Subgroups — #13124879

Order 1664.

454 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×#529	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₄×D₅₂	C₂	1	1	yes	maximal, characteristic, complemented
3	#11718727	C₂	1	2	yes	maximal, complemented
4	#11718718	C₂	1	2	yes	maximal, complemented
5	#11718707	C₂	1	1	yes	maximal, characteristic, complemented
6	#11718705	C₂	1	1	yes	maximal, characteristic, complemented
7	#11718697	C₂	1	1	yes	maximal, characteristic
8	#11718695	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄×#6106	C₂	1	2	yes	maximal, complemented
10	C₄×#6104	C₂	1	2	yes	maximal, complemented
11	#11718586	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×D₅₂	C₂²	1	1	yes	characteristic
13	C₂²×C₄×D₁₃	—	2	1	no	—
14	C₁₃×#126	—	2	1	no	—
15	C₁₃×#125	C₂²	1	1	yes	characteristic, complemented
16	C₁₃×#124	C₂²	1	1	yes	characteristic, complemented
17	C₁₃×#123	—	2	1	no	—
18	C₁₃×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
19	C₁₃×#121	C₂²	1	1	yes	characteristic
20	C₄×Q₈×C₁₃	C₂²	1	1	yes	characteristic, complemented
21	C₄×D₄×C₁₃	—	2	1	no	—
22	C₄×D₄×C₁₃	C₂²	1	1	yes	characteristic, complemented
23	C₁₃×#117	—	2	1	no	—
24	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
25	C₂×#6106	C₂²	1	2	yes	—
26	C₂×#6104	C₂²	1	2	yes	—
27	C₂×#6103	—	2	1	no	—
28	C₂×C₄⋊Dic₁₃	C₂²	1	1	yes	characteristic
29	C₄×D₅₂	—	2	1	no	—
30	C₄×D₅₂	—	2	1	no	—
31	C₄×D₅₂	C₂²	1	2	yes	—
32	C₄×D₅₂	C₂²	1	2	yes	—
33	#88075	C₂²	1	2	yes	complemented
34	#88072	C₂²	1	2	yes	complemented
35	C₂²⋊#1055	C₂²	1	1	yes	characteristic
36	C₂²⋊#1055	C₂²	1	1	yes	characteristic
37	#88049	C₂²	1	2	yes	—
38	#88049	C₂²	1	2	yes	—
39	#88048	C₂²	1	2	yes	complemented
40	#88047	C₂²	1	2	yes	complemented
41	C₄⋊#1055	C₂²	1	2	yes	complemented
42	C₄×#1055	C₂²	1	2	yes	complemented
43	C₂³×D₁₃	—	2	1	no	—
44	C₂×Q₈×C₁₃	C₂³	1	1	yes	characteristic
45	C₂×D₄×C₁₃	—	2	1	no	—
46	C₂×D₄×C₁₃	C₂³	1	1	yes	characteristic
47	C₂²×C₄×C₁₃	—	2	1	no	—
48	C₂²×C₄×C₁₃	D₄	1	1	yes	characteristic
49	C₂²×C₄×C₁₃	D₄	1	1	yes	characteristic
50	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
51	C₂²×Dic₁₃	—	2	1	no	—
52	C₂×D₅₂	—	2	1	no	—
53	C₂×D₅₂	—	2	4	no	—
54	C₂×D₅₂	C₂³	1	2	yes	—
55	C₂×C₄×D₁₃	—	2	2	no	—
56	C₂×C₄×D₁₃	—	2	2	no	—
57	C₂×C₄×D₁₃	—	4	4	no	—
58	C₁₃×C₄⋊C₄	—	2	1	no	—
59	C₁₃×C₄⋊C₄	—	2	1	no	—
60	C₁₃×C₄⋊C₄	—	2	1	no	—
61	C₁₃×C₄⋊C₄	—	2	1	no	—
62	C₁₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
63	C₁₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
64	C₁₃×C₂²⋊C₄	—	2	1	no	—
65	C₁₃×C₂²⋊C₄	—	2	1	no	—
66	C₁₃×C₂²⋊C₄	—	2	1	no	—
67	C₁₃×C₂²⋊C₄	—	2	1	no	—
68	C₁₃×C₂²⋊C₄	—	2	1	no	—
69	C₄²×C₁₃	—	2	1	no	—
70	C₄²×C₁₃	D₄	1	1	yes	characteristic
71	C₄²×C₁₃	D₄	1	1	yes	characteristic
72	C₄²×C₁₃	C₂³	1	1	yes	characteristic
73	C₄²×C₁₃	C₂³	1	1	yes	characteristic
74	#6106	—	2	4	no	—
75	#6104	—	2	4	no	—
76	#6103	—	2	2	no	—
77	#6103	—	2	2	no	—
78	C₄⋊Dic₁₃	—	2	1	no	—
79	C₄⋊Dic₁₃	C₂³	1	2	yes	—
80	C₂×#1055	C₂³	1	2	yes	—
81	C₂×#1055	C₂³	1	2	yes	—
82	#3105	—	13	1	no	maximal, Sylow
83	C₂³×C₁₃	—	2	1	no	—
84	C₂³×C₁₃	C₂×D₄	1	1	yes	characteristic
85	C₂²×D₁₃	—	2	4	no	—
86	C₂²×D₁₃	—	2	2	no	—
87	C₂²×D₁₃	—	4	2	no	—
88	C₂²×D₁₃	—	4	2	no	—
89	Q₈×C₁₃	—	2	2	no	—
90	D₄×C₁₃	—	2	2	no	—
91	D₄×C₁₃	—	4	2	no	—
92	C₂×C₄×C₁₃	—	2	2	no	—
93	C₂×C₄×C₁₃	—	2	2	no	—
94	C₂×C₄×C₁₃	—	2	2	no	—
95	C₂×C₄×C₁₃	—	2	1	no	—
96	C₂×C₄×C₁₃	—	2	1	no	—
97	C₂×C₄×C₁₃	—	2	1	no	—
98	C₂×C₄×C₁₃	—	4	2	no	—
99	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
100	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
101	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
102	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
103	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
104	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
105	C₂×Dic₁₃	—	2	2	no	—
106	C₂×Dic₁₃	—	4	2	no	—
107	D₅₂	—	2	4	no	—
108	D₅₂	—	2	2	no	—
109	D₅₂	#41	1	4	yes	—
110	C₄×D₁₃	—	4	4	no	—
111	C₄×D₁₃	—	4	4	no	—
112	#1055	—	2	2	no	—
113	#1055	—	2	2	no	—
114	#529	D₁₃	1	1	yes	characteristic, complemented
115	C₂×C₄×D₄	—	13	1	no	—
116	#484	—	13	1	no	—
117	#481	—	13	1	no	—
118	#478	—	13	2	no	—
119	#463	—	13	2	no	—
120	#450	—	13	1	no	—
121	#447	—	13	1	no	—
122	C₄×SD₁₆	—	13	2	no	—
123	C₄×D₈	—	13	2	no	—
124	#431	—	13	1	no	—
125	C₂²×C₁₃	—	2	1	no	—
126	C₂²×C₁₃	—	2	1	no	—
127	C₂²×C₁₃	—	4	1	no	—
128	C₂²×C₁₃	—	4	1	no	—
129	C₂²×C₁₃	—	4	1	no	—
130	C₂²×C₁₃	#135	1	2	yes	—
131	C₂²×C₁₃	C₂²×D₄	1	1	yes	characteristic
132	C₂×D₁₃	—	2	4	no	—
133	C₂×D₁₃	—	4	8	no	—
134	C₂×D₁₃	—	4	2	no	—
135	C₂×D₁₃	—	4	2	no	—
136	C₄×C₁₃	—	2	1	no	—
137	C₄×C₁₃	—	2	1	no	—
138	C₄×C₁₃	—	2	1	no	—
139	C₄×C₁₃	—	2	1	no	—
140	C₄×C₁₃	—	4	1	no	—
141	C₄×C₁₃	—	4	1	no	—
142	C₄×C₁₃	—	4	1	no	—
143	C₄×C₁₃	AGL(1,ℤ/8)	1	2	yes	—
144	C₄×C₁₃	C₂×#41	1	2	yes	—
145	Dic₁₃	—	4	2	no	—
146	C₂²×D₄	—	13	1	no	—
147	C₂³×C₄	—	26	1	no	—
148	C₂×SD₁₆	—	13	2	no	—
149	C₂×D₈	—	13	2	no	—
150	#126	—	2	1	no	—
151	#125	C₂×D₁₃	1	1	yes	characteristic, complemented
152	#124	C₂×D₁₃	1	1	yes	characteristic, complemented
153	#123	—	2	1	no	—
154	C₂²⋊Q₈	C₂×D₁₃	1	1	yes	characteristic
155	#121	C₂×D₁₃	1	1	yes	characteristic
156	C₄×Q₈	C₂×D₁₃	1	1	yes	characteristic, complemented
157	C₄×D₄	—	2	1	no	—
158	C₄×D₄	—	13	2	no	—
159	C₄×D₄	—	13	2	no	—
160	C₄×D₄	—	26	1	no	—
161	C₄×D₄	—	26	1	no	—
162	C₄×D₄	C₂×D₁₃	1	1	yes	characteristic, complemented
163	#117	—	2	1	no	—
164	C₂×C₄⋊C₄	—	13	1	no	—
165	C₂×C₂²⋊C₄	—	26	1	no	—
166	C₂×C₄²	C₂×D₁₃	1	1	yes	characteristic
167	#107	—	13	2	no	—
168	#106	—	13	2	no	—
169	C₄⋊C₈	—	13	2	no	—
170	#103	—	13	2	no	—
171	#102	—	13	2	no	—
172	#102	—	13	2	no	—
173	#102	—	13	2	no	—
174	C₂²⋊C₈	—	13	1	no	—
175	C₂²⋊C₈	—	13	1	no	—
176	C₄×C₈	—	13	2	no	—
177	C₂×C₁₃	—	2	2	no	—
178	C₂×C₁₃	—	8	1	no	—
179	C₂×C₁₃	#525	1	1	yes	characteristic
180	C₂×C₁₃	C₂×#135	1	1	yes	characteristic
181	C₂×C₁₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
182	D₁₃	—	4	4	no	—
183	C₂⁴	—	26	1	no	—
184	C₂×Q₈	C₂²×D₁₃	1	1	yes	characteristic
185	C₂×D₄	—	2	1	no	—
186	C₂×D₄	—	13	2	no	—
187	C₂×D₄	—	26	4	no	—
188	C₂×D₄	—	26	1	no	—
189	C₂×D₄	C₂²×D₁₃	1	1	yes	characteristic
190	C₂²×C₄	—	2	1	no	—
191	C₂²×C₄	—	26	1	no	—
192	C₂²×C₄	—	26	2	no	—
193	C₂²×C₄	—	26	2	no	—
194	C₂²×C₄	—	52	4	no	—
195	C₂²×C₄	C₂²⋊D₁₃	1	1	yes	characteristic
196	C₂²×C₄	C₂²⋊D₁₃	1	1	yes	characteristic
197	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
198	SD₁₆	—	26	4	no	—
199	D₈	—	26	4	no	—
200	C₂×C₈	—	13	2	no	—
201	C₂×C₈	—	13	2	no	—
202	C₄⋊C₄	—	2	1	no	—
203	C₄⋊C₄	—	2	1	no	—
204	C₄⋊C₄	—	2	1	no	—
205	C₄⋊C₄	—	2	1	no	—
206	C₄⋊C₄	—	13	2	no	—
207	C₄⋊C₄	—	26	1	no	—
208	C₄⋊C₄	C₂²×D₁₃	1	1	yes	characteristic
209	C₄⋊C₄	C₂²×D₁₃	1	1	yes	characteristic
210	C₂²⋊C₄	—	2	1	no	—
211	C₂²⋊C₄	—	2	1	no	—
212	C₂²⋊C₄	—	2	1	no	—
213	C₂²⋊C₄	—	2	1	no	—
214	C₂²⋊C₄	—	2	1	no	—
215	C₂²⋊C₄	—	26	2	no	—
216	C₂²⋊C₄	—	26	2	no	—
217	C₄²	—	2	1	no	—
218	C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
219	C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
220	C₄²	C₂²×D₁₃	1	1	yes	characteristic
221	C₄²	C₂²×D₁₃	1	1	yes	characteristic
222	C₁₃	#3105	1	1	yes	characteristic, complemented, Sylow
223	C₂³	—	2	1	no	—
224	C₂³	—	26	4	no	—
225	C₂³	—	26	2	no	—
226	C₂³	—	52	2	no	—
227	C₂³	—	52	2	no	—
228	C₂³	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
229	Q₈	—	2	2	no	—
230	D₄	—	2	2	no	—
231	D₄	—	4	2	no	—
232	D₄	—	13	4	no	—
233	D₄	—	26	4	no	—
234	D₄	—	26	2	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	2	1	no	—
239	C₂×C₄	—	2	1	no	—
240	C₂×C₄	—	2	1	no	—
241	C₂×C₄	—	4	2	no	—
242	C₂×C₄	—	26	2	no	—
243	C₂×C₄	—	52	4	no	—
244	C₂×C₄	—	52	2	no	—
245	C₂×C₄	—	52	4	no	—
246	C₂×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
247	C₂×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
248	C₂×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
249	C₂×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
250	C₂×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
251	C₂×C₄	C₂³×D₁₃	1	1	yes	characteristic
252	C₈	—	26	2	no	—
253	C₈	—	26	2	no	—
254	C₂²	—	2	1	no	—
255	C₂²	—	2	1	no	—
256	C₂²	—	4	1	no	—
257	C₂²	—	4	1	no	—
258	C₂²	—	4	1	no	—
259	C₂²	—	26	4	no	—
260	C₂²	—	52	8	no	—
261	C₂²	—	52	2	no	—
262	C₂²	—	52	2	no	—
263	C₂²	#88204	1	2	yes	—
264	C₂²	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
265	C₄	—	2	1	no	—
266	C₄	—	2	1	no	—
267	C₄	—	2	1	no	—
268	C₄	—	2	1	no	—
269	C₄	—	4	1	no	—
270	C₄	—	4	1	no	—
271	C₄	—	4	1	no	—
272	C₄	—	52	2	no	—
273	C₄	#88203	1	2	yes	—
274	C₄	#41×D₁₃	1	2	yes	—
275	C₂	—	2	2	no	—
276	C₂	—	8	1	no	—
277	C₂	—	52	4	no	—
278	C₂	C₂×#88203	1	1	yes	characteristic
279	C₂	C₂×#88204	1	1	yes	characteristic
280	C₂	#11719533	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₁₃
commutator, cosocle	C₂×C₄×C₁₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₃
Fitting	C₁₃×#529	C₂
socle	C₂²×C₁₃	C₂²×D₄
prime radical	#11718697	C₂
hypercenter	#529	D₁₃
hypocenter	C₁₃	#3105
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #529

Complements

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

N	H ≅ G/N	type
1. C₁₃×#529	C₂	semidirect
2. C₂×C₄×D₅₂	C₂	semidirect
3. #11718727	C₂	semidirect
4. #11718718	C₂	semidirect
5. #11718707	C₂	semidirect
6. #11718705	C₂	semidirect
8. #11718695	C₂	semidirect
9. C₄×#6106	C₂	semidirect
10. C₄×#6104	C₂	semidirect
11. #11718586	C₂	semidirect
15. C₁₃×#125	C₂²	semidirect
16. C₁₃×#124	C₂²	semidirect
20. C₄×Q₈×C₁₃	C₂²	semidirect
22. C₄×D₄×C₁₃	C₂²	semidirect
33. #88075	C₂²	semidirect
34. #88072	C₂²	semidirect
39. #88048	C₂²	semidirect
40. #88047	C₂²	semidirect
41. C₄⋊#1055	C₂²	semidirect
42. C₄×#1055	C₂²	semidirect
114. #529	D₁₃	semidirect
151. #125	C₂×D₁₃	semidirect
152. #124	C₂×D₁₃	semidirect
156. C₄×Q₈	C₂×D₁₃	semidirect
162. C₄×D₄	C₂×D₁₃	semidirect
222. C₁₃	#3105	semidirect

Maximal subgroups

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

subgroup	index
C₁₃×#529	2
C₂×C₄×D₅₂	2
#11718727	2
#11718718	2
#11718707	2
#11718705	2
#11718697	2
#11718695	2
C₄×#6106	2
C₄×#6104	2
#11718586	2
#3105	13