Subgroups — C₂²×C₁₃×#498 #15746991

Order 3328.

2398 conjugacy classes of subgroups.

The subgroup lattice has 146 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₁₃×C₄⋊Q₈	C₂	1	2	yes	maximal
2	C₂×C₁₃×#498	C₂	1	24	yes	maximal, direct product factor
3	C₂²×C₁₃×#106	C₂	1	4	yes	maximal
4	C₂²×C₄×C₈×C₁₃	C₂	1	1	yes	maximal, characteristic
5	C₂³×Q₈×C₁₃	—	2	2	no	—
6	C₂³×C₈×C₁₃	C₂²	1	2	yes	—
7	C₂×C₁₃×C₄⋊Q₈	—	2	12	no	—
8	C₂×C₁₃×C₄⋊Q₈	C₂²	1	24	yes	—
9	C₂²×C₁₃×C₄⋊C₄	—	2	2	no	—
10	C₂²×C₁₃×C₄⋊C₄	C₂²	1	4	yes	—
11	C₂²×C₄²×C₁₃	C₂²	1	1	yes	characteristic
12	C₁₃×#498	C₂²	1	64	yes	direct product factor
13	C₂×C₁₃×#106	C₂²	1	48	yes	—
14	C₂×C₄×C₈×C₁₃	C₂²	1	12	yes	—
15	C₂²×Q₈×C₁₃	—	2	24	no	—
16	C₂²×Q₈×C₁₃	—	4	16	no	—
17	C₂³×C₄×C₁₃	—	2	4	no	—
18	C₂³×C₄×C₁₃	D₄	1	2	yes	—
19	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
20	C₂²×C₈×C₁₃	Q₈	1	16	yes	complemented
21	C₂²×C₈×C₁₃	C₂³	1	12	yes	—
22	C₁₃×C₄⋊Q₈	—	2	48	no	—
23	C₁₃×C₄⋊Q₈	C₂³	1	32	yes	—
24	C₂×C₁₃×C₄⋊C₄	—	2	24	no	—
25	C₂×C₁₃×C₄⋊C₄	—	2	12	no	—
26	C₂×C₁₃×C₄⋊C₄	C₂³	1	24	yes	—
27	C₂×C₄²×C₁₃	D₄	1	6	yes	—
28	C₂×C₄²×C₁₃	C₂³	1	6	yes	—
29	C₁₃×#106	C₂³	1	64	yes	—
30	C₄×C₈×C₁₃	C₂³	1	16	yes	—
31	C₂²×#498	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
32	C₂⁴×C₁₃	C₂×D₄	1	1	yes	characteristic
33	C₂×Q₈×C₁₃	—	2	32	no	—
34	C₂×Q₈×C₁₃	—	4	96	no	—
35	C₂²×C₄×C₁₃	—	2	24	no	—
36	C₂²×C₄×C₁₃	—	4	16	no	—
37	C₂²×C₄×C₁₃	SD₁₆	1	16	yes	—
38	C₂²×C₄×C₁₃	C₂×D₄	1	3	yes	—
39	C₂²×C₄×C₁₃	C₂×D₄	1	12	yes	—
40	C₂²×C₄×C₁₃	C₂×Q₈	1	8	yes	—
41	C₂²×C₄×C₁₃	C₂⁴	1	3	yes	—
42	C₂×C₈×C₁₃	C₂×Q₈	1	48	yes	complemented
43	C₂×C₈×C₁₃	C₂⁴	1	8	yes	—
44	C₁₃×C₄⋊C₄	—	2	32	no	—
45	C₁₃×C₄⋊C₄	—	2	24	no	—
46	C₁₃×C₄⋊C₄	C₂⁴	1	16	yes	—
47	C₄²×C₁₃	C₂×D₄	1	12	yes	—
48	C₄²×C₁₃	C₂⁴	1	4	yes	—
49	C₂²×C₄⋊Q₈	C₂×C₁₃	1	2	yes	—
50	C₂×#498	C₂×C₁₃	1	24	yes	direct product factor
51	C₂²×#106	C₂×C₁₃	1	4	yes	—
52	C₂²×C₄×C₈	C₂×C₁₃	1	1	yes	characteristic
53	C₂³×C₁₃	C₄⋊Q₈	1	4	yes	—
54	C₂³×C₁₃	C₂×SD₁₆	1	8	yes	—
55	C₂³×C₁₃	C₂²×D₄	1	3	yes	—
56	Q₈×C₁₃	—	4	64	no	—
57	C₂×C₄×C₁₃	—	2	16	no	—
58	C₂×C₄×C₁₃	—	4	48	no	—
59	C₂×C₄×C₁₃	C₄⋊Q₈	1	12	yes	—
60	C₂×C₄×C₁₃	C₂×SD₁₆	1	48	yes	—
61	C₂×C₄×C₁₃	C₂²×D₄	1	3	yes	—
62	C₂×C₄×C₁₃	C₂²×D₄	1	8	yes	—
63	C₂×C₄×C₁₃	C₂²×Q₈	1	12	yes	—
64	C₂×C₄×C₁₃	C₂⁵	1	1	yes	characteristic
65	C₈×C₁₃	C₂²×Q₈	1	16	yes	complemented
66	C₂³×Q₈	—	2	2	no	—
67	C₂³×C₈	C₂²×C₁₃	1	2	yes	—
68	C₂×C₄⋊Q₈	—	2	12	no	—
69	C₂×C₄⋊Q₈	C₂²×C₁₃	1	24	yes	—
70	C₂²×C₄⋊C₄	—	2	2	no	—
71	C₂²×C₄⋊C₄	C₂²×C₁₃	1	4	yes	—
72	C₂²×C₄²	C₂²×C₁₃	1	1	yes	characteristic
73	#498	C₂²×C₁₃	1	64	yes	direct product factor
74	C₂×#106	C₂²×C₁₃	1	48	yes	—
75	C₂×C₄×C₈	C₂²×C₁₃	1	12	yes	—
76	C₂²×C₁₃	#498	1	16	yes	direct product factor
77	C₂²×C₁₃	C₂×C₄⋊Q₈	1	6	yes	—
78	C₂²×C₁₃	C₂²×SD₁₆	1	12	yes	—
79	C₂²×C₁₃	C₂³×D₄	1	1	yes	characteristic
80	C₄×C₁₃	—	4	16	no	—
81	C₄×C₁₃	C₂×C₄⋊Q₈	1	6	yes	—
82	C₄×C₁₃	C₂²×SD₁₆	1	16	yes	—
83	C₄×C₁₃	C₂³×Q₈	1	2	yes	—
84	C₂²×Q₈	—	2	24	no	—
85	C₂²×Q₈	—	4	16	no	—
86	C₂³×C₄	—	2	4	no	—
87	C₂³×C₄	D₄×C₁₃	1	2	yes	—
88	C₂³×C₄	C₂³×C₁₃	1	1	yes	characteristic
89	C₂²×C₈	Q₈×C₁₃	1	16	yes	complemented
90	C₂²×C₈	C₂³×C₁₃	1	12	yes	—
91	C₄⋊Q₈	—	2	48	no	—
92	C₄⋊Q₈	C₂³×C₁₃	1	32	yes	—
93	C₂×C₄⋊C₄	—	2	24	no	—
94	C₂×C₄⋊C₄	—	2	12	no	—
95	C₂×C₄⋊C₄	C₂³×C₁₃	1	24	yes	—
96	C₂×C₄²	D₄×C₁₃	1	6	yes	—
97	C₂×C₄²	C₂³×C₁₃	1	6	yes	—
98	#106	C₂³×C₁₃	1	64	yes	—
99	C₄×C₈	C₂³×C₁₃	1	16	yes	—
100	C₂×C₁₃	C₂×#498	1	12	yes	direct product factor
101	C₂×C₁₃	C₂²×C₄⋊Q₈	1	1	yes	characteristic
102	C₂×C₁₃	C₂³×SD₁₆	1	2	yes	—
103	C₂⁴	C₂×D₄×C₁₃	1	1	yes	characteristic
104	C₂×Q₈	—	2	32	no	—
105	C₂×Q₈	—	4	96	no	—
106	C₂²×C₄	—	2	24	no	—
107	C₂²×C₄	—	4	16	no	—
108	C₂²×C₄	C₁₃×SD₁₆	1	16	yes	—
109	C₂²×C₄	C₂×D₄×C₁₃	1	3	yes	—
110	C₂²×C₄	C₂×D₄×C₁₃	1	12	yes	—
111	C₂²×C₄	C₂×Q₈×C₁₃	1	8	yes	—
112	C₂²×C₄	C₂⁴×C₁₃	1	3	yes	—
113	C₂×C₈	C₂×Q₈×C₁₃	1	48	yes	complemented
114	C₂×C₈	C₂⁴×C₁₃	1	8	yes	—
115	C₄⋊C₄	—	2	32	no	—
116	C₄⋊C₄	—	2	24	no	—
117	C₄⋊C₄	C₂⁴×C₁₃	1	16	yes	—
118	C₄²	C₂×D₄×C₁₃	1	12	yes	—
119	C₄²	C₂⁴×C₁₃	1	4	yes	—
120	C₁₃	C₂²×#498	1	1	yes	characteristic, direct product factor, Sylow
121	C₂³	C₁₃×C₄⋊Q₈	1	4	yes	—
122	C₂³	C₂×C₁₃×SD₁₆	1	8	yes	—
123	C₂³	C₂²×D₄×C₁₃	1	3	yes	—
124	Q₈	—	4	64	no	—
125	C₂×C₄	—	2	16	no	—
126	C₂×C₄	—	4	48	no	—
127	C₂×C₄	C₁₃×C₄⋊Q₈	1	12	yes	—
128	C₂×C₄	C₂×C₁₃×SD₁₆	1	48	yes	—
129	C₂×C₄	C₂²×D₄×C₁₃	1	3	yes	—
130	C₂×C₄	C₂²×D₄×C₁₃	1	8	yes	—
131	C₂×C₄	C₂²×Q₈×C₁₃	1	12	yes	—
132	C₂×C₄	C₂⁵×C₁₃	1	1	yes	characteristic
133	C₈	C₂²×Q₈×C₁₃	1	16	yes	complemented
134	C₂²	C₁₃×#498	1	16	yes	direct product factor
135	C₂²	C₂×C₁₃×C₄⋊Q₈	1	6	yes	—
136	C₂²	C₂²×C₁₃×SD₁₆	1	12	yes	—
137	C₂²	C₂³×D₄×C₁₃	1	1	yes	characteristic
138	C₄	—	4	16	no	—
139	C₄	C₂×C₁₃×C₄⋊Q₈	1	6	yes	—
140	C₄	C₂²×C₁₃×SD₁₆	1	16	yes	—
141	C₄	C₂³×Q₈×C₁₃	1	2	yes	—
142	C₂	C₂×C₁₃×#498	1	12	yes	direct product factor
143	C₂	C₂²×C₁₃×C₄⋊Q₈	1	1	yes	characteristic
144	C₂	C₂³×C₁₃×SD₁₆	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂⁴×C₁₃	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₁₃
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₁₃ ◁ C₂²×C₄²×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₁₃×#498	C₂	direct (N × H)
12. C₁₃×#498	C₂²	direct (N × H)
20. C₂²×C₈×C₁₃	Q₈	semidirect
31. C₂²×#498	C₁₃	direct (N × H)
42. C₂×C₈×C₁₃	C₂×Q₈	semidirect
50. C₂×#498	C₂×C₁₃	direct (N × H)
65. C₈×C₁₃	C₂²×Q₈	semidirect
73. #498	C₂²×C₁₃	direct (N × H)
76. C₂²×C₁₃	#498	direct (N × H)
89. C₂²×C₈	Q₈×C₁₃	semidirect
100. C₂×C₁₃	C₂×#498	direct (N × H)
113. C₂×C₈	C₂×Q₈×C₁₃	semidirect
120. C₁₃	C₂²×#498	direct (N × H)
133. C₈	C₂²×Q₈×C₁₃	semidirect
134. C₂²	C₁₃×#498	direct (N × H)
142. C₂	C₂×C₁₃×#498	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×C₄⋊Q₈	2
C₂×C₁₃×#498	2
C₂²×C₁₃×#106	2
C₂²×C₄×C₈×C₁₃	2
C₂²×#498	13

Subgroups — C22×C13×#498 #15746991

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₁₃×#498 #15746991