Subgroups — C₁₃×#20572 #15706911

Order 3328.

914 conjugacy classes of subgroups.

The subgroup lattice has 180 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×Q₈×C₁₃	C₂	1	2	yes	maximal
2	C₄²×Q₈×C₁₃	C₂	1	1	yes	maximal, characteristic
3	C₁₃×#1770	C₂	1	6	yes	maximal
4	C₁₃×#1767	C₂	1	3	yes	maximal
5	C₁₃×#1744	C₂	1	3	yes	maximal
6	C₂×C₄×Q₈×C₁₃	C₄	1	2	yes	—
7	C₂×C₄×Q₈×C₁₃	C₂²	1	1	yes	characteristic
8	C₈×Q₈×C₁₃	C₄	1	16	yes	complemented
9	C₂×C₁₃×C₄⋊C₈	C₂²	1	6	yes	—
10	C₂×C₁₃×#97	C₂²	1	6	yes	—
11	C₂×C₄×C₈×C₁₃	C₂²	1	6	yes	—
12	C₄×C₁₃×C₄⋊C₄	C₄	1	6	yes	—
13	C₄×C₁₃×C₄⋊C₄	C₂²	1	3	yes	—
14	C₄³×C₁₃	C₂²	1	3	yes	—
15	C₁₃×#336	C₂²	1	1	yes	characteristic
16	C₁₃×#336	C₂²	1	3	yes	—
17	C₁₃×#336	C₂²	1	6	yes	—
18	C₂²×Q₈×C₁₃	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
20	C₂²×C₈×C₁₃	C₂³	1	6	yes	—
21	C₄×Q₈×C₁₃	—	2	8	no	—
22	C₄×Q₈×C₁₃	C₂×C₄	1	8	yes	—
23	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	6	yes	—
24	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	3	yes	—
25	C₂×C₄²×C₁₃	C₂×C₄	1	6	yes	—
26	C₂×C₄²×C₁₃	C₂×C₄	1	6	yes	—
27	C₂×C₄²×C₁₃	C₂³	1	3	yes	—
28	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
29	C₂×C₄²×C₁₃	C₂³	1	3	yes	—
30	C₁₃×C₄⋊C₈	C₂×C₄	1	24	yes	—
31	C₁₃×#97	—	4	6	no	—
32	C₄×C₈×C₁₃	C₂×C₄	1	24	yes	—
33	C₁₃×#95	C₂×C₄	1	6	yes	—
34	#20572	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
35	C₂×Q₈×C₁₃	C₄²	1	4	yes	—
36	C₂×Q₈×C₁₃	M₁₆	1	8	yes	—
37	C₂²×C₄×C₁₃	C₂²×C₄	1	6	yes	—
38	C₂²×C₄×C₁₃	C₂²×C₄	1	3	yes	—
39	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
40	C₂²×C₄×C₁₃	C₂²×C₄	1	3	yes	—
41	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
42	C₂×C₈×C₁₃	—	2	6	no	—
43	C₂×C₈×C₁₃	—	2	2	no	—
44	C₂×C₈×C₁₃	—	2	2	no	—
45	C₂×C₈×C₁₃	—	2	6	no	—
46	C₂×C₈×C₁₃	C₂²×C₄	1	4	yes	—
47	C₂×C₈×C₁₃	C₂²×C₄	1	12	yes	—
48	C₁₃×C₄⋊C₄	—	2	12	no	—
49	C₁₃×C₄⋊C₄	C₄²	1	12	yes	—
50	C₄²×C₁₃	—	2	12	no	—
51	C₄²×C₁₃	—	2	12	no	—
52	C₄²×C₁₃	—	2	3	no	—
53	C₄²×C₁₃	—	2	2	no	—
54	C₄²×C₁₃	—	2	3	no	—
55	C₄²×C₁₃	C₂²×C₄	1	12	yes	—
56	C₂×C₈×Q₈	C₂×C₁₃	1	2	yes	—
57	C₄²×Q₈	C₂×C₁₃	1	1	yes	characteristic
58	#1770	C₂×C₁₃	1	6	yes	—
59	#1767	C₂×C₁₃	1	3	yes	—
60	#1744	C₂×C₁₃	1	3	yes	—
61	C₂³×C₁₃	C₂³×C₄	1	1	yes	characteristic
62	Q₈×C₁₃	#97	1	16	yes	complemented
63	C₂×C₄×C₁₃	—	2	2	no	—
64	C₂×C₄×C₁₃	—	2	2	no	—
65	C₂×C₄×C₁₃	—	2	6	no	—
66	C₂×C₄×C₁₃	—	2	6	no	—
67	C₂×C₄×C₁₃	—	2	6	no	—
68	C₂×C₄×C₁₃	—	2	6	no	—
69	C₂×C₄×C₁₃	C₂×C₄²	1	6	yes	—
70	C₂×C₄×C₁₃	C₂×C₄²	1	6	yes	—
71	C₂×C₄×C₁₃	C₂×M₁₆	1	12	yes	—
72	C₂×C₄×C₁₃	#131	1	4	yes	—
73	C₂×C₄×C₁₃	C₂³×C₄	1	2	yes	—
74	C₂×C₄×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
75	C₂×C₄×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
76	C₂×C₄×C₁₃	2₋¹⁺⁴	1	2	yes	—
77	C₈×C₁₃	—	2	4	no	—
78	C₈×C₁₃	—	2	12	no	—
79	C₂×C₄×Q₈	C₄×C₁₃	1	2	yes	—
80	C₂×C₄×Q₈	C₂²×C₁₃	1	1	yes	characteristic
81	C₈×Q₈	C₄×C₁₃	1	16	yes	complemented
82	C₂×C₄⋊C₈	C₂²×C₁₃	1	6	yes	—
83	C₂×#97	C₂²×C₁₃	1	6	yes	—
84	C₂×C₄×C₈	C₂²×C₁₃	1	6	yes	—
85	C₄×C₄⋊C₄	C₄×C₁₃	1	6	yes	—
86	C₄×C₄⋊C₄	C₂²×C₁₃	1	3	yes	—
87	C₄³	C₂²×C₁₃	1	3	yes	—
88	#336	C₂²×C₁₃	1	1	yes	characteristic
89	#336	C₂²×C₁₃	1	3	yes	—
90	#336	C₂²×C₁₃	1	6	yes	—
91	C₂²×C₁₃	C₂²×C₄²	1	1	yes	characteristic
92	C₂²×C₁₃	#519	1	1	yes	characteristic
93	C₂²×C₁₃	#520	1	1	yes	characteristic
94	C₂²×C₁₃	C₂²×M₁₆	1	2	yes	—
95	C₂²×C₁₃	C₂×#131	1	2	yes	—
96	C₄×C₁₃	—	2	6	no	—
97	C₄×C₁₃	—	2	4	no	—
98	C₄×C₁₃	—	4	6	no	—
99	C₄×C₁₃	C₂×#97	1	12	yes	—
100	C₄×C₁₃	#519	1	2	yes	—
101	C₄×C₁₃	#520	1	2	yes	—
102	C₂²×Q₈	C₂×C₄×C₁₃	1	1	yes	characteristic
103	C₂²×C₈	C₂³×C₁₃	1	2	yes	—
104	C₂²×C₈	C₂³×C₁₃	1	6	yes	—
105	C₄×Q₈	—	2	8	no	—
106	C₄×Q₈	C₂×C₄×C₁₃	1	8	yes	—
107	C₂×C₄⋊C₄	C₂×C₄×C₁₃	1	6	yes	—
108	C₂×C₄⋊C₄	C₂×C₄×C₁₃	1	3	yes	—
109	C₂×C₄²	C₂×C₄×C₁₃	1	6	yes	—
110	C₂×C₄²	C₂×C₄×C₁₃	1	6	yes	—
111	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
112	C₂×C₄²	C₂³×C₁₃	1	3	yes	—
113	C₂×C₄²	C₂³×C₁₃	1	3	yes	—
114	C₄⋊C₈	C₂×C₄×C₁₃	1	24	yes	—
115	#97	—	4	6	no	—
116	C₄×C₈	C₂×C₄×C₁₃	1	24	yes	—
117	#95	C₂×C₄×C₁₃	1	6	yes	—
118	C₂×C₁₃	#2276	1	1	yes	characteristic
119	C₂×C₁₃	C₂²×#97	1	1	yes	characteristic
120	C₂×C₁₃	C₄×#131	1	1	yes	characteristic
121	C₂×C₁₃	#2984	1	2	yes	—
122	C₂×C₁₃	#2991	1	2	yes	—
123	C₂×Q₈	C₄²×C₁₃	1	4	yes	—
124	C₂×Q₈	C₁₃×M₁₆	1	8	yes	—
125	C₂²×C₄	C₂²×C₄×C₁₃	1	6	yes	—
126	C₂²×C₄	C₂²×C₄×C₁₃	1	3	yes	—
127	C₂²×C₄	C₂²×C₄×C₁₃	1	2	yes	—
128	C₂²×C₄	C₂²×C₄×C₁₃	1	3	yes	—
129	C₂²×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
130	C₂×C₈	—	2	6	no	—
131	C₂×C₈	—	2	2	no	—
132	C₂×C₈	—	2	2	no	—
133	C₂×C₈	—	2	6	no	—
134	C₂×C₈	C₂²×C₄×C₁₃	1	4	yes	—
135	C₂×C₈	C₂²×C₄×C₁₃	1	12	yes	—
136	C₄⋊C₄	—	2	12	no	—
137	C₄⋊C₄	C₄²×C₁₃	1	12	yes	—
138	C₄²	—	2	12	no	—
139	C₄²	—	2	12	no	—
140	C₄²	—	2	3	no	—
141	C₄²	—	2	2	no	—
142	C₄²	—	2	3	no	—
143	C₄²	C₂²×C₄×C₁₃	1	12	yes	—
144	C₁₃	#20572	1	1	yes	characteristic, direct product factor, Sylow
145	C₂³	C₂³×C₄×C₁₃	1	1	yes	characteristic
146	Q₈	C₁₃×#97	1	16	yes	complemented
147	C₂×C₄	—	2	2	no	—
148	C₂×C₄	—	2	2	no	—
149	C₂×C₄	—	2	6	no	—
150	C₂×C₄	—	2	6	no	—
151	C₂×C₄	—	2	6	no	—
152	C₂×C₄	—	2	6	no	—
153	C₂×C₄	C₂×C₄²×C₁₃	1	6	yes	—
154	C₂×C₄	C₂×C₄²×C₁₃	1	6	yes	—
155	C₂×C₄	C₂×C₁₃×M₁₆	1	12	yes	—
156	C₂×C₄	C₁₃×#131	1	4	yes	—
157	C₂×C₄	C₂³×C₄×C₁₃	1	2	yes	—
158	C₂×C₄	C₁₃×H(5,𝔽₂)	1	1	yes	characteristic
159	C₂×C₄	C₁₃×2₋¹⁺⁴	1	1	yes	characteristic
160	C₂×C₄	C₁₃×2₋¹⁺⁴	1	2	yes	—
161	C₈	—	2	4	no	—
162	C₈	—	2	12	no	—
163	C₂²	C₂²×C₄²×C₁₃	1	1	yes	characteristic
164	C₂²	C₁₃×#519	1	1	yes	characteristic
165	C₂²	C₁₃×#520	1	1	yes	characteristic
166	C₂²	C₂²×C₁₃×M₁₆	1	2	yes	—
167	C₂²	C₂×C₁₃×#131	1	2	yes	—
168	C₄	—	2	6	no	—
169	C₄	—	2	4	no	—
170	C₄	—	4	6	no	—
171	C₄	C₂×C₁₃×#97	1	12	yes	—
172	C₄	C₁₃×#519	1	2	yes	—
173	C₄	C₁₃×#520	1	2	yes	—
174	C₂	C₁₃×#2276	1	1	yes	characteristic
175	C₂	C₂²×C₁₃×#97	1	1	yes	characteristic
176	C₂	C₄×C₁₃×#131	1	1	yes	characteristic
177	C₂	C₁₃×#2984	1	2	yes	—
178	C₂	C₁₃×#2991	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₁₃	C₂⁴
commutator	C₂²	C₂²×C₄²×C₁₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃
socle, prime radical	C₂³×C₁₃	C₂³×C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
8. C₈×Q₈×C₁₃	C₄	semidirect
34. #20572	C₁₃	direct (N × H)
62. Q₈×C₁₃	#97	semidirect
81. C₈×Q₈	C₄×C₁₃	semidirect
144. C₁₃	#20572	direct (N × H)
146. Q₈	C₁₃×#97	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₈×Q₈×C₁₃	2
C₄²×Q₈×C₁₃	2
C₁₃×#1770	2
C₁₃×#1767	2
C₁₃×#1744	2
#20572	13

Subgroups — C13×#20572 #15706911

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#20572 #15706911