Subgroups — C₂×C₁₃×#2250 #15720053

Order 3328.

930 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₁₃×SD₃₂	C₂	1	2	yes	maximal, complemented
2	C₂²×C₁₃×D₁₆	C₂	1	2	yes	maximal, complemented
3	C₂×C₁₃×#495	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₁₃×#493	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₃×#2250	C₂	1	8	yes	maximal, direct product factor
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	—	2	1	no	—
10	C₂²×C₁₃×D₈	C₂²	1	2	yes	—
11	C₂²×C₁₃×D₈	C₂²	1	1	yes	characteristic
12	C₂×C₁₃×#127	—	2	1	no	—
13	C₂×C₁₃×#124	—	2	1	no	—
14	C₂×C₁₃×SD₃₂	—	2	8	no	—
15	C₂×C₁₃×SD₃₂	C₂²	1	8	yes	complemented
16	C₂×C₁₃×D₁₆	—	2	8	no	—
17	C₂×C₁₃×D₁₆	C₂²	1	8	yes	complemented
18	C₂²×C₁₃×C₁₆	C₂²	1	2	yes	—
19	C₁₃×#495	—	2	2	no	—
20	C₁₃×#495	C₂²	1	4	yes	complemented
21	C₁₃×#493	—	2	2	no	—
22	C₁₃×#493	C₂²	1	4	yes	complemented
23	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
24	C₁₃×#346	C₂²	1	4	yes	complemented
25	C₂²×Q₈×C₁₃	—	2	1	no	—
26	C₂²×D₄×C₁₃	—	2	1	no	—
27	C₂²×D₄×C₁₃	—	2	2	no	—
28	C₂²×D₄×C₁₃	—	4	2	no	—
29	C₂×C₁₃×Q₁₆	—	2	4	no	—
30	C₂×C₁₃×Q₁₆	—	2	1	no	—
31	C₂×C₁₃×Q₁₆	C₂³	1	2	yes	—
32	C₂×C₁₃×SD₁₆	—	2	4	no	—
33	C₂×C₁₃×SD₁₆	—	4	4	no	—
34	C₂×C₁₃×D₈	—	2	4	no	—
35	C₂×C₁₃×D₈	—	2	4	no	—
36	C₂×C₁₃×D₈	—	2	4	no	—
37	C₂×C₁₃×D₈	—	2	2	no	—
38	C₂×C₁₃×D₈	—	2	4	no	—
39	C₂×C₁₃×D₈	—	2	1	no	—
40	C₂×C₁₃×D₈	—	4	4	no	—
41	C₂×C₁₃×D₈	C₂³	1	4	yes	—
42	C₂×C₁₃×D₈	C₂³	1	2	yes	—
43	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
44	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
45	C₁₃×#127	—	2	4	no	—
46	C₁₃×#127	—	4	2	no	—
47	C₁₃×#124	—	2	4	no	—
48	C₁₃×#124	—	4	2	no	—
49	C₂×C₁₃×C₂²⋊C₄	—	4	2	no	—
50	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
51	C₁₃×SD₃₂	—	2	16	no	—
52	C₁₃×D₁₆	—	2	16	no	—
53	C₂×C₁₃×C₁₆	D₄	1	8	yes	complemented
54	C₂×C₁₃×C₁₆	C₂³	1	4	yes	—
55	C₄×C₈×C₁₃	D₄	1	2	yes	—
56	C₄×C₈×C₁₃	C₂³	1	2	yes	—
57	C₂×#2250	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
58	C₂⁴×C₁₃	—	4	1	no	—
59	C₂⁴×C₁₃	—	4	2	no	—
60	C₂×Q₈×C₁₃	—	2	2	no	—
61	C₂×Q₈×C₁₃	—	4	1	no	—
62	C₂×Q₈×C₁₃	—	4	2	no	—
63	C₂×Q₈×C₁₃	—	4	2	no	—
64	C₂×D₄×C₁₃	—	2	2	no	—
65	C₂×D₄×C₁₃	—	2	4	no	—
66	C₂×D₄×C₁₃	—	4	2	no	—
67	C₂×D₄×C₁₃	—	4	4	no	—
68	C₂×D₄×C₁₃	—	4	4	no	—
69	C₂×D₄×C₁₃	—	4	2	no	—
70	C₂×D₄×C₁₃	—	4	1	no	—
71	C₂×D₄×C₁₃	—	4	8	no	—
72	C₂×D₄×C₁₃	—	4	2	no	—
73	C₂×D₄×C₁₃	—	8	8	no	—
74	C₂²×C₄×C₁₃	—	4	1	no	—
75	C₂²×C₄×C₁₃	D₈	1	2	yes	—
76	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
77	C₁₃×Q₁₆	—	2	4	no	—
78	C₁₃×Q₁₆	—	4	2	no	—
79	C₁₃×SD₁₆	—	4	8	no	—
80	C₁₃×D₈	—	2	4	no	—
81	C₁₃×D₈	—	2	4	no	—
82	C₁₃×D₈	—	2	4	no	—
83	C₁₃×D₈	—	4	2	no	—
84	C₁₃×D₈	—	4	4	no	—
85	C₁₃×D₈	—	4	8	no	—
86	C₂×C₈×C₁₃	D₈	1	4	yes	—
87	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
88	C₂×C₈×C₁₃	C₂×D₄	1	4	yes	—
89	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
90	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
91	C₁₃×C₂²⋊C₄	—	4	8	no	—
92	C₄²×C₁₃	D₈	1	2	yes	—
93	C₄²×C₁₃	C₂×D₄	1	2	yes	—
94	C₁₃×C₁₆	C₂×D₄	1	8	yes	complemented
95	C₂²×SD₃₂	C₂×C₁₃	1	2	yes	complemented
96	C₂²×D₁₆	C₂×C₁₃	1	2	yes	complemented
97	C₂×#495	C₂×C₁₃	1	1	yes	characteristic, complemented
98	C₂×#493	C₂×C₁₃	1	1	yes	characteristic, complemented
99	#2250	C₂×C₁₃	1	8	yes	direct product factor
100	C₂×#346	C₂×C₁₃	1	1	yes	characteristic, complemented
101	C₂³×C₁₃	—	4	2	no	—
102	C₂³×C₁₃	—	4	4	no	—
103	C₂³×C₁₃	—	8	4	no	—
104	C₂³×C₁₃	—	8	2	no	—
105	C₂³×C₁₃	—	8	8	no	—
106	C₂³×C₁₃	—	8	4	no	—
107	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
108	Q₈×C₁₃	—	4	2	no	—
109	Q₈×C₁₃	—	4	2	no	—
110	Q₈×C₁₃	—	8	2	no	—
111	D₄×C₁₃	—	4	2	no	—
112	D₄×C₁₃	—	4	4	no	—
113	D₄×C₁₃	—	4	4	no	—
114	D₄×C₁₃	—	4	2	no	—
115	D₄×C₁₃	—	8	16	no	—
116	D₄×C₁₃	—	8	4	no	—
117	D₄×C₁₃	—	8	2	no	—
118	C₂×C₄×C₁₃	—	2	4	no	—
119	C₂×C₄×C₁₃	—	4	2	no	—
120	C₂×C₄×C₁₃	—	8	2	no	—
121	C₂×C₄×C₁₃	#127	1	2	yes	—
122	C₂×C₄×C₁₃	C₂×D₈	1	4	yes	—
123	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
124	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
125	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
126	C₈×C₁₃	#127	1	2	yes	—
127	C₈×C₁₃	C₂×D₈	1	4	yes	—
128	C₈×C₁₃	C₂²×D₄	1	2	yes	—
129	C₂²×Q₁₆	C₂²×C₁₃	1	1	yes	characteristic
130	C₂²×SD₁₆	—	2	1	no	—
131	C₂²×D₈	—	2	1	no	—
132	C₂²×D₈	C₂²×C₁₃	1	2	yes	—
133	C₂²×D₈	C₂²×C₁₃	1	1	yes	characteristic
134	C₂×#127	—	2	1	no	—
135	C₂×#124	—	2	1	no	—
136	C₂×SD₃₂	—	2	8	no	—
137	C₂×SD₃₂	C₂²×C₁₃	1	8	yes	complemented
138	C₂×D₁₆	—	2	8	no	—
139	C₂×D₁₆	C₂²×C₁₃	1	8	yes	complemented
140	C₂²×C₁₆	C₂²×C₁₃	1	2	yes	—
141	#495	—	2	2	no	—
142	#495	C₂²×C₁₃	1	4	yes	complemented
143	#493	—	2	2	no	—
144	#493	C₂²×C₁₃	1	4	yes	complemented
145	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
146	#346	C₂²×C₁₃	1	4	yes	complemented
147	C₂²×C₁₃	—	8	4	no	—
148	C₂²×C₁₃	—	8	2	no	—
149	C₂²×C₁₃	—	8	8	no	—
150	C₂²×C₁₃	—	8	4	no	—
151	C₂²×C₁₃	—	16	8	no	—
152	C₂²×C₁₃	—	16	4	no	—
153	C₂²×C₁₃	#493	1	2	yes	—
154	C₂²×C₁₃	#509	1	4	yes	—
155	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
156	C₄×C₁₃	—	2	4	no	—
157	C₄×C₁₃	—	8	2	no	—
158	C₄×C₁₃	#493	1	2	yes	—
159	C₄×C₁₃	C₂×#127	1	1	yes	characteristic
160	C₄×C₁₃	C₂²×D₈	1	1	yes	characteristic
161	C₂²×Q₈	—	2	1	no	—
162	C₂²×D₄	—	2	1	no	—
163	C₂²×D₄	—	2	2	no	—
164	C₂²×D₄	—	4	2	no	—
165	C₂×Q₁₆	—	2	4	no	—
166	C₂×Q₁₆	—	2	1	no	—
167	C₂×Q₁₆	C₂³×C₁₃	1	2	yes	—
168	C₂×SD₁₆	—	2	4	no	—
169	C₂×SD₁₆	—	4	4	no	—
170	C₂×D₈	—	2	4	no	—
171	C₂×D₈	—	2	4	no	—
172	C₂×D₈	—	2	4	no	—
173	C₂×D₈	—	2	4	no	—
174	C₂×D₈	—	2	2	no	—
175	C₂×D₈	—	2	1	no	—
176	C₂×D₈	—	4	4	no	—
177	C₂×D₈	C₂³×C₁₃	1	4	yes	—
178	C₂×D₈	C₂³×C₁₃	1	2	yes	—
179	C₂²×C₈	D₄×C₁₃	1	1	yes	characteristic
180	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
181	#127	—	2	4	no	—
182	#127	—	4	2	no	—
183	#124	—	2	4	no	—
184	#124	—	4	2	no	—
185	C₂×C₂²⋊C₄	—	4	2	no	—
186	C₂×C₄²	D₄×C₁₃	1	1	yes	characteristic
187	SD₃₂	—	2	16	no	—
188	D₁₆	—	2	16	no	—
189	C₂×C₁₆	D₄×C₁₃	1	8	yes	complemented
190	C₂×C₁₆	C₂³×C₁₃	1	4	yes	—
191	C₄×C₈	D₄×C₁₃	1	2	yes	—
192	C₄×C₈	C₂³×C₁₃	1	2	yes	—
193	C₂×C₁₃	—	16	4	no	—
194	C₂×C₁₃	—	16	2	no	—
195	C₂×C₁₃	#2250	1	4	yes	direct product factor
196	C₂×C₁₃	C₂×#493	1	1	yes	characteristic
197	C₂×C₁₃	C₂×#509	1	2	yes	—
198	C₂⁴	—	4	1	no	—
199	C₂⁴	—	4	2	no	—
200	C₂×Q₈	—	2	2	no	—
201	C₂×Q₈	—	4	1	no	—
202	C₂×Q₈	—	4	2	no	—
203	C₂×Q₈	—	4	2	no	—
204	C₂×D₄	—	2	2	no	—
205	C₂×D₄	—	2	4	no	—
206	C₂×D₄	—	4	2	no	—
207	C₂×D₄	—	4	4	no	—
208	C₂×D₄	—	4	4	no	—
209	C₂×D₄	—	4	2	no	—
210	C₂×D₄	—	4	1	no	—
211	C₂×D₄	—	4	8	no	—
212	C₂×D₄	—	4	2	no	—
213	C₂×D₄	—	8	8	no	—
214	C₂²×C₄	—	4	1	no	—
215	C₂²×C₄	C₁₃×D₈	1	2	yes	—
216	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
217	Q₁₆	—	2	4	no	—
218	Q₁₆	—	4	2	no	—
219	SD₁₆	—	4	8	no	—
220	D₈	—	2	4	no	—
221	D₈	—	2	4	no	—
222	D₈	—	2	4	no	—
223	D₈	—	4	2	no	—
224	D₈	—	4	4	no	—
225	D₈	—	4	8	no	—
226	C₂×C₈	C₁₃×D₈	1	4	yes	—
227	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
228	C₂×C₈	C₂×D₄×C₁₃	1	4	yes	—
229	C₂×C₈	C₂×D₄×C₁₃	1	2	yes	—
230	C₂×C₈	C₂⁴×C₁₃	1	1	yes	characteristic
231	C₂²⋊C₄	—	4	8	no	—
232	C₄²	C₁₃×D₈	1	2	yes	—
233	C₄²	C₂×D₄×C₁₃	1	2	yes	—
234	C₁₆	C₂×D₄×C₁₃	1	8	yes	complemented
235	C₁₃	C₂×#2250	1	1	yes	characteristic, direct product factor, Sylow
236	C₂³	—	4	2	no	—
237	C₂³	—	4	4	no	—
238	C₂³	—	8	4	no	—
239	C₂³	—	8	2	no	—
240	C₂³	—	8	8	no	—
241	C₂³	—	8	4	no	—
242	C₂³	C₂×C₁₃×D₈	1	1	yes	characteristic
243	Q₈	—	4	2	no	—
244	Q₈	—	4	2	no	—
245	Q₈	—	8	2	no	—
246	D₄	—	4	2	no	—
247	D₄	—	4	4	no	—
248	D₄	—	4	4	no	—
249	D₄	—	4	2	no	—
250	D₄	—	8	16	no	—
251	D₄	—	8	4	no	—
252	D₄	—	8	2	no	—
253	C₂×C₄	—	2	4	no	—
254	C₂×C₄	—	4	2	no	—
255	C₂×C₄	—	8	2	no	—
256	C₂×C₄	C₁₃×#127	1	2	yes	—
257	C₂×C₄	C₂×C₁₃×D₈	1	4	yes	—
258	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
259	C₂×C₄	C₂×C₁₃×D₈	1	2	yes	—
260	C₂×C₄	C₂²×D₄×C₁₃	1	1	yes	characteristic
261	C₈	C₁₃×#127	1	2	yes	—
262	C₈	C₂×C₁₃×D₈	1	4	yes	—
263	C₈	C₂²×D₄×C₁₃	1	2	yes	—
264	C₂²	—	8	4	no	—
265	C₂²	—	8	2	no	—
266	C₂²	—	8	8	no	—
267	C₂²	—	8	4	no	—
268	C₂²	—	16	8	no	—
269	C₂²	—	16	4	no	—
270	C₂²	C₁₃×#493	1	2	yes	—
271	C₂²	C₁₃×#509	1	4	yes	—
272	C₂²	C₂²×C₁₃×D₈	1	1	yes	characteristic
273	C₄	—	2	4	no	—
274	C₄	—	8	2	no	—
275	C₄	C₁₃×#493	1	2	yes	—
276	C₄	C₂×C₁₃×#127	1	1	yes	characteristic
277	C₄	C₂²×C₁₃×D₈	1	1	yes	characteristic
278	C₂	—	16	4	no	—
279	C₂	—	16	2	no	—
280	C₂	C₁₃×#2250	1	4	yes	direct product factor
281	C₂	C₂×C₁₃×#493	1	1	yes	characteristic
282	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 ◁ 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. C₂²×C₁₃×SD₃₂	C₂	semidirect
2. C₂²×C₁₃×D₁₆	C₂	semidirect
3. C₂×C₁₃×#495	C₂	semidirect
4. C₂×C₁₃×#493	C₂	semidirect
5. C₁₃×#2250	C₂	direct (N × H)
6. C₂×C₁₃×#346	C₂	semidirect
15. C₂×C₁₃×SD₃₂	C₂²	semidirect
17. C₂×C₁₃×D₁₆	C₂²	semidirect
20. C₁₃×#495	C₂²	semidirect
22. C₁₃×#493	C₂²	semidirect
24. C₁₃×#346	C₂²	semidirect
53. C₂×C₁₃×C₁₆	D₄	semidirect
57. C₂×#2250	C₁₃	direct (N × H)
94. C₁₃×C₁₆	C₂×D₄	semidirect
95. C₂²×SD₃₂	C₂×C₁₃	semidirect
96. C₂²×D₁₆	C₂×C₁₃	semidirect
97. C₂×#495	C₂×C₁₃	semidirect
98. C₂×#493	C₂×C₁₃	semidirect
99. #2250	C₂×C₁₃	direct (N × H)
100. C₂×#346	C₂×C₁₃	semidirect
137. C₂×SD₃₂	C₂²×C₁₃	semidirect
139. C₂×D₁₆	C₂²×C₁₃	semidirect
142. #495	C₂²×C₁₃	semidirect
144. #493	C₂²×C₁₃	semidirect
146. #346	C₂²×C₁₃	semidirect
189. C₂×C₁₆	D₄×C₁₃	semidirect
195. C₂×C₁₃	#2250	direct (N × H)
234. C₁₆	C₂×D₄×C₁₃	semidirect
235. C₁₃	C₂×#2250	direct (N × H)
280. C₂	C₁₃×#2250	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×SD₃₂	2
C₂²×C₁₃×D₁₆	2
C₂×C₁₃×#495	2
C₂×C₁₃×#493	2
C₁₃×#2250	2
C₂×C₁₃×#346	2
C₂×#2250	13

Subgroups — C2×C13×#2250 #15720053

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₁₃×#2250 #15720053