Subgroups — C₂×2₋¹⁺⁴×Dic₁₃ #16769529

Order 3328.

11210 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	2₋¹⁺⁴×Dic₁₃	C₂	1	32	yes	maximal, direct product factor
2	C₂×#11719532	C₂	1	10	yes	maximal, complemented
3	C₂×#11719517	C₂	1	5	yes	maximal, complemented
4	C₂×#41×Dic₁₃	C₂	1	10	yes	maximal, complemented
5	C₂²×Q₈×Dic₁₃	C₂	1	5	yes	maximal, complemented
6	C₂²×C₁₃×2₋¹⁺⁴	C₂	1	1	yes	maximal, characteristic
7	C₂×C₁₃×2₋¹⁺⁴	C₄	1	32	yes	complemented
8	C₂×C₁₃×2₋¹⁺⁴	C₂²	1	16	yes	—
9	C₂²×C₁₃×#41	C₂²	1	10	yes	—
10	C₂³×Q₈×C₁₃	C₂²	1	5	yes	—
11	#11719532	C₂²	1	160	yes	complemented
12	#41×Dic₁₃	C₂²	1	160	yes	complemented
13	#11719517	C₂²	1	80	yes	complemented
14	C₂×Q₈×Dic₁₃	C₂²	1	10	yes	—
15	C₂×Q₈×Dic₁₃	C₂²	1	20	yes	—
16	C₂×Q₈×Dic₁₃	C₂²	1	10	yes	—
17	C₂×Q₈×Dic₁₃	C₂²	1	80	yes	complemented
18	C₂×D₄×Dic₁₃	C₂²	1	10	yes	—
19	C₂×D₄×Dic₁₃	C₂²	1	30	yes	—
20	C₂×#88180	C₂²	1	30	yes	—
21	C₂²×C₄⋊Dic₁₃	C₂²	1	15	yes	—
22	C₂²×C₄×Dic₁₃	C₂²	1	15	yes	—
23	C₁₃×2₋¹⁺⁴	C₂×C₄	1	256	yes	complemented
24	C₂×C₁₃×#41	C₂×C₄	1	160	yes	complemented
25	C₂×C₁₃×#41	C₂³	1	80	yes	—
26	C₂²×Q₈×C₁₃	C₂×C₄	1	80	yes	complemented
27	C₂²×Q₈×C₁₃	C₂³	1	10	yes	—
28	C₂²×Q₈×C₁₃	C₂³	1	40	yes	—
29	C₂²×D₄×C₁₃	C₂³	1	10	yes	—
30	C₂³×C₄×C₁₃	C₂³	1	15	yes	—
31	C₂³×Dic₁₃	C₂³	1	5	yes	—
32	C₂³×Dic₁₃	C₂³	1	15	yes	—
33	C₂×C₂²⋊Dic₁₃	C₂³	1	20	yes	—
34	Q₈×Dic₁₃	C₂³	1	80	yes	—
35	Q₈×Dic₁₃	C₂³	1	160	yes	—
36	Q₈×Dic₁₃	C₂³	1	80	yes	—
37	D₄×Dic₁₃	C₂³	1	80	yes	—
38	D₄×Dic₁₃	C₂³	1	240	yes	—
39	#88180	C₂³	1	240	yes	—
40	C₂×C₄⋊Dic₁₃	C₂³	1	30	yes	—
41	C₂×C₄⋊Dic₁₃	C₂³	1	120	yes	—
42	C₂×C₄⋊Dic₁₃	C₂³	1	10	yes	—
43	C₂×C₄×Dic₁₃	C₂³	1	10	yes	—
44	C₂×C₄×Dic₁₃	C₂³	1	30	yes	—
45	C₂×C₄×Dic₁₃	C₂³	1	120	yes	—
46	C₂×C₄×2₋¹⁺⁴	—	13	1	no	maximal, Sylow
47	C₂⁴×C₁₃	C₂⁴	1	5	yes	—
48	C₁₃×#41	C₂²×C₄	1	640	yes	complemented
49	C₂×Q₈×C₁₃	C₂²×C₄	1	80	yes	—
50	C₂×Q₈×C₁₃	C₂²×C₄	1	320	yes	complemented
51	C₂×Q₈×C₁₃	C₂⁴	1	40	yes	—
52	C₂×D₄×C₁₃	C₂²×C₄	1	80	yes	—
53	C₂×D₄×C₁₃	C₂⁴	1	40	yes	—
54	C₂²×C₄×C₁₃	C₂²×C₄	1	120	yes	—
55	C₂²×C₄×C₁₃	C₂⁴	1	10	yes	—
56	C₂²×C₄×C₁₃	C₂⁴	1	60	yes	—
57	C₂²×Dic₁₃	—	2	20	no	—
58	C₂²×Dic₁₃	—	4	30	no	—
59	C₂²×Dic₁₃	C₂⁴	1	5	yes	—
60	C₂²×Dic₁₃	C₂⁴	1	20	yes	—
61	C₂²×Dic₁₃	C₂⁴	1	10	yes	—
62	C₂²×Dic₁₃	C₂⁴	1	60	yes	—
63	C₂²×Dic₁₃	C₂⁴	1	1	yes	characteristic
64	C₂²⋊Dic₁₃	C₂⁴	1	80	yes	—
65	C₄⋊Dic₁₃	C₂⁴	1	120	yes	—
66	C₄⋊Dic₁₃	C₂⁴	1	40	yes	—
67	C₄×Dic₁₃	C₂⁴	1	40	yes	—
68	C₄×Dic₁₃	C₂⁴	1	120	yes	—
69	C₂²×2₋¹⁺⁴	D₁₃	1	1	yes	characteristic
70	C₄×2₋¹⁺⁴	—	13	32	no	—
71	C₂×#520	—	13	10	no	—
72	C₂×#519	—	13	5	no	—
73	C₂×C₄×#41	—	13	10	no	—
74	C₂²×C₄×Q₈	—	13	5	no	—
75	C₂³×C₁₃	—	2	10	no	—
76	C₂³×C₁₃	—	2	10	no	—
77	C₂³×C₁₃	C₂³×C₄	1	20	yes	—
78	C₂³×C₁₃	C₂⁵	1	10	yes	—
79	C₂³×C₁₃	C₂⁵	1	1	yes	characteristic
80	Q₈×C₁₃	C₂³×C₄	1	160	yes	—
81	D₄×C₁₃	C₂³×C₄	1	160	yes	—
82	C₂×C₄×C₁₃	C₂³×C₄	1	40	yes	—
83	C₂×C₄×C₁₃	C₂³×C₄	1	240	yes	—
84	C₂×C₄×C₁₃	C₂⁵	1	20	yes	—
85	C₂×Dic₁₃	—	2	10	no	—
86	C₂×Dic₁₃	—	2	20	no	—
87	C₂×Dic₁₃	—	2	40	no	—
88	C₂×Dic₁₃	—	4	60	no	—
89	C₂×Dic₁₃	2₋¹⁺⁴	1	4	yes	direct product factor
90	C₂×Dic₁₃	C₂⁵	1	10	yes	—
91	C₂×Dic₁₃	C₂⁵	1	20	yes	—
92	C₂×Dic₁₃	C₂⁵	1	2	yes	—
93	C₂×2₋¹⁺⁴	Dic₁₃	1	32	yes	direct product factor
94	C₂×2₋¹⁺⁴	C₂×D₁₃	1	16	yes	—
95	C₂²×#41	C₂×D₁₃	1	10	yes	—
96	C₂³×Q₈	C₂×D₁₃	1	5	yes	—
97	#520	—	13	160	no	—
98	#519	—	13	80	no	—
99	C₄×#41	—	13	160	no	—
100	C₂×C₄×Q₈	—	13	10	no	—
101	C₂×C₄×Q₈	—	13	20	no	—
102	C₂×C₄×Q₈	—	13	10	no	—
103	C₂×C₄×Q₈	—	13	80	no	—
104	C₂×C₄×D₄	—	13	10	no	—
105	C₂×C₄×D₄	—	13	30	no	—
106	C₂×#117	—	13	30	no	—
107	C₂²×C₄⋊C₄	—	13	15	no	—
108	C₂²×C₄²	—	13	15	no	—
109	C₂²×C₁₃	—	2	10	no	—
110	C₂²×C₁₃	—	2	10	no	—
111	C₂²×C₁₃	—	2	40	no	—
112	C₂²×C₁₃	C₂⁴×C₄	1	20	yes	—
113	C₂²×C₁₃	C₂⁴×C₄	1	2	yes	—
114	C₂²×C₁₃	C₂×2₋¹⁺⁴	1	2	yes	—
115	C₂²×C₁₃	#585	1	2	yes	—
116	C₂²×C₁₃	C₂⁶	1	1	yes	characteristic
117	C₄×C₁₃	C₂⁴×C₄	1	40	yes	—
118	Dic₁₃	—	2	10	no	—
119	Dic₁₃	—	2	20	no	—
120	Dic₁₃	C₂×2₋¹⁺⁴	1	4	yes	direct product factor
121	2₋¹⁺⁴	C₂×Dic₁₃	1	256	yes	direct product factor
122	C₂×#41	C₂×Dic₁₃	1	160	yes	complemented
123	C₂×#41	C₂²×D₁₃	1	80	yes	—
124	C₂²×Q₈	C₂×Dic₁₃	1	80	yes	complemented
125	C₂²×Q₈	C₂²×D₁₃	1	10	yes	—
126	C₂²×Q₈	C₂²×D₁₃	1	40	yes	—
127	C₂²×D₄	C₂²×D₁₃	1	10	yes	—
128	C₂³×C₄	—	13	5	no	—
129	C₂³×C₄	—	13	15	no	—
130	C₂³×C₄	C₂²×D₁₃	1	15	yes	—
131	C₄×Q₈	—	13	80	no	—
132	C₄×Q₈	—	13	160	no	—
133	C₄×Q₈	—	13	80	no	—
134	C₄×D₄	—	13	80	no	—
135	C₄×D₄	—	13	240	no	—
136	#117	—	13	240	no	—
137	C₂×C₄⋊C₄	—	13	30	no	—
138	C₂×C₄⋊C₄	—	13	120	no	—
139	C₂×C₄⋊C₄	—	13	10	no	—
140	C₂×C₂²⋊C₄	—	13	20	no	—
141	C₂×C₄²	—	13	30	no	—
142	C₂×C₄²	—	13	10	no	—
143	C₂×C₄²	—	13	120	no	—
144	C₂×C₁₃	—	2	20	no	—
145	C₂×C₁₃	C₄×2₋¹⁺⁴	1	4	yes	complemented
146	C₂×C₁₃	C₂⁵×C₄	1	1	yes	characteristic
147	C₂×C₁₃	C₂²×2₋¹⁺⁴	1	1	yes	characteristic
148	C₂×C₁₃	C₂×#585	1	1	yes	characteristic
149	C₂⁴	C₂³×D₁₃	1	5	yes	—
150	#41	C₂²×Dic₁₃	1	640	yes	complemented
151	C₂×Q₈	C₂²×Dic₁₃	1	80	yes	—
152	C₂×Q₈	C₂²×Dic₁₃	1	320	yes	complemented
153	C₂×Q₈	C₂³×D₁₃	1	40	yes	—
154	C₂×D₄	C₂²×Dic₁₃	1	80	yes	—
155	C₂×D₄	C₂³×D₁₃	1	40	yes	—
156	C₂²×C₄	—	13	1	no	—
157	C₂²×C₄	—	13	20	no	—
158	C₂²×C₄	—	13	5	no	—
159	C₂²×C₄	—	13	10	no	—
160	C₂²×C₄	—	13	60	no	—
161	C₂²×C₄	—	26	20	no	—
162	C₂²×C₄	—	52	30	no	—
163	C₂²×C₄	C₂²×Dic₁₃	1	120	yes	—
164	C₂²×C₄	C₂³×D₁₃	1	60	yes	—
165	C₂²×C₄	C₂³×D₁₃	1	10	yes	—
166	C₄⋊C₄	—	13	120	no	—
167	C₄⋊C₄	—	13	40	no	—
168	C₂²⋊C₄	—	13	80	no	—
169	C₄²	—	13	40	no	—
170	C₄²	—	13	120	no	—
171	C₁₃	C₂×C₄×2₋¹⁺⁴	1	1	yes	characteristic, complemented, Sylow
172	C₂³	—	2	10	no	—
173	C₂³	—	2	10	no	—
174	C₂³	C₂³×Dic₁₃	1	20	yes	—
175	C₂³	C₂⁴×D₁₃	1	10	yes	—
176	C₂³	C₂⁴×D₁₃	1	1	yes	characteristic
177	Q₈	C₂³×Dic₁₃	1	160	yes	—
178	D₄	C₂³×Dic₁₃	1	160	yes	—
179	C₂×C₄	—	13	4	no	—
180	C₂×C₄	—	13	2	no	—
181	C₂×C₄	—	13	10	no	—
182	C₂×C₄	—	13	20	no	—
183	C₂×C₄	—	26	40	no	—
184	C₂×C₄	—	26	10	no	—
185	C₂×C₄	—	26	20	no	—
186	C₂×C₄	—	52	60	no	—
187	C₂×C₄	C₂³×Dic₁₃	1	40	yes	—
188	C₂×C₄	C₂³×Dic₁₃	1	240	yes	—
189	C₂×C₄	C₂⁴×D₁₃	1	20	yes	—
190	C₂²	—	2	40	no	—
191	C₂²	—	2	10	no	—
192	C₂²	—	2	10	no	—
193	C₂²	D₁₃×2₋¹⁺⁴	1	2	yes	—
194	C₂²	#11719652	1	2	yes	—
195	C₂²	C₂⁴×Dic₁₃	1	20	yes	—
196	C₂²	C₂⁴×Dic₁₃	1	2	yes	—
197	C₂²	C₂⁵×D₁₃	1	1	yes	characteristic
198	C₄	—	13	4	no	—
199	C₄	—	26	10	no	—
200	C₄	—	26	20	no	—
201	C₄	C₂⁴×Dic₁₃	1	40	yes	—
202	C₂	—	2	20	no	—
203	C₂	2₋¹⁺⁴×Dic₁₃	1	4	yes	direct product factor
204	C₂	C₂⁵×Dic₁₃	1	1	yes	characteristic
205	C₂	C₂×D₁₃×2₋¹⁺⁴	1	1	yes	characteristic
206	C₂	C₂×#11719652	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂⁴×D₁₃
commutator	C₂×C₁₃	C₂⁵×C₄
Frattini	C₂²	C₂⁵×D₁₃
Fitting, prime radical	C₂²×C₁₃×2₋¹⁺⁴	C₂
socle	C₂³×C₁₃	C₂⁵
cosocle	C₂²×C₁₃	C₂⁶
hypercenter	C₂²×2₋¹⁺⁴	D₁₃
hypocenter	C₁₃	C₂×C₄×2₋¹⁺⁴
LCS	G ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂²×2₋¹⁺⁴

Complements

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

N	H ≅ G/N	type
1. 2₋¹⁺⁴×Dic₁₃	C₂	direct (N × H)
2. C₂×#11719532	C₂	semidirect
3. C₂×#11719517	C₂	semidirect
4. C₂×#41×Dic₁₃	C₂	semidirect
5. C₂²×Q₈×Dic₁₃	C₂	semidirect
7. C₂×C₁₃×2₋¹⁺⁴	C₄	semidirect
11. #11719532	C₂²	semidirect
12. #41×Dic₁₃	C₂²	semidirect
13. #11719517	C₂²	semidirect
17. C₂×Q₈×Dic₁₃	C₂²	semidirect
23. C₁₃×2₋¹⁺⁴	C₂×C₄	semidirect
24. C₂×C₁₃×#41	C₂×C₄	semidirect
26. C₂²×Q₈×C₁₃	C₂×C₄	semidirect
48. C₁₃×#41	C₂²×C₄	semidirect
50. C₂×Q₈×C₁₃	C₂²×C₄	semidirect
89. C₂×Dic₁₃	2₋¹⁺⁴	direct (N × H)
93. C₂×2₋¹⁺⁴	Dic₁₃	direct (N × H)
120. Dic₁₃	C₂×2₋¹⁺⁴	direct (N × H)
121. 2₋¹⁺⁴	C₂×Dic₁₃	direct (N × H)
122. C₂×#41	C₂×Dic₁₃	semidirect
124. C₂²×Q₈	C₂×Dic₁₃	semidirect
145. C₂×C₁₃	C₄×2₋¹⁺⁴	semidirect
150. #41	C₂²×Dic₁₃	semidirect
152. C₂×Q₈	C₂²×Dic₁₃	semidirect
171. C₁₃	C₂×C₄×2₋¹⁺⁴	semidirect
203. C₂	2₋¹⁺⁴×Dic₁₃	direct (N × H)

Maximal subgroups

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

subgroup	index
2₋¹⁺⁴×Dic₁₃	2
C₂×#11719532	2
C₂×#11719517	2
C₂×#41×Dic₁₃	2
C₂²×Q₈×Dic₁₃	2
C₂²×C₁₃×2₋¹⁺⁴	2
C₂×C₄×2₋¹⁺⁴	13

Subgroups — C2×2−1+4×Dic13 #16769529

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×2₋¹⁺⁴×Dic₁₃ #16769529