Subgroups — C₂×C₁₃×#2246 #15720048

Order 3328.

1022 conjugacy classes of subgroups.

The subgroup lattice has 168 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₁₃×D₁₆	C₂	1	4	yes	maximal, complemented
2	C₂×C₁₃×#493	C₂	1	2	yes	maximal, complemented
3	C₁₃×#2246	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₄×C₁₃×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×C₁₃×D₈	—	2	2	no	—
6	C₂²×C₁₃×D₈	C₂²	1	4	yes	—
7	C₂×C₁₃×#127	—	2	2	no	—
8	C₂×C₁₃×D₁₆	—	2	16	no	—
9	C₂×C₁₃×D₁₆	C₂²	1	16	yes	complemented
10	C₂²×C₁₃×C₁₆	C₂²	1	2	yes	—
11	C₁₃×#493	—	2	4	no	—
12	C₁₃×#493	C₂²	1	8	yes	complemented
13	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
14	C₄×C₁₃×C₁₆	C₂²	1	4	yes	complemented
15	C₂²×D₄×C₁₃	—	2	4	no	—
16	C₂²×D₄×C₁₃	—	4	4	no	—
17	C₂×C₁₃×D₈	—	2	16	no	—
18	C₂×C₁₃×D₈	—	2	4	no	—
19	C₂×C₁₃×D₈	—	2	8	no	—
20	C₂×C₁₃×D₈	—	4	8	no	—
21	C₂×C₁₃×D₈	C₂³	1	8	yes	—
22	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
23	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
24	C₁₃×#127	—	2	8	no	—
25	C₁₃×#127	—	4	4	no	—
26	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
27	C₁₃×D₁₆	—	2	32	no	—
28	C₂×C₁₃×C₁₆	D₄	1	8	yes	complemented
29	C₂×C₁₃×C₁₆	C₂³	1	4	yes	—
30	C₄×C₈×C₁₃	D₄	1	2	yes	—
31	C₄×C₈×C₁₃	C₂³	1	2	yes	—
32	C₂×#2246	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
33	C₂⁴×C₁₃	—	4	4	no	—
34	C₂×D₄×C₁₃	—	2	8	no	—
35	C₂×D₄×C₁₃	—	4	8	no	—
36	C₂×D₄×C₁₃	—	4	8	no	—
37	C₂×D₄×C₁₃	—	4	16	no	—
38	C₂×D₄×C₁₃	—	4	4	no	—
39	C₂×D₄×C₁₃	—	8	16	no	—
40	C₂²×C₄×C₁₃	D₈	1	2	yes	—
41	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
42	C₁₃×D₈	—	2	16	no	—
43	C₁₃×D₈	—	4	8	no	—
44	C₁₃×D₈	—	4	16	no	—
45	C₂×C₈×C₁₃	D₈	1	4	yes	—
46	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
47	C₂×C₈×C₁₃	C₂×D₄	1	4	yes	—
48	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
49	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
50	C₄²×C₁₃	D₈	1	2	yes	—
51	C₄²×C₁₃	C₂×D₄	1	2	yes	—
52	C₁₃×C₁₆	C₂×D₄	1	8	yes	complemented
53	C₂²×D₁₆	C₂×C₁₃	1	4	yes	complemented
54	C₂×#493	C₂×C₁₃	1	2	yes	complemented
55	#2246	C₂×C₁₃	1	8	yes	direct product factor
56	C₂×C₄×C₁₆	C₂×C₁₃	1	1	yes	characteristic, complemented
57	C₂³×C₁₃	—	4	8	no	—
58	C₂³×C₁₃	—	8	8	no	—
59	C₂³×C₁₃	—	8	16	no	—
60	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
61	D₄×C₁₃	—	4	8	no	—
62	D₄×C₁₃	—	4	8	no	—
63	D₄×C₁₃	—	8	8	no	—
64	D₄×C₁₃	—	8	32	no	—
65	C₂×C₄×C₁₃	D₁₆	1	8	yes	complemented
66	C₂×C₄×C₁₃	#127	1	2	yes	—
67	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
68	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
69	C₂×C₄×C₁₃	C₂×D₈	1	4	yes	—
70	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
71	C₈×C₁₃	#127	1	2	yes	—
72	C₈×C₁₃	C₂×D₈	1	4	yes	—
73	C₈×C₁₃	C₂²×D₄	1	2	yes	—
74	C₂²×D₈	—	2	2	no	—
75	C₂²×D₈	C₂²×C₁₃	1	4	yes	—
76	C₂×#127	—	2	2	no	—
77	C₂×D₁₆	—	2	16	no	—
78	C₂×D₁₆	C₂²×C₁₃	1	16	yes	complemented
79	C₂²×C₁₆	C₂²×C₁₃	1	2	yes	—
80	#493	—	2	4	no	—
81	#493	C₂²×C₁₃	1	8	yes	complemented
82	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
83	C₄×C₁₆	C₂²×C₁₃	1	4	yes	complemented
84	C₂²×C₁₃	—	8	8	no	—
85	C₂²×C₁₃	—	8	16	no	—
86	C₂²×C₁₃	—	16	16	no	—
87	C₂²×C₁₃	#493	1	2	yes	—
88	C₂²×C₁₃	C₂×D₁₆	1	4	yes	—
89	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
90	C₄×C₁₃	#493	1	2	yes	—
91	C₄×C₁₃	C₂×D₁₆	1	8	yes	complemented
92	C₄×C₁₃	C₂×#127	1	1	yes	characteristic
93	C₄×C₁₃	C₂²×D₈	1	1	yes	characteristic
94	C₂²×D₄	—	2	4	no	—
95	C₂²×D₄	—	4	4	no	—
96	C₂×D₈	—	2	16	no	—
97	C₂×D₈	—	2	8	no	—
98	C₂×D₈	—	2	4	no	—
99	C₂×D₈	—	4	8	no	—
100	C₂×D₈	C₂³×C₁₃	1	8	yes	—
101	C₂²×C₈	D₄×C₁₃	1	1	yes	characteristic
102	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
103	#127	—	2	8	no	—
104	#127	—	4	4	no	—
105	C₂×C₄²	D₄×C₁₃	1	1	yes	characteristic
106	D₁₆	—	2	32	no	—
107	C₂×C₁₆	D₄×C₁₃	1	8	yes	complemented
108	C₂×C₁₆	C₂³×C₁₃	1	4	yes	—
109	C₄×C₈	D₄×C₁₃	1	2	yes	—
110	C₄×C₈	C₂³×C₁₃	1	2	yes	—
111	C₂×C₁₃	—	16	8	no	—
112	C₂×C₁₃	#2246	1	4	yes	direct product factor
113	C₂×C₁₃	C₂×#493	1	1	yes	characteristic
114	C₂×C₁₃	C₂²×D₁₆	1	2	yes	—
115	C₂⁴	—	4	4	no	—
116	C₂×D₄	—	2	8	no	—
117	C₂×D₄	—	4	8	no	—
118	C₂×D₄	—	4	8	no	—
119	C₂×D₄	—	4	16	no	—
120	C₂×D₄	—	4	4	no	—
121	C₂×D₄	—	8	16	no	—
122	C₂²×C₄	C₁₃×D₈	1	2	yes	—
123	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
124	D₈	—	2	16	no	—
125	D₈	—	4	8	no	—
126	D₈	—	4	16	no	—
127	C₂×C₈	C₁₃×D₈	1	4	yes	—
128	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
129	C₂×C₈	C₂×D₄×C₁₃	1	4	yes	—
130	C₂×C₈	C₂×D₄×C₁₃	1	2	yes	—
131	C₂×C₈	C₂⁴×C₁₃	1	1	yes	characteristic
132	C₄²	C₁₃×D₈	1	2	yes	—
133	C₄²	C₂×D₄×C₁₃	1	2	yes	—
134	C₁₆	C₂×D₄×C₁₃	1	8	yes	complemented
135	C₁₃	C₂×#2246	1	1	yes	characteristic, direct product factor, Sylow
136	C₂³	—	4	8	no	—
137	C₂³	—	8	8	no	—
138	C₂³	—	8	16	no	—
139	C₂³	C₂×C₁₃×D₈	1	1	yes	characteristic
140	D₄	—	4	8	no	—
141	D₄	—	4	8	no	—
142	D₄	—	8	8	no	—
143	D₄	—	8	32	no	—
144	C₂×C₄	C₁₃×D₁₆	1	8	yes	complemented
145	C₂×C₄	C₁₃×#127	1	2	yes	—
146	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
147	C₂×C₄	C₂×C₁₃×D₈	1	2	yes	—
148	C₂×C₄	C₂×C₁₃×D₈	1	4	yes	—
149	C₂×C₄	C₂²×D₄×C₁₃	1	1	yes	characteristic
150	C₈	C₁₃×#127	1	2	yes	—
151	C₈	C₂×C₁₃×D₈	1	4	yes	—
152	C₈	C₂²×D₄×C₁₃	1	2	yes	—
153	C₂²	—	8	8	no	—
154	C₂²	—	8	16	no	—
155	C₂²	—	16	16	no	—
156	C₂²	C₁₃×#493	1	2	yes	—
157	C₂²	C₂×C₁₃×D₁₆	1	4	yes	—
158	C₂²	C₂²×C₁₃×D₈	1	1	yes	characteristic
159	C₄	C₁₃×#493	1	2	yes	—
160	C₄	C₂×C₁₃×D₁₆	1	8	yes	complemented
161	C₄	C₂×C₁₃×#127	1	1	yes	characteristic
162	C₄	C₂²×C₁₃×D₈	1	1	yes	characteristic
163	C₂	—	16	8	no	—
164	C₂	C₁₃×#2246	1	4	yes	direct product factor
165	C₂	C₂×C₁₃×#493	1	1	yes	characteristic
166	C₂	C₂²×C₁₃×D₁₆	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₁₃	C₂×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₁₃
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×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
1. C₂²×C₁₃×D₁₆	C₂	semidirect
2. C₂×C₁₃×#493	C₂	semidirect
3. C₁₃×#2246	C₂	direct (N × H)
4. C₂×C₄×C₁₃×C₁₆	C₂	semidirect
9. C₂×C₁₃×D₁₆	C₂²	semidirect
12. C₁₃×#493	C₂²	semidirect
14. C₄×C₁₃×C₁₆	C₂²	semidirect
28. C₂×C₁₃×C₁₆	D₄	semidirect
32. C₂×#2246	C₁₃	direct (N × H)
52. C₁₃×C₁₆	C₂×D₄	semidirect
53. C₂²×D₁₆	C₂×C₁₃	semidirect
54. C₂×#493	C₂×C₁₃	semidirect
55. #2246	C₂×C₁₃	direct (N × H)
56. C₂×C₄×C₁₆	C₂×C₁₃	semidirect
65. C₂×C₄×C₁₃	D₁₆	semidirect
78. C₂×D₁₆	C₂²×C₁₃	semidirect
81. #493	C₂²×C₁₃	semidirect
83. C₄×C₁₆	C₂²×C₁₃	semidirect
91. C₄×C₁₃	C₂×D₁₆	semidirect
107. C₂×C₁₆	D₄×C₁₃	semidirect
112. C₂×C₁₃	#2246	direct (N × H)
134. C₁₆	C₂×D₄×C₁₃	semidirect
135. C₁₃	C₂×#2246	direct (N × H)
144. C₂×C₄	C₁₃×D₁₆	semidirect
160. C₄	C₂×C₁₃×D₁₆	semidirect
164. C₂	C₁₃×#2246	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₁₃×D₁₆	2
C₂×C₁₃×#493	2
C₁₃×#2246	2
C₂×C₄×C₁₃×C₁₆	2
C₂×#2246	13

Subgroups — C2×C13×#2246 #15720048

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₁₃×#2246 #15720048