Subgroups — C₄×Q₁₆×D₁₃ #13123597

Order 1664.

488 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₁₆×D₁₃	C₂	1	1	yes	maximal, characteristic
2	C₄×Q₈×D₁₃	C₂	1	2	yes	maximal
3	C₄×C₁₃×Q₁₆	C₂	1	1	yes	maximal, characteristic, complemented
4	Q₁₆×Dic₁₃	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄×#6107	C₂	1	2	yes	maximal, complemented
6	D₁₃×#107	C₂	1	1	yes	maximal, characteristic
7	#11718534	C₂	1	1	yes	maximal, characteristic, complemented
8	D₁₃×#103	C₂	1	2	yes	maximal
9	#11718451	C₂	1	2	yes	maximal, complemented
10	C₄×Dic₅₂	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×C₈×D₁₃	C₂	1	1	yes	maximal, characteristic
12	C₂×Q₈×D₁₃	C₂²	1	2	yes	—
13	C₂×C₁₃×Q₁₆	C₂²	1	1	yes	characteristic
14	C₄×Q₈×C₁₃	C₂²	1	2	yes	—
15	Q₈×Dic₁₃	C₂²	1	2	yes	—
16	C₂×#6107	C₂²	1	2	yes	—
17	Q₁₆×D₁₃	C₄	1	8	yes	direct product factor
18	C₂×Dic₅₂	C₂²	1	1	yes	characteristic
19	C₂×C₈×D₁₃	C₂²	1	1	yes	characteristic
20	C₂×C₈×D₁₃	C₂²	1	1	yes	characteristic
21	C₄⋊C₄×D₁₃	—	2	2	no	—
22	C₄⋊C₄×D₁₃	C₂²	1	2	yes	—
23	#88141	—	2	2	no	—
24	#88141	C₂²	1	2	yes	—
25	C₄²×D₁₃	—	2	2	no	—
26	C₄²×D₁₃	C₂²	1	1	yes	characteristic
27	C₄×Dic₂₆	—	2	2	no	—
28	C₄×Dic₂₆	C₂²	1	2	yes	—
29	C₁₃×#107	C₂²	1	1	yes	characteristic
30	C₁₃×#103	C₂²	1	2	yes	—
31	C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
32	#88075	C₂²	1	2	yes	—
33	#88058	C₂²	1	1	yes	characteristic
34	#88056	C₂²	1	2	yes	—
35	C₈×Dic₁₃	C₂²	1	1	yes	characteristic
36	C₈×Dic₁₃	C₂²	1	1	yes	characteristic
37	#88050	C₂²	1	2	yes	—
38	#88047	C₂²	1	2	yes	—
39	C₄×#1055	C₂²	1	1	yes	characteristic
40	C₂×Q₈×C₁₃	C₂³	1	2	yes	—
41	Q₈×D₁₃	—	2	4	no	—
42	Q₈×D₁₃	C₂×C₄	1	8	yes	—
43	C₂×C₄×D₁₃	—	2	2	no	—
44	C₂×C₄×D₁₃	—	2	2	no	—
45	C₂×C₄×D₁₃	D₄	1	1	yes	characteristic
46	C₂×C₄×D₁₃	D₄	1	1	yes	characteristic
47	C₂×C₄×D₁₃	C₂³	1	1	yes	characteristic
48	C₂×Dic₂₆	—	2	2	no	—
49	C₂×Dic₂₆	C₂³	1	2	yes	—
50	C₁₃×Q₁₆	C₂×C₄	1	4	yes	complemented
51	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
52	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
53	C₁₃×C₄⋊C₄	—	2	2	no	—
54	C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
55	C₄²×C₁₃	—	2	2	no	—
56	C₄²×C₁₃	C₂³	1	1	yes	characteristic
57	#6107	C₂×C₄	1	8	yes	complemented
58	C₄⋊Dic₁₃	—	2	2	no	—
59	C₄⋊Dic₁₃	C₂³	1	2	yes	—
60	#6101	—	2	2	no	—
61	#6101	—	2	2	no	—
62	#6101	—	2	2	no	—
63	C₄×Dic₁₃	—	2	2	no	—
64	C₄×Dic₁₃	—	2	2	no	—
65	C₄×Dic₁₃	—	2	2	no	—
66	C₄×Dic₁₃	D₄	1	1	yes	characteristic
67	C₄×Dic₁₃	D₄	1	1	yes	characteristic
68	C₄×Dic₁₃	C₂³	1	1	yes	characteristic
69	C₂×#1055	C₂³	1	1	yes	characteristic
70	C₂×#1055	C₂³	1	1	yes	characteristic
71	Dic₅₂	C₂×C₄	1	4	yes	complemented
72	C₈×D₁₃	—	2	2	no	—
73	C₈×D₁₃	C₂×C₄	1	4	yes	—
74	C₂×C₄×Q₁₆	—	13	1	no	maximal, Sylow
75	C₂²×D₁₃	C₂×D₄	1	1	yes	characteristic
76	Q₈×C₁₃	—	2	2	no	—
77	Q₈×C₁₃	C₂²×C₄	1	4	yes	—
78	C₂×C₄×C₁₃	—	2	2	no	—
79	C₂×C₄×C₁₃	—	2	2	no	—
80	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
81	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
82	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
83	C₂×Dic₁₃	—	2	2	no	—
84	C₂×Dic₁₃	—	2	2	no	—
85	C₂×Dic₁₃	C₂×D₄	1	1	yes	characteristic
86	C₂×Dic₁₃	C₂×D₄	1	1	yes	characteristic
87	C₂×Dic₁₃	C₂×D₄	1	1	yes	characteristic
88	C₄×D₁₃	—	2	8	no	—
89	C₄×D₁₃	—	2	2	no	—
90	C₄×D₁₃	—	4	4	no	—
91	C₄×D₁₃	Q₁₆	1	4	yes	direct product factor
92	C₄×D₁₃	C₂²×C₄	1	2	yes	—
93	C₄×D₁₃	#41	1	2	yes	—
94	Dic₂₆	—	2	2	no	—
95	Dic₂₆	—	2	8	no	—
96	Dic₂₆	C₂²×C₄	1	4	yes	—
97	C₈×C₁₃	—	2	1	no	—
98	C₈×C₁₃	C₂²×C₄	1	2	yes	—
99	#1055	—	2	1	no	—
100	#1055	C₂²×C₄	1	2	yes	—
101	C₂²×Q₁₆	—	13	1	no	—
102	C₂×C₄×Q₈	—	13	2	no	—
103	C₄×Q₁₆	—	13	1	no	—
104	C₄×Q₁₆	—	13	2	no	—
105	C₄×Q₁₆	—	13	1	no	—
106	C₄×Q₁₆	—	13	2	no	—
107	C₄×Q₁₆	—	13	1	no	—
108	C₄×Q₁₆	D₁₃	1	1	yes	characteristic, direct product factor
109	C₂×#107	—	13	1	no	—
110	C₂×#103	—	13	2	no	—
111	C₂×C₄×C₈	—	13	1	no	—
112	C₂²×C₁₃	C₂²×D₄	1	1	yes	characteristic
113	C₂×D₁₃	C₄×D₄	1	2	yes	—
114	C₂×D₁₃	C₂×Q₁₆	1	2	yes	—
115	C₂×D₁₃	#135	1	2	yes	—
116	C₄×C₁₃	—	2	4	no	—
117	C₄×C₁₃	—	2	1	no	—
118	C₄×C₁₃	—	4	2	no	—
119	C₄×C₁₃	C₂×Q₁₆	1	2	yes	complemented
120	C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
121	C₄×C₁₃	C₂×#41	1	1	yes	characteristic
122	Dic₁₃	—	2	4	no	—
123	Dic₁₃	—	2	1	no	—
124	Dic₁₃	—	4	2	no	—
125	Dic₁₃	C₄×D₄	1	2	yes	—
126	Dic₁₃	C₂×Q₁₆	1	2	yes	complemented
127	C₂²×Q₈	—	13	2	no	—
128	C₂×Q₁₆	—	13	2	no	—
129	C₂×Q₁₆	—	13	8	no	—
130	C₂×Q₁₆	—	13	1	no	—
131	C₂×Q₁₆	C₂×D₁₃	1	1	yes	characteristic
132	C₂²×C₈	—	13	1	no	—
133	C₂²×C₈	—	13	1	no	—
134	C₄×Q₈	—	13	2	no	—
135	C₄×Q₈	—	13	2	no	—
136	C₄×Q₈	—	13	2	no	—
137	C₄×Q₈	—	26	2	no	—
138	C₄×Q₈	—	26	2	no	—
139	C₄×Q₈	C₂×D₁₃	1	2	yes	—
140	C₂×C₄⋊C₄	—	13	2	no	—
141	C₂×C₄⋊C₄	—	26	2	no	—
142	C₂×C₄²	—	13	1	no	—
143	C₂×C₄²	—	26	2	no	—
144	#107	—	13	1	no	—
145	#107	—	13	2	no	—
146	#107	C₂×D₁₃	1	1	yes	characteristic
147	#103	—	13	2	no	—
148	#103	—	13	2	no	—
149	#103	—	13	2	no	—
150	#103	C₂×D₁₃	1	2	yes	—
151	C₄×C₈	—	13	1	no	—
152	C₄×C₈	—	13	1	no	—
153	C₄×C₈	—	13	1	no	—
154	C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
155	C₂×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
156	C₂×C₁₃	C₂²×Q₁₆	1	1	yes	characteristic
157	C₂×C₁₃	C₂×#135	1	1	yes	characteristic
158	D₁₃	C₄×Q₁₆	1	4	yes	direct product factor
159	C₂×Q₈	—	13	8	no	—
160	C₂×Q₈	—	13	2	no	—
161	C₂×Q₈	—	26	2	no	—
162	C₂×Q₈	—	26	4	no	—
163	C₂×Q₈	C₂²×D₁₃	1	2	yes	—
164	C₂²×C₄	—	13	1	no	—
165	C₂²×C₄	—	13	1	no	—
166	C₂²×C₄	—	13	1	no	—
167	C₂²×C₄	—	26	2	no	—
168	C₂²×C₄	—	26	2	no	—
169	Q₁₆	—	13	8	no	—
170	Q₁₆	—	13	4	no	—
171	Q₁₆	C₄×D₁₃	1	4	yes	direct product factor
172	C₂×C₈	—	13	1	no	—
173	C₂×C₈	—	13	4	no	—
174	C₂×C₈	—	13	1	no	—
175	C₂×C₈	—	26	2	no	—
176	C₂×C₈	C₂²×D₁₃	1	1	yes	characteristic
177	C₂×C₈	C₂²×D₁₃	1	1	yes	characteristic
178	C₄⋊C₄	—	2	2	no	—
179	C₄⋊C₄	—	13	2	no	—
180	C₄⋊C₄	—	26	2	no	—
181	C₄⋊C₄	—	26	2	no	—
182	C₄⋊C₄	—	26	2	no	—
183	C₄⋊C₄	—	26	2	no	—
184	C₄⋊C₄	C₂²×D₁₃	1	2	yes	—
185	C₄²	—	2	2	no	—
186	C₄²	—	13	1	no	—
187	C₄²	—	13	1	no	—
188	C₄²	—	13	1	no	—
189	C₄²	—	26	2	no	—
190	C₄²	—	26	2	no	—
191	C₄²	—	26	2	no	—
192	C₄²	C₂²×D₁₃	1	1	yes	characteristic
193	C₁₃	C₂×C₄×Q₁₆	1	1	yes	characteristic, complemented, Sylow
194	C₂³	—	13	1	no	—
195	Q₈	—	2	2	no	—
196	Q₈	—	13	4	no	—
197	Q₈	—	26	8	no	—
198	Q₈	—	26	2	no	—
199	Q₈	C₂×C₄×D₁₃	1	4	yes	—
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	—	2	2	no	—
202	C₂×C₄	—	13	1	no	—
203	C₂×C₄	—	13	1	no	—
204	C₂×C₄	—	13	4	no	—
205	C₂×C₄	—	13	2	no	—
206	C₂×C₄	—	13	2	no	—
207	C₂×C₄	—	13	1	no	—
208	C₂×C₄	—	26	2	no	—
209	C₂×C₄	—	26	8	no	—
210	C₂×C₄	—	26	2	no	—
211	C₂×C₄	—	26	2	no	—
212	C₂×C₄	—	52	4	no	—
213	C₂×C₄	D₄×D₁₃	1	1	yes	characteristic
214	C₂×C₄	D₄×D₁₃	1	1	yes	characteristic
215	C₂×C₄	C₂³×D₁₃	1	1	yes	characteristic
216	C₈	—	2	1	no	—
217	C₈	—	13	2	no	—
218	C₈	—	26	1	no	—
219	C₈	C₂×C₄×D₁₃	1	2	yes	—
220	C₂²	—	13	2	no	—
221	C₂²	—	13	2	no	—
222	C₂²	—	13	2	no	—
223	C₂²	C₂×D₄×D₁₃	1	1	yes	characteristic
224	C₄	—	2	1	no	—
225	C₄	—	2	4	no	—
226	C₄	—	4	2	no	—
227	C₄	—	13	2	no	—
228	C₄	—	13	2	no	—
229	C₄	—	26	1	no	—
230	C₄	—	26	4	no	—
231	C₄	—	52	2	no	—
232	C₄	Q₁₆×D₁₃	1	2	yes	direct product factor
233	C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
234	C₄	#41×D₁₃	1	1	yes	characteristic
235	C₂	—	13	4	no	—
236	C₂	C₄×D₄×D₁₃	1	1	yes	characteristic
237	C₂	C₂×Q₁₆×D₁₃	1	1	yes	characteristic
238	C₂	D₁₃×#135	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₄×D₁₃
commutator	C₄×C₁₃	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₃
Fitting	C₄×C₁₃×Q₁₆	C₂
socle	C₂²×C₁₃	C₂²×D₄
prime radical	C₂²×D₁₃	C₂×D₄
cosocle	C₂×C₄×C₁₃	C₂⁴
hypercenter	C₄×Q₁₆	D₁₃
hypocenter	C₁₃	C₂×C₄×Q₁₆
LCS	G ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×Q₁₆

Complements

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

N	H ≅ G/N	type
3. C₄×C₁₃×Q₁₆	C₂	semidirect
4. Q₁₆×Dic₁₃	C₂	semidirect
5. C₄×#6107	C₂	semidirect
7. #11718534	C₂	semidirect
9. #11718451	C₂	semidirect
10. C₄×Dic₅₂	C₂	semidirect
17. Q₁₆×D₁₃	C₄	direct (N × H)
50. C₁₃×Q₁₆	C₂×C₄	semidirect
57. #6107	C₂×C₄	semidirect
71. Dic₅₂	C₂×C₄	semidirect
91. C₄×D₁₃	Q₁₆	direct (N × H)
108. C₄×Q₁₆	D₁₃	direct (N × H)
119. C₄×C₁₃	C₂×Q₁₆	semidirect
126. Dic₁₃	C₂×Q₁₆	semidirect
158. D₁₃	C₄×Q₁₆	direct (N × H)
171. Q₁₆	C₄×D₁₃	direct (N × H)
193. C₁₃	C₂×C₄×Q₁₆	semidirect
232. C₄	Q₁₆×D₁₃	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×Q₁₆×D₁₃	2
C₄×Q₈×D₁₃	2
C₄×C₁₃×Q₁₆	2
Q₁₆×Dic₁₃	2
C₄×#6107	2
D₁₃×#107	2
#11718534	2
D₁₃×#103	2
#11718451	2
C₄×Dic₅₂	2
C₄×C₈×D₁₃	2
C₂×C₄×Q₁₆	13

Subgroups — C4×Q16×D13 #13123597

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×Q₁₆×D₁₃ #13123597