Subgroups — D₁₃×#1376 #15777770

Order 3328.

670 conjugacy classes of subgroups.

The subgroup lattice has 310 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×C₄×C₈×D₁₃	C₂	1	1	yes	maximal, characteristic
3	#13112918	C₂	1	1	yes	maximal, characteristic, complemented
4	#13112917	C₂	1	2	yes	maximal, complemented
5	C₁₃×#1376	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₈×D₁₃	C₄	1	1	yes	characteristic
7	C₂²×C₈×D₁₃	C₂²	1	1	yes	characteristic
8	C₂×C₄²×D₁₃	C₄	1	1	yes	characteristic
9	C₂×C₁₃×M₃₂	C₂²	1	2	yes	—
10	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
11	C₂×C₈×Dic₁₃	C₄	1	1	yes	characteristic
12	C₂×C₈×Dic₁₃	C₂²	1	1	yes	characteristic
13	C₂×#88052	C₂²	1	2	yes	—
14	C₂×C₄×#1055	C₄	1	1	yes	characteristic
15	D₁₃×M₃₂	—	2	4	no	—
16	D₁₃×M₃₂	C₄	1	8	yes	complemented
17	C₂×#88038	—	2	2	no	—
18	C₂×C₁₆×D₁₃	—	2	2	no	—
19	C₄×C₈×D₁₃	—	2	4	no	—
20	C₂²×C₄×D₁₃	—	2	1	no	—
21	C₂²×C₄×D₁₃	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
24	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄×Dic₁₃	—	2	1	no	—
26	C₂×C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
27	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
28	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
29	C₂×C₈×D₁₃	—	2	2	no	—
30	C₂×C₈×D₁₃	—	2	2	no	—
31	C₂×C₈×D₁₃	C₈	1	2	yes	—
32	C₂×C₈×D₁₃	C₈	1	2	yes	—
33	C₂×C₈×D₁₃	C₂×C₄	1	4	yes	—
34	C₂×C₈×D₁₃	D₄	1	1	yes	characteristic
35	C₂×C₈×D₁₃	D₄	1	1	yes	characteristic
36	C₂×C₈×D₁₃	D₄	1	4	yes	—
37	C₂×C₈×D₁₃	Q₈	1	1	yes	characteristic
38	C₂×C₈×D₁₃	Q₈	1	1	yes	characteristic
39	C₄²×D₁₃	—	2	2	no	—
40	C₄²×D₁₃	C₈	1	4	yes	—
41	C₁₃×M₃₂	—	2	2	no	—
42	C₁₃×M₃₂	C₂×C₄	1	4	yes	complemented
43	C₂×C₁₃×C₁₆	—	2	2	no	—
44	C₄×C₈×C₁₃	—	2	2	no	—
45	C₈×Dic₁₃	—	2	2	no	—
46	C₈×Dic₁₃	—	2	2	no	—
47	#88052	—	2	2	no	—
48	#88052	C₂×C₄	1	4	yes	complemented
49	C₂×#6090	—	2	2	no	—
50	C₄×#1055	—	2	2	no	—
51	#88038	—	2	8	no	—
52	C₁₆×D₁₃	—	2	8	no	—
53	C₂×#1376	—	13	1	no	maximal, Sylow
54	C₂³×D₁₃	M₁₆	1	1	yes	characteristic
55	C₂²×C₄×C₁₃	—	2	1	no	—
56	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
57	C₂²×Dic₁₃	—	2	1	no	—
58	C₂²×Dic₁₃	M₁₆	1	1	yes	characteristic
59	C₂×C₄×D₁₃	—	2	8	no	—
60	C₂×C₄×D₁₃	—	2	4	no	—
61	C₂×C₄×D₁₃	C₄²	1	2	yes	—
62	C₂×C₄×D₁₃	C₂²⋊C₄	1	2	yes	—
63	C₂×C₄×D₁₃	C₄⋊C₄	1	1	yes	characteristic
64	C₂×C₄×D₁₃	C₄⋊C₄	1	1	yes	characteristic
65	C₂×C₄×D₁₃	C₂×C₈	1	1	yes	characteristic
66	C₂×C₄×D₁₃	C₂×C₈	1	1	yes	characteristic
67	C₂×C₄×D₁₃	C₂×C₈	1	2	yes	—
68	C₂×C₄×D₁₃	M₁₆	1	2	yes	—
69	C₂×C₈×C₁₃	—	2	1	no	—
70	C₂×C₈×C₁₃	—	2	1	no	—
71	C₂×C₈×C₁₃	C₂×C₈	1	2	yes	—
72	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
73	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
74	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
75	C₂×C₈×C₁₃	C₂×Q₈	1	1	yes	characteristic
76	C₄²×C₁₃	—	2	1	no	—
77	C₄²×C₁₃	C₂×C₈	1	2	yes	—
78	C₄×Dic₁₃	—	2	1	no	—
79	C₄×Dic₁₃	—	2	4	no	—
80	C₄×Dic₁₃	C₂×C₈	1	2	yes	—
81	C₂×#1055	—	2	1	no	—
82	C₂×#1055	—	2	1	no	—
83	C₂×#1055	C₄²	1	2	yes	—
84	C₂×#1055	C₂²⋊C₄	1	2	yes	—
85	C₂×#1055	C₄⋊C₄	1	1	yes	characteristic
86	C₂×#1055	C₄⋊C₄	1	1	yes	characteristic
87	C₂×#1055	C₂×C₈	1	2	yes	—
88	C₈×D₁₃	—	2	4	no	—
89	C₈×D₁₃	—	2	4	no	—
90	C₈×D₁₃	C₂²⋊C₄	1	4	yes	—
91	C₈×D₁₃	C₂²⋊C₄	1	4	yes	—
92	C₈×D₁₃	C₄⋊C₄	1	4	yes	—
93	C₈×D₁₃	C₄⋊C₄	1	4	yes	—
94	C₁₃×C₁₆	—	2	4	no	—
95	#6090	—	2	4	no	—
96	C₂²×M₃₂	—	13	2	no	—
97	C₂²×C₄×C₈	—	13	1	no	—
98	#1376	—	13	2	no	—
99	#1376	—	13	1	no	—
100	#1376	D₁₃	1	1	yes	characteristic, direct product factor
101	C₂³×C₁₃	C₂×M₁₆	1	1	yes	characteristic
102	C₂²×D₁₃	—	2	4	no	—
103	C₂²×D₁₃	#97	1	2	yes	—
104	C₂²×D₁₃	C₂²⋊C₈	1	2	yes	—
105	C₂²×D₁₃	C₄⋊C₈	1	1	yes	characteristic
106	C₂²×D₁₃	C₄⋊C₈	1	1	yes	characteristic
107	C₂×C₄×C₁₃	—	2	4	no	—
108	C₂×C₄×C₁₃	—	2	2	no	—
109	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
110	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
111	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
112	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
113	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
114	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
115	C₂×Dic₁₃	—	2	4	no	—
116	C₂×Dic₁₃	—	2	2	no	—
117	C₂×Dic₁₃	#97	1	2	yes	—
118	C₂×Dic₁₃	C₂²⋊C₈	1	2	yes	—
119	C₂×Dic₁₃	C₄⋊C₈	1	1	yes	characteristic
120	C₂×Dic₁₃	C₄⋊C₈	1	1	yes	characteristic
121	C₄×D₁₃	—	2	16	no	—
122	C₄×D₁₃	#95	1	2	yes	—
123	C₄×D₁₃	#95	1	2	yes	—
124	C₄×D₁₃	C₄×C₈	1	2	yes	—
125	C₄×D₁₃	C₄×C₈	1	2	yes	—
126	C₄×D₁₃	C₂²⋊C₈	1	2	yes	—
127	C₄×D₁₃	C₂²⋊C₈	1	2	yes	—
128	C₄×D₁₃	C₄⋊C₈	1	2	yes	—
129	C₄×D₁₃	C₄⋊C₈	1	2	yes	—
130	C₈×C₁₃	—	2	2	no	—
131	C₈×C₁₃	C₂×C₂²⋊C₄	1	2	yes	—
132	C₈×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
133	#1055	—	2	2	no	—
134	#1055	#95	1	2	yes	—
135	#1055	#95	1	2	yes	—
136	C₂³×C₈	—	13	1	no	—
137	C₂³×C₈	—	13	1	no	—
138	C₂²×C₄²	—	13	1	no	—
139	C₂×M₃₂	—	13	2	no	—
140	C₂×M₃₂	—	13	8	no	—
141	C₂×M₃₂	—	26	2	no	—
142	C₂×M₃₂	—	26	4	no	—
143	C₂×M₃₂	C₂×D₁₃	1	2	yes	—
144	C₂²×C₁₆	—	26	2	no	—
145	C₂×C₄×C₈	—	13	1	no	—
146	C₂×C₄×C₈	—	13	1	no	—
147	C₂×C₄×C₈	—	13	1	no	—
148	C₂×C₄×C₈	—	26	4	no	—
149	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
150	C₂²×C₁₃	—	2	2	no	—
151	C₂²×C₁₃	C₂×#97	1	1	yes	characteristic
152	C₂²×C₁₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
153	C₂²×C₁₃	C₂×C₄⋊C₈	1	1	yes	characteristic
154	C₂×D₁₃	—	2	2	no	—
155	C₂×D₁₃	—	2	4	no	—
156	C₂×D₁₃	—	2	4	no	—
157	C₂×D₁₃	#336	1	2	yes	—
158	C₂×D₁₃	#336	1	2	yes	—
159	C₂×D₁₃	#364	1	4	yes	—
160	C₄×C₁₃	—	2	4	no	—
161	C₄×C₁₃	C₂×#95	1	1	yes	characteristic
162	C₄×C₁₃	C₂×C₄×C₈	1	1	yes	characteristic
163	C₄×C₁₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
164	C₄×C₁₃	C₂×C₄⋊C₈	1	1	yes	characteristic
165	Dic₁₃	—	2	4	no	—
166	Dic₁₃	#336	1	2	yes	—
167	Dic₁₃	#336	1	2	yes	—
168	C₂³×C₄	—	13	1	no	—
169	C₂³×C₄	—	26	1	no	—
170	C₂²×C₈	—	13	1	no	—
171	C₂²×C₈	—	13	4	no	—
172	C₂²×C₈	—	13	4	no	—
173	C₂²×C₈	—	13	1	no	—
174	C₂²×C₈	—	13	1	no	—
175	C₂²×C₈	—	13	1	no	—
176	C₂²×C₈	—	13	1	no	—
177	C₂²×C₈	—	13	1	no	—
178	C₂²×C₈	—	13	2	no	—
179	C₂²×C₈	—	13	2	no	—
180	C₂²×C₈	—	26	2	no	—
181	C₂²×C₈	—	26	2	no	—
182	C₂²×C₈	C₄×D₁₃	1	1	yes	characteristic
183	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
184	C₂×C₄²	—	13	1	no	—
185	C₂×C₄²	—	13	4	no	—
186	C₂×C₄²	—	26	2	no	—
187	C₂×C₄²	—	26	1	no	—
188	C₂×C₄²	C₄×D₁₃	1	1	yes	characteristic
189	M₃₂	—	2	2	no	—
190	M₃₂	—	13	4	no	—
191	M₃₂	—	26	8	no	—
192	M₃₂	—	26	2	no	—
193	M₃₂	C₄×D₁₃	1	4	yes	complemented
194	C₂×C₁₆	—	2	2	no	—
195	C₂×C₁₆	—	26	8	no	—
196	C₂×C₁₆	—	26	2	no	—
197	C₄×C₈	—	2	2	no	—
198	C₄×C₈	—	26	2	no	—
199	C₄×C₈	—	26	2	no	—
200	C₄×C₈	—	26	2	no	—
201	C₂×C₁₃	—	2	1	no	—
202	C₂×C₁₃	—	2	1	no	—
203	C₂×C₁₃	C₂×#336	1	1	yes	characteristic
204	C₂×C₁₃	C₂×#364	1	2	yes	—
205	D₁₃	—	2	2	no	—
206	D₁₃	#1376	1	4	yes	direct product factor
207	C₂⁴	—	13	1	no	—
208	C₂²×C₄	—	2	1	no	—
209	C₂²×C₄	—	13	1	no	—
210	C₂²×C₄	—	13	2	no	—
211	C₂²×C₄	—	13	2	no	—
212	C₂²×C₄	—	13	1	no	—
213	C₂²×C₄	—	13	1	no	—
214	C₂²×C₄	—	13	2	no	—
215	C₂²×C₄	—	13	1	no	—
216	C₂²×C₄	—	13	1	no	—
217	C₂²×C₄	—	13	2	no	—
218	C₂²×C₄	—	26	4	no	—
219	C₂²×C₄	—	26	8	no	—
220	C₂²×C₄	—	26	1	no	—
221	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
222	C₂×C₈	—	2	1	no	—
223	C₂×C₈	—	2	1	no	—
224	C₂×C₈	—	13	2	no	—
225	C₂×C₈	—	13	2	no	—
226	C₂×C₈	—	13	1	no	—
227	C₂×C₈	—	13	1	no	—
228	C₂×C₈	—	13	4	no	—
229	C₂×C₈	—	13	4	no	—
230	C₂×C₈	—	13	4	no	—
231	C₂×C₈	—	13	4	no	—
232	C₂×C₈	—	13	2	no	—
233	C₂×C₈	—	26	4	no	—
234	C₂×C₈	—	26	4	no	—
235	C₂×C₈	—	26	1	no	—
236	C₂×C₈	—	26	1	no	—
237	C₂×C₈	C₈×D₁₃	1	2	yes	—
238	C₂×C₈	C₂×C₄×D₁₃	1	2	yes	—
239	C₂×C₈	D₄×D₁₃	1	2	yes	—
240	C₂×C₈	D₄×D₁₃	1	1	yes	characteristic
241	C₂×C₈	Q₈×D₁₃	1	1	yes	characteristic
242	C₄²	—	2	1	no	—
243	C₄²	—	13	2	no	—
244	C₄²	—	26	4	no	—
245	C₄²	—	26	1	no	—
246	C₄²	C₈×D₁₃	1	2	yes	—
247	C₁₆	—	2	4	no	—
248	C₁₆	—	26	4	no	—
249	C₁₃	C₂×#1376	1	1	yes	characteristic, complemented, Sylow
250	C₂³	—	13	1	no	—
251	C₂³	—	13	1	no	—
252	C₂³	—	13	2	no	—
253	C₂³	—	13	2	no	—
254	C₂³	—	26	4	no	—
255	C₂³	M₁₆×D₁₃	1	1	yes	characteristic
256	C₂×C₄	—	2	4	no	—
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	13	1	no	—
259	C₂×C₄	—	13	1	no	—
260	C₂×C₄	—	13	2	no	—
261	C₂×C₄	—	13	2	no	—
262	C₂×C₄	—	13	2	no	—
263	C₂×C₄	—	13	2	no	—
264	C₂×C₄	—	13	2	no	—
265	C₂×C₄	—	13	2	no	—
266	C₂×C₄	—	13	2	no	—
267	C₂×C₄	—	13	2	no	—
268	C₂×C₄	—	13	2	no	—
269	C₂×C₄	—	13	2	no	—
270	C₂×C₄	—	26	16	no	—
271	C₂×C₄	—	26	4	no	—
272	C₂×C₄	—	26	2	no	—
273	C₂×C₄	C₄²×D₁₃	1	1	yes	characteristic
274	C₂×C₄	C₂²⋊C₄×D₁₃	1	1	yes	characteristic
275	C₂×C₄	C₄⋊C₄×D₁₃	1	1	yes	characteristic
276	C₂×C₄	C₂×C₈×D₁₃	1	1	yes	characteristic
277	C₂×C₄	C₂×C₈×D₁₃	1	1	yes	characteristic
278	C₂×C₄	M₁₆×D₁₃	1	1	yes	characteristic
279	C₈	—	2	2	no	—
280	C₈	—	13	2	no	—
281	C₈	—	13	2	no	—
282	C₈	—	26	2	no	—
283	C₈	C₂²⋊C₄×D₁₃	1	2	yes	—
284	C₈	C₄⋊C₄×D₁₃	1	2	yes	—
285	C₂²	—	2	2	no	—
286	C₂²	—	13	2	no	—
287	C₂²	—	13	4	no	—
288	C₂²	—	13	2	no	—
289	C₂²	—	26	4	no	—
290	C₂²	—	26	4	no	—
291	C₂²	—	26	2	no	—
292	C₂²	D₁₃×#97	1	1	yes	characteristic
293	C₂²	D₁₃×C₂²⋊C₈	1	1	yes	characteristic
294	C₂²	D₁₃×C₄⋊C₈	1	1	yes	characteristic
295	C₄	—	2	4	no	—
296	C₄	—	13	2	no	—
297	C₄	—	13	2	no	—
298	C₄	—	26	4	no	—
299	C₄	D₁₃×#95	1	1	yes	characteristic
300	C₄	C₄×C₈×D₁₃	1	1	yes	characteristic
301	C₄	D₁₃×C₂²⋊C₈	1	1	yes	characteristic
302	C₄	D₁₃×C₄⋊C₈	1	1	yes	characteristic
303	C₂	—	2	1	no	—
304	C₂	—	2	1	no	—
305	C₂	—	13	4	no	—
306	C₂	—	26	2	no	—
307	C₂	D₁₃×#336	1	1	yes	characteristic
308	C₂	D₁₃×#364	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	D₄×D₁₃
commutator	C₄×C₁₃	C₂×C₄×C₈
Frattini	C₂²×C₈	C₂²×D₁₃
Fitting	C₁₃×#1376	C₂
socle	C₂²×C₁₃	C₂×C₄⋊C₈
prime radical	C₂³×D₁₃	M₁₆
cosocle	C₂²×C₈×C₁₃	C₂³
hypercenter	#1376	D₁₃
hypocenter	C₁₃	C₂×#1376
LCS	G ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ #1376

Complements

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

N	H ≅ G/N	type
3. #13112918	C₂	semidirect
4. #13112917	C₂	semidirect
5. C₁₃×#1376	C₂	semidirect
16. D₁₃×M₃₂	C₄	semidirect
42. C₁₃×M₃₂	C₂×C₄	semidirect
48. #88052	C₂×C₄	semidirect
100. #1376	D₁₃	direct (N × H)
193. M₃₂	C₄×D₁₃	semidirect
206. D₁₃	#1376	direct (N × H)
249. C₁₃	C₂×#1376	semidirect

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×C₄×C₈×D₁₃	2
#13112918	2
#13112917	2
C₁₃×#1376	2
C₂×#1376	13

Subgroups — D13×#1376 #15777770

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₁₃×#1376 #15777770