Subgroups — C₁₃×#12909 #15699248

Order 3328.

314 conjugacy classes of subgroups.

The subgroup lattice has 146 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₃×#368	C₂	1	4	yes	maximal
2	C₂×C₄×C₁₃×C₁₆	C₂	1	1	yes	maximal, characteristic
3	C₁₃×#1848	C₂	1	2	yes	maximal
4	C₂²×C₁₃×C₁₆	C₂²	1	2	yes	—
5	C₂×C₁₃×#108	—	2	2	no	—
6	C₂×C₁₃×#108	C₂²	1	4	yes	—
7	C₁₃×#423	—	2	2	no	—
8	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
9	C₁₃×#368	—	2	8	no	—
10	C₄×C₁₃×C₁₆	C₄	1	2	yes	—
11	C₄×C₁₃×C₁₆	C₄	1	2	yes	—
12	C₂×C₁₃×M₁₆	—	2	4	no	—
13	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
14	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
15	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
16	C₂×C₁₃×C₁₆	C₂×C₄	1	4	yes	—
17	C₂×C₁₃×C₁₆	D₄	1	4	yes	—
18	C₂×C₁₃×C₁₆	Q₈	1	4	yes	—
19	C₁₃×#108	—	2	8	no	—
20	C₁₃×#108	—	4	4	no	—
21	C₁₃×C₄⋊C₈	—	4	4	no	—
22	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
23	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
24	C₄×C₈×C₁₃	Q₈	1	1	yes	characteristic
25	C₄×C₈×C₁₃	Q₈	1	1	yes	characteristic
26	#12909	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
27	C₂²×C₄×C₁₃	Q₁₆	1	2	yes	—
28	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
29	C₁₃×M₁₆	—	4	4	no	—
30	C₁₃×M₁₆	—	4	4	no	—
31	C₂×C₈×C₁₃	—	4	4	no	—
32	C₂×C₈×C₁₃	C₄⋊C₄	1	1	yes	characteristic
33	C₂×C₈×C₁₃	C₄⋊C₄	1	1	yes	characteristic
34	C₂×C₈×C₁₃	D₈	1	2	yes	—
35	C₂×C₈×C₁₃	Q₁₆	1	2	yes	—
36	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
37	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
38	C₂×C₈×C₁₃	C₂×Q₈	1	2	yes	—
39	C₂×C₈×C₁₃	C₂×Q₈	1	1	yes	characteristic
40	C₄²×C₁₃	C₄⋊C₄	1	1	yes	characteristic
41	C₄²×C₁₃	C₄⋊C₄	1	1	yes	characteristic
42	C₄²×C₁₃	D₈	1	2	yes	—
43	C₁₃×C₁₆	C₄⋊C₄	1	8	yes	—
44	C₂×#368	C₂×C₁₃	1	4	yes	—
45	C₂×C₄×C₁₆	C₂×C₁₃	1	1	yes	characteristic
46	#1848	C₂×C₁₃	1	2	yes	—
47	C₂³×C₁₃	C₂×Q₁₆	1	1	yes	characteristic
48	C₂×C₄×C₁₃	—	2	4	no	—
49	C₂×C₄×C₁₃	#107	1	2	yes	—
50	C₂×C₄×C₁₃	#107	1	2	yes	—
51	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
52	C₂×C₄×C₁₃	#127	1	1	yes	characteristic
53	C₂×C₄×C₁₃	C₄⋊Q₈	1	1	yes	characteristic
54	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
55	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
56	C₂×C₄×C₁₃	C₂×Q₁₆	1	1	yes	characteristic
57	C₈×C₁₃	—	8	4	no	—
58	C₈×C₁₃	#107	1	2	yes	—
59	C₈×C₁₃	#107	1	2	yes	—
60	C₈×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
61	C₈×C₁₃	C₄⋊Q₈	1	2	yes	—
62	C₂²×C₁₆	C₂²×C₁₃	1	2	yes	—
63	C₂×#108	—	2	2	no	—
64	C₂×#108	C₂²×C₁₃	1	4	yes	—
65	#423	—	2	2	no	—
66	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
67	#368	—	2	8	no	—
68	C₄×C₁₆	C₄×C₁₃	1	2	yes	—
69	C₄×C₁₆	C₄×C₁₃	1	2	yes	—
70	C₂²×C₁₃	—	2	2	no	—
71	C₂²×C₁₃	C₂×#107	1	1	yes	characteristic
72	C₂²×C₁₃	#494	1	1	yes	characteristic
73	C₂²×C₁₃	#500	1	1	yes	characteristic
74	C₄×C₁₃	—	2	2	no	—
75	C₄×C₁₃	#368	1	4	yes	complemented
76	C₄×C₁₃	#384	1	1	yes	characteristic
77	C₄×C₁₃	C₂×#107	1	1	yes	characteristic
78	C₄×C₁₃	#493	1	1	yes	characteristic
79	C₄×C₁₃	#500	1	1	yes	characteristic
80	C₂×M₁₆	—	2	4	no	—
81	C₂²×C₈	D₄×C₁₃	1	1	yes	characteristic
82	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
83	C₂×C₄²	D₄×C₁₃	1	1	yes	characteristic
84	C₂×C₁₆	C₂×C₄×C₁₃	1	4	yes	—
85	C₂×C₁₆	D₄×C₁₃	1	4	yes	—
86	C₂×C₁₆	Q₈×C₁₃	1	4	yes	—
87	#108	—	2	8	no	—
88	#108	—	4	4	no	—
89	C₄⋊C₈	—	4	4	no	—
90	C₄×C₈	C₂×C₄×C₁₃	1	1	yes	characteristic
91	C₄×C₈	C₂×C₄×C₁₃	1	1	yes	characteristic
92	C₄×C₈	Q₈×C₁₃	1	1	yes	characteristic
93	C₄×C₈	Q₈×C₁₃	1	1	yes	characteristic
94	C₂×C₁₃	—	2	1	no	—
95	C₂×C₁₃	—	2	1	no	—
96	C₂×C₁₃	#1845	1	1	yes	characteristic
97	C₂×C₁₃	C₂×#368	1	2	yes	—
98	C₂²×C₄	C₁₃×Q₁₆	1	2	yes	—
99	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
100	M₁₆	—	4	4	no	—
101	M₁₆	—	4	4	no	—
102	C₂×C₈	—	4	4	no	—
103	C₂×C₈	C₁₃×C₄⋊C₄	1	1	yes	characteristic
104	C₂×C₈	C₁₃×C₄⋊C₄	1	1	yes	characteristic
105	C₂×C₈	C₁₃×D₈	1	2	yes	—
106	C₂×C₈	C₁₃×Q₁₆	1	2	yes	—
107	C₂×C₈	C₂²×C₄×C₁₃	1	1	yes	characteristic
108	C₂×C₈	C₂×D₄×C₁₃	1	2	yes	—
109	C₂×C₈	C₂×Q₈×C₁₃	1	2	yes	—
110	C₂×C₈	C₂×Q₈×C₁₃	1	1	yes	characteristic
111	C₄²	C₁₃×C₄⋊C₄	1	1	yes	characteristic
112	C₄²	C₁₃×C₄⋊C₄	1	1	yes	characteristic
113	C₄²	C₁₃×D₈	1	2	yes	—
114	C₁₆	C₁₃×C₄⋊C₄	1	8	yes	—
115	C₁₃	#12909	1	1	yes	characteristic, direct product factor, Sylow
116	C₂³	C₂×C₁₃×Q₁₆	1	1	yes	characteristic
117	C₂×C₄	—	2	4	no	—
118	C₂×C₄	C₁₃×#107	1	2	yes	—
119	C₂×C₄	C₁₃×#107	1	2	yes	—
120	C₂×C₄	C₂×C₁₃×C₄⋊C₄	1	1	yes	characteristic
121	C₂×C₄	C₁₃×#127	1	1	yes	characteristic
122	C₂×C₄	C₁₃×C₄⋊Q₈	1	1	yes	characteristic
123	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
124	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
125	C₂×C₄	C₂×C₁₃×Q₁₆	1	1	yes	characteristic
126	C₈	—	8	4	no	—
127	C₈	C₁₃×#107	1	2	yes	—
128	C₈	C₁₃×#107	1	2	yes	—
129	C₈	C₂×C₁₃×C₄⋊C₄	1	2	yes	—
130	C₈	C₁₃×C₄⋊Q₈	1	2	yes	—
131	C₂²	—	2	2	no	—
132	C₂²	C₂×C₁₃×#107	1	1	yes	characteristic
133	C₂²	C₁₃×#494	1	1	yes	characteristic
134	C₂²	C₁₃×#500	1	1	yes	characteristic
135	C₄	—	2	2	no	—
136	C₄	C₁₃×#368	1	4	yes	complemented
137	C₄	C₁₃×#384	1	1	yes	characteristic
138	C₄	C₂×C₁₃×#107	1	1	yes	characteristic
139	C₄	C₁₃×#493	1	1	yes	characteristic
140	C₄	C₁₃×#500	1	1	yes	characteristic
141	C₂	—	2	1	no	—
142	C₂	—	2	1	no	—
143	C₂	C₁₃×#1845	1	1	yes	characteristic
144	C₂	C₂×C₁₃×#368	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₁₃	C₂×D₈
commutator	C₂×C₈	C₂²×C₄×C₁₃
Frattini, cosocle	C₂²×C₈	C₂³×C₁₃
socle	C₂²×C₁₃	C₂×#107
prime radical	C₂³×C₁₃	C₂×Q₁₆
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×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
26. #12909	C₁₃	direct (N × H)
75. C₄×C₁₃	#368	semidirect
115. C₁₃	#12909	direct (N × H)
136. C₄	C₁₃×#368	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₁₃×#368	2
C₂×C₄×C₁₃×C₁₆	2
C₁₃×#1848	2
#12909	13

Subgroups — C13×#12909 #15699248

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#12909 #15699248