Subgroups — C₁₃×#2568 #13111626

Order 1664.

468 conjugacy classes of subgroups.

The subgroup lattice has 162 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₃×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×Q₈×C₁₃	C₂	1	1	yes	maximal, characteristic
3	C₁₃×#400	C₂	1	4	yes	maximal
4	C₁₃×#391	C₂	1	4	yes	maximal
5	C₁₃×#391	C₂	1	2	yes	maximal
6	C₁₃×#382	C₂	1	2	yes	maximal
7	C₄×C₁₃×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
8	C₂²×Q₈×C₁₃	C₂²	1	2	yes	—
9	C₂²×Q₈×C₁₃	C₂²	1	1	yes	characteristic
10	C₁₃×C₄⋊Q₈	—	2	4	no	—
11	C₄×Q₈×C₁₃	—	2	4	no	—
12	C₂×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
13	C₂×C₁₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
14	C₂×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
15	C₂×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
16	C₂×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
17	C₂×C₄²×C₁₃	C₂²	1	2	yes	—
18	C₂×C₄²×C₁₃	C₂²	1	2	yes	—
19	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
20	C₁₃×#95	C₂²	1	2	yes	—
21	C₁₃×#95	C₂²	1	8	yes	—
22	C₁₃×#95	C₂²	1	8	yes	—
23	C₂×Q₈×C₁₃	—	2	2	no	—
24	C₂×Q₈×C₁₃	—	2	4	no	—
25	C₂×Q₈×C₁₃	—	2	2	no	—
26	C₂×Q₈×C₁₃	—	2	8	no	—
27	C₂×Q₈×C₁₃	Q₈	1	4	yes	complemented
28	C₂²×C₄×C₁₃	C₂³	1	4	yes	—
29	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
30	C₂²×C₄×C₁₃	C₂³	1	4	yes	—
31	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
32	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
33	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
34	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
35	C₁₃×C₄⋊C₄	—	2	4	no	—
36	C₁₃×C₄⋊C₄	—	2	2	no	—
37	C₁₃×C₄⋊C₄	—	2	4	no	—
38	C₁₃×C₄⋊C₄	—	2	4	no	—
39	C₁₃×C₄⋊C₄	—	2	4	no	—
40	C₄²×C₁₃	—	2	4	no	—
41	C₄²×C₁₃	—	2	4	no	—
42	C₄²×C₁₃	D₄	1	4	yes	—
43	#2568	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
44	C₂³×C₁₃	C₂⁴	1	1	yes	characteristic
45	Q₈×C₁₃	—	2	8	no	—
46	Q₈×C₁₃	—	4	8	no	—
47	C₂×C₄×C₁₃	—	2	2	no	—
48	C₂×C₄×C₁₃	—	2	2	no	—
49	C₂×C₄×C₁₃	—	2	2	no	—
50	C₂×C₄×C₁₃	—	2	2	no	—
51	C₂×C₄×C₁₃	—	2	4	no	—
52	C₂×C₄×C₁₃	—	2	4	no	—
53	C₂×C₄×C₁₃	—	2	4	no	—
54	C₂×C₄×C₁₃	—	2	8	no	—
55	C₂×C₄×C₁₃	—	2	4	no	—
56	C₂×C₄×C₁₃	—	2	4	no	—
57	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
58	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
59	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
60	C₂×C₄×C₁₃	C₂×Q₈	1	4	yes	—
61	C₂×C₄×C₁₃	C₂×Q₈	1	2	yes	—
62	C₂×C₄×C₁₃	#41	1	2	yes	—
63	C₂×C₄×C₁₃	#41	1	4	yes	—
64	C₂×C₄⋊Q₈	C₂×C₁₃	1	1	yes	characteristic
65	C₂×C₄×Q₈	C₂×C₁₃	1	1	yes	characteristic
66	#400	C₂×C₁₃	1	4	yes	—
67	#391	C₂×C₁₃	1	4	yes	—
68	#391	C₂×C₁₃	1	2	yes	—
69	#382	C₂×C₁₃	1	2	yes	—
70	C₄×C₄⋊C₄	C₂×C₁₃	1	1	yes	characteristic
71	C₂²×C₁₃	C₂²×D₄	1	1	yes	characteristic
72	C₂²×C₁₃	C₂²×Q₈	1	1	yes	characteristic
73	C₂²×C₁₃	C₂×#41	1	1	yes	characteristic
74	C₂²×C₁₃	C₂×#41	1	2	yes	—
75	C₂²×C₁₃	2₋¹⁺⁴	1	2	yes	—
76	C₄×C₁₃	—	2	4	no	—
77	C₄×C₁₃	—	2	2	no	—
78	C₄×C₁₃	—	4	4	no	—
79	C₄×C₁₃	—	4	2	no	—
80	C₄×C₁₃	—	4	4	no	—
81	C₄×C₁₃	C₂²⋊Q₈	1	4	yes	—
82	C₄×C₁₃	#124	1	4	yes	—
83	C₂²×Q₈	C₂²×C₁₃	1	2	yes	—
84	C₂²×Q₈	C₂²×C₁₃	1	1	yes	characteristic
85	C₄⋊Q₈	—	2	4	no	—
86	C₄×Q₈	—	2	4	no	—
87	C₂×C₄⋊C₄	C₂²×C₁₃	1	1	yes	characteristic
88	C₂×C₄⋊C₄	C₂²×C₁₃	1	2	yes	—
89	C₂×C₄⋊C₄	C₂²×C₁₃	1	2	yes	—
90	C₂×C₄⋊C₄	C₂²×C₁₃	1	2	yes	—
91	C₂×C₄⋊C₄	C₂²×C₁₃	1	2	yes	—
92	C₂×C₄²	C₂²×C₁₃	1	2	yes	—
93	C₂×C₄²	C₂²×C₁₃	1	2	yes	—
94	C₂×C₄²	C₂²×C₁₃	1	1	yes	characteristic
95	#95	C₂²×C₁₃	1	2	yes	—
96	#95	C₂²×C₁₃	1	8	yes	—
97	#95	C₂²×C₁₃	1	8	yes	—
98	C₂×C₁₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
99	C₂×C₁₃	C₂×#124	1	1	yes	characteristic
100	C₂×C₁₃	#536	1	1	yes	characteristic
101	C₂×C₁₃	#556	1	2	yes	—
102	C₂×C₁₃	#557	1	2	yes	—
103	C₂×Q₈	—	2	2	no	—
104	C₂×Q₈	—	2	4	no	—
105	C₂×Q₈	—	2	2	no	—
106	C₂×Q₈	—	2	8	no	—
107	C₂×Q₈	Q₈×C₁₃	1	4	yes	complemented
108	C₂²×C₄	C₂³×C₁₃	1	4	yes	—
109	C₂²×C₄	C₂³×C₁₃	1	2	yes	—
110	C₂²×C₄	C₂³×C₁₃	1	4	yes	—
111	C₂²×C₄	C₂³×C₁₃	1	1	yes	characteristic
112	C₂²×C₄	C₂³×C₁₃	1	2	yes	—
113	C₂²×C₄	C₂³×C₁₃	1	1	yes	characteristic
114	C₂²×C₄	C₂³×C₁₃	1	1	yes	characteristic
115	C₄⋊C₄	—	2	4	no	—
116	C₄⋊C₄	—	2	2	no	—
117	C₄⋊C₄	—	2	4	no	—
118	C₄⋊C₄	—	2	4	no	—
119	C₄⋊C₄	—	2	4	no	—
120	C₄²	—	2	4	no	—
121	C₄²	—	2	4	no	—
122	C₄²	D₄×C₁₃	1	4	yes	—
123	C₁₃	#2568	1	1	yes	characteristic, direct product factor, Sylow
124	C₂³	C₂⁴×C₁₃	1	1	yes	characteristic
125	Q₈	—	2	8	no	—
126	Q₈	—	4	8	no	—
127	C₂×C₄	—	2	2	no	—
128	C₂×C₄	—	2	2	no	—
129	C₂×C₄	—	2	2	no	—
130	C₂×C₄	—	2	2	no	—
131	C₂×C₄	—	2	4	no	—
132	C₂×C₄	—	2	4	no	—
133	C₂×C₄	—	2	4	no	—
134	C₂×C₄	—	2	8	no	—
135	C₂×C₄	—	2	4	no	—
136	C₂×C₄	—	2	4	no	—
137	C₂×C₄	C₂×D₄×C₁₃	1	2	yes	—
138	C₂×C₄	C₂×D₄×C₁₃	1	2	yes	—
139	C₂×C₄	C₂×D₄×C₁₃	1	2	yes	—
140	C₂×C₄	C₂×Q₈×C₁₃	1	4	yes	—
141	C₂×C₄	C₂×Q₈×C₁₃	1	2	yes	—
142	C₂×C₄	C₁₃×#41	1	2	yes	—
143	C₂×C₄	C₁₃×#41	1	4	yes	—
144	C₂²	C₂²×D₄×C₁₃	1	1	yes	characteristic
145	C₂²	C₂²×Q₈×C₁₃	1	1	yes	characteristic
146	C₂²	C₂×C₁₃×#41	1	1	yes	characteristic
147	C₂²	C₂×C₁₃×#41	1	2	yes	—
148	C₂²	C₁₃×2₋¹⁺⁴	1	2	yes	—
149	C₄	—	2	4	no	—
150	C₄	—	2	2	no	—
151	C₄	—	4	4	no	—
152	C₄	—	4	2	no	—
153	C₄	—	4	4	no	—
154	C₄	C₁₃×C₂²⋊Q₈	1	4	yes	—
155	C₄	C₁₃×#124	1	4	yes	—
156	C₂	C₂×C₁₃×C₂²⋊Q₈	1	1	yes	characteristic
157	C₂	C₂×C₁₃×#124	1	1	yes	characteristic
158	C₂	C₁₃×#536	1	1	yes	characteristic
159	C₂	C₁₃×#556	1	2	yes	—
160	C₂	C₁₃×#557	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₁₃	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₁₃
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
27. C₂×Q₈×C₁₃	Q₈	semidirect
43. #2568	C₁₃	direct (N × H)
107. C₂×Q₈	Q₈×C₁₃	semidirect
123. C₁₃	#2568	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×C₄⋊Q₈	2
C₂×C₄×Q₈×C₁₃	2
C₁₃×#400	2
C₁₃×#391	2
C₁₃×#382	2
C₄×C₁₃×C₄⋊C₄	2
#2568	13

Subgroups — C13×#2568 #13111626

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#2568 #13111626