Subgroups — C₁₃×#12890 #15699229

Order 3328.

318 conjugacy classes of subgroups.

The subgroup lattice has 170 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₃×#346	C₂	1	1	yes	maximal, characteristic
2	C₁₃×#1846	C₂	1	1	yes	maximal, characteristic
3	C₁₃×#1845	C₂	1	1	yes	maximal, characteristic
4	C₁₃×#1380	C₂	1	4	yes	maximal
5	C₂²×C₁₃×C₁₆	C₂²	1	2	yes	—
6	C₂×C₁₃×#107	—	2	1	no	—
7	C₂×C₁₃×#107	C₂²	1	2	yes	—
8	C₂×C₁₃×#107	C₂²	1	2	yes	—
9	C₂×C₁₃×#106	—	2	1	no	—
10	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
11	C₁₃×#384	—	2	1	no	—
12	C₁₃×#382	—	2	1	no	—
13	C₁₃×#346	C₄	1	4	yes	complemented
14	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
15	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
16	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
17	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
18	C₂×C₁₃×C₄⋊C₄	—	4	2	no	—
19	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
20	C₂×C₁₃×C₁₆	—	2	2	no	—
21	C₂×C₁₃×C₁₆	—	2	2	no	—
22	C₂×C₁₃×C₁₆	C₂×C₄	1	4	yes	—
23	C₁₃×#107	—	2	4	no	—
24	C₁₃×#107	—	2	4	no	—
25	C₁₃×#107	—	4	2	no	—
26	C₁₃×#106	—	4	2	no	—
27	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
28	C₄×C₈×C₁₃	Q₈	1	2	yes	—
29	C₁₃×#95	—	4	2	no	—
30	#12890	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
31	C₂²×C₄×C₁₃	—	4	2	no	—
32	C₂²×C₄×C₁₃	—	4	2	no	—
33	C₂²×C₄×C₁₃	D₈	1	2	yes	—
34	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
35	C₂×C₈×C₁₃	C₄⋊C₄	1	2	yes	—
36	C₂×C₈×C₁₃	SD₁₆	1	2	yes	—
37	C₂×C₈×C₁₃	SD₁₆	1	2	yes	—
38	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
39	C₂×C₈×C₁₃	C₂×Q₈	1	1	yes	characteristic
40	C₂×C₈×C₁₃	#41	1	2	yes	—
41	C₂×C₈×C₁₃	#41	1	2	yes	—
42	C₁₃×C₄⋊C₄	—	4	2	no	—
43	C₁₃×C₄⋊C₄	—	4	2	no	—
44	C₁₃×C₄⋊C₄	—	4	2	no	—
45	C₁₃×C₄⋊C₄	—	4	2	no	—
46	C₁₃×C₄⋊C₄	—	8	4	no	—
47	C₄²×C₁₃	C₄⋊C₄	1	2	yes	—
48	C₄²×C₁₃	Q₁₆	1	2	yes	—
49	C₁₃×C₁₆	—	2	4	no	—
50	C₂×#346	C₂×C₁₃	1	1	yes	characteristic
51	#1846	C₂×C₁₃	1	1	yes	characteristic
52	#1845	C₂×C₁₃	1	1	yes	characteristic
53	#1380	C₂×C₁₃	1	4	yes	—
54	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
55	C₂×C₄×C₁₃	—	2	2	no	—
56	C₂×C₄×C₁₃	—	2	2	no	—
57	C₂×C₄×C₁₃	—	8	2	no	—
58	C₂×C₄×C₁₃	—	8	2	no	—
59	C₂×C₄×C₁₃	—	8	4	no	—
60	C₂×C₄×C₁₃	—	8	2	no	—
61	C₂×C₄×C₁₃	—	8	2	no	—
62	C₂×C₄×C₁₃	#107	1	4	yes	—
63	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
64	C₂×C₄×C₁₃	#124	1	1	yes	characteristic
65	C₂×C₄×C₁₃	#124	1	1	yes	characteristic
66	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
67	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
68	C₂×C₄×C₁₃	C₂×Q₁₆	1	1	yes	characteristic
69	C₈×C₁₃	#106	1	4	yes	—
70	C₈×C₁₃	#117	1	2	yes	—
71	C₈×C₁₃	#125	1	2	yes	—
72	C₂²×C₁₆	C₂²×C₁₃	1	2	yes	—
73	C₂×#107	—	2	1	no	—
74	C₂×#107	C₂²×C₁₃	1	2	yes	—
75	C₂×#107	C₂²×C₁₃	1	2	yes	—
76	C₂×#106	—	2	1	no	—
77	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
78	#384	—	2	1	no	—
79	#382	—	2	1	no	—
80	#346	C₄×C₁₃	1	4	yes	complemented
81	C₂²×C₁₃	C₂×#107	1	1	yes	characteristic
82	C₂²×C₁₃	#486	1	1	yes	characteristic
83	C₂²×C₁₃	#486	1	1	yes	characteristic
84	C₂²×C₁₃	#509	1	2	yes	—
85	C₂²×C₁₃	#510	1	2	yes	—
86	C₄×C₁₃	—	2	4	no	—
87	C₄×C₁₃	—	16	2	no	—
88	C₄×C₁₃	—	16	2	no	—
89	C₄×C₁₃	#382	1	1	yes	characteristic
90	C₄×C₁₃	C₂×#106	1	1	yes	characteristic
91	C₄×C₁₃	#487	1	1	yes	characteristic
92	C₄×C₁₃	#487	1	1	yes	characteristic
93	C₂²×C₈	D₄×C₁₃	1	1	yes	characteristic
94	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
95	C₂×C₄⋊C₄	—	2	2	no	—
96	C₂×C₄⋊C₄	—	2	2	no	—
97	C₂×C₄⋊C₄	—	4	2	no	—
98	C₂×C₄²	D₄×C₁₃	1	1	yes	characteristic
99	C₂×C₁₆	—	2	2	no	—
100	C₂×C₁₆	—	2	2	no	—
101	C₂×C₁₆	C₂×C₄×C₁₃	1	4	yes	—
102	#107	—	2	4	no	—
103	#107	—	2	4	no	—
104	#107	—	4	2	no	—
105	#106	—	4	2	no	—
106	C₄×C₈	C₂×C₄×C₁₃	1	2	yes	—
107	C₄×C₈	Q₈×C₁₃	1	2	yes	—
108	#95	—	4	2	no	—
109	C₂×C₁₃	#1834	1	1	yes	characteristic
110	C₂×C₁₃	#2159	1	2	yes	—
111	C₂×C₁₃	#2243	1	2	yes	—
112	C₂×C₁₃	#2244	1	1	yes	characteristic
113	C₂×C₁₃	#2245	1	1	yes	characteristic
114	C₂²×C₄	—	4	2	no	—
115	C₂²×C₄	—	4	2	no	—
116	C₂²×C₄	C₁₃×D₈	1	2	yes	—
117	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
118	C₂×C₈	C₁₃×C₄⋊C₄	1	2	yes	—
119	C₂×C₈	C₁₃×SD₁₆	1	2	yes	—
120	C₂×C₈	C₁₃×SD₁₆	1	2	yes	—
121	C₂×C₈	C₂²×C₄×C₁₃	1	1	yes	characteristic
122	C₂×C₈	C₂×Q₈×C₁₃	1	1	yes	characteristic
123	C₂×C₈	C₁₃×#41	1	2	yes	—
124	C₂×C₈	C₁₃×#41	1	2	yes	—
125	C₄⋊C₄	—	4	2	no	—
126	C₄⋊C₄	—	4	2	no	—
127	C₄⋊C₄	—	4	2	no	—
128	C₄⋊C₄	—	4	2	no	—
129	C₄⋊C₄	—	8	4	no	—
130	C₄²	C₁₃×C₄⋊C₄	1	2	yes	—
131	C₄²	C₁₃×Q₁₆	1	2	yes	—
132	C₁₆	—	2	4	no	—
133	C₁₃	#12890	1	1	yes	characteristic, direct product factor, Sylow
134	C₂³	C₂×C₁₃×D₈	1	1	yes	characteristic
135	C₂×C₄	—	2	2	no	—
136	C₂×C₄	—	2	2	no	—
137	C₂×C₄	—	8	2	no	—
138	C₂×C₄	—	8	2	no	—
139	C₂×C₄	—	8	4	no	—
140	C₂×C₄	—	8	2	no	—
141	C₂×C₄	—	8	2	no	—
142	C₂×C₄	C₁₃×#107	1	4	yes	—
143	C₂×C₄	C₂×C₁₃×C₄⋊C₄	1	1	yes	characteristic
144	C₂×C₄	C₁₃×#124	1	1	yes	characteristic
145	C₂×C₄	C₁₃×#124	1	1	yes	characteristic
146	C₂×C₄	C₂×C₁₃×SD₁₆	1	1	yes	characteristic
147	C₂×C₄	C₂×C₁₃×SD₁₆	1	1	yes	characteristic
148	C₂×C₄	C₂×C₁₃×Q₁₆	1	1	yes	characteristic
149	C₈	C₁₃×#106	1	4	yes	—
150	C₈	C₁₃×#117	1	2	yes	—
151	C₈	C₁₃×#125	1	2	yes	—
152	C₂²	C₂×C₁₃×#107	1	1	yes	characteristic
153	C₂²	C₁₃×#486	1	1	yes	characteristic
154	C₂²	C₁₃×#486	1	1	yes	characteristic
155	C₂²	C₁₃×#509	1	2	yes	—
156	C₂²	C₁₃×#510	1	2	yes	—
157	C₄	—	2	4	no	—
158	C₄	—	16	2	no	—
159	C₄	—	16	2	no	—
160	C₄	C₁₃×#382	1	1	yes	characteristic
161	C₄	C₂×C₁₃×#106	1	1	yes	characteristic
162	C₄	C₁₃×#487	1	1	yes	characteristic
163	C₄	C₁₃×#487	1	1	yes	characteristic
164	C₂	C₁₃×#1834	1	1	yes	characteristic
165	C₂	C₁₃×#2159	1	2	yes	—
166	C₂	C₁₃×#2243	1	2	yes	—
167	C₂	C₁₃×#2244	1	1	yes	characteristic
168	C₂	C₁₃×#2245	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₁₃	C₂×D₈
commutator	C₂×C₈	C₂²×C₄×C₁₃
Frattini, cosocle	C₂²×C₈	C₂³×C₁₃
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×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
13. C₁₃×#346	C₄	semidirect
30. #12890	C₁₃	direct (N × H)
80. #346	C₄×C₁₃	semidirect
133. C₁₃	#12890	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×#346	2
C₁₃×#1846	2
C₁₃×#1845	2
C₁₃×#1380	2
#12890	13

Subgroups — C13×#12890 #15699229

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#12890 #15699229