Subgroups — C₂×C₁₃×#2262 #15720317

Order 3328.

470 conjugacy classes of subgroups.

The subgroup lattice has 174 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₃×#508	C₂	1	2	yes	maximal, complemented
2	C₁₃×#2262	C₂	1	8	yes	maximal, direct product factor
3	C₂×C₁₃×Q₆₄	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₁₃×SD₆₄	C₂	1	2	yes	maximal, complemented
5	C₂×C₁₃×D₃₂	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₁₃×C₃₂	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₁₃×#135	—	2	2	no	—
8	C₁₃×#508	—	2	4	no	—
9	C₁₃×#508	C₂²	1	8	yes	complemented
10	C₂×C₁₃×Q₃₂	C₂²	1	2	yes	—
11	C₂×C₁₃×SD₃₂	—	2	2	no	—
12	C₂×C₁₃×D₁₆	C₂²	1	2	yes	—
13	C₂²×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
14	C₁₃×Q₆₄	C₂²	1	4	yes	complemented
15	C₁₃×SD₆₄	C₂²	1	8	yes	complemented
16	C₁₃×D₃₂	C₂²	1	4	yes	complemented
17	C₂×C₁₃×C₃₂	C₂²	1	1	yes	characteristic
18	C₂×C₁₃×C₃₂	C₂²	1	1	yes	characteristic
19	C₂×C₁₃×C₃₂	C₂²	1	4	yes	complemented
20	C₂×C₁₃×#41	—	4	2	no	—
21	C₁₃×#135	—	2	8	no	—
22	C₁₃×#135	—	4	4	no	—
23	C₂×C₁₃×Q₁₆	—	2	2	no	—
24	C₂×C₁₃×SD₁₆	—	4	2	no	—
25	C₂×C₁₃×D₈	—	2	2	no	—
26	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
27	C₁₃×Q₃₂	—	2	2	no	—
28	C₁₃×Q₃₂	C₂³	1	4	yes	—
29	C₁₃×SD₃₂	—	2	8	no	—
30	C₁₃×D₁₆	—	2	2	no	—
31	C₁₃×D₁₆	C₂³	1	4	yes	—
32	C₂×C₁₃×C₁₆	D₄	1	1	yes	characteristic
33	C₂×C₁₃×C₁₆	D₄	1	2	yes	—
34	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
35	C₂×C₁₃×C₁₆	C₂³	1	2	yes	—
36	C₁₃×C₃₂	C₂³	1	2	yes	—
37	C₁₃×C₃₂	C₂³	1	2	yes	—
38	C₂×#2262	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
39	C₁₃×#41	—	4	8	no	—
40	C₁₃×#41	—	8	4	no	—
41	C₂×Q₈×C₁₃	—	4	2	no	—
42	C₂×D₄×C₁₃	—	4	2	no	—
43	C₂×D₄×C₁₃	—	8	2	no	—
44	C₂²×C₄×C₁₃	—	8	2	no	—
45	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
46	C₁₃×Q₁₆	—	2	4	no	—
47	C₁₃×Q₁₆	—	4	2	no	—
48	C₁₃×SD₁₆	—	4	8	no	—
49	C₁₃×D₈	—	2	4	no	—
50	C₁₃×D₈	—	4	2	no	—
51	C₂×C₈×C₁₃	D₈	1	1	yes	characteristic
52	C₂×C₈×C₁₃	D₈	1	2	yes	—
53	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
54	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
55	C₁₃×C₁₆	C₂×D₄	1	1	yes	characteristic
56	C₁₃×C₁₆	C₂×D₄	1	2	yes	—
57	C₁₃×C₁₆	C₂⁴	1	1	yes	characteristic
58	C₂×#508	C₂×C₁₃	1	2	yes	complemented
59	#2262	C₂×C₁₃	1	8	yes	direct product factor
60	C₂×Q₆₄	C₂×C₁₃	1	1	yes	characteristic, complemented
61	C₂×SD₆₄	C₂×C₁₃	1	2	yes	complemented
62	C₂×D₃₂	C₂×C₁₃	1	1	yes	characteristic, complemented
63	C₂²×C₃₂	C₂×C₁₃	1	1	yes	characteristic, complemented
64	C₂³×C₁₃	—	8	2	no	—
65	C₂³×C₁₃	D₁₆	1	1	yes	characteristic
66	Q₈×C₁₃	—	4	4	no	—
67	Q₈×C₁₃	—	8	2	no	—
68	D₄×C₁₃	—	4	4	no	—
69	D₄×C₁₃	—	8	8	no	—
70	D₄×C₁₃	—	8	2	no	—
71	C₂×C₄×C₁₃	—	8	2	no	—
72	C₂×C₄×C₁₃	—	8	8	no	—
73	C₂×C₄×C₁₃	D₁₆	1	2	yes	—
74	C₂×C₄×C₁₃	D₁₆	1	1	yes	characteristic
75	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
76	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
77	C₈×C₁₃	C₂×D₈	1	1	yes	characteristic
78	C₈×C₁₃	C₂×D₈	1	2	yes	—
79	C₈×C₁₃	C₂²×D₄	1	1	yes	characteristic
80	C₂×#135	—	2	2	no	—
81	#508	—	2	4	no	—
82	#508	C₂²×C₁₃	1	8	yes	complemented
83	C₂×Q₃₂	C₂²×C₁₃	1	2	yes	—
84	C₂×SD₃₂	—	2	2	no	—
85	C₂×D₁₆	C₂²×C₁₃	1	2	yes	—
86	C₂²×C₁₆	C₂²×C₁₃	1	1	yes	characteristic
87	Q₆₄	C₂²×C₁₃	1	4	yes	complemented
88	SD₆₄	C₂²×C₁₃	1	8	yes	complemented
89	D₃₂	C₂²×C₁₃	1	4	yes	complemented
90	C₂×C₃₂	C₂²×C₁₃	1	1	yes	characteristic
91	C₂×C₃₂	C₂²×C₁₃	1	1	yes	characteristic
92	C₂×C₃₂	C₂²×C₁₃	1	4	yes	complemented
93	C₂²×C₁₃	—	2	2	no	—
94	C₂²×C₁₃	—	8	4	no	—
95	C₂²×C₁₃	—	16	4	no	—
96	C₂²×C₁₃	C₂×D₁₆	1	2	yes	—
97	C₂²×C₁₃	C₂×D₁₆	1	1	yes	characteristic
98	C₄×C₁₃	—	8	4	no	—
99	C₄×C₁₃	C₂×D₁₆	1	2	yes	—
100	C₄×C₁₃	C₂×D₁₆	1	1	yes	characteristic
101	C₄×C₁₃	C₂²×D₈	1	1	yes	characteristic
102	C₂×#41	—	4	2	no	—
103	#135	—	2	8	no	—
104	#135	—	4	4	no	—
105	C₂×Q₁₆	—	2	2	no	—
106	C₂×SD₁₆	—	4	2	no	—
107	C₂×D₈	—	2	2	no	—
108	C₂²×C₈	D₄×C₁₃	1	1	yes	characteristic
109	Q₃₂	—	2	2	no	—
110	Q₃₂	C₂³×C₁₃	1	4	yes	—
111	SD₃₂	—	2	8	no	—
112	D₁₆	—	2	2	no	—
113	D₁₆	C₂³×C₁₃	1	4	yes	—
114	C₂×C₁₆	D₄×C₁₃	1	1	yes	characteristic
115	C₂×C₁₆	D₄×C₁₃	1	2	yes	—
116	C₂×C₁₆	C₂³×C₁₃	1	1	yes	characteristic
117	C₂×C₁₆	C₂³×C₁₃	1	2	yes	—
118	C₃₂	C₂³×C₁₃	1	2	yes	—
119	C₃₂	C₂³×C₁₃	1	2	yes	—
120	C₂×C₁₃	—	2	2	no	—
121	C₂×C₁₃	—	16	4	no	—
122	C₂×C₁₃	#2262	1	2	yes	direct product factor
123	C₂×C₁₃	C₂²×D₁₆	1	1	yes	characteristic
124	#41	—	4	8	no	—
125	#41	—	8	4	no	—
126	C₂×Q₈	—	4	2	no	—
127	C₂×D₄	—	4	2	no	—
128	C₂×D₄	—	8	2	no	—
129	C₂²×C₄	—	8	2	no	—
130	C₂²×C₄	C₁₃×D₈	1	1	yes	characteristic
131	Q₁₆	—	2	4	no	—
132	Q₁₆	—	4	2	no	—
133	SD₁₆	—	4	8	no	—
134	D₈	—	2	4	no	—
135	D₈	—	4	2	no	—
136	C₂×C₈	C₁₃×D₈	1	1	yes	characteristic
137	C₂×C₈	C₁₃×D₈	1	2	yes	—
138	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
139	C₂×C₈	C₂×D₄×C₁₃	1	2	yes	—
140	C₁₆	C₂×D₄×C₁₃	1	1	yes	characteristic
141	C₁₆	C₂×D₄×C₁₃	1	2	yes	—
142	C₁₆	C₂⁴×C₁₃	1	1	yes	characteristic
143	C₁₃	C₂×#2262	1	1	yes	characteristic, direct product factor, Sylow
144	C₂³	—	8	2	no	—
145	C₂³	C₁₃×D₁₆	1	1	yes	characteristic
146	Q₈	—	4	4	no	—
147	Q₈	—	8	2	no	—
148	D₄	—	4	4	no	—
149	D₄	—	8	8	no	—
150	D₄	—	8	2	no	—
151	C₂×C₄	—	8	2	no	—
152	C₂×C₄	—	8	8	no	—
153	C₂×C₄	C₁₃×D₁₆	1	1	yes	characteristic
154	C₂×C₄	C₁₃×D₁₆	1	2	yes	—
155	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
156	C₂×C₄	C₂×C₁₃×D₈	1	2	yes	—
157	C₈	C₂×C₁₃×D₈	1	1	yes	characteristic
158	C₈	C₂×C₁₃×D₈	1	2	yes	—
159	C₈	C₂²×D₄×C₁₃	1	1	yes	characteristic
160	C₂²	—	2	2	no	—
161	C₂²	—	8	4	no	—
162	C₂²	—	16	4	no	—
163	C₂²	C₂×C₁₃×D₁₆	1	2	yes	—
164	C₂²	C₂×C₁₃×D₁₆	1	1	yes	characteristic
165	C₄	—	8	4	no	—
166	C₄	C₂×C₁₃×D₁₆	1	2	yes	—
167	C₄	C₂×C₁₃×D₁₆	1	1	yes	characteristic
168	C₄	C₂²×C₁₃×D₈	1	1	yes	characteristic
169	C₂	—	2	2	no	—
170	C₂	—	16	4	no	—
171	C₂	C₁₃×#2262	1	2	yes	direct product factor
172	C₂	C₂²×C₁₃×D₁₆	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₁₃	D₁₆
commutator, Frattini, cosocle	C₁₆	C₂⁴×C₁₃
socle	C₂²×C₁₃	C₂×D₁₆
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₃ ◁ C₂²×C₄×C₁₃ ◁ C₂²×C₈×C₁₃ ◁ C₂²×C₁₃×C₁₆ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₁₃×#508	C₂	semidirect
2. C₁₃×#2262	C₂	direct (N × H)
3. C₂×C₁₃×Q₆₄	C₂	semidirect
4. C₂×C₁₃×SD₆₄	C₂	semidirect
5. C₂×C₁₃×D₃₂	C₂	semidirect
6. C₂²×C₁₃×C₃₂	C₂	semidirect
9. C₁₃×#508	C₂²	semidirect
14. C₁₃×Q₆₄	C₂²	semidirect
15. C₁₃×SD₆₄	C₂²	semidirect
16. C₁₃×D₃₂	C₂²	semidirect
19. C₂×C₁₃×C₃₂	C₂²	semidirect
38. C₂×#2262	C₁₃	direct (N × H)
58. C₂×#508	C₂×C₁₃	semidirect
59. #2262	C₂×C₁₃	direct (N × H)
60. C₂×Q₆₄	C₂×C₁₃	semidirect
61. C₂×SD₆₄	C₂×C₁₃	semidirect
62. C₂×D₃₂	C₂×C₁₃	semidirect
63. C₂²×C₃₂	C₂×C₁₃	semidirect
82. #508	C₂²×C₁₃	semidirect
87. Q₆₄	C₂²×C₁₃	semidirect
88. SD₆₄	C₂²×C₁₃	semidirect
89. D₃₂	C₂²×C₁₃	semidirect
92. C₂×C₃₂	C₂²×C₁₃	semidirect
122. C₂×C₁₃	#2262	direct (N × H)
143. C₁₃	C₂×#2262	direct (N × H)
171. C₂	C₁₃×#2262	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×#508	2
C₁₃×#2262	2
C₂×C₁₃×Q₆₄	2
C₂×C₁₃×SD₆₄	2
C₂×C₁₃×D₃₂	2
C₂²×C₁₃×C₃₂	2
C₂×#2262	13

Subgroups — C2×C13×#2262 #15720317

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₁₃×#2262 #15720317