Subgroups — #13123683

Order 1664.

528 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₈×D₁₃	C₂	1	1	yes	maximal, characteristic
2	#11719250	C₂	1	2	yes	maximal, complemented
3	C₄×C₁₃×D₈	C₂	1	1	yes	maximal, characteristic, complemented
4	D₈×Dic₁₃	C₂	1	1	yes	maximal, characteristic, complemented
5	#11718675	C₂	1	2	yes	maximal, complemented
6	#11718541	C₂	1	1	yes	maximal, characteristic, complemented
7	#11718533	C₂	1	1	yes	maximal, characteristic, complemented
8	#11718446	C₂	1	2	yes	maximal, complemented
9	#11718431	C₂	1	2	yes	maximal, complemented
10	C₄×D₁₀₄	C₂	1	1	yes	maximal, characteristic, complemented
11	#11718350	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×D₄×D₁₃	C₂²	1	2	yes	—
13	C₂×C₁₃×D₈	C₂²	1	1	yes	characteristic
14	C₄×D₄×C₁₃	C₂²	1	2	yes	complemented
15	D₄×Dic₁₃	C₂²	1	2	yes	complemented
16	C₂×#6104	C₂²	1	2	yes	—
17	C₄×C₂²⋊D₁₃	—	2	2	no	—
18	C₂×#6103	—	2	2	no	—
19	D₈×D₁₃	C₄	1	8	yes	complemented
20	C₂×D₁₀₄	C₂²	1	1	yes	characteristic
21	C₂×#6094	C₂²	1	1	yes	characteristic
22	C₂×C₈×D₁₃	C₂²	1	1	yes	characteristic
23	#88147	C₂²	1	2	yes	complemented
24	#88146	C₂²	1	2	yes	—
25	#88135	—	2	2	no	—
26	C₂²⋊C₄×D₁₃	—	2	2	no	—
27	C₄×D₅₂	C₂²	1	2	yes	complemented
28	#88126	C₂²	1	1	yes	characteristic
29	C₁₃×#107	C₂²	1	1	yes	characteristic, complemented
30	C₁₃×#102	C₂²	1	2	yes	complemented
31	C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic, complemented
32	#88072	C₂²	1	2	yes	complemented
33	#88061	C₂²	1	2	yes	complemented
34	#88058	C₂²	1	1	yes	characteristic, complemented
35	#88055	C₂²	1	1	yes	characteristic, complemented
36	C₈×Dic₁₃	C₂²	1	1	yes	characteristic, complemented
37	#88049	C₂²	1	2	yes	complemented
38	#88048	C₂²	1	2	yes	complemented
39	#88043	C₂²	1	1	yes	characteristic, complemented
40	C₂³×D₁₃	—	2	2	no	—
41	C₂×D₄×C₁₃	C₂³	1	2	yes	—
42	C₂²×C₄×C₁₃	—	2	2	no	—
43	C₂×C₂²⋊D₁₃	—	2	2	no	—
44	C₂²×Dic₁₃	—	2	2	no	—
45	D₄×D₁₃	—	2	4	no	—
46	D₄×D₁₃	C₂×C₄	1	8	yes	complemented
47	C₂×D₅₂	C₂³	1	2	yes	—
48	C₂×C₄×D₁₃	—	2	2	no	—
49	C₂×C₄×D₁₃	—	2	2	no	—
50	C₂×C₄×D₁₃	C₂³	1	1	yes	characteristic
51	C₁₃×D₈	C₂×C₄	1	4	yes	complemented
52	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
53	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
54	C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
55	C₁₃×C₂²⋊C₄	—	2	2	no	—
56	C₄²×C₁₃	C₂³	1	1	yes	characteristic
57	C₂²⋊Dic₁₃	—	2	2	no	—
58	#6104	C₂×C₄	1	8	yes	complemented
59	#6103	—	2	2	no	—
60	#6103	—	2	2	no	—
61	#6103	—	2	2	no	—
62	#6103	D₄	1	1	yes	characteristic
63	#6103	D₄	1	1	yes	characteristic
64	C₄⋊Dic₁₃	C₂³	1	2	yes	—
65	#6101	D₄	1	1	yes	characteristic
66	#6101	D₄	1	1	yes	characteristic
67	C₄×Dic₁₃	—	2	2	no	—
68	C₄×Dic₁₃	C₂³	1	1	yes	characteristic
69	C₂×#1055	C₂³	1	1	yes	characteristic
70	C₂×#1055	C₂³	1	1	yes	characteristic
71	D₁₀₄	C₂×C₄	1	4	yes	complemented
72	#6094	—	2	2	no	—
73	C₈×D₁₃	C₂×C₄	1	4	yes	complemented
74	#2948	—	13	1	no	maximal, Sylow
75	C₂³×C₁₃	—	2	2	no	—
76	C₂²×D₁₃	—	2	2	no	—
77	C₂²×D₁₃	—	2	8	no	—
78	C₂²×D₁₃	—	8	2	no	—
79	C₂²×D₁₃	—	8	2	no	—
80	C₂²×D₁₃	C₂×D₄	1	1	yes	characteristic
81	D₄×C₁₃	—	2	2	no	—
82	D₄×C₁₃	C₂²×C₄	1	4	yes	—
83	C₂×C₄×C₁₃	—	2	2	no	—
84	C₂×C₄×C₁₃	—	8	2	no	—
85	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
86	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
87	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
88	C₂²⋊D₁₃	—	2	8	no	—
89	C₂×Dic₁₃	—	2	2	no	—
90	C₂×Dic₁₃	—	8	2	no	—
91	C₂×Dic₁₃	C₂×D₄	1	1	yes	characteristic
92	C₂×Dic₁₃	C₂×D₄	1	1	yes	characteristic
93	C₂×Dic₁₃	C₂×D₄	1	1	yes	characteristic
94	D₅₂	—	2	2	no	—
95	D₅₂	C₂²×C₄	1	4	yes	—
96	C₄×D₁₃	—	8	2	no	—
97	C₄×D₁₃	—	8	2	no	—
98	C₄×D₁₃	C₂²×C₄	1	2	yes	—
99	C₄×D₁₃	#41	1	2	yes	—
100	C₈×C₁₃	—	2	1	no	—
101	C₈×C₁₃	C₂²×C₄	1	2	yes	—
102	#1055	—	2	1	no	—
103	#1055	C₂²×C₄	1	2	yes	—
104	C₂²×D₈	—	13	1	no	—
105	#518	—	13	2	no	—
106	#442	—	13	2	no	—
107	#442	—	13	2	no	—
108	C₄×D₈	—	13	1	no	—
109	C₄×D₈	—	13	1	no	—
110	C₄×D₈	—	13	1	no	—
111	C₄×D₈	D₁₃	1	1	yes	characteristic, complemented
112	#427	—	13	1	no	—
113	#418	—	13	2	no	—
114	#405	—	13	1	no	—
115	C₂²×C₁₃	—	2	4	no	—
116	C₂²×C₁₃	—	4	2	no	—
117	C₂²×C₁₃	—	4	2	no	—
118	C₂²×C₁₃	C₂²×D₄	1	1	yes	characteristic
119	C₂×D₁₃	—	2	1	no	—
120	C₂×D₁₃	—	2	1	no	—
121	C₂×D₁₃	—	2	4	no	—
122	C₂×D₁₃	—	4	2	no	—
123	C₂×D₁₃	—	4	2	no	—
124	C₂×D₁₃	—	8	8	no	—
125	C₂×D₁₃	C₄×D₄	1	2	yes	—
126	C₄×C₁₃	—	2	1	no	—
127	C₄×C₁₃	—	2	1	no	—
128	C₄×C₁₃	—	4	2	no	—
129	C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
130	C₄×C₁₃	C₂×#41	1	1	yes	characteristic
131	Dic₁₃	—	2	1	no	—
132	Dic₁₃	—	2	1	no	—
133	Dic₁₃	—	4	2	no	—
134	Dic₁₃	C₄×D₄	1	2	yes	—
135	C₂²×D₄	—	13	2	no	—
136	C₂×D₈	—	13	2	no	—
137	C₂×D₈	—	13	8	no	—
138	C₂×D₈	—	13	1	no	—
139	C₂×D₈	C₂×D₁₃	1	1	yes	characteristic
140	C₂×M₁₆	—	13	1	no	—
141	C₂²×C₈	—	13	1	no	—
142	C₄×D₄	—	13	2	no	—
143	C₄×D₄	—	13	2	no	—
144	C₄×D₄	—	13	2	no	—
145	C₄×D₄	—	26	2	no	—
146	C₄×D₄	—	26	2	no	—
147	C₄×D₄	C₂×D₁₃	1	2	yes	complemented
148	#117	—	13	1	no	—
149	#117	—	13	2	no	—
150	C₂×C₂²⋊C₄	—	26	2	no	—
151	C₂×C₂²⋊C₄	—	26	2	no	—
152	#107	—	13	1	no	—
153	#107	C₂×D₁₃	1	1	yes	characteristic, complemented
154	#106	—	13	2	no	—
155	#102	—	13	2	no	—
156	#102	—	13	2	no	—
157	#102	—	13	2	no	—
158	#102	C₂×D₁₃	1	2	yes	complemented
159	#97	—	13	1	no	—
160	#97	—	13	1	no	—
161	C₄×C₈	—	13	1	no	—
162	C₄×C₈	C₂×D₁₃	1	1	yes	characteristic, complemented
163	C₂×C₁₃	—	4	4	no	—
164	C₂×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
165	C₂×C₁₃	#576	1	1	yes	characteristic
166	C₂×C₁₃	#576	1	1	yes	characteristic
167	D₁₃	—	2	2	no	—
168	D₁₃	—	4	4	no	—
169	C₂⁴	—	26	2	no	—
170	C₂×D₄	—	13	8	no	—
171	C₂×D₄	—	13	2	no	—
172	C₂×D₄	—	26	4	no	—
173	C₂×D₄	—	26	2	no	—
174	C₂×D₄	C₂²×D₁₃	1	2	yes	—
175	C₂²×C₄	—	2	2	no	—
176	C₂²×C₄	—	13	1	no	—
177	C₂²×C₄	—	26	2	no	—
178	C₂²×C₄	—	26	2	no	—
179	C₂²×C₄	—	26	2	no	—
180	D₈	—	13	8	no	—
181	D₈	—	13	4	no	—
182	D₈	C₄×D₁₃	1	4	yes	complemented
183	M₁₆	—	26	2	no	—
184	C₂×C₈	—	13	1	no	—
185	C₂×C₈	—	13	4	no	—
186	C₂×C₈	—	13	1	no	—
187	C₂×C₈	C₂²×D₁₃	1	1	yes	characteristic
188	C₂×C₈	C₂²×D₁₃	1	1	yes	characteristic
189	C₄⋊C₄	—	13	1	no	—
190	C₄⋊C₄	—	13	1	no	—
191	C₄⋊C₄	—	13	2	no	—
192	C₄⋊C₄	C₂²×D₁₃	1	2	yes	—
193	C₂²⋊C₄	—	2	2	no	—
194	C₂²⋊C₄	—	13	1	no	—
195	C₂²⋊C₄	—	13	1	no	—
196	C₂²⋊C₄	—	26	2	no	—
197	C₂²⋊C₄	—	26	2	no	—
198	C₂²⋊C₄	—	26	2	no	—
199	C₂²⋊C₄	—	26	2	no	—
200	C₄²	—	13	1	no	—
201	C₄²	—	26	2	no	—
202	C₄²	C₂²×D₁₃	1	1	yes	characteristic
203	C₁₃	#2948	1	1	yes	characteristic, complemented, Sylow
204	C₂³	—	2	2	no	—
205	C₂³	—	13	1	no	—
206	C₂³	—	26	8	no	—
207	C₂³	—	26	2	no	—
208	C₂³	—	104	2	no	—
209	C₂³	—	104	2	no	—
210	D₄	—	2	2	no	—
211	D₄	—	13	4	no	—
212	D₄	—	26	8	no	—
213	D₄	—	26	2	no	—
214	D₄	C₂×C₄×D₁₃	1	4	yes	—
215	C₂×C₄	—	2	2	no	—
216	C₂×C₄	—	8	2	no	—
217	C₂×C₄	—	13	1	no	—
218	C₂×C₄	—	13	2	no	—
219	C₂×C₄	—	13	2	no	—
220	C₂×C₄	—	13	1	no	—
221	C₂×C₄	—	13	1	no	—
222	C₂×C₄	—	26	2	no	—
223	C₂×C₄	—	104	2	no	—
224	C₂×C₄	—	104	2	no	—
225	C₂×C₄	—	104	2	no	—
226	C₂×C₄	D₄×D₁₃	1	1	yes	characteristic
227	C₂×C₄	D₄×D₁₃	1	1	yes	characteristic
228	C₂×C₄	C₂³×D₁₃	1	1	yes	characteristic
229	C₈	—	2	1	no	—
230	C₈	—	13	2	no	—
231	C₈	—	26	1	no	—
232	C₈	C₂×C₄×D₁₃	1	2	yes	—
233	C₂²	—	2	4	no	—
234	C₂²	—	4	2	no	—
235	C₂²	—	4	2	no	—
236	C₂²	—	13	2	no	—
237	C₂²	—	26	1	no	—
238	C₂²	—	26	1	no	—
239	C₂²	—	26	4	no	—
240	C₂²	—	52	2	no	—
241	C₂²	—	52	2	no	—
242	C₂²	—	104	8	no	—
243	C₂²	C₂×D₄×D₁₃	1	1	yes	characteristic
244	C₄	—	2	1	no	—
245	C₄	—	2	1	no	—
246	C₄	—	4	2	no	—
247	C₄	—	13	2	no	—
248	C₄	—	26	1	no	—
249	C₄	—	26	1	no	—
250	C₄	—	52	2	no	—
251	C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
252	C₄	#41×D₁₃	1	1	yes	characteristic
253	C₂	—	4	4	no	—
254	C₂	—	26	2	no	—
255	C₂	—	52	4	no	—
256	C₂	C₄×D₄×D₁₃	1	1	yes	characteristic
257	C₂	#11719462	1	1	yes	characteristic
258	C₂	#11719474	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₃
commutator	C₄×C₁₃	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₃
Fitting	C₄×C₁₃×D₈	C₂
socle	C₂²×C₁₃	C₂²×D₄
prime radical	C₂×D₈×D₁₃	C₂
cosocle	C₂×C₄×C₁₃	C₂⁴
hypercenter	C₄×D₈	D₁₃
hypocenter	C₁₃	#2948
LCS	G ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×D₈

Complements

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

N	H ≅ G/N	type
2. #11719250	C₂	semidirect
3. C₄×C₁₃×D₈	C₂	semidirect
4. D₈×Dic₁₃	C₂	semidirect
5. #11718675	C₂	semidirect
6. #11718541	C₂	semidirect
7. #11718533	C₂	semidirect
8. #11718446	C₂	semidirect
9. #11718431	C₂	semidirect
10. C₄×D₁₀₄	C₂	semidirect
11. #11718350	C₂	semidirect
14. C₄×D₄×C₁₃	C₂²	semidirect
15. D₄×Dic₁₃	C₂²	semidirect
19. D₈×D₁₃	C₄	semidirect
23. #88147	C₂²	semidirect
27. C₄×D₅₂	C₂²	semidirect
29. C₁₃×#107	C₂²	semidirect
30. C₁₃×#102	C₂²	semidirect
31. C₄×C₈×C₁₃	C₂²	semidirect
32. #88072	C₂²	semidirect
33. #88061	C₂²	semidirect
34. #88058	C₂²	semidirect
35. #88055	C₂²	semidirect
36. C₈×Dic₁₃	C₂²	semidirect
37. #88049	C₂²	semidirect
38. #88048	C₂²	semidirect
39. #88043	C₂²	semidirect
46. D₄×D₁₃	C₂×C₄	semidirect
51. C₁₃×D₈	C₂×C₄	semidirect
58. #6104	C₂×C₄	semidirect
71. D₁₀₄	C₂×C₄	semidirect
73. C₈×D₁₃	C₂×C₄	semidirect
111. C₄×D₈	D₁₃	semidirect
147. C₄×D₄	C₂×D₁₃	semidirect
153. #107	C₂×D₁₃	semidirect
158. #102	C₂×D₁₃	semidirect
162. C₄×C₈	C₂×D₁₃	semidirect
182. D₈	C₄×D₁₃	semidirect
203. C₁₃	#2948	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₈×D₁₃	2
#11719250	2
C₄×C₁₃×D₈	2
D₈×Dic₁₃	2
#11718675	2
#11718541	2
#11718533	2
#11718446	2
#11718431	2
C₄×D₁₀₄	2
#11718350	2
#2948	13