Subgroups — #13123741

Order 1664.

458 conjugacy classes of subgroups.

The subgroup lattice has 278 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11719473	C₂	1	1	yes	maximal, characteristic
2	#11719455	C₂	1	2	yes	maximal, complemented
3	C₁₃×#443	C₂	1	1	yes	maximal, characteristic, complemented
4	#11718858	C₂	1	1	yes	maximal, characteristic, complemented
5	#11718806	C₂	1	2	yes	maximal, complemented
6	#11718555	C₂	1	1	yes	maximal, characteristic, complemented
7	#11718554	C₂	1	1	yes	maximal, characteristic, complemented
8	#11718485	C₂	1	2	yes	maximal, complemented
9	D₁₃×#104	C₂	1	2	yes	maximal, complemented
10	#11718361	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×#6094	C₂	1	1	yes	maximal, characteristic, complemented
12	#41×D₁₃	C₂²	1	2	yes	—
13	C₁₃×#135	C₂²	1	1	yes	characteristic
14	C₁₃×#131	C₂²	1	2	yes	complemented
15	#88204	C₂²	1	2	yes	—
16	#88202	C₂²	1	2	yes	complemented
17	#88172	C₄	1	2	yes	complemented
18	#88169	C₄	1	2	yes	complemented
19	#88168	C₄	1	2	yes	complemented
20	#88165	C₄	1	2	yes	complemented
21	#88161	—	2	2	no	—
22	#88161	C₂²	1	2	yes	complemented
23	M₁₆×D₁₃	—	2	2	no	—
24	M₁₆×D₁₃	C₂²	1	2	yes	—
25	#88158	C₂²	1	1	yes	characteristic
26	#88155	—	2	2	no	—
27	#88155	C₂²	1	2	yes	complemented
28	C₂×#6094	—	2	2	no	—
29	C₂×#6094	C₂²	1	1	yes	characteristic
30	C₂×#6094	C₂²	1	1	yes	characteristic
31	C₄²×D₁₃	C₂²	1	1	yes	characteristic
32	C₁₃×#108	C₂²	1	1	yes	characteristic, complemented
33	C₁₃×#104	C₂²	1	2	yes	complemented
34	C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic, complemented
35	#88077	C₂²	1	2	yes	complemented
36	#88065	C₂²	1	2	yes	complemented
37	#88062	C₂²	1	2	yes	complemented
38	#88059	C₂²	1	1	yes	characteristic, complemented
39	#88055	C₂²	1	1	yes	characteristic, complemented
40	#88055	C₂²	1	1	yes	characteristic, complemented
41	#88045	C₂²	1	2	yes	complemented
42	C₄×#1055	C₂²	1	1	yes	characteristic, complemented
43	C₁₃×#41	C₂³	1	2	yes	—
44	#6131	C₂×C₄	1	2	yes	—
45	Q₈×D₁₃	C₂×C₄	1	2	yes	—
46	#6129	—	2	2	no	—
47	#6129	C₂×C₄	1	2	yes	—
48	D₄×D₁₃	—	2	2	no	—
49	D₄×D₁₃	C₂×C₄	1	2	yes	—
50	#6127	—	2	2	no	—
51	#6127	C₂³	1	2	yes	—
52	C₂×C₄×D₁₃	—	2	2	no	—
53	C₂×C₄×D₁₃	—	2	1	no	—
54	C₂×C₄×D₁₃	C₂³	1	1	yes	characteristic
55	C₁₃×Q₁₆	C₂×C₄	1	1	yes	characteristic, complemented
56	C₁₃×SD₁₆	C₂×C₄	1	2	yes	complemented
57	C₁₃×D₈	C₂×C₄	1	1	yes	characteristic, complemented
58	C₁₃×M₁₆	—	2	2	no	—
59	C₁₃×M₁₆	C₂³	1	2	yes	—
60	C₂×C₈×C₁₃	—	2	2	no	—
61	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
62	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
63	C₄²×C₁₃	C₂³	1	1	yes	characteristic
64	#6107	C₂×C₄	1	2	yes	complemented
65	#6106	C₂×C₄	1	2	yes	complemented
66	#6105	C₂×C₄	1	2	yes	complemented
67	#6104	C₂×C₄	1	2	yes	complemented
68	C₄×Dic₁₃	—	2	1	no	—
69	C₄×Dic₁₃	C₂³	1	1	yes	characteristic
70	#6099	—	2	2	no	—
71	#6099	C₂³	1	2	yes	—
72	C₂×#1055	—	2	2	no	—
73	C₂×#1055	C₂³	1	1	yes	characteristic
74	C₂×#1055	C₂³	1	1	yes	characteristic
75	Dic₅₂	C₂×C₄	1	1	yes	characteristic, complemented
76	D₁₀₄	C₂×C₄	1	1	yes	characteristic, complemented
77	#6095	C₂×C₄	1	2	yes	complemented
78	#6094	—	2	2	no	—
79	#6094	—	2	2	no	—
80	#6094	—	2	2	no	—
81	#6094	C₂×C₄	1	2	yes	—
82	#6094	C₂×C₄	1	2	yes	—
83	#6094	D₄	1	2	yes	—
84	#6094	D₄	1	2	yes	—
85	C₈×D₁₃	—	2	2	no	—
86	C₈×D₁₃	—	2	2	no	—
87	C₈×D₁₃	—	2	2	no	—
88	#2956	—	13	1	no	maximal, Sylow
89	C₂²×D₁₃	—	2	2	no	—
90	C₂²×D₁₃	#41	1	1	yes	characteristic
91	Q₈×C₁₃	C₂²×C₄	1	2	yes	—
92	D₄×C₁₃	—	2	2	no	—
93	D₄×C₁₃	C₂²×C₄	1	2	yes	—
94	C₂×C₄×C₁₃	—	2	1	no	—
95	C₂×C₄×C₁₃	—	2	2	no	—
96	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
97	C₂²⋊D₁₃	—	2	2	no	—
98	C₂²⋊D₁₃	—	2	2	no	—
99	C₂²⋊D₁₃	—	2	2	no	—
100	C₂×Dic₁₃	—	2	2	no	—
101	C₂×Dic₁₃	—	2	1	no	—
102	C₂×Dic₁₃	#41	1	1	yes	characteristic
103	D₅₂	—	2	2	no	—
104	D₅₂	C₂²×C₄	1	2	yes	—
105	C₄×D₁₃	—	2	2	no	—
106	C₄×D₁₃	—	2	2	no	—
107	C₄×D₁₃	—	2	2	no	—
108	C₄×D₁₃	—	4	2	no	—
109	C₄×D₁₃	C₂²×C₄	1	2	yes	—
110	C₄×D₁₃	C₂×D₄	1	2	yes	—
111	Dic₂₆	—	2	2	no	—
112	Dic₂₆	C₂²×C₄	1	2	yes	—
113	C₈×C₁₃	—	2	2	no	—
114	C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
115	C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
116	C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
117	C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
118	#1055	—	2	2	no	—
119	#1055	C₂²×C₄	1	2	yes	—
120	#1055	C₂×D₄	1	2	yes	—
121	#575	—	13	1	no	—
122	#568	—	13	2	no	—
123	#444	—	13	1	no	—
124	#444	—	13	2	no	—
125	#444	—	13	1	no	—
126	#443	—	13	2	no	—
127	#443	—	13	1	no	—
128	#443	D₁₃	1	1	yes	characteristic, complemented
129	#430	—	13	1	no	—
130	C₂×#104	—	13	2	no	—
131	C₄×M₁₆	—	13	1	no	—
132	C₂²×C₁₃	—	2	2	no	—
133	C₂²×C₁₃	C₂×#41	1	1	yes	characteristic
134	C₂×D₁₃	—	2	2	no	—
135	C₂×D₁₃	—	4	1	no	—
136	C₂×D₁₃	—	8	2	no	—
137	C₂×D₁₃	C₄×D₄	1	2	yes	—
138	C₄×C₁₃	—	2	2	no	—
139	C₄×C₁₃	—	2	2	no	—
140	C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
141	C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
142	Dic₁₃	—	2	2	no	—
143	Dic₁₃	—	4	1	no	—
144	Dic₁₃	C₄×D₄	1	2	yes	—
145	C₂×#41	—	13	2	no	—
146	#137	—	13	2	no	—
147	#137	—	13	2	no	—
148	AGL(1,ℤ/8)	—	13	2	no	—
149	AGL(1,ℤ/8)	—	13	2	no	—
150	#135	—	13	2	no	—
151	#135	—	13	1	no	—
152	#135	C₂×D₁₃	1	1	yes	characteristic
153	#131	—	13	2	no	—
154	#131	—	13	2	no	—
155	#131	—	13	2	no	—
156	#131	—	26	2	no	—
157	#131	—	26	2	no	—
158	#131	C₂×D₁₃	1	2	yes	complemented
159	C₂×M₁₆	—	13	1	no	—
160	C₂×M₁₆	—	13	1	no	—
161	C₂×M₁₆	—	13	2	no	—
162	C₂×M₁₆	—	26	2	no	—
163	C₂×M₁₆	—	26	2	no	—
164	C₂×C₄²	—	13	1	no	—
165	#108	—	13	2	no	—
166	#108	—	13	1	no	—
167	#108	C₂×D₁₃	1	1	yes	characteristic, complemented
168	#104	—	13	2	no	—
169	#104	—	13	2	no	—
170	#104	—	13	2	no	—
171	#104	C₂×D₁₃	1	2	yes	complemented
172	#97	—	13	1	no	—
173	#97	—	13	1	no	—
174	C₄×C₈	—	13	1	no	—
175	C₄×C₈	C₂×D₁₃	1	1	yes	characteristic, complemented
176	C₂×C₁₃	—	2	1	no	—
177	C₂×C₁₃	—	4	2	no	—
178	C₂×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
179	D₁₃	—	2	2	no	—
180	D₁₃	—	4	2	no	—
181	#41	—	13	2	no	—
182	#41	—	13	2	no	—
183	#41	—	13	2	no	—
184	#41	—	26	2	no	—
185	#41	—	26	2	no	—
186	#41	C₂²×D₁₃	1	2	yes	—
187	C₂×Q₈	—	13	2	no	—
188	C₂×D₄	—	13	2	no	—
189	C₂×D₄	—	26	2	no	—
190	C₂²×C₄	—	13	1	no	—
191	C₂²×C₄	—	26	2	no	—
192	C₂²×C₄	—	26	1	no	—
193	Q₁₆	—	13	2	no	—
194	Q₁₆	—	13	1	no	—
195	Q₁₆	C₄×D₁₃	1	1	yes	characteristic, complemented
196	SD₁₆	—	13	2	no	—
197	SD₁₆	—	13	2	no	—
198	SD₁₆	—	13	2	no	—
199	SD₁₆	C₄×D₁₃	1	2	yes	complemented
200	D₈	—	13	2	no	—
201	D₈	—	13	1	no	—
202	D₈	C₄×D₁₃	1	1	yes	characteristic, complemented
203	M₁₆	—	2	2	no	—
204	M₁₆	—	13	2	no	—
205	M₁₆	—	13	2	no	—
206	M₁₆	—	13	2	no	—
207	M₁₆	—	13	2	no	—
208	M₁₆	—	13	2	no	—
209	M₁₆	—	26	2	no	—
210	M₁₆	—	26	2	no	—
211	M₁₆	—	26	2	no	—
212	M₁₆	—	26	2	no	—
213	M₁₆	C₂²×D₁₃	1	2	yes	—
214	C₂×C₈	—	2	2	no	—
215	C₂×C₈	—	13	1	no	—
216	C₂×C₈	—	13	1	no	—
217	C₂×C₈	—	26	2	no	—
218	C₂×C₈	—	26	2	no	—
219	C₂×C₈	—	26	2	no	—
220	C₂×C₈	—	26	2	no	—
221	C₂×C₈	C₂²×D₁₃	1	1	yes	characteristic
222	C₂×C₈	C₂²×D₁₃	1	1	yes	characteristic
223	C₄²	—	13	1	no	—
224	C₄²	—	26	1	no	—
225	C₄²	C₂²×D₁₃	1	1	yes	characteristic
226	C₁₃	#2956	1	1	yes	characteristic, complemented, Sylow
227	C₂³	—	13	1	no	—
228	C₂³	—	26	2	no	—
229	Q₈	—	13	2	no	—
230	Q₈	—	26	2	no	—
231	Q₈	C₂×C₄×D₁₃	1	2	yes	—
232	D₄	—	2	2	no	—
233	D₄	—	13	2	no	—
234	D₄	—	26	2	no	—
235	D₄	—	26	2	no	—
236	D₄	—	26	2	no	—
237	D₄	—	26	2	no	—
238	D₄	C₂×C₄×D₁₃	1	2	yes	—
239	C₂×C₄	—	2	1	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	13	1	no	—
242	C₂×C₄	—	13	2	no	—
243	C₂×C₄	—	13	2	no	—
244	C₂×C₄	—	26	2	no	—
245	C₂×C₄	—	26	2	no	—
246	C₂×C₄	—	26	2	no	—
247	C₂×C₄	—	26	2	no	—
248	C₂×C₄	—	26	1	no	—
249	C₂×C₄	—	52	2	no	—
250	C₂×C₄	C₂³×D₁₃	1	1	yes	characteristic
251	C₈	—	2	2	no	—
252	C₈	—	13	2	no	—
253	C₈	—	13	2	no	—
254	C₈	—	26	2	no	—
255	C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
256	C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
257	C₈	D₄×D₁₃	1	1	yes	characteristic
258	C₈	D₄×D₁₃	1	1	yes	characteristic
259	C₂²	—	2	2	no	—
260	C₂²	—	13	2	no	—
261	C₂²	—	26	2	no	—
262	C₂²	—	52	1	no	—
263	C₂²	—	104	2	no	—
264	C₂²	#41×D₁₃	1	1	yes	characteristic
265	C₄	—	2	2	no	—
266	C₄	—	2	2	no	—
267	C₄	—	13	2	no	—
268	C₄	—	26	2	no	—
269	C₄	—	52	1	no	—
270	C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
271	C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
272	C₂	—	2	1	no	—
273	C₂	—	4	2	no	—
274	C₂	—	26	2	no	—
275	C₂	—	52	2	no	—
276	C₂	C₄×D₄×D₁₃	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×D₄×D₁₃
commutator	C₄×C₁₃	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₃
Fitting	C₁₃×#443	C₂
socle	C₂×C₁₃	C₂×C₄×D₄
prime radical	#11719473	C₂
cosocle	C₂×C₄×C₁₃	C₂⁴
hypercenter	#443	D₁₃
hypocenter	C₁₃	#2956
LCS	G ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ #443

Complements

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

N	H ≅ G/N	type
2. #11719455	C₂	semidirect
3. C₁₃×#443	C₂	semidirect
4. #11718858	C₂	semidirect
5. #11718806	C₂	semidirect
6. #11718555	C₂	semidirect
7. #11718554	C₂	semidirect
8. #11718485	C₂	semidirect
9. D₁₃×#104	C₂	semidirect
10. #11718361	C₂	semidirect
11. C₄×#6094	C₂	semidirect
14. C₁₃×#131	C₂²	semidirect
16. #88202	C₂²	semidirect
17. #88172	C₄	semidirect
18. #88169	C₄	semidirect
19. #88168	C₄	semidirect
20. #88165	C₄	semidirect
22. #88161	C₂²	semidirect
27. #88155	C₂²	semidirect
32. C₁₃×#108	C₂²	semidirect
33. C₁₃×#104	C₂²	semidirect
34. C₄×C₈×C₁₃	C₂²	semidirect
35. #88077	C₂²	semidirect
36. #88065	C₂²	semidirect
37. #88062	C₂²	semidirect
38. #88059	C₂²	semidirect
39. #88055	C₂²	semidirect
40. #88055	C₂²	semidirect
41. #88045	C₂²	semidirect
42. C₄×#1055	C₂²	semidirect
55. C₁₃×Q₁₆	C₂×C₄	semidirect
56. C₁₃×SD₁₆	C₂×C₄	semidirect
57. C₁₃×D₈	C₂×C₄	semidirect
64. #6107	C₂×C₄	semidirect
65. #6106	C₂×C₄	semidirect
66. #6105	C₂×C₄	semidirect
67. #6104	C₂×C₄	semidirect
75. Dic₅₂	C₂×C₄	semidirect
76. D₁₀₄	C₂×C₄	semidirect
77. #6095	C₂×C₄	semidirect
128. #443	D₁₃	semidirect
158. #131	C₂×D₁₃	semidirect
167. #108	C₂×D₁₃	semidirect
171. #104	C₂×D₁₃	semidirect
175. C₄×C₈	C₂×D₁₃	semidirect
195. Q₁₆	C₄×D₁₃	semidirect
199. SD₁₆	C₄×D₁₃	semidirect
202. D₈	C₄×D₁₃	semidirect
226. C₁₃	#2956	semidirect

Maximal subgroups

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

subgroup	index
#11719473	2
#11719455	2
C₁₃×#443	2
#11718858	2
#11718806	2
#11718555	2
#11718554	2
#11718485	2
D₁₃×#104	2
#11718361	2
C₄×#6094	2
#2956	13