Subgroups — C₂×C₁₃×#2263 #15720318

Order 3328.

502 conjugacy classes of subgroups.

The subgroup lattice has 230 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₃×#508	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₁₃×D₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₃×#2263	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₁₃×SD₆₄	C₂	1	2	yes	maximal, complemented
5	C₂×C₁₃×D₃₂	C₂	1	2	yes	maximal, complemented
6	C₂×C₁₃×M₆₄	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₁₃×#135	—	2	1	no	—
8	C₂²×C₁₃×D₈	—	2	1	no	—
9	C₁₃×#508	—	2	2	no	—
10	C₁₃×#508	C₂²	1	4	yes	complemented
11	C₂×C₁₃×Q₃₂	C₂²	1	1	yes	characteristic, complemented
12	C₂×C₁₃×SD₃₂	—	2	1	no	—
13	C₂×C₁₃×D₁₆	—	2	1	no	—
14	C₂×C₁₃×D₁₆	—	2	2	no	—
15	C₂×C₁₃×D₁₆	C₂²	1	2	yes	—
16	C₂×C₁₃×D₁₆	C₂²	1	4	yes	complemented
17	C₂×C₁₃×D₁₆	C₂²	1	1	yes	characteristic, complemented
18	C₂²×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
19	C₁₃×SD₆₄	C₂²	1	8	yes	complemented
20	C₁₃×D₃₂	C₂²	1	8	yes	complemented
21	C₁₃×M₆₄	C₂²	1	4	yes	complemented
22	C₂×C₁₃×C₃₂	C₂²	1	2	yes	complemented
23	C₂×C₁₃×#41	—	4	1	no	—
24	C₂²×D₄×C₁₃	—	4	1	no	—
25	C₁₃×#135	—	2	4	no	—
26	C₁₃×#135	—	4	2	no	—
27	C₂×C₁₃×Q₁₆	—	2	1	no	—
28	C₂×C₁₃×SD₁₆	—	4	1	no	—
29	C₂×C₁₃×D₈	—	2	2	no	—
30	C₂×C₁₃×D₈	—	2	4	no	—
31	C₂×C₁₃×D₈	—	2	1	no	—
32	C₂×C₁₃×D₈	—	4	1	no	—
33	C₂×C₁₃×D₈	—	4	2	no	—
34	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
35	C₁₃×Q₃₂	—	2	1	no	—
36	C₁₃×Q₃₂	C₂³	1	2	yes	complemented
37	C₁₃×SD₃₂	—	2	4	no	—
38	C₁₃×D₁₆	—	2	2	no	—
39	C₁₃×D₁₆	—	2	4	no	—
40	C₁₃×D₁₆	—	2	1	no	—
41	C₁₃×D₁₆	C₂³	1	4	yes	—
42	C₁₃×D₁₆	C₂³	1	2	yes	complemented
43	C₂×C₁₃×C₁₆	D₄	1	1	yes	characteristic
44	C₂×C₁₃×C₁₆	D₄	1	2	yes	—
45	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
46	C₂×C₁₃×C₁₆	C₂³	1	2	yes	—
47	C₁₃×C₃₂	C₂³	1	4	yes	complemented
48	C₂×#2263	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
49	C₂⁴×C₁₃	—	8	1	no	—
50	C₁₃×#41	—	4	4	no	—
51	C₁₃×#41	—	8	2	no	—
52	C₂×Q₈×C₁₃	—	4	1	no	—
53	C₂×D₄×C₁₃	—	4	4	no	—
54	C₂×D₄×C₁₃	—	4	2	no	—
55	C₂×D₄×C₁₃	—	4	1	no	—
56	C₂×D₄×C₁₃	—	8	1	no	—
57	C₂×D₄×C₁₃	—	8	2	no	—
58	C₂×D₄×C₁₃	—	8	1	no	—
59	C₂²×C₄×C₁₃	—	8	1	no	—
60	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
61	C₁₃×Q₁₆	—	2	2	no	—
62	C₁₃×Q₁₆	—	4	1	no	—
63	C₁₃×SD₁₆	—	4	4	no	—
64	C₁₃×D₈	—	2	4	no	—
65	C₁₃×D₈	—	2	2	no	—
66	C₁₃×D₈	—	4	2	no	—
67	C₁₃×D₈	—	4	4	no	—
68	C₁₃×D₈	—	4	1	no	—
69	C₂×C₈×C₁₃	D₈	1	1	yes	characteristic
70	C₂×C₈×C₁₃	D₈	1	2	yes	—
71	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
72	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
73	C₁₃×C₁₆	C₂×D₄	1	1	yes	characteristic
74	C₁₃×C₁₆	C₂×D₄	1	2	yes	—
75	C₁₃×C₁₆	C₂⁴	1	1	yes	characteristic
76	C₂×#508	C₂×C₁₃	1	1	yes	characteristic, complemented
77	C₂²×D₁₆	C₂×C₁₃	1	1	yes	characteristic, complemented
78	#2263	C₂×C₁₃	1	8	yes	direct product factor
79	C₂×SD₆₄	C₂×C₁₃	1	2	yes	complemented
80	C₂×D₃₂	C₂×C₁₃	1	2	yes	complemented
81	C₂×M₆₄	C₂×C₁₃	1	1	yes	characteristic, complemented
82	C₂³×C₁₃	—	8	4	no	—
83	C₂³×C₁₃	—	8	1	no	—
84	C₂³×C₁₃	—	8	2	no	—
85	C₂³×C₁₃	—	16	4	no	—
86	C₂³×C₁₃	D₁₆	1	1	yes	characteristic
87	Q₈×C₁₃	—	4	2	no	—
88	Q₈×C₁₃	—	8	1	no	—
89	D₄×C₁₃	—	4	4	no	—
90	D₄×C₁₃	—	4	2	no	—
91	D₄×C₁₃	—	8	4	no	—
92	D₄×C₁₃	—	8	4	no	—
93	D₄×C₁₃	—	8	2	no	—
94	D₄×C₁₃	—	8	1	no	—
95	C₂×C₄×C₁₃	—	8	1	no	—
96	C₂×C₄×C₁₃	—	8	4	no	—
97	C₂×C₄×C₁₃	D₁₆	1	2	yes	—
98	C₂×C₄×C₁₃	D₁₆	1	1	yes	characteristic
99	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
100	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
101	C₈×C₁₃	C₂×D₈	1	1	yes	characteristic
102	C₈×C₁₃	C₂×D₈	1	2	yes	—
103	C₈×C₁₃	C₂²×D₄	1	1	yes	characteristic
104	C₂×#135	—	2	1	no	—
105	C₂²×D₈	—	2	1	no	—
106	#508	—	2	2	no	—
107	#508	C₂²×C₁₃	1	4	yes	complemented
108	C₂×Q₃₂	C₂²×C₁₃	1	1	yes	characteristic, complemented
109	C₂×SD₃₂	—	2	1	no	—
110	C₂×D₁₆	—	2	1	no	—
111	C₂×D₁₆	—	2	2	no	—
112	C₂×D₁₆	C₂²×C₁₃	1	2	yes	—
113	C₂×D₁₆	C₂²×C₁₃	1	4	yes	complemented
114	C₂×D₁₆	C₂²×C₁₃	1	1	yes	characteristic, complemented
115	C₂²×C₁₆	C₂²×C₁₃	1	1	yes	characteristic
116	SD₆₄	C₂²×C₁₃	1	8	yes	complemented
117	D₃₂	C₂²×C₁₃	1	8	yes	complemented
118	M₆₄	C₂²×C₁₃	1	4	yes	complemented
119	C₂×C₃₂	C₂²×C₁₃	1	2	yes	complemented
120	C₂²×C₁₃	—	2	2	no	—
121	C₂²×C₁₃	—	8	2	no	—
122	C₂²×C₁₃	—	8	4	no	—
123	C₂²×C₁₃	—	16	8	no	—
124	C₂²×C₁₃	—	16	2	no	—
125	C₂²×C₁₃	—	16	4	no	—
126	C₂²×C₁₃	C₂×D₁₆	1	2	yes	—
127	C₂²×C₁₃	C₂×D₁₆	1	1	yes	characteristic
128	C₄×C₁₃	—	8	2	no	—
129	C₄×C₁₃	C₂×D₁₆	1	2	yes	—
130	C₄×C₁₃	C₂×D₁₆	1	1	yes	characteristic
131	C₄×C₁₃	C₂²×D₈	1	1	yes	characteristic
132	C₂×#41	—	4	1	no	—
133	C₂²×D₄	—	4	1	no	—
134	#135	—	2	4	no	—
135	#135	—	4	2	no	—
136	C₂×Q₁₆	—	2	1	no	—
137	C₂×SD₁₆	—	4	1	no	—
138	C₂×D₈	—	2	2	no	—
139	C₂×D₈	—	2	4	no	—
140	C₂×D₈	—	2	1	no	—
141	C₂×D₈	—	4	1	no	—
142	C₂×D₈	—	4	2	no	—
143	C₂²×C₈	D₄×C₁₃	1	1	yes	characteristic
144	Q₃₂	—	2	1	no	—
145	Q₃₂	C₂³×C₁₃	1	2	yes	complemented
146	SD₃₂	—	2	4	no	—
147	D₁₆	—	2	2	no	—
148	D₁₆	—	2	4	no	—
149	D₁₆	—	2	1	no	—
150	D₁₆	C₂³×C₁₃	1	4	yes	—
151	D₁₆	C₂³×C₁₃	1	2	yes	complemented
152	C₂×C₁₆	D₄×C₁₃	1	1	yes	characteristic
153	C₂×C₁₆	D₄×C₁₃	1	2	yes	—
154	C₂×C₁₆	C₂³×C₁₃	1	1	yes	characteristic
155	C₂×C₁₆	C₂³×C₁₃	1	2	yes	—
156	C₃₂	C₂³×C₁₃	1	4	yes	complemented
157	C₂×C₁₃	—	2	2	no	—
158	C₂×C₁₃	—	16	2	no	—
159	C₂×C₁₃	—	16	4	no	—
160	C₂×C₁₃	#2263	1	2	yes	direct product factor
161	C₂×C₁₃	C₂²×D₁₆	1	1	yes	characteristic
162	C₂⁴	—	8	1	no	—
163	#41	—	4	4	no	—
164	#41	—	8	2	no	—
165	C₂×Q₈	—	4	1	no	—
166	C₂×D₄	—	4	2	no	—
167	C₂×D₄	—	4	4	no	—
168	C₂×D₄	—	4	1	no	—
169	C₂×D₄	—	8	1	no	—
170	C₂×D₄	—	8	1	no	—
171	C₂×D₄	—	8	2	no	—
172	C₂²×C₄	—	8	1	no	—
173	C₂²×C₄	C₁₃×D₈	1	1	yes	characteristic
174	Q₁₆	—	2	2	no	—
175	Q₁₆	—	4	1	no	—
176	SD₁₆	—	4	4	no	—
177	D₈	—	2	4	no	—
178	D₈	—	2	2	no	—
179	D₈	—	4	2	no	—
180	D₈	—	4	4	no	—
181	D₈	—	4	1	no	—
182	C₂×C₈	C₁₃×D₈	1	1	yes	characteristic
183	C₂×C₈	C₁₃×D₈	1	2	yes	—
184	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
185	C₂×C₈	C₂×D₄×C₁₃	1	2	yes	—
186	C₁₆	C₂×D₄×C₁₃	1	1	yes	characteristic
187	C₁₆	C₂×D₄×C₁₃	1	2	yes	—
188	C₁₆	C₂⁴×C₁₃	1	1	yes	characteristic
189	C₁₃	C₂×#2263	1	1	yes	characteristic, direct product factor, Sylow
190	C₂³	—	8	1	no	—
191	C₂³	—	8	4	no	—
192	C₂³	—	8	2	no	—
193	C₂³	—	16	4	no	—
194	C₂³	C₁₃×D₁₆	1	1	yes	characteristic
195	Q₈	—	4	2	no	—
196	Q₈	—	8	1	no	—
197	D₄	—	4	4	no	—
198	D₄	—	4	2	no	—
199	D₄	—	8	4	no	—
200	D₄	—	8	4	no	—
201	D₄	—	8	2	no	—
202	D₄	—	8	1	no	—
203	C₂×C₄	—	8	1	no	—
204	C₂×C₄	—	8	4	no	—
205	C₂×C₄	C₁₃×D₁₆	1	1	yes	characteristic
206	C₂×C₄	C₁₃×D₁₆	1	2	yes	—
207	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
208	C₂×C₄	C₂×C₁₃×D₈	1	2	yes	—
209	C₈	C₂×C₁₃×D₈	1	1	yes	characteristic
210	C₈	C₂×C₁₃×D₈	1	2	yes	—
211	C₈	C₂²×D₄×C₁₃	1	1	yes	characteristic
212	C₂²	—	2	2	no	—
213	C₂²	—	8	2	no	—
214	C₂²	—	8	4	no	—
215	C₂²	—	16	8	no	—
216	C₂²	—	16	2	no	—
217	C₂²	—	16	4	no	—
218	C₂²	C₂×C₁₃×D₁₆	1	2	yes	—
219	C₂²	C₂×C₁₃×D₁₆	1	1	yes	characteristic
220	C₄	—	8	2	no	—
221	C₄	C₂×C₁₃×D₁₆	1	2	yes	—
222	C₄	C₂×C₁₃×D₁₆	1	1	yes	characteristic
223	C₄	C₂²×C₁₃×D₈	1	1	yes	characteristic
224	C₂	—	2	2	no	—
225	C₂	—	16	4	no	—
226	C₂	—	16	2	no	—
227	C₂	C₁₃×#2263	1	2	yes	direct product factor
228	C₂	C₂²×C₁₃×D₁₆	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₃	C₂×D₁₆
commutator, Frattini, cosocle	C₁₆	C₂⁴×C₁₃
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃ ◁ C₂²×C₄×C₁₃ ◁ C₂²×C₈×C₁₃ ◁ C₂²×C₁₃×C₁₆ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₁₃×#508	C₂	semidirect
2. C₂²×C₁₃×D₁₆	C₂	semidirect
3. C₁₃×#2263	C₂	direct (N × H)
4. C₂×C₁₃×SD₆₄	C₂	semidirect
5. C₂×C₁₃×D₃₂	C₂	semidirect
6. C₂×C₁₃×M₆₄	C₂	semidirect
10. C₁₃×#508	C₂²	semidirect
11. C₂×C₁₃×Q₃₂	C₂²	semidirect
16. C₂×C₁₃×D₁₆	C₂²	semidirect
17. C₂×C₁₃×D₁₆	C₂²	semidirect
19. C₁₃×SD₆₄	C₂²	semidirect
20. C₁₃×D₃₂	C₂²	semidirect
21. C₁₃×M₆₄	C₂²	semidirect
22. C₂×C₁₃×C₃₂	C₂²	semidirect
36. C₁₃×Q₃₂	C₂³	semidirect
42. C₁₃×D₁₆	C₂³	semidirect
47. C₁₃×C₃₂	C₂³	semidirect
48. C₂×#2263	C₁₃	direct (N × H)
76. C₂×#508	C₂×C₁₃	semidirect
77. C₂²×D₁₆	C₂×C₁₃	semidirect
78. #2263	C₂×C₁₃	direct (N × H)
79. C₂×SD₆₄	C₂×C₁₃	semidirect
80. C₂×D₃₂	C₂×C₁₃	semidirect
81. C₂×M₆₄	C₂×C₁₃	semidirect
107. #508	C₂²×C₁₃	semidirect
108. C₂×Q₃₂	C₂²×C₁₃	semidirect
113. C₂×D₁₆	C₂²×C₁₃	semidirect
114. C₂×D₁₆	C₂²×C₁₃	semidirect
116. SD₆₄	C₂²×C₁₃	semidirect
117. D₃₂	C₂²×C₁₃	semidirect
118. M₆₄	C₂²×C₁₃	semidirect
119. C₂×C₃₂	C₂²×C₁₃	semidirect
145. Q₃₂	C₂³×C₁₃	semidirect
151. D₁₆	C₂³×C₁₃	semidirect
156. C₃₂	C₂³×C₁₃	semidirect
160. C₂×C₁₃	#2263	direct (N × H)
189. C₁₃	C₂×#2263	direct (N × H)
227. C₂	C₁₃×#2263	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×#508	2
C₂²×C₁₃×D₁₆	2
C₁₃×#2263	2
C₂×C₁₃×SD₆₄	2
C₂×C₁₃×D₃₂	2
C₂×C₁₃×M₆₄	2
C₂×#2263	13

Subgroups — C2×C13×#2263 #15720318

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₁₃×#2263 #15720318