Subgroups — C₂²×C₁₃×#366 #15719744

Order 3328.

782 conjugacy classes of subgroups.

The subgroup lattice has 132 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₁₃×C₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₁₃×#107	C₂	1	2	yes	maximal
3	C₂×C₁₃×#366	C₂	1	12	yes	maximal, direct product factor
4	C₂³×C₈×C₁₃	C₂²	1	1	yes	characteristic
5	C₂²×C₁₃×C₄⋊C₄	—	2	2	no	—
6	C₂²×C₁₃×C₁₆	C₄	1	8	yes	complemented
7	C₂²×C₁₃×C₁₆	C₂²	1	6	yes	—
8	C₂×C₁₃×#107	—	2	6	no	—
9	C₂×C₁₃×#107	C₂²	1	12	yes	—
10	C₁₃×#366	C₂²	1	16	yes	direct product factor
11	C₂³×C₄×C₁₃	—	4	2	no	—
12	C₂³×C₄×C₁₃	D₄	1	1	yes	characteristic
13	C₂²×C₈×C₁₃	C₂×C₄	1	4	yes	—
14	C₂²×C₈×C₁₃	D₄	1	3	yes	—
15	C₂²×C₈×C₁₃	Q₈	1	4	yes	—
16	C₂²×C₈×C₁₃	C₂³	1	3	yes	—
17	C₂×C₁₃×C₄⋊C₄	—	2	12	no	—
18	C₂×C₁₃×C₄⋊C₄	—	4	6	no	—
19	C₂×C₁₃×C₁₆	C₂×C₄	1	24	yes	complemented
20	C₂×C₁₃×C₁₆	C₂³	1	4	yes	—
21	C₁₃×#107	—	2	12	no	—
22	C₁₃×#107	C₂³	1	8	yes	—
23	C₂²×#366	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
24	C₂⁴×C₁₃	D₈	1	1	yes	characteristic
25	C₂²×C₄×C₁₃	—	4	12	no	—
26	C₂²×C₄×C₁₃	—	8	8	no	—
27	C₂²×C₄×C₁₃	C₄⋊C₄	1	4	yes	—
28	C₂²×C₄×C₁₃	D₈	1	3	yes	—
29	C₂²×C₄×C₁₃	Q₁₆	1	4	yes	—
30	C₂²×C₄×C₁₃	C₂×D₄	1	3	yes	—
31	C₂×C₈×C₁₃	C₄⋊C₄	1	12	yes	—
32	C₂×C₈×C₁₃	C₂²×C₄	1	6	yes	—
33	C₂×C₈×C₁₃	C₂×D₄	1	3	yes	—
34	C₂×C₈×C₁₃	C₂×Q₈	1	6	yes	—
35	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
36	C₁₃×C₄⋊C₄	—	2	8	no	—
37	C₁₃×C₄⋊C₄	—	4	12	no	—
38	C₁₃×C₁₆	C₂²×C₄	1	8	yes	complemented
39	C₂³×C₁₆	C₂×C₁₃	1	1	yes	characteristic
40	C₂²×#107	C₂×C₁₃	1	2	yes	—
41	C₂×#366	C₂×C₁₃	1	12	yes	direct product factor
42	C₂³×C₁₃	#107	1	4	yes	—
43	C₂³×C₁₃	D₁₆	1	4	yes	—
44	C₂³×C₁₃	Q₃₂	1	4	yes	—
45	C₂³×C₁₃	C₂×D₈	1	3	yes	—
46	C₂×C₄×C₁₃	—	4	8	no	—
47	C₂×C₄×C₁₃	—	8	24	no	—
48	C₂×C₄×C₁₃	#107	1	12	yes	—
49	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	6	yes	—
50	C₂×C₄×C₁₃	C₂×D₈	1	3	yes	—
51	C₂×C₄×C₁₃	C₂×Q₁₆	1	6	yes	—
52	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
53	C₈×C₁₃	C₂×C₄⋊C₄	1	6	yes	—
54	C₈×C₁₃	C₂³×C₄	1	1	yes	characteristic
55	C₈×C₁₃	C₂²×Q₈	1	1	yes	characteristic
56	C₂³×C₈	C₂²×C₁₃	1	1	yes	characteristic
57	C₂²×C₄⋊C₄	—	2	2	no	—
58	C₂²×C₁₆	C₄×C₁₃	1	8	yes	complemented
59	C₂²×C₁₆	C₂²×C₁₃	1	6	yes	—
60	C₂×#107	—	2	6	no	—
61	C₂×#107	C₂²×C₁₃	1	12	yes	—
62	#366	C₂²×C₁₃	1	16	yes	direct product factor
63	C₂²×C₁₃	#366	1	16	yes	direct product factor
64	C₂²×C₁₃	C₂×#107	1	6	yes	—
65	C₂²×C₁₃	C₂×D₁₆	1	6	yes	—
66	C₂²×C₁₃	C₂×Q₃₂	1	6	yes	—
67	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
68	C₄×C₁₃	—	8	8	no	—
69	C₄×C₁₃	C₂×#107	1	6	yes	—
70	C₄×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
71	C₄×C₁₃	C₂²×Q₁₆	1	1	yes	characteristic
72	C₂³×C₄	—	4	2	no	—
73	C₂³×C₄	D₄×C₁₃	1	1	yes	characteristic
74	C₂²×C₈	C₂×C₄×C₁₃	1	4	yes	—
75	C₂²×C₈	D₄×C₁₃	1	3	yes	—
76	C₂²×C₈	Q₈×C₁₃	1	4	yes	—
77	C₂²×C₈	C₂³×C₁₃	1	3	yes	—
78	C₂×C₄⋊C₄	—	2	12	no	—
79	C₂×C₄⋊C₄	—	4	6	no	—
80	C₂×C₁₆	C₂×C₄×C₁₃	1	24	yes	complemented
81	C₂×C₁₆	C₂³×C₁₃	1	4	yes	—
82	#107	—	2	12	no	—
83	#107	C₂³×C₁₃	1	8	yes	—
84	C₂×C₁₃	C₂×#366	1	12	yes	direct product factor
85	C₂×C₁₃	C₂²×#107	1	1	yes	characteristic
86	C₂×C₁₃	C₂²×D₁₆	1	1	yes	characteristic
87	C₂×C₁₃	C₂²×Q₃₂	1	1	yes	characteristic
88	C₂⁴	C₁₃×D₈	1	1	yes	characteristic
89	C₂²×C₄	—	4	12	no	—
90	C₂²×C₄	—	8	8	no	—
91	C₂²×C₄	C₁₃×C₄⋊C₄	1	4	yes	—
92	C₂²×C₄	C₁₃×D₈	1	3	yes	—
93	C₂²×C₄	C₁₃×Q₁₆	1	4	yes	—
94	C₂²×C₄	C₂×D₄×C₁₃	1	3	yes	—
95	C₂×C₈	C₁₃×C₄⋊C₄	1	12	yes	—
96	C₂×C₈	C₂²×C₄×C₁₃	1	6	yes	—
97	C₂×C₈	C₂×D₄×C₁₃	1	3	yes	—
98	C₂×C₈	C₂×Q₈×C₁₃	1	6	yes	—
99	C₂×C₈	C₂⁴×C₁₃	1	1	yes	characteristic
100	C₄⋊C₄	—	2	8	no	—
101	C₄⋊C₄	—	4	12	no	—
102	C₁₆	C₂²×C₄×C₁₃	1	8	yes	complemented
103	C₁₃	C₂²×#366	1	1	yes	characteristic, direct product factor, Sylow
104	C₂³	C₁₃×#107	1	4	yes	—
105	C₂³	C₁₃×D₁₆	1	4	yes	—
106	C₂³	C₁₃×Q₃₂	1	4	yes	—
107	C₂³	C₂×C₁₃×D₈	1	3	yes	—
108	C₂×C₄	—	4	8	no	—
109	C₂×C₄	—	8	24	no	—
110	C₂×C₄	C₁₃×#107	1	12	yes	—
111	C₂×C₄	C₂×C₁₃×C₄⋊C₄	1	6	yes	—
112	C₂×C₄	C₂×C₁₃×D₈	1	3	yes	—
113	C₂×C₄	C₂×C₁₃×Q₁₆	1	6	yes	—
114	C₂×C₄	C₂²×D₄×C₁₃	1	1	yes	characteristic
115	C₈	C₂×C₁₃×C₄⋊C₄	1	6	yes	—
116	C₈	C₂³×C₄×C₁₃	1	1	yes	characteristic
117	C₈	C₂²×Q₈×C₁₃	1	1	yes	characteristic
118	C₂²	C₁₃×#366	1	16	yes	direct product factor
119	C₂²	C₂×C₁₃×#107	1	6	yes	—
120	C₂²	C₂×C₁₃×D₁₆	1	6	yes	—
121	C₂²	C₂×C₁₃×Q₃₂	1	6	yes	—
122	C₂²	C₂²×C₁₃×D₈	1	1	yes	characteristic
123	C₄	—	8	8	no	—
124	C₄	C₂×C₁₃×#107	1	6	yes	—
125	C₄	C₂²×C₁₃×C₄⋊C₄	1	1	yes	characteristic
126	C₄	C₂²×C₁₃×Q₁₆	1	1	yes	characteristic
127	C₂	C₂×C₁₃×#366	1	12	yes	direct product factor
128	C₂	C₂²×C₁₃×#107	1	1	yes	characteristic
129	C₂	C₂²×C₁₃×D₁₆	1	1	yes	characteristic
130	C₂	C₂²×C₁₃×Q₃₂	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂⁴×C₁₃	D₈
commutator	C₈	C₂³×C₄×C₁₃
Frattini, cosocle	C₂×C₈	C₂⁴×C₁₃
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₁₃ ◁ C₂³×C₄×C₁₃ ◁ C₂³×C₈×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂×C₁₃×#366	C₂	direct (N × H)
6. C₂²×C₁₃×C₁₆	C₄	semidirect
10. C₁₃×#366	C₂²	direct (N × H)
19. C₂×C₁₃×C₁₆	C₂×C₄	semidirect
23. C₂²×#366	C₁₃	direct (N × H)
38. C₁₃×C₁₆	C₂²×C₄	semidirect
41. C₂×#366	C₂×C₁₃	direct (N × H)
58. C₂²×C₁₆	C₄×C₁₃	semidirect
62. #366	C₂²×C₁₃	direct (N × H)
63. C₂²×C₁₃	#366	direct (N × H)
80. C₂×C₁₆	C₂×C₄×C₁₃	semidirect
84. C₂×C₁₃	C₂×#366	direct (N × H)
102. C₁₆	C₂²×C₄×C₁₃	semidirect
103. C₁₃	C₂²×#366	direct (N × H)
118. C₂²	C₁₃×#366	direct (N × H)
127. C₂	C₂×C₁₃×#366	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₁₃×C₁₆	2
C₂²×C₁₃×#107	2
C₂×C₁₃×#366	2
C₂²×#366	13

Subgroups — C22×C13×#366 #15719744

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₁₃×#366 #15719744