Subgroups — C₂×D₁₃×#347 #16015571

Order 3328.

782 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₁₃×M₃₂	C₂	1	2	yes	maximal
2	C₂×D₁₃×#97	C₂	1	1	yes	maximal, characteristic
3	C₂×#11718058	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#11718104	C₂	1	2	yes	maximal, complemented
5	D₁₃×#347	C₂	1	8	yes	maximal, direct product factor
6	C₂×C₁₃×#347	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×C₈×D₁₃	C₄	1	1	yes	characteristic
8	C₂²×C₈×D₁₃	C₂²	1	1	yes	characteristic
9	C₂×C₄²×D₁₃	C₄	1	1	yes	characteristic
10	C₂×C₁₃×M₃₂	C₂²	1	2	yes	—
11	C₂×C₁₃×#97	C₂²	1	1	yes	characteristic
12	C₂×#88055	C₄	1	1	yes	characteristic
13	C₂×#88055	C₂²	1	1	yes	characteristic
14	C₂×#88052	C₂²	1	2	yes	—
15	C₂×#88043	C₄	1	1	yes	characteristic
16	D₁₃×M₃₂	C₄	1	8	yes	complemented
17	D₁₃×M₃₂	C₂²	1	8	yes	—
18	C₂×#88038	C₄	1	4	yes	complemented
19	C₂×C₁₆×D₁₃	C₄	1	4	yes	complemented
20	D₁₃×#97	C₄	1	4	yes	—
21	D₁₃×#97	C₂²	1	4	yes	—
22	C₁₃×#347	C₂²	1	4	yes	complemented
23	#11718104	C₂²	1	8	yes	complemented
24	#11718058	C₂²	1	4	yes	complemented
25	C₂²×C₄×D₁₃	—	2	1	no	—
26	C₂²×C₄×D₁₃	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
29	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄×Dic₁₃	—	2	1	no	—
31	C₂×C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
32	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
33	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
34	C₂×C₈×D₁₃	—	2	1	no	—
35	C₂×C₈×D₁₃	—	2	2	no	—
36	C₂×C₈×D₁₃	—	2	1	no	—
37	C₂×C₈×D₁₃	C₂×C₄	1	4	yes	—
38	C₂×C₈×D₁₃	C₂×C₄	1	2	yes	—
39	C₂×C₈×D₁₃	C₂×C₄	1	2	yes	—
40	C₂×C₈×D₁₃	C₂×C₄	1	2	yes	—
41	C₂×C₈×D₁₃	C₂×C₄	1	4	yes	—
42	C₂×C₈×D₁₃	C₂³	1	2	yes	—
43	C₄²×D₁₃	—	2	2	no	—
44	C₄²×D₁₃	C₂×C₄	1	4	yes	—
45	C₁₃×M₃₂	C₂×C₄	1	4	yes	complemented
46	C₁₃×M₃₂	C₂³	1	4	yes	—
47	C₂×C₁₃×C₁₆	C₂×C₄	1	4	yes	complemented
48	C₁₃×#97	C₂×C₄	1	2	yes	—
49	C₁₃×#97	C₂³	1	2	yes	—
50	#88055	C₂×C₄	1	4	yes	—
51	#88055	C₂×C₄	1	2	yes	—
52	#88055	C₂³	1	2	yes	—
53	#88052	C₂×C₄	1	4	yes	complemented
54	#88052	C₂³	1	4	yes	—
55	C₂×#6090	C₂×C₄	1	4	yes	complemented
56	#88043	C₂×C₄	1	4	yes	—
57	#88038	C₂×C₄	1	16	yes	complemented
58	C₁₆×D₁₃	C₂×C₄	1	16	yes	complemented
59	C₂²×#347	—	13	1	no	maximal, Sylow
60	C₂³×D₁₃	M₁₆	1	1	yes	characteristic
61	C₂²×C₄×C₁₃	—	2	1	no	—
62	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
63	C₂²×Dic₁₃	—	2	1	no	—
64	C₂²×Dic₁₃	M₁₆	1	1	yes	characteristic
65	C₂×C₄×D₁₃	—	2	4	no	—
66	C₂×C₄×D₁₃	—	4	4	no	—
67	C₂×C₄×D₁₃	C₄²	1	2	yes	—
68	C₂×C₄×D₁₃	C₄²	1	1	yes	characteristic
69	C₂×C₄×D₁₃	C₄²	1	1	yes	characteristic
70	C₂×C₄×D₁₃	M₁₆	1	2	yes	—
71	C₂×C₄×D₁₃	M₁₆	1	1	yes	characteristic
72	C₂×C₄×D₁₃	M₁₆	1	2	yes	—
73	C₂×C₄×D₁₃	M₁₆	1	1	yes	characteristic
74	C₂×C₄×D₁₃	C₂²×C₄	1	2	yes	—
75	C₂×C₈×C₁₃	—	2	1	no	—
76	C₂×C₈×C₁₃	—	2	1	no	—
77	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
78	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
80	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
81	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
82	C₄²×C₁₃	—	2	1	no	—
83	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
84	C₄×Dic₁₃	—	2	1	no	—
85	C₄×Dic₁₃	—	2	4	no	—
86	C₄×Dic₁₃	C₂²×C₄	1	2	yes	—
87	C₂×#1055	—	2	1	no	—
88	C₂×#1055	—	2	1	no	—
89	C₂×#1055	C₄²	1	2	yes	—
90	C₂×#1055	C₄²	1	2	yes	—
91	C₂×#1055	C₂²×C₄	1	2	yes	—
92	C₂×#1055	C₂²×C₄	1	2	yes	—
93	C₈×D₁₃	—	2	4	no	—
94	C₈×D₁₃	—	2	4	no	—
95	C₈×D₁₃	C₄²	1	4	yes	—
96	C₈×D₁₃	C₄²	1	4	yes	—
97	C₈×D₁₃	C₂²×C₄	1	4	yes	—
98	C₈×D₁₃	C₂²×C₄	1	4	yes	—
99	C₁₃×C₁₆	C₂²×C₄	1	8	yes	complemented
100	#6090	C₂²×C₄	1	8	yes	complemented
101	C₂²×M₃₂	—	13	2	no	—
102	C₂²×#97	—	13	1	no	—
103	C₂×#347	—	13	2	no	—
104	C₂×#347	—	13	1	no	—
105	C₂×#347	—	13	8	no	—
106	C₂×#347	D₁₃	1	1	yes	characteristic, direct product factor
107	C₂³×C₁₃	C₂×M₁₆	1	1	yes	characteristic
108	C₂²×D₁₃	—	2	4	no	—
109	C₂²×D₁₃	#97	1	1	yes	characteristic
110	C₂²×D₁₃	#97	1	2	yes	—
111	C₂²×D₁₃	#97	1	1	yes	characteristic
112	C₂²×D₁₃	C₂×M₁₆	1	2	yes	—
113	C₂×C₄×C₁₃	—	2	2	no	—
114	C₂×C₄×C₁₃	—	4	2	no	—
115	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
116	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
117	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
118	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
119	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
120	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
121	C₂×Dic₁₃	—	2	2	no	—
122	C₂×Dic₁₃	—	4	2	no	—
123	C₂×Dic₁₃	#97	1	2	yes	—
124	C₂×Dic₁₃	#97	1	1	yes	characteristic
125	C₂×Dic₁₃	#97	1	1	yes	characteristic
126	C₂×Dic₁₃	C₂×M₁₆	1	2	yes	—
127	C₄×D₁₃	—	4	8	no	—
128	C₄×D₁₃	#97	1	2	yes	—
129	C₄×D₁₃	#97	1	2	yes	—
130	C₄×D₁₃	#97	1	2	yes	—
131	C₄×D₁₃	#97	1	2	yes	—
132	C₄×D₁₃	C₂×C₄²	1	2	yes	—
133	C₄×D₁₃	C₂×C₄²	1	2	yes	—
134	C₄×D₁₃	C₂×M₁₆	1	2	yes	—
135	C₄×D₁₃	C₂×M₁₆	1	2	yes	—
136	C₈×C₁₃	—	2	2	no	—
137	C₈×C₁₃	C₂×C₄²	1	2	yes	—
138	C₈×C₁₃	C₂³×C₄	1	2	yes	—
139	#1055	—	2	2	no	—
140	#1055	C₂×C₄²	1	4	yes	—
141	C₂³×C₈	—	13	1	no	—
142	C₂³×C₈	—	13	1	no	—
143	C₂²×C₄²	—	13	1	no	—
144	C₂×M₃₂	—	13	2	no	—
145	C₂×M₃₂	—	13	8	no	—
146	C₂×M₃₂	—	13	8	no	—
147	C₂×M₃₂	—	13	4	no	—
148	C₂×M₃₂	C₂×D₁₃	1	2	yes	—
149	C₂²×C₁₆	—	13	4	no	—
150	C₂×#97	—	13	1	no	—
151	C₂×#97	—	13	1	no	—
152	C₂×#97	—	13	1	no	—
153	C₂×#97	—	13	4	no	—
154	C₂×#97	—	13	4	no	—
155	C₂×#97	C₂×D₁₃	1	1	yes	characteristic
156	#347	—	13	8	no	—
157	#347	—	13	4	no	—
158	#347	C₂×D₁₃	1	4	yes	direct product factor
159	C₂²×C₁₃	—	2	2	no	—
160	C₂²×C₁₃	C₂×#97	1	1	yes	characteristic
161	C₂²×C₁₃	C₂×#97	1	1	yes	characteristic
162	C₂²×C₁₃	C₂²×M₁₆	1	1	yes	characteristic
163	C₂×D₁₃	—	2	2	no	—
164	C₂×D₁₃	—	2	4	no	—
165	C₂×D₁₃	—	2	4	no	—
166	C₂×D₁₃	#347	1	4	yes	direct product factor
167	C₂×D₁₃	C₂×#97	1	2	yes	—
168	C₂×D₁₃	C₂×#97	1	2	yes	—
169	C₄×C₁₃	—	4	2	no	—
170	C₄×C₁₃	C₂×#97	1	1	yes	characteristic
171	C₄×C₁₃	C₂×#97	1	1	yes	characteristic
172	C₄×C₁₃	C₂²×C₄²	1	1	yes	characteristic
173	C₄×C₁₃	C₂²×M₁₆	1	1	yes	characteristic
174	Dic₁₃	—	4	2	no	—
175	Dic₁₃	C₂×#97	1	2	yes	—
176	Dic₁₃	C₂×#97	1	2	yes	—
177	C₂³×C₄	—	13	1	no	—
178	C₂³×C₄	—	26	1	no	—
179	C₂²×C₈	—	13	1	no	—
180	C₂²×C₈	—	13	2	no	—
181	C₂²×C₈	—	13	2	no	—
182	C₂²×C₈	—	13	4	no	—
183	C₂²×C₈	—	13	2	no	—
184	C₂²×C₈	—	13	2	no	—
185	C₂²×C₈	—	13	1	no	—
186	C₂²×C₈	—	13	4	no	—
187	C₂²×C₈	—	26	1	no	—
188	C₂²×C₈	—	26	1	no	—
189	C₂²×C₈	—	26	2	no	—
190	C₂²×C₈	C₄×D₁₃	1	1	yes	characteristic
191	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
192	C₂×C₄²	—	13	1	no	—
193	C₂×C₄²	—	13	4	no	—
194	C₂×C₄²	—	26	2	no	—
195	C₂×C₄²	—	26	1	no	—
196	C₂×C₄²	C₄×D₁₃	1	1	yes	characteristic
197	M₃₂	—	13	4	no	—
198	M₃₂	—	13	4	no	—
199	M₃₂	—	13	16	no	—
200	M₃₂	C₄×D₁₃	1	4	yes	complemented
201	M₃₂	C₂²×D₁₃	1	4	yes	—
202	C₂×C₁₆	—	13	4	no	—
203	C₂×C₁₆	—	13	16	no	—
204	C₂×C₁₆	C₄×D₁₃	1	4	yes	complemented
205	#97	—	13	4	no	—
206	#97	—	13	2	no	—
207	#97	—	13	2	no	—
208	#97	—	13	4	no	—
209	#97	C₄×D₁₃	1	2	yes	—
210	#97	C₂²×D₁₃	1	2	yes	—
211	C₂×C₁₃	—	2	1	no	—
212	C₂×C₁₃	—	2	1	no	—
213	C₂×C₁₃	C₂×#347	1	2	yes	complemented
214	C₂×C₁₃	C₂²×#97	1	1	yes	characteristic
215	D₁₃	—	2	2	no	—
216	D₁₃	C₂×#347	1	4	yes	direct product factor
217	C₂⁴	—	13	1	no	—
218	C₂²×C₄	—	2	1	no	—
219	C₂²×C₄	—	13	1	no	—
220	C₂²×C₄	—	13	2	no	—
221	C₂²×C₄	—	13	1	no	—
222	C₂²×C₄	—	13	1	no	—
223	C₂²×C₄	—	13	2	no	—
224	C₂²×C₄	—	13	1	no	—
225	C₂²×C₄	—	13	2	no	—
226	C₂²×C₄	—	13	1	no	—
227	C₂²×C₄	—	13	2	no	—
228	C₂²×C₄	—	26	4	no	—
229	C₂²×C₄	—	26	1	no	—
230	C₂²×C₄	—	52	4	no	—
231	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
232	C₂×C₈	—	2	1	no	—
233	C₂×C₈	—	2	1	no	—
234	C₂×C₈	—	13	2	no	—
235	C₂×C₈	—	13	2	no	—
236	C₂×C₈	—	13	2	no	—
237	C₂×C₈	—	13	4	no	—
238	C₂×C₈	—	13	4	no	—
239	C₂×C₈	—	13	4	no	—
240	C₂×C₈	—	13	4	no	—
241	C₂×C₈	—	13	2	no	—
242	C₂×C₈	—	26	4	no	—
243	C₂×C₈	—	26	4	no	—
244	C₂×C₈	—	26	1	no	—
245	C₂×C₈	—	26	1	no	—
246	C₂×C₈	C₂×C₄×D₁₃	1	2	yes	—
247	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
248	C₂×C₈	C₂×C₄×D₁₃	1	2	yes	—
249	C₂×C₈	C₂×C₄×D₁₃	1	2	yes	—
250	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
251	C₄²	—	2	1	no	—
252	C₄²	—	13	2	no	—
253	C₄²	—	26	4	no	—
254	C₄²	—	26	1	no	—
255	C₄²	C₂×C₄×D₁₃	1	2	yes	—
256	C₁₆	—	13	8	no	—
257	C₁₆	C₂×C₄×D₁₃	1	8	yes	complemented
258	C₁₃	C₂²×#347	1	1	yes	characteristic, complemented, Sylow
259	C₂³	—	13	1	no	—
260	C₂³	—	13	2	no	—
261	C₂³	—	13	2	no	—
262	C₂³	—	13	1	no	—
263	C₂³	—	26	4	no	—
264	C₂³	M₁₆×D₁₃	1	1	yes	characteristic
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	4	2	no	—
267	C₂×C₄	—	13	2	no	—
268	C₂×C₄	—	13	2	no	—
269	C₂×C₄	—	13	1	no	—
270	C₂×C₄	—	13	1	no	—
271	C₂×C₄	—	13	2	no	—
272	C₂×C₄	—	13	2	no	—
273	C₂×C₄	—	13	2	no	—
274	C₂×C₄	—	13	2	no	—
275	C₂×C₄	—	13	2	no	—
276	C₂×C₄	—	13	2	no	—
277	C₂×C₄	—	13	2	no	—
278	C₂×C₄	—	13	2	no	—
279	C₂×C₄	—	26	2	no	—
280	C₂×C₄	—	52	8	no	—
281	C₂×C₄	—	52	2	no	—
282	C₂×C₄	C₄²×D₁₃	1	1	yes	characteristic
283	C₂×C₄	C₄²×D₁₃	1	1	yes	characteristic
284	C₂×C₄	M₁₆×D₁₃	1	1	yes	characteristic
285	C₂×C₄	M₁₆×D₁₃	1	1	yes	characteristic
286	C₂×C₄	M₁₆×D₁₃	1	1	yes	characteristic
287	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
288	C₈	—	2	2	no	—
289	C₈	—	13	4	no	—
290	C₈	—	26	2	no	—
291	C₈	C₄²×D₁₃	1	2	yes	—
292	C₈	C₂²×C₄×D₁₃	1	2	yes	—
293	C₂²	—	2	2	no	—
294	C₂²	—	13	4	no	—
295	C₂²	—	13	2	no	—
296	C₂²	—	13	2	no	—
297	C₂²	—	26	4	no	—
298	C₂²	—	26	4	no	—
299	C₂²	—	26	2	no	—
300	C₂²	D₁₃×#97	1	1	yes	characteristic
301	C₂²	D₁₃×#97	1	1	yes	characteristic
302	C₂²	C₂×M₁₆×D₁₃	1	1	yes	characteristic
303	C₄	—	4	2	no	—
304	C₄	—	13	2	no	—
305	C₄	—	13	2	no	—
306	C₄	—	52	2	no	—
307	C₄	D₁₃×#97	1	1	yes	characteristic
308	C₄	D₁₃×#97	1	1	yes	characteristic
309	C₄	C₂×C₄²×D₁₃	1	1	yes	characteristic
310	C₄	C₂×M₁₆×D₁₃	1	1	yes	characteristic
311	C₂	—	2	1	no	—
312	C₂	—	2	1	no	—
313	C₂	—	13	4	no	—
314	C₂	—	26	2	no	—
315	C₂	D₁₃×#347	1	2	yes	direct product factor
316	C₂	C₂×D₁₃×#97	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₄²×D₁₃
commutator	C₄×C₁₃	C₂²×C₄²
Frattini	C₂×C₈	C₂³×D₁₃
Fitting	C₂×C₁₃×#347	C₂
socle	C₂²×C₁₃	C₂×#97
prime radical	C₂³×D₁₃	M₁₆
cosocle	C₂×C₈×C₁₃	C₂⁴
hypercenter	C₂×#347	D₁₃
hypocenter	C₁₃	C₂²×#347
LCS	G ▷ C₄×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#347

Complements

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

N	H ≅ G/N	type
3. C₂×#11718058	C₂	semidirect
4. C₂×#11718104	C₂	semidirect
5. D₁₃×#347	C₂	direct (N × H)
6. C₂×C₁₃×#347	C₂	semidirect
16. D₁₃×M₃₂	C₄	semidirect
18. C₂×#88038	C₄	semidirect
19. C₂×C₁₆×D₁₃	C₄	semidirect
22. C₁₃×#347	C₂²	semidirect
23. #11718104	C₂²	semidirect
24. #11718058	C₂²	semidirect
45. C₁₃×M₃₂	C₂×C₄	semidirect
47. C₂×C₁₃×C₁₆	C₂×C₄	semidirect
53. #88052	C₂×C₄	semidirect
55. C₂×#6090	C₂×C₄	semidirect
57. #88038	C₂×C₄	semidirect
58. C₁₆×D₁₃	C₂×C₄	semidirect
99. C₁₃×C₁₆	C₂²×C₄	semidirect
100. #6090	C₂²×C₄	semidirect
106. C₂×#347	D₁₃	direct (N × H)
158. #347	C₂×D₁₃	direct (N × H)
166. C₂×D₁₃	#347	direct (N × H)
200. M₃₂	C₄×D₁₃	semidirect
204. C₂×C₁₆	C₄×D₁₃	semidirect
213. C₂×C₁₃	C₂×#347	semidirect
216. D₁₃	C₂×#347	direct (N × H)
257. C₁₆	C₂×C₄×D₁₃	semidirect
258. C₁₃	C₂²×#347	semidirect
315. C₂	D₁₃×#347	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×D₁₃×M₃₂	2
C₂×D₁₃×#97	2
C₂×#11718058	2
C₂×#11718104	2
D₁₃×#347	2
C₂×C₁₃×#347	2
C₂²×#347	13

Subgroups — C2×D13×#347 #16015571

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×D₁₃×#347 #16015571