Subgroups — C₂×C₁₃×#2218 #15719940

Order 3328.

786 conjugacy classes of subgroups.

The subgroup lattice has 256 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₁₃×D₁₆	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₁₃×M₃₂	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₁₃×#466	C₂	1	2	yes	maximal, complemented
4	C₁₃×#2218	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₁₃×#367	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₁₃×#357	C₂	1	2	yes	maximal, complemented
7	C₂²×C₁₃×D₈	C₂²	1	2	yes	—
8	C₂³×C₈×C₁₃	C₂²	1	1	yes	characteristic
9	C₂×C₁₃×#121	—	2	2	no	—
10	C₂×C₁₃×D₁₆	—	2	4	no	—
11	C₂×C₁₃×D₁₆	C₂²	1	4	yes	complemented
12	C₂×C₁₃×M₃₂	—	2	4	no	—
13	C₂×C₁₃×M₃₂	C₂²	1	4	yes	complemented
14	C₂²×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
15	C₂²×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
16	C₁₃×#466	—	2	4	no	—
17	C₁₃×#466	C₂²	1	8	yes	complemented
18	C₂×C₁₃×#107	C₂²	1	2	yes	—
19	C₂×C₁₃×#102	—	2	2	no	—
20	C₁₃×#367	C₂²	1	4	yes	complemented
21	C₁₃×#357	C₂²	1	8	yes	complemented
22	C₂²×D₄×C₁₃	—	2	2	no	—
23	C₂²×D₄×C₁₃	—	4	2	no	—
24	C₂³×C₄×C₁₃	D₄	1	1	yes	characteristic
25	C₂×C₁₃×D₈	—	2	4	no	—
26	C₂×C₁₃×D₈	—	2	2	no	—
27	C₂×C₁₃×D₈	—	2	4	no	—
28	C₂×C₁₃×D₈	C₂³	1	4	yes	—
29	C₂²×C₈×C₁₃	—	2	2	no	—
30	C₂²×C₈×C₁₃	—	2	2	no	—
31	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
32	C₂²×C₈×C₁₃	D₄	1	2	yes	—
33	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
34	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
35	C₁₃×#121	—	2	8	no	—
36	C₁₃×#121	—	4	4	no	—
37	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
38	C₂×C₁₃×C₂²⋊C₄	—	4	2	no	—
39	C₁₃×D₁₆	—	2	8	no	—
40	C₁₃×M₃₂	—	2	8	no	—
41	C₂×C₁₃×C₁₆	—	2	2	no	—
42	C₂×C₁₃×C₁₆	D₄	1	4	yes	complemented
43	C₂×C₁₃×C₁₆	C₂³	1	2	yes	—
44	C₂×C₁₃×C₁₆	C₂³	1	2	yes	—
45	C₁₃×#107	—	2	2	no	—
46	C₁₃×#107	C₂³	1	4	yes	—
47	C₁₃×#102	—	2	8	no	—
48	C₂×#2218	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
49	C₂⁴×C₁₃	—	4	2	no	—
50	C₂⁴×C₁₃	D₈	1	1	yes	characteristic
51	C₂×D₄×C₁₃	—	2	4	no	—
52	C₂×D₄×C₁₃	—	4	8	no	—
53	C₂×D₄×C₁₃	—	4	4	no	—
54	C₂×D₄×C₁₃	—	4	2	no	—
55	C₂×D₄×C₁₃	—	4	4	no	—
56	C₂×D₄×C₁₃	—	8	8	no	—
57	C₂²×C₄×C₁₃	—	2	2	no	—
58	C₂²×C₄×C₁₃	—	2	2	no	—
59	C₂²×C₄×C₁₃	—	4	2	no	—
60	C₂²×C₄×C₁₃	D₈	1	2	yes	—
61	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
62	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
63	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
64	C₁₃×D₈	—	2	4	no	—
65	C₁₃×D₈	—	2	4	no	—
66	C₁₃×D₈	—	4	4	no	—
67	C₂×C₈×C₁₃	—	2	2	no	—
68	C₂×C₈×C₁₃	—	2	4	no	—
69	C₂×C₈×C₁₃	—	2	2	no	—
70	C₂×C₈×C₁₃	—	2	2	no	—
71	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
72	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
73	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
74	C₂×C₈×C₁₃	#41	1	2	yes	—
75	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
76	C₁₃×C₄⋊C₄	—	2	4	no	—
77	C₁₃×C₄⋊C₄	—	4	2	no	—
78	C₁₃×C₂²⋊C₄	—	4	8	no	—
79	C₁₃×C₁₆	—	2	2	no	—
80	C₁₃×C₁₆	C₂×D₄	1	4	yes	complemented
81	C₂²×D₁₆	C₂×C₁₃	1	1	yes	characteristic, complemented
82	C₂²×M₃₂	C₂×C₁₃	1	1	yes	characteristic, complemented
83	C₂×#466	C₂×C₁₃	1	2	yes	complemented
84	#2218	C₂×C₁₃	1	8	yes	direct product factor
85	C₂×#367	C₂×C₁₃	1	1	yes	characteristic, complemented
86	C₂×#357	C₂×C₁₃	1	2	yes	complemented
87	C₂³×C₁₃	—	2	2	no	—
88	C₂³×C₁₃	—	2	4	no	—
89	C₂³×C₁₃	—	4	4	no	—
90	C₂³×C₁₃	—	8	4	no	—
91	C₂³×C₁₃	—	8	4	no	—
92	C₂³×C₁₃	—	8	4	no	—
93	C₂³×C₁₃	C₂×D₈	1	2	yes	—
94	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
95	D₄×C₁₃	—	4	4	no	—
96	D₄×C₁₃	—	4	4	no	—
97	D₄×C₁₃	—	8	16	no	—
98	D₄×C₁₃	—	8	4	no	—
99	C₂×C₄×C₁₃	—	2	2	no	—
100	C₂×C₄×C₁₃	—	2	2	no	—
101	C₂×C₄×C₁₃	—	2	2	no	—
102	C₂×C₄×C₁₃	—	2	4	no	—
103	C₂×C₄×C₁₃	—	4	4	no	—
104	C₂×C₄×C₁₃	—	8	4	no	—
105	C₂×C₄×C₁₃	#121	1	2	yes	—
106	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
107	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
108	C₂×C₄×C₁₃	#135	1	2	yes	—
109	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
110	C₈×C₁₃	—	2	2	no	—
111	C₈×C₁₃	#121	1	2	yes	—
112	C₈×C₁₃	C₂²×D₄	1	1	yes	characteristic
113	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
114	C₂²×D₈	C₂²×C₁₃	1	2	yes	—
115	C₂³×C₈	C₂²×C₁₃	1	1	yes	characteristic
116	C₂×#121	—	2	2	no	—
117	C₂×D₁₆	—	2	4	no	—
118	C₂×D₁₆	C₂²×C₁₃	1	4	yes	complemented
119	C₂×M₃₂	—	2	4	no	—
120	C₂×M₃₂	C₂²×C₁₃	1	4	yes	complemented
121	C₂²×C₁₆	C₂²×C₁₃	1	1	yes	characteristic
122	C₂²×C₁₆	C₂²×C₁₃	1	1	yes	characteristic
123	#466	—	2	4	no	—
124	#466	C₂²×C₁₃	1	8	yes	complemented
125	C₂×#107	C₂²×C₁₃	1	2	yes	—
126	C₂×#102	—	2	2	no	—
127	#367	C₂²×C₁₃	1	4	yes	complemented
128	#357	C₂²×C₁₃	1	8	yes	complemented
129	C₂²×C₁₃	—	2	4	no	—
130	C₂²×C₁₃	—	2	2	no	—
131	C₂²×C₁₃	—	4	4	no	—
132	C₂²×C₁₃	—	8	4	no	—
133	C₂²×C₁₃	—	8	4	no	—
134	C₂²×C₁₃	—	8	4	no	—
135	C₂²×C₁₃	—	16	8	no	—
136	C₂²×C₁₃	#466	1	2	yes	—
137	C₂²×C₁₃	#509	1	4	yes	—
138	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
139	C₄×C₁₃	—	2	2	no	—
140	C₄×C₁₃	—	8	4	no	—
141	C₄×C₁₃	#466	1	2	yes	—
142	C₄×C₁₃	C₂×#121	1	1	yes	characteristic
143	C₄×C₁₃	C₂×#135	1	1	yes	characteristic
144	C₂²×D₄	—	2	2	no	—
145	C₂²×D₄	—	4	2	no	—
146	C₂³×C₄	D₄×C₁₃	1	1	yes	characteristic
147	C₂×D₈	—	2	4	no	—
148	C₂×D₈	—	2	2	no	—
149	C₂×D₈	—	2	4	no	—
150	C₂×D₈	C₂³×C₁₃	1	4	yes	—
151	C₂²×C₈	—	2	2	no	—
152	C₂²×C₈	—	2	2	no	—
153	C₂²×C₈	D₄×C₁₃	1	1	yes	characteristic
154	C₂²×C₈	D₄×C₁₃	1	2	yes	—
155	C₂²×C₈	C₂³×C₁₃	1	2	yes	—
156	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
157	#121	—	2	8	no	—
158	#121	—	4	4	no	—
159	C₂×C₄⋊C₄	—	2	2	no	—
160	C₂×C₂²⋊C₄	—	4	2	no	—
161	D₁₆	—	2	8	no	—
162	M₃₂	—	2	8	no	—
163	C₂×C₁₆	—	2	2	no	—
164	C₂×C₁₆	D₄×C₁₃	1	4	yes	complemented
165	C₂×C₁₆	C₂³×C₁₃	1	2	yes	—
166	C₂×C₁₆	C₂³×C₁₃	1	2	yes	—
167	#107	—	2	2	no	—
168	#107	C₂³×C₁₃	1	4	yes	—
169	#102	—	2	8	no	—
170	C₂×C₁₃	—	4	2	no	—
171	C₂×C₁₃	—	16	4	no	—
172	C₂×C₁₃	#2218	1	4	yes	direct product factor
173	C₂×C₁₃	C₂×#466	1	1	yes	characteristic
174	C₂×C₁₃	C₂×#509	1	2	yes	—
175	C₂⁴	—	4	2	no	—
176	C₂⁴	C₁₃×D₈	1	1	yes	characteristic
177	C₂×D₄	—	2	4	no	—
178	C₂×D₄	—	4	8	no	—
179	C₂×D₄	—	4	4	no	—
180	C₂×D₄	—	4	2	no	—
181	C₂×D₄	—	4	4	no	—
182	C₂×D₄	—	8	8	no	—
183	C₂²×C₄	—	2	2	no	—
184	C₂²×C₄	—	2	2	no	—
185	C₂²×C₄	—	4	2	no	—
186	C₂²×C₄	C₁₃×D₈	1	1	yes	characteristic
187	C₂²×C₄	C₁₃×D₈	1	2	yes	—
188	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
189	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
190	D₈	—	2	4	no	—
191	D₈	—	2	4	no	—
192	D₈	—	4	4	no	—
193	C₂×C₈	—	2	2	no	—
194	C₂×C₈	—	2	4	no	—
195	C₂×C₈	—	2	2	no	—
196	C₂×C₈	—	2	2	no	—
197	C₂×C₈	C₂×D₄×C₁₃	1	2	yes	—
198	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
199	C₂×C₈	C₂×D₄×C₁₃	1	2	yes	—
200	C₂×C₈	C₁₃×#41	1	2	yes	—
201	C₂×C₈	C₂⁴×C₁₃	1	1	yes	characteristic
202	C₄⋊C₄	—	2	4	no	—
203	C₄⋊C₄	—	4	2	no	—
204	C₂²⋊C₄	—	4	8	no	—
205	C₁₆	—	2	2	no	—
206	C₁₆	C₂×D₄×C₁₃	1	4	yes	complemented
207	C₁₃	C₂×#2218	1	1	yes	characteristic, direct product factor, Sylow
208	C₂³	—	2	2	no	—
209	C₂³	—	2	4	no	—
210	C₂³	—	4	4	no	—
211	C₂³	—	8	4	no	—
212	C₂³	—	8	4	no	—
213	C₂³	—	8	4	no	—
214	C₂³	C₂×C₁₃×D₈	1	2	yes	—
215	C₂³	C₂×C₁₃×D₈	1	1	yes	characteristic
216	D₄	—	4	4	no	—
217	D₄	—	4	4	no	—
218	D₄	—	8	16	no	—
219	D₄	—	8	4	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	2	4	no	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	4	4	no	—
225	C₂×C₄	—	8	4	no	—
226	C₂×C₄	C₁₃×#121	1	2	yes	—
227	C₂×C₄	C₂×C₁₃×D₈	1	2	yes	—
228	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
229	C₂×C₄	C₁₃×#135	1	2	yes	—
230	C₂×C₄	C₂²×D₄×C₁₃	1	1	yes	characteristic
231	C₈	—	2	2	no	—
232	C₈	C₁₃×#121	1	2	yes	—
233	C₈	C₂²×D₄×C₁₃	1	1	yes	characteristic
234	C₈	C₂×C₁₃×#41	1	1	yes	characteristic
235	C₂²	—	2	2	no	—
236	C₂²	—	2	4	no	—
237	C₂²	—	4	4	no	—
238	C₂²	—	8	4	no	—
239	C₂²	—	8	4	no	—
240	C₂²	—	8	4	no	—
241	C₂²	—	16	8	no	—
242	C₂²	C₁₃×#466	1	2	yes	—
243	C₂²	C₁₃×#509	1	4	yes	—
244	C₂²	C₂²×C₁₃×D₈	1	1	yes	characteristic
245	C₄	—	2	2	no	—
246	C₄	—	8	4	no	—
247	C₄	C₁₃×#466	1	2	yes	—
248	C₄	C₂×C₁₃×#121	1	1	yes	characteristic
249	C₄	C₂×C₁₃×#135	1	1	yes	characteristic
250	C₂	—	4	2	no	—
251	C₂	—	16	4	no	—
252	C₂	C₁₃×#2218	1	4	yes	direct product factor
253	C₂	C₂×C₁₃×#466	1	1	yes	characteristic
254	C₂	C₂×C₁₃×#509	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
1. C₂²×C₁₃×D₁₆	C₂	semidirect
2. C₂²×C₁₃×M₃₂	C₂	semidirect
3. C₂×C₁₃×#466	C₂	semidirect
4. C₁₃×#2218	C₂	direct (N × H)
5. C₂×C₁₃×#367	C₂	semidirect
6. C₂×C₁₃×#357	C₂	semidirect
11. C₂×C₁₃×D₁₆	C₂²	semidirect
13. C₂×C₁₃×M₃₂	C₂²	semidirect
17. C₁₃×#466	C₂²	semidirect
20. C₁₃×#367	C₂²	semidirect
21. C₁₃×#357	C₂²	semidirect
42. C₂×C₁₃×C₁₆	D₄	semidirect
48. C₂×#2218	C₁₃	direct (N × H)
80. C₁₃×C₁₆	C₂×D₄	semidirect
81. C₂²×D₁₆	C₂×C₁₃	semidirect
82. C₂²×M₃₂	C₂×C₁₃	semidirect
83. C₂×#466	C₂×C₁₃	semidirect
84. #2218	C₂×C₁₃	direct (N × H)
85. C₂×#367	C₂×C₁₃	semidirect
86. C₂×#357	C₂×C₁₃	semidirect
118. C₂×D₁₆	C₂²×C₁₃	semidirect
120. C₂×M₃₂	C₂²×C₁₃	semidirect
124. #466	C₂²×C₁₃	semidirect
127. #367	C₂²×C₁₃	semidirect
128. #357	C₂²×C₁₃	semidirect
164. C₂×C₁₆	D₄×C₁₃	semidirect
172. C₂×C₁₃	#2218	direct (N × H)
206. C₁₆	C₂×D₄×C₁₃	semidirect
207. C₁₃	C₂×#2218	direct (N × H)
252. C₂	C₁₃×#2218	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₁₃×D₁₆	2
C₂²×C₁₃×M₃₂	2
C₂×C₁₃×#466	2
C₁₃×#2218	2
C₂×C₁₃×#367	2
C₂×C₁₃×#357	2
C₂×#2218	13

Subgroups — C2×C13×#2218 #15719940

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₁₃×#2218 #15719940