Subgroups — C₂×C₁₃×#2244 #15720038

Order 3328.

686 conjugacy classes of subgroups.

The subgroup lattice has 204 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₃×#499	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₁₃×#493	C₂	1	1	yes	maximal, characteristic
3	C₁₃×#2244	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₁₃×#357	C₂	1	4	yes	maximal, complemented
5	C₂×C₁₃×#346	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₁₃×D₈	—	2	1	no	—
7	C₂²×C₁₃×D₈	C₂²	1	2	yes	—
8	C₂×C₁₃×#127	—	2	1	no	—
9	C₂×C₁₃×#125	—	2	1	no	—
10	C₂²×C₁₃×C₁₆	C₂²	1	2	yes	—
11	C₁₃×#499	—	2	2	no	—
12	C₁₃×#499	C₂²	1	4	yes	complemented
13	C₁₃×#493	—	2	2	no	—
14	C₁₃×#493	C₂²	1	4	yes	—
15	C₂×C₁₃×#107	C₂²	1	2	yes	—
16	C₂×C₁₃×#106	—	2	1	no	—
17	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
18	C₁₃×#357	C₂²	1	16	yes	complemented
19	C₁₃×#346	C₂²	1	4	yes	complemented
20	C₂²×D₄×C₁₃	—	2	2	no	—
21	C₂²×D₄×C₁₃	—	4	2	no	—
22	C₂×C₁₃×D₈	—	2	2	no	—
23	C₂×C₁₃×D₈	—	2	4	no	—
24	C₂×C₁₃×D₈	—	2	8	no	—
25	C₂×C₁₃×D₈	—	4	4	no	—
26	C₂×C₁₃×D₈	C₂³	1	4	yes	—
27	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
28	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
29	C₁₃×#127	—	2	4	no	—
30	C₁₃×#127	—	4	2	no	—
31	C₁₃×#125	—	2	4	no	—
32	C₁₃×#125	—	4	2	no	—
33	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
34	C₂×C₁₃×C₄⋊C₄	—	4	2	no	—
35	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
36	C₂×C₁₃×C₁₆	—	2	4	no	—
37	C₂×C₁₃×C₁₆	C₂³	1	4	yes	—
38	C₁₃×#107	—	2	2	no	—
39	C₁₃×#107	C₂³	1	4	yes	—
40	C₁₃×#106	—	2	4	no	—
41	C₄×C₈×C₁₃	D₄	1	2	yes	—
42	C₄×C₈×C₁₃	C₂³	1	2	yes	—
43	C₂×#2244	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
44	C₂⁴×C₁₃	—	4	2	no	—
45	C₂×D₄×C₁₃	—	2	4	no	—
46	C₂×D₄×C₁₃	—	4	4	no	—
47	C₂×D₄×C₁₃	—	4	8	no	—
48	C₂×D₄×C₁₃	—	4	2	no	—
49	C₂×D₄×C₁₃	—	4	4	no	—
50	C₂×D₄×C₁₃	—	8	8	no	—
51	C₂²×C₄×C₁₃	—	4	2	no	—
52	C₂²×C₄×C₁₃	D₈	1	2	yes	—
53	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
54	C₁₃×D₈	—	2	8	no	—
55	C₁₃×D₈	—	4	4	no	—
56	C₁₃×D₈	—	4	8	no	—
57	C₂×C₈×C₁₃	SD₁₆	1	4	yes	—
58	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
59	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
60	C₂×C₈×C₁₃	#41	1	4	yes	—
61	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
62	C₁₃×C₄⋊C₄	—	2	4	no	—
63	C₁₃×C₄⋊C₄	—	4	2	no	—
64	C₁₃×C₄⋊C₄	—	4	8	no	—
65	C₄²×C₁₃	D₈	1	2	yes	—
66	C₄²×C₁₃	C₂×D₄	1	2	yes	—
67	C₁₃×C₁₆	—	2	4	no	—
68	C₂×#499	C₂×C₁₃	1	1	yes	characteristic, complemented
69	C₂×#493	C₂×C₁₃	1	1	yes	characteristic
70	#2244	C₂×C₁₃	1	8	yes	direct product factor
71	C₂×#357	C₂×C₁₃	1	4	yes	complemented
72	C₂×#346	C₂×C₁₃	1	1	yes	characteristic, complemented
73	C₂³×C₁₃	—	4	4	no	—
74	C₂³×C₁₃	—	8	4	no	—
75	C₂³×C₁₃	—	8	8	no	—
76	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
77	D₄×C₁₃	—	4	4	no	—
78	D₄×C₁₃	—	4	4	no	—
79	D₄×C₁₃	—	8	16	no	—
80	D₄×C₁₃	—	8	4	no	—
81	C₂×C₄×C₁₃	—	2	4	no	—
82	C₂×C₄×C₁₃	—	4	4	no	—
83	C₂×C₄×C₁₃	—	8	4	no	—
84	C₂×C₄×C₁₃	#124	1	2	yes	—
85	C₂×C₄×C₁₃	C₂×D₈	1	4	yes	—
86	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
87	C₂×C₄×C₁₃	C₂×SD₁₆	1	2	yes	—
88	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
89	C₈×C₁₃	#124	1	2	yes	—
90	C₈×C₁₃	C₂×SD₁₆	1	4	yes	—
91	C₈×C₁₃	C₂×#41	1	2	yes	—
92	C₂²×D₈	—	2	1	no	—
93	C₂²×D₈	C₂²×C₁₃	1	2	yes	—
94	C₂×#127	—	2	1	no	—
95	C₂×#125	—	2	1	no	—
96	C₂²×C₁₆	C₂²×C₁₃	1	2	yes	—
97	#499	—	2	2	no	—
98	#499	C₂²×C₁₃	1	4	yes	complemented
99	#493	—	2	2	no	—
100	#493	C₂²×C₁₃	1	4	yes	—
101	C₂×#107	C₂²×C₁₃	1	2	yes	—
102	C₂×#106	—	2	1	no	—
103	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
104	#357	C₂²×C₁₃	1	16	yes	complemented
105	#346	C₂²×C₁₃	1	4	yes	complemented
106	C₂²×C₁₃	—	8	4	no	—
107	C₂²×C₁₃	—	8	8	no	—
108	C₂²×C₁₃	—	16	8	no	—
109	C₂²×C₁₃	#486	1	2	yes	—
110	C₂²×C₁₃	#509	1	4	yes	—
111	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
112	C₄×C₁₃	—	2	4	no	—
113	C₄×C₁₃	—	8	4	no	—
114	C₄×C₁₃	#486	1	2	yes	—
115	C₄×C₁₃	C₂×#124	1	1	yes	characteristic
116	C₄×C₁₃	C₂²×SD₁₆	1	1	yes	characteristic
117	C₂²×D₄	—	2	2	no	—
118	C₂²×D₄	—	4	2	no	—
119	C₂×D₈	—	2	4	no	—
120	C₂×D₈	—	2	2	no	—
121	C₂×D₈	—	2	8	no	—
122	C₂×D₈	—	4	4	no	—
123	C₂×D₈	C₂³×C₁₃	1	4	yes	—
124	C₂²×C₈	D₄×C₁₃	1	1	yes	characteristic
125	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
126	#127	—	2	4	no	—
127	#127	—	4	2	no	—
128	#125	—	2	4	no	—
129	#125	—	4	2	no	—
130	C₂×C₄⋊C₄	—	2	2	no	—
131	C₂×C₄⋊C₄	—	4	2	no	—
132	C₂×C₄²	D₄×C₁₃	1	1	yes	characteristic
133	C₂×C₁₆	—	2	4	no	—
134	C₂×C₁₆	C₂³×C₁₃	1	4	yes	—
135	#107	—	2	2	no	—
136	#107	C₂³×C₁₃	1	4	yes	—
137	#106	—	2	4	no	—
138	C₄×C₈	D₄×C₁₃	1	2	yes	—
139	C₄×C₈	C₂³×C₁₃	1	2	yes	—
140	C₂×C₁₃	—	16	4	no	—
141	C₂×C₁₃	#2244	1	4	yes	direct product factor
142	C₂×C₁₃	C₂×#486	1	1	yes	characteristic
143	C₂×C₁₃	C₂×#509	1	2	yes	—
144	C₂⁴	—	4	2	no	—
145	C₂×D₄	—	2	4	no	—
146	C₂×D₄	—	4	4	no	—
147	C₂×D₄	—	4	8	no	—
148	C₂×D₄	—	4	2	no	—
149	C₂×D₄	—	4	4	no	—
150	C₂×D₄	—	8	8	no	—
151	C₂²×C₄	—	4	2	no	—
152	C₂²×C₄	C₁₃×D₈	1	2	yes	—
153	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
154	D₈	—	2	8	no	—
155	D₈	—	4	4	no	—
156	D₈	—	4	8	no	—
157	C₂×C₈	C₁₃×SD₁₆	1	4	yes	—
158	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
159	C₂×C₈	C₂×D₄×C₁₃	1	2	yes	—
160	C₂×C₈	C₁₃×#41	1	4	yes	—
161	C₂×C₈	C₂⁴×C₁₃	1	1	yes	characteristic
162	C₄⋊C₄	—	2	4	no	—
163	C₄⋊C₄	—	4	2	no	—
164	C₄⋊C₄	—	4	8	no	—
165	C₄²	C₁₃×D₈	1	2	yes	—
166	C₄²	C₂×D₄×C₁₃	1	2	yes	—
167	C₁₆	—	2	4	no	—
168	C₁₃	C₂×#2244	1	1	yes	characteristic, direct product factor, Sylow
169	C₂³	—	4	4	no	—
170	C₂³	—	8	4	no	—
171	C₂³	—	8	8	no	—
172	C₂³	C₂×C₁₃×D₈	1	1	yes	characteristic
173	D₄	—	4	4	no	—
174	D₄	—	4	4	no	—
175	D₄	—	8	16	no	—
176	D₄	—	8	4	no	—
177	C₂×C₄	—	2	4	no	—
178	C₂×C₄	—	4	4	no	—
179	C₂×C₄	—	8	4	no	—
180	C₂×C₄	C₁₃×#124	1	2	yes	—
181	C₂×C₄	C₂×C₁₃×D₈	1	4	yes	—
182	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
183	C₂×C₄	C₂×C₁₃×SD₁₆	1	2	yes	—
184	C₂×C₄	C₂²×D₄×C₁₃	1	1	yes	characteristic
185	C₈	C₁₃×#124	1	2	yes	—
186	C₈	C₂×C₁₃×SD₁₆	1	4	yes	—
187	C₈	C₂×C₁₃×#41	1	2	yes	—
188	C₂²	—	8	4	no	—
189	C₂²	—	8	8	no	—
190	C₂²	—	16	8	no	—
191	C₂²	C₁₃×#486	1	2	yes	—
192	C₂²	C₁₃×#509	1	4	yes	—
193	C₂²	C₂²×C₁₃×D₈	1	1	yes	characteristic
194	C₄	—	2	4	no	—
195	C₄	—	8	4	no	—
196	C₄	C₁₃×#486	1	2	yes	—
197	C₄	C₂×C₁₃×#124	1	1	yes	characteristic
198	C₄	C₂²×C₁₃×SD₁₆	1	1	yes	characteristic
199	C₂	—	16	4	no	—
200	C₂	C₁₃×#2244	1	4	yes	direct product factor
201	C₂	C₂×C₁₃×#486	1	1	yes	characteristic
202	C₂	C₂×C₁₃×#509	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₁₃	C₂×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₁₃
prime radical	C₂×C₁₃×#493	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₃ ◁ C₂×C₄²×C₁₃ ◁ C₂×C₄×C₈×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₁₃×#499	C₂	semidirect
3. C₁₃×#2244	C₂	direct (N × H)
4. C₂×C₁₃×#357	C₂	semidirect
5. C₂×C₁₃×#346	C₂	semidirect
12. C₁₃×#499	C₂²	semidirect
18. C₁₃×#357	C₂²	semidirect
19. C₁₃×#346	C₂²	semidirect
43. C₂×#2244	C₁₃	direct (N × H)
68. C₂×#499	C₂×C₁₃	semidirect
70. #2244	C₂×C₁₃	direct (N × H)
71. C₂×#357	C₂×C₁₃	semidirect
72. C₂×#346	C₂×C₁₃	semidirect
98. #499	C₂²×C₁₃	semidirect
104. #357	C₂²×C₁₃	semidirect
105. #346	C₂²×C₁₃	semidirect
141. C₂×C₁₃	#2244	direct (N × H)
168. C₁₃	C₂×#2244	direct (N × H)
200. C₂	C₁₃×#2244	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₁₃×#499	2
C₂×C₁₃×#493	2
C₁₃×#2244	2
C₂×C₁₃×#357	2
C₂×C₁₃×#346	2
C₂×#2244	13

Subgroups — C2×C13×#2244 #15720038

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₁₃×#2244 #15720038