Subgroups — #13123734

Order 1664.

456 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11719473	C₂	1	1	yes	maximal, characteristic
2	D₁₃×#131	C₂	1	2	yes	maximal, complemented
3	C₁₃×#443	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	#11718554	C₂	1	1	yes	maximal, characteristic, complemented
8	#11718484	C₂	1	2	yes	maximal, complemented
9	#11718482	C₂	1	2	yes	maximal, complemented
10	#11718361	C₂	1	1	yes	maximal, characteristic, complemented
11	#11718350	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	#88172	C₄	1	2	yes	complemented
18	#88169	C₄	1	2	yes	complemented
19	#88168	C₄	1	2	yes	complemented
20	#88165	C₄	1	2	yes	complemented
21	#88161	—	2	2	no	—
22	#88161	C₂²	1	2	yes	complemented
23	M₁₆×D₁₃	—	2	2	no	—
24	M₁₆×D₁₃	C₂²	1	2	yes	—
25	#88158	C₂²	1	1	yes	characteristic
26	#88155	—	2	2	no	—
27	#88155	C₂²	1	2	yes	complemented
28	C₂×#6094	C₂²	1	1	yes	characteristic
29	C₂×C₈×D₁₃	—	2	2	no	—
30	C₂×C₈×D₁₃	C₂²	1	1	yes	characteristic
31	#88126	C₂²	1	1	yes	characteristic
32	C₁₃×#108	C₂²	1	1	yes	characteristic, complemented
33	C₁₃×#104	C₂²	1	2	yes	complemented
34	C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic, complemented
35	#88077	C₂²	1	2	yes	complemented
36	#88065	C₂²	1	2	yes	complemented
37	#88062	C₂²	1	2	yes	complemented
38	#88059	C₂²	1	1	yes	characteristic, complemented
39	#88055	C₂²	1	1	yes	characteristic, complemented
40	C₈×Dic₁₃	C₂²	1	1	yes	characteristic, complemented
41	#88045	C₂²	1	2	yes	complemented
42	#88043	C₂²	1	1	yes	characteristic, complemented
43	C₁₃×#41	C₂³	1	2	yes	—
44	#6131	C₂×C₄	1	2	yes	—
45	Q₈×D₁₃	C₂×C₄	1	2	yes	—
46	#6129	—	2	2	no	—
47	#6129	C₂×C₄	1	2	yes	—
48	D₄×D₁₃	—	2	2	no	—
49	D₄×D₁₃	C₂×C₄	1	2	yes	—
50	#6127	—	2	2	no	—
51	#6127	C₂³	1	2	yes	—
52	C₂×C₄×D₁₃	—	2	2	no	—
53	C₂×C₄×D₁₃	C₂³	1	1	yes	characteristic
54	C₁₃×Q₁₆	C₂×C₄	1	1	yes	characteristic
55	C₁₃×SD₁₆	C₂×C₄	1	2	yes	—
56	C₁₃×D₈	C₂×C₄	1	1	yes	characteristic
57	C₁₃×M₁₆	—	2	2	no	—
58	C₁₃×M₁₆	C₂³	1	2	yes	—
59	C₂×C₈×C₁₃	—	2	2	no	—
60	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
61	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
62	C₄²×C₁₃	C₂³	1	1	yes	characteristic
63	#6107	C₂×C₄	1	2	yes	—
64	#6106	C₂×C₄	1	2	yes	—
65	#6105	C₂×C₄	1	2	yes	—
66	#6104	C₂×C₄	1	2	yes	—
67	#6103	—	2	1	no	—
68	#6101	—	2	1	no	—
69	C₄×Dic₁₃	C₂³	1	1	yes	characteristic
70	#6099	—	2	2	no	—
71	#6099	C₂³	1	2	yes	—
72	C₂×#1055	—	2	2	no	—
73	C₂×#1055	C₂³	1	1	yes	characteristic
74	C₂×#1055	C₂³	1	1	yes	characteristic
75	Dic₅₂	C₂×C₄	1	1	yes	characteristic
76	D₁₀₄	C₂×C₄	1	1	yes	characteristic
77	#6095	C₂×C₄	1	2	yes	—
78	#6094	—	2	2	no	—
79	#6094	—	2	2	no	—
80	#6094	—	2	2	no	—
81	#6094	C₂×C₄	1	2	yes	—
82	#6094	C₂×C₄	1	2	yes	—
83	C₈×D₁₃	—	2	2	no	—
84	C₈×D₁₃	—	2	2	no	—
85	C₈×D₁₃	—	2	2	no	—
86	C₈×D₁₃	D₄	1	2	yes	—
87	C₈×D₁₃	D₄	1	2	yes	—
88	#2955	—	13	1	no	maximal, Sylow
89	C₂²×D₁₃	—	2	2	no	—
90	C₂²×D₁₃	#41	1	1	yes	characteristic
91	Q₈×C₁₃	C₂²×C₄	1	2	yes	—
92	D₄×C₁₃	—	2	2	no	—
93	D₄×C₁₃	C₂²×C₄	1	2	yes	—
94	C₂×C₄×C₁₃	—	2	2	no	—
95	C₂×C₄×C₁₃	—	2	1	no	—
96	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
97	C₂²⋊D₁₃	—	2	2	no	—
98	C₂²⋊D₁₃	—	2	2	no	—
99	C₂²⋊D₁₃	—	2	2	no	—
100	C₂×Dic₁₃	—	2	2	no	—
101	C₂×Dic₁₃	—	2	1	no	—
102	C₂×Dic₁₃	#41	1	1	yes	characteristic
103	D₅₂	—	2	2	no	—
104	D₅₂	C₂²×C₄	1	2	yes	—
105	C₄×D₁₃	—	2	2	no	—
106	C₄×D₁₃	—	2	2	no	—
107	C₄×D₁₃	—	2	2	no	—
108	C₄×D₁₃	C₂²×C₄	1	2	yes	—
109	C₄×D₁₃	C₂×D₄	1	2	yes	—
110	Dic₂₆	—	2	2	no	—
111	Dic₂₆	C₂²×C₄	1	2	yes	—
112	C₈×C₁₃	—	2	2	no	—
113	C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
114	C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
115	C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
116	C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
117	#1055	—	2	2	no	—
118	#1055	C₂²×C₄	1	2	yes	—
119	#1055	C₂×D₄	1	2	yes	—
120	#575	—	13	1	no	—
121	C₂×#131	—	13	2	no	—
122	#444	—	13	1	no	—
123	#444	—	13	1	no	—
124	#444	—	13	2	no	—
125	#443	—	13	2	no	—
126	#443	—	13	1	no	—
127	#443	D₁₃	1	1	yes	characteristic, complemented
128	#430	—	13	1	no	—
129	#421	—	13	2	no	—
130	#405	—	13	1	no	—
131	C₂²×C₁₃	—	2	2	no	—
132	C₂²×C₁₃	C₂×#41	1	1	yes	characteristic
133	C₂×D₁₃	—	2	2	no	—
134	C₂×D₁₃	—	4	1	no	—
135	C₂×D₁₃	—	4	4	no	—
136	C₂×D₁₃	C₄×D₄	1	2	yes	—
137	C₄×C₁₃	—	2	2	no	—
138	C₄×C₁₃	—	4	1	no	—
139	C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
140	C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
141	Dic₁₃	—	2	2	no	—
142	Dic₁₃	—	4	1	no	—
143	Dic₁₃	C₄×D₄	1	2	yes	—
144	C₂×#41	—	13	2	no	—
145	#137	—	13	2	no	—
146	#137	—	13	2	no	—
147	AGL(1,ℤ/8)	—	13	2	no	—
148	AGL(1,ℤ/8)	—	13	2	no	—
149	#135	—	13	2	no	—
150	#135	—	13	1	no	—
151	#135	C₂×D₁₃	1	1	yes	characteristic
152	#131	—	13	2	no	—
153	#131	—	13	2	no	—
154	#131	—	13	2	no	—
155	#131	—	26	2	no	—
156	#131	—	26	2	no	—
157	#131	C₂×D₁₃	1	2	yes	complemented
158	C₂×M₁₆	—	13	1	no	—
159	C₂×M₁₆	—	13	2	no	—
160	C₂×M₁₆	—	26	2	no	—
161	C₂²×C₈	—	13	1	no	—
162	C₂²×C₈	—	26	2	no	—
163	#117	—	13	1	no	—
164	#108	—	13	2	no	—
165	#108	—	13	1	no	—
166	#108	C₂×D₁₃	1	1	yes	characteristic, complemented
167	#104	—	13	2	no	—
168	#104	—	13	2	no	—
169	#104	—	13	2	no	—
170	#104	C₂×D₁₃	1	2	yes	complemented
171	#97	—	13	1	no	—
172	#97	—	13	1	no	—
173	C₄×C₈	—	13	1	no	—
174	C₄×C₈	C₂×D₁₃	1	1	yes	characteristic, complemented
175	C₂×C₁₃	—	2	1	no	—
176	C₂×C₁₃	—	4	2	no	—
177	C₂×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
178	D₁₃	—	2	2	no	—
179	D₁₃	—	4	2	no	—
180	#41	—	13	2	no	—
181	#41	—	13	2	no	—
182	#41	—	13	2	no	—
183	#41	—	26	2	no	—
184	#41	—	26	2	no	—
185	#41	C₂²×D₁₃	1	2	yes	—
186	C₂×Q₈	—	13	2	no	—
187	C₂×D₄	—	13	2	no	—
188	C₂×D₄	—	26	2	no	—
189	C₂²×C₄	—	13	1	no	—
190	C₂²×C₄	—	26	2	no	—
191	Q₁₆	—	13	2	no	—
192	Q₁₆	—	13	1	no	—
193	Q₁₆	C₄×D₁₃	1	1	yes	characteristic
194	SD₁₆	—	13	2	no	—
195	SD₁₆	—	13	2	no	—
196	SD₁₆	—	13	2	no	—
197	SD₁₆	C₄×D₁₃	1	2	yes	—
198	D₈	—	13	2	no	—
199	D₈	—	13	1	no	—
200	D₈	C₄×D₁₃	1	1	yes	characteristic
201	M₁₆	—	2	2	no	—
202	M₁₆	—	13	2	no	—
203	M₁₆	—	13	2	no	—
204	M₁₆	—	13	2	no	—
205	M₁₆	—	26	2	no	—
206	M₁₆	—	26	2	no	—
207	M₁₆	—	26	2	no	—
208	M₁₆	—	26	2	no	—
209	M₁₆	C₂²×D₁₃	1	2	yes	—
210	C₂×C₈	—	2	2	no	—
211	C₂×C₈	—	13	1	no	—
212	C₂×C₈	—	13	1	no	—
213	C₂×C₈	—	13	2	no	—
214	C₂×C₈	—	13	2	no	—
215	C₂×C₈	—	26	2	no	—
216	C₂×C₈	—	26	2	no	—
217	C₂×C₈	—	26	2	no	—
218	C₂×C₈	—	26	2	no	—
219	C₂×C₈	C₂²×D₁₃	1	1	yes	characteristic
220	C₂×C₈	C₂²×D₁₃	1	1	yes	characteristic
221	C₄⋊C₄	—	26	1	no	—
222	C₂²⋊C₄	—	26	1	no	—
223	C₄²	—	13	1	no	—
224	C₄²	C₂²×D₁₃	1	1	yes	characteristic
225	C₁₃	#2955	1	1	yes	characteristic, complemented, Sylow
226	C₂³	—	13	1	no	—
227	C₂³	—	26	2	no	—
228	Q₈	—	13	2	no	—
229	Q₈	—	26	2	no	—
230	Q₈	C₂×C₄×D₁₃	1	2	yes	—
231	D₄	—	2	2	no	—
232	D₄	—	13	2	no	—
233	D₄	—	26	2	no	—
234	D₄	—	26	2	no	—
235	D₄	—	26	2	no	—
236	D₄	—	26	2	no	—
237	D₄	C₂×C₄×D₁₃	1	2	yes	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	1	no	—
240	C₂×C₄	—	13	1	no	—
241	C₂×C₄	—	13	2	no	—
242	C₂×C₄	—	13	2	no	—
243	C₂×C₄	—	26	2	no	—
244	C₂×C₄	—	26	2	no	—
245	C₂×C₄	—	26	2	no	—
246	C₂×C₄	—	26	2	no	—
247	C₂×C₄	—	26	1	no	—
248	C₂×C₄	C₂³×D₁₃	1	1	yes	characteristic
249	C₈	—	2	2	no	—
250	C₈	—	13	2	no	—
251	C₈	—	13	2	no	—
252	C₈	—	26	2	no	—
253	C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
254	C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
255	C₈	D₄×D₁₃	1	1	yes	characteristic
256	C₈	D₄×D₁₃	1	1	yes	characteristic
257	C₂²	—	2	2	no	—
258	C₂²	—	13	2	no	—
259	C₂²	—	26	2	no	—
260	C₂²	—	52	1	no	—
261	C₂²	—	52	4	no	—
262	C₂²	#41×D₁₃	1	1	yes	characteristic
263	C₄	—	2	2	no	—
264	C₄	—	4	1	no	—
265	C₄	—	13	2	no	—
266	C₄	—	26	2	no	—
267	C₄	—	52	1	no	—
268	C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
269	C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
270	C₂	—	2	1	no	—
271	C₂	—	4	2	no	—
272	C₂	—	26	2	no	—
273	C₂	—	52	2	no	—
274	C₂	C₄×D₄×D₁₃	1	1	yes	characteristic

	subgroup	G/H
center	C₈	D₄×D₁₃
commutator	C₄×C₁₃	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₃
Fitting	C₁₃×#443	C₂
socle	C₂×C₁₃	C₂×C₄×D₄
prime radical	#11719473	C₂
cosocle	C₂×C₄×C₁₃	C₂⁴
hypercenter	#443	D₁₃
hypocenter	C₁₃	#2955
LCS	G ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₈ ◁ C₂×C₈ ◁ #443

Complements

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

N	H ≅ G/N	type
2. D₁₃×#131	C₂	semidirect
3. C₁₃×#443	C₂	semidirect
4. #11718858	C₂	semidirect
5. #11718805	C₂	semidirect
6. #11718555	C₂	semidirect
7. #11718554	C₂	semidirect
8. #11718484	C₂	semidirect
9. #11718482	C₂	semidirect
10. #11718361	C₂	semidirect
11. #11718350	C₂	semidirect
14. C₁₃×#131	C₂²	semidirect
16. #88202	C₂²	semidirect
17. #88172	C₄	semidirect
18. #88169	C₄	semidirect
19. #88168	C₄	semidirect
20. #88165	C₄	semidirect
22. #88161	C₂²	semidirect
27. #88155	C₂²	semidirect
32. C₁₃×#108	C₂²	semidirect
33. C₁₃×#104	C₂²	semidirect
34. C₄×C₈×C₁₃	C₂²	semidirect
35. #88077	C₂²	semidirect
36. #88065	C₂²	semidirect
37. #88062	C₂²	semidirect
38. #88059	C₂²	semidirect
39. #88055	C₂²	semidirect
40. C₈×Dic₁₃	C₂²	semidirect
41. #88045	C₂²	semidirect
42. #88043	C₂²	semidirect
127. #443	D₁₃	semidirect
157. #131	C₂×D₁₃	semidirect
166. #108	C₂×D₁₃	semidirect
170. #104	C₂×D₁₃	semidirect
174. C₄×C₈	C₂×D₁₃	semidirect
225. C₁₃	#2955	semidirect

Maximal subgroups

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

subgroup	index
#11719473	2
D₁₃×#131	2
C₁₃×#443	2
#11718858	2
#11718805	2
#11718555	2
#11718554	2
#11718484	2
#11718482	2
#11718361	2
#11718350	2
#2955	13