Subgroups — Dic₁₃×#366 #15778320

Order 3328.

366 conjugacy classes of subgroups.

The subgroup lattice has 210 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718096	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₆×Dic₁₃	C₂	1	1	yes	maximal, characteristic
3	#107×Dic₁₃	C₂	1	2	yes	maximal
4	#13112927	C₂	1	2	yes	maximal
5	C₂×C₁₃×#366	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
7	C₂×C₁₃×#107	C₂²	1	2	yes	—
8	C₂×#88058	C₂²	1	2	yes	—
9	C₂×C₈×Dic₁₃	C₂²	1	1	yes	characteristic
10	C₂²×#6090	C₂²	1	1	yes	characteristic
11	C₄⋊C₄×Dic₁₃	—	2	2	no	—
12	C₁₃×#366	C₄	1	4	yes	complemented
13	#11718121	—	2	2	no	—
14	#11718096	C₄	1	4	yes	complemented
15	C₁₆×Dic₁₃	C₄	1	4	yes	complemented
16	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
17	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
18	C₂×C₄⋊Dic₁₃	—	2	2	no	—
19	C₂×C₄×Dic₁₃	—	4	2	no	—
20	C₂×C₄×Dic₁₃	D₄	1	1	yes	characteristic
21	C₂²×#1055	D₄	1	1	yes	characteristic
22	C₂×C₁₃×C₁₆	C₂×C₄	1	2	yes	—
23	C₂×C₁₃×C₁₆	C₂×C₄	1	2	yes	—
24	C₂×C₁₃×C₁₆	C₂×C₄	1	2	yes	—
25	C₁₃×#107	—	2	2	no	—
26	C₁₃×#107	C₂×C₄	1	4	yes	—
27	#88071	—	4	2	no	—
28	#88058	—	2	2	no	—
29	#88058	C₂×C₄	1	4	yes	—
30	C₈×Dic₁₃	C₂×C₄	1	2	yes	—
31	C₈×Dic₁₃	Q₈	1	2	yes	—
32	C₂×#6090	—	2	1	no	—
33	C₂×#6090	—	2	1	no	—
34	C₂×#6090	C₂×C₄	1	2	yes	—
35	C₄×#366	—	13	1	no	maximal, Sylow
36	C₂²×C₄×C₁₃	—	4	2	no	—
37	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
38	C₂²×Dic₁₃	—	4	2	no	—
39	C₂²×Dic₁₃	D₈	1	1	yes	characteristic
40	C₂²×Dic₁₃	D₈	1	1	yes	characteristic
41	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
42	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
43	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
44	C₂×C₈×C₁₃	C₂×Q₈	1	1	yes	characteristic
45	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
46	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
47	C₁₃×C₄⋊C₄	—	2	4	no	—
48	C₁₃×C₄⋊C₄	—	4	2	no	—
49	C₄⋊Dic₁₃	—	2	4	no	—
50	C₄⋊Dic₁₃	—	4	2	no	—
51	C₄×Dic₁₃	—	8	4	no	—
52	C₄×Dic₁₃	C₄⋊C₄	1	2	yes	—
53	C₄×Dic₁₃	Q₁₆	1	2	yes	—
54	C₂×#1055	—	2	1	no	—
55	C₂×#1055	—	2	1	no	—
56	C₂×#1055	C₄⋊C₄	1	2	yes	—
57	C₁₃×C₁₆	C₄²	1	4	yes	complemented
58	#6090	—	2	2	no	—
59	C₂×#366	—	13	1	no	—
60	C₂×#366	D₁₃	1	1	yes	characteristic
61	C₂×C₄×C₁₆	—	13	1	no	—
62	C₄×#107	—	13	2	no	—
63	#1380	—	13	2	no	—
64	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
65	C₂×C₄×C₁₃	—	4	4	no	—
66	C₂×C₄×C₁₃	—	8	2	no	—
67	C₂×C₄×C₁₃	—	8	2	no	—
68	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
69	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
70	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
71	C₂×C₄×C₁₃	C₂×Q₁₆	1	1	yes	characteristic
72	C₂×C₄×C₁₃	#135	1	1	yes	characteristic
73	C₂×C₄×C₁₃	#135	1	1	yes	characteristic
74	C₂×Dic₁₃	—	2	1	no	—
75	C₂×Dic₁₃	—	2	1	no	—
76	C₂×Dic₁₃	—	4	4	no	—
77	C₂×Dic₁₃	—	8	2	no	—
78	C₂×Dic₁₃	—	8	2	no	—
79	C₂×Dic₁₃	#107	1	2	yes	—
80	C₂×Dic₁₃	#107	1	2	yes	—
81	C₂×Dic₁₃	D₁₆	1	2	yes	—
82	C₂×Dic₁₃	Q₃₂	1	2	yes	—
83	C₈×C₁₃	C₂×C₄²	1	1	yes	characteristic
84	C₈×C₁₃	#117	1	1	yes	characteristic
85	C₈×C₁₃	C₄×Q₈	1	1	yes	characteristic
86	C₈×C₁₃	C₄×Q₈	1	1	yes	characteristic
87	#1055	—	2	2	no	—
88	C₂²×C₁₆	—	13	1	no	—
89	C₂²×C₁₆	C₂×D₁₃	1	1	yes	characteristic
90	C₂×#107	—	13	2	no	—
91	C₂×#107	C₂×D₁₃	1	2	yes	—
92	C₂×C₄×C₈	—	13	1	no	—
93	C₄×C₄⋊C₄	—	26	2	no	—
94	#366	—	13	4	no	—
95	#366	Dic₁₃	1	4	yes	direct product factor
96	C₄×C₁₆	—	13	4	no	—
97	#340	—	26	2	no	—
98	C₂²×C₁₃	C₂×#107	1	1	yes	characteristic
99	C₂²×C₁₃	C₄×D₈	1	1	yes	characteristic
100	C₂²×C₁₃	C₄×D₈	1	1	yes	characteristic
101	C₂²×C₁₃	C₂×D₁₆	1	1	yes	characteristic
102	C₂²×C₁₃	C₂×Q₃₂	1	1	yes	characteristic
103	C₂²×C₁₃	#508	1	1	yes	characteristic
104	C₂²×C₁₃	#508	1	1	yes	characteristic
105	C₄×C₁₃	—	8	4	no	—
106	C₄×C₁₃	C₄×C₄⋊C₄	1	1	yes	characteristic
107	C₄×C₁₃	#427	1	1	yes	characteristic
108	C₄×C₁₃	C₄×Q₁₆	1	1	yes	characteristic
109	C₄×C₁₃	C₄×Q₁₆	1	1	yes	characteristic
110	Dic₁₃	—	2	2	no	—
111	Dic₁₃	—	8	4	no	—
112	Dic₁₃	#366	1	4	yes	direct product factor
113	C₂²×C₈	—	13	1	no	—
114	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
115	C₂×C₄⋊C₄	—	2	2	no	—
116	C₂×C₄⋊C₄	—	26	2	no	—
117	C₂×C₄²	—	13	1	no	—
118	C₂×C₄²	—	52	2	no	—
119	C₂×C₁₆	—	13	2	no	—
120	C₂×C₁₆	—	26	1	no	—
121	C₂×C₁₆	—	26	1	no	—
122	C₂×C₁₆	C₄×D₁₃	1	2	yes	—
123	C₂×C₁₆	C₄×D₁₃	1	2	yes	—
124	C₂×C₁₆	C₂×Dic₁₃	1	2	yes	—
125	#107	—	2	2	no	—
126	#107	—	13	4	no	—
127	#107	—	26	2	no	—
128	#107	C₂×Dic₁₃	1	4	yes	—
129	C₄×C₈	—	13	2	no	—
130	C₄×C₈	—	13	2	no	—
131	#95	—	52	2	no	—
132	C₂×C₁₃	C₄×#107	1	1	yes	characteristic
133	C₂×C₁₃	C₂×#366	1	1	yes	characteristic
134	C₂×C₁₃	#2158	1	1	yes	characteristic
135	C₂×C₁₃	C₄×D₁₆	1	1	yes	characteristic
136	C₂×C₁₃	C₄×D₁₆	1	1	yes	characteristic
137	C₂×C₁₃	C₄×Q₃₂	1	1	yes	characteristic
138	C₂×C₁₃	C₄×Q₃₂	1	1	yes	characteristic
139	C₂²×C₄	—	4	2	no	—
140	C₂²×C₄	—	13	1	no	—
141	C₂²×C₄	—	13	1	no	—
142	C₂²×C₄	—	52	2	no	—
143	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
144	C₂×C₈	—	13	2	no	—
145	C₂×C₈	—	26	1	no	—
146	C₂×C₈	—	26	1	no	—
147	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
148	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
149	C₂×C₈	#6129	1	1	yes	characteristic
150	C₂×C₈	Q₈×D₁₃	1	1	yes	characteristic
151	C₂×C₈	#6131	1	1	yes	characteristic
152	C₂×C₈	C₂²×Dic₁₃	1	1	yes	characteristic
153	C₄⋊C₄	—	2	4	no	—
154	C₄⋊C₄	—	4	2	no	—
155	C₄⋊C₄	—	26	4	no	—
156	C₄⋊C₄	—	52	2	no	—
157	C₄²	—	13	2	no	—
158	C₄²	—	13	2	no	—
159	C₄²	—	104	4	no	—
160	C₁₆	—	26	2	no	—
161	C₁₆	C₄×Dic₁₃	1	4	yes	complemented
162	C₁₃	C₄×#366	1	1	yes	characteristic, complemented, Sylow
163	C₂³	D₈×D₁₃	1	1	yes	characteristic
164	C₂×C₄	—	4	4	no	—
165	C₂×C₄	—	8	2	no	—
166	C₂×C₄	—	8	2	no	—
167	C₂×C₄	—	13	2	no	—
168	C₂×C₄	—	13	2	no	—
169	C₂×C₄	—	13	2	no	—
170	C₂×C₄	—	13	2	no	—
171	C₂×C₄	—	26	1	no	—
172	C₂×C₄	—	26	1	no	—
173	C₂×C₄	—	52	4	no	—
174	C₂×C₄	—	104	2	no	—
175	C₂×C₄	—	104	2	no	—
176	C₂×C₄	C₄⋊C₄×D₁₃	1	1	yes	characteristic
177	C₂×C₄	#88147	1	1	yes	characteristic
178	C₂×C₄	#88166	1	1	yes	characteristic
179	C₂×C₄	Q₁₆×D₁₃	1	1	yes	characteristic
180	C₂×C₄	#88173	1	1	yes	characteristic
181	C₂×C₄	D₄×Dic₁₃	1	1	yes	characteristic
182	C₈	—	26	2	no	—
183	C₈	#88141	1	1	yes	characteristic
184	C₈	#88146	1	1	yes	characteristic
185	C₈	C₂×C₄×Dic₁₃	1	1	yes	characteristic
186	C₈	Q₈×Dic₁₃	1	1	yes	characteristic
187	C₂²	#11718533	1	1	yes	characteristic
188	C₂²	D₁₃×#107	1	1	yes	characteristic
189	C₂²	D₁₃×D₁₆	1	1	yes	characteristic
190	C₂²	#11718573	1	1	yes	characteristic
191	C₂²	D₁₃×Q₃₂	1	1	yes	characteristic
192	C₂²	#11718580	1	1	yes	characteristic
193	C₂²	D₈×Dic₁₃	1	1	yes	characteristic
194	C₄	—	8	4	no	—
195	C₄	—	13	4	no	—
196	C₄	—	26	2	no	—
197	C₄	—	104	4	no	—
198	C₄	#11718534	1	1	yes	characteristic
199	C₄	#11718541	1	1	yes	characteristic
200	C₄	C₄⋊C₄×Dic₁₃	1	1	yes	characteristic
201	C₄	Q₁₆×Dic₁₃	1	1	yes	characteristic
202	C₂	#107×Dic₁₃	1	1	yes	characteristic
203	C₂	D₁₃×#366	1	1	yes	characteristic
204	C₂	#13116617	1	1	yes	characteristic
205	C₂	#13116682	1	1	yes	characteristic
206	C₂	D₁₆×Dic₁₃	1	1	yes	characteristic
207	C₂	#13116694	1	1	yes	characteristic
208	C₂	Q₃₂×Dic₁₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	D₈×D₁₃
commutator	C₈×C₁₃	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₁₃
Fitting	C₂×C₁₃×#366	C₂
socle, prime radical	C₂³×C₁₃	C₂×D₈
cosocle	C₂²×C₈×C₁₃	C₂³
hypercenter	C₂×#366	D₁₃
hypocenter	C₁₃	C₄×#366
LCS	G ▷ C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#366

Complements

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

N	H ≅ G/N	type
12. C₁₃×#366	C₄	semidirect
14. #11718096	C₄	semidirect
15. C₁₆×Dic₁₃	C₄	semidirect
57. C₁₃×C₁₆	C₄²	semidirect
95. #366	Dic₁₃	direct (N × H)
112. Dic₁₃	#366	direct (N × H)
161. C₁₆	C₄×Dic₁₃	semidirect
162. C₁₃	C₄×#366	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11718096	2
C₂×C₁₆×Dic₁₃	2
#107×Dic₁₃	2
#13112927	2
C₂×C₁₃×#366	2
C₄×#366	13

Subgroups — Dic13×#366 #15778320

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Dic₁₃×#366 #15778320