Subgroups — #13123572

Order 1664.

544 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	C₂×M₁₆×D₁₃	C₂	1	2	yes	maximal, complemented
2	C₄×D₄×D₁₃	C₂	1	1	yes	maximal, characteristic
3	C₁₃×#436	C₂	1	1	yes	maximal, characteristic, complemented
4	#11718788	C₂	1	2	yes	maximal, complemented
5	#11718673	C₂	1	1	yes	maximal, characteristic, complemented
6	#11718498	C₂	1	1	yes	maximal, characteristic, complemented
7	#11718495	C₂	1	1	yes	maximal, characteristic, complemented
8	#11718389	C₂	1	2	yes	maximal, complemented
9	#11718387	C₂	1	2	yes	maximal, complemented
10	C₄×#6094	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈×D₅₂	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×D₄×D₁₃	C₄	1	1	yes	characteristic
13	C₂²×C₄×D₁₃	C₂²	1	2	yes	—
14	C₂×C₁₃×M₁₆	C₂²	1	2	yes	complemented
15	C₄×D₄×C₁₃	C₂²	1	1	yes	characteristic
16	D₄×Dic₁₃	C₄	1	1	yes	characteristic
17	C₄×C₂²⋊D₁₃	C₂²	1	2	yes	—
18	C₂×#6099	C₂²	1	2	yes	complemented
19	M₁₆×D₁₃	—	2	8	no	—
20	C₂×#6094	C₂²	1	2	yes	—
21	C₂×#6094	C₂²	1	1	yes	characteristic
22	C₂×#6094	C₂²	1	1	yes	characteristic
23	C₂×C₈×D₁₃	C₂²	1	2	yes	complemented
24	C₂×C₈×D₁₃	C₂²	1	2	yes	complemented
25	#88147	C₄	1	1	yes	characteristic
26	C₄⋊C₄×D₁₃	C₄	1	1	yes	characteristic
27	#88135	C₄	1	2	yes	—
28	C₂²⋊C₄×D₁₃	C₄	1	2	yes	—
29	C₄×D₅₂	C₂²	1	1	yes	characteristic
30	C₄²×D₁₃	C₂²	1	1	yes	characteristic
31	C₁₃×C₄⋊C₈	C₂²	1	1	yes	characteristic, complemented
32	C₁₃×C₂²⋊C₈	C₂²	1	2	yes	complemented
33	C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic, complemented
34	C₂²⋊#1055	C₂²	1	2	yes	complemented
35	#88060	C₂²	1	2	yes	complemented
36	#88060	C₂²	1	2	yes	complemented
37	#88055	C₂²	1	1	yes	characteristic, complemented
38	#88055	C₂²	1	1	yes	characteristic, complemented
39	#88054	C₂²	1	2	yes	complemented
40	C₄⋊#1055	C₂²	1	1	yes	characteristic, complemented
41	C₄×#1055	C₂²	1	1	yes	characteristic, complemented
42	C₂³×D₁₃	C₂×C₄	1	2	yes	—
43	C₂×D₄×C₁₃	C₂×C₄	1	1	yes	characteristic
44	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
45	C₂×C₂²⋊D₁₃	C₂×C₄	1	2	yes	—
46	C₂²×Dic₁₃	C₂×C₄	1	2	yes	—
47	D₄×D₁₃	—	2	4	no	—
48	C₂×D₅₂	C₂×C₄	1	1	yes	characteristic
49	C₂×C₄×D₁₃	—	2	4	no	—
50	C₂×C₄×D₁₃	—	2	4	no	—
51	C₂×C₄×D₁₃	C₂×C₄	1	1	yes	characteristic
52	C₂×C₄×D₁₃	C₂×C₄	1	2	yes	—
53	C₂×C₄×D₁₃	C₂×C₄	1	2	yes	—
54	C₂×C₄×D₁₃	C₂×C₄	1	1	yes	characteristic
55	C₂×C₄×D₁₃	C₂³	1	2	yes	—
56	C₂×C₄×D₁₃	C₂³	1	1	yes	characteristic
57	C₁₃×M₁₆	—	2	4	no	—
58	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
59	C₂×C₈×C₁₃	C₂³	1	2	yes	—
60	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
61	C₁₃×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
62	C₁₃×C₂²⋊C₄	C₂×C₄	1	2	yes	—
63	C₄²×C₁₃	C₂³	1	1	yes	characteristic
64	C₂²⋊Dic₁₃	C₂×C₄	1	2	yes	—
65	#6103	C₂×C₄	1	2	yes	—
66	#6103	C₂×C₄	1	2	yes	—
67	C₄⋊Dic₁₃	C₂×C₄	1	1	yes	characteristic
68	#6101	C₂×C₄	1	2	yes	—
69	C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
70	C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
71	C₄×Dic₁₃	C₂³	1	1	yes	characteristic
72	#6099	—	2	4	no	—
73	C₂×#1055	C₂³	1	1	yes	characteristic
74	C₂×#1055	C₂³	1	2	yes	—
75	C₂×#1055	C₂³	1	1	yes	characteristic
76	#6094	—	2	4	no	—
77	#6094	—	2	2	no	—
78	#6094	D₄	1	4	yes	complemented
79	C₈×D₁₃	—	2	4	no	—
80	C₈×D₁₃	—	2	4	no	—
81	D₄×M₁₆	—	13	1	no	maximal, Sylow
82	C₂³×C₁₃	C₂²×C₄	1	2	yes	—
83	C₂²×D₁₃	—	2	4	no	—
84	C₂²×D₁₃	—	2	4	no	—
85	C₂²×D₁₃	—	4	2	no	—
86	C₂²×D₁₃	C₂²×C₄	1	1	yes	characteristic
87	C₂²×D₁₃	C₂²×C₄	1	2	yes	—
88	D₄×C₁₃	—	2	2	no	—
89	C₂×C₄×C₁₃	—	2	2	no	—
90	C₂×C₄×C₁₃	—	2	2	no	—
91	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
92	C₂×C₄×C₁₃	C₂²×C₄	1	2	yes	—
93	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
94	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
95	C₂²⋊D₁₃	—	2	4	no	—
96	C₂×Dic₁₃	—	2	4	no	—
97	C₂×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
98	C₂×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
99	C₂×Dic₁₃	C₂²×C₄	1	2	yes	—
100	C₂×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
101	D₅₂	M₁₆	1	4	yes	complemented
102	C₄×D₁₃	—	2	4	no	—
103	C₄×D₁₃	—	2	2	no	—
104	C₄×D₁₃	—	2	2	no	—
105	C₄×D₁₃	—	2	2	no	—
106	C₄×D₁₃	—	2	4	no	—
107	C₄×D₁₃	—	4	1	no	—
108	C₄×D₁₃	C₂×D₄	1	2	yes	—
109	C₄×D₁₃	#41	1	2	yes	—
110	C₈×C₁₃	—	2	2	no	—
111	C₈×C₁₃	—	2	1	no	—
112	C₈×C₁₃	C₂×D₄	1	2	yes	complemented
113	#1055	—	2	2	no	—
114	#1055	—	2	1	no	—
115	#1055	C₂×D₄	1	2	yes	complemented
116	C₂²×M₁₆	—	13	2	no	—
117	C₂×C₄×D₄	—	13	1	no	—
118	#436	—	13	1	no	—
119	#436	D₁₃	1	1	yes	characteristic, complemented
120	#435	—	13	2	no	—
121	#435	—	13	2	no	—
122	C₈×D₄	—	13	1	no	—
123	C₈×D₄	—	13	1	no	—
124	#423	—	13	1	no	—
125	#407	—	13	2	no	—
126	C₄×M₁₆	—	13	1	no	—
127	C₂²×C₁₃	—	2	2	no	—
128	C₂²×C₁₃	—	2	2	no	—
129	C₂²×C₁₃	—	2	2	no	—
130	C₂²×C₁₃	C₂³×C₄	1	1	yes	characteristic
131	C₂×D₁₃	—	2	2	no	—
132	C₂×D₁₃	—	2	1	no	—
133	C₂×D₁₃	—	2	1	no	—
134	C₂×D₁₃	—	2	2	no	—
135	C₂×D₁₃	—	4	8	no	—
136	C₂×D₁₃	C₄×D₄	1	2	yes	—
137	C₂×D₁₃	C₂×M₁₆	1	4	yes	complemented
138	C₄×C₁₃	—	2	2	no	—
139	C₄×C₁₃	—	2	1	no	—
140	C₄×C₁₃	C₂×M₁₆	1	2	yes	complemented
141	C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
142	C₄×C₁₃	C₂×#41	1	1	yes	characteristic
143	Dic₁₃	—	2	2	no	—
144	Dic₁₃	—	2	1	no	—
145	Dic₁₃	—	2	1	no	—
146	Dic₁₃	C₄×D₄	1	2	yes	—
147	C₂²×D₄	—	13	1	no	—
148	C₂³×C₄	—	13	2	no	—
149	C₂×M₁₆	—	13	2	no	—
150	C₂×M₁₆	—	13	1	no	—
151	C₂×M₁₆	—	13	2	no	—
152	C₂×M₁₆	—	13	1	no	—
153	C₂×M₁₆	—	26	8	no	—
154	C₂×M₁₆	C₂×D₁₃	1	2	yes	complemented
155	C₂²×C₈	—	13	2	no	—
156	C₂²×C₈	—	13	2	no	—
157	C₄×D₄	—	13	1	no	—
158	C₄×D₄	—	13	2	no	—
159	C₄×D₄	—	13	1	no	—
160	C₄×D₄	—	13	2	no	—
161	C₄×D₄	—	13	1	no	—
162	C₄×D₄	C₂×D₁₃	1	1	yes	characteristic
163	C₂×C₄⋊C₄	—	13	1	no	—
164	C₂×C₂²⋊C₄	—	13	2	no	—
165	C₂×C₄²	—	13	1	no	—
166	C₄⋊C₈	—	13	2	no	—
167	C₄⋊C₈	—	13	1	no	—
168	C₄⋊C₈	C₂×D₁₃	1	1	yes	characteristic, complemented
169	C₂²⋊C₈	—	13	2	no	—
170	C₂²⋊C₈	—	13	2	no	—
171	C₂²⋊C₈	—	13	2	no	—
172	C₂²⋊C₈	C₂×D₁₃	1	2	yes	complemented
173	#97	—	13	1	no	—
174	#97	—	13	1	no	—
175	C₄×C₈	—	13	1	no	—
176	C₄×C₈	C₂×D₁₃	1	1	yes	characteristic, complemented
177	C₂×C₁₃	—	4	2	no	—
178	C₂×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
179	C₂×C₁₃	C₂²×M₁₆	1	1	yes	characteristic
180	C₂×C₁₃	#568	1	1	yes	characteristic
181	D₁₃	—	2	2	no	—
182	D₁₃	—	2	4	no	—
183	C₂⁴	—	13	2	no	—
184	C₂×D₄	—	13	2	no	—
185	C₂×D₄	—	13	1	no	—
186	C₂×D₄	—	26	4	no	—
187	C₂×D₄	C₄×D₁₃	1	1	yes	characteristic
188	C₂²×C₄	—	13	1	no	—
189	C₂²×C₄	—	13	2	no	—
190	C₂²×C₄	—	13	1	no	—
191	C₂²×C₄	—	13	2	no	—
192	C₂²×C₄	—	13	2	no	—
193	C₂²×C₄	—	13	2	no	—
194	C₂²×C₄	—	13	1	no	—
195	C₂²×C₄	—	26	4	no	—
196	C₂²×C₄	—	26	4	no	—
197	C₂²×C₄	C₂²×D₁₃	1	2	yes	—
198	M₁₆	—	2	4	no	—
199	M₁₆	—	13	4	no	—
200	M₁₆	—	26	4	no	—
201	M₁₆	—	26	2	no	—
202	M₁₆	—	26	4	no	—
203	C₂×C₈	—	13	1	no	—
204	C₂×C₈	—	13	1	no	—
205	C₂×C₈	—	13	2	no	—
206	C₂×C₈	—	26	4	no	—
207	C₂×C₈	—	26	4	no	—
208	C₂×C₈	C₂²×D₁₃	1	1	yes	characteristic
209	C₂×C₈	C₂²×D₁₃	1	2	yes	—
210	C₂×C₈	C₂²×D₁₃	1	1	yes	characteristic
211	C₄⋊C₄	—	13	2	no	—
212	C₄⋊C₄	—	13	1	no	—
213	C₄⋊C₄	C₄×D₁₃	1	1	yes	characteristic
214	C₂²⋊C₄	—	13	2	no	—
215	C₂²⋊C₄	—	13	2	no	—
216	C₂²⋊C₄	—	13	2	no	—
217	C₂²⋊C₄	C₄×D₁₃	1	2	yes	—
218	C₄²	—	13	1	no	—
219	C₄²	—	13	1	no	—
220	C₄²	—	13	1	no	—
221	C₄²	C₂²×D₁₃	1	1	yes	characteristic
222	C₁₃	D₄×M₁₆	1	1	yes	characteristic, complemented, Sylow
223	C₂³	—	13	2	no	—
224	C₂³	—	13	1	no	—
225	C₂³	—	26	4	no	—
226	C₂³	—	26	4	no	—
227	C₂³	—	52	2	no	—
228	C₂³	C₂×C₄×D₁₃	1	2	yes	—
229	D₄	—	2	2	no	—
230	D₄	—	13	4	no	—
231	D₄	—	26	4	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	13	1	no	—
235	C₂×C₄	—	13	2	no	—
236	C₂×C₄	—	13	2	no	—
237	C₂×C₄	—	13	1	no	—
238	C₂×C₄	—	13	1	no	—
239	C₂×C₄	—	13	2	no	—
240	C₂×C₄	—	26	4	no	—
241	C₂×C₄	—	26	4	no	—
242	C₂×C₄	—	26	4	no	—
243	C₂×C₄	—	26	2	no	—
244	C₂×C₄	—	26	2	no	—
245	C₂×C₄	—	26	2	no	—
246	C₂×C₄	—	52	1	no	—
247	C₂×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
248	C₂×C₄	C₂×C₄×D₁₃	1	2	yes	—
249	C₂×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
250	C₂×C₄	C₂³×D₁₃	1	1	yes	characteristic
251	C₈	—	2	2	no	—
252	C₈	—	2	1	no	—
253	C₈	—	13	2	no	—
254	C₈	—	26	2	no	—
255	C₈	—	26	1	no	—
256	C₈	D₄×D₁₃	1	2	yes	complemented
257	C₂²	—	2	2	no	—
258	C₂²	—	2	2	no	—
259	C₂²	—	2	2	no	—
260	C₂²	—	13	4	no	—
261	C₂²	—	13	2	no	—
262	C₂²	—	26	1	no	—
263	C₂²	—	26	1	no	—
264	C₂²	—	26	2	no	—
265	C₂²	—	26	2	no	—
266	C₂²	—	52	8	no	—
267	C₂²	C₂²×C₄×D₁₃	1	1	yes	characteristic
268	C₄	—	2	1	no	—
269	C₄	—	2	2	no	—
270	C₄	—	13	2	no	—
271	C₄	—	26	1	no	—
272	C₄	—	26	1	no	—
273	C₄	—	26	2	no	—
274	C₄	M₁₆×D₁₃	1	2	yes	complemented
275	C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
276	C₄	#41×D₁₃	1	1	yes	characteristic
277	C₂	—	4	2	no	—
278	C₂	—	26	2	no	—
279	C₂	—	26	4	no	—
280	C₂	C₄×D₄×D₁₃	1	1	yes	characteristic
281	C₂	C₂×M₁₆×D₁₃	1	1	yes	characteristic
282	C₂	#11719455	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₃
commutator, socle	C₂²×C₁₃	C₂³×C₄
Fitting	C₁₃×#436	C₂
prime radical	C₂×D₄×D₁₃	C₄
cosocle	C₂×C₄×C₁₃	C₂⁴
hypercenter	#436	D₁₃
hypocenter	C₁₃	D₄×M₁₆
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ #436

Complements

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

N	H ≅ G/N	type
1. C₂×M₁₆×D₁₃	C₂	semidirect
3. C₁₃×#436	C₂	semidirect
4. #11718788	C₂	semidirect
5. #11718673	C₂	semidirect
6. #11718498	C₂	semidirect
7. #11718495	C₂	semidirect
8. #11718389	C₂	semidirect
9. #11718387	C₂	semidirect
10. C₄×#6094	C₂	semidirect
11. C₈×D₅₂	C₂	semidirect
14. C₂×C₁₃×M₁₆	C₂²	semidirect
18. C₂×#6099	C₂²	semidirect
23. C₂×C₈×D₁₃	C₂²	semidirect
24. C₂×C₈×D₁₃	C₂²	semidirect
31. C₁₃×C₄⋊C₈	C₂²	semidirect
32. C₁₃×C₂²⋊C₈	C₂²	semidirect
33. C₄×C₈×C₁₃	C₂²	semidirect
34. C₂²⋊#1055	C₂²	semidirect
35. #88060	C₂²	semidirect
36. #88060	C₂²	semidirect
37. #88055	C₂²	semidirect
38. #88055	C₂²	semidirect
39. #88054	C₂²	semidirect
40. C₄⋊#1055	C₂²	semidirect
41. C₄×#1055	C₂²	semidirect
78. #6094	D₄	semidirect
101. D₅₂	M₁₆	semidirect
112. C₈×C₁₃	C₂×D₄	semidirect
115. #1055	C₂×D₄	semidirect
119. #436	D₁₃	semidirect
137. C₂×D₁₃	C₂×M₁₆	semidirect
140. C₄×C₁₃	C₂×M₁₆	semidirect
154. C₂×M₁₆	C₂×D₁₃	semidirect
168. C₄⋊C₈	C₂×D₁₃	semidirect
172. C₂²⋊C₈	C₂×D₁₃	semidirect
176. C₄×C₈	C₂×D₁₃	semidirect
222. C₁₃	D₄×M₁₆	semidirect
256. C₈	D₄×D₁₃	semidirect
274. C₄	M₁₆×D₁₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×M₁₆×D₁₃	2
C₄×D₄×D₁₃	2
C₁₃×#436	2
#11718788	2
#11718673	2
#11718498	2
#11718495	2
#11718389	2
#11718387	2
C₄×#6094	2
C₈×D₅₂	2
D₄×M₁₆	13