Subgroups — #13123750

Order 1664.

452 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₁₃×#135	C₂	1	1	yes	maximal, characteristic
2	#11719455	C₂	1	2	yes	maximal, complemented
3	C₁₃×#444	C₂	1	1	yes	maximal, characteristic, complemented
4	#11718858	C₂	1	1	yes	maximal, characteristic, complemented
5	#11718805	C₂	1	2	yes	maximal, complemented
6	#11718555	C₂	1	1	yes	maximal, characteristic, complemented
7	D₁₃×#108	C₂	1	1	yes	maximal, characteristic, complemented
8	#11718485	C₂	1	2	yes	maximal, complemented
9	#11718482	C₂	1	2	yes	maximal, complemented
10	#11718378	C₂	1	1	yes	maximal, characteristic, complemented
11	#11718369	C₂	1	1	yes	maximal, characteristic, complemented
12	#41×D₁₃	C₂²	1	2	yes	—
13	C₁₃×#135	C₂²	1	1	yes	characteristic
14	C₁₃×#131	C₂²	1	2	yes	complemented
15	#88204	C₂²	1	2	yes	—
16	#88202	C₂²	1	2	yes	complemented
17	#88173	C₄	1	1	yes	characteristic, complemented
18	Q₁₆×D₁₃	C₄	1	1	yes	characteristic, complemented
19	#88170	C₄	1	2	yes	complemented
20	SD₁₆×D₁₃	C₄	1	2	yes	complemented
21	#88166	C₄	1	1	yes	characteristic, complemented
22	D₈×D₁₃	C₄	1	1	yes	characteristic, complemented
23	#88161	—	2	2	no	—
24	#88161	C₂²	1	2	yes	complemented
25	M₁₆×D₁₃	—	2	2	no	—
26	M₁₆×D₁₃	C₂²	1	2	yes	—
27	#88158	C₂²	1	1	yes	characteristic
28	#88155	—	2	2	no	—
29	#88155	C₂²	1	2	yes	complemented
30	C₂×#6094	—	2	2	no	—
31	C₂×#6094	C₂²	1	1	yes	characteristic
32	C₂×C₈×D₁₃	C₂²	1	1	yes	characteristic
33	#88126	C₂²	1	1	yes	characteristic
34	C₁₃×#108	C₂²	1	1	yes	characteristic, complemented
35	C₁₃×#104	C₂²	1	2	yes	complemented
36	C₁₃×#97	C₂²	1	1	yes	characteristic, complemented
37	#88077	C₂²	1	2	yes	complemented
38	#88065	C₂²	1	2	yes	complemented
39	#88062	C₂²	1	2	yes	complemented
40	#88059	C₂²	1	1	yes	characteristic, complemented
41	#88055	C₂²	1	1	yes	characteristic, complemented
42	C₈×Dic₁₃	C₂²	1	1	yes	characteristic, complemented
43	#88045	C₂²	1	2	yes	complemented
44	C₄×#1055	C₂²	1	1	yes	characteristic, complemented
45	C₁₃×#41	C₂³	1	2	yes	—
46	#6131	C₂×C₄	1	2	yes	—
47	Q₈×D₁₃	C₂×C₄	1	2	yes	—
48	#6129	—	2	2	no	—
49	#6129	C₂×C₄	1	2	yes	—
50	D₄×D₁₃	—	2	2	no	—
51	D₄×D₁₃	C₂×C₄	1	2	yes	—
52	#6127	—	2	2	no	—
53	#6127	C₂³	1	2	yes	—
54	C₂×C₄×D₁₃	—	2	2	no	—
55	C₂×C₄×D₁₃	C₂³	1	1	yes	characteristic
56	C₁₃×Q₁₆	C₂×C₄	1	1	yes	characteristic
57	C₁₃×SD₁₆	C₂×C₄	1	2	yes	—
58	C₁₃×D₈	C₂×C₄	1	1	yes	characteristic
59	C₁₃×M₁₆	—	2	2	no	—
60	C₁₃×M₁₆	C₂³	1	2	yes	—
61	C₂×C₈×C₁₃	—	2	2	no	—
62	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
63	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
64	C₄²×C₁₃	C₂³	1	1	yes	characteristic
65	#6107	C₂×C₄	1	2	yes	—
66	#6106	C₂×C₄	1	2	yes	—
67	#6105	C₂×C₄	1	2	yes	—
68	#6104	C₂×C₄	1	2	yes	—
69	#6103	—	2	1	no	—
70	#6101	—	2	1	no	—
71	C₄×Dic₁₃	C₂³	1	1	yes	characteristic
72	#6099	—	2	2	no	—
73	#6099	C₂³	1	2	yes	—
74	C₂×#1055	—	2	2	no	—
75	C₂×#1055	C₂³	1	1	yes	characteristic
76	C₂×#1055	C₂³	1	1	yes	characteristic
77	Dic₅₂	C₂×C₄	1	1	yes	characteristic
78	D₁₀₄	C₂×C₄	1	1	yes	characteristic
79	#6095	C₂×C₄	1	2	yes	—
80	#6094	—	2	2	no	—
81	#6094	—	2	2	no	—
82	#6094	—	2	2	no	—
83	#6094	D₄	1	2	yes	—
84	#6094	D₄	1	2	yes	—
85	C₈×D₁₃	—	2	2	no	—
86	C₈×D₁₃	—	2	2	no	—
87	C₈×D₁₃	—	2	2	no	—
88	C₈×D₁₃	C₂×C₄	1	1	yes	characteristic
89	C₈×D₁₃	C₂×C₄	1	1	yes	characteristic
90	C₈×D₁₃	C₂×C₄	1	1	yes	characteristic
91	C₈×D₁₃	C₂×C₄	1	1	yes	characteristic
92	#2957	—	13	1	no	maximal, Sylow
93	C₂²×D₁₃	—	2	2	no	—
94	C₂²×D₁₃	#41	1	1	yes	characteristic
95	Q₈×C₁₃	C₂²×C₄	1	2	yes	—
96	D₄×C₁₃	—	2	2	no	—
97	D₄×C₁₃	C₂²×C₄	1	2	yes	—
98	C₂×C₄×C₁₃	—	2	2	no	—
99	C₂×C₄×C₁₃	—	2	1	no	—
100	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
101	C₂²⋊D₁₃	—	2	2	no	—
102	C₂²⋊D₁₃	—	2	2	no	—
103	C₂²⋊D₁₃	—	2	2	no	—
104	C₂×Dic₁₃	—	2	2	no	—
105	C₂×Dic₁₃	—	2	1	no	—
106	C₂×Dic₁₃	#41	1	1	yes	characteristic
107	D₅₂	—	2	2	no	—
108	D₅₂	C₂²×C₄	1	2	yes	—
109	C₄×D₁₃	—	2	2	no	—
110	C₄×D₁₃	—	2	2	no	—
111	C₄×D₁₃	—	2	2	no	—
112	C₄×D₁₃	C₂²×C₄	1	1	yes	characteristic
113	C₄×D₁₃	C₂²×C₄	1	1	yes	characteristic
114	C₄×D₁₃	C₂×D₄	1	1	yes	characteristic
115	C₄×D₁₃	C₂×D₄	1	1	yes	characteristic
116	Dic₂₆	—	2	2	no	—
117	Dic₂₆	C₂²×C₄	1	2	yes	—
118	C₈×C₁₃	—	2	2	no	—
119	C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
120	C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
121	C₈×C₁₃	C₂×D₄	1	2	yes	—
122	#1055	—	2	2	no	—
123	#1055	C₂²×C₄	1	1	yes	characteristic
124	#1055	C₂²×C₄	1	1	yes	characteristic
125	#1055	C₂×D₄	1	2	yes	—
126	C₂×#135	—	13	1	no	—
127	#568	—	13	2	no	—
128	#444	—	13	1	no	—
129	#444	—	13	1	no	—
130	#444	—	13	1	no	—
131	#444	D₁₃	1	1	yes	characteristic, complemented
132	#443	—	13	2	no	—
133	#443	—	13	2	no	—
134	C₂×#108	—	13	1	no	—
135	#421	—	13	2	no	—
136	#405	—	13	1	no	—
137	C₂²×C₁₃	—	2	2	no	—
138	C₂²×C₁₃	C₂×#41	1	1	yes	characteristic
139	C₂×D₁₃	—	2	2	no	—
140	C₂×D₁₃	—	4	1	no	—
141	C₂×D₁₃	—	8	2	no	—
142	C₂×D₁₃	C₄×D₄	1	1	yes	characteristic
143	C₂×D₁₃	C₄×D₄	1	1	yes	characteristic
144	C₄×C₁₃	—	2	2	no	—
145	C₄×C₁₃	—	4	1	no	—
146	C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
147	C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
148	Dic₁₃	—	2	2	no	—
149	Dic₁₃	—	4	1	no	—
150	Dic₁₃	C₄×D₄	1	1	yes	characteristic
151	Dic₁₃	C₄×D₄	1	1	yes	characteristic
152	C₂×#41	—	13	2	no	—
153	#135	—	13	2	no	—
154	#135	—	13	1	no	—
155	#135	—	13	2	no	—
156	#135	—	13	1	no	—
157	#135	—	13	1	no	—
158	#135	C₂×D₁₃	1	1	yes	characteristic
159	C₂×Q₁₆	—	13	1	no	—
160	C₂×SD₁₆	—	13	2	no	—
161	C₂×D₈	—	13	1	no	—
162	#131	—	13	2	no	—
163	#131	—	13	2	no	—
164	#131	—	13	2	no	—
165	#131	—	26	2	no	—
166	#131	—	26	2	no	—
167	#131	C₂×D₁₃	1	2	yes	complemented
168	C₂×M₁₆	—	13	1	no	—
169	C₂×M₁₆	—	13	2	no	—
170	C₂×M₁₆	—	26	2	no	—
171	C₂×M₁₆	—	26	2	no	—
172	C₂²×C₈	—	13	1	no	—
173	#117	—	13	1	no	—
174	#108	—	13	2	no	—
175	#108	—	13	1	no	—
176	#108	C₂×D₁₃	1	1	yes	characteristic, complemented
177	#104	—	13	2	no	—
178	#104	—	13	2	no	—
179	#104	—	13	2	no	—
180	#104	C₂×D₁₃	1	2	yes	complemented
181	#97	—	13	1	no	—
182	#97	C₂×D₁₃	1	1	yes	characteristic, complemented
183	C₄×C₈	—	13	1	no	—
184	C₄×C₈	—	13	1	no	—
185	C₂×C₁₃	—	2	1	no	—
186	C₂×C₁₃	—	4	2	no	—
187	C₂×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
188	D₁₃	—	2	1	no	—
189	D₁₃	—	2	1	no	—
190	D₁₃	—	4	2	no	—
191	#41	—	13	2	no	—
192	#41	—	13	2	no	—
193	#41	—	13	2	no	—
194	#41	—	26	2	no	—
195	#41	—	26	2	no	—
196	#41	C₂²×D₁₃	1	2	yes	—
197	C₂×Q₈	—	13	2	no	—
198	C₂×D₄	—	13	2	no	—
199	C₂×D₄	—	26	2	no	—
200	C₂²×C₄	—	13	1	no	—
201	C₂²×C₄	—	26	2	no	—
202	Q₁₆	—	13	2	no	—
203	Q₁₆	—	13	1	no	—
204	Q₁₆	C₄×D₁₃	1	1	yes	characteristic
205	SD₁₆	—	13	2	no	—
206	SD₁₆	—	13	2	no	—
207	SD₁₆	—	13	2	no	—
208	SD₁₆	C₄×D₁₃	1	2	yes	—
209	D₈	—	13	2	no	—
210	D₈	—	13	1	no	—
211	D₈	C₄×D₁₃	1	1	yes	characteristic
212	M₁₆	—	2	2	no	—
213	M₁₆	—	13	2	no	—
214	M₁₆	—	13	2	no	—
215	M₁₆	—	13	2	no	—
216	M₁₆	—	26	2	no	—
217	M₁₆	—	26	2	no	—
218	M₁₆	—	26	2	no	—
219	M₁₆	—	26	2	no	—
220	M₁₆	C₂²×D₁₃	1	2	yes	—
221	C₂×C₈	—	2	2	no	—
222	C₂×C₈	—	13	1	no	—
223	C₂×C₈	—	13	1	no	—
224	C₂×C₈	—	13	1	no	—
225	C₂×C₈	—	13	1	no	—
226	C₂×C₈	—	13	1	no	—
227	C₂×C₈	—	13	1	no	—
228	C₂×C₈	—	26	2	no	—
229	C₂×C₈	—	26	2	no	—
230	C₂×C₈	—	26	2	no	—
231	C₂×C₈	—	26	2	no	—
232	C₂×C₈	C₂²×D₁₃	1	1	yes	characteristic
233	C₂×C₈	C₂²×D₁₃	1	1	yes	characteristic
234	C₄⋊C₄	—	26	1	no	—
235	C₂²⋊C₄	—	26	1	no	—
236	C₄²	—	13	1	no	—
237	C₄²	C₂²×D₁₃	1	1	yes	characteristic
238	C₁₃	#2957	1	1	yes	characteristic, complemented, Sylow
239	C₂³	—	13	1	no	—
240	C₂³	—	26	2	no	—
241	Q₈	—	13	2	no	—
242	Q₈	—	26	2	no	—
243	Q₈	C₂×C₄×D₁₃	1	2	yes	—
244	D₄	—	2	2	no	—
245	D₄	—	13	2	no	—
246	D₄	—	26	2	no	—
247	D₄	—	26	2	no	—
248	D₄	—	26	2	no	—
249	D₄	—	26	2	no	—
250	D₄	C₂×C₄×D₁₃	1	2	yes	—
251	C₂×C₄	—	2	2	no	—
252	C₂×C₄	—	2	1	no	—
253	C₂×C₄	—	13	1	no	—
254	C₂×C₄	—	13	1	no	—
255	C₂×C₄	—	13	1	no	—
256	C₂×C₄	—	13	1	no	—
257	C₂×C₄	—	13	1	no	—
258	C₂×C₄	—	26	2	no	—
259	C₂×C₄	—	26	2	no	—
260	C₂×C₄	—	26	2	no	—
261	C₂×C₄	—	26	2	no	—
262	C₂×C₄	—	26	1	no	—
263	C₂×C₄	C₂³×D₁₃	1	1	yes	characteristic
264	C₈	—	2	2	no	—
265	C₈	—	13	1	no	—
266	C₈	—	13	1	no	—
267	C₈	—	13	2	no	—
268	C₈	—	26	2	no	—
269	C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
270	C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
271	C₈	D₄×D₁₃	1	2	yes	—
272	C₂²	—	2	2	no	—
273	C₂²	—	13	1	no	—
274	C₂²	—	13	1	no	—
275	C₂²	—	26	2	no	—
276	C₂²	—	52	1	no	—
277	C₂²	—	104	2	no	—
278	C₂²	#41×D₁₃	1	1	yes	characteristic
279	C₄	—	2	2	no	—
280	C₄	—	4	1	no	—
281	C₄	—	13	1	no	—
282	C₄	—	13	1	no	—
283	C₄	—	26	2	no	—
284	C₄	—	52	1	no	—
285	C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
286	C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
287	C₂	—	2	1	no	—
288	C₂	—	4	2	no	—
289	C₂	—	26	1	no	—
290	C₂	—	26	1	no	—
291	C₂	—	52	2	no	—
292	C₂	C₄×D₄×D₁₃	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×D₄×D₁₃
commutator	C₄×C₁₃	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₃
Fitting	C₁₃×#444	C₂
socle	C₂×C₁₃	C₂×C₄×D₄
prime radical	D₁₃×#135	C₂
cosocle	C₂×C₄×C₁₃	C₂⁴
hypercenter	#444	D₁₃
hypocenter	C₁₃	#2957
LCS	G ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ #444

Complements

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

N	H ≅ G/N	type
2. #11719455	C₂	semidirect
3. C₁₃×#444	C₂	semidirect
4. #11718858	C₂	semidirect
5. #11718805	C₂	semidirect
6. #11718555	C₂	semidirect
7. D₁₃×#108	C₂	semidirect
8. #11718485	C₂	semidirect
9. #11718482	C₂	semidirect
10. #11718378	C₂	semidirect
11. #11718369	C₂	semidirect
14. C₁₃×#131	C₂²	semidirect
16. #88202	C₂²	semidirect
17. #88173	C₄	semidirect
18. Q₁₆×D₁₃	C₄	semidirect
19. #88170	C₄	semidirect
20. SD₁₆×D₁₃	C₄	semidirect
21. #88166	C₄	semidirect
22. D₈×D₁₃	C₄	semidirect
24. #88161	C₂²	semidirect
29. #88155	C₂²	semidirect
34. C₁₃×#108	C₂²	semidirect
35. C₁₃×#104	C₂²	semidirect
36. C₁₃×#97	C₂²	semidirect
37. #88077	C₂²	semidirect
38. #88065	C₂²	semidirect
39. #88062	C₂²	semidirect
40. #88059	C₂²	semidirect
41. #88055	C₂²	semidirect
42. C₈×Dic₁₃	C₂²	semidirect
43. #88045	C₂²	semidirect
44. C₄×#1055	C₂²	semidirect
131. #444	D₁₃	semidirect
167. #131	C₂×D₁₃	semidirect
176. #108	C₂×D₁₃	semidirect
180. #104	C₂×D₁₃	semidirect
182. #97	C₂×D₁₃	semidirect
238. C₁₃	#2957	semidirect

Maximal subgroups

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

subgroup	index
D₁₃×#135	2
#11719455	2
C₁₃×#444	2
#11718858	2
#11718805	2
#11718555	2
D₁₃×#108	2
#11718485	2
#11718482	2
#11718378	2
#11718369	2
#2957	13