Subgroups — C₁₃×#21965 #15708304

Order 3328.

786 conjugacy classes of subgroups.

The subgroup lattice has 208 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₃×#423	C₂	1	1	yes	maximal, characteristic
2	C₁₃×#2283	C₂	1	2	yes	maximal
3	C₁₃×#1949	C₂	1	8	yes	maximal, complemented
4	C₁₃×#1809	C₂	1	4	yes	maximal
5	C₂²×C₁₃×M₁₆	C₂²	1	2	yes	—
6	C₂²×C₄²×C₁₃	C₂²	1	1	yes	characteristic
7	C₂×C₁₃×#107	C₂²	1	4	yes	—
8	C₂×C₁₃×#106	C₂²	1	4	yes	—
9	C₁₃×#423	C₄	1	8	yes	complemented
10	C₂×C₁₃×C₄⋊C₈	C₂²	1	4	yes	—
11	C₁₃×#383	—	2	2	no	—
12	C₁₃×#382	C₂²	1	4	yes	—
13	C₁₃×#381	—	2	2	no	—
14	C₁₃×#380	C₂²	1	4	yes	—
15	C₄×C₁₃×C₄⋊C₄	C₂²	1	4	yes	—
16	C₄×C₁₃×C₂²⋊C₄	—	2	2	no	—
17	C₁₃×#376	—	2	2	no	—
18	C₁₃×#340	C₂²	1	8	yes	—
19	C₂³×C₄×C₁₃	D₄	1	2	yes	—
20	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
21	C₂×C₁₃×M₁₆	—	4	2	no	—
22	C₂×C₁₃×M₁₆	—	4	2	no	—
23	C₂×C₁₃×M₁₆	C₂×C₄	1	8	yes	—
24	C₂²×C₈×C₁₃	C₂³	1	4	yes	—
25	C₂×C₁₃×C₄⋊C₄	C₂³	1	8	yes	—
26	C₂×C₁₃×C₂²⋊C₄	—	2	4	no	—
27	C₂×C₄²×C₁₃	—	2	4	no	—
28	C₂×C₄²×C₁₃	C₂×C₄	1	4	yes	—
29	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
30	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
31	C₂×C₄²×C₁₃	Q₈	1	4	yes	—
32	C₂×C₄²×C₁₃	C₂³	1	2	yes	—
33	C₁₃×#107	—	4	4	no	—
34	C₁₃×#106	—	4	4	no	—
35	C₁₃×C₄⋊C₈	C₂×C₄	1	16	yes	—
36	C₁₃×#95	—	2	4	no	—
37	C₁₃×#95	—	2	4	no	—
38	C₁₃×#95	—	2	4	no	—
39	C₁₃×#95	—	2	4	no	—
40	#21965	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
41	C₂⁴×C₁₃	C₂×D₄	1	1	yes	characteristic
42	C₂²×C₄×C₁₃	—	2	2	no	—
43	C₂²×C₄×C₁₃	—	2	2	no	—
44	C₂²×C₄×C₁₃	—	2	8	no	—
45	C₂²×C₄×C₁₃	—	4	4	no	—
46	C₂²×C₄×C₁₃	C₄⋊C₄	1	8	yes	—
47	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
48	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
49	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
50	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
51	C₂²×C₄×C₁₃	C₂×Q₈	1	2	yes	—
52	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
53	C₁₃×M₁₆	—	4	8	no	—
54	C₂×C₈×C₁₃	—	2	4	no	—
55	C₂×C₈×C₁₃	—	2	4	no	—
56	C₂×C₈×C₁₃	C₂²×C₄	1	8	yes	—
57	C₁₃×C₄⋊C₄	—	2	8	no	—
58	C₁₃×C₄⋊C₄	—	2	8	no	—
59	C₁₃×C₂²⋊C₄	—	4	8	no	—
60	C₄²×C₁₃	—	4	8	no	—
61	C₄²×C₁₃	C₄⋊C₄	1	4	yes	—
62	C₄²×C₁₃	C₄⋊C₄	1	4	yes	—
63	C₄²×C₁₃	C₂²×C₄	1	4	yes	—
64	C₄²×C₁₃	C₂×Q₈	1	4	yes	—
65	C₂×#423	C₂×C₁₃	1	1	yes	characteristic
66	#2283	C₂×C₁₃	1	2	yes	—
67	#1949	C₂×C₁₃	1	8	yes	complemented
68	#1809	C₂×C₁₃	1	4	yes	—
69	C₂³×C₁₃	—	2	2	no	—
70	C₂³×C₁₃	—	2	4	no	—
71	C₂³×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
72	C₂³×C₁₃	C₂²×D₄	1	1	yes	characteristic
73	C₂×C₄×C₁₃	—	2	2	no	—
74	C₂×C₄×C₁₃	—	2	4	no	—
75	C₂×C₄×C₁₃	—	2	4	no	—
76	C₂×C₄×C₁₃	—	2	2	no	—
77	C₂×C₄×C₁₃	—	2	2	no	—
78	C₂×C₄×C₁₃	—	2	4	no	—
79	C₂×C₄×C₁₃	—	4	8	no	—
80	C₂×C₄×C₁₃	—	4	8	no	—
81	C₂×C₄×C₁₃	—	4	8	no	—
82	C₂×C₄×C₁₃	—	4	8	no	—
83	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
84	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	4	yes	—
85	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	4	yes	—
86	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
87	C₂×C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
88	C₂×C₄×C₁₃	H(5,𝔽₂)	1	2	yes	—
89	C₂×C₄×C₁₃	2₋¹⁺⁴	1	2	yes	—
90	C₈×C₁₃	—	2	8	no	—
91	C₂²×M₁₆	C₂²×C₁₃	1	2	yes	—
92	C₂²×C₄²	C₂²×C₁₃	1	1	yes	characteristic
93	C₂×#107	C₂²×C₁₃	1	4	yes	—
94	C₂×#106	C₂²×C₁₃	1	4	yes	—
95	#423	C₄×C₁₃	1	8	yes	complemented
96	C₂×C₄⋊C₈	C₂²×C₁₃	1	4	yes	—
97	#383	—	2	2	no	—
98	#382	C₂²×C₁₃	1	4	yes	—
99	#381	—	2	2	no	—
100	#380	C₂²×C₁₃	1	4	yes	—
101	C₄×C₄⋊C₄	C₂²×C₁₃	1	4	yes	—
102	C₄×C₂²⋊C₄	—	2	2	no	—
103	#376	—	2	2	no	—
104	#340	C₂²×C₁₃	1	8	yes	—
105	C₂²×C₁₃	—	2	2	no	—
106	C₂²×C₁₃	—	2	4	no	—
107	C₂²×C₁₃	—	4	4	no	—
108	C₂²×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
109	C₂²×C₁₃	#537	1	2	yes	—
110	C₂²×C₁₃	#575	1	4	yes	—
111	C₄×C₁₃	—	2	2	no	—
112	C₄×C₁₃	—	2	4	no	—
113	C₄×C₁₃	—	2	4	no	—
114	C₄×C₁₃	—	8	8	no	—
115	C₄×C₁₃	#518	1	1	yes	characteristic
116	C₄×C₁₃	#519	1	1	yes	characteristic
117	C₄×C₁₃	#544	1	2	yes	—
118	C₂³×C₄	D₄×C₁₃	1	2	yes	—
119	C₂³×C₄	C₂³×C₁₃	1	1	yes	characteristic
120	C₂×M₁₆	—	4	2	no	—
121	C₂×M₁₆	—	4	2	no	—
122	C₂×M₁₆	C₂×C₄×C₁₃	1	8	yes	—
123	C₂²×C₈	C₂³×C₁₃	1	4	yes	—
124	C₂×C₄⋊C₄	C₂³×C₁₃	1	8	yes	—
125	C₂×C₂²⋊C₄	—	2	4	no	—
126	C₂×C₄²	—	2	4	no	—
127	C₂×C₄²	C₂×C₄×C₁₃	1	4	yes	—
128	C₂×C₄²	D₄×C₁₃	1	1	yes	characteristic
129	C₂×C₄²	D₄×C₁₃	1	1	yes	characteristic
130	C₂×C₄²	Q₈×C₁₃	1	4	yes	—
131	C₂×C₄²	C₂³×C₁₃	1	2	yes	—
132	#107	—	4	4	no	—
133	#106	—	4	4	no	—
134	C₄⋊C₈	C₂×C₄×C₁₃	1	16	yes	—
135	#95	—	2	4	no	—
136	#95	—	2	4	no	—
137	#95	—	2	4	no	—
138	#95	—	2	4	no	—
139	C₂×C₁₃	—	4	2	no	—
140	C₂×C₁₃	#2316	1	1	yes	characteristic
141	C₂×C₁₃	#2911	1	2	yes	—
142	C₂×C₁₃	#3256	1	4	yes	—
143	C₂⁴	C₂×D₄×C₁₃	1	1	yes	characteristic
144	C₂²×C₄	—	2	2	no	—
145	C₂²×C₄	—	2	2	no	—
146	C₂²×C₄	—	2	8	no	—
147	C₂²×C₄	—	4	4	no	—
148	C₂²×C₄	C₁₃×C₄⋊C₄	1	8	yes	—
149	C₂²×C₄	C₂²×C₄×C₁₃	1	2	yes	—
150	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
151	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
152	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
153	C₂²×C₄	C₂×Q₈×C₁₃	1	2	yes	—
154	C₂²×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
155	M₁₆	—	4	8	no	—
156	C₂×C₈	—	2	4	no	—
157	C₂×C₈	—	2	4	no	—
158	C₂×C₈	C₂²×C₄×C₁₃	1	8	yes	—
159	C₄⋊C₄	—	2	8	no	—
160	C₄⋊C₄	—	2	8	no	—
161	C₂²⋊C₄	—	4	8	no	—
162	C₄²	—	4	8	no	—
163	C₄²	C₁₃×C₄⋊C₄	1	4	yes	—
164	C₄²	C₁₃×C₄⋊C₄	1	4	yes	—
165	C₄²	C₂²×C₄×C₁₃	1	4	yes	—
166	C₄²	C₂×Q₈×C₁₃	1	4	yes	—
167	C₁₃	#21965	1	1	yes	characteristic, direct product factor, Sylow
168	C₂³	—	2	2	no	—
169	C₂³	—	2	4	no	—
170	C₂³	C₂×C₁₃×C₄⋊C₄	1	2	yes	—
171	C₂³	C₂²×D₄×C₁₃	1	1	yes	characteristic
172	C₂×C₄	—	2	2	no	—
173	C₂×C₄	—	2	4	no	—
174	C₂×C₄	—	2	4	no	—
175	C₂×C₄	—	2	2	no	—
176	C₂×C₄	—	2	2	no	—
177	C₂×C₄	—	2	4	no	—
178	C₂×C₄	—	4	8	no	—
179	C₂×C₄	—	4	8	no	—
180	C₂×C₄	—	4	8	no	—
181	C₂×C₄	—	4	8	no	—
182	C₂×C₄	C₂×C₁₃×C₄⋊C₄	1	2	yes	—
183	C₂×C₄	C₂×C₁₃×C₄⋊C₄	1	4	yes	—
184	C₂×C₄	C₂×C₁₃×C₄⋊C₄	1	4	yes	—
185	C₂×C₄	C₂³×C₄×C₁₃	1	1	yes	characteristic
186	C₂×C₄	C₂²×Q₈×C₁₃	1	1	yes	characteristic
187	C₂×C₄	C₁₃×H(5,𝔽₂)	1	2	yes	—
188	C₂×C₄	C₁₃×2₋¹⁺⁴	1	2	yes	—
189	C₈	—	2	8	no	—
190	C₂²	—	2	2	no	—
191	C₂²	—	2	4	no	—
192	C₂²	—	4	4	no	—
193	C₂²	C₂²×C₁₃×C₄⋊C₄	1	1	yes	characteristic
194	C₂²	C₁₃×#537	1	2	yes	—
195	C₂²	C₁₃×#575	1	4	yes	—
196	C₄	—	2	2	no	—
197	C₄	—	2	4	no	—
198	C₄	—	2	4	no	—
199	C₄	—	8	8	no	—
200	C₄	C₁₃×#518	1	1	yes	characteristic
201	C₄	C₁₃×#519	1	1	yes	characteristic
202	C₄	C₁₃×#544	1	2	yes	—
203	C₂	—	4	2	no	—
204	C₂	C₁₃×#2316	1	1	yes	characteristic
205	C₂	C₁₃×#2911	1	2	yes	—
206	C₂	C₁₃×#3256	1	4	yes	—

	subgroup	G/H
center, socle	C₂³×C₁₃	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₁₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃
prime radical	C₂⁴×C₁₃	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₃ ◁ C₂²×C₄²×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₁₃×#1949	C₂	semidirect
9. C₁₃×#423	C₄	semidirect
40. #21965	C₁₃	direct (N × H)
67. #1949	C₂×C₁₃	semidirect
95. #423	C₄×C₁₃	semidirect
167. C₁₃	#21965	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×#423	2
C₁₃×#2283	2
C₁₃×#1949	2
C₁₃×#1809	2
#21965	13

Subgroups — C13×#21965 #15708304

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#21965 #15708304