Subgroups — C₂×C₁₃×#2264 #15720319

Order 3328.

462 conjugacy classes of subgroups.

The subgroup lattice has 222 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
2	C₂²×C₁₃×Q₃₂	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₃×#2264	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₁₃×Q₆₄	C₂	1	2	yes	maximal, complemented
5	C₂×C₁₃×SD₆₄	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×Q₁₆	—	2	1	no	—
9	C₁₃×#508	—	2	2	no	—
10	C₁₃×#508	C₂²	1	4	yes	—
11	C₂×C₁₃×Q₃₂	—	2	1	no	—
12	C₂×C₁₃×Q₃₂	—	2	2	no	—
13	C₂×C₁₃×Q₃₂	C₂²	1	2	yes	—
14	C₂×C₁₃×Q₃₂	C₂²	1	1	yes	characteristic
15	C₂×C₁₃×Q₃₂	C₂²	1	4	yes	complemented
16	C₂×C₁₃×SD₃₂	—	2	1	no	—
17	C₂×C₁₃×D₁₆	C₂²	1	1	yes	characteristic
18	C₂²×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
19	C₁₃×Q₆₄	C₂²	1	8	yes	complemented
20	C₁₃×SD₆₄	C₂²	1	8	yes	complemented
21	C₁₃×M₆₄	C₂²	1	4	yes	complemented
22	C₂×C₁₃×C₃₂	C₂²	1	2	yes	—
23	C₂×C₁₃×#41	—	4	1	no	—
24	C₂²×Q₈×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×Q₁₆	—	2	2	no	—
29	C₂×C₁₃×Q₁₆	—	2	4	no	—
30	C₂×C₁₃×Q₁₆	—	4	1	no	—
31	C₂×C₁₃×Q₁₆	—	4	2	no	—
32	C₂×C₁₃×SD₁₆	—	4	1	no	—
33	C₂×C₁₃×D₈	—	2	1	no	—
34	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
35	C₁₃×Q₃₂	—	2	2	no	—
36	C₁₃×Q₃₂	—	2	4	no	—
37	C₁₃×Q₃₂	—	2	1	no	—
38	C₁₃×Q₃₂	C₂³	1	4	yes	—
39	C₁₃×Q₃₂	C₂³	1	2	yes	—
40	C₁₃×SD₃₂	—	2	4	no	—
41	C₁₃×D₁₆	—	2	1	no	—
42	C₁₃×D₁₆	C₂³	1	2	yes	—
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	—
48	C₂×#2264	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
49	C₁₃×#41	—	4	4	no	—
50	C₁₃×#41	—	8	2	no	—
51	C₂×Q₈×C₁₃	—	4	1	no	—
52	C₂×Q₈×C₁₃	—	4	4	no	—
53	C₂×Q₈×C₁₃	—	4	2	no	—
54	C₂×Q₈×C₁₃	—	8	2	no	—
55	C₂×Q₈×C₁₃	—	8	1	no	—
56	C₂×D₄×C₁₃	—	4	1	no	—
57	C₂×D₄×C₁₃	—	8	1	no	—
58	C₂²×C₄×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	—	2	4	no	—
63	C₁₃×Q₁₆	—	4	2	no	—
64	C₁₃×Q₁₆	—	4	4	no	—
65	C₁₃×Q₁₆	—	4	1	no	—
66	C₁₃×SD₁₆	—	4	4	no	—
67	C₁₃×D₈	—	2	2	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
77	C₂²×Q₃₂	C₂×C₁₃	1	1	yes	characteristic, complemented
78	#2264	C₂×C₁₃	1	8	yes	direct product factor
79	C₂×Q₆₄	C₂×C₁₃	1	2	yes	complemented
80	C₂×SD₆₄	C₂×C₁₃	1	2	yes	complemented
81	C₂×M₆₄	C₂×C₁₃	1	1	yes	characteristic, complemented
82	C₂³×C₁₃	—	8	1	no	—
83	C₂³×C₁₃	D₁₆	1	1	yes	characteristic
84	Q₈×C₁₃	—	4	2	no	—
85	Q₈×C₁₃	—	4	4	no	—
86	Q₈×C₁₃	—	8	4	no	—
87	Q₈×C₁₃	—	8	2	no	—
88	Q₈×C₁₃	—	8	1	no	—
89	D₄×C₁₃	—	4	2	no	—
90	D₄×C₁₃	—	8	4	no	—
91	D₄×C₁₃	—	8	1	no	—
92	C₂×C₄×C₁₃	—	8	4	no	—
93	C₂×C₄×C₁₃	—	8	1	no	—
94	C₂×C₄×C₁₃	—	8	2	no	—
95	C₂×C₄×C₁₃	—	8	4	no	—
96	C₂×C₄×C₁₃	D₁₆	1	2	yes	—
97	C₂×C₄×C₁₃	D₁₆	1	1	yes	characteristic
98	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
99	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
100	C₈×C₁₃	C₂×D₈	1	1	yes	characteristic
101	C₈×C₁₃	C₂×D₈	1	2	yes	—
102	C₈×C₁₃	C₂²×D₄	1	1	yes	characteristic
103	C₂×#135	—	2	1	no	—
104	C₂²×Q₁₆	—	2	1	no	—
105	#508	—	2	2	no	—
106	#508	C₂²×C₁₃	1	4	yes	—
107	C₂×Q₃₂	—	2	1	no	—
108	C₂×Q₃₂	—	2	2	no	—
109	C₂×Q₃₂	C₂²×C₁₃	1	2	yes	—
110	C₂×Q₃₂	C₂²×C₁₃	1	1	yes	characteristic
111	C₂×Q₃₂	C₂²×C₁₃	1	4	yes	complemented
112	C₂×SD₃₂	—	2	1	no	—
113	C₂×D₁₆	C₂²×C₁₃	1	1	yes	characteristic
114	C₂²×C₁₆	C₂²×C₁₃	1	1	yes	characteristic
115	Q₆₄	C₂²×C₁₃	1	8	yes	complemented
116	SD₆₄	C₂²×C₁₃	1	8	yes	complemented
117	M₆₄	C₂²×C₁₃	1	4	yes	complemented
118	C₂×C₃₂	C₂²×C₁₃	1	2	yes	—
119	C₂²×C₁₃	—	2	2	no	—
120	C₂²×C₁₃	—	8	2	no	—
121	C₂²×C₁₃	—	16	2	no	—
122	C₂²×C₁₃	C₂×D₁₆	1	2	yes	—
123	C₂²×C₁₃	C₂×D₁₆	1	1	yes	characteristic
124	C₄×C₁₃	—	8	2	no	—
125	C₄×C₁₃	—	8	4	no	—
126	C₄×C₁₃	C₂×D₁₆	1	2	yes	—
127	C₄×C₁₃	C₂×D₁₆	1	1	yes	characteristic
128	C₄×C₁₃	C₂²×D₈	1	1	yes	characteristic
129	C₂×#41	—	4	1	no	—
130	C₂²×Q₈	—	4	1	no	—
131	#135	—	2	4	no	—
132	#135	—	4	2	no	—
133	C₂×Q₁₆	—	2	2	no	—
134	C₂×Q₁₆	—	2	4	no	—
135	C₂×Q₁₆	—	2	1	no	—
136	C₂×Q₁₆	—	4	1	no	—
137	C₂×Q₁₆	—	4	2	no	—
138	C₂×SD₁₆	—	4	1	no	—
139	C₂×D₈	—	2	1	no	—
140	C₂²×C₈	D₄×C₁₃	1	1	yes	characteristic
141	Q₃₂	—	2	2	no	—
142	Q₃₂	—	2	4	no	—
143	Q₃₂	—	2	1	no	—
144	Q₃₂	C₂³×C₁₃	1	4	yes	—
145	Q₃₂	C₂³×C₁₃	1	2	yes	—
146	SD₃₂	—	2	4	no	—
147	D₁₆	—	2	1	no	—
148	D₁₆	C₂³×C₁₃	1	2	yes	—
149	C₂×C₁₆	D₄×C₁₃	1	1	yes	characteristic
150	C₂×C₁₆	D₄×C₁₃	1	2	yes	—
151	C₂×C₁₆	C₂³×C₁₃	1	1	yes	characteristic
152	C₂×C₁₆	C₂³×C₁₃	1	2	yes	—
153	C₃₂	C₂³×C₁₃	1	4	yes	—
154	C₂×C₁₃	—	2	2	no	—
155	C₂×C₁₃	—	16	2	no	—
156	C₂×C₁₃	#2264	1	2	yes	direct product factor
157	C₂×C₁₃	C₂²×D₁₆	1	1	yes	characteristic
158	#41	—	4	4	no	—
159	#41	—	8	2	no	—
160	C₂×Q₈	—	4	1	no	—
161	C₂×Q₈	—	4	2	no	—
162	C₂×Q₈	—	4	4	no	—
163	C₂×Q₈	—	8	1	no	—
164	C₂×Q₈	—	8	2	no	—
165	C₂×D₄	—	4	1	no	—
166	C₂×D₄	—	8	1	no	—
167	C₂²×C₄	—	8	1	no	—
168	C₂²×C₄	—	8	1	no	—
169	C₂²×C₄	C₁₃×D₈	1	1	yes	characteristic
170	Q₁₆	—	2	2	no	—
171	Q₁₆	—	2	4	no	—
172	Q₁₆	—	4	2	no	—
173	Q₁₆	—	4	4	no	—
174	Q₁₆	—	4	1	no	—
175	SD₁₆	—	4	4	no	—
176	D₈	—	2	2	no	—
177	D₈	—	4	1	no	—
178	C₂×C₈	C₁₃×D₈	1	1	yes	characteristic
179	C₂×C₈	C₁₃×D₈	1	2	yes	—
180	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
181	C₂×C₈	C₂×D₄×C₁₃	1	2	yes	—
182	C₁₆	C₂×D₄×C₁₃	1	1	yes	characteristic
183	C₁₆	C₂×D₄×C₁₃	1	2	yes	—
184	C₁₆	C₂⁴×C₁₃	1	1	yes	characteristic
185	C₁₃	C₂×#2264	1	1	yes	characteristic, direct product factor, Sylow
186	C₂³	—	8	1	no	—
187	C₂³	C₁₃×D₁₆	1	1	yes	characteristic
188	Q₈	—	4	2	no	—
189	Q₈	—	4	4	no	—
190	Q₈	—	8	4	no	—
191	Q₈	—	8	2	no	—
192	Q₈	—	8	1	no	—
193	D₄	—	4	2	no	—
194	D₄	—	8	4	no	—
195	D₄	—	8	1	no	—
196	C₂×C₄	—	8	1	no	—
197	C₂×C₄	—	8	4	no	—
198	C₂×C₄	—	8	2	no	—
199	C₂×C₄	—	8	4	no	—
200	C₂×C₄	C₁₃×D₁₆	1	1	yes	characteristic
201	C₂×C₄	C₁₃×D₁₆	1	2	yes	—
202	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
203	C₂×C₄	C₂×C₁₃×D₈	1	2	yes	—
204	C₈	C₂×C₁₃×D₈	1	1	yes	characteristic
205	C₈	C₂×C₁₃×D₈	1	2	yes	—
206	C₈	C₂²×D₄×C₁₃	1	1	yes	characteristic
207	C₂²	—	2	2	no	—
208	C₂²	—	8	2	no	—
209	C₂²	—	16	2	no	—
210	C₂²	C₂×C₁₃×D₁₆	1	2	yes	—
211	C₂²	C₂×C₁₃×D₁₆	1	1	yes	characteristic
212	C₄	—	8	2	no	—
213	C₄	—	8	4	no	—
214	C₄	C₂×C₁₃×D₁₆	1	2	yes	—
215	C₄	C₂×C₁₃×D₁₆	1	1	yes	characteristic
216	C₄	C₂²×C₁₃×D₈	1	1	yes	characteristic
217	C₂	—	2	2	no	—
218	C₂	—	16	2	no	—
219	C₂	C₁₃×#2264	1	2	yes	direct product factor
220	C₂	C₂²×C₁₃×D₁₆	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₃	C₂×D₁₆
commutator, Frattini, cosocle	C₁₆	C₂⁴×C₁₃
prime radical	C₂×C₁₃×#508	C₂
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
2. C₂²×C₁₃×Q₃₂	C₂	semidirect
3. C₁₃×#2264	C₂	direct (N × H)
4. C₂×C₁₃×Q₆₄	C₂	semidirect
5. C₂×C₁₃×SD₆₄	C₂	semidirect
6. C₂×C₁₃×M₆₄	C₂	semidirect
15. C₂×C₁₃×Q₃₂	C₂²	semidirect
19. C₁₃×Q₆₄	C₂²	semidirect
20. C₁₃×SD₆₄	C₂²	semidirect
21. C₁₃×M₆₄	C₂²	semidirect
48. C₂×#2264	C₁₃	direct (N × H)
77. C₂²×Q₃₂	C₂×C₁₃	semidirect
78. #2264	C₂×C₁₃	direct (N × H)
79. C₂×Q₆₄	C₂×C₁₃	semidirect
80. C₂×SD₆₄	C₂×C₁₃	semidirect
81. C₂×M₆₄	C₂×C₁₃	semidirect
111. C₂×Q₃₂	C₂²×C₁₃	semidirect
115. Q₆₄	C₂²×C₁₃	semidirect
116. SD₆₄	C₂²×C₁₃	semidirect
117. M₆₄	C₂²×C₁₃	semidirect
156. C₂×C₁₃	#2264	direct (N × H)
185. C₁₃	C₂×#2264	direct (N × H)
219. C₂	C₁₃×#2264	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×Q₃₂	2
C₁₃×#2264	2
C₂×C₁₃×Q₆₄	2
C₂×C₁₃×SD₆₄	2
C₂×C₁₃×M₆₄	2
C₂×#2264	13

Subgroups — C2×C13×#2264 #15720319

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₁₃×#2264 #15720319