Subgroups — C₁₃×#13645 #15699984

Order 3328.

204 conjugacy classes of subgroups.

The subgroup lattice has 192 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₃×M₆₄	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₁₃×C₃₂	C₂	1	1	yes	maximal, characteristic, complemented
3	D₄×C₁₃×C₁₆	C₂	1	1	yes	maximal, characteristic
4	C₁₃×C₄⋊C₃₂	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₃×C₂²⋊C₃₂	C₂	1	1	yes	maximal, characteristic, complemented
6	C₁₃×C₂²⋊C₃₂	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₃×#1397	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
9	C₂²×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
10	C₈×D₄×C₁₃	C₄	1	1	yes	characteristic
11	C₁₃×M₆₄	—	2	2	no	—
12	C₂×C₁₃×C₃₂	—	2	2	no	—
13	C₂×C₁₃×C₃₂	C₂²	1	1	yes	characteristic
14	C₂×C₁₃×C₃₂	C₂²	1	1	yes	characteristic
15	C₂×C₁₃×C₃₂	C₂²	1	1	yes	characteristic
16	C₂×C₁₃×C₃₂	C₂²	1	1	yes	characteristic
17	C₁₃×C₄⋊C₁₆	C₄	1	1	yes	characteristic
18	C₁₃×C₂²⋊C₁₆	C₄	1	1	yes	characteristic
19	C₁₃×C₂²⋊C₁₆	C₄	1	1	yes	characteristic
20	C₄×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
21	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
23	C₄×D₄×C₁₃	C₈	1	1	yes	characteristic
24	C₂×C₁₃×C₁₆	—	2	1	no	—
25	C₂×C₁₃×C₁₆	—	2	1	no	—
26	C₂×C₁₃×C₁₆	—	2	1	no	—
27	C₂×C₁₃×C₁₆	—	2	1	no	—
28	C₂×C₁₃×C₁₆	C₂×C₄	1	1	yes	characteristic
29	C₂×C₁₃×C₁₆	C₂×C₄	1	1	yes	characteristic
30	C₂×C₁₃×C₁₆	C₂×C₄	1	1	yes	characteristic
31	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
32	C₁₃×C₄⋊C₈	C₈	1	1	yes	characteristic
33	C₁₃×C₂²⋊C₈	C₈	1	1	yes	characteristic
34	C₁₃×C₂²⋊C₈	C₈	1	1	yes	characteristic
35	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
36	C₁₃×C₃₂	—	2	1	no	—
37	C₁₃×C₃₂	—	2	1	no	—
38	C₁₃×C₃₂	—	2	1	no	—
39	C₁₃×C₃₂	D₄	1	2	yes	complemented
40	#13645	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
41	C₂×D₄×C₁₃	C₁₆	1	1	yes	characteristic
42	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
43	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
44	C₂×C₈×C₁₃	—	2	1	no	—
45	C₂×C₈×C₁₃	—	2	1	no	—
46	C₂×C₈×C₁₃	—	2	1	no	—
47	C₂×C₈×C₁₃	—	2	1	no	—
48	C₂×C₈×C₁₃	C₂×C₈	1	1	yes	characteristic
49	C₂×C₈×C₁₃	C₂×C₈	1	1	yes	characteristic
50	C₂×C₈×C₁₃	C₂×C₈	1	1	yes	characteristic
51	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
52	C₁₃×C₄⋊C₄	C₁₆	1	1	yes	characteristic
53	C₁₃×C₂²⋊C₄	C₁₆	1	1	yes	characteristic
54	C₁₃×C₂²⋊C₄	C₁₆	1	1	yes	characteristic
55	C₄²×C₁₃	C₂×C₈	1	1	yes	characteristic
56	C₁₃×C₁₆	—	2	1	no	—
57	C₁₃×C₁₆	—	2	1	no	—
58	C₁₃×C₁₆	—	2	1	no	—
59	C₁₃×C₁₆	C₂×D₄	1	1	yes	characteristic
60	C₁₃×C₁₆	#41	1	1	yes	characteristic
61	C₂×M₆₄	C₂×C₁₃	1	1	yes	characteristic, complemented
62	C₂²×C₃₂	C₂×C₁₃	1	1	yes	characteristic, complemented
63	D₄×C₁₆	C₂×C₁₃	1	1	yes	characteristic
64	C₄⋊C₃₂	C₂×C₁₃	1	1	yes	characteristic, complemented
65	C₂²⋊C₃₂	C₂×C₁₃	1	1	yes	characteristic, complemented
66	C₂²⋊C₃₂	C₂×C₁₃	1	1	yes	characteristic, complemented
67	#1397	C₂×C₁₃	1	1	yes	characteristic, complemented
68	C₂³×C₁₃	C₂×C₁₆	1	1	yes	characteristic
69	C₂³×C₁₃	C₂×C₁₆	1	1	yes	characteristic
70	D₄×C₁₃	—	2	2	no	—
71	C₂×C₄×C₁₃	—	2	1	no	—
72	C₂×C₄×C₁₃	—	2	1	no	—
73	C₂×C₄×C₁₃	—	2	1	no	—
74	C₂×C₄×C₁₃	—	2	1	no	—
75	C₂×C₄×C₁₃	C₂×C₁₆	1	1	yes	characteristic
76	C₂×C₄×C₁₃	C₂×C₁₆	1	1	yes	characteristic
77	C₂×C₄×C₁₃	C₂×C₁₆	1	1	yes	characteristic
78	C₂×C₄×C₁₃	C₂×C₁₆	1	1	yes	characteristic
79	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
80	C₈×C₁₃	—	2	1	no	—
81	C₈×C₁₃	—	2	1	no	—
82	C₈×C₁₃	—	2	1	no	—
83	C₈×C₁₃	C₄×D₄	1	1	yes	characteristic
84	C₈×C₁₃	#131	1	1	yes	characteristic
85	C₂²×C₁₆	C₂²×C₁₃	1	1	yes	characteristic
86	C₂²×C₁₆	C₂²×C₁₃	1	1	yes	characteristic
87	C₈×D₄	C₄×C₁₃	1	1	yes	characteristic
88	M₆₄	—	2	2	no	—
89	C₂×C₃₂	—	2	2	no	—
90	C₂×C₃₂	C₂²×C₁₃	1	1	yes	characteristic
91	C₂×C₃₂	C₂²×C₁₃	1	1	yes	characteristic
92	C₂×C₃₂	C₂²×C₁₃	1	1	yes	characteristic
93	C₂×C₃₂	C₂²×C₁₃	1	1	yes	characteristic
94	C₄⋊C₁₆	C₄×C₁₃	1	1	yes	characteristic
95	C₂²⋊C₁₆	C₄×C₁₃	1	1	yes	characteristic
96	C₂²⋊C₁₆	C₄×C₁₃	1	1	yes	characteristic
97	C₄×C₁₆	C₂²×C₁₃	1	1	yes	characteristic
98	C₂²×C₁₃	—	2	1	no	—
99	C₂²×C₁₃	—	2	1	no	—
100	C₂²×C₁₃	—	2	1	no	—
101	C₂²×C₁₃	—	2	1	no	—
102	C₂²×C₁₃	—	2	1	no	—
103	C₂²×C₁₃	M₆₄	1	2	yes	complemented
104	C₂²×C₁₃	C₂²×C₁₆	1	1	yes	characteristic
105	C₄×C₁₃	—	2	1	no	—
106	C₄×C₁₃	—	2	1	no	—
107	C₄×C₁₃	—	2	1	no	—
108	C₄×C₁₃	—	2	1	no	—
109	C₄×C₁₃	C₈×D₄	1	1	yes	characteristic
110	C₄×C₁₃	#504	1	1	yes	characteristic
111	C₂²×C₈	C₂×C₄×C₁₃	1	1	yes	characteristic
112	C₂²×C₈	C₂×C₄×C₁₃	1	1	yes	characteristic
113	C₄×D₄	C₈×C₁₃	1	1	yes	characteristic
114	C₂×C₁₆	—	2	1	no	—
115	C₂×C₁₆	—	2	1	no	—
116	C₂×C₁₆	—	2	1	no	—
117	C₂×C₁₆	—	2	1	no	—
118	C₂×C₁₆	C₂×C₄×C₁₃	1	1	yes	characteristic
119	C₂×C₁₆	C₂×C₄×C₁₃	1	1	yes	characteristic
120	C₂×C₁₆	C₂×C₄×C₁₃	1	1	yes	characteristic
121	C₂×C₁₆	C₂³×C₁₃	1	1	yes	characteristic
122	C₄⋊C₈	C₈×C₁₃	1	1	yes	characteristic
123	C₂²⋊C₈	C₈×C₁₃	1	1	yes	characteristic
124	C₂²⋊C₈	C₈×C₁₃	1	1	yes	characteristic
125	C₄×C₈	C₂×C₄×C₁₃	1	1	yes	characteristic
126	C₃₂	—	2	1	no	—
127	C₃₂	—	2	1	no	—
128	C₃₂	—	2	1	no	—
129	C₃₂	D₄×C₁₃	1	2	yes	complemented
130	C₂×C₁₃	—	2	2	no	—
131	C₂×C₁₃	—	4	1	no	—
132	C₂×C₁₃	D₄×C₁₆	1	1	yes	characteristic
133	C₂×C₁₃	C₂×M₆₄	1	1	yes	characteristic
134	C₂×C₁₃	#2258	1	1	yes	characteristic
135	C₂×D₄	C₁₃×C₁₆	1	1	yes	characteristic
136	C₂²×C₄	C₂×C₈×C₁₃	1	1	yes	characteristic
137	C₂²×C₄	C₂×C₈×C₁₃	1	1	yes	characteristic
138	C₂×C₈	—	2	1	no	—
139	C₂×C₈	—	2	1	no	—
140	C₂×C₈	—	2	1	no	—
141	C₂×C₈	—	2	1	no	—
142	C₂×C₈	C₂×C₈×C₁₃	1	1	yes	characteristic
143	C₂×C₈	C₂×C₈×C₁₃	1	1	yes	characteristic
144	C₂×C₈	C₂×C₈×C₁₃	1	1	yes	characteristic
145	C₂×C₈	C₂²×C₄×C₁₃	1	1	yes	characteristic
146	C₄⋊C₄	C₁₃×C₁₆	1	1	yes	characteristic
147	C₂²⋊C₄	C₁₃×C₁₆	1	1	yes	characteristic
148	C₂²⋊C₄	C₁₃×C₁₆	1	1	yes	characteristic
149	C₄²	C₂×C₈×C₁₃	1	1	yes	characteristic
150	C₁₆	—	2	1	no	—
151	C₁₆	—	2	1	no	—
152	C₁₆	—	2	1	no	—
153	C₁₆	C₂×D₄×C₁₃	1	1	yes	characteristic
154	C₁₆	C₁₃×#41	1	1	yes	characteristic
155	C₁₃	#13645	1	1	yes	characteristic, direct product factor, Sylow
156	C₂³	C₂×C₁₃×C₁₆	1	1	yes	characteristic
157	C₂³	C₂×C₁₃×C₁₆	1	1	yes	characteristic
158	D₄	—	2	2	no	—
159	C₂×C₄	—	2	1	no	—
160	C₂×C₄	—	2	1	no	—
161	C₂×C₄	—	2	1	no	—
162	C₂×C₄	—	2	1	no	—
163	C₂×C₄	C₂×C₁₃×C₁₆	1	1	yes	characteristic
164	C₂×C₄	C₂×C₁₃×C₁₆	1	1	yes	characteristic
165	C₂×C₄	C₂×C₁₃×C₁₆	1	1	yes	characteristic
166	C₂×C₄	C₂×C₁₃×C₁₆	1	1	yes	characteristic
167	C₂×C₄	C₂²×C₈×C₁₃	1	1	yes	characteristic
168	C₈	—	2	1	no	—
169	C₈	—	2	1	no	—
170	C₈	—	2	1	no	—
171	C₈	C₄×D₄×C₁₃	1	1	yes	characteristic
172	C₈	C₁₃×#131	1	1	yes	characteristic
173	C₂²	—	2	1	no	—
174	C₂²	—	2	1	no	—
175	C₂²	—	2	1	no	—
176	C₂²	—	2	1	no	—
177	C₂²	—	2	1	no	—
178	C₂²	C₁₃×M₆₄	1	2	yes	complemented
179	C₂²	C₂²×C₁₃×C₁₆	1	1	yes	characteristic
180	C₄	—	2	1	no	—
181	C₄	—	2	1	no	—
182	C₄	—	2	1	no	—
183	C₄	—	2	1	no	—
184	C₄	C₈×D₄×C₁₃	1	1	yes	characteristic
185	C₄	C₁₃×#504	1	1	yes	characteristic
186	C₂	—	2	2	no	—
187	C₂	—	4	1	no	—
188	C₂	D₄×C₁₃×C₁₆	1	1	yes	characteristic
189	C₂	C₂×C₁₃×M₆₄	1	1	yes	characteristic
190	C₂	C₁₃×#2258	1	1	yes	characteristic

	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	C₂×D₄×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
1. C₂×C₁₃×M₆₄	C₂	semidirect
2. C₂²×C₁₃×C₃₂	C₂	semidirect
4. C₁₃×C₄⋊C₃₂	C₂	semidirect
5. C₁₃×C₂²⋊C₃₂	C₂	semidirect
6. C₁₃×C₂²⋊C₃₂	C₂	semidirect
7. C₁₃×#1397	C₂	semidirect
39. C₁₃×C₃₂	D₄	semidirect
40. #13645	C₁₃	direct (N × H)
61. C₂×M₆₄	C₂×C₁₃	semidirect
62. C₂²×C₃₂	C₂×C₁₃	semidirect
64. C₄⋊C₃₂	C₂×C₁₃	semidirect
65. C₂²⋊C₃₂	C₂×C₁₃	semidirect
66. C₂²⋊C₃₂	C₂×C₁₃	semidirect
67. #1397	C₂×C₁₃	semidirect
103. C₂²×C₁₃	M₆₄	semidirect
129. C₃₂	D₄×C₁₃	semidirect
155. C₁₃	#13645	direct (N × H)
178. 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
C₂×C₁₃×M₆₄	2
C₂²×C₁₃×C₃₂	2
D₄×C₁₃×C₁₆	2
C₁₃×C₄⋊C₃₂	2
C₁₃×C₂²⋊C₃₂	2
C₁₃×#1397	2
#13645	13

Subgroups — C13×#13645 #15699984

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#13645 #15699984