Subgroups — #13123751

Order 1664.

452 conjugacy classes of subgroups.

The subgroup lattice has 282 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#568	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#88204	C₂	1	1	yes	maximal, characteristic
3	#11719450	C₂	1	1	yes	maximal, characteristic, complemented
4	#11718897	C₂	1	1	yes	maximal, characteristic, complemented
5	#11718806	C₂	1	4	yes	maximal, complemented
6	#11718805	C₂	1	4	yes	maximal, complemented
7	#11718800	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#88062	C₂	1	1	yes	maximal, characteristic, complemented
9	#11718783	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₁₃×#41	C₂²	1	1	yes	characteristic
11	C₂×#6127	C₂²	1	1	yes	characteristic
12	C₁₃×#131	—	2	2	no	—
13	C₁₃×#131	C₂²	1	4	yes	complemented
14	C₂×C₁₃×M₁₆	—	2	2	no	—
15	C₂×C₁₃×M₁₆	C₂²	1	1	yes	characteristic
16	C₂×C₁₃×M₁₆	C₂²	1	1	yes	characteristic
17	#88204	C₄	1	2	yes	complemented
18	#88204	C₄	1	2	yes	complemented
19	#88204	C₂²	1	2	yes	—
20	#88204	C₂²	1	2	yes	—
21	C₂×#6107	C₄	1	1	yes	characteristic, complemented
22	C₂×#6106	C₄	1	1	yes	characteristic, complemented
23	C₂×#6105	C₄	1	1	yes	characteristic, complemented
24	C₂×#6104	C₄	1	1	yes	characteristic, complemented
25	#88180	C₂²	1	1	yes	characteristic
26	C₂×#6099	C₂²	1	1	yes	characteristic
27	C₂²×#1055	C₂²	1	1	yes	characteristic
28	#88161	—	2	2	no	—
29	M₁₆×D₁₃	—	2	2	no	—
30	#88155	C₂²	1	4	yes	complemented
31	#88077	C₂²	1	4	yes	complemented
32	#88065	C₂²	1	4	yes	complemented
33	#88062	C₂²	1	2	yes	complemented
34	#88062	C₂²	1	2	yes	complemented
35	#88055	C₂²	1	2	yes	complemented
36	C₈×Dic₁₃	C₂²	1	2	yes	complemented
37	C₁₃×#41	—	2	1	no	—
38	C₁₃×#41	—	2	1	no	—
39	C₁₃×#41	C₂×C₄	1	2	yes	—
40	C₁₃×#41	C₂³	1	2	yes	—
41	C₂×Q₈×C₁₃	C₂×C₄	1	1	yes	characteristic
42	C₂×D₄×C₁₃	—	2	1	no	—
43	C₂×D₄×C₁₃	C₂×C₄	1	1	yes	characteristic
44	C₂²×C₄×C₁₃	—	2	1	no	—
45	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
46	C₂×C₂²⋊D₁₃	—	2	1	no	—
47	#6127	—	2	1	no	—
48	#6127	—	2	1	no	—
49	#6127	C₂×C₄	1	2	yes	—
50	#6127	C₂³	1	2	yes	—
51	C₂×D₅₂	C₂×C₄	1	1	yes	characteristic
52	C₂×C₄×D₁₃	—	2	1	no	—
53	C₂×Dic₂₆	C₂×C₄	1	1	yes	characteristic
54	C₁₃×M₁₆	—	2	1	no	—
55	C₁₃×M₁₆	—	2	4	no	—
56	C₁₃×M₁₆	D₄	1	2	yes	—
57	C₁₃×M₁₆	D₄	1	2	yes	—
58	C₁₃×M₁₆	C₂³	1	2	yes	—
59	C₂×C₈×C₁₃	—	2	1	no	—
60	C₂×C₈×C₁₃	—	2	4	no	—
61	C₂×C₈×C₁₃	C₂³	1	2	yes	—
62	C₂²⋊Dic₁₃	—	2	1	no	—
63	#6107	C₂×C₄	1	2	yes	—
64	#6107	C₂×C₄	1	2	yes	—
65	#6106	C₂×C₄	1	2	yes	—
66	#6106	C₂×C₄	1	2	yes	—
67	#6105	C₂×C₄	1	2	yes	—
68	#6105	C₂×C₄	1	2	yes	—
69	#6104	C₂×C₄	1	2	yes	—
70	#6104	C₂×C₄	1	2	yes	—
71	C₄⋊Dic₁₃	—	2	1	no	—
72	C₄×Dic₁₃	C₂³	1	2	yes	—
73	#6099	—	2	1	no	—
74	#6099	C₂³	1	2	yes	—
75	C₂×#1055	—	2	1	no	—
76	C₂×#1055	C₂×C₄	1	1	yes	characteristic
77	C₂×#1055	C₂×C₄	1	1	yes	characteristic
78	C₂×#1055	C₂×C₄	1	1	yes	characteristic
79	C₂×#1055	C₂×C₄	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	#2957	—	13	1	no	maximal, Sylow
85	C₂³×C₁₃	—	2	1	no	—
86	C₂³×C₁₃	#41	1	1	yes	characteristic
87	C₂²×D₁₃	—	2	1	no	—
88	Q₈×C₁₃	—	2	1	no	—
89	Q₈×C₁₃	C₂²×C₄	1	2	yes	—
90	D₄×C₁₃	—	2	2	no	—
91	D₄×C₁₃	—	2	2	no	—
92	D₄×C₁₃	—	2	1	no	—
93	D₄×C₁₃	C₂²×C₄	1	2	yes	—
94	C₂×C₄×C₁₃	—	2	1	no	—
95	C₂×C₄×C₁₃	—	2	2	no	—
96	C₂×C₄×C₁₃	—	2	2	no	—
97	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
98	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
99	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
100	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
101	C₂×C₄×C₁₃	#41	1	1	yes	characteristic
102	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
103	C₂²⋊D₁₃	—	2	2	no	—
104	C₂²⋊D₁₃	—	2	2	no	—
105	C₂×Dic₁₃	—	2	1	no	—
106	C₂×Dic₁₃	—	2	2	no	—
107	D₅₂	—	2	1	no	—
108	D₅₂	C₂²×C₄	1	2	yes	—
109	C₄×D₁₃	—	2	2	no	—
110	C₄×D₁₃	—	2	2	no	—
111	Dic₂₆	—	2	1	no	—
112	Dic₂₆	C₂²×C₄	1	2	yes	—
113	C₈×C₁₃	—	2	2	no	—
114	C₈×C₁₃	C₂×D₄	1	4	yes	—
115	#1055	—	2	2	no	—
116	#1055	C₂²×C₄	1	1	yes	characteristic
117	#1055	C₂²×C₄	1	1	yes	characteristic
118	#1055	C₂²×C₄	1	1	yes	characteristic
119	#1055	C₂²×C₄	1	1	yes	characteristic
120	C₂×#135	—	13	1	no	—
121	#568	—	13	1	no	—
122	#568	D₁₃	1	1	yes	characteristic, complemented
123	#444	—	13	4	no	—
124	#443	—	13	4	no	—
125	C₂×#108	—	13	1	no	—
126	#421	—	13	1	no	—
127	#421	—	13	1	no	—
128	#405	—	13	1	no	—
129	C₂²×C₁₃	—	2	2	no	—
130	C₂²×C₁₃	—	4	1	no	—
131	C₂²×C₁₃	—	8	1	no	—
132	C₂²×C₁₃	C₄×D₄	1	1	yes	characteristic
133	C₂²×C₁₃	C₄×D₄	1	1	yes	characteristic
134	C₂²×C₁₃	C₂×#41	1	1	yes	characteristic
135	C₂×D₁₃	—	2	2	no	—
136	C₂×D₁₃	—	8	1	no	—
137	C₄×C₁₃	—	2	2	no	—
138	C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
139	C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
140	C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
141	C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
142	Dic₁₃	—	2	2	no	—
143	Dic₁₃	—	4	2	no	—
144	C₂×#41	—	13	1	no	—
145	C₂×#41	C₂×D₁₃	1	1	yes	characteristic
146	#135	—	13	2	no	—
147	#135	—	13	2	no	—
148	#135	—	13	2	no	—
149	#135	—	13	2	no	—
150	C₂×Q₁₆	—	13	1	no	—
151	C₂×SD₁₆	—	13	1	no	—
152	C₂×SD₁₆	—	13	1	no	—
153	C₂×D₈	—	13	1	no	—
154	#131	—	2	2	no	—
155	#131	—	13	4	no	—
156	#131	—	26	2	no	—
157	#131	C₂×D₁₃	1	4	yes	complemented
158	C₂×M₁₆	—	2	2	no	—
159	C₂×M₁₆	—	13	1	no	—
160	C₂×M₁₆	—	26	2	no	—
161	C₂×M₁₆	C₂×D₁₃	1	1	yes	characteristic
162	C₂×M₁₆	C₂×D₁₃	1	1	yes	characteristic
163	C₂²×C₈	—	13	1	no	—
164	#117	—	13	1	no	—
165	#108	—	13	2	no	—
166	#108	—	13	2	no	—
167	#104	—	13	4	no	—
168	#104	—	13	4	no	—
169	#97	—	13	2	no	—
170	C₄×C₈	—	13	2	no	—
171	C₂×C₁₃	—	2	1	no	—
172	C₂×C₁₃	—	2	1	no	—
173	C₂×C₁₃	—	2	1	no	—
174	C₂×C₁₃	—	4	2	no	—
175	C₂×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
176	D₁₃	—	4	2	no	—
177	#41	—	2	1	no	—
178	#41	—	2	1	no	—
179	#41	—	13	2	no	—
180	#41	—	13	2	no	—
181	#41	—	26	1	no	—
182	#41	—	26	1	no	—
183	#41	C₄×D₁₃	1	2	yes	—
184	#41	C₂²×D₁₃	1	2	yes	—
185	C₂×Q₈	—	13	1	no	—
186	C₂×Q₈	C₄×D₁₃	1	1	yes	characteristic
187	C₂×D₄	—	2	1	no	—
188	C₂×D₄	—	13	1	no	—
189	C₂×D₄	—	26	1	no	—
190	C₂×D₄	C₄×D₁₃	1	1	yes	characteristic
191	C₂²×C₄	—	2	1	no	—
192	C₂²×C₄	—	26	1	no	—
193	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
194	Q₁₆	—	13	2	no	—
195	Q₁₆	—	13	2	no	—
196	SD₁₆	—	13	2	no	—
197	SD₁₆	—	13	2	no	—
198	SD₁₆	—	13	2	no	—
199	SD₁₆	—	13	2	no	—
200	D₈	—	13	2	no	—
201	D₈	—	13	2	no	—
202	M₁₆	—	2	1	no	—
203	M₁₆	—	2	4	no	—
204	M₁₆	—	13	2	no	—
205	M₁₆	—	26	1	no	—
206	M₁₆	—	26	4	no	—
207	M₁₆	C₂²⋊D₁₃	1	2	yes	—
208	M₁₆	C₂²⋊D₁₃	1	2	yes	—
209	M₁₆	C₂²×D₁₃	1	2	yes	—
210	C₂×C₈	—	2	1	no	—
211	C₂×C₈	—	2	4	no	—
212	C₂×C₈	—	13	1	no	—
213	C₂×C₈	—	13	1	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	1	no	—
219	C₂×C₈	—	26	4	no	—
220	C₂×C₈	C₂²×D₁₃	1	2	yes	—
221	C₄⋊C₄	—	26	1	no	—
222	C₂²⋊C₄	—	26	1	no	—
223	C₄²	—	13	2	no	—
224	C₁₃	#2957	1	1	yes	characteristic, complemented, Sylow
225	C₂³	—	2	1	no	—
226	C₂³	—	26	1	no	—
227	C₂³	#6127	1	1	yes	characteristic
228	Q₈	—	2	1	no	—
229	Q₈	—	13	2	no	—
230	Q₈	—	26	1	no	—
231	Q₈	C₂×C₄×D₁₃	1	2	yes	—
232	D₄	—	2	2	no	—
233	D₄	—	2	2	no	—
234	D₄	—	2	1	no	—
235	D₄	—	13	2	no	—
236	D₄	—	26	2	no	—
237	D₄	—	26	2	no	—
238	D₄	—	26	1	no	—
239	D₄	C₂×C₄×D₁₃	1	2	yes	—
240	C₂×C₄	—	2	1	no	—
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	2	2	no	—
243	C₂×C₄	—	26	1	no	—
244	C₂×C₄	—	26	2	no	—
245	C₂×C₄	—	26	2	no	—
246	C₂×C₄	—	26	2	no	—
247	C₂×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
248	C₂×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
249	C₂×C₄	#6127	1	1	yes	characteristic
250	C₂×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
251	C₂×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
252	C₂×C₄	C₂³×D₁₃	1	1	yes	characteristic
253	C₈	—	2	2	no	—
254	C₈	—	13	1	no	—
255	C₈	—	13	1	no	—
256	C₈	—	13	1	no	—
257	C₈	—	13	1	no	—
258	C₈	—	26	2	no	—
259	C₈	C₂×C₂²⋊D₁₃	1	4	yes	—
260	C₂²	—	2	2	no	—
261	C₂²	—	4	1	no	—
262	C₂²	—	8	1	no	—
263	C₂²	—	26	2	no	—
264	C₂²	—	104	1	no	—
265	C₂²	C₄×C₂²⋊D₁₃	1	1	yes	characteristic
266	C₂²	C₄×C₂²⋊D₁₃	1	1	yes	characteristic
267	C₂²	C₂×#6127	1	1	yes	characteristic
268	C₄	—	2	2	no	—
269	C₄	—	26	2	no	—
270	C₄	—	52	2	no	—
271	C₄	C₄×C₂²⋊D₁₃	1	1	yes	characteristic
272	C₄	C₄×C₂²⋊D₁₃	1	1	yes	characteristic
273	C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
274	C₄	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
275	C₂	—	2	1	no	—
276	C₂	—	2	1	no	—
277	C₂	—	2	1	no	—
278	C₂	—	4	2	no	—
279	C₂	—	52	2	no	—
280	C₂	C₂×C₄×C₂²⋊D₁₃	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂²×C₂²⋊D₁₃
commutator	C₄×C₁₃	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₃
Fitting	C₁₃×#568	C₂
socle	C₂×C₁₃	C₂×C₄×D₄
prime radical	C₂×#88204	C₂
cosocle	C₂×C₄×C₁₃	C₂⁴
hypercenter	#568	D₁₃
hypocenter	C₁₃	#2957
LCS	G ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₄ ◁ C₂×M₁₆ ◁ #568

Complements

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

N	H ≅ G/N	type
1. C₁₃×#568	C₂	semidirect
3. #11719450	C₂	semidirect
4. #11718897	C₂	semidirect
5. #11718806	C₂	semidirect
6. #11718805	C₂	semidirect
7. #11718800	C₂	semidirect
8. C₂×#88062	C₂	semidirect
9. #11718783	C₂	semidirect
13. C₁₃×#131	C₂²	semidirect
17. #88204	C₄	semidirect
18. #88204	C₄	semidirect
21. C₂×#6107	C₄	semidirect
22. C₂×#6106	C₄	semidirect
23. C₂×#6105	C₄	semidirect
24. C₂×#6104	C₄	semidirect
30. #88155	C₂²	semidirect
31. #88077	C₂²	semidirect
32. #88065	C₂²	semidirect
33. #88062	C₂²	semidirect
34. #88062	C₂²	semidirect
35. #88055	C₂²	semidirect
36. C₈×Dic₁₃	C₂²	semidirect
122. #568	D₁₃	semidirect
157. #131	C₂×D₁₃	semidirect
224. C₁₃	#2957	semidirect

Maximal subgroups

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

subgroup	index
C₁₃×#568	2
C₂×#88204	2
#11719450	2
#11718897	2
#11718806	2
#11718805	2
#11718800	2
C₂×#88062	2
#11718783	2
#2957	13