Subgroups — #13123561

Order 1664.

502 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₁₃×M₁₆	C₂	1	1	yes	maximal, characteristic, complemented
2	#11719496	C₂	1	1	yes	maximal, characteristic
3	C₂×#88161	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₂²⋊#1055	C₂	1	1	yes	maximal, characteristic, complemented
5	#11718790	C₂	1	1	yes	maximal, characteristic, complemented
6	#11718789	C₂	1	2	yes	maximal, complemented
7	#11718788	C₂	1	2	yes	maximal, complemented
8	#11718783	C₂	1	1	yes	maximal, characteristic, complemented
9	#11718780	C₂	1	1	yes	maximal, characteristic, complemented
10	#11718772	C₂	1	2	yes	maximal, complemented
11	C₈×C₂²⋊D₁₃	C₂	1	2	yes	maximal, complemented
12	C₂³×C₄×C₁₃	C₂²	1	1	yes	characteristic
13	C₂×#6127	C₂²	1	1	yes	characteristic
14	C₂×C₁₃×M₁₆	—	2	4	no	—
15	C₂×C₁₃×M₁₆	C₂²	1	1	yes	characteristic
16	C₂×C₁₃×M₁₆	C₂²	1	2	yes	—
17	C₂×C₁₃×M₁₆	C₂²	1	1	yes	characteristic
18	C₂²×C₈×C₁₃	C₂²	1	2	yes	—
19	#88207	C₄	1	2	yes	—
20	#88184	C₄	1	1	yes	characteristic
21	#88184	C₄	1	1	yes	characteristic
22	#88183	C₄	1	2	yes	—
23	C₄×C₂²⋊D₁₃	C₂²	1	2	yes	—
24	C₄×C₂²⋊D₁₃	C₂²	1	2	yes	—
25	#88180	C₂²	1	1	yes	characteristic
26	#88178	C₄	1	1	yes	characteristic
27	#88178	C₄	1	1	yes	characteristic
28	C₂²×#1055	C₂²	1	1	yes	characteristic
29	C₂²×#1055	C₂²	1	1	yes	characteristic
30	#88161	—	2	4	no	—
31	C₂×#6094	C₂²	1	2	yes	complemented
32	C₂×C₈×D₁₃	C₂²	1	2	yes	complemented
33	C₂²⋊#1055	C₂²	1	1	yes	characteristic
34	C₂²⋊#1055	C₂²	1	1	yes	characteristic
35	C₂²⋊#1055	C₂²	1	1	yes	characteristic
36	C₂²⋊#1055	C₂²	1	1	yes	characteristic
37	#88060	C₂²	1	2	yes	complemented
38	#88060	C₂²	1	2	yes	complemented
39	#88055	C₂²	1	2	yes	complemented
40	#88054	C₂²	1	2	yes	complemented
41	#88054	C₂²	1	2	yes	complemented
42	C₈×Dic₁₃	C₂²	1	2	yes	complemented
43	C₂⁴×C₁₃	C₂×C₄	1	1	yes	characteristic
44	C₂²×C₄×C₁₃	—	2	2	no	—
45	C₂²×C₄×C₁₃	—	2	2	no	—
46	C₂²×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
47	C₂²×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
48	C₂²×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
49	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
50	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
51	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
52	C₂×C₂²⋊D₁₃	C₂×C₄	1	2	yes	—
53	C₂×C₂²⋊D₁₃	C₂×C₄	1	2	yes	—
54	C₂×C₂²⋊D₁₃	C₂×C₄	1	2	yes	—
55	#6127	—	2	2	no	—
56	#6127	—	2	2	no	—
57	C₂×D₅₂	C₂×C₄	1	1	yes	characteristic
58	C₂×C₄×D₁₃	C₂³	1	2	yes	—
59	C₂×Dic₂₆	C₂×C₄	1	1	yes	characteristic
60	C₁₃×M₁₆	—	2	2	no	—
61	C₁₃×M₁₆	—	2	4	no	—
62	C₁₃×M₁₆	D₄	1	4	yes	complemented
63	C₂×C₈×C₁₃	—	2	4	no	—
64	C₂×C₈×C₁₃	C₂³	1	2	yes	—
65	C₂×C₈×C₁₃	C₂³	1	2	yes	—
66	C₂²⋊Dic₁₃	C₂×C₄	1	2	yes	—
67	C₂²⋊Dic₁₃	C₂×C₄	1	2	yes	—
68	C₂²⋊Dic₁₃	C₂×C₄	1	2	yes	—
69	#6103	C₂×C₄	1	2	yes	—
70	#6103	C₂×C₄	1	2	yes	—
71	C₄⋊Dic₁₃	C₂×C₄	1	1	yes	characteristic
72	C₄⋊Dic₁₃	C₂×C₄	1	1	yes	characteristic
73	#6101	C₂×C₄	1	2	yes	—
74	#6101	C₂×C₄	1	2	yes	—
75	C₄×Dic₁₃	C₂³	1	2	yes	—
76	C₂×#1055	—	2	2	no	—
77	C₂×#1055	—	2	2	no	—
78	C₂×#1055	C₂³	1	1	yes	characteristic
79	C₂×#1055	C₂³	1	1	yes	characteristic
80	C₂×#1055	C₂³	1	1	yes	characteristic
81	C₂×#1055	C₂³	1	1	yes	characteristic
82	#6094	—	2	4	no	—
83	C₈×D₁₃	—	2	4	no	—
84	#2933	—	13	1	no	maximal, Sylow
85	C₂³×C₁₃	—	2	2	no	—
86	C₂³×C₁₃	—	2	2	no	—
87	C₂³×C₁₃	—	4	1	no	—
88	C₂³×C₁₃	C₂²×C₄	1	1	yes	characteristic
89	C₂³×C₁₃	C₂²×C₄	1	1	yes	characteristic
90	C₂³×C₁₃	C₂²×C₄	1	1	yes	characteristic
91	C₂²×D₁₃	C₂²×C₄	1	2	yes	—
92	C₂×C₄×C₁₃	—	2	2	no	—
93	C₂×C₄×C₁₃	—	2	2	no	—
94	C₂×C₄×C₁₃	—	2	2	no	—
95	C₂×C₄×C₁₃	—	2	1	no	—
96	C₂×C₄×C₁₃	—	2	1	no	—
97	C₂×C₄×C₁₃	—	2	1	no	—
98	C₂×C₄×C₁₃	—	2	1	no	—
99	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
100	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
101	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
102	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
103	C₂×C₄×C₁₃	#41	1	2	yes	—
104	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
105	C₂²⋊D₁₃	—	2	4	no	—
106	C₂²⋊D₁₃	—	2	4	no	—
107	C₂²⋊D₁₃	—	4	2	no	—
108	C₂×Dic₁₃	C₂²×C₄	1	2	yes	—
109	C₂×Dic₁₃	C₂²×C₄	1	2	yes	—
110	C₂×Dic₁₃	C₂²×C₄	1	2	yes	—
111	D₅₂	—	2	2	no	—
112	C₄×D₁₃	—	2	2	no	—
113	C₄×D₁₃	—	2	2	no	—
114	Dic₂₆	—	2	2	no	—
115	C₈×C₁₃	—	2	2	no	—
116	C₈×C₁₃	C₂×D₄	1	4	yes	—
117	#1055	—	2	1	no	—
118	#1055	—	2	1	no	—
119	#1055	—	2	1	no	—
120	#1055	—	2	1	no	—
121	C₂×#131	—	13	1	no	—
122	C₂²×M₁₆	D₁₃	1	1	yes	characteristic, complemented
123	#525	—	13	1	no	—
124	#436	—	13	2	no	—
125	#435	—	13	2	no	—
126	#435	—	13	2	no	—
127	C₈×D₄	—	13	2	no	—
128	#424	—	13	1	no	—
129	#408	—	13	1	no	—
130	C₂×C₂²⋊C₈	—	13	1	no	—
131	#405	—	13	1	no	—
132	C₂²×C₁₃	—	2	1	no	—
133	C₂²×C₁₃	—	2	1	no	—
134	C₂²×C₁₃	—	2	1	no	—
135	C₂²×C₁₃	—	2	1	no	—
136	C₂²×C₁₃	—	2	1	no	—
137	C₂²×C₁₃	—	2	1	no	—
138	C₂²×C₁₃	—	2	1	no	—
139	C₂²×C₁₃	—	4	4	no	—
140	C₂²×C₁₃	C₄×D₄	1	2	yes	—
141	C₂²×C₁₃	#131	1	2	yes	complemented
142	C₂²×C₁₃	C₂³×C₄	1	1	yes	characteristic
143	C₂×D₁₃	—	2	2	no	—
144	C₂×D₁₃	—	2	2	no	—
145	C₂×D₁₃	—	2	2	no	—
146	C₄×C₁₃	—	2	1	no	—
147	C₄×C₁₃	—	2	1	no	—
148	C₄×C₁₃	C₄×D₄	1	2	yes	—
149	C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
150	C₄×C₁₃	C₂×#41	1	1	yes	characteristic
151	Dic₁₃	—	2	2	no	—
152	Dic₁₃	—	2	2	no	—
153	Dic₁₃	—	2	2	no	—
154	C₂×#41	—	13	1	no	—
155	C₂³×C₄	C₂×D₁₃	1	1	yes	characteristic
156	#131	—	26	4	no	—
157	C₂×M₁₆	—	2	4	no	—
158	C₂×M₁₆	—	13	2	no	—
159	C₂×M₁₆	C₂×D₁₃	1	1	yes	characteristic
160	C₂×M₁₆	C₂×D₁₃	1	2	yes	—
161	C₂×M₁₆	C₂×D₁₃	1	1	yes	characteristic
162	C₂²×C₈	—	13	1	no	—
163	C₂²×C₈	—	13	1	no	—
164	C₂²×C₈	—	13	2	no	—
165	C₂²×C₈	C₂×D₁₃	1	2	yes	—
166	#123	—	13	2	no	—
167	C₂²⋊Q₈	—	13	1	no	—
168	C₂²⋊Q₈	—	13	1	no	—
169	#121	—	13	1	no	—
170	#121	—	13	1	no	—
171	#120	—	13	2	no	—
172	C₄×D₄	—	13	2	no	—
173	C₄×D₄	—	13	2	no	—
174	#117	—	13	1	no	—
175	C₄⋊C₈	—	13	2	no	—
176	C₄⋊C₈	—	13	2	no	—
177	C₂²⋊C₈	—	13	1	no	—
178	C₂²⋊C₈	—	13	1	no	—
179	C₂²⋊C₈	—	13	1	no	—
180	C₂²⋊C₈	—	13	1	no	—
181	C₂²⋊C₈	—	13	2	no	—
182	C₂²⋊C₈	—	13	2	no	—
183	#97	—	13	2	no	—
184	C₄×C₈	—	13	2	no	—
185	C₂×C₁₃	—	2	2	no	—
186	C₂×C₁₃	—	2	2	no	—
187	C₂×C₁₃	—	4	1	no	—
188	C₂×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
189	C₂×C₁₃	C₂×#131	1	1	yes	characteristic
190	C₂×C₁₃	#568	1	1	yes	characteristic
191	D₁₃	—	4	2	no	—
192	C₂⁴	C₄×D₁₃	1	1	yes	characteristic
193	#41	—	26	2	no	—
194	#41	—	26	2	no	—
195	C₂×Q₈	—	13	1	no	—
196	C₂×D₄	—	13	2	no	—
197	C₂×D₄	—	13	2	no	—
198	C₂×D₄	—	13	2	no	—
199	C₂×D₄	—	13	1	no	—
200	C₂²×C₄	—	2	2	no	—
201	C₂²×C₄	—	2	2	no	—
202	C₂²×C₄	—	13	2	no	—
203	C₂²×C₄	C₄×D₁₃	1	1	yes	characteristic
204	C₂²×C₄	C₄×D₁₃	1	1	yes	characteristic
205	C₂²×C₄	C₄×D₁₃	1	1	yes	characteristic
206	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
207	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
208	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
209	M₁₆	—	2	2	no	—
210	M₁₆	—	2	4	no	—
211	M₁₆	—	26	4	no	—
212	M₁₆	C₂²⋊D₁₃	1	4	yes	complemented
213	C₂×C₈	—	2	4	no	—
214	C₂×C₈	—	13	1	no	—
215	C₂×C₈	—	13	1	no	—
216	C₂×C₈	—	13	1	no	—
217	C₂×C₈	—	13	1	no	—
218	C₂×C₈	—	26	2	no	—
219	C₂×C₈	—	26	2	no	—
220	C₂×C₈	—	26	4	no	—
221	C₂×C₈	C₂²×D₁₃	1	2	yes	—
222	C₂×C₈	C₂²×D₁₃	1	2	yes	—
223	C₄⋊C₄	—	13	1	no	—
224	C₄⋊C₄	—	13	1	no	—
225	C₄⋊C₄	—	13	2	no	—
226	C₄⋊C₄	—	13	2	no	—
227	C₂²⋊C₄	—	13	2	no	—
228	C₂²⋊C₄	—	13	2	no	—
229	C₂²⋊C₄	—	13	2	no	—
230	C₂²⋊C₄	—	13	2	no	—
231	C₂²⋊C₄	—	13	2	no	—
232	C₄²	—	13	2	no	—
233	C₁₃	#2933	1	1	yes	characteristic, complemented, Sylow
234	C₂³	—	2	2	no	—
235	C₂³	—	2	2	no	—
236	C₂³	—	4	1	no	—
237	C₂³	—	13	2	no	—
238	C₂³	C₂×C₄×D₁₃	1	1	yes	characteristic
239	C₂³	C₂×C₄×D₁₃	1	1	yes	characteristic
240	C₂³	C₂×C₄×D₁₃	1	1	yes	characteristic
241	Q₈	—	26	2	no	—
242	D₄	—	26	4	no	—
243	D₄	—	26	4	no	—
244	D₄	—	26	2	no	—
245	D₄	—	52	2	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	2	2	no	—
248	C₂×C₄	—	2	2	no	—
249	C₂×C₄	—	2	1	no	—
250	C₂×C₄	—	2	1	no	—
251	C₂×C₄	—	2	1	no	—
252	C₂×C₄	—	2	1	no	—
253	C₂×C₄	—	13	2	no	—
254	C₂×C₄	—	13	2	no	—
255	C₂×C₄	—	13	2	no	—
256	C₂×C₄	—	26	2	no	—
257	C₂×C₄	—	26	2	no	—
258	C₂×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
259	C₂×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
260	C₂×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
261	C₂×C₄	#6127	1	2	yes	—
262	C₂×C₄	C₂×C₂²⋊D₁₃	1	2	yes	—
263	C₂×C₄	C₂³×D₁₃	1	1	yes	characteristic
264	C₈	—	2	2	no	—
265	C₈	—	26	1	no	—
266	C₈	—	26	1	no	—
267	C₈	—	26	1	no	—
268	C₈	—	26	1	no	—
269	C₈	C₂×C₂²⋊D₁₃	1	4	yes	—
270	C₂²	—	2	1	no	—
271	C₂²	—	2	1	no	—
272	C₂²	—	2	1	no	—
273	C₂²	—	2	1	no	—
274	C₂²	—	2	1	no	—
275	C₂²	—	2	1	no	—
276	C₂²	—	2	1	no	—
277	C₂²	—	4	4	no	—
278	C₂²	—	26	2	no	—
279	C₂²	—	26	2	no	—
280	C₂²	—	26	2	no	—
281	C₂²	#88161	1	2	yes	complemented
282	C₂²	C₄×C₂²⋊D₁₃	1	2	yes	—
283	C₂²	C₂²×C₄×D₁₃	1	1	yes	characteristic
284	C₄	—	2	1	no	—
285	C₄	—	2	1	no	—
286	C₄	—	26	2	no	—
287	C₄	—	26	2	no	—
288	C₄	—	26	2	no	—
289	C₄	C₄×C₂²⋊D₁₃	1	2	yes	—
290	C₄	C₂×#6127	1	1	yes	characteristic
291	C₄	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
292	C₂	—	2	2	no	—
293	C₂	—	2	2	no	—
294	C₂	—	4	1	no	—
295	C₂	—	52	2	no	—
296	C₂	C₂×#88161	1	1	yes	characteristic
297	C₂	#11719450	1	1	yes	characteristic
298	C₂	C₂×C₄×C₂²⋊D₁₃	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₃
commutator, socle	C₂²×C₁₃	C₂³×C₄
Fitting	C₂²×C₁₃×M₁₆	C₂
prime radical	#11719496	C₂
cosocle	C₂×C₄×C₁₃	C₂⁴
hypercenter	C₂²×M₁₆	D₁₃
hypocenter	C₁₃	#2933
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×M₁₆

Complements

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

N	H ≅ G/N	type
1. C₂²×C₁₃×M₁₆	C₂	semidirect
3. C₂×#88161	C₂	semidirect
4. C₂×C₂²⋊#1055	C₂	semidirect
5. #11718790	C₂	semidirect
6. #11718789	C₂	semidirect
7. #11718788	C₂	semidirect
8. #11718783	C₂	semidirect
9. #11718780	C₂	semidirect
10. #11718772	C₂	semidirect
11. C₈×C₂²⋊D₁₃	C₂	semidirect
31. C₂×#6094	C₂²	semidirect
32. C₂×C₈×D₁₃	C₂²	semidirect
37. #88060	C₂²	semidirect
38. #88060	C₂²	semidirect
39. #88055	C₂²	semidirect
40. #88054	C₂²	semidirect
41. #88054	C₂²	semidirect
42. C₈×Dic₁₃	C₂²	semidirect
62. C₁₃×M₁₆	D₄	semidirect
122. C₂²×M₁₆	D₁₃	semidirect
141. C₂²×C₁₃	#131	semidirect
212. M₁₆	C₂²⋊D₁₃	semidirect
233. C₁₃	#2933	semidirect
281. C₂²	#88161	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₁₃×M₁₆	2
#11719496	2
C₂×#88161	2
C₂×C₂²⋊#1055	2
#11718790	2
#11718789	2
#11718788	2
#11718783	2
#11718780	2
#11718772	2
C₈×C₂²⋊D₁₃	2
#2933	13