×C₄×C₁₃	—	2	2	no	—
79	C₂×C₄×C₁₃	—	2	2	no	—
80	C₂×C₄×C₁₃	—	2	2	no	—
81	C₂×C₄×C₁₃	—	2	2	no	—
82	C₂×C₄×C₁₃	—	2	2	no	—
83	C₂×C₄×C₁₃	—	4	4	no	—
84	C₂×C₄×C₁₃	C₂×D₄	1	4	yes	—
85	C₂×C₄×C₁₃	#41	1	4	yes	—
86	C₂×#126	C₂×C₁₃	1	2	yes	—
87	#398	C₂×C₁₃	1	2	yes	complemented
88	#398	C₂×C₁₃	1	2	yes	complemented
89	#397	C₂×C₁₃	1	1	yes	characteristic
90	#397	C₂×C₁₃	1	1	yes	characteristic
91	#389	C₂×C₁₃	1	2	yes	complemented
92	#387	C₂×C₁₃	1	2	yes	complemented
93	#385	C₂×C₁₃	1	1	yes	characteristic
94	#380	C₂×C₁₃	1	2	yes	—
95	C₂²×C₁₃	—	2	2	no	—
96	C₂²×C₁₃	—	2	4	no	—
97	C₂²×C₁₃	—	4	2	no	—
98	C₂²×C₁₃	—	4	1	no	—
99	C₂²×C₁₃	—	4	1	no	—
100	C₂²×C₁₃	C₂²×D₄	1	1	yes	characteristic
101	C₂²×C₁₃	C₂×#41	1	2	yes	—
102	C₂²×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
103	C₂²×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
104	C₂²×C₁₃	2₋¹⁺⁴	1	2	yes	—
105	C₄×C₁₃	—	2	4	no	—
106	C₄×C₁₃	—	2	4	no	—
107	C₄×C₁₃	—	4	2	no	—
108	C₄×C₁₃	—	4	2	no	—
109	C₄×C₁₃	—	4	2	no	—
110	C₄×C₁₃	—	4	2	no	—
111	C₄×C₁₃	—	4	1	no	—
112	C₄×C₁₃	—	4	1	no	—
113	C₂³×C₄	C₂²×C₁₃	1	1	yes	characteristic
114	#126	—	2	8	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	2	yes	—
118	C₂×C₄⋊C₄	C₂²×C₁₃	1	2	yes	—
119	C₂×C₄⋊C₄	C₂²×C₁₃	1	2	yes	—
120	C₂×C₄⋊C₄	C₂²×C₁₃	1	2	yes	—
121	C₂×C₄⋊C₄	C₂²×C₁₃	1	1	yes	characteristic
122	C₂×C₄⋊C₄	C₂²×C₁₃	1	1	yes	characteristic
123	C₂×C₂²⋊C₄	C₂²×C₁₃	1	2	yes	—
124	C₂×C₂²⋊C₄	C₂²×C₁₃	1	2	yes	—
125	C₂×C₂²⋊C₄	C₂²×C₁₃	1	1	yes	characteristic
126	C₂×C₂²⋊C₄	C₂²×C₁₃	1	1	yes	characteristic
127	C₂×C₄²	C₂²×C₁₃	1	2	yes	—
128	#95	C₂²×C₁₃	1	2	yes	—
129	#95	C₂²×C₁₃	1	2	yes	—
130	#95	C₂²×C₁₃	1	2	yes	—
131	#95	C₂²×C₁₃	1	1	yes	characteristic
132	#95	C₂²×C₁₃	1	2	yes	—
133	#95	C₂²×C₁₃	1	2	yes	—
134	#95	C₂²×C₁₃	1	1	yes	characteristic
135	C₂×C₁₃	—	4	2	no	—
136	C₂×C₁₃	#537	1	1	yes	characteristic
137	C₂×C₁₃	#541	1	2	yes	—
138	C₂×C₁₃	#547	1	2	yes	—
139	C₂×C₁₃	#551	1	1	yes	characteristic
140	C₂×C₁₃	#563	1	1	yes	characteristic
141	C₂⁴	C₂³×C₁₃	1	1	yes	characteristic
142	C₂²×C₄	—	2	4	no	—
143	C₂²×C₄	—	4	1	no	—
144	C₂²×C₄	C₂³×C₁₃	1	1	yes	characteristic
145	C₂²×C₄	C₂³×C₁₃	1	2	yes	—
146	C₂²×C₄	C₂³×C₁₃	1	2	yes	—
147	C₂²×C₄	C₂³×C₁₃	1	1	yes	characteristic
148	C₂²×C₄	C₂³×C₁₃	1	2	yes	—
149	C₂²×C₄	C₂³×C₁₃	1	2	yes	—
150	C₂²×C₄	C₂³×C₁₃	1	2	yes	—
151	C₂²×C₄	C₂³×C₁₃	1	2	yes	—
152	C₄⋊C₄	—	2	4	no	—
153	C₄⋊C₄	—	2	4	no	—
154	C₄⋊C₄	—	2	4	no	—
155	C₄⋊C₄	—	2	4	no	—
156	C₄⋊C₄	—	2	4	no	—
157	C₄⋊C₄	—	4	2	no	—
158	C₄⋊C₄	—	4	1	no	—
159	C₄⋊C₄	—	4	1	no	—
160	C₂²⋊C₄	—	2	4	no	—
161	C₂²⋊C₄	—	2	4	no	—
162	C₂²⋊C₄	—	4	1	no	—
163	C₂²⋊C₄	D₄×C₁₃	1	4	yes	—
164	C₄²	—	2	4	no	—
165	C₁₃	#2722	1	1	yes	characteristic, direct product factor, Sylow
166	C₂³	—	2	2	no	—
167	C₂³	—	2	1	no	—
168	C₂³	—	2	1	no	—
169	C₂³	—	2	2	no	—
170	C₂³	C₂×D₄×C₁₃	1	2	yes	—
171	C₂³	C₂⁴×C₁₃	1	1	yes	characteristic
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	2	no	—
178	C₂×C₄	—	2	2	no	—
179	C₂×C₄	—	2	2	no	—
180	C₂×C₄	—	2	2	no	—
181	C₂×C₄	—	2	2	no	—
182	C₂×C₄	—	2	2	no	—
183	C₂×C₄	—	2	1	no	—
184	C₂×C₄	—	2	1	no	—
185	C₂×C₄	—	2	2	no	—
186	C₂×C₄	—	2	2	no	—
187	C₂×C₄	—	2	2	no	—
188	C₂×C₄	—	2	2	no	—
189	C₂×C₄	—	2	2	no	—
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	2	2	no	—
192	C₂×C₄	—	4	4	no	—
193	C₂×C₄	C₂×D₄×C₁₃	1	4	yes	—
194	C₂×C₄	C₁₃×#41	1	4	yes	—
195	C₂²	—	2	2	no	—
196	C₂²	—	2	4	no	—
197	C₂²	—	4	2	no	—
198	C₂²	—	4	1	no	—
199	C₂²	—	4	1	no	—
200	C₂²	C₂²×D₄×C₁₃	1	1	yes	characteristic
201	C₂²	C₂×C₁₃×#41	1	2	yes	—
202	C₂²	C₁₃×H(5,𝔽₂)	1	1	yes	characteristic
203	C₂²	C₁₃×2₋¹⁺⁴	1	1	yes	characteristic
204	C₂²	C₁₃×2₋¹⁺⁴	1	2	yes	—
205	C₄	—	2	4	no	—
206	C₄	—	2	4	no	—
207	C₄	—	4	2	no	—
208	C₄	—	4	2	no	—
209	C₄	—	4	2	no	—
210	C₄	—	4	2	no	—
211	C₄	—	4	1	no	—
212	C₄	—	4	1	no	—
213	C₂	—	4	2	no	—
214	C₂	C₁₃×#537	1	1	yes	characteristic
215	C₂	C₁₃×#541	1	2	yes	—
216	C₂	C₁₃×#547	1	2	yes	—
217	C₂	C₁₃×#551	1	1	yes	characteristic
218	C₂	C₁₃×#563	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₁₃	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₁₃
prime radical	C₂⁴×C₁₃	C₂³
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. C₁₃×#398	C₂	semidirect
3. C₁₃×#398	C₂	semidirect
6. C₁₃×#389	C₂	semidirect
7. C₁₃×#387	C₂	semidirect
56. #2722	C₁₃	direct (N × H)
87. #398	C₂×C₁₃	semidirect
88. #398	C₂×C₁₃	semidirect
91. #389	C₂×C₁₃	semidirect
92. #387	C₂×C₁₃	semidirect
165. C₁₃	#2722	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×#126	2
C₁₃×#398	2
C₁₃×#397	2
C₁₃×#389	2
C₁₃×#387	2
C₁₃×#385	2
C₁₃×#380	2
#2722	13

Subgroups — C13×#2722 #13111780

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#2722 #13111780