Subgroups — C₁₃×#8266 #15694605

Order 3328.

414 conjugacy classes of subgroups.

The subgroup lattice has 158 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄²×C₈×C₁₃	C₂	1	1	yes	maximal, characteristic
2	C₁₃×#1564	C₂	1	2	yes	maximal
3	C₁₃×#1450	C₂	1	2	yes	maximal
4	C₁₃×#1276	C₂	1	2	yes	maximal
5	C₁₃×#431	C₂²	1	4	yes	—
6	C₂×C₄×C₈×C₁₃	C₄	1	2	yes	—
7	C₂×C₄×C₈×C₁₃	C₄	1	2	yes	—
8	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
9	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
10	C₄³×C₁₃	C₂²	1	1	yes	characteristic
11	C₁₃×#335	C₄	1	4	yes	complemented
12	C₁₃×#334	C₄	1	4	yes	complemented
13	C₁₃×#322	—	2	2	no	—
14	C₈²×C₁₃	—	2	2	no	—
15	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
16	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
17	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
18	C₂×C₄²×C₁₃	—	2	2	no	—
19	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
20	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
21	C₁₃×C₄⋊C₈	C₂×C₄	1	8	yes	—
22	C₁₃×C₂²⋊C₈	—	2	4	no	—
23	C₄×C₈×C₁₃	—	2	4	no	—
24	C₄×C₈×C₁₃	—	2	4	no	—
25	C₄×C₈×C₁₃	C₈	1	8	yes	complemented
26	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
27	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
28	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
29	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
30	C₄×C₈×C₁₃	Q₈	1	2	yes	—
31	#8266	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
32	C₂²×C₄×C₁₃	—	2	2	no	—
33	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
34	C₂²×C₄×C₁₃	#41	1	2	yes	—
35	C₂×C₈×C₁₃	—	2	2	no	—
36	C₂×C₈×C₁₃	—	2	8	no	—
37	C₂×C₈×C₁₃	C₄²	1	2	yes	—
38	C₂×C₈×C₁₃	C₄²	1	1	yes	characteristic
39	C₂×C₈×C₁₃	C₄²	1	1	yes	characteristic
40	C₂×C₈×C₁₃	C₄⋊C₄	1	2	yes	—
41	C₂×C₈×C₁₃	C₄⋊C₄	1	2	yes	—
42	C₂×C₈×C₁₃	C₂×C₈	1	4	yes	—
43	C₂×C₈×C₁₃	C₂×C₈	1	4	yes	—
44	C₂×C₈×C₁₃	M₁₆	1	4	yes	—
45	C₄²×C₁₃	—	2	8	no	—
46	C₄²×C₁₃	—	2	2	no	—
47	C₄²×C₁₃	C₂×C₈	1	4	yes	—
48	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
49	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
50	C₄²×C₁₃	C₂×Q₈	1	1	yes	characteristic
51	C₄²×C₈	C₂×C₁₃	1	1	yes	characteristic
52	#1564	C₂×C₁₃	1	2	yes	—
53	#1450	C₂×C₁₃	1	2	yes	—
54	#1276	C₂×C₁₃	1	2	yes	—
55	C₂³×C₁₃	#117	1	1	yes	characteristic
56	C₂×C₄×C₁₃	—	2	8	no	—
57	C₂×C₄×C₁₃	—	2	4	no	—
58	C₂×C₄×C₁₃	—	2	4	no	—
59	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
60	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
61	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
62	C₂×C₄×C₁₃	C₄×Q₈	1	2	yes	—
63	C₂×C₄×C₁₃	C₂²×C₈	1	2	yes	—
64	C₂×C₄×C₁₃	C₂×M₁₆	1	2	yes	—
65	C₈×C₁₃	—	2	2	no	—
66	C₈×C₁₃	—	4	8	no	—
67	C₈×C₁₃	C₄×C₈	1	2	yes	—
68	C₈×C₁₃	C₄×C₈	1	2	yes	—
69	C₈×C₁₃	C₄⋊C₈	1	4	yes	—
70	C₈×C₁₃	C₄⋊C₈	1	4	yes	—
71	#431	C₂²×C₁₃	1	4	yes	—
72	C₂×C₄×C₈	C₄×C₁₃	1	2	yes	—
73	C₂×C₄×C₈	C₄×C₁₃	1	2	yes	—
74	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
75	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
76	C₄³	C₂²×C₁₃	1	1	yes	characteristic
77	#335	C₄×C₁₃	1	4	yes	complemented
78	#334	C₄×C₁₃	1	4	yes	complemented
79	#322	—	2	2	no	—
80	C₈²	—	2	2	no	—
81	C₂²×C₁₃	—	2	2	no	—
82	C₂²×C₁₃	C₄×C₄⋊C₄	1	1	yes	characteristic
83	C₂²×C₁₃	#431	1	2	yes	—
84	C₄×C₁₃	—	2	8	no	—
85	C₄×C₁₃	—	2	2	no	—
86	C₄×C₁₃	C₂×C₄×C₈	1	1	yes	characteristic
87	C₄×C₁₃	C₄×M₁₆	1	1	yes	characteristic
88	C₄×C₁₃	C₂×C₄⋊C₈	1	2	yes	—
89	C₄×C₁₃	#443	1	4	yes	—
90	C₂²×C₈	C₂×C₄×C₁₃	1	1	yes	characteristic
91	C₂²×C₈	C₂×C₄×C₁₃	1	1	yes	characteristic
92	C₂²×C₈	C₂×C₄×C₁₃	1	2	yes	—
93	C₂×C₄²	—	2	2	no	—
94	C₂×C₄²	C₂×C₄×C₁₃	1	2	yes	—
95	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
96	C₄⋊C₈	C₂×C₄×C₁₃	1	8	yes	—
97	C₂²⋊C₈	—	2	4	no	—
98	C₄×C₈	—	2	4	no	—
99	C₄×C₈	—	2	4	no	—
100	C₄×C₈	C₈×C₁₃	1	8	yes	complemented
101	C₄×C₈	C₂×C₄×C₁₃	1	2	yes	—
102	C₄×C₈	C₂×C₄×C₁₃	1	2	yes	—
103	C₄×C₈	D₄×C₁₃	1	1	yes	characteristic
104	C₄×C₈	D₄×C₁₃	1	1	yes	characteristic
105	C₄×C₈	Q₈×C₁₃	1	2	yes	—
106	C₂×C₁₃	—	2	2	no	—
107	C₂×C₁₃	C₄×C₄⋊C₈	1	1	yes	characteristic
108	C₂×C₁₃	#1772	1	2	yes	—
109	C₂²×C₄	—	2	2	no	—
110	C₂²×C₄	C₂²×C₄×C₁₃	1	1	yes	characteristic
111	C₂²×C₄	C₁₃×#41	1	2	yes	—
112	C₂×C₈	—	2	2	no	—
113	C₂×C₈	—	2	8	no	—
114	C₂×C₈	C₄²×C₁₃	1	2	yes	—
115	C₂×C₈	C₄²×C₁₃	1	1	yes	characteristic
116	C₂×C₈	C₄²×C₁₃	1	1	yes	characteristic
117	C₂×C₈	C₁₃×C₄⋊C₄	1	2	yes	—
118	C₂×C₈	C₁₃×C₄⋊C₄	1	2	yes	—
119	C₂×C₈	C₂×C₈×C₁₃	1	4	yes	—
120	C₂×C₈	C₂×C₈×C₁₃	1	4	yes	—
121	C₂×C₈	C₁₃×M₁₆	1	4	yes	—
122	C₄²	—	2	8	no	—
123	C₄²	—	2	2	no	—
124	C₄²	C₂×C₈×C₁₃	1	4	yes	—
125	C₄²	C₂²×C₄×C₁₃	1	2	yes	—
126	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
127	C₄²	C₂×Q₈×C₁₃	1	1	yes	characteristic
128	C₁₃	#8266	1	1	yes	characteristic, direct product factor, Sylow
129	C₂³	C₁₃×#117	1	1	yes	characteristic
130	C₂×C₄	—	2	8	no	—
131	C₂×C₄	—	2	4	no	—
132	C₂×C₄	—	2	4	no	—
133	C₂×C₄	C₂×C₄²×C₁₃	1	1	yes	characteristic
134	C₂×C₄	C₂×C₁₃×C₄⋊C₄	1	1	yes	characteristic
135	C₂×C₄	C₄×D₄×C₁₃	1	2	yes	—
136	C₂×C₄	C₄×Q₈×C₁₃	1	2	yes	—
137	C₂×C₄	C₂²×C₈×C₁₃	1	2	yes	—
138	C₂×C₄	C₂×C₁₃×M₁₆	1	2	yes	—
139	C₈	—	2	2	no	—
140	C₈	—	4	8	no	—
141	C₈	C₄×C₈×C₁₃	1	2	yes	—
142	C₈	C₄×C₈×C₁₃	1	2	yes	—
143	C₈	C₁₃×C₄⋊C₈	1	4	yes	—
144	C₈	C₁₃×C₄⋊C₈	1	4	yes	—
145	C₂²	—	2	2	no	—
146	C₂²	C₄×C₁₃×C₄⋊C₄	1	1	yes	characteristic
147	C₂²	C₁₃×#431	1	2	yes	—
148	C₄	—	2	8	no	—
149	C₄	—	2	2	no	—
150	C₄	C₂×C₄×C₈×C₁₃	1	1	yes	characteristic
151	C₄	C₄×C₁₃×M₁₆	1	1	yes	characteristic
152	C₄	C₂×C₁₃×C₄⋊C₈	1	2	yes	—
153	C₄	C₁₃×#443	1	4	yes	—
154	C₂	—	2	2	no	—
155	C₂	C₄×C₁₃×C₄⋊C₈	1	1	yes	characteristic
156	C₂	C₁₃×#1772	1	2	yes	—

	subgroup	G/H
center	C₄×C₈×C₁₃	D₄
commutator	C₄	C₂×C₄×C₈×C₁₃
Frattini, cosocle	C₂×C₄²	C₂³×C₁₃
socle	C₂²×C₁₃	C₄×C₄⋊C₄
prime radical	C₂³×C₁₃	#117
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₈×C₁₃ ◁ C₂×C₄×C₈×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
11. C₁₃×#335	C₄	semidirect
12. C₁₃×#334	C₄	semidirect
25. C₄×C₈×C₁₃	C₈	semidirect
31. #8266	C₁₃	direct (N × H)
77. #335	C₄×C₁₃	semidirect
78. #334	C₄×C₁₃	semidirect
100. C₄×C₈	C₈×C₁₃	semidirect
128. C₁₃	#8266	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄²×C₈×C₁₃	2
C₁₃×#1564	2
C₁₃×#1450	2
C₁₃×#1276	2
#8266	13

Subgroups — C13×#8266 #15694605

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#8266 #15694605