Subgroups — #13128734

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	C₁₃×#533	C₂	1	1	yes	maximal, characteristic, complemented
2	#11719438	C₂	1	4	yes	maximal, complemented
3	#11719396	C₂	1	4	yes	maximal, complemented
4	#11719342	C₂	1	8	yes	maximal, complemented
5	#11719290	C₂	1	4	yes	maximal, complemented
6	#11719289	C₂	1	4	yes	maximal, complemented
7	#11719231	C₂	1	1	yes	maximal, characteristic, complemented
8	#11719230	C₂	1	1	yes	maximal, characteristic
9	#11719226	C₂	1	1	yes	maximal, characteristic, complemented
10	#11719221	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×#88129	C₂	1	1	yes	maximal, characteristic, complemented
12	C₄×#6127	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂×#6127	C₂²	1	1	yes	characteristic
14	C₂²×D₅₂	C₂²	1	1	yes	characteristic
15	C₂²×Dic₂₆	C₂²	1	1	yes	characteristic
16	C₁₃×C₄⋊Q₈	C₂²	1	2	yes	complemented
17	C₁₃×#125	C₂²	1	2	yes	complemented
18	C₁₃×C₂²⋊Q₈	C₂²	1	4	yes	—
19	C₄×Q₈×C₁₃	C₂²	1	4	yes	complemented
20	C₁₃×#117	C₂²	1	1	yes	characteristic
21	C₁₃×#117	C₂²	1	1	yes	characteristic
22	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
23	#88201	C₂²	1	8	yes	complemented
24	#88200	C₂²	1	8	yes	complemented
25	C₄×C₂²⋊D₁₃	C₂²	1	4	yes	—
26	C₂×#6103	C₂²	1	4	yes	—
27	#88180	C₂²	1	1	yes	characteristic
28	#88152	C₂²	1	8	yes	complemented
29	#88152	C₂²	1	8	yes	complemented
30	#88151	C₂²	1	4	yes	complemented
31	#88149	C₂²	1	4	yes	—
32	#88148	C₂²	1	8	yes	complemented
33	#88148	C₂²	1	8	yes	complemented
34	#88147	C₂²	1	4	yes	—
35	#88147	C₂²	1	8	yes	complemented
36	#88146	C₂²	1	8	yes	complemented
37	#88141	C₂²	1	4	yes	complemented
38	#88140	C₂²	1	4	yes	—
39	#88136	C₂²	1	4	yes	—
40	C₂²⋊C₄×D₁₃	C₂²	1	4	yes	—
41	#88131	C₂²	1	4	yes	—
42	#88129	C₂²	1	2	yes	—
43	#88129	C₂²	1	4	yes	—
44	#88129	C₂²	1	2	yes	—
45	#88129	C₂²	1	2	yes	—
46	#88129	C₂²	1	2	yes	complemented
47	#88128	C₂²	1	2	yes	—
48	C₄×D₅₂	C₂²	1	4	yes	—
49	C₄×D₅₂	C₂²	1	2	yes	complemented
50	#88126	C₂²	1	2	yes	complemented
51	C₄²×D₁₃	C₂²	1	2	yes	complemented
52	#88123	C₂²	1	2	yes	complemented
53	C₄×Dic₂₆	C₂²	1	2	yes	complemented
54	C₄×Dic₂₆	C₂²	1	4	yes	complemented
55	C₂³×D₁₃	C₂³	1	2	yes	—
56	C₂×Q₈×C₁₃	C₂³	1	4	yes	—
57	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
58	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
59	C₂×C₂²⋊D₁₃	C₂³	1	2	yes	—
60	C₂²×Dic₁₃	C₂³	1	2	yes	—
61	#6127	—	2	4	no	—
62	C₂×D₅₂	—	2	4	no	—
63	C₂×D₅₂	C₂³	1	2	yes	—
64	C₂×D₅₂	C₂³	1	1	yes	characteristic
65	C₂×D₅₂	C₂³	1	2	yes	—
66	C₂×D₅₂	C₂³	1	8	yes	—
67	C₂×C₄×D₁₃	C₂³	1	2	yes	—
68	C₂×C₄×D₁₃	C₂³	1	4	yes	—
69	C₂×C₄×D₁₃	C₂³	1	8	yes	—
70	C₂×Dic₂₆	—	2	4	no	—
71	C₂×Dic₂₆	C₂³	1	1	yes	characteristic
72	C₂×Dic₂₆	C₂³	1	2	yes	—
73	C₂×Dic₂₆	C₂³	1	2	yes	—
74	C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
75	C₁₃×C₄⋊C₄	C₂³	1	4	yes	—
76	C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
77	C₁₃×C₄⋊C₄	C₂³	1	4	yes	—
78	C₁₃×C₄⋊C₄	C₂³	1	4	yes	—
79	C₁₃×C₂²⋊C₄	C₂³	1	2	yes	—
80	C₁₃×C₂²⋊C₄	C₂³	1	2	yes	—
81	C₄²×C₁₃	C₂³	1	2	yes	—
82	C₄²×C₁₃	C₂³	1	2	yes	—
83	C₄²×C₁₃	C₂³	1	2	yes	—
84	C₄²×C₁₃	C₂³	1	1	yes	characteristic
85	C₄²×C₁₃	C₂³	1	1	yes	characteristic
86	C₂²⋊Dic₁₃	C₂³	1	2	yes	—
87	#6103	C₂³	1	4	yes	—
88	#6103	C₂³	1	8	yes	—
89	#6103	C₂³	1	8	yes	—
90	#6103	C₂³	1	8	yes	—
91	#6103	C₂³	1	8	yes	—
92	#6103	C₂³	1	8	yes	—
93	C₄⋊Dic₁₃	C₂³	1	2	yes	—
94	C₄⋊Dic₁₃	C₂³	1	8	yes	—
95	#6101	C₂³	1	4	yes	—
96	#6101	C₂³	1	8	yes	—
97	C₄×Dic₁₃	C₂³	1	4	yes	—
98	C₄×Dic₁₃	C₂³	1	8	yes	—
99	C₄×Dic₁₃	C₂³	1	2	yes	—
100	#3510	—	13	1	no	maximal, Sylow
101	C₂³×C₁₃	C₂⁴	1	1	yes	characteristic
102	C₂²×D₁₃	—	2	4	no	—
103	C₂²×D₁₃	—	4	4	no	—
104	C₂²×D₁₃	C₂⁴	1	4	yes	—
105	C₂²×D₁₃	C₂⁴	1	2	yes	—
106	Q₈×C₁₃	—	4	4	no	—
107	C₂×C₄×C₁₃	—	4	2	no	—
108	C₂×C₄×C₁₃	#41	1	4	yes	—
109	C₂×C₄×C₁₃	C₂⁴	1	4	yes	—
110	C₂×C₄×C₁₃	C₂⁴	1	2	yes	—
111	C₂×C₄×C₁₃	C₂⁴	1	2	yes	—
112	C₂×C₄×C₁₃	C₂⁴	1	4	yes	—
113	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
114	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
115	C₂²⋊D₁₃	—	2	4	no	—
116	C₂×Dic₁₃	—	2	4	no	—
117	C₂×Dic₁₃	C₂⁴	1	4	yes	—
118	C₂×Dic₁₃	C₂⁴	1	2	yes	—
119	C₂×Dic₁₃	C₂⁴	1	4	yes	—
120	D₅₂	—	2	4	no	—
121	D₅₂	—	2	4	no	—
122	D₅₂	—	2	2	no	—
123	D₅₂	—	4	8	no	—
124	C₄×D₁₃	—	2	4	no	—
125	C₄×D₁₃	—	4	4	no	—
126	C₄×D₁₃	—	4	8	no	—
127	Dic₂₆	—	2	4	no	—
128	Dic₂₆	—	2	4	no	—
129	Dic₂₆	—	2	2	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	D₁₃	1	1	yes	characteristic, complemented
137	#529	—	13	4	no	—
138	C₂×#124	—	13	1	no	—
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	C₂×#41	1	2	yes	—
144	C₂²×C₁₃	C₂⁵	1	1	yes	characteristic
145	C₂×D₁₃	—	2	2	no	—
146	C₂×D₁₃	—	2	4	no	—
147	C₂×D₁₃	—	2	4	no	—
148	C₂×D₁₃	—	2	8	no	—
149	C₂×D₁₃	—	8	4	no	—
150	C₄×C₁₃	—	2	2	no	—
151	C₄×C₁₃	—	2	4	no	—
152	C₄×C₁₃	—	2	4	no	—
153	C₄×C₁₃	—	2	2	no	—
154	C₄×C₁₃	C₂×#41	1	2	yes	—
155	C₄×C₁₃	C₂×#41	1	2	yes	—
156	Dic₁₃	—	2	4	no	—
157	Dic₁₃	—	2	2	no	—
158	Dic₁₃	—	2	4	no	—
159	C₂×#41	—	13	1	no	—
160	C₂²×Q₈	—	13	1	no	—
161	C₂²×D₄	—	13	1	no	—
162	C₄⋊Q₈	—	13	2	no	—
163	C₄⋊Q₈	C₂×D₁₃	1	2	yes	complemented
164	#127	—	13	2	no	—
165	#126	—	13	8	no	—
166	#126	—	13	8	no	—
167	#125	C₂×D₁₃	1	2	yes	complemented
168	#124	—	13	8	no	—
169	#124	—	13	2	no	—
170	#124	—	13	2	no	—
171	#124	—	13	4	no	—
172	#124	—	13	2	no	—
173	#124	—	13	2	no	—
174	#123	—	13	8	no	—
175	#123	—	13	8	no	—
176	#123	—	13	4	no	—
177	C₂²⋊Q₈	—	13	8	no	—
178	C₂²⋊Q₈	—	13	4	no	—
179	C₂²⋊Q₈	C₂×D₁₃	1	4	yes	—
180	#121	—	13	4	no	—
181	#120	—	13	4	no	—
182	C₄×Q₈	—	13	4	no	—
183	C₄×Q₈	—	13	2	no	—
184	C₄×Q₈	—	13	4	no	—
185	C₄×Q₈	C₂×D₁₃	1	4	yes	complemented
186	C₄×D₄	—	13	4	no	—
187	C₄×D₄	—	13	4	no	—
188	C₄×D₄	—	13	8	no	—
189	C₄×D₄	—	13	2	no	—
190	C₄×D₄	—	13	4	no	—
191	#117	—	13	1	no	—
192	#117	—	13	8	no	—
193	#117	—	13	4	no	—
194	#117	—	13	2	no	—
195	#117	C₂×D₁₃	1	1	yes	characteristic
196	#117	C₂×D₁₃	1	1	yes	characteristic
197	C₂×C₂²⋊C₄	—	13	4	no	—
198	C₂×C₂²⋊C₄	—	13	4	no	—
199	C₂×C₄²	—	13	2	no	—
200	C₂×C₄²	C₂×D₁₃	1	1	yes	characteristic
201	C₂×C₁₃	—	2	2	no	—
202	C₂×C₁₃	C₂²×#41	1	1	yes	characteristic
203	C₂×C₁₃	#585	1	2	yes	—
204	D₁₃	—	4	2	no	—
205	D₁₃	—	4	4	no	—
206	C₂⁴	—	13	2	no	—
207	#41	—	26	4	no	—
208	C₂×Q₈	—	13	2	no	—
209	C₂×Q₈	—	13	2	no	—
210	C₂×Q₈	—	13	1	no	—
211	C₂×Q₈	—	26	4	no	—
212	C₂×Q₈	C₂²×D₁₃	1	4	yes	—
213	C₂×D₄	—	13	2	no	—
214	C₂×D₄	—	13	2	no	—
215	C₂×D₄	—	13	2	no	—
216	C₂×D₄	—	13	1	no	—
217	C₂×D₄	—	13	8	no	—
218	C₂×D₄	—	26	4	no	—
219	C₂²×C₄	—	13	2	no	—
220	C₂²×C₄	—	13	2	no	—
221	C₂²×C₄	—	13	4	no	—
222	C₂²×C₄	—	13	8	no	—
223	C₂²×C₄	C₂²×D₁₃	1	2	yes	—
224	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
225	C₄⋊C₄	—	13	2	no	—
226	C₄⋊C₄	—	13	4	no	—
227	C₄⋊C₄	—	13	8	no	—
228	C₄⋊C₄	—	13	8	no	—
229	C₄⋊C₄	C₂²×D₁₃	1	4	yes	—
230	C₄⋊C₄	C₂²×D₁₃	1	2	yes	—
231	C₄⋊C₄	C₂²×D₁₃	1	2	yes	—
232	C₄⋊C₄	C₂²×D₁₃	1	4	yes	—
233	C₄⋊C₄	C₂²×D₁₃	1	4	yes	—
234	C₂²⋊C₄	—	13	2	no	—
235	C₂²⋊C₄	—	13	8	no	—
236	C₂²⋊C₄	—	13	8	no	—
237	C₂²⋊C₄	—	13	8	no	—
238	C₂²⋊C₄	—	13	4	no	—
239	C₂²⋊C₄	—	13	8	no	—
240	C₂²⋊C₄	—	13	8	no	—
241	C₂²⋊C₄	C₂²×D₁₃	1	2	yes	—
242	C₂²⋊C₄	C₂²×D₁₃	1	2	yes	—
243	C₄²	—	13	2	no	—
244	C₄²	—	13	4	no	—
245	C₄²	—	13	8	no	—
246	C₄²	C₂²×D₁₃	1	1	yes	characteristic
247	C₄²	C₂²×D₁₃	1	1	yes	characteristic
248	C₄²	C₂²×D₁₃	1	2	yes	—
249	C₄²	C₂²×D₁₃	1	2	yes	—
250	C₄²	C₂²×D₁₃	1	2	yes	—
251	C₁₃	#3510	1	1	yes	characteristic, complemented, Sylow
252	C₂³	—	13	4	no	—
253	C₂³	—	13	2	no	—
254	C₂³	—	26	4	no	—
255	C₂³	—	52	4	no	—
256	C₂³	C₂³×D₁₃	1	1	yes	characteristic
257	Q₈	—	4	4	no	—
258	Q₈	—	26	4	no	—
259	Q₈	—	26	4	no	—
260	Q₈	—	26	2	no	—
261	D₄	—	26	4	no	—
262	D₄	—	26	4	no	—
263	D₄	—	26	4	no	—
264	D₄	—	26	2	no	—
265	D₄	—	52	8	no	—
266	C₂×C₄	—	4	2	no	—
267	C₂×C₄	—	13	4	no	—
268	C₂×C₄	—	13	4	no	—
269	C₂×C₄	—	13	2	no	—
270	C₂×C₄	—	26	4	no	—
271	C₂×C₄	—	26	4	no	—
272	C₂×C₄	—	52	4	no	—
273	C₂×C₄	—	52	8	no	—
274	C₂×C₄	#6131	1	4	yes	—
275	C₂×C₄	C₂³×D₁₃	1	2	yes	—
276	C₂×C₄	C₂³×D₁₃	1	4	yes	—
277	C₂×C₄	C₂³×D₁₃	1	2	yes	—
278	C₂×C₄	C₂³×D₁₃	1	4	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	—	26	2	no	—
283	C₂²	—	26	4	no	—
284	C₂²	—	26	4	no	—
285	C₂²	—	26	8	no	—
286	C₂²	—	104	4	no	—
287	C₂²	C₂×#6131	1	2	yes	—
288	C₂²	C₂⁴×D₁₃	1	1	yes	characteristic
289	C₄	—	2	2	no	—
290	C₄	—	2	2	no	—
291	C₄	—	2	4	no	—
292	C₄	—	2	4	no	—
293	C₄	—	26	2	no	—
294	C₄	—	26	4	no	—
295	C₄	—	26	4	no	—
296	C₄	C₂×#6131	1	2	yes	—
297	C₄	C₂×#6131	1	2	yes	—
298	C₂	—	2	2	no	—
299	C₂	—	52	4	no	—
300	C₂	—	52	2	no	—
301	C₂	C₂²×#6131	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	C₁₃×#533	C₂
prime radical	#11719230	C₂
hypercenter	#533	D₁₃
hypocenter	C₁₃	#3510
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ #533

Complements

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

N	H ≅ G/N	type
1. C₁₃×#533	C₂	semidirect
2. #11719438	C₂	semidirect
3. #11719396	C₂	semidirect
4. #11719342	C₂	semidirect
5. #11719290	C₂	semidirect
6. #11719289	C₂	semidirect
7. #11719231	C₂	semidirect
9. #11719226	C₂	semidirect
10. #11719221	C₂	semidirect
11. C₂×#88129	C₂	semidirect
12. C₄×#6127	C₂	semidirect
16. C₁₃×C₄⋊Q₈	C₂²	semidirect
17. C₁₃×#125	C₂²	semidirect
19. C₄×Q₈×C₁₃	C₂²	semidirect
23. #88201	C₂²	semidirect
24. #88200	C₂²	semidirect
28. #88152	C₂²	semidirect
29. #88152	C₂²	semidirect
30. #88151	C₂²	semidirect
32. #88148	C₂²	semidirect
33. #88148	C₂²	semidirect
35. #88147	C₂²	semidirect
36. #88146	C₂²	semidirect
37. #88141	C₂²	semidirect
46. #88129	C₂²	semidirect
49. C₄×D₅₂	C₂²	semidirect
50. #88126	C₂²	semidirect
51. C₄²×D₁₃	C₂²	semidirect
52. #88123	C₂²	semidirect
53. C₄×Dic₂₆	C₂²	semidirect
54. C₄×Dic₂₆	C₂²	semidirect
136. #533	D₁₃	semidirect
163. C₄⋊Q₈	C₂×D₁₃	semidirect
167. #125	C₂×D₁₃	semidirect
185. C₄×Q₈	C₂×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
C₁₃×#533	2
#11719438	2
#11719396	2
#11719342	2
#11719290	2
#11719289	2
#11719231	2
#11719230	2
#11719226	2
#11719221	2
C₂×#88129	2
C₄×#6127	2
#3510	13