Subgroups — C₁₃×#2592 #13111650

Order 1664.

520 conjugacy classes of subgroups.

The subgroup lattice has 204 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₃×#124	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×Q₈×C₁₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₃×#397	C₂	1	2	yes	maximal, complemented
4	C₁₃×#393	C₂	1	2	yes	maximal, complemented
5	C₁₃×#391	C₂	1	2	yes	maximal, complemented
6	C₁₃×#390	C₂	1	2	yes	maximal
7	C₁₃×#388	C₂	1	4	yes	maximal, complemented
8	C₄×C₁₃×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×Q₈×C₁₃	C₂²	1	1	yes	characteristic
10	C₂²×Q₈×C₁₃	C₂²	1	1	yes	characteristic
11	C₂²×D₄×C₁₃	C₂²	1	1	yes	characteristic
12	C₁₃×#124	—	2	4	no	—
13	C₄×Q₈×C₁₃	—	2	4	no	—
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	1	yes	characteristic
17	C₂×C₁₃×C₂²⋊C₄	C₂²	1	2	yes	—
18	C₂×C₁₃×C₂²⋊C₄	C₂²	1	4	yes	—
19	C₂×C₁₃×C₂²⋊C₄	C₂²	1	2	yes	—
20	C₂×C₁₃×C₂²⋊C₄	C₂²	1	4	yes	—
21	C₂×C₄²×C₁₃	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	C₁₃×#95	C₂²	1	4	yes	—
26	C₁₃×#95	C₂²	1	4	yes	—
27	C₂⁴×C₁₃	C₂³	1	2	yes	—
28	C₂×Q₈×C₁₃	—	2	2	no	—
29	C₂×Q₈×C₁₃	—	2	2	no	—
30	C₂×Q₈×C₁₃	—	2	4	no	—
31	C₂×Q₈×C₁₃	—	2	4	no	—
32	C₂×D₄×C₁₃	—	2	2	no	—
33	C₂×D₄×C₁₃	—	2	4	no	—
34	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
35	C₂²×C₄×C₁₃	C₂³	1	4	yes	—
36	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
37	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
38	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
39	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
40	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
41	C₁₃×C₄⋊C₄	—	2	4	no	—
42	C₁₃×C₄⋊C₄	—	2	2	no	—
43	C₁₃×C₄⋊C₄	—	2	4	no	—
44	C₁₃×C₂²⋊C₄	—	2	4	no	—
45	C₁₃×C₂²⋊C₄	—	2	4	no	—
46	C₁₃×C₂²⋊C₄	—	2	8	no	—
47	C₁₃×C₂²⋊C₄	—	4	4	no	—
48	C₄²×C₁₃	—	2	4	no	—
49	C₄²×C₁₃	—	4	2	no	—
50	C₄²×C₁₃	D₄	1	4	yes	—
51	#2592	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
52	C₂³×C₁₃	—	2	2	no	—
53	C₂³×C₁₃	—	2	4	no	—
54	C₂³×C₁₃	—	2	4	no	—
55	C₂³×C₁₃	—	2	2	no	—
56	C₂³×C₁₃	—	2	2	no	—
57	C₂³×C₁₃	C₂⁴	1	1	yes	characteristic
58	Q₈×C₁₃	—	2	8	no	—
59	Q₈×C₁₃	—	4	4	no	—
60	D₄×C₁₃	—	4	4	no	—
61	C₂×C₄×C₁₃	—	2	2	no	—
62	C₂×C₄×C₁₃	—	2	2	no	—
63	C₂×C₄×C₁₃	—	2	2	no	—
64	C₂×C₄×C₁₃	—	2	1	no	—
65	C₂×C₄×C₁₃	—	2	1	no	—
66	C₂×C₄×C₁₃	—	2	2	no	—
67	C₂×C₄×C₁₃	—	2	2	no	—
68	C₂×C₄×C₁₃	—	2	4	no	—
69	C₂×C₄×C₁₃	—	2	4	no	—
70	C₂×C₄×C₁₃	—	2	4	no	—
71	C₂×C₄×C₁₃	—	2	4	no	—
72	C₂×C₄×C₁₃	—	2	2	no	—
73	C₂×C₄×C₁₃	—	2	2	no	—
74	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
75	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
76	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
77	C₂×C₄×C₁₃	#41	1	4	yes	—
78	C₂×C₄×C₁₃	#41	1	4	yes	—
79	C₂×#124	C₂×C₁₃	1	1	yes	characteristic
80	C₂×C₄×Q₈	C₂×C₁₃	1	1	yes	characteristic, complemented
81	#397	C₂×C₁₃	1	2	yes	complemented
82	#393	C₂×C₁₃	1	2	yes	complemented
83	#391	C₂×C₁₃	1	2	yes	complemented
84	#390	C₂×C₁₃	1	2	yes	—
85	#388	C₂×C₁₃	1	4	yes	complemented
86	C₄×C₄⋊C₄	C₂×C₁₃	1	1	yes	characteristic, complemented
87	C₂²×C₁₃	—	4	2	no	—
88	C₂²×C₁₃	—	4	2	no	—
89	C₂²×C₁₃	—	4	2	no	—
90	C₂²×C₁₃	—	4	4	no	—
91	C₂²×C₁₃	—	4	4	no	—
92	C₂²×C₁₃	C₂²×D₄	1	1	yes	characteristic
93	C₂²×C₁₃	C₂×#41	1	2	yes	—
94	C₂²×C₁₃	C₂×#41	1	2	yes	—
95	C₂²×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
96	C₂²×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
97	C₄×C₁₃	—	2	2	no	—
98	C₄×C₁₃	—	2	4	no	—
99	C₄×C₁₃	—	2	2	no	—
100	C₄×C₁₃	—	4	2	no	—
101	C₄×C₁₃	—	4	2	no	—
102	C₄×C₁₃	—	4	4	no	—
103	C₄×C₁₃	#124	1	4	yes	—
104	C₂²×Q₈	C₂²×C₁₃	1	1	yes	characteristic
105	C₂²×Q₈	C₂²×C₁₃	1	1	yes	characteristic
106	C₂²×D₄	C₂²×C₁₃	1	1	yes	characteristic
107	#124	—	2	4	no	—
108	C₄×Q₈	—	2	4	no	—
109	C₂×C₄⋊C₄	C₂²×C₁₃	1	2	yes	—
110	C₂×C₄⋊C₄	C₂²×C₁₃	1	1	yes	characteristic
111	C₂×C₄⋊C₄	C₂²×C₁₃	1	2	yes	—
112	C₂×C₂²⋊C₄	C₂²×C₁₃	1	2	yes	—
113	C₂×C₂²⋊C₄	C₂²×C₁₃	1	2	yes	—
114	C₂×C₂²⋊C₄	C₂²×C₁₃	1	4	yes	—
115	C₂×C₂²⋊C₄	C₂²×C₁₃	1	4	yes	—
116	C₂×C₄²	C₂²×C₁₃	1	2	yes	—
117	C₂×C₄²	C₂²×C₁₃	1	2	yes	—
118	C₂×C₄²	C₂²×C₁₃	1	1	yes	characteristic
119	#95	C₂²×C₁₃	1	2	yes	—
120	#95	C₂²×C₁₃	1	4	yes	—
121	#95	C₂²×C₁₃	1	4	yes	—
122	C₂×C₁₃	—	8	2	no	—
123	C₂×C₁₃	#525	1	1	yes	characteristic
124	C₂×C₁₃	C₂×#124	1	1	yes	characteristic
125	C₂×C₁₃	#537	1	1	yes	characteristic
126	C₂×C₁₃	#556	1	2	yes	—
127	C₂×C₁₃	#559	1	2	yes	—
128	C₂⁴	C₂³×C₁₃	1	2	yes	—
129	C₂×Q₈	—	2	2	no	—
130	C₂×Q₈	—	2	2	no	—
131	C₂×Q₈	—	2	4	no	—
132	C₂×Q₈	—	2	4	no	—
133	C₂×D₄	—	2	2	no	—
134	C₂×D₄	—	2	4	no	—
135	C₂²×C₄	C₂³×C₁₃	1	2	yes	—
136	C₂²×C₄	C₂³×C₁₃	1	4	yes	—
137	C₂²×C₄	C₂³×C₁₃	1	2	yes	—
138	C₂²×C₄	C₂³×C₁₃	1	1	yes	characteristic
139	C₂²×C₄	C₂³×C₁₃	1	1	yes	characteristic
140	C₂²×C₄	C₂³×C₁₃	1	2	yes	—
141	C₂²×C₄	C₂³×C₁₃	1	1	yes	characteristic
142	C₄⋊C₄	—	2	4	no	—
143	C₄⋊C₄	—	2	2	no	—
144	C₄⋊C₄	—	2	4	no	—
145	C₂²⋊C₄	—	2	4	no	—
146	C₂²⋊C₄	—	2	4	no	—
147	C₂²⋊C₄	—	2	8	no	—
148	C₂²⋊C₄	—	4	4	no	—
149	C₄²	—	2	4	no	—
150	C₄²	—	4	2	no	—
151	C₄²	D₄×C₁₃	1	4	yes	—
152	C₁₃	#2592	1	1	yes	characteristic, direct product factor, Sylow
153	C₂³	—	2	2	no	—
154	C₂³	—	2	4	no	—
155	C₂³	—	2	4	no	—
156	C₂³	—	2	2	no	—
157	C₂³	—	2	2	no	—
158	C₂³	C₂⁴×C₁₃	1	1	yes	characteristic
159	Q₈	—	2	8	no	—
160	Q₈	—	4	4	no	—
161	D₄	—	4	4	no	—
162	C₂×C₄	—	2	2	no	—
163	C₂×C₄	—	2	2	no	—
164	C₂×C₄	—	2	2	no	—
165	C₂×C₄	—	2	1	no	—
166	C₂×C₄	—	2	1	no	—
167	C₂×C₄	—	2	2	no	—
168	C₂×C₄	—	2	2	no	—
169	C₂×C₄	—	2	4	no	—
170	C₂×C₄	—	2	4	no	—
171	C₂×C₄	—	2	4	no	—
172	C₂×C₄	—	2	4	no	—
173	C₂×C₄	—	2	2	no	—
174	C₂×C₄	—	2	2	no	—
175	C₂×C₄	C₂×D₄×C₁₃	1	2	yes	—
176	C₂×C₄	C₂×D₄×C₁₃	1	2	yes	—
177	C₂×C₄	C₂×D₄×C₁₃	1	2	yes	—
178	C₂×C₄	C₁₃×#41	1	4	yes	—
179	C₂×C₄	C₁₃×#41	1	4	yes	—
180	C₂²	—	4	2	no	—
181	C₂²	—	4	2	no	—
182	C₂²	—	4	2	no	—
183	C₂²	—	4	4	no	—
184	C₂²	—	4	4	no	—
185	C₂²	C₂²×D₄×C₁₃	1	1	yes	characteristic
186	C₂²	C₂×C₁₃×#41	1	2	yes	—
187	C₂²	C₂×C₁₃×#41	1	2	yes	—
188	C₂²	C₁₃×H(5,𝔽₂)	1	1	yes	characteristic
189	C₂²	C₁₃×2₋¹⁺⁴	1	1	yes	characteristic
190	C₄	—	2	2	no	—
191	C₄	—	2	4	no	—
192	C₄	—	2	2	no	—
193	C₄	—	4	2	no	—
194	C₄	—	4	2	no	—
195	C₄	—	4	4	no	—
196	C₄	C₁₃×#124	1	4	yes	—
197	C₂	—	8	2	no	—
198	C₂	C₁₃×#525	1	1	yes	characteristic
199	C₂	C₂×C₁₃×#124	1	1	yes	characteristic
200	C₂	C₁₃×#537	1	1	yes	characteristic
201	C₂	C₁₃×#556	1	2	yes	—
202	C₂	C₁₃×#559	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₁₃	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₁₃
prime radical	C₂²×D₄×C₁₃	C₂²
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. C₂×C₄×Q₈×C₁₃	C₂	semidirect
3. C₁₃×#397	C₂	semidirect
4. C₁₃×#393	C₂	semidirect
5. C₁₃×#391	C₂	semidirect
7. C₁₃×#388	C₂	semidirect
8. C₄×C₁₃×C₄⋊C₄	C₂	semidirect
51. #2592	C₁₃	direct (N × H)
80. C₂×C₄×Q₈	C₂×C₁₃	semidirect
81. #397	C₂×C₁₃	semidirect
82. #393	C₂×C₁₃	semidirect
83. #391	C₂×C₁₃	semidirect
85. #388	C₂×C₁₃	semidirect
86. C₄×C₄⋊C₄	C₂×C₁₃	semidirect
152. C₁₃	#2592	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×#124	2
C₂×C₄×Q₈×C₁₃	2
C₁₃×#397	2
C₁₃×#393	2
C₁₃×#391	2
C₁₃×#390	2
C₁₃×#388	2
C₄×C₁₃×C₄⋊C₄	2
#2592	13

Subgroups — C13×#2592 #13111650

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#2592 #13111650