Subgroups — D₁₃×#2158 #16015944

Order 3328.

694 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₁₆×D₁₃	C₂	1	1	yes	maximal, characteristic
2	D₁₃×#427	C₂	1	2	yes	maximal
3	#13116620	C₂	1	1	yes	maximal, characteristic, complemented
4	#13116619	C₂	1	2	yes	maximal, complemented
5	#13116618	C₂	1	2	yes	maximal, complemented
6	#13116617	C₂	1	2	yes	maximal, complemented
7	D₁₃×#367	C₂	1	2	yes	maximal, complemented
8	D₁₃×#366	C₂	1	2	yes	maximal, complemented
9	C₁₃×#2158	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×C₈×D₁₃	C₂²	1	1	yes	characteristic
11	D₁₃×#117	—	2	2	no	—
12	C₂²×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
13	C₁₃×#427	C₂²	1	2	yes	—
14	#11718767	C₂²	1	2	yes	—
15	C₂²×#6090	C₂²	1	1	yes	characteristic
16	#11718659	—	2	2	no	—
17	C₂×C₁₆×D₁₃	C₄	1	8	yes	complemented
18	C₂×C₁₆×D₁₃	C₂²	1	1	yes	characteristic
19	C₂×C₁₆×D₁₃	C₂²	1	1	yes	characteristic
20	C₂×C₁₆×D₁₃	C₂²	1	1	yes	characteristic
21	C₂×C₁₆×D₁₃	C₂²	1	1	yes	characteristic
22	#11718541	C₂²	1	4	yes	—
23	D₁₃×#107	C₂²	1	4	yes	—
24	#11718521	—	2	2	no	—
25	D₁₃×#106	—	2	2	no	—
26	C₁₃×#367	C₂²	1	2	yes	complemented
27	C₁₃×#366	C₂²	1	2	yes	complemented
28	#11718097	C₂²	1	2	yes	complemented
29	#11718096	C₂²	1	2	yes	complemented
30	#11718082	C₂²	1	4	yes	complemented
31	#11718081	C₂²	1	4	yes	complemented
32	C₂²×C₄×D₁₃	D₄	1	1	yes	characteristic
33	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
34	C₁₃×#117	—	2	2	no	—
35	#88180	—	2	2	no	—
36	C₂²×#1055	D₄	1	1	yes	characteristic
37	C₂×C₈×D₁₃	C₂×C₄	1	4	yes	—
38	C₂×C₈×D₁₃	D₄	1	1	yes	characteristic
39	C₂×C₈×D₁₃	D₄	1	1	yes	characteristic
40	C₂×C₈×D₁₃	Q₈	1	4	yes	—
41	C₂×C₈×D₁₃	C₂³	1	1	yes	characteristic
42	C₂×C₈×D₁₃	C₂³	1	1	yes	characteristic
43	#88146	—	2	4	no	—
44	C₄⋊C₄×D₁₃	—	2	4	no	—
45	C₂²⋊C₄×D₁₃	—	4	2	no	—
46	#88131	—	4	2	no	—
47	#88126	—	4	2	no	—
48	C₄²×D₁₃	—	4	2	no	—
49	C₂×C₁₃×C₁₆	C₂×C₄	1	4	yes	complemented
50	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
51	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
52	C₁₃×#107	C₂³	1	4	yes	—
53	C₁₃×#106	—	2	2	no	—
54	#88058	C₂³	1	4	yes	—
55	#88057	—	2	2	no	—
56	C₂×#6090	C₂×C₄	1	4	yes	complemented
57	C₂×#6090	C₂³	1	1	yes	characteristic
58	C₂×#6090	C₂³	1	1	yes	characteristic
59	#88048	—	2	4	no	—
60	#88047	—	2	4	no	—
61	C₁₆×D₁₃	C₂×C₄	1	4	yes	—
62	C₁₆×D₁₃	C₂×C₄	1	4	yes	—
63	C₁₆×D₁₃	C₂×C₄	1	4	yes	complemented
64	C₁₆×D₁₃	C₂×C₄	1	4	yes	complemented
65	C₂×#2158	—	13	1	no	maximal, Sylow
66	C₂³×D₁₃	D₈	1	1	yes	characteristic
67	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
68	C₂²×Dic₁₃	D₈	1	1	yes	characteristic
69	C₂×C₄×D₁₃	—	4	4	no	—
70	C₂×C₄×D₁₃	C₄⋊C₄	1	4	yes	—
71	C₂×C₄×D₁₃	D₈	1	1	yes	characteristic
72	C₂×C₄×D₁₃	D₈	1	1	yes	characteristic
73	C₂×C₄×D₁₃	Q₁₆	1	4	yes	—
74	C₂×C₄×D₁₃	C₂×D₄	1	1	yes	characteristic
75	C₂×C₄×D₁₃	C₂×D₄	1	1	yes	characteristic
76	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
77	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
78	C₂×C₈×C₁₃	C₂×Q₈	1	2	yes	—
79	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
80	C₁₃×C₄⋊C₄	—	2	4	no	—
81	C₁₃×C₂²⋊C₄	—	4	2	no	—
82	C₄²×C₁₃	—	4	2	no	—
83	C₂²⋊Dic₁₃	—	4	2	no	—
84	#6103	—	4	4	no	—
85	C₄⋊Dic₁₃	—	2	4	no	—
86	#6101	—	4	4	no	—
87	C₄×Dic₁₃	—	4	2	no	—
88	C₄×Dic₁₃	—	4	4	no	—
89	C₂×#1055	C₄⋊C₄	1	4	yes	—
90	C₂×#1055	C₂×D₄	1	1	yes	characteristic
91	C₂×#1055	C₂×D₄	1	1	yes	characteristic
92	C₈×D₁₃	C₄⋊C₄	1	4	yes	—
93	C₈×D₁₃	C₄⋊C₄	1	4	yes	—
94	C₈×D₁₃	C₂²×C₄	1	2	yes	—
95	C₈×D₁₃	C₂²×C₄	1	2	yes	—
96	C₈×D₁₃	C₂×Q₈	1	2	yes	—
97	C₈×D₁₃	C₂×Q₈	1	2	yes	—
98	C₁₃×C₁₆	C₂²×C₄	1	2	yes	—
99	C₁₃×C₁₆	C₂²×C₄	1	2	yes	—
100	#6090	C₂²×C₄	1	2	yes	—
101	#6090	C₂²×C₄	1	2	yes	—
102	C₂³×C₁₆	—	13	1	no	—
103	C₂×#427	—	13	2	no	—
104	#2158	—	13	1	no	—
105	#2158	—	13	2	no	—
106	#2158	—	13	2	no	—
107	#2158	—	13	2	no	—
108	#2158	D₁₃	1	1	yes	characteristic, direct product factor
109	C₂×#367	—	13	2	no	—
110	C₂×#366	—	13	2	no	—
111	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
112	C₂²×D₁₃	—	2	4	no	—
113	C₂²×D₁₃	#107	1	4	yes	—
114	C₂²×D₁₃	C₂×D₈	1	1	yes	characteristic
115	C₂²×D₁₃	C₂×D₈	1	1	yes	characteristic
116	C₂×C₄×C₁₃	—	4	4	no	—
117	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
118	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
119	C₂×C₄×C₁₃	C₂×Q₁₆	1	2	yes	—
120	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
121	C₂×Dic₁₃	—	4	4	no	—
122	C₂×Dic₁₃	#107	1	4	yes	—
123	C₂×Dic₁₃	C₂×D₈	1	1	yes	characteristic
124	C₂×Dic₁₃	C₂×D₈	1	1	yes	characteristic
125	C₄×D₁₃	—	8	8	no	—
126	C₄×D₁₃	#107	1	4	yes	—
127	C₄×D₁₃	#107	1	4	yes	—
128	C₄×D₁₃	C₂×C₄⋊C₄	1	2	yes	—
129	C₄×D₁₃	C₂×C₄⋊C₄	1	2	yes	—
130	C₄×D₁₃	C₂×Q₁₆	1	2	yes	—
131	C₄×D₁₃	C₂×Q₁₆	1	2	yes	—
132	C₈×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
133	C₈×C₁₃	C₂³×C₄	1	1	yes	characteristic
134	C₈×C₁₃	C₂²×Q₈	1	1	yes	characteristic
135	#1055	C₂×C₄⋊C₄	1	2	yes	—
136	#1055	C₂×C₄⋊C₄	1	2	yes	—
137	C₂³×C₈	—	13	1	no	—
138	C₂×#117	—	26	2	no	—
139	C₂²×C₁₆	—	13	1	no	—
140	C₂²×C₁₆	—	13	8	no	—
141	C₂²×C₁₆	—	13	1	no	—
142	C₂²×C₁₆	—	13	1	no	—
143	C₂²×C₁₆	—	13	1	no	—
144	C₂²×C₁₆	—	13	1	no	—
145	C₂²×C₁₆	C₂×D₁₃	1	1	yes	characteristic
146	#427	—	13	2	no	—
147	#427	—	13	4	no	—
148	#427	—	26	2	no	—
149	#427	—	26	2	no	—
150	#427	C₂×D₁₃	1	2	yes	—
151	C₂×#107	—	13	4	no	—
152	C₂×#106	—	26	2	no	—
153	#367	—	13	2	no	—
154	#367	—	13	4	no	—
155	#367	C₂×D₁₃	1	2	yes	complemented
156	#366	—	13	2	no	—
157	#366	—	13	4	no	—
158	#366	C₂×D₁₃	1	2	yes	complemented
159	C₂²×C₁₃	—	2	2	no	—
160	C₂²×C₁₃	C₂×#107	1	2	yes	—
161	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
162	C₂×D₁₃	—	2	2	no	—
163	C₂×D₁₃	—	2	8	no	—
164	C₂×D₁₃	C₂×#107	1	2	yes	—
165	C₂×D₁₃	C₂×#107	1	2	yes	—
166	C₂×D₁₃	#508	1	4	yes	—
167	C₄×C₁₃	—	8	4	no	—
168	C₄×C₁₃	C₂×#107	1	2	yes	—
169	C₄×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
170	C₄×C₁₃	C₂²×Q₁₆	1	1	yes	characteristic
171	Dic₁₃	—	8	4	no	—
172	Dic₁₃	C₂×#107	1	2	yes	—
173	Dic₁₃	C₂×#107	1	2	yes	—
174	C₂³×C₄	—	13	1	no	—
175	C₂²×C₈	—	13	1	no	—
176	C₂²×C₈	—	13	4	no	—
177	C₂²×C₈	—	13	1	no	—
178	C₂²×C₈	—	13	1	no	—
179	C₂²×C₈	—	13	4	no	—
180	C₂²×C₈	—	13	1	no	—
181	C₂²×C₈	—	13	1	no	—
182	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
183	#117	—	2	2	no	—
184	#117	—	26	4	no	—
185	#117	—	26	2	no	—
186	#117	—	52	2	no	—
187	#117	—	52	2	no	—
188	C₂×C₄⋊C₄	—	26	4	no	—
189	C₂×C₂²⋊C₄	—	52	2	no	—
190	C₂×C₄²	—	52	2	no	—
191	C₂×C₁₆	—	13	4	no	—
192	C₂×C₁₆	—	13	1	no	—
193	C₂×C₁₆	—	13	1	no	—
194	C₂×C₁₆	—	13	4	no	—
195	C₂×C₁₆	—	13	4	no	—
196	C₂×C₁₆	—	13	4	no	—
197	C₂×C₁₆	—	13	4	no	—
198	C₂×C₁₆	C₄×D₁₃	1	4	yes	complemented
199	C₂×C₁₆	C₂²×D₁₃	1	1	yes	characteristic
200	C₂×C₁₆	C₂²×D₁₃	1	1	yes	characteristic
201	#107	—	13	4	no	—
202	#107	—	26	4	no	—
203	#107	C₂²×D₁₃	1	4	yes	—
204	#106	—	2	2	no	—
205	#106	—	26	4	no	—
206	#106	—	26	2	no	—
207	C₂×C₁₃	—	2	2	no	—
208	C₂×C₁₃	C₂²×#107	1	1	yes	characteristic
209	C₂×C₁₃	C₂×#508	1	2	yes	—
210	D₁₃	—	2	2	no	—
211	D₁₃	#2158	1	4	yes	direct product factor
212	C₂⁴	—	13	1	no	—
213	C₂²×C₄	—	13	1	no	—
214	C₂²×C₄	—	13	4	no	—
215	C₂²×C₄	—	13	1	no	—
216	C₂²×C₄	—	13	1	no	—
217	C₂²×C₄	—	13	4	no	—
218	C₂²×C₄	—	13	1	no	—
219	C₂²×C₄	—	13	1	no	—
220	C₂²×C₄	—	52	4	no	—
221	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
222	C₂×C₈	—	13	4	no	—
223	C₂×C₈	—	13	1	no	—
224	C₂×C₈	—	13	1	no	—
225	C₂×C₈	—	13	4	no	—
226	C₂×C₈	—	13	4	no	—
227	C₂×C₈	—	13	2	no	—
228	C₂×C₈	—	13	2	no	—
229	C₂×C₈	—	13	2	no	—
230	C₂×C₈	—	13	2	no	—
231	C₂×C₈	C₂×C₄×D₁₃	1	2	yes	—
232	C₂×C₈	D₄×D₁₃	1	1	yes	characteristic
233	C₂×C₈	Q₈×D₁₃	1	2	yes	—
234	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
235	C₄⋊C₄	—	2	4	no	—
236	C₄⋊C₄	—	26	4	no	—
237	C₄⋊C₄	—	52	4	no	—
238	C₂²⋊C₄	—	4	2	no	—
239	C₂²⋊C₄	—	52	4	no	—
240	C₂²⋊C₄	—	52	2	no	—
241	C₄²	—	4	2	no	—
242	C₄²	—	52	4	no	—
243	C₄²	—	52	2	no	—
244	C₁₆	—	13	2	no	—
245	C₁₆	—	13	2	no	—
246	C₁₆	C₂×C₄×D₁₃	1	2	yes	—
247	C₁₆	C₂×C₄×D₁₃	1	2	yes	—
248	C₁₃	C₂×#2158	1	1	yes	characteristic, complemented, Sylow
249	C₂³	—	13	1	no	—
250	C₂³	—	13	1	no	—
251	C₂³	—	13	4	no	—
252	C₂³	—	26	4	no	—
253	C₂³	D₈×D₁₃	1	1	yes	characteristic
254	C₂×C₄	—	4	4	no	—
255	C₂×C₄	—	13	1	no	—
256	C₂×C₄	—	13	1	no	—
257	C₂×C₄	—	13	4	no	—
258	C₂×C₄	—	13	2	no	—
259	C₂×C₄	—	13	2	no	—
260	C₂×C₄	—	13	2	no	—
261	C₂×C₄	—	13	2	no	—
262	C₂×C₄	—	13	4	no	—
263	C₂×C₄	—	13	4	no	—
264	C₂×C₄	—	52	4	no	—
265	C₂×C₄	—	104	8	no	—
266	C₂×C₄	C₄⋊C₄×D₁₃	1	2	yes	—
267	C₂×C₄	D₈×D₁₃	1	1	yes	characteristic
268	C₂×C₄	Q₁₆×D₁₃	1	2	yes	—
269	C₂×C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
270	C₈	—	13	2	no	—
271	C₈	—	13	2	no	—
272	C₈	C₄⋊C₄×D₁₃	1	2	yes	—
273	C₈	C₂²×C₄×D₁₃	1	1	yes	characteristic
274	C₈	C₂×Q₈×D₁₃	1	1	yes	characteristic
275	C₂²	—	2	2	no	—
276	C₂²	—	13	4	no	—
277	C₂²	—	13	2	no	—
278	C₂²	—	13	2	no	—
279	C₂²	—	26	8	no	—
280	C₂²	—	26	2	no	—
281	C₂²	D₁₃×#107	1	2	yes	—
282	C₂²	C₂×D₈×D₁₃	1	1	yes	characteristic
283	C₄	—	8	4	no	—
284	C₄	—	13	2	no	—
285	C₄	—	13	2	no	—
286	C₄	—	104	4	no	—
287	C₄	D₁₃×#107	1	2	yes	—
288	C₄	C₂×C₄⋊C₄×D₁₃	1	1	yes	characteristic
289	C₄	C₂×Q₁₆×D₁₃	1	1	yes	characteristic
290	C₂	—	2	2	no	—
291	C₂	—	13	4	no	—
292	C₂	—	26	2	no	—
293	C₂	C₂×D₁₃×#107	1	1	yes	characteristic
294	C₂	D₁₃×#508	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	D₈×D₁₃
commutator	C₈×C₁₃	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₁₃
Fitting	C₁₃×#2158	C₂
socle	C₂²×C₁₃	C₂²×D₈
prime radical	C₂³×D₁₃	D₈
cosocle	C₂×C₈×C₁₃	C₂⁴
hypercenter	#2158	D₁₃
hypocenter	C₁₃	C₂×#2158
LCS	G ▷ C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #2158

Complements

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

N	H ≅ G/N	type
3. #13116620	C₂	semidirect
4. #13116619	C₂	semidirect
5. #13116618	C₂	semidirect
6. #13116617	C₂	semidirect
7. D₁₃×#367	C₂	semidirect
8. D₁₃×#366	C₂	semidirect
9. C₁₃×#2158	C₂	semidirect
17. C₂×C₁₆×D₁₃	C₄	semidirect
26. C₁₃×#367	C₂²	semidirect
27. C₁₃×#366	C₂²	semidirect
28. #11718097	C₂²	semidirect
29. #11718096	C₂²	semidirect
30. #11718082	C₂²	semidirect
31. #11718081	C₂²	semidirect
49. C₂×C₁₃×C₁₆	C₂×C₄	semidirect
56. C₂×#6090	C₂×C₄	semidirect
63. C₁₆×D₁₃	C₂×C₄	semidirect
64. C₁₆×D₁₃	C₂×C₄	semidirect
108. #2158	D₁₃	direct (N × H)
155. #367	C₂×D₁₃	semidirect
158. #366	C₂×D₁₃	semidirect
198. C₂×C₁₆	C₄×D₁₃	semidirect
211. D₁₃	#2158	direct (N × H)
248. C₁₃	C₂×#2158	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²×C₁₆×D₁₃	2
D₁₃×#427	2
#13116620	2
#13116619	2
#13116618	2
#13116617	2
D₁₃×#367	2
D₁₃×#366	2
C₁₃×#2158	2
C₂×#2158	13

Subgroups — D13×#2158 #16015944

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₁₃×#2158 #16015944