Subgroups — C₂×C₁₃×#2238 #15719993

Order 3328.

604 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₃×#469	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₃×#430	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₃×#2238	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₁₃×#365	C₂	1	2	yes	maximal, complemented
5	C₂×C₁₃×#361	C₂	1	2	yes	maximal, complemented
6	C₂×C₁₃×#349	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×C₁₃×D₈	C₂²	1	1	yes	characteristic
8	C₂²×C₁₃×M₁₆	—	2	1	no	—
9	C₂²×C₁₃×M₁₆	C₂²	1	1	yes	characteristic
10	C₂×C₁₃×#121	—	2	1	no	—
11	C₂×C₁₃×M₃₂	C₂²	1	2	yes	—
12	C₁₃×#469	—	2	2	no	—
13	C₁₃×#469	C₂²	1	4	yes	—
14	C₁₃×#430	—	2	2	no	—
15	C₁₃×#430	C₂²	1	4	yes	complemented
16	C₂×C₁₃×#108	—	2	1	no	—
17	C₂×C₁₃×#108	C₂²	1	2	yes	—
18	C₂×C₁₃×#106	C₂²	1	1	yes	characteristic
19	C₂×C₁₃×#102	—	2	1	no	—
20	C₁₃×#365	C₂²	1	8	yes	complemented
21	C₁₃×#361	C₂²	1	8	yes	complemented
22	C₁₃×#349	C₂²	1	4	yes	complemented
23	C₂²×D₄×C₁₃	—	2	1	no	—
24	C₂²×D₄×C₁₃	—	4	1	no	—
25	C₂³×C₄×C₁₃	D₄	1	1	yes	characteristic
26	C₂×C₁₃×D₈	—	2	1	no	—
27	C₂×C₁₃×D₈	—	2	2	no	—
28	C₂×C₁₃×D₈	—	2	2	no	—
29	C₂×C₁₃×D₈	C₂³	1	2	yes	—
30	C₂×C₁₃×M₁₆	—	2	2	no	—
31	C₂×C₁₃×M₁₆	—	2	2	no	—
32	C₂×C₁₃×M₁₆	—	2	4	no	—
33	C₂×C₁₃×M₁₆	—	2	2	no	—
34	C₂×C₁₃×M₁₆	—	4	2	no	—
35	C₂×C₁₃×M₁₆	—	4	1	no	—
36	C₂×C₁₃×M₁₆	D₄	1	2	yes	—
37	C₂×C₁₃×M₁₆	C₂³	1	2	yes	—
38	C₂²×C₈×C₁₃	—	4	1	no	—
39	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
40	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
41	C₁₃×#121	—	2	4	no	—
42	C₁₃×#121	—	4	2	no	—
43	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
44	C₂×C₁₃×C₂²⋊C₄	—	4	1	no	—
45	C₁₃×M₃₂	—	2	2	no	—
46	C₁₃×M₃₂	C₂³	1	4	yes	—
47	C₂×C₁₃×C₁₆	—	2	2	no	—
48	C₁₃×#108	—	2	4	no	—
49	C₁₃×#108	—	2	2	no	—
50	C₁₃×#108	C₂³	1	4	yes	—
51	C₁₃×#106	—	2	1	no	—
52	C₁₃×#106	C₂³	1	2	yes	—
53	C₁₃×#102	—	2	4	no	—
54	C₂×#2238	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
55	C₂⁴×C₁₃	—	4	1	no	—
56	C₂⁴×C₁₃	SD₁₆	1	1	yes	characteristic
57	C₂×D₄×C₁₃	—	2	2	no	—
58	C₂×D₄×C₁₃	—	4	4	no	—
59	C₂×D₄×C₁₃	—	4	1	no	—
60	C₂×D₄×C₁₃	—	4	1	no	—
61	C₂×D₄×C₁₃	—	4	1	no	—
62	C₂×D₄×C₁₃	—	4	1	no	—
63	C₂×D₄×C₁₃	—	4	1	no	—
64	C₂×D₄×C₁₃	—	8	4	no	—
65	C₂²×C₄×C₁₃	—	2	1	no	—
66	C₂²×C₄×C₁₃	—	2	1	no	—
67	C₂²×C₄×C₁₃	—	2	1	no	—
68	C₂²×C₄×C₁₃	—	2	1	no	—
69	C₂²×C₄×C₁₃	—	4	1	no	—
70	C₂²×C₄×C₁₃	SD₁₆	1	1	yes	characteristic
71	C₂²×C₄×C₁₃	SD₁₆	1	2	yes	—
72	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
73	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
74	C₁₃×D₈	—	2	4	no	—
75	C₁₃×D₈	—	4	2	no	—
76	C₁₃×M₁₆	—	2	4	no	—
77	C₁₃×M₁₆	—	2	4	no	—
78	C₁₃×M₁₆	—	2	4	no	—
79	C₁₃×M₁₆	—	4	4	no	—
80	C₁₃×M₁₆	—	4	2	no	—
81	C₂×C₈×C₁₃	—	2	1	no	—
82	C₂×C₈×C₁₃	—	2	1	no	—
83	C₂×C₈×C₁₃	—	4	4	no	—
84	C₂×C₈×C₁₃	—	4	2	no	—
85	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
86	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
87	C₂×C₈×C₁₃	#41	1	2	yes	—
88	C₂×C₈×C₁₃	#41	1	2	yes	—
89	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
90	C₁₃×C₄⋊C₄	—	2	2	no	—
91	C₁₃×C₄⋊C₄	—	4	1	no	—
92	C₁₃×C₂²⋊C₄	—	4	4	no	—
93	C₁₃×C₁₆	—	2	4	no	—
94	C₂×#469	C₂×C₁₃	1	1	yes	characteristic
95	C₂×#430	C₂×C₁₃	1	1	yes	characteristic, complemented
96	#2238	C₂×C₁₃	1	8	yes	direct product factor
97	C₂×#365	C₂×C₁₃	1	2	yes	complemented
98	C₂×#361	C₂×C₁₃	1	2	yes	complemented
99	C₂×#349	C₂×C₁₃	1	1	yes	characteristic, complemented
100	C₂³×C₁₃	—	2	1	no	—
101	C₂³×C₁₃	—	2	1	no	—
102	C₂³×C₁₃	—	4	2	no	—
103	C₂³×C₁₃	—	4	2	no	—
104	C₂³×C₁₃	—	8	2	no	—
105	C₂³×C₁₃	—	8	2	no	—
106	C₂³×C₁₃	—	8	1	no	—
107	C₂³×C₁₃	—	8	1	no	—
108	C₂³×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
109	C₂³×C₁₃	C₂×SD₁₆	1	2	yes	—
110	D₄×C₁₃	—	4	2	no	—
111	D₄×C₁₃	—	4	2	no	—
112	D₄×C₁₃	—	8	8	no	—
113	D₄×C₁₃	—	8	1	no	—
114	D₄×C₁₃	—	8	1	no	—
115	C₂×C₄×C₁₃	—	2	1	no	—
116	C₂×C₄×C₁₃	—	2	1	no	—
117	C₂×C₄×C₁₃	—	2	2	no	—
118	C₂×C₄×C₁₃	—	2	2	no	—
119	C₂×C₄×C₁₃	—	2	2	no	—
120	C₂×C₄×C₁₃	—	2	2	no	—
121	C₂×C₄×C₁₃	—	4	2	no	—
122	C₂×C₄×C₁₃	—	8	2	no	—
123	C₂×C₄×C₁₃	#123	1	1	yes	characteristic
124	C₂×C₄×C₁₃	#123	1	1	yes	characteristic
125	C₂×C₄×C₁₃	C₂×SD₁₆	1	2	yes	—
126	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
127	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
128	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
129	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
130	C₈×C₁₃	—	2	2	no	—
131	C₈×C₁₃	—	4	4	no	—
132	C₈×C₁₃	#123	1	2	yes	—
133	C₈×C₁₃	C₂×#41	1	2	yes	—
134	C₂²×D₈	C₂²×C₁₃	1	1	yes	characteristic
135	C₂²×M₁₆	—	2	1	no	—
136	C₂²×M₁₆	C₂²×C₁₃	1	1	yes	characteristic
137	C₂×#121	—	2	1	no	—
138	C₂×M₃₂	C₂²×C₁₃	1	2	yes	—
139	#469	—	2	2	no	—
140	#469	C₂²×C₁₃	1	4	yes	—
141	#430	—	2	2	no	—
142	#430	C₂²×C₁₃	1	4	yes	complemented
143	C₂×#108	—	2	1	no	—
144	C₂×#108	C₂²×C₁₃	1	2	yes	—
145	C₂×#106	C₂²×C₁₃	1	1	yes	characteristic
146	C₂×#102	—	2	1	no	—
147	#365	C₂²×C₁₃	1	8	yes	complemented
148	#361	C₂²×C₁₃	1	8	yes	complemented
149	#349	C₂²×C₁₃	1	4	yes	complemented
150	C₂²×C₁₃	—	2	2	no	—
151	C₂²×C₁₃	—	2	2	no	—
152	C₂²×C₁₃	—	4	2	no	—
153	C₂²×C₁₃	—	4	2	no	—
154	C₂²×C₁₃	—	8	1	no	—
155	C₂²×C₁₃	—	8	2	no	—
156	C₂²×C₁₃	—	8	2	no	—
157	C₂²×C₁₃	—	8	2	no	—
158	C₂²×C₁₃	—	16	2	no	—
159	C₂²×C₁₃	—	16	2	no	—
160	C₂²×C₁₃	#481	1	1	yes	characteristic
161	C₂²×C₁₃	#481	1	1	yes	characteristic
162	C₂²×C₁₃	C₂²×SD₁₆	1	1	yes	characteristic
163	C₄×C₁₃	—	2	2	no	—
164	C₄×C₁₃	—	8	2	no	—
165	C₄×C₁₃	#481	1	1	yes	characteristic
166	C₄×C₁₃	#481	1	1	yes	characteristic
167	C₄×C₁₃	C₂×#123	1	1	yes	characteristic
168	C₄×C₁₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
169	C₂²×D₄	—	2	1	no	—
170	C₂²×D₄	—	4	1	no	—
171	C₂³×C₄	D₄×C₁₃	1	1	yes	characteristic
172	C₂×D₈	—	2	1	no	—
173	C₂×D₈	—	2	2	no	—
174	C₂×D₈	—	2	2	no	—
175	C₂×D₈	C₂³×C₁₃	1	2	yes	—
176	C₂×M₁₆	—	2	2	no	—
177	C₂×M₁₆	—	2	2	no	—
178	C₂×M₁₆	—	2	2	no	—
179	C₂×M₁₆	—	2	4	no	—
180	C₂×M₁₆	—	4	2	no	—
181	C₂×M₁₆	—	4	1	no	—
182	C₂×M₁₆	D₄×C₁₃	1	2	yes	—
183	C₂×M₁₆	C₂³×C₁₃	1	2	yes	—
184	C₂²×C₈	—	4	1	no	—
185	C₂²×C₈	D₄×C₁₃	1	1	yes	characteristic
186	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
187	#121	—	2	4	no	—
188	#121	—	4	2	no	—
189	C₂×C₄⋊C₄	—	2	1	no	—
190	C₂×C₂²⋊C₄	—	4	1	no	—
191	M₃₂	—	2	2	no	—
192	M₃₂	C₂³×C₁₃	1	4	yes	—
193	C₂×C₁₆	—	2	2	no	—
194	#108	—	2	4	no	—
195	#108	—	2	2	no	—
196	#108	C₂³×C₁₃	1	4	yes	—
197	#106	—	2	1	no	—
198	#106	C₂³×C₁₃	1	2	yes	—
199	#102	—	2	4	no	—
200	C₂×C₁₃	—	2	1	no	—
201	C₂×C₁₃	—	2	1	no	—
202	C₂×C₁₃	—	4	2	no	—
203	C₂×C₁₃	—	16	2	no	—
204	C₂×C₁₃	#2238	1	2	yes	direct product factor
205	C₂×C₁₃	C₂×#481	1	1	yes	characteristic
206	C₂⁴	—	4	1	no	—
207	C₂⁴	C₁₃×SD₁₆	1	1	yes	characteristic
208	C₂×D₄	—	2	2	no	—
209	C₂×D₄	—	4	4	no	—
210	C₂×D₄	—	4	1	no	—
211	C₂×D₄	—	4	1	no	—
212	C₂×D₄	—	4	1	no	—
213	C₂×D₄	—	4	1	no	—
214	C₂×D₄	—	4	1	no	—
215	C₂×D₄	—	8	4	no	—
216	C₂²×C₄	—	2	1	no	—
217	C₂²×C₄	—	2	1	no	—
218	C₂²×C₄	—	2	1	no	—
219	C₂²×C₄	—	2	1	no	—
220	C₂²×C₄	—	4	1	no	—
221	C₂²×C₄	C₁₃×SD₁₆	1	1	yes	characteristic
222	C₂²×C₄	C₁₃×SD₁₆	1	2	yes	—
223	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
224	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
225	D₈	—	2	4	no	—
226	D₈	—	4	2	no	—
227	M₁₆	—	2	4	no	—
228	M₁₆	—	2	4	no	—
229	M₁₆	—	2	4	no	—
230	M₁₆	—	4	4	no	—
231	M₁₆	—	4	2	no	—
232	C₂×C₈	—	2	1	no	—
233	C₂×C₈	—	2	1	no	—
234	C₂×C₈	—	4	4	no	—
235	C₂×C₈	—	4	2	no	—
236	C₂×C₈	C₂×D₄×C₁₃	1	2	yes	—
237	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
238	C₂×C₈	C₁₃×#41	1	2	yes	—
239	C₂×C₈	C₁₃×#41	1	2	yes	—
240	C₂×C₈	C₂⁴×C₁₃	1	1	yes	characteristic
241	C₄⋊C₄	—	2	2	no	—
242	C₄⋊C₄	—	4	1	no	—
243	C₂²⋊C₄	—	4	4	no	—
244	C₁₆	—	2	4	no	—
245	C₁₃	C₂×#2238	1	1	yes	characteristic, direct product factor, Sylow
246	C₂³	—	2	1	no	—
247	C₂³	—	2	1	no	—
248	C₂³	—	4	2	no	—
249	C₂³	—	4	2	no	—
250	C₂³	—	8	2	no	—
251	C₂³	—	8	2	no	—
252	C₂³	—	8	1	no	—
253	C₂³	—	8	1	no	—
254	C₂³	C₂×C₁₃×SD₁₆	1	1	yes	characteristic
255	C₂³	C₂×C₁₃×SD₁₆	1	2	yes	—
256	D₄	—	4	2	no	—
257	D₄	—	4	2	no	—
258	D₄	—	8	8	no	—
259	D₄	—	8	1	no	—
260	D₄	—	8	1	no	—
261	C₂×C₄	—	2	1	no	—
262	C₂×C₄	—	2	1	no	—
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	—	2	2	no	—
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	—	4	2	no	—
268	C₂×C₄	—	8	2	no	—
269	C₂×C₄	C₁₃×#123	1	1	yes	characteristic
270	C₂×C₄	C₁₃×#123	1	1	yes	characteristic
271	C₂×C₄	C₂×C₁₃×SD₁₆	1	2	yes	—
272	C₂×C₄	C₂×C₁₃×SD₁₆	1	1	yes	characteristic
273	C₂×C₄	C₁₃×AGL(1,ℤ/8)	1	1	yes	characteristic
274	C₂×C₄	C₁₃×AGL(1,ℤ/8)	1	1	yes	characteristic
275	C₂×C₄	C₂²×D₄×C₁₃	1	1	yes	characteristic
276	C₈	—	2	2	no	—
277	C₈	—	4	4	no	—
278	C₈	C₁₃×#123	1	2	yes	—
279	C₈	C₂×C₁₃×#41	1	2	yes	—
280	C₂²	—	2	2	no	—
281	C₂²	—	2	2	no	—
282	C₂²	—	4	2	no	—
283	C₂²	—	4	2	no	—
284	C₂²	—	8	1	no	—
285	C₂²	—	8	2	no	—
286	C₂²	—	8	2	no	—
287	C₂²	—	8	2	no	—
288	C₂²	—	16	2	no	—
289	C₂²	—	16	2	no	—
290	C₂²	C₁₃×#481	1	1	yes	characteristic
291	C₂²	C₁₃×#481	1	1	yes	characteristic
292	C₂²	C₂²×C₁₃×SD₁₆	1	1	yes	characteristic
293	C₄	—	2	2	no	—
294	C₄	—	8	2	no	—
295	C₄	C₁₃×#481	1	1	yes	characteristic
296	C₄	C₁₃×#481	1	1	yes	characteristic
297	C₄	C₂×C₁₃×#123	1	1	yes	characteristic
298	C₄	C₂×C₁₃×AGL(1,ℤ/8)	1	1	yes	characteristic
299	C₂	—	2	1	no	—
300	C₂	—	2	1	no	—
301	C₂	—	4	2	no	—
302	C₂	—	16	2	no	—
303	C₂	C₁₃×#2238	1	2	yes	direct product factor
304	C₂	C₂×C₁₃×#481	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₃	#481
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₁₃
prime radical	C₂×C₁₃×#469	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃ ◁ C₂²×C₄×C₁₃ ◁ C₂²×C₁₃×M₁₆ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₁₃×#430	C₂	semidirect
3. C₁₃×#2238	C₂	direct (N × H)
4. C₂×C₁₃×#365	C₂	semidirect
5. C₂×C₁₃×#361	C₂	semidirect
6. C₂×C₁₃×#349	C₂	semidirect
15. C₁₃×#430	C₂²	semidirect
20. C₁₃×#365	C₂²	semidirect
21. C₁₃×#361	C₂²	semidirect
22. C₁₃×#349	C₂²	semidirect
54. C₂×#2238	C₁₃	direct (N × H)
95. C₂×#430	C₂×C₁₃	semidirect
96. #2238	C₂×C₁₃	direct (N × H)
97. C₂×#365	C₂×C₁₃	semidirect
98. C₂×#361	C₂×C₁₃	semidirect
99. C₂×#349	C₂×C₁₃	semidirect
142. #430	C₂²×C₁₃	semidirect
147. #365	C₂²×C₁₃	semidirect
148. #361	C₂²×C₁₃	semidirect
149. #349	C₂²×C₁₃	semidirect
204. C₂×C₁₃	#2238	direct (N × H)
245. C₁₃	C₂×#2238	direct (N × H)
303. C₂	C₁₃×#2238	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×#469	2
C₂×C₁₃×#430	2
C₁₃×#2238	2
C₂×C₁₃×#365	2
C₂×C₁₃×#361	2
C₂×C₁₃×#349	2
C₂×#2238	13

Subgroups — C2×C13×#2238 #15719993

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₁₃×#2238 #15719993