Subgroups — #13123748

Order 1664.

452 conjugacy classes of subgroups.

The subgroup lattice has 302 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×#443	C₂	1	1	yes	maximal, characteristic, complemented
4	#11718857	C₂	1	1	yes	maximal, characteristic, complemented
5	#11718806	C₂	1	2	yes	maximal, complemented
6	#11718556	C₂	1	1	yes	maximal, characteristic, complemented
7	D₁₃×#108	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	#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	C₄×C₈×C₁₃	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	#88043	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	1	yes	characteristic
84	#6094	D₄	1	1	yes	characteristic
85	#6094	D₄	1	1	yes	characteristic
86	#6094	D₄	1	1	yes	characteristic
87	C₈×D₁₃	—	2	2	no	—
88	C₈×D₁₃	—	2	2	no	—
89	C₈×D₁₃	—	2	2	no	—
90	C₈×D₁₃	C₂×C₄	1	1	yes	characteristic
91	C₈×D₁₃	C₂×C₄	1	1	yes	characteristic
92	C₈×D₁₃	C₂×C₄	1	1	yes	characteristic
93	C₈×D₁₃	C₂×C₄	1	1	yes	characteristic
94	#2957	—	13	1	no	maximal, Sylow
95	C₂²×D₁₃	—	2	2	no	—
96	C₂²×D₁₃	#41	1	1	yes	characteristic
97	Q₈×C₁₃	C₂²×C₄	1	2	yes	—
98	D₄×C₁₃	—	2	2	no	—
99	D₄×C₁₃	C₂²×C₄	1	2	yes	—
100	C₂×C₄×C₁₃	—	2	2	no	—
101	C₂×C₄×C₁₃	—	2	1	no	—
102	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
103	C₂²⋊D₁₃	—	2	2	no	—
104	C₂²⋊D₁₃	—	2	2	no	—
105	C₂²⋊D₁₃	—	2	2	no	—
106	C₂×Dic₁₃	—	2	2	no	—
107	C₂×Dic₁₃	—	2	1	no	—
108	C₂×Dic₁₃	#41	1	1	yes	characteristic
109	D₅₂	—	2	2	no	—
110	D₅₂	C₂²×C₄	1	2	yes	—
111	C₄×D₁₃	—	2	2	no	—
112	C₄×D₁₃	—	2	2	no	—
113	C₄×D₁₃	—	2	2	no	—
114	C₄×D₁₃	C₂²×C₄	1	1	yes	characteristic
115	C₄×D₁₃	C₂²×C₄	1	1	yes	characteristic
116	C₄×D₁₃	C₂×D₄	1	1	yes	characteristic
117	C₄×D₁₃	C₂×D₄	1	1	yes	characteristic
118	Dic₂₆	—	2	2	no	—
119	Dic₂₆	C₂²×C₄	1	2	yes	—
120	C₈×C₁₃	—	2	2	no	—
121	C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
122	C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
123	C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
124	C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
125	#1055	—	2	2	no	—
126	#1055	C₂²×C₄	1	1	yes	characteristic
127	#1055	C₂²×C₄	1	1	yes	characteristic
128	#1055	C₂×D₄	1	1	yes	characteristic
129	#1055	C₂×D₄	1	1	yes	characteristic
130	C₂×#135	—	13	1	no	—
131	#568	—	13	2	no	—
132	#444	—	13	2	no	—
133	#444	—	13	2	no	—
134	#443	—	13	1	no	—
135	#443	—	13	1	no	—
136	#443	—	13	1	no	—
137	#443	D₁₃	1	1	yes	characteristic, complemented
138	C₂×#108	—	13	1	no	—
139	#421	—	13	2	no	—
140	#405	—	13	1	no	—
141	C₂²×C₁₃	—	2	2	no	—
142	C₂²×C₁₃	C₂×#41	1	1	yes	characteristic
143	C₂×D₁₃	—	2	2	no	—
144	C₂×D₁₃	—	4	1	no	—
145	C₂×D₁₃	—	8	2	no	—
146	C₂×D₁₃	C₄×D₄	1	1	yes	characteristic
147	C₂×D₁₃	C₄×D₄	1	1	yes	characteristic
148	C₄×C₁₃	—	2	2	no	—
149	C₄×C₁₃	—	4	1	no	—
150	C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
151	C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
152	Dic₁₃	—	2	2	no	—
153	Dic₁₃	—	4	1	no	—
154	Dic₁₃	C₄×D₄	1	1	yes	characteristic
155	Dic₁₃	C₄×D₄	1	1	yes	characteristic
156	C₂×#41	—	13	2	no	—
157	#135	—	13	2	no	—
158	#135	—	13	1	no	—
159	#135	—	13	2	no	—
160	#135	—	13	1	no	—
161	#135	—	13	1	no	—
162	#135	C₂×D₁₃	1	1	yes	characteristic
163	C₂×Q₁₆	—	13	1	no	—
164	C₂×SD₁₆	—	13	2	no	—
165	C₂×D₈	—	13	1	no	—
166	#131	—	13	2	no	—
167	#131	—	13	2	no	—
168	#131	—	13	2	no	—
169	#131	—	26	2	no	—
170	#131	—	26	2	no	—
171	#131	C₂×D₁₃	1	2	yes	complemented
172	C₂×M₁₆	—	13	1	no	—
173	C₂×M₁₆	—	13	2	no	—
174	C₂×M₁₆	—	26	2	no	—
175	C₂×M₁₆	—	26	2	no	—
176	C₂²×C₈	—	13	1	no	—
177	#117	—	13	1	no	—
178	#108	—	13	2	no	—
179	#108	—	13	1	no	—
180	#108	C₂×D₁₃	1	1	yes	characteristic, complemented
181	#104	—	13	2	no	—
182	#104	—	13	2	no	—
183	#104	—	13	2	no	—
184	#104	C₂×D₁₃	1	2	yes	complemented
185	#97	—	13	1	no	—
186	#97	—	13	1	no	—
187	C₄×C₈	—	13	1	no	—
188	C₄×C₈	C₂×D₁₃	1	1	yes	characteristic, complemented
189	C₂×C₁₃	—	2	1	no	—
190	C₂×C₁₃	—	4	2	no	—
191	C₂×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
192	D₁₃	—	2	1	no	—
193	D₁₃	—	2	1	no	—
194	D₁₃	—	4	2	no	—
195	#41	—	13	2	no	—
196	#41	—	13	2	no	—
197	#41	—	13	2	no	—
198	#41	—	26	2	no	—
199	#41	—	26	2	no	—
200	#41	C₂²×D₁₃	1	2	yes	—
201	C₂×Q₈	—	13	2	no	—
202	C₂×D₄	—	13	2	no	—
203	C₂×D₄	—	26	2	no	—
204	C₂²×C₄	—	13	1	no	—
205	C₂²×C₄	—	26	2	no	—
206	Q₁₆	—	13	2	no	—
207	Q₁₆	—	13	1	no	—
208	Q₁₆	C₄×D₁₃	1	1	yes	characteristic
209	SD₁₆	—	13	2	no	—
210	SD₁₆	—	13	2	no	—
211	SD₁₆	—	13	2	no	—
212	SD₁₆	C₄×D₁₃	1	2	yes	—
213	D₈	—	13	2	no	—
214	D₈	—	13	1	no	—
215	D₈	C₄×D₁₃	1	1	yes	characteristic
216	M₁₆	—	2	2	no	—
217	M₁₆	—	13	1	no	—
218	M₁₆	—	13	1	no	—
219	M₁₆	—	13	1	no	—
220	M₁₆	—	13	1	no	—
221	M₁₆	—	13	2	no	—
222	M₁₆	—	26	2	no	—
223	M₁₆	—	26	2	no	—
224	M₁₆	—	26	2	no	—
225	M₁₆	—	26	2	no	—
226	M₁₆	C₂²×D₁₃	1	2	yes	—
227	C₂×C₈	—	2	2	no	—
228	C₂×C₈	—	13	1	no	—
229	C₂×C₈	—	13	1	no	—
230	C₂×C₈	—	13	1	no	—
231	C₂×C₈	—	13	1	no	—
232	C₂×C₈	—	13	1	no	—
233	C₂×C₈	—	13	1	no	—
234	C₂×C₈	—	26	2	no	—
235	C₂×C₈	—	26	2	no	—
236	C₂×C₈	—	26	2	no	—
237	C₂×C₈	—	26	2	no	—
238	C₂×C₈	C₂²×D₁₃	1	1	yes	characteristic
239	C₂×C₈	C₂²×D₁₃	1	1	yes	characteristic
240	C₄⋊C₄	—	26	1	no	—
241	C₂²⋊C₄	—	26	1	no	—
242	C₄²	—	13	1	no	—
243	C₄²	C₂²×D₁₃	1	1	yes	characteristic
244	C₁₃	#2957	1	1	yes	characteristic, complemented, Sylow
245	C₂³	—	13	1	no	—
246	C₂³	—	26	2	no	—
247	Q₈	—	13	2	no	—
248	Q₈	—	26	2	no	—
249	Q₈	C₂×C₄×D₁₃	1	2	yes	—
250	D₄	—	2	2	no	—
251	D₄	—	13	2	no	—
252	D₄	—	26	2	no	—
253	D₄	—	26	2	no	—
254	D₄	—	26	2	no	—
255	D₄	—	26	2	no	—
256	D₄	C₂×C₄×D₁₃	1	2	yes	—
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	2	1	no	—
259	C₂×C₄	—	13	1	no	—
260	C₂×C₄	—	13	1	no	—
261	C₂×C₄	—	13	1	no	—
262	C₂×C₄	—	13	1	no	—
263	C₂×C₄	—	13	1	no	—
264	C₂×C₄	—	26	2	no	—
265	C₂×C₄	—	26	2	no	—
266	C₂×C₄	—	26	2	no	—
267	C₂×C₄	—	26	2	no	—
268	C₂×C₄	—	26	1	no	—
269	C₂×C₄	C₂³×D₁₃	1	1	yes	characteristic
270	C₈	—	2	2	no	—
271	C₈	—	13	1	no	—
272	C₈	—	13	1	no	—
273	C₈	—	13	1	no	—
274	C₈	—	13	1	no	—
275	C₈	—	26	2	no	—
276	C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
277	C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
278	C₈	D₄×D₁₃	1	1	yes	characteristic
279	C₈	D₄×D₁₃	1	1	yes	characteristic
280	C₂²	—	2	2	no	—
281	C₂²	—	13	1	no	—
282	C₂²	—	13	1	no	—
283	C₂²	—	26	2	no	—
284	C₂²	—	52	1	no	—
285	C₂²	—	104	2	no	—
286	C₂²	#41×D₁₃	1	1	yes	characteristic
287	C₄	—	2	2	no	—
288	C₄	—	4	1	no	—
289	C₄	—	13	1	no	—
290	C₄	—	13	1	no	—
291	C₄	—	26	2	no	—
292	C₄	—	52	1	no	—
293	C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
294	C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
295	C₂	—	2	1	no	—
296	C₂	—	4	2	no	—
297	C₂	—	26	1	no	—
298	C₂	—	26	1	no	—
299	C₂	—	52	2	no	—
300	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×#443	C₂
socle	C₂×C₁₃	C₂×C₄×D₄
prime radical	D₁₃×#135	C₂
cosocle	C₂×C₄×C₁₃	C₂⁴
hypercenter	#443	D₁₃
hypocenter	C₁₃	#2957
LCS	G ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ #443

Complements

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

N	H ≅ G/N	type
2. #11719455	C₂	semidirect
3. C₁₃×#443	C₂	semidirect
4. #11718857	C₂	semidirect
5. #11718806	C₂	semidirect
6. #11718556	C₂	semidirect
7. D₁₃×#108	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. #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. C₄×C₈×C₁₃	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. #88043	C₂²	semidirect
137. #443	D₁₃	semidirect
171. #131	C₂×D₁₃	semidirect
180. #108	C₂×D₁₃	semidirect
184. #104	C₂×D₁₃	semidirect
188. C₄×C₈	C₂×D₁₃	semidirect
244. 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×#443	2
#11718857	2
#11718806	2
#11718556	2
D₁₃×#108	2
#11718484	2
#11718482	2
#11718361	2
#11718350	2
#2957	13