Subgroups — C₂×C₁₃×#2243 #15720037

Order 3328.

634 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₃×#500	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₁₃×#495	C₂	1	1	yes	maximal, characteristic
3	C₁₃×#2243	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₁₃×#358	C₂	1	2	yes	maximal, complemented
5	C₂×C₁₃×#357	C₂	1	2	yes	maximal
6	C₂×C₁₃×#346	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×C₁₃×Q₁₆	C₂²	1	1	yes	characteristic
8	C₂²×C₁₃×SD₁₆	—	2	1	no	—
9	C₂²×C₁₃×D₈	C₂²	1	1	yes	characteristic
10	C₂×C₁₃×C₄⋊Q₈	—	2	1	no	—
11	C₂×C₁₃×#124	—	2	1	no	—
12	C₂²×C₁₃×C₁₆	C₂²	1	2	yes	—
13	C₁₃×#500	—	2	2	no	—
14	C₁₃×#500	C₂²	1	4	yes	complemented
15	C₁₃×#495	—	2	2	no	—
16	C₁₃×#495	C₂²	1	4	yes	—
17	C₂×C₁₃×#107	—	2	1	no	—
18	C₂×C₁₃×#107	C₂²	1	2	yes	—
19	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
20	C₁₃×#358	C₂²	1	8	yes	complemented
21	C₁₃×#357	C₂²	1	8	yes	—
22	C₁₃×#346	C₂²	1	4	yes	complemented
23	C₂²×Q₈×C₁₃	—	2	1	no	—
24	C₂²×Q₈×C₁₃	—	4	1	no	—
25	C₂²×D₄×C₁₃	—	2	1	no	—
26	C₂×C₁₃×Q₁₆	—	2	1	no	—
27	C₂×C₁₃×Q₁₆	—	2	4	no	—
28	C₂×C₁₃×Q₁₆	C₂³	1	2	yes	—
29	C₂×C₁₃×SD₁₆	—	2	4	no	—
30	C₂×C₁₃×SD₁₆	—	4	4	no	—
31	C₂×C₁₃×D₈	—	2	1	no	—
32	C₂×C₁₃×D₈	—	2	4	no	—
33	C₂×C₁₃×D₈	C₂³	1	2	yes	—
34	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
35	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
36	C₁₃×C₄⋊Q₈	—	2	4	no	—
37	C₁₃×C₄⋊Q₈	—	4	2	no	—
38	C₁₃×#124	—	2	4	no	—
39	C₁₃×#124	—	4	2	no	—
40	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
41	C₂×C₁₃×C₄⋊C₄	—	4	1	no	—
42	C₂×C₁₃×C₂²⋊C₄	—	4	1	no	—
43	C₂×C₁₃×C₂²⋊C₄	—	4	1	no	—
44	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
45	C₂×C₁₃×C₁₆	—	2	4	no	—
46	C₂×C₁₃×C₁₆	C₂³	1	4	yes	—
47	C₁₃×#107	—	2	4	no	—
48	C₁₃×#107	—	2	2	no	—
49	C₁₃×#107	C₂³	1	4	yes	—
50	C₄×C₈×C₁₃	D₄	1	2	yes	—
51	C₄×C₈×C₁₃	C₂³	1	2	yes	—
52	C₂×#2243	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
53	C₂⁴×C₁₃	—	4	1	no	—
54	C₂×Q₈×C₁₃	—	2	2	no	—
55	C₂×Q₈×C₁₃	—	4	1	no	—
56	C₂×Q₈×C₁₃	—	4	4	no	—
57	C₂×Q₈×C₁₃	—	4	2	no	—
58	C₂×Q₈×C₁₃	—	4	2	no	—
59	C₂×Q₈×C₁₃	—	8	4	no	—
60	C₂×D₄×C₁₃	—	2	2	no	—
61	C₂×D₄×C₁₃	—	4	2	no	—
62	C₂×D₄×C₁₃	—	4	1	no	—
63	C₂×D₄×C₁₃	—	4	2	no	—
64	C₂²×C₄×C₁₃	—	4	1	no	—
65	C₂²×C₄×C₁₃	—	4	2	no	—
66	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
67	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
68	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
69	C₁₃×Q₁₆	—	2	4	no	—
70	C₁₃×Q₁₆	—	4	2	no	—
71	C₁₃×SD₁₆	—	4	8	no	—
72	C₁₃×D₈	—	2	4	no	—
73	C₁₃×D₈	—	4	2	no	—
74	C₂×C₈×C₁₃	SD₁₆	1	4	yes	—
75	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
76	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
77	C₂×C₈×C₁₃	#41	1	4	yes	—
78	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
79	C₁₃×C₄⋊C₄	—	2	4	no	—
80	C₁₃×C₄⋊C₄	—	4	2	no	—
81	C₁₃×C₄⋊C₄	—	4	4	no	—
82	C₁₃×C₂²⋊C₄	—	4	4	no	—
83	C₁₃×C₂²⋊C₄	—	4	4	no	—
84	C₄²×C₁₃	D₈	1	2	yes	—
85	C₄²×C₁₃	C₂×D₄	1	2	yes	—
86	C₁₃×C₁₆	—	2	4	no	—
87	C₂×#500	C₂×C₁₃	1	1	yes	characteristic, complemented
88	C₂×#495	C₂×C₁₃	1	1	yes	characteristic
89	#2243	C₂×C₁₃	1	8	yes	direct product factor
90	C₂×#358	C₂×C₁₃	1	2	yes	complemented
91	C₂×#357	C₂×C₁₃	1	2	yes	—
92	C₂×#346	C₂×C₁₃	1	1	yes	characteristic, complemented
93	C₂³×C₁₃	—	4	2	no	—
94	C₂³×C₁₃	—	8	2	no	—
95	C₂³×C₁₃	—	8	2	no	—
96	C₂³×C₁₃	—	8	2	no	—
97	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
98	Q₈×C₁₃	—	4	2	no	—
99	Q₈×C₁₃	—	4	2	no	—
100	Q₈×C₁₃	—	8	8	no	—
101	Q₈×C₁₃	—	8	2	no	—
102	D₄×C₁₃	—	4	2	no	—
103	D₄×C₁₃	—	4	2	no	—
104	D₄×C₁₃	—	8	2	no	—
105	C₂×C₄×C₁₃	—	2	2	no	—
106	C₂×C₄×C₁₃	—	2	2	no	—
107	C₂×C₄×C₁₃	—	4	2	no	—
108	C₂×C₄×C₁₃	—	4	4	no	—
109	C₂×C₄×C₁₃	—	8	2	no	—
110	C₂×C₄×C₁₃	—	8	4	no	—
111	C₂×C₄×C₁₃	#124	1	2	yes	—
112	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
113	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
114	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
115	C₂×C₄×C₁₃	C₂×SD₁₆	1	2	yes	—
116	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
117	C₈×C₁₃	#124	1	2	yes	—
118	C₈×C₁₃	C₂×SD₁₆	1	4	yes	—
119	C₈×C₁₃	C₂×#41	1	2	yes	—
120	C₂²×Q₁₆	C₂²×C₁₃	1	1	yes	characteristic
121	C₂²×SD₁₆	—	2	1	no	—
122	C₂²×D₈	C₂²×C₁₃	1	1	yes	characteristic
123	C₂×C₄⋊Q₈	—	2	1	no	—
124	C₂×#124	—	2	1	no	—
125	C₂²×C₁₆	C₂²×C₁₃	1	2	yes	—
126	#500	—	2	2	no	—
127	#500	C₂²×C₁₃	1	4	yes	complemented
128	#495	—	2	2	no	—
129	#495	C₂²×C₁₃	1	4	yes	—
130	C₂×#107	—	2	1	no	—
131	C₂×#107	C₂²×C₁₃	1	2	yes	—
132	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
133	#358	C₂²×C₁₃	1	8	yes	complemented
134	#357	C₂²×C₁₃	1	8	yes	—
135	#346	C₂²×C₁₃	1	4	yes	complemented
136	C₂²×C₁₃	—	8	2	no	—
137	C₂²×C₁₃	—	8	2	no	—
138	C₂²×C₁₃	—	8	2	no	—
139	C₂²×C₁₃	—	16	4	no	—
140	C₂²×C₁₃	#486	1	2	yes	—
141	C₂²×C₁₃	#509	1	2	yes	—
142	C₂²×C₁₃	#510	1	2	yes	—
143	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
144	C₄×C₁₃	—	2	2	no	—
145	C₄×C₁₃	—	2	2	no	—
146	C₄×C₁₃	—	8	2	no	—
147	C₄×C₁₃	—	8	4	no	—
148	C₄×C₁₃	#486	1	2	yes	—
149	C₄×C₁₃	C₂×#124	1	1	yes	characteristic
150	C₄×C₁₃	C₂²×SD₁₆	1	1	yes	characteristic
151	C₂²×Q₈	—	2	1	no	—
152	C₂²×Q₈	—	4	1	no	—
153	C₂²×D₄	—	2	1	no	—
154	C₂×Q₁₆	—	2	1	no	—
155	C₂×Q₁₆	—	2	4	no	—
156	C₂×Q₁₆	C₂³×C₁₃	1	2	yes	—
157	C₂×SD₁₆	—	2	4	no	—
158	C₂×SD₁₆	—	4	4	no	—
159	C₂×D₈	—	2	1	no	—
160	C₂×D₈	—	2	4	no	—
161	C₂×D₈	C₂³×C₁₃	1	2	yes	—
162	C₂²×C₈	D₄×C₁₃	1	1	yes	characteristic
163	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
164	C₄⋊Q₈	—	2	4	no	—
165	C₄⋊Q₈	—	4	2	no	—
166	#124	—	2	4	no	—
167	#124	—	4	2	no	—
168	C₂×C₄⋊C₄	—	2	2	no	—
169	C₂×C₄⋊C₄	—	4	1	no	—
170	C₂×C₂²⋊C₄	—	4	1	no	—
171	C₂×C₂²⋊C₄	—	4	1	no	—
172	C₂×C₄²	D₄×C₁₃	1	1	yes	characteristic
173	C₂×C₁₆	—	2	4	no	—
174	C₂×C₁₆	C₂³×C₁₃	1	4	yes	—
175	#107	—	2	4	no	—
176	#107	—	2	2	no	—
177	#107	C₂³×C₁₃	1	4	yes	—
178	C₄×C₈	D₄×C₁₃	1	2	yes	—
179	C₄×C₈	C₂³×C₁₃	1	2	yes	—
180	C₂×C₁₃	—	16	2	no	—
181	C₂×C₁₃	#2243	1	4	yes	direct product factor
182	C₂×C₁₃	C₂×#486	1	1	yes	characteristic
183	C₂×C₁₃	C₂×#509	1	1	yes	characteristic
184	C₂×C₁₃	C₂×#510	1	1	yes	characteristic
185	C₂⁴	—	4	1	no	—
186	C₂×Q₈	—	2	2	no	—
187	C₂×Q₈	—	4	1	no	—
188	C₂×Q₈	—	4	4	no	—
189	C₂×Q₈	—	4	2	no	—
190	C₂×Q₈	—	4	2	no	—
191	C₂×Q₈	—	8	4	no	—
192	C₂×D₄	—	2	2	no	—
193	C₂×D₄	—	4	2	no	—
194	C₂×D₄	—	4	1	no	—
195	C₂×D₄	—	4	2	no	—
196	C₂²×C₄	—	4	1	no	—
197	C₂²×C₄	—	4	2	no	—
198	C₂²×C₄	C₁₃×D₈	1	1	yes	characteristic
199	C₂²×C₄	C₁₃×D₈	1	1	yes	characteristic
200	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
201	Q₁₆	—	2	4	no	—
202	Q₁₆	—	4	2	no	—
203	SD₁₆	—	4	8	no	—
204	D₈	—	2	4	no	—
205	D₈	—	4	2	no	—
206	C₂×C₈	C₁₃×SD₁₆	1	4	yes	—
207	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
208	C₂×C₈	C₂×D₄×C₁₃	1	2	yes	—
209	C₂×C₈	C₁₃×#41	1	4	yes	—
210	C₂×C₈	C₂⁴×C₁₃	1	1	yes	characteristic
211	C₄⋊C₄	—	2	4	no	—
212	C₄⋊C₄	—	4	2	no	—
213	C₄⋊C₄	—	4	4	no	—
214	C₂²⋊C₄	—	4	4	no	—
215	C₂²⋊C₄	—	4	4	no	—
216	C₄²	C₁₃×D₈	1	2	yes	—
217	C₄²	C₂×D₄×C₁₃	1	2	yes	—
218	C₁₆	—	2	4	no	—
219	C₁₃	C₂×#2243	1	1	yes	characteristic, direct product factor, Sylow
220	C₂³	—	4	2	no	—
221	C₂³	—	8	2	no	—
222	C₂³	—	8	2	no	—
223	C₂³	—	8	2	no	—
224	C₂³	C₂×C₁₃×D₈	1	1	yes	characteristic
225	Q₈	—	4	2	no	—
226	Q₈	—	4	2	no	—
227	Q₈	—	8	8	no	—
228	Q₈	—	8	2	no	—
229	D₄	—	4	2	no	—
230	D₄	—	4	2	no	—
231	D₄	—	8	2	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	4	2	no	—
235	C₂×C₄	—	4	4	no	—
236	C₂×C₄	—	8	2	no	—
237	C₂×C₄	—	8	4	no	—
238	C₂×C₄	C₁₃×#124	1	2	yes	—
239	C₂×C₄	C₂×C₁₃×D₈	1	2	yes	—
240	C₂×C₄	C₂×C₁₃×D₈	1	2	yes	—
241	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
242	C₂×C₄	C₂×C₁₃×SD₁₆	1	2	yes	—
243	C₂×C₄	C₂²×D₄×C₁₃	1	1	yes	characteristic
244	C₈	C₁₃×#124	1	2	yes	—
245	C₈	C₂×C₁₃×SD₁₆	1	4	yes	—
246	C₈	C₂×C₁₃×#41	1	2	yes	—
247	C₂²	—	8	2	no	—
248	C₂²	—	8	2	no	—
249	C₂²	—	8	2	no	—
250	C₂²	—	16	4	no	—
251	C₂²	C₁₃×#486	1	2	yes	—
252	C₂²	C₁₃×#509	1	2	yes	—
253	C₂²	C₁₃×#510	1	2	yes	—
254	C₂²	C₂²×C₁₃×D₈	1	1	yes	characteristic
255	C₄	—	2	2	no	—
256	C₄	—	2	2	no	—
257	C₄	—	8	2	no	—
258	C₄	—	8	4	no	—
259	C₄	C₁₃×#486	1	2	yes	—
260	C₄	C₂×C₁₃×#124	1	1	yes	characteristic
261	C₄	C₂²×C₁₃×SD₁₆	1	1	yes	characteristic
262	C₂	—	16	2	no	—
263	C₂	C₁₃×#2243	1	4	yes	direct product factor
264	C₂	C₂×C₁₃×#486	1	1	yes	characteristic
265	C₂	C₂×C₁₃×#509	1	1	yes	characteristic
266	C₂	C₂×C₁₃×#510	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₁₃	C₂×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₁₃
prime radical	C₂²×C₁₃×D₈	C₂²
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×#500	C₂	semidirect
3. C₁₃×#2243	C₂	direct (N × H)
4. C₂×C₁₃×#358	C₂	semidirect
6. C₂×C₁₃×#346	C₂	semidirect
14. C₁₃×#500	C₂²	semidirect
20. C₁₃×#358	C₂²	semidirect
22. C₁₃×#346	C₂²	semidirect
52. C₂×#2243	C₁₃	direct (N × H)
87. C₂×#500	C₂×C₁₃	semidirect
89. #2243	C₂×C₁₃	direct (N × H)
90. C₂×#358	C₂×C₁₃	semidirect
92. C₂×#346	C₂×C₁₃	semidirect
127. #500	C₂²×C₁₃	semidirect
133. #358	C₂²×C₁₃	semidirect
135. #346	C₂²×C₁₃	semidirect
181. C₂×C₁₃	#2243	direct (N × H)
219. C₁₃	C₂×#2243	direct (N × H)
263. C₂	C₁₃×#2243	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×#500	2
C₂×C₁₃×#495	2
C₁₃×#2243	2
C₂×C₁₃×#358	2
C₂×C₁₃×#357	2
C₂×C₁₃×#346	2
C₂×#2243	13

Subgroups — C2×C13×#2243 #15720037

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₁₃×#2243 #15720037