Subgroups — C₁₃×#13553 #15699892

Order 3328.

330 conjugacy classes of subgroups.

The subgroup lattice has 158 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	C₂×C₁₃×M₆₄	C₂	1	2	yes	maximal, complemented
3	C₁₃×C₂²⋊C₃₂	C₂	1	4	yes	maximal, complemented
4	C₂³×C₈×C₁₃	C₄	1	1	yes	characteristic
5	C₂²×C₁₃×C₁₆	—	2	4	no	—
6	C₂²×C₁₃×C₁₆	C₄	1	1	yes	characteristic
7	C₂²×C₁₃×C₁₆	C₄	1	2	yes	—
8	C₂²×C₁₃×C₁₆	C₂²	1	2	yes	—
9	C₂²×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
10	C₁₃×M₆₄	—	2	4	no	—
11	C₂×C₁₃×C₃₂	C₂²	1	4	yes	complemented
12	C₂³×C₄×C₁₃	C₈	1	1	yes	characteristic
13	C₂²×C₈×C₁₃	—	2	2	no	—
14	C₂²×C₈×C₁₃	—	2	2	no	—
15	C₂²×C₈×C₁₃	C₈	1	1	yes	characteristic
16	C₂²×C₈×C₁₃	C₈	1	2	yes	—
17	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
18	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
19	C₂×C₁₃×C₁₆	—	2	2	no	—
20	C₂×C₁₃×C₁₆	—	2	4	no	—
21	C₂×C₁₃×C₁₆	—	2	4	no	—
22	C₂×C₁₃×C₁₆	C₂×C₄	1	2	yes	—
23	C₂×C₁₃×C₁₆	C₂×C₄	1	1	yes	characteristic
24	C₂×C₁₃×C₁₆	D₄	1	4	yes	—
25	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
26	C₁₃×C₃₂	—	2	4	no	—
27	#13553	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
28	C₂⁴×C₁₃	C₁₆	1	1	yes	characteristic
29	C₂²×C₄×C₁₃	—	2	2	no	—
30	C₂²×C₄×C₁₃	—	2	2	no	—
31	C₂²×C₄×C₁₃	C₁₆	1	1	yes	characteristic
32	C₂²×C₄×C₁₃	C₁₆	1	2	yes	—
33	C₂²×C₄×C₁₃	C₂×C₈	1	2	yes	—
34	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
35	C₂×C₈×C₁₃	—	2	2	no	—
36	C₂×C₈×C₁₃	—	2	4	no	—
37	C₂×C₈×C₁₃	—	2	2	no	—
38	C₂×C₈×C₁₃	—	2	2	no	—
39	C₂×C₈×C₁₃	C₂²⋊C₄	1	2	yes	—
40	C₂×C₈×C₁₃	C₂×C₈	1	2	yes	—
41	C₂×C₈×C₁₃	C₂×C₈	1	1	yes	characteristic
42	C₂×C₈×C₁₃	M₁₆	1	2	yes	—
43	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
44	C₁₃×C₁₆	—	2	2	no	—
45	C₁₃×C₁₆	C₂²⋊C₄	1	2	yes	—
46	C₁₃×C₁₆	C₂×D₄	1	2	yes	—
47	C₂³×C₁₆	C₂×C₁₃	1	1	yes	characteristic
48	C₂×M₆₄	C₂×C₁₃	1	2	yes	complemented
49	C₂²⋊C₃₂	C₂×C₁₃	1	4	yes	complemented
50	C₂³×C₁₃	—	2	2	no	—
51	C₂³×C₁₃	—	2	4	no	—
52	C₂³×C₁₃	C₂×C₁₆	1	1	yes	characteristic
53	C₂³×C₁₃	C₂×C₁₆	1	2	yes	—
54	C₂×C₄×C₁₃	—	2	2	no	—
55	C₂×C₄×C₁₃	—	2	4	no	—
56	C₂×C₄×C₁₃	—	2	2	no	—
57	C₂×C₄×C₁₃	—	2	2	no	—
58	C₂×C₄×C₁₃	C₂²⋊C₈	1	2	yes	—
59	C₂×C₄×C₁₃	C₂×C₁₆	1	2	yes	—
60	C₂×C₄×C₁₃	C₂×C₁₆	1	1	yes	characteristic
61	C₂×C₄×C₁₃	M₃₂	1	2	yes	—
62	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
63	C₈×C₁₃	—	2	2	no	—
64	C₈×C₁₃	C₂²⋊C₈	1	2	yes	—
65	C₈×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
66	C₈×C₁₃	C₂×M₁₆	1	1	yes	characteristic
67	C₂³×C₈	C₄×C₁₃	1	1	yes	characteristic
68	C₂²×C₁₆	—	2	4	no	—
69	C₂²×C₁₆	C₄×C₁₃	1	1	yes	characteristic
70	C₂²×C₁₆	C₄×C₁₃	1	2	yes	—
71	C₂²×C₁₆	C₂²×C₁₃	1	2	yes	—
72	C₂²×C₁₆	C₂²×C₁₃	1	1	yes	characteristic
73	M₆₄	—	2	4	no	—
74	C₂×C₃₂	C₂²×C₁₃	1	4	yes	complemented
75	C₂²×C₁₃	—	2	8	no	—
76	C₂²×C₁₃	—	2	2	no	—
77	C₂²×C₁₃	—	2	2	no	—
78	C₂²×C₁₃	—	2	2	no	—
79	C₂²×C₁₃	C₂²⋊C₁₆	1	2	yes	—
80	C₂²×C₁₃	M₆₄	1	4	yes	complemented
81	C₂²×C₁₃	C₂²×C₁₆	1	1	yes	characteristic
82	C₄×C₁₃	—	2	2	no	—
83	C₄×C₁₃	C₂²⋊C₁₆	1	2	yes	—
84	C₄×C₁₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
85	C₄×C₁₃	C₂×M₃₂	1	1	yes	characteristic
86	C₂³×C₄	C₈×C₁₃	1	1	yes	characteristic
87	C₂²×C₈	—	2	2	no	—
88	C₂²×C₈	—	2	2	no	—
89	C₂²×C₈	C₈×C₁₃	1	1	yes	characteristic
90	C₂²×C₈	C₈×C₁₃	1	2	yes	—
91	C₂²×C₈	C₂×C₄×C₁₃	1	2	yes	—
92	C₂²×C₈	C₂×C₄×C₁₃	1	1	yes	characteristic
93	C₂×C₁₆	—	2	2	no	—
94	C₂×C₁₆	—	2	4	no	—
95	C₂×C₁₆	—	2	4	no	—
96	C₂×C₁₆	C₂×C₄×C₁₃	1	2	yes	—
97	C₂×C₁₆	C₂×C₄×C₁₃	1	1	yes	characteristic
98	C₂×C₁₆	D₄×C₁₃	1	4	yes	—
99	C₂×C₁₆	C₂³×C₁₃	1	1	yes	characteristic
100	C₃₂	—	2	4	no	—
101	C₂×C₁₃	—	2	2	no	—
102	C₂×C₁₃	—	2	4	no	—
103	C₂×C₁₃	C₂×C₂²⋊C₁₆	1	1	yes	characteristic
104	C₂×C₁₃	C₂×M₆₄	1	2	yes	—
105	C₂⁴	C₁₃×C₁₆	1	1	yes	characteristic
106	C₂²×C₄	—	2	2	no	—
107	C₂²×C₄	—	2	2	no	—
108	C₂²×C₄	C₁₃×C₁₆	1	1	yes	characteristic
109	C₂²×C₄	C₁₃×C₁₆	1	2	yes	—
110	C₂²×C₄	C₂×C₈×C₁₃	1	2	yes	—
111	C₂²×C₄	C₂×C₈×C₁₃	1	1	yes	characteristic
112	C₂×C₈	—	2	2	no	—
113	C₂×C₈	—	2	4	no	—
114	C₂×C₈	—	2	2	no	—
115	C₂×C₈	—	2	2	no	—
116	C₂×C₈	C₁₃×C₂²⋊C₄	1	2	yes	—
117	C₂×C₈	C₂×C₈×C₁₃	1	2	yes	—
118	C₂×C₈	C₂×C₈×C₁₃	1	1	yes	characteristic
119	C₂×C₈	C₁₃×M₁₆	1	2	yes	—
120	C₂×C₈	C₂²×C₄×C₁₃	1	1	yes	characteristic
121	C₁₆	—	2	2	no	—
122	C₁₆	C₁₃×C₂²⋊C₄	1	2	yes	—
123	C₁₆	C₂×D₄×C₁₃	1	2	yes	—
124	C₁₃	#13553	1	1	yes	characteristic, direct product factor, Sylow
125	C₂³	—	2	2	no	—
126	C₂³	—	2	4	no	—
127	C₂³	C₂×C₁₃×C₁₆	1	1	yes	characteristic
128	C₂³	C₂×C₁₃×C₁₆	1	2	yes	—
129	C₂×C₄	—	2	2	no	—
130	C₂×C₄	—	2	2	no	—
131	C₂×C₄	—	2	4	no	—
132	C₂×C₄	—	2	2	no	—
133	C₂×C₄	C₁₃×C₂²⋊C₈	1	2	yes	—
134	C₂×C₄	C₂×C₁₃×C₁₆	1	2	yes	—
135	C₂×C₄	C₂×C₁₃×C₁₆	1	1	yes	characteristic
136	C₂×C₄	C₁₃×M₃₂	1	2	yes	—
137	C₂×C₄	C₂²×C₈×C₁₃	1	1	yes	characteristic
138	C₈	—	2	2	no	—
139	C₈	C₁₃×C₂²⋊C₈	1	2	yes	—
140	C₈	C₂×C₁₃×C₂²⋊C₄	1	1	yes	characteristic
141	C₈	C₂×C₁₃×M₁₆	1	1	yes	characteristic
142	C₂²	—	2	8	no	—
143	C₂²	—	2	2	no	—
144	C₂²	—	2	2	no	—
145	C₂²	—	2	2	no	—
146	C₂²	C₁₃×C₂²⋊C₁₆	1	2	yes	—
147	C₂²	C₁₃×M₆₄	1	4	yes	complemented
148	C₂²	C₂²×C₁₃×C₁₆	1	1	yes	characteristic
149	C₄	—	2	2	no	—
150	C₄	C₁₃×C₂²⋊C₁₆	1	2	yes	—
151	C₄	C₂×C₁₃×C₂²⋊C₈	1	1	yes	characteristic
152	C₄	C₂×C₁₃×M₃₂	1	1	yes	characteristic
153	C₂	—	2	2	no	—
154	C₂	—	2	4	no	—
155	C₂	C₂×C₁₃×C₂²⋊C₁₆	1	1	yes	characteristic
156	C₂	C₂×C₁₃×M₆₄	1	2	yes	—

	subgroup	G/H
center	C₂×C₁₃×C₁₆	C₂³
commutator	C₂²	C₂²×C₁₃×C₁₆
Frattini, cosocle	C₂×C₁₆	C₂³×C₁₃
socle	C₂²×C₁₃	C₂²×C₁₆
prime radical	C₂⁴×C₁₃	C₁₆
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₁₃×C₁₆ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₁₃×M₆₄	C₂	semidirect
3. C₁₃×C₂²⋊C₃₂	C₂	semidirect
11. C₂×C₁₃×C₃₂	C₂²	semidirect
27. #13553	C₁₃	direct (N × H)
48. C₂×M₆₄	C₂×C₁₃	semidirect
49. C₂²⋊C₃₂	C₂×C₁₃	semidirect
74. C₂×C₃₂	C₂²×C₁₃	semidirect
80. C₂²×C₁₃	M₆₄	semidirect
124. C₁₃	#13553	direct (N × H)
147. C₂²	C₁₃×M₆₄	semidirect

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
C₂×C₁₃×M₆₄	2
C₁₃×C₂²⋊C₃₂	2
#13553	13

Subgroups — C13×#13553 #15699892

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#13553 #15699892