Subgroups — C₁₃×#12668 #15699007

Order 3328.

308 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	C₁₃×#2155	C₂	1	2	yes	maximal
2	C₄×C₁₃×M₃₂	C₂	1	1	yes	maximal, characteristic
3	C₁₃×#1772	C₂	1	2	yes	maximal
4	C₁₃×#1385	C₂	1	1	yes	maximal, characteristic
5	C₁₃×#1384	C₂	1	1	yes	maximal, characteristic
6	C₂×C₁₃×M₃₂	C₂²	1	1	yes	characteristic
7	C₂×C₁₃×M₃₂	C₂²	1	1	yes	characteristic
8	C₂×C₁₃×#108	C₂²	1	2	yes	—
9	C₁₃×#427	C₂²	1	2	yes	—
10	C₁₃×#405	—	2	2	no	—
11	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
12	C₁₃×#365	C₄	1	8	yes	complemented
13	C₁₃×#346	C₄	1	2	yes	complemented
14	C₄×C₁₃×C₁₆	C₄	1	2	yes	complemented
15	C₁₃×#339	—	2	2	no	—
16	C₂×C₁₃×M₁₆	—	2	2	no	—
17	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
18	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
19	C₁₃×#117	—	2	2	no	—
20	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
21	C₁₃×M₃₂	—	2	2	no	—
22	C₁₃×M₃₂	C₂×C₄	1	4	yes	—
23	C₂×C₁₃×C₁₆	C₂×C₄	1	2	yes	—
24	C₂×C₁₃×C₁₆	C₂×C₄	1	2	yes	—
25	C₁₃×#108	—	2	2	no	—
26	C₁₃×#108	C₂×C₄	1	4	yes	—
27	C₁₃×#107	—	2	2	no	—
28	C₁₃×#106	C₂×C₄	1	4	yes	—
29	C₁₃×#97	—	4	2	no	—
30	C₄×C₈×C₁₃	—	4	2	no	—
31	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
32	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
33	C₄×C₈×C₁₃	Q₈	1	1	yes	characteristic
34	C₄×C₈×C₁₃	Q₈	1	1	yes	characteristic
35	#12668	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
36	C₂²×C₄×C₁₃	—	2	1	no	—
37	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
38	C₁₃×M₁₆	—	2	4	no	—
39	C₁₃×M₁₆	—	4	2	no	—
40	C₂×C₈×C₁₃	—	4	2	no	—
41	C₂×C₈×C₁₃	C₄⋊C₄	1	1	yes	characteristic
42	C₂×C₈×C₁₃	C₄⋊C₄	1	1	yes	characteristic
43	C₂×C₈×C₁₃	SD₁₆	1	2	yes	—
44	C₂×C₈×C₁₃	SD₁₆	1	2	yes	—
45	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
46	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
47	C₂×C₈×C₁₃	C₂×Q₈	1	1	yes	characteristic
48	C₂×C₈×C₁₃	#41	1	2	yes	—
49	C₁₃×C₄⋊C₄	—	2	4	no	—
50	C₁₃×C₂²⋊C₄	—	4	2	no	—
51	C₄²×C₁₃	—	2	1	no	—
52	C₄²×C₁₃	—	4	2	no	—
53	C₄²×C₁₃	C₄⋊C₄	1	1	yes	characteristic
54	C₄²×C₁₃	C₄⋊C₄	1	1	yes	characteristic
55	C₁₃×C₁₆	—	2	2	no	—
56	C₁₃×C₁₆	C₄²	1	4	yes	—
57	#2155	C₂×C₁₃	1	2	yes	—
58	C₄×M₃₂	C₂×C₁₃	1	1	yes	characteristic
59	#1772	C₂×C₁₃	1	2	yes	—
60	#1385	C₂×C₁₃	1	1	yes	characteristic
61	#1384	C₂×C₁₃	1	1	yes	characteristic
62	C₂³×C₁₃	#135	1	1	yes	characteristic
63	C₂×C₄×C₁₃	—	2	1	no	—
64	C₂×C₄×C₁₃	—	2	1	no	—
65	C₂×C₄×C₁₃	—	4	2	no	—
66	C₂×C₄×C₁₃	—	4	4	no	—
67	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
68	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
69	C₂×C₄×C₁₃	C₂×SD₁₆	1	2	yes	—
70	C₂×C₄×C₁₃	#135	1	1	yes	characteristic
71	C₈×C₁₃	—	4	4	no	—
72	C₈×C₁₃	#106	1	2	yes	—
73	C₈×C₁₃	#106	1	1	yes	characteristic
74	C₈×C₁₃	#106	1	1	yes	characteristic
75	C₈×C₁₃	C₂×C₄²	1	1	yes	characteristic
76	C₈×C₁₃	#117	1	1	yes	characteristic
77	C₈×C₁₃	C₄×Q₈	1	2	yes	—
78	C₂×M₃₂	C₂²×C₁₃	1	1	yes	characteristic
79	C₂×M₃₂	C₂²×C₁₃	1	1	yes	characteristic
80	C₂×#108	C₂²×C₁₃	1	2	yes	—
81	#427	C₂²×C₁₃	1	2	yes	—
82	#405	—	2	2	no	—
83	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
84	#365	C₄×C₁₃	1	8	yes	complemented
85	#346	C₄×C₁₃	1	2	yes	complemented
86	C₄×C₁₆	C₄×C₁₃	1	2	yes	complemented
87	#339	—	2	2	no	—
88	C₂²×C₁₃	—	4	1	no	—
89	C₂²×C₁₃	#427	1	1	yes	characteristic
90	C₂²×C₁₃	C₄×SD₁₆	1	2	yes	—
91	C₄×C₁₃	—	2	2	no	—
92	C₄×C₁₃	—	4	1	no	—
93	C₄×C₁₃	—	8	4	no	—
94	C₄×C₁₃	C₄×C₄⋊C₄	1	1	yes	characteristic
95	C₄×C₁₃	C₂×#106	1	1	yes	characteristic
96	C₄×C₁₃	C₄×SD₁₆	1	2	yes	—
97	C₂×M₁₆	—	2	2	no	—
98	C₂²×C₈	D₄×C₁₃	1	1	yes	characteristic
99	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
100	#117	—	2	2	no	—
101	C₂×C₄²	D₄×C₁₃	1	1	yes	characteristic
102	M₃₂	—	2	2	no	—
103	M₃₂	C₂×C₄×C₁₃	1	4	yes	—
104	C₂×C₁₆	C₂×C₄×C₁₃	1	2	yes	—
105	C₂×C₁₆	C₂×C₄×C₁₃	1	2	yes	—
106	#108	—	2	2	no	—
107	#108	C₂×C₄×C₁₃	1	4	yes	—
108	#107	—	2	2	no	—
109	#106	C₂×C₄×C₁₃	1	4	yes	—
110	#97	—	4	2	no	—
111	C₄×C₈	—	4	2	no	—
112	C₄×C₈	C₂×C₄×C₁₃	1	1	yes	characteristic
113	C₄×C₈	C₂×C₄×C₁₃	1	1	yes	characteristic
114	C₄×C₈	Q₈×C₁₃	1	1	yes	characteristic
115	C₄×C₈	Q₈×C₁₃	1	1	yes	characteristic
116	C₂×C₁₃	—	2	1	no	—
117	C₂×C₁₃	—	2	2	no	—
118	C₂×C₁₃	C₄×#106	1	1	yes	characteristic
119	C₂²×C₄	—	2	1	no	—
120	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
121	M₁₆	—	2	4	no	—
122	M₁₆	—	4	2	no	—
123	C₂×C₈	—	4	2	no	—
124	C₂×C₈	C₁₃×C₄⋊C₄	1	1	yes	characteristic
125	C₂×C₈	C₁₃×C₄⋊C₄	1	1	yes	characteristic
126	C₂×C₈	C₁₃×SD₁₆	1	2	yes	—
127	C₂×C₈	C₁₃×SD₁₆	1	2	yes	—
128	C₂×C₈	C₂²×C₄×C₁₃	1	1	yes	characteristic
129	C₂×C₈	C₂²×C₄×C₁₃	1	2	yes	—
130	C₂×C₈	C₂×Q₈×C₁₃	1	1	yes	characteristic
131	C₂×C₈	C₁₃×#41	1	2	yes	—
132	C₄⋊C₄	—	2	4	no	—
133	C₂²⋊C₄	—	4	2	no	—
134	C₄²	—	2	1	no	—
135	C₄²	—	4	2	no	—
136	C₄²	C₁₃×C₄⋊C₄	1	1	yes	characteristic
137	C₄²	C₁₃×C₄⋊C₄	1	1	yes	characteristic
138	C₁₆	—	2	2	no	—
139	C₁₆	C₄²×C₁₃	1	4	yes	—
140	C₁₃	#12668	1	1	yes	characteristic, direct product factor, Sylow
141	C₂³	C₁₃×#135	1	1	yes	characteristic
142	C₂×C₄	—	2	1	no	—
143	C₂×C₄	—	2	1	no	—
144	C₂×C₄	—	4	2	no	—
145	C₂×C₄	—	4	4	no	—
146	C₂×C₄	C₂×C₁₃×C₄⋊C₄	1	1	yes	characteristic
147	C₂×C₄	C₄×D₄×C₁₃	1	2	yes	—
148	C₂×C₄	C₂×C₁₃×SD₁₆	1	2	yes	—
149	C₂×C₄	C₁₃×#135	1	1	yes	characteristic
150	C₈	—	4	4	no	—
151	C₈	C₁₃×#106	1	2	yes	—
152	C₈	C₁₃×#106	1	1	yes	characteristic
153	C₈	C₁₃×#106	1	1	yes	characteristic
154	C₈	C₂×C₄²×C₁₃	1	1	yes	characteristic
155	C₈	C₁₃×#117	1	1	yes	characteristic
156	C₈	C₄×Q₈×C₁₃	1	2	yes	—
157	C₂²	—	4	1	no	—
158	C₂²	C₁₃×#427	1	1	yes	characteristic
159	C₂²	C₄×C₁₃×SD₁₆	1	2	yes	—
160	C₄	—	2	2	no	—
161	C₄	—	4	1	no	—
162	C₄	—	8	4	no	—
163	C₄	C₄×C₁₃×C₄⋊C₄	1	1	yes	characteristic
164	C₄	C₂×C₁₃×#106	1	1	yes	characteristic
165	C₄	C₄×C₁₃×SD₁₆	1	2	yes	—
166	C₂	—	2	1	no	—
167	C₂	—	2	2	no	—
168	C₂	C₄×C₁₃×#106	1	1	yes	characteristic

	subgroup	G/H
center	C₈×C₁₃	#106
commutator	C₈	C₂×C₄²×C₁₃
Frattini, cosocle	C₂²×C₈	C₂³×C₁₃
socle	C₂×C₁₃	C₄×#106
prime radical	C₂³×C₁₃	#135
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈×C₁₃ ◁ C₂²×C₈×C₁₃ ◁ C₂×C₄×C₈×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
12. C₁₃×#365	C₄	semidirect
13. C₁₃×#346	C₄	semidirect
14. C₄×C₁₃×C₁₆	C₄	semidirect
35. #12668	C₁₃	direct (N × H)
84. #365	C₄×C₁₃	semidirect
85. #346	C₄×C₁₃	semidirect
86. C₄×C₁₆	C₄×C₁₃	semidirect
140. C₁₃	#12668	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×#2155	2
C₄×C₁₃×M₃₂	2
C₁₃×#1772	2
C₁₃×#1385	2
C₁₃×#1384	2
#12668	13

Subgroups — C13×#12668 #15699007

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#12668 #15699007