Subgroups — C₁₃×#2956 #13112014

Order 1664.

458 conjugacy classes of subgroups.

The subgroup lattice has 172 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#575	C₂	1	1	yes	maximal, characteristic
2	C₁₃×#568	C₂	1	2	yes	maximal, complemented
3	C₁₃×#444	C₂	1	4	yes	maximal, complemented
4	C₁₃×#443	C₂	1	4	yes	maximal, complemented
5	C₁₃×#430	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₁₃×#104	C₂	1	2	yes	maximal, complemented
7	C₄×C₁₃×M₁₆	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₁₃×#41	C₂²	1	2	yes	—
9	C₁₃×#137	C₄	1	4	yes	complemented
10	C₁₃×AGL(1,ℤ/8)	C₄	1	4	yes	complemented
11	C₁₃×#135	C₂²	1	4	yes	—
12	C₁₃×#131	—	2	4	no	—
13	C₁₃×#131	C₂²	1	8	yes	complemented
14	C₂×C₁₃×M₁₆	—	2	4	no	—
15	C₂×C₁₃×M₁₆	C₂²	1	2	yes	—
16	C₂×C₁₃×M₁₆	C₂²	1	1	yes	characteristic
17	C₂×C₁₃×M₁₆	C₂²	1	1	yes	characteristic
18	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
19	C₁₃×#108	C₂²	1	4	yes	complemented
20	C₁₃×#104	C₂²	1	4	yes	complemented
21	C₁₃×#104	C₂²	1	4	yes	complemented
22	C₁₃×#97	C₂²	1	2	yes	complemented
23	C₄×C₈×C₁₃	C₂²	1	2	yes	complemented
24	C₁₃×#41	—	2	2	no	—
25	C₁₃×#41	—	2	2	no	—
26	C₁₃×#41	C₂×C₄	1	4	yes	—
27	C₁₃×#41	C₂³	1	4	yes	—
28	C₂×Q₈×C₁₃	C₂×C₄	1	2	yes	—
29	C₂×D₄×C₁₃	—	2	2	no	—
30	C₂×D₄×C₁₃	C₂×C₄	1	2	yes	—
31	C₂²×C₄×C₁₃	—	2	2	no	—
32	C₂²×C₄×C₁₃	—	2	1	no	—
33	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
34	C₁₃×Q₁₆	C₂×C₄	1	4	yes	complemented
35	C₁₃×SD₁₆	C₂×C₄	1	8	yes	complemented
36	C₁₃×D₈	C₂×C₄	1	4	yes	complemented
37	C₁₃×M₁₆	—	2	2	no	—
38	C₁₃×M₁₆	—	2	8	no	—
39	C₁₃×M₁₆	C₂×C₄	1	4	yes	—
40	C₁₃×M₁₆	D₄	1	4	yes	—
41	C₁₃×M₁₆	C₂³	1	4	yes	—
42	C₂×C₈×C₁₃	—	2	2	no	—
43	C₂×C₈×C₁₃	—	2	8	no	—
44	C₂×C₈×C₁₃	C₂³	1	2	yes	—
45	C₂×C₈×C₁₃	C₂³	1	2	yes	—
46	C₄²×C₁₃	—	2	1	no	—
47	C₄²×C₁₃	C₂³	1	1	yes	characteristic
48	C₄²×C₁₃	C₂³	1	1	yes	characteristic
49	#2956	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
50	C₂³×C₁₃	—	2	2	no	—
51	C₂³×C₁₃	#41	1	1	yes	characteristic
52	Q₈×C₁₃	—	2	2	no	—
53	Q₈×C₁₃	C₂²×C₄	1	4	yes	—
54	D₄×C₁₃	—	2	4	no	—
55	D₄×C₁₃	—	2	4	no	—
56	D₄×C₁₃	—	2	2	no	—
57	D₄×C₁₃	C₂²×C₄	1	4	yes	—
58	C₂×C₄×C₁₃	—	2	2	no	—
59	C₂×C₄×C₁₃	—	2	1	no	—
60	C₂×C₄×C₁₃	—	2	4	no	—
61	C₂×C₄×C₁₃	—	2	4	no	—
62	C₂×C₄×C₁₃	—	2	1	no	—
63	C₂×C₄×C₁₃	—	4	2	no	—
64	C₂×C₄×C₁₃	C₂²×C₄	1	2	yes	—
65	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
66	C₂×C₄×C₁₃	#41	1	1	yes	characteristic
67	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
68	C₈×C₁₃	—	2	4	no	—
69	C₈×C₁₃	C₂²×C₄	1	4	yes	—
70	C₈×C₁₃	C₂×D₄	1	4	yes	—
71	#575	C₂×C₁₃	1	1	yes	characteristic
72	#568	C₂×C₁₃	1	2	yes	complemented
73	#444	C₂×C₁₃	1	4	yes	complemented
74	#443	C₂×C₁₃	1	4	yes	complemented
75	#430	C₂×C₁₃	1	1	yes	characteristic, complemented
76	C₂×#104	C₂×C₁₃	1	2	yes	complemented
77	C₄×M₁₆	C₂×C₁₃	1	1	yes	characteristic, complemented
78	C₂²×C₁₃	—	2	4	no	—
79	C₂²×C₁₃	—	4	1	no	—
80	C₂²×C₁₃	—	8	2	no	—
81	C₂²×C₁₃	C₄×D₄	1	2	yes	—
82	C₂²×C₁₃	C₂×#41	1	1	yes	characteristic
83	C₄×C₁₃	—	2	2	no	—
84	C₄×C₁₃	—	2	4	no	—
85	C₄×C₁₃	—	4	1	no	—
86	C₄×C₁₃	C₄×D₄	1	2	yes	—
87	C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
88	C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
89	C₂×#41	C₂²×C₁₃	1	2	yes	—
90	#137	C₄×C₁₃	1	4	yes	complemented
91	AGL(1,ℤ/8)	C₄×C₁₃	1	4	yes	complemented
92	#135	C₂²×C₁₃	1	4	yes	—
93	#131	—	2	4	no	—
94	#131	C₂²×C₁₃	1	8	yes	complemented
95	C₂×M₁₆	—	2	4	no	—
96	C₂×M₁₆	C₂²×C₁₃	1	2	yes	—
97	C₂×M₁₆	C₂²×C₁₃	1	1	yes	characteristic
98	C₂×M₁₆	C₂²×C₁₃	1	1	yes	characteristic
99	C₂×C₄²	C₂²×C₁₃	1	1	yes	characteristic
100	#108	C₂²×C₁₃	1	4	yes	complemented
101	#104	C₂²×C₁₃	1	4	yes	complemented
102	#104	C₂²×C₁₃	1	4	yes	complemented
103	#97	C₂²×C₁₃	1	2	yes	complemented
104	C₄×C₈	C₂²×C₁₃	1	2	yes	complemented
105	C₂×C₁₃	—	2	1	no	—
106	C₂×C₁₃	—	2	2	no	—
107	C₂×C₁₃	—	4	4	no	—
108	C₂×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
109	#41	—	2	2	no	—
110	#41	—	2	2	no	—
111	#41	C₂×C₄×C₁₃	1	4	yes	—
112	#41	C₂³×C₁₃	1	4	yes	—
113	C₂×Q₈	C₂×C₄×C₁₃	1	2	yes	—
114	C₂×D₄	—	2	2	no	—
115	C₂×D₄	C₂×C₄×C₁₃	1	2	yes	—
116	C₂²×C₄	—	2	2	no	—
117	C₂²×C₄	—	2	1	no	—
118	C₂²×C₄	C₂³×C₁₃	1	1	yes	characteristic
119	Q₁₆	C₂×C₄×C₁₃	1	4	yes	complemented
120	SD₁₆	C₂×C₄×C₁₃	1	8	yes	complemented
121	D₈	C₂×C₄×C₁₃	1	4	yes	complemented
122	M₁₆	—	2	2	no	—
123	M₁₆	—	2	8	no	—
124	M₁₆	C₂×C₄×C₁₃	1	4	yes	—
125	M₁₆	D₄×C₁₃	1	4	yes	—
126	M₁₆	C₂³×C₁₃	1	4	yes	—
127	C₂×C₈	—	2	2	no	—
128	C₂×C₈	—	2	8	no	—
129	C₂×C₈	C₂³×C₁₃	1	2	yes	—
130	C₂×C₈	C₂³×C₁₃	1	2	yes	—
131	C₄²	—	2	1	no	—
132	C₄²	C₂³×C₁₃	1	1	yes	characteristic
133	C₄²	C₂³×C₁₃	1	1	yes	characteristic
134	C₁₃	#2956	1	1	yes	characteristic, direct product factor, Sylow
135	C₂³	—	2	2	no	—
136	C₂³	C₁₃×#41	1	1	yes	characteristic
137	Q₈	—	2	2	no	—
138	Q₈	C₂²×C₄×C₁₃	1	4	yes	—
139	D₄	—	2	4	no	—
140	D₄	—	2	4	no	—
141	D₄	—	2	2	no	—
142	D₄	C₂²×C₄×C₁₃	1	4	yes	—
143	C₂×C₄	—	2	2	no	—
144	C₂×C₄	—	2	1	no	—
145	C₂×C₄	—	2	4	no	—
146	C₂×C₄	—	2	4	no	—
147	C₂×C₄	—	2	1	no	—
148	C₂×C₄	—	4	2	no	—
149	C₂×C₄	C₂²×C₄×C₁₃	1	2	yes	—
150	C₂×C₄	C₂×D₄×C₁₃	1	2	yes	—
151	C₂×C₄	C₁₃×#41	1	1	yes	characteristic
152	C₂×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
153	C₈	—	2	4	no	—
154	C₈	C₂²×C₄×C₁₃	1	4	yes	—
155	C₈	C₂×D₄×C₁₃	1	4	yes	—
156	C₂²	—	2	4	no	—
157	C₂²	—	4	1	no	—
158	C₂²	—	8	2	no	—
159	C₂²	C₄×D₄×C₁₃	1	2	yes	—
160	C₂²	C₂×C₁₃×#41	1	1	yes	characteristic
161	C₄	—	2	2	no	—
162	C₄	—	2	4	no	—
163	C₄	—	4	1	no	—
164	C₄	C₄×D₄×C₁₃	1	2	yes	—
165	C₄	C₂³×C₄×C₁₃	1	1	yes	characteristic
166	C₄	C₂²×D₄×C₁₃	1	1	yes	characteristic
167	C₂	—	2	1	no	—
168	C₂	—	2	2	no	—
169	C₂	—	4	4	no	—
170	C₂	C₂×C₄×D₄×C₁₃	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₁₃	C₂²×D₄
commutator	C₄	C₂³×C₄×C₁₃
Frattini, cosocle	C₂×C₄	C₂⁴×C₁₃
socle	C₂×C₁₃	C₂×C₄×D₄
prime radical	C₁₃×#575	C₂
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₁₃ ◁ C₂×C₁₃×M₁₆ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₁₃×#568	C₂	semidirect
3. C₁₃×#444	C₂	semidirect
4. C₁₃×#443	C₂	semidirect
5. C₁₃×#430	C₂	semidirect
6. C₂×C₁₃×#104	C₂	semidirect
7. C₄×C₁₃×M₁₆	C₂	semidirect
9. C₁₃×#137	C₄	semidirect
10. C₁₃×AGL(1,ℤ/8)	C₄	semidirect
13. C₁₃×#131	C₂²	semidirect
19. C₁₃×#108	C₂²	semidirect
20. C₁₃×#104	C₂²	semidirect
21. C₁₃×#104	C₂²	semidirect
22. C₁₃×#97	C₂²	semidirect
23. C₄×C₈×C₁₃	C₂²	semidirect
34. C₁₃×Q₁₆	C₂×C₄	semidirect
35. C₁₃×SD₁₆	C₂×C₄	semidirect
36. C₁₃×D₈	C₂×C₄	semidirect
49. #2956	C₁₃	direct (N × H)
72. #568	C₂×C₁₃	semidirect
73. #444	C₂×C₁₃	semidirect
74. #443	C₂×C₁₃	semidirect
75. #430	C₂×C₁₃	semidirect
76. C₂×#104	C₂×C₁₃	semidirect
77. C₄×M₁₆	C₂×C₁₃	semidirect
90. #137	C₄×C₁₃	semidirect
91. AGL(1,ℤ/8)	C₄×C₁₃	semidirect
94. #131	C₂²×C₁₃	semidirect
100. #108	C₂²×C₁₃	semidirect
101. #104	C₂²×C₁₃	semidirect
102. #104	C₂²×C₁₃	semidirect
103. #97	C₂²×C₁₃	semidirect
104. C₄×C₈	C₂²×C₁₃	semidirect
119. Q₁₆	C₂×C₄×C₁₃	semidirect
120. SD₁₆	C₂×C₄×C₁₃	semidirect
121. D₈	C₂×C₄×C₁₃	semidirect
134. C₁₃	#2956	direct (N × H)

Maximal subgroups

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

subgroup	index
C₁₃×#575	2
C₁₃×#568	2
C₁₃×#444	2
C₁₃×#443	2
C₁₃×#430	2
C₂×C₁₃×#104	2
C₄×C₁₃×M₁₆	2
#2956	13

Subgroups — C13×#2956 #13112014

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#2956 #13112014