Subgroups — C₂×C₄×C₁₃×D₁₆ #15719783

Order 3328.

806 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₁₃×D₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×C₁₃×D₈	C₂	1	2	yes	maximal, complemented
3	C₄×C₁₃×D₁₆	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₁₃×#366	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₁₃×#357	C₂	1	2	yes	maximal, complemented
6	C₂×C₄×C₁₃×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×C₁₃×D₈	C₂²	1	2	yes	—
8	C₂×C₄×D₄×C₁₃	—	2	2	no	—
9	C₂×C₁₃×D₁₆	C₄	1	8	yes	direct product factor
10	C₂×C₁₃×D₁₆	C₂²	1	4	yes	—
11	C₂²×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
12	C₂²×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
13	C₄×C₁₃×D₈	—	2	4	no	—
14	C₄×C₁₃×D₈	C₂²	1	8	yes	complemented
15	C₂×C₁₃×#107	C₂²	1	2	yes	—
16	C₂×C₁₃×#102	—	2	2	no	—
17	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
18	C₁₃×#366	C₂²	1	4	yes	complemented
19	C₁₃×#357	C₂²	1	8	yes	complemented
20	C₄×C₁₃×C₁₆	C₂²	1	4	yes	complemented
21	C₂²×D₄×C₁₃	—	2	2	no	—
22	C₂³×C₄×C₁₃	—	4	2	no	—
23	C₂×C₁₃×D₈	—	2	2	no	—
24	C₂×C₁₃×D₈	—	2	4	no	—
25	C₂×C₁₃×D₈	C₂×C₄	1	8	yes	complemented
26	C₂×C₁₃×D₈	C₂³	1	4	yes	—
27	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
28	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
29	C₄×D₄×C₁₃	—	2	8	no	—
30	C₄×D₄×C₁₃	—	4	4	no	—
31	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
32	C₂×C₁₃×C₂²⋊C₄	—	4	2	no	—
33	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
34	C₁₃×D₁₆	C₂×C₄	1	16	yes	direct product factor
35	C₂×C₁₃×C₁₆	—	2	2	no	—
36	C₂×C₁₃×C₁₆	C₂×C₄	1	4	yes	complemented
37	C₂×C₁₃×C₁₆	C₂³	1	2	yes	—
38	C₂×C₁₃×C₁₆	C₂³	1	2	yes	—
39	C₁₃×#107	—	2	2	no	—
40	C₁₃×#107	C₂³	1	4	yes	—
41	C₁₃×#102	—	2	8	no	—
42	C₄×C₈×C₁₃	D₄	1	2	yes	—
43	C₄×C₈×C₁₃	C₂³	1	2	yes	—
44	C₂×C₄×D₁₆	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
45	C₂⁴×C₁₃	—	4	2	no	—
46	C₂×D₄×C₁₃	—	2	8	no	—
47	C₂×D₄×C₁₃	—	2	4	no	—
48	C₂×D₄×C₁₃	—	4	2	no	—
49	C₂×D₄×C₁₃	—	4	4	no	—
50	C₂²×C₄×C₁₃	—	4	2	no	—
51	C₂²×C₄×C₁₃	—	4	8	no	—
52	C₂²×C₄×C₁₃	—	8	8	no	—
53	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
54	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
55	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
56	C₁₃×D₈	—	2	8	no	—
57	C₁₃×D₈	—	2	4	no	—
58	C₁₃×D₈	C₂²×C₄	1	8	yes	complemented
59	C₂×C₈×C₁₃	—	2	2	no	—
60	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
61	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
62	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
63	C₂×C₈×C₁₃	#41	1	2	yes	—
64	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
65	C₁₃×C₄⋊C₄	—	2	4	no	—
66	C₁₃×C₄⋊C₄	—	4	2	no	—
67	C₁₃×C₂²⋊C₄	—	4	8	no	—
68	C₄²×C₁₃	D₈	1	2	yes	—
69	C₄²×C₁₃	C₂×D₄	1	2	yes	—
70	C₁₃×C₁₆	—	2	2	no	—
71	C₁₃×C₁₆	C₂²×C₄	1	4	yes	complemented
72	C₂²×D₁₆	C₂×C₁₃	1	1	yes	characteristic
73	C₂×C₄×D₈	C₂×C₁₃	1	2	yes	complemented
74	C₄×D₁₆	C₂×C₁₃	1	8	yes	direct product factor
75	C₂×#366	C₂×C₁₃	1	1	yes	characteristic, complemented
76	C₂×#357	C₂×C₁₃	1	2	yes	complemented
77	C₂×C₄×C₁₆	C₂×C₁₃	1	1	yes	characteristic, complemented
78	C₂³×C₁₃	—	4	4	no	—
79	C₂³×C₁₃	—	4	8	no	—
80	C₂³×C₁₃	—	8	4	no	—
81	C₂³×C₁₃	—	8	4	no	—
82	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
83	D₄×C₁₃	—	2	8	no	—
84	D₄×C₁₃	—	4	4	no	—
85	D₄×C₁₃	—	4	8	no	—
86	C₂×C₄×C₁₃	—	2	2	no	—
87	C₂×C₄×C₁₃	—	4	4	no	—
88	C₂×C₄×C₁₃	—	8	4	no	—
89	C₂×C₄×C₁₃	—	8	16	no	—
90	C₂×C₄×C₁₃	D₁₆	1	4	yes	direct product factor
91	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
92	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
93	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
94	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
95	C₂×C₄×C₁₃	#135	1	2	yes	—
96	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
97	C₈×C₁₃	—	2	2	no	—
98	C₈×C₁₃	C₄×D₄	1	2	yes	—
99	C₈×C₁₃	C₂³×C₄	1	1	yes	characteristic
100	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
101	C₂²×D₈	C₂²×C₁₃	1	2	yes	—
102	C₂×C₄×D₄	—	2	2	no	—
103	C₂×D₁₆	C₄×C₁₃	1	8	yes	direct product factor
104	C₂×D₁₆	C₂²×C₁₃	1	4	yes	—
105	C₂²×C₁₆	C₂²×C₁₃	1	1	yes	characteristic
106	C₂²×C₁₆	C₂²×C₁₃	1	1	yes	characteristic
107	C₄×D₈	—	2	4	no	—
108	C₄×D₈	C₂²×C₁₃	1	8	yes	complemented
109	C₂×#107	C₂²×C₁₃	1	2	yes	—
110	C₂×#102	—	2	2	no	—
111	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
112	#366	C₂²×C₁₃	1	4	yes	complemented
113	#357	C₂²×C₁₃	1	8	yes	complemented
114	C₄×C₁₆	C₂²×C₁₃	1	4	yes	complemented
115	C₂²×C₁₃	—	4	8	no	—
116	C₂²×C₁₃	—	8	4	no	—
117	C₂²×C₁₃	—	8	4	no	—
118	C₂²×C₁₃	—	8	16	no	—
119	C₂²×C₁₃	C₄×D₈	1	2	yes	—
120	C₂²×C₁₃	C₂×D₁₆	1	2	yes	—
121	C₂²×C₁₃	#508	1	2	yes	—
122	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
123	C₄×C₁₃	—	2	2	no	—
124	C₄×C₁₃	—	8	4	no	—
125	C₄×C₁₃	C₄×D₈	1	2	yes	—
126	C₄×C₁₃	C₂×D₁₆	1	4	yes	direct product factor
127	C₄×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
128	C₄×C₁₃	C₂×#135	1	1	yes	characteristic
129	C₂²×D₄	—	2	2	no	—
130	C₂³×C₄	—	4	2	no	—
131	C₂×D₈	—	2	2	no	—
132	C₂×D₈	—	2	4	no	—
133	C₂×D₈	C₂×C₄×C₁₃	1	8	yes	complemented
134	C₂×D₈	C₂³×C₁₃	1	4	yes	—
135	C₂²×C₈	D₄×C₁₃	1	1	yes	characteristic
136	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
137	C₄×D₄	—	2	8	no	—
138	C₄×D₄	—	4	4	no	—
139	C₂×C₄⋊C₄	—	2	2	no	—
140	C₂×C₂²⋊C₄	—	4	2	no	—
141	C₂×C₄²	D₄×C₁₃	1	1	yes	characteristic
142	D₁₆	C₂×C₄×C₁₃	1	16	yes	direct product factor
143	C₂×C₁₆	—	2	2	no	—
144	C₂×C₁₆	C₂×C₄×C₁₃	1	4	yes	complemented
145	C₂×C₁₆	C₂³×C₁₃	1	2	yes	—
146	C₂×C₁₆	C₂³×C₁₃	1	2	yes	—
147	#107	—	2	2	no	—
148	#107	C₂³×C₁₃	1	4	yes	—
149	#102	—	2	8	no	—
150	C₄×C₈	D₄×C₁₃	1	2	yes	—
151	C₄×C₈	C₂³×C₁₃	1	2	yes	—
152	C₂×C₁₃	—	8	8	no	—
153	C₂×C₁₃	C₄×D₁₆	1	4	yes	direct product factor
154	C₂×C₁₃	C₂×C₄×D₈	1	1	yes	characteristic
155	C₂×C₁₃	C₂²×D₁₆	1	1	yes	characteristic
156	C₂×C₁₃	C₂×#508	1	1	yes	characteristic
157	C₂⁴	—	4	2	no	—
158	C₂×D₄	—	2	8	no	—
159	C₂×D₄	—	2	4	no	—
160	C₂×D₄	—	4	2	no	—
161	C₂×D₄	—	4	4	no	—
162	C₂²×C₄	—	4	2	no	—
163	C₂²×C₄	—	4	8	no	—
164	C₂²×C₄	—	8	8	no	—
165	C₂²×C₄	C₁₃×D₈	1	1	yes	characteristic
166	C₂²×C₄	C₁₃×D₈	1	1	yes	characteristic
167	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
168	D₈	—	2	8	no	—
169	D₈	—	2	4	no	—
170	D₈	C₂²×C₄×C₁₃	1	8	yes	complemented
171	C₂×C₈	—	2	2	no	—
172	C₂×C₈	C₂²×C₄×C₁₃	1	2	yes	—
173	C₂×C₈	C₂×D₄×C₁₃	1	2	yes	—
174	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
175	C₂×C₈	C₁₃×#41	1	2	yes	—
176	C₂×C₈	C₂⁴×C₁₃	1	1	yes	characteristic
177	C₄⋊C₄	—	2	4	no	—
178	C₄⋊C₄	—	4	2	no	—
179	C₂²⋊C₄	—	4	8	no	—
180	C₄²	C₁₃×D₈	1	2	yes	—
181	C₄²	C₂×D₄×C₁₃	1	2	yes	—
182	C₁₆	—	2	2	no	—
183	C₁₆	C₂²×C₄×C₁₃	1	4	yes	complemented
184	C₁₃	C₂×C₄×D₁₆	1	1	yes	characteristic, direct product factor, Sylow
185	C₂³	—	4	8	no	—
186	C₂³	—	4	4	no	—
187	C₂³	—	8	4	no	—
188	C₂³	—	8	4	no	—
189	C₂³	C₂×C₁₃×D₈	1	1	yes	characteristic
190	D₄	—	2	8	no	—
191	D₄	—	4	8	no	—
192	D₄	—	4	4	no	—
193	C₂×C₄	—	2	2	no	—
194	C₂×C₄	—	4	4	no	—
195	C₂×C₄	—	8	4	no	—
196	C₂×C₄	—	8	16	no	—
197	C₂×C₄	C₁₃×D₁₆	1	4	yes	direct product factor
198	C₂×C₄	C₄×D₄×C₁₃	1	2	yes	—
199	C₂×C₄	C₂×C₁₃×D₈	1	2	yes	—
200	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
201	C₂×C₄	C₂×C₁₃×D₈	1	2	yes	—
202	C₂×C₄	C₁₃×#135	1	2	yes	—
203	C₂×C₄	C₂²×D₄×C₁₃	1	1	yes	characteristic
204	C₈	—	2	2	no	—
205	C₈	C₄×D₄×C₁₃	1	2	yes	—
206	C₈	C₂³×C₄×C₁₃	1	1	yes	characteristic
207	C₈	C₂×C₁₃×#41	1	1	yes	characteristic
208	C₂²	—	4	8	no	—
209	C₂²	—	8	4	no	—
210	C₂²	—	8	4	no	—
211	C₂²	—	8	16	no	—
212	C₂²	C₄×C₁₃×D₈	1	2	yes	—
213	C₂²	C₂×C₁₃×D₁₆	1	2	yes	—
214	C₂²	C₁₃×#508	1	2	yes	—
215	C₂²	C₂²×C₁₃×D₈	1	1	yes	characteristic
216	C₄	—	2	2	no	—
217	C₄	—	8	4	no	—
218	C₄	C₄×C₁₃×D₈	1	2	yes	—
219	C₄	C₂×C₁₃×D₁₆	1	4	yes	direct product factor
220	C₄	C₂×C₄×D₄×C₁₃	1	1	yes	characteristic
221	C₄	C₂×C₁₃×#135	1	1	yes	characteristic
222	C₂	—	8	8	no	—
223	C₂	C₄×C₁₃×D₁₆	1	4	yes	direct product factor
224	C₂	C₂×C₄×C₁₃×D₈	1	1	yes	characteristic
225	C₂	C₂²×C₁₃×D₁₆	1	1	yes	characteristic
226	C₂	C₂×C₁₃×#508	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₁₃	D₈
commutator	C₈	C₂³×C₄×C₁₃
Frattini, cosocle	C₂×C₈	C₂⁴×C₁₃
socle	C₂³×C₁₃	C₂×D₈
prime radical	C₂²×C₁₃×D₁₆	C₂
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₁₃ ◁ C₂×C₄²×C₁₃ ◁ C₂×C₄×C₈×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₄×C₁₃×D₈	C₂	semidirect
3. C₄×C₁₃×D₁₆	C₂	direct (N × H)
4. C₂×C₁₃×#366	C₂	semidirect
5. C₂×C₁₃×#357	C₂	semidirect
6. C₂×C₄×C₁₃×C₁₆	C₂	semidirect
9. C₂×C₁₃×D₁₆	C₄	direct (N × H)
14. C₄×C₁₃×D₈	C₂²	semidirect
18. C₁₃×#366	C₂²	semidirect
19. C₁₃×#357	C₂²	semidirect
20. C₄×C₁₃×C₁₆	C₂²	semidirect
25. C₂×C₁₃×D₈	C₂×C₄	semidirect
34. C₁₃×D₁₆	C₂×C₄	direct (N × H)
36. C₂×C₁₃×C₁₆	C₂×C₄	semidirect
44. C₂×C₄×D₁₆	C₁₃	direct (N × H)
58. C₁₃×D₈	C₂²×C₄	semidirect
71. C₁₃×C₁₆	C₂²×C₄	semidirect
73. C₂×C₄×D₈	C₂×C₁₃	semidirect
74. C₄×D₁₆	C₂×C₁₃	direct (N × H)
75. C₂×#366	C₂×C₁₃	semidirect
76. C₂×#357	C₂×C₁₃	semidirect
77. C₂×C₄×C₁₆	C₂×C₁₃	semidirect
90. C₂×C₄×C₁₃	D₁₆	direct (N × H)
103. C₂×D₁₆	C₄×C₁₃	direct (N × H)
108. C₄×D₈	C₂²×C₁₃	semidirect
112. #366	C₂²×C₁₃	semidirect
113. #357	C₂²×C₁₃	semidirect
114. C₄×C₁₆	C₂²×C₁₃	semidirect
126. C₄×C₁₃	C₂×D₁₆	direct (N × H)
133. C₂×D₈	C₂×C₄×C₁₃	semidirect
142. D₁₆	C₂×C₄×C₁₃	direct (N × H)
144. C₂×C₁₆	C₂×C₄×C₁₃	semidirect
153. C₂×C₁₃	C₄×D₁₆	direct (N × H)
170. D₈	C₂²×C₄×C₁₃	semidirect
183. C₁₆	C₂²×C₄×C₁₃	semidirect
184. C₁₃	C₂×C₄×D₁₆	direct (N × H)
197. C₂×C₄	C₁₃×D₁₆	direct (N × H)
219. C₄	C₂×C₁₃×D₁₆	direct (N × H)
223. C₂	C₄×C₁₃×D₁₆	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₁₃×D₁₆	2
C₂×C₄×C₁₃×D₈	2
C₄×C₁₃×D₁₆	2
C₂×C₁₃×#366	2
C₂×C₁₃×#357	2
C₂×C₄×C₁₃×C₁₆	2
C₂×C₄×D₁₆	13

Subgroups — C2×C4×C13×D16 #15719783

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄×C₁₃×D₁₆ #15719783