Subgroups — #13128730

Order 1664.

1040 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₃×#124	C₂	1	1	yes	maximal, characteristic, complemented
2	#11719521	C₂	1	1	yes	maximal, characteristic, complemented
3	#11719517	C₂	1	1	yes	maximal, characteristic, complemented
4	#11719510	C₂	1	1	yes	maximal, characteristic
5	#11719503	C₂	1	1	yes	maximal, characteristic, complemented
6	#11719387	C₂	1	4	yes	maximal, complemented
7	#11719386	C₂	1	4	yes	maximal, complemented
8	#11719385	C₂	1	4	yes	maximal, complemented
9	#11719376	C₂	1	4	yes	maximal, complemented
10	#11719200	C₂	1	8	yes	maximal, complemented
11	C₄×#6127	C₂	1	1	yes	maximal, characteristic, complemented
12	#11719175	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂²×Q₈×C₁₃	C₂²	1	1	yes	characteristic
14	C₂²×D₄×C₁₃	C₂²	1	1	yes	characteristic
15	C₂×#6127	C₂²	1	1	yes	characteristic
16	C₁₃×#124	C₂²	1	4	yes	—
17	C₁₃×#124	C₂²	1	2	yes	—
18	C₁₃×#124	C₂²	1	2	yes	—
19	C₂×C₁₃×C₂²⋊C₄	C₂²	1	4	yes	—
20	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
21	#88207	C₂²	1	4	yes	—
22	C₂×C₂²⋊Dic₁₃	C₂²	1	4	yes	—
23	#88201	C₂²	1	2	yes	complemented
24	#88200	C₂²	1	4	yes	complemented
25	Q₈×Dic₁₃	C₂²	1	4	yes	complemented
26	Q₈×Dic₁₃	C₂²	1	4	yes	complemented
27	#88198	C₂²	1	2	yes	complemented
28	#88194	C₂²	1	2	yes	—
29	#88192	C₂²	1	4	yes	—
30	#88190	C₂²	1	2	yes	—
31	D₄×Dic₁₃	C₂²	1	4	yes	—
32	D₄×Dic₁₃	C₂²	1	4	yes	—
33	#88183	C₂²	1	4	yes	—
34	C₄×C₂²⋊D₁₃	C₂²	1	4	yes	—
35	#88180	C₂²	1	1	yes	characteristic
36	#88180	C₂²	1	1	yes	characteristic
37	#88180	C₂²	1	1	yes	characteristic
38	#88180	C₂²	1	4	yes	—
39	#88178	C₂²	1	4	yes	—
40	#88140	C₂²	1	8	yes	complemented
41	#88139	C₂²	1	8	yes	complemented
42	#88137	C₂²	1	8	yes	complemented
43	#88135	C₂²	1	8	yes	complemented
44	#88133	C₂²	1	8	yes	complemented
45	#88133	C₂²	1	8	yes	complemented
46	#88132	C₂²	1	8	yes	complemented
47	#88131	C₂²	1	8	yes	complemented
48	C₄×D₅₂	C₂²	1	2	yes	complemented
49	#88126	C₂²	1	2	yes	complemented
50	C₄²×D₁₃	C₂²	1	2	yes	complemented
51	#88124	C₂²	1	2	yes	complemented
52	#88123	C₂²	1	2	yes	complemented
53	C₄×Dic₂₆	C₂²	1	4	yes	complemented
54	C₄×Dic₂₆	C₂²	1	2	yes	complemented
55	C₂⁴×C₁₃	C₂³	1	2	yes	—
56	C₂×Q₈×C₁₃	—	2	4	no	—
57	C₂×Q₈×C₁₃	C₂³	1	2	yes	—
58	C₂×Q₈×C₁₃	C₂³	1	2	yes	—
59	C₂×D₄×C₁₃	—	2	4	no	—
60	C₂×D₄×C₁₃	C₂³	1	2	yes	—
61	C₂×D₄×C₁₃	C₂³	1	2	yes	—
62	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
63	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
64	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
65	C₂×C₂²⋊D₁₃	C₂³	1	2	yes	—
66	C₂×C₂²⋊D₁₃	C₂³	1	8	yes	—
67	C₂²×Dic₁₃	C₂³	1	8	yes	—
68	#6127	—	2	4	no	—
69	C₂×D₅₂	C₂³	1	1	yes	characteristic
70	C₂×C₄×D₁₃	C₂³	1	2	yes	—
71	C₂×C₄×D₁₃	C₂³	1	4	yes	—
72	C₂×Dic₂₆	C₂³	1	1	yes	characteristic
73	C₂×Dic₂₆	C₂³	1	4	yes	—
74	C₁₃×C₂²⋊C₄	C₂³	1	8	yes	—
75	C₁₃×C₂²⋊C₄	C₂³	1	8	yes	—
76	C₄²×C₁₃	C₂³	1	2	yes	—
77	C₄²×C₁₃	C₂³	1	1	yes	characteristic
78	C₄²×C₁₃	C₂³	1	1	yes	characteristic
79	C₂²⋊Dic₁₃	C₂³	1	8	yes	—
80	C₂²⋊Dic₁₃	C₂³	1	8	yes	—
81	C₂²⋊Dic₁₃	C₂³	1	2	yes	—
82	C₂²⋊Dic₁₃	C₂³	1	2	yes	—
83	C₂²⋊Dic₁₃	C₂³	1	2	yes	—
84	#6103	C₂³	1	4	yes	—
85	#6103	C₂³	1	8	yes	—
86	C₄⋊Dic₁₃	C₂³	1	4	yes	—
87	C₄⋊Dic₁₃	C₂³	1	2	yes	—
88	C₄⋊Dic₁₃	C₂³	1	2	yes	—
89	C₄⋊Dic₁₃	C₂³	1	2	yes	—
90	C₄⋊Dic₁₃	C₂³	1	8	yes	—
91	#6101	C₂³	1	4	yes	—
92	#6101	C₂³	1	4	yes	—
93	#6101	C₂³	1	4	yes	—
94	#6101	C₂³	1	8	yes	—
95	C₄×Dic₁₃	C₂³	1	2	yes	—
96	C₄×Dic₁₃	C₂³	1	2	yes	—
97	C₄×Dic₁₃	C₂³	1	4	yes	—
98	C₄×Dic₁₃	C₂³	1	2	yes	—
99	C₄×Dic₁₃	C₂³	1	8	yes	—
100	#3510	—	13	1	no	maximal, Sylow
101	C₂³×C₁₃	—	2	4	no	—
102	C₂³×C₁₃	—	4	4	no	—
103	C₂³×C₁₃	C₂⁴	1	4	yes	—
104	C₂³×C₁₃	C₂⁴	1	1	yes	characteristic
105	C₂²×D₁₃	C₂⁴	1	2	yes	—
106	Q₈×C₁₃	—	2	4	no	—
107	Q₈×C₁₃	—	2	4	no	—
108	D₄×C₁₃	—	2	4	no	—
109	D₄×C₁₃	—	2	4	no	—
110	C₂×C₄×C₁₃	—	2	4	no	—
111	C₂×C₄×C₁₃	—	4	2	no	—
112	C₂×C₄×C₁₃	#41	1	4	yes	—
113	C₂×C₄×C₁₃	C₂⁴	1	2	yes	—
114	C₂×C₄×C₁₃	C₂⁴	1	4	yes	—
115	C₂×C₄×C₁₃	C₂⁴	1	2	yes	—
116	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
117	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
118	C₂²⋊D₁₃	—	2	4	no	—
119	C₂²⋊D₁₃	—	4	8	no	—
120	C₂×Dic₁₃	—	4	8	no	—
121	C₂×Dic₁₃	C₂⁴	1	4	yes	—
122	C₂×Dic₁₃	C₂⁴	1	4	yes	—
123	C₂×Dic₁₃	C₂⁴	1	2	yes	—
124	C₂×Dic₁₃	C₂⁴	1	4	yes	—
125	D₅₂	—	2	2	no	—
126	C₄×D₁₃	—	2	4	no	—
127	C₄×D₁₃	—	4	4	no	—
128	Dic₂₆	—	2	2	no	—
129	Dic₂₆	—	4	4	no	—
130	#559	—	13	4	no	—
131	#556	—	13	4	no	—
132	#551	—	13	8	no	—
133	#542	—	13	4	no	—
134	#536	—	13	1	no	—
135	#535	—	13	1	no	—
136	#533	—	13	1	no	—
137	#529	—	13	4	no	—
138	C₂×#124	D₁₃	1	1	yes	characteristic, complemented
139	#519	—	13	1	no	—
140	#518	—	13	1	no	—
141	C₄×#41	—	13	1	no	—
142	C₂²×C₁₃	—	2	2	no	—
143	C₂²×C₁₃	—	2	4	no	—
144	C₂²×C₁₃	—	2	8	no	—
145	C₂²×C₁₃	—	8	4	no	—
146	C₂²×C₁₃	C₂×#41	1	2	yes	—
147	C₂²×C₁₃	C₂⁵	1	1	yes	characteristic
148	C₂×D₁₃	—	2	2	no	—
149	C₂×D₁₃	—	2	4	no	—
150	C₄×C₁₃	—	2	4	no	—
151	C₄×C₁₃	—	2	2	no	—
152	C₄×C₁₃	—	2	2	no	—
153	C₄×C₁₃	C₂×#41	1	2	yes	—
154	C₄×C₁₃	C₂×#41	1	2	yes	—
155	Dic₁₃	—	2	2	no	—
156	Dic₁₃	—	2	4	no	—
157	Dic₁₃	—	2	4	no	—
158	Dic₁₃	—	2	4	no	—
159	C₂×#41	—	13	1	no	—
160	C₂²×Q₈	C₂×D₁₃	1	1	yes	characteristic
161	C₂²×D₄	C₂×D₁₃	1	1	yes	characteristic
162	C₄⋊Q₈	—	13	2	no	—
163	C₄⋊Q₈	—	13	2	no	—
164	#127	—	13	2	no	—
165	#126	—	13	8	no	—
166	#126	—	13	8	no	—
167	#125	—	13	2	no	—
168	#124	—	13	2	no	—
169	#124	—	13	2	no	—
170	#124	—	13	8	no	—
171	#124	C₂×D₁₃	1	2	yes	—
172	#124	C₂×D₁₃	1	2	yes	—
173	#124	C₂×D₁₃	1	4	yes	—
174	#123	—	13	4	no	—
175	#123	—	13	8	no	—
176	#123	—	13	8	no	—
177	C₂²⋊Q₈	—	13	4	no	—
178	C₂²⋊Q₈	—	13	4	no	—
179	C₂²⋊Q₈	—	13	8	no	—
180	#121	—	13	4	no	—
181	#120	—	13	4	no	—
182	C₄×Q₈	—	13	4	no	—
183	C₄×Q₈	—	13	4	no	—
184	C₄×Q₈	—	13	2	no	—
185	C₄×Q₈	—	13	4	no	—
186	C₄×D₄	—	13	4	no	—
187	C₄×D₄	—	13	4	no	—
188	C₄×D₄	—	13	4	no	—
189	C₄×D₄	—	13	8	no	—
190	C₄×D₄	—	13	2	no	—
191	#117	—	13	1	no	—
192	#117	—	13	1	no	—
193	#117	—	13	1	no	—
194	#117	—	13	4	no	—
195	#117	—	13	8	no	—
196	#117	—	13	2	no	—
197	C₂×C₂²⋊C₄	—	13	4	no	—
198	C₂×C₂²⋊C₄	C₂×D₁₃	1	4	yes	—
199	C₂×C₄²	—	13	2	no	—
200	C₂×C₄²	C₂×D₁₃	1	1	yes	characteristic
201	C₂×C₁₃	—	2	2	no	—
202	C₂×C₁₃	—	4	4	no	—
203	C₂×C₁₃	C₂²×#41	1	1	yes	characteristic
204	C₂×C₁₃	#585	1	2	yes	—
205	D₁₃	—	4	2	no	—
206	C₂⁴	C₂²×D₁₃	1	2	yes	—
207	#41	—	26	4	no	—
208	C₂×Q₈	—	2	4	no	—
209	C₂×Q₈	—	13	1	no	—
210	C₂×Q₈	—	13	4	no	—
211	C₂×Q₈	C₂²×D₁₃	1	2	yes	—
212	C₂×Q₈	C₂²×D₁₃	1	2	yes	—
213	C₂×D₄	—	2	4	no	—
214	C₂×D₄	—	13	2	no	—
215	C₂×D₄	—	13	8	no	—
216	C₂×D₄	—	13	1	no	—
217	C₂×D₄	C₂²×D₁₃	1	2	yes	—
218	C₂×D₄	C₂²×D₁₃	1	2	yes	—
219	C₂²×C₄	—	13	8	no	—
220	C₂²×C₄	—	13	2	no	—
221	C₂²×C₄	—	13	4	no	—
222	C₂²×C₄	C₂²×D₁₃	1	2	yes	—
223	C₂²×C₄	C₂²×D₁₃	1	2	yes	—
224	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
225	C₄⋊C₄	—	13	4	no	—
226	C₄⋊C₄	—	13	2	no	—
227	C₄⋊C₄	—	13	2	no	—
228	C₄⋊C₄	—	13	4	no	—
229	C₄⋊C₄	—	13	2	no	—
230	C₄⋊C₄	—	13	8	no	—
231	C₄⋊C₄	—	13	8	no	—
232	C₄⋊C₄	—	13	4	no	—
233	C₄⋊C₄	—	13	4	no	—
234	C₂²⋊C₄	—	13	2	no	—
235	C₂²⋊C₄	—	13	2	no	—
236	C₂²⋊C₄	—	13	8	no	—
237	C₂²⋊C₄	—	13	2	no	—
238	C₂²⋊C₄	—	13	8	no	—
239	C₂²⋊C₄	—	13	4	no	—
240	C₂²⋊C₄	—	13	8	no	—
241	C₂²⋊C₄	C₂²×D₁₃	1	8	yes	—
242	C₂²⋊C₄	C₂²×D₁₃	1	8	yes	—
243	C₄²	—	13	2	no	—
244	C₄²	—	13	4	no	—
245	C₄²	—	13	8	no	—
246	C₄²	—	13	2	no	—
247	C₄²	—	13	2	no	—
248	C₄²	C₂²×D₁₃	1	1	yes	characteristic
249	C₄²	C₂²×D₁₃	1	1	yes	characteristic
250	C₄²	C₂²×D₁₃	1	2	yes	—
251	C₁₃	#3510	1	1	yes	characteristic, complemented, Sylow
252	C₂³	—	2	4	no	—
253	C₂³	—	4	4	no	—
254	C₂³	—	13	2	no	—
255	C₂³	C₂³×D₁₃	1	4	yes	—
256	C₂³	C₂³×D₁₃	1	1	yes	characteristic
257	Q₈	—	2	4	no	—
258	Q₈	—	2	4	no	—
259	Q₈	—	26	2	no	—
260	Q₈	—	52	4	no	—
261	D₄	—	2	4	no	—
262	D₄	—	2	4	no	—
263	D₄	—	26	4	no	—
264	D₄	—	26	2	no	—
265	D₄	—	52	8	no	—
266	C₂×C₄	—	2	4	no	—
267	C₂×C₄	—	4	2	no	—
268	C₂×C₄	—	13	4	no	—
269	C₂×C₄	—	13	4	no	—
270	C₂×C₄	—	13	2	no	—
271	C₂×C₄	—	13	4	no	—
272	C₂×C₄	—	26	4	no	—
273	C₂×C₄	—	52	4	no	—
274	C₂×C₄	—	52	8	no	—
275	C₂×C₄	#6129	1	4	yes	—
276	C₂×C₄	C₂³×D₁₃	1	2	yes	—
277	C₂×C₄	C₂³×D₁₃	1	4	yes	—
278	C₂×C₄	C₂³×D₁₃	1	2	yes	—
279	C₂×C₄	C₂³×D₁₃	1	1	yes	characteristic
280	C₂×C₄	C₂³×D₁₃	1	1	yes	characteristic
281	C₂²	—	2	2	no	—
282	C₂²	—	2	4	no	—
283	C₂²	—	2	8	no	—
284	C₂²	—	8	4	no	—
285	C₂²	—	26	2	no	—
286	C₂²	—	26	4	no	—
287	C₂²	C₂×#6129	1	2	yes	—
288	C₂²	C₂⁴×D₁₃	1	1	yes	characteristic
289	C₄	—	2	4	no	—
290	C₄	—	2	2	no	—
291	C₄	—	2	2	no	—
292	C₄	—	26	2	no	—
293	C₄	—	26	4	no	—
294	C₄	—	26	4	no	—
295	C₄	—	26	4	no	—
296	C₄	C₂×#6129	1	2	yes	—
297	C₄	C₂×#6129	1	2	yes	—
298	C₂	—	2	2	no	—
299	C₂	—	4	4	no	—
300	C₂	—	52	2	no	—
301	C₂	C₂²×#6129	1	1	yes	characteristic
302	C₂	#11719648	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₁₃
commutator, socle, cosocle	C₂²×C₁₃	C₂⁵
Fitting	C₂×C₁₃×#124	C₂
prime radical	#11719510	C₂
hypercenter	C₂×#124	D₁₃
hypocenter	C₁₃	#3510
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂×#124

Complements

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

N	H ≅ G/N	type
1. C₂×C₁₃×#124	C₂	semidirect
2. #11719521	C₂	semidirect
3. #11719517	C₂	semidirect
5. #11719503	C₂	semidirect
6. #11719387	C₂	semidirect
7. #11719386	C₂	semidirect
8. #11719385	C₂	semidirect
9. #11719376	C₂	semidirect
10. #11719200	C₂	semidirect
11. C₄×#6127	C₂	semidirect
12. #11719175	C₂	semidirect
23. #88201	C₂²	semidirect
24. #88200	C₂²	semidirect
25. Q₈×Dic₁₃	C₂²	semidirect
26. Q₈×Dic₁₃	C₂²	semidirect
27. #88198	C₂²	semidirect
40. #88140	C₂²	semidirect
41. #88139	C₂²	semidirect
42. #88137	C₂²	semidirect
43. #88135	C₂²	semidirect
44. #88133	C₂²	semidirect
45. #88133	C₂²	semidirect
46. #88132	C₂²	semidirect
47. #88131	C₂²	semidirect
48. C₄×D₅₂	C₂²	semidirect
49. #88126	C₂²	semidirect
50. C₄²×D₁₃	C₂²	semidirect
51. #88124	C₂²	semidirect
52. #88123	C₂²	semidirect
53. C₄×Dic₂₆	C₂²	semidirect
54. C₄×Dic₂₆	C₂²	semidirect
138. C₂×#124	D₁₃	semidirect
251. C₁₃	#3510	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₁₃×#124	2
#11719521	2
#11719517	2
#11719510	2
#11719503	2
#11719387	2
#11719386	2
#11719385	2
#11719376	2
#11719200	2
C₄×#6127	2
#11719175	2
#3510	13