Subgroups — C₁₃×#12000 #15698339

Order 3328.

242 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	C₄×C₁₃×M₃₂	C₂	1	1	yes	maximal, characteristic
2	C₁₃×#1564	C₂	1	1	yes	maximal, characteristic
3	C₂×C₁₃×#334	C₂	1	1	yes	maximal, characteristic
4	C₁₃×#1368	C₂	1	4	yes	maximal, complemented
5	C₂×C₁₃×M₃₂	C₄	1	1	yes	characteristic
6	C₂×C₁₃×M₃₂	C₄	1	1	yes	characteristic
7	C₁₃×#431	—	2	1	no	—
8	C₂×C₁₃×C₄⋊C₈	—	2	1	no	—
9	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
10	C₁₃×#346	C₂²	1	2	yes	—
11	C₄×C₁₃×C₁₆	C₄	1	2	yes	—
12	C₁₃×#335	—	2	1	no	—
13	C₁₃×#334	—	2	1	no	—
14	C₁₃×#334	C₂²	1	2	yes	—
15	C₁₃×#334	C₂²	1	2	yes	—
16	C₂²×C₈×C₁₃	—	4	1	no	—
17	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₈×C₁₃	Q₈	1	1	yes	characteristic
19	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
20	C₁₃×M₃₂	—	2	2	no	—
21	C₁₃×M₃₂	C₈	1	4	yes	complemented
22	C₂×C₁₃×C₁₆	C₂×C₄	1	2	yes	—
23	C₂×C₁₃×C₁₆	C₂×C₄	1	2	yes	—
24	C₁₃×C₄⋊C₈	—	2	2	no	—
25	C₁₃×C₄⋊C₈	—	2	2	no	—
26	C₁₃×C₄⋊C₈	—	4	1	no	—
27	C₁₃×C₂²⋊C₈	—	4	1	no	—
28	C₄×C₈×C₁₃	—	4	1	no	—
29	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
30	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
31	C₄×C₈×C₁₃	Q₈	1	1	yes	characteristic
32	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
33	#12000	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
34	C₂²×C₄×C₁₃	C₄⋊C₄	1	1	yes	characteristic
35	C₂²×C₄×C₁₃	SD₁₆	1	1	yes	characteristic
36	C₂²×C₄×C₁₃	SD₁₆	1	1	yes	characteristic
37	C₂×C₈×C₁₃	—	2	2	no	—
38	C₂×C₈×C₁₃	—	4	2	no	—
39	C₂×C₈×C₁₃	—	4	2	no	—
40	C₂×C₈×C₁₃	—	8	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	C₂×C₈	1	2	yes	—
44	C₂×C₈×C₁₃	M₁₆	1	2	yes	—
45	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
46	C₂×C₈×C₁₃	C₂×Q₈	1	1	yes	characteristic
47	C₄²×C₁₃	C₄⋊C₄	1	1	yes	characteristic
48	C₄²×C₁₃	SD₁₆	1	1	yes	characteristic
49	C₄²×C₁₃	SD₁₆	1	1	yes	characteristic
50	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
51	C₁₃×C₁₆	—	2	2	no	—
52	C₁₃×C₁₆	C₂×C₈	1	4	yes	complemented
53	C₄×M₃₂	C₂×C₁₃	1	1	yes	characteristic
54	#1564	C₂×C₁₃	1	1	yes	characteristic
55	C₂×#334	C₂×C₁₃	1	1	yes	characteristic
56	#1368	C₂×C₁₃	1	4	yes	complemented
57	C₂³×C₁₃	#106	1	1	yes	characteristic
58	C₂×C₄×C₁₃	—	2	2	no	—
59	C₂×C₄×C₁₃	—	4	1	no	—
60	C₂×C₄×C₁₃	C₄⋊C₈	1	2	yes	—
61	C₂×C₄×C₁₃	#106	1	1	yes	characteristic
62	C₂×C₄×C₁₃	#106	1	1	yes	characteristic
63	C₂×C₄×C₁₃	#106	1	1	yes	characteristic
64	C₂×C₄×C₁₃	#108	1	2	yes	—
65	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
66	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
67	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
68	C₈×C₁₃	—	2	1	no	—
69	C₈×C₁₃	—	2	1	no	—
70	C₈×C₁₃	—	8	2	no	—
71	C₈×C₁₃	—	8	2	no	—
72	C₈×C₁₃	C₄⋊C₈	1	2	yes	—
73	C₈×C₁₃	C₂²×C₈	1	1	yes	characteristic
74	C₈×C₁₃	C₂×M₁₆	1	1	yes	characteristic
75	C₂×M₃₂	C₄×C₁₃	1	1	yes	characteristic
76	C₂×M₃₂	C₄×C₁₃	1	1	yes	characteristic
77	#431	—	2	1	no	—
78	C₂×C₄⋊C₈	—	2	1	no	—
79	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
80	#346	C₂²×C₁₃	1	2	yes	—
81	C₄×C₁₆	C₄×C₁₃	1	2	yes	—
82	#335	—	2	1	no	—
83	#334	—	2	1	no	—
84	#334	C₂²×C₁₃	1	2	yes	—
85	#334	C₂²×C₁₃	1	2	yes	—
86	C₂²×C₁₃	—	2	1	no	—
87	C₂²×C₁₃	—	2	1	no	—
88	C₂²×C₁₃	#334	1	2	yes	—
89	C₂²×C₁₃	C₂×#106	1	1	yes	characteristic
90	C₄×C₁₃	—	2	1	no	—
91	C₄×C₁₃	—	2	1	no	—
92	C₄×C₁₃	—	2	1	no	—
93	C₄×C₁₃	#334	1	2	yes	—
94	C₄×C₁₃	#365	1	2	yes	—
95	C₄×C₁₃	C₂×C₄⋊C₈	1	1	yes	characteristic
96	C₄×C₁₃	C₂×#108	1	1	yes	characteristic
97	C₂²×C₈	—	4	1	no	—
98	C₂²×C₈	C₂×C₄×C₁₃	1	1	yes	characteristic
99	C₂²×C₈	Q₈×C₁₃	1	1	yes	characteristic
100	C₂×C₄²	D₄×C₁₃	1	1	yes	characteristic
101	M₃₂	—	2	2	no	—
102	M₃₂	C₈×C₁₃	1	4	yes	complemented
103	C₂×C₁₆	C₂×C₄×C₁₃	1	2	yes	—
104	C₂×C₁₆	C₂×C₄×C₁₃	1	2	yes	—
105	C₄⋊C₈	—	2	2	no	—
106	C₄⋊C₈	—	2	2	no	—
107	C₄⋊C₈	—	4	1	no	—
108	C₂²⋊C₈	—	4	1	no	—
109	C₄×C₈	—	4	1	no	—
110	C₄×C₈	C₂×C₄×C₁₃	1	1	yes	characteristic
111	C₄×C₈	D₄×C₁₃	1	1	yes	characteristic
112	C₄×C₈	Q₈×C₁₃	1	1	yes	characteristic
113	C₄×C₈	C₂³×C₁₃	1	1	yes	characteristic
114	C₂×C₁₃	—	2	2	no	—
115	C₂×C₁₃	C₂×#334	1	1	yes	characteristic
116	C₂×C₁₃	C₂×#365	1	1	yes	characteristic
117	C₂×C₁₃	#2155	1	1	yes	characteristic
118	C₂²×C₄	C₁₃×C₄⋊C₄	1	1	yes	characteristic
119	C₂²×C₄	C₁₃×SD₁₆	1	1	yes	characteristic
120	C₂²×C₄	C₁₃×SD₁₆	1	1	yes	characteristic
121	C₂×C₈	—	2	2	no	—
122	C₂×C₈	—	4	2	no	—
123	C₂×C₈	—	4	2	no	—
124	C₂×C₈	—	8	2	no	—
125	C₂×C₈	C₁₃×C₄⋊C₄	1	1	yes	characteristic
126	C₂×C₈	C₁₃×C₄⋊C₄	1	1	yes	characteristic
127	C₂×C₈	C₂×C₈×C₁₃	1	2	yes	—
128	C₂×C₈	C₁₃×M₁₆	1	2	yes	—
129	C₂×C₈	C₂²×C₄×C₁₃	1	1	yes	characteristic
130	C₂×C₈	C₂×Q₈×C₁₃	1	1	yes	characteristic
131	C₄²	C₁₃×C₄⋊C₄	1	1	yes	characteristic
132	C₄²	C₁₃×SD₁₆	1	1	yes	characteristic
133	C₄²	C₁₃×SD₁₆	1	1	yes	characteristic
134	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
135	C₁₆	—	2	2	no	—
136	C₁₆	C₂×C₈×C₁₃	1	4	yes	complemented
137	C₁₃	#12000	1	1	yes	characteristic, direct product factor, Sylow
138	C₂³	C₁₃×#106	1	1	yes	characteristic
139	C₂×C₄	—	2	2	no	—
140	C₂×C₄	—	4	1	no	—
141	C₂×C₄	C₁₃×C₄⋊C₈	1	2	yes	—
142	C₂×C₄	C₁₃×#106	1	1	yes	characteristic
143	C₂×C₄	C₁₃×#106	1	1	yes	characteristic
144	C₂×C₄	C₁₃×#106	1	1	yes	characteristic
145	C₂×C₄	C₁₃×#108	1	2	yes	—
146	C₂×C₄	C₂×C₁₃×C₄⋊C₄	1	1	yes	characteristic
147	C₂×C₄	C₂×C₁₃×SD₁₆	1	1	yes	characteristic
148	C₂×C₄	C₂×C₁₃×SD₁₆	1	1	yes	characteristic
149	C₈	—	2	1	no	—
150	C₈	—	2	1	no	—
151	C₈	—	8	2	no	—
152	C₈	—	8	2	no	—
153	C₈	C₁₃×C₄⋊C₈	1	2	yes	—
154	C₈	C₂²×C₈×C₁₃	1	1	yes	characteristic
155	C₈	C₂×C₁₃×M₁₆	1	1	yes	characteristic
156	C₂²	—	2	1	no	—
157	C₂²	—	2	1	no	—
158	C₂²	C₁₃×#334	1	2	yes	—
159	C₂²	C₂×C₁₃×#106	1	1	yes	characteristic
160	C₄	—	2	1	no	—
161	C₄	—	2	1	no	—
162	C₄	—	2	1	no	—
163	C₄	C₁₃×#334	1	2	yes	—
164	C₄	C₁₃×#365	1	2	yes	—
165	C₄	C₂×C₁₃×C₄⋊C₈	1	1	yes	characteristic
166	C₄	C₂×C₁₃×#108	1	1	yes	characteristic
167	C₂	—	2	2	no	—
168	C₂	C₂×C₁₃×#334	1	1	yes	characteristic
169	C₂	C₂×C₁₃×#365	1	1	yes	characteristic
170	C₂	C₁₃×#2155	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₁₃	#106
commutator	C₈	C₂²×C₈×C₁₃
Frattini, cosocle	C₄×C₈	C₂³×C₁₃
socle	C₂²×C₁₃	C₂×#106
prime radical	C₂³×C₁₃	#106
LCS	G ▷ 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
4. C₁₃×#1368	C₂	semidirect
21. C₁₃×M₃₂	C₈	semidirect
33. #12000	C₁₃	direct (N × H)
52. C₁₃×C₁₆	C₂×C₈	semidirect
56. #1368	C₂×C₁₃	semidirect
102. M₃₂	C₈×C₁₃	semidirect
136. C₁₆	C₂×C₈×C₁₃	semidirect
137. C₁₃	#12000	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄×C₁₃×M₃₂	2
C₁₃×#1564	2
C₂×C₁₃×#334	2
C₁₃×#1368	2
#12000	13

Subgroups — C13×#12000 #15698339

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#12000 #15698339