Subgroups — #13128525

Order 1664.

1328 conjugacy classes of subgroups.

The subgroup lattice has 286 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₁₃×C₂²⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#88207	C₂	1	2	yes	maximal, complemented
3	#11719496	C₂	1	4	yes	maximal, complemented
4	#11719200	C₂	1	8	yes	maximal, complemented
5	#11719199	C₂	1	4	yes	maximal, complemented
6	#11719198	C₂	1	1	yes	maximal, characteristic, complemented
7	#11719195	C₂	1	8	yes	maximal, complemented
8	#11719191	C₂	1	2	yes	maximal, complemented
9	#11719188	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂⁵×C₁₃	C₂²	1	1	yes	characteristic
11	C₂³×C₄×C₁₃	C₂²	1	2	yes	complemented
12	C₂²×C₂²⋊D₁₃	C₂²	1	2	yes	—
13	C₂²×C₂²⋊D₁₃	C₂²	1	4	yes	complemented
14	C₂×#6127	C₂²	1	4	yes	complemented
15	C₂×C₁₃×C₂²⋊C₄	C₂²	1	1	yes	characteristic
16	C₂×C₁₃×C₂²⋊C₄	C₂²	1	1	yes	characteristic
17	C₂×C₁₃×C₂²⋊C₄	C₂²	1	8	yes	complemented
18	C₂×C₁₃×C₂²⋊C₄	C₂²	1	2	yes	complemented
19	#88207	C₂²	1	8	yes	—
20	#88207	C₂²	1	4	yes	complemented
21	#88207	C₂²	1	4	yes	complemented
22	C₂×C₂²⋊Dic₁₃	C₂²	1	2	yes	—
23	C₂×C₂²⋊Dic₁₃	C₂²	1	4	yes	complemented
24	#88184	C₂²	1	4	yes	complemented
25	#88184	C₂²	1	4	yes	complemented
26	#88183	C₂²	1	8	yes	complemented
27	C₄×C₂²⋊D₁₃	C₂²	1	8	yes	complemented
28	C₄×C₂²⋊D₁₃	C₂²	1	8	yes	complemented
29	#88180	C₂²	1	4	yes	complemented
30	#88178	C₂²	1	4	yes	complemented
31	#88178	C₂²	1	4	yes	complemented
32	#88140	C₂²	1	4	yes	complemented
33	#88139	C₂²	1	8	yes	complemented
34	#88138	C₂²	1	8	yes	complemented
35	#88137	C₂²	1	8	yes	complemented
36	#88136	C₂²	1	4	yes	complemented
37	#88135	C₂²	1	8	yes	complemented
38	C₂²⋊C₄×D₁₃	C₂²	1	4	yes	complemented
39	#88133	C₂²	1	8	yes	complemented
40	#88132	C₂²	1	8	yes	complemented
41	#88131	C₂²	1	4	yes	complemented
42	C₂⁴×C₁₃	—	2	4	no	—
43	C₂⁴×C₁₃	—	4	4	no	—
44	C₂⁴×C₁₃	C₂³	1	1	yes	characteristic
45	C₂⁴×C₁₃	C₂³	1	4	yes	—
46	C₂⁴×C₁₃	C₂³	1	1	yes	characteristic
47	C₂⁴×C₁₃	C₂³	1	1	yes	characteristic
48	C₂³×D₁₃	C₂³	1	2	yes	—
49	C₂²×C₄×C₁₃	—	4	4	no	—
50	C₂²×C₄×C₁₃	C₂³	1	4	yes	—
51	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
52	C₂²×C₄×C₁₃	C₂³	1	4	yes	—
53	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
54	C₂×C₂²⋊D₁₃	—	2	8	no	—
55	C₂×C₂²⋊D₁₃	—	4	8	no	—
56	C₂×C₂²⋊D₁₃	C₂³	1	4	yes	—
57	C₂×C₂²⋊D₁₃	C₂³	1	4	yes	—
58	C₂×C₂²⋊D₁₃	C₂³	1	8	yes	—
59	C₂×C₂²⋊D₁₃	C₂³	1	8	yes	—
60	C₂²×Dic₁₃	C₂³	1	2	yes	—
61	C₂²×Dic₁₃	C₂³	1	4	yes	—
62	#6127	—	4	8	no	—
63	C₂×D₅₂	C₂³	1	4	yes	complemented
64	C₂×C₄×D₁₃	C₂³	1	8	yes	complemented
65	C₂×Dic₂₆	C₂³	1	4	yes	complemented
66	C₁₃×C₂²⋊C₄	C₂³	1	4	yes	—
67	C₁₃×C₂²⋊C₄	C₂³	1	4	yes	—
68	C₁₃×C₂²⋊C₄	C₂³	1	8	yes	—
69	C₂²⋊Dic₁₃	C₂³	1	4	yes	—
70	C₂²⋊Dic₁₃	C₂³	1	8	yes	—
71	C₂²⋊Dic₁₃	C₂³	1	4	yes	—
72	C₂²⋊Dic₁₃	C₂³	1	8	yes	—
73	#6103	C₂³	1	8	yes	complemented
74	#6103	C₂³	1	8	yes	complemented
75	C₄⋊Dic₁₃	C₂³	1	4	yes	complemented
76	C₄⋊Dic₁₃	C₂³	1	4	yes	complemented
77	#6101	C₂³	1	8	yes	complemented
78	#6101	C₂³	1	8	yes	complemented
79	C₄×Dic₁₃	C₂³	1	8	yes	complemented
80	#3490	—	13	1	no	maximal, Sylow
81	C₂³×C₁₃	—	2	2	no	—
82	C₂³×C₁₃	—	2	4	no	—
83	C₂³×C₁₃	—	2	8	no	—
84	C₂³×C₁₃	—	2	2	no	—
85	C₂³×C₁₃	—	2	8	no	—
86	C₂³×C₁₃	—	4	2	no	—
87	C₂³×C₁₃	—	4	4	no	—
88	C₂³×C₁₃	—	4	2	no	—
89	C₂³×C₁₃	—	4	16	no	—
90	C₂³×C₁₃	#41	1	4	yes	complemented
91	C₂³×C₁₃	C₂⁴	1	2	yes	—
92	C₂³×C₁₃	C₂⁴	1	1	yes	characteristic
93	C₂³×C₁₃	C₂⁴	1	2	yes	—
94	C₂³×C₁₃	C₂⁴	1	2	yes	—
95	C₂²×D₁₃	—	2	4	no	—
96	C₂²×D₁₃	—	4	4	no	—
97	C₂²×D₁₃	C₂⁴	1	4	yes	—
98	C₂×C₄×C₁₃	—	2	8	no	—
99	C₂×C₄×C₁₃	—	4	2	no	—
100	C₂×C₄×C₁₃	—	4	4	no	—
101	C₂×C₄×C₁₃	—	4	2	no	—
102	C₂×C₄×C₁₃	C₂⁴	1	4	yes	—
103	C₂×C₄×C₁₃	C₂⁴	1	4	yes	—
104	C₂²⋊D₁₃	—	2	8	no	—
105	C₂²⋊D₁₃	—	2	8	no	—
106	C₂²⋊D₁₃	—	2	16	no	—
107	C₂²⋊D₁₃	—	4	8	no	—
108	C₂×Dic₁₃	—	2	4	no	—
109	C₂×Dic₁₃	—	4	4	no	—
110	C₂×Dic₁₃	C₂⁴	1	4	yes	—
111	C₂×Dic₁₃	C₂⁴	1	8	yes	—
112	D₅₂	—	4	4	no	—
113	C₄×D₁₃	—	4	8	no	—
114	Dic₂₆	—	4	4	no	—
115	#551	—	13	8	no	—
116	#546	—	13	8	no	—
117	#543	—	13	1	no	—
118	#538	—	13	4	no	—
119	#534	—	13	1	no	—
120	#525	—	13	4	no	—
121	C₂×#120	—	13	2	no	—
122	#518	—	13	2	no	—
123	C₂²×C₂²⋊C₄	D₁₃	1	1	yes	characteristic, complemented
124	C₂²×C₁₃	—	2	8	no	—
125	C₂²×C₁₃	—	2	2	no	—
126	C₂²×C₁₃	—	2	4	no	—
127	C₂²×C₁₃	—	2	2	no	—
128	C₂²×C₁₃	—	2	2	no	—
129	C₂²×C₁₃	—	2	2	no	—
130	C₂²×C₁₃	—	2	4	no	—
131	C₂²×C₁₃	—	4	4	no	—
132	C₂²×C₁₃	—	4	16	no	—
133	C₂²×C₁₃	—	4	4	no	—
134	C₂²×C₁₃	C₂×#41	1	4	yes	—
135	C₂²×C₁₃	C₂×#41	1	2	yes	—
136	C₂²×C₁₃	H(5,𝔽₂)	1	4	yes	—
137	C₂²×C₁₃	C₂⁵	1	1	yes	characteristic
138	C₂×D₁₃	—	2	4	no	—
139	C₂×D₁₃	—	2	8	no	—
140	C₂×D₁₃	—	8	4	no	—
141	C₄×C₁₃	—	2	4	no	—
142	C₄×C₁₃	—	2	4	no	—
143	Dic₁₃	—	2	4	no	—
144	Dic₁₃	—	2	8	no	—
145	C₂⁵	C₂×D₁₃	1	1	yes	characteristic
146	C₂×#41	—	13	4	no	—
147	C₂²×D₄	—	13	2	no	—
148	C₂²×D₄	—	13	4	no	—
149	C₂³×C₄	C₂×D₁₃	1	2	yes	complemented
150	#126	—	13	8	no	—
151	#124	—	13	8	no	—
152	#123	—	13	8	no	—
153	#123	—	13	4	no	—
154	#123	—	13	8	no	—
155	C₂²⋊Q₈	—	13	4	no	—
156	C₂²⋊Q₈	—	13	4	no	—
157	C₂²⋊Q₈	—	13	8	no	—
158	#121	—	13	4	no	—
159	#121	—	13	4	no	—
160	#121	—	13	8	no	—
161	#120	—	13	4	no	—
162	#120	—	13	4	no	—
163	#120	—	13	8	no	—
164	#120	—	13	4	no	—
165	C₄×D₄	—	13	8	no	—
166	C₄×D₄	—	13	8	no	—
167	C₄×D₄	—	13	8	no	—
168	#117	—	13	4	no	—
169	#117	—	13	4	no	—
170	C₂×C₂²⋊C₄	—	13	2	no	—
171	C₂×C₂²⋊C₄	—	13	4	no	—
172	C₂×C₂²⋊C₄	—	13	4	no	—
173	C₂×C₂²⋊C₄	C₂×D₁₃	1	1	yes	characteristic
174	C₂×C₂²⋊C₄	C₂×D₁₃	1	1	yes	characteristic
175	C₂×C₂²⋊C₄	C₂×D₁₃	1	2	yes	complemented
176	C₂×C₂²⋊C₄	C₂×D₁₃	1	8	yes	complemented
177	C₂×C₁₃	—	2	4	no	—
178	C₂×C₁₃	—	2	2	no	—
179	C₂×C₁₃	—	2	4	no	—
180	C₂×C₁₃	—	4	2	no	—
181	C₂×C₁₃	C₂²×#41	1	1	yes	characteristic
182	C₂×C₁₃	C₂×H(5,𝔽₂)	1	2	yes	—
183	D₁₃	—	4	4	no	—
184	C₂⁴	—	2	4	no	—
185	C₂⁴	—	4	4	no	—
186	C₂⁴	—	13	2	no	—
187	C₂⁴	C₂²×D₁₃	1	1	yes	characteristic
188	C₂⁴	C₂²×D₁₃	1	4	yes	—
189	C₂⁴	C₂²×D₁₃	1	1	yes	characteristic
190	C₂⁴	C₂²×D₁₃	1	1	yes	characteristic
191	#41	—	52	8	no	—
192	C₂×Q₈	—	13	4	no	—
193	C₂×D₄	—	13	4	no	—
194	C₂×D₄	—	13	4	no	—
195	C₂×D₄	—	13	8	no	—
196	C₂×D₄	—	13	8	no	—
197	C₂×D₄	—	13	4	no	—
198	C₂×D₄	—	26	8	no	—
199	C₂×D₄	—	52	8	no	—
200	C₂²×C₄	—	4	4	no	—
201	C₂²×C₄	—	13	2	no	—
202	C₂²×C₄	—	13	4	no	—
203	C₂²×C₄	—	13	8	no	—
204	C₂²×C₄	C₂²×D₁₃	1	4	yes	—
205	C₂²×C₄	C₂²×D₁₃	1	2	yes	—
206	C₂²×C₄	C₂²×D₁₃	1	2	yes	—
207	C₂²×C₄	C₂²×D₁₃	1	4	yes	—
208	C₄⋊C₄	—	13	8	no	—
209	C₄⋊C₄	—	13	4	no	—
210	C₄⋊C₄	—	13	4	no	—
211	C₄⋊C₄	—	13	8	no	—
212	C₂²⋊C₄	—	13	4	no	—
213	C₂²⋊C₄	—	13	8	no	—
214	C₂²⋊C₄	—	13	4	no	—
215	C₂²⋊C₄	—	13	8	no	—
216	C₂²⋊C₄	—	13	8	no	—
217	C₂²⋊C₄	—	13	8	no	—
218	C₂²⋊C₄	C₂²×D₁₃	1	4	yes	—
219	C₂²⋊C₄	C₂²×D₁₃	1	4	yes	—
220	C₂²⋊C₄	C₂²×D₁₃	1	8	yes	—
221	C₄²	—	13	8	no	—
222	C₁₃	#3490	1	1	yes	characteristic, complemented, Sylow
223	C₂³	—	2	2	no	—
224	C₂³	—	2	4	no	—
225	C₂³	—	2	8	no	—
226	C₂³	—	2	2	no	—
227	C₂³	—	2	8	no	—
228	C₂³	—	4	2	no	—
229	C₂³	—	4	4	no	—
230	C₂³	—	4	2	no	—
231	C₂³	—	4	16	no	—
232	C₂³	—	13	4	no	—
233	C₂³	—	26	4	no	—
234	C₂³	—	52	4	no	—
235	C₂³	#6127	1	4	yes	complemented
236	C₂³	C₂³×D₁₃	1	2	yes	—
237	C₂³	C₂³×D₁₃	1	1	yes	characteristic
238	C₂³	C₂³×D₁₃	1	2	yes	—
239	C₂³	C₂³×D₁₃	1	2	yes	—
240	Q₈	—	52	4	no	—
241	D₄	—	26	8	no	—
242	D₄	—	26	8	no	—
243	D₄	—	26	16	no	—
244	D₄	—	52	4	no	—
245	D₄	—	52	8	no	—
246	C₂×C₄	—	2	8	no	—
247	C₂×C₄	—	4	2	no	—
248	C₂×C₄	—	4	4	no	—
249	C₂×C₄	—	4	2	no	—
250	C₂×C₄	—	13	4	no	—
251	C₂×C₄	—	13	8	no	—
252	C₂×C₄	—	26	4	no	—
253	C₂×C₄	—	52	8	no	—
254	C₂×C₄	—	52	4	no	—
255	C₂×C₄	C₂³×D₁₃	1	4	yes	—
256	C₂×C₄	C₂³×D₁₃	1	4	yes	—
257	C₂²	—	2	8	no	—
258	C₂²	—	2	2	no	—
259	C₂²	—	2	2	no	—
260	C₂²	—	2	2	no	—
261	C₂²	—	2	4	no	—
262	C₂²	—	2	2	no	—
263	C₂²	—	2	4	no	—
264	C₂²	—	4	4	no	—
265	C₂²	—	4	16	no	—
266	C₂²	—	4	4	no	—
267	C₂²	—	26	4	no	—
268	C₂²	—	26	8	no	—
269	C₂²	—	104	4	no	—
270	C₂²	C₂×#6127	1	4	yes	—
271	C₂²	C₂×#6127	1	2	yes	—
272	C₂²	#88251	1	4	yes	—
273	C₂²	C₂⁴×D₁₃	1	1	yes	characteristic
274	C₄	—	2	4	no	—
275	C₄	—	2	4	no	—
276	C₄	—	26	4	no	—
277	C₄	—	26	8	no	—
278	C₂	—	2	2	no	—
279	C₂	—	2	4	no	—
280	C₂	—	2	4	no	—
281	C₂	—	4	2	no	—
282	C₂	—	52	4	no	—
283	C₂	C₂²×#6127	1	1	yes	characteristic
284	C₂	C₂×#88251	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₁₃
commutator, socle, cosocle	C₂²×C₁₃	C₂⁵
Fitting	C₂²×C₁₃×C₂²⋊C₄	C₂
hypercenter	C₂²×C₂²⋊C₄	D₁₃
hypocenter	C₁₃	#3490
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₂²⋊C₄

Complements

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

N	H ≅ G/N	type
1. C₂²×C₁₃×C₂²⋊C₄	C₂	semidirect
2. C₂×#88207	C₂	semidirect
3. #11719496	C₂	semidirect
4. #11719200	C₂	semidirect
5. #11719199	C₂	semidirect
6. #11719198	C₂	semidirect
7. #11719195	C₂	semidirect
8. #11719191	C₂	semidirect
9. #11719188	C₂	semidirect
11. C₂³×C₄×C₁₃	C₂²	semidirect
13. C₂²×C₂²⋊D₁₃	C₂²	semidirect
14. C₂×#6127	C₂²	semidirect
17. C₂×C₁₃×C₂²⋊C₄	C₂²	semidirect
18. C₂×C₁₃×C₂²⋊C₄	C₂²	semidirect
20. #88207	C₂²	semidirect
21. #88207	C₂²	semidirect
23. C₂×C₂²⋊Dic₁₃	C₂²	semidirect
24. #88184	C₂²	semidirect
25. #88184	C₂²	semidirect
26. #88183	C₂²	semidirect
27. C₄×C₂²⋊D₁₃	C₂²	semidirect
28. C₄×C₂²⋊D₁₃	C₂²	semidirect
29. #88180	C₂²	semidirect
30. #88178	C₂²	semidirect
31. #88178	C₂²	semidirect
32. #88140	C₂²	semidirect
33. #88139	C₂²	semidirect
34. #88138	C₂²	semidirect
35. #88137	C₂²	semidirect
36. #88136	C₂²	semidirect
37. #88135	C₂²	semidirect
38. C₂²⋊C₄×D₁₃	C₂²	semidirect
39. #88133	C₂²	semidirect
40. #88132	C₂²	semidirect
41. #88131	C₂²	semidirect
63. C₂×D₅₂	C₂³	semidirect
64. C₂×C₄×D₁₃	C₂³	semidirect
65. C₂×Dic₂₆	C₂³	semidirect
73. #6103	C₂³	semidirect
74. #6103	C₂³	semidirect
75. C₄⋊Dic₁₃	C₂³	semidirect
76. C₄⋊Dic₁₃	C₂³	semidirect
77. #6101	C₂³	semidirect
78. #6101	C₂³	semidirect
79. C₄×Dic₁₃	C₂³	semidirect
90. C₂³×C₁₃	#41	semidirect
123. C₂²×C₂²⋊C₄	D₁₃	semidirect
149. C₂³×C₄	C₂×D₁₃	semidirect
175. C₂×C₂²⋊C₄	C₂×D₁₃	semidirect
176. C₂×C₂²⋊C₄	C₂×D₁₃	semidirect
222. C₁₃	#3490	semidirect
235. C₂³	#6127	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₁₃×C₂²⋊C₄	2
C₂×#88207	2
#11719496	2
#11719200	2
#11719199	2
#11719198	2
#11719195	2
#11719191	2
#11719188	2
#3490	13