Subgroups — #13123874

Order 1664.

452 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×D₄×D₁₃	C₂	1	1	yes	maximal, characteristic
2	C₁₃×#436	C₂	1	1	yes	maximal, characteristic, complemented
3	#11718789	C₂	1	2	yes	maximal, complemented
4	#11718787	C₂	1	2	yes	maximal, complemented
5	#11718673	C₂	1	1	yes	maximal, characteristic, complemented
6	#11718499	C₂	1	1	yes	maximal, characteristic, complemented
7	#11718494	C₂	1	1	yes	maximal, characteristic, complemented
8	#11718390	C₂	1	2	yes	maximal, complemented
9	#11718387	C₂	1	2	yes	maximal, complemented
10	#11718349	C₂	1	1	yes	maximal, characteristic, complemented
11	#11718346	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×D₄×D₁₃	C₄	1	1	yes	characteristic
13	C₂²×C₄×D₁₃	C₂²	1	2	yes	—
14	C₂×C₁₃×M₁₆	C₂²	1	2	yes	complemented
15	C₄×D₄×C₁₃	C₂²	1	1	yes	characteristic
16	D₄×Dic₁₃	C₄	1	1	yes	characteristic
17	C₄×C₂²⋊D₁₃	C₂²	1	2	yes	—
18	C₂×#6099	C₂²	1	2	yes	complemented
19	C₂×#6094	C₂²	1	2	yes	complemented
20	C₂×#6094	C₂²	1	2	yes	complemented
21	#88147	C₄	1	1	yes	characteristic
22	C₄⋊C₄×D₁₃	C₄	1	1	yes	characteristic
23	#88135	C₄	1	2	yes	—
24	C₂²⋊C₄×D₁₃	C₄	1	2	yes	—
25	C₄×D₅₂	C₂²	1	1	yes	characteristic
26	C₄²×D₁₃	C₂²	1	1	yes	characteristic
27	C₁₃×C₄⋊C₈	C₂²	1	1	yes	characteristic, complemented
28	C₁₃×C₂²⋊C₈	C₂²	1	2	yes	complemented
29	C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic, complemented
30	C₂²⋊#1055	C₂²	1	2	yes	complemented
31	#88060	C₂²	1	2	yes	—
32	#88060	C₂²	1	1	yes	characteristic
33	#88060	C₂²	1	1	yes	characteristic
34	#88060	C₂²	1	2	yes	complemented
35	#88060	C₂²	1	2	yes	complemented
36	#88054	C₂²	1	1	yes	characteristic, complemented
37	#88054	C₂²	1	1	yes	characteristic, complemented
38	C₈×Dic₁₃	C₂²	1	2	yes	complemented
39	C₄⋊#1055	C₂²	1	1	yes	characteristic, complemented
40	C₄×#1055	C₂²	1	1	yes	characteristic, complemented
41	C₂³×D₁₃	C₂×C₄	1	2	yes	—
42	C₂×D₄×C₁₃	C₂×C₄	1	1	yes	characteristic
43	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
44	C₂×C₂²⋊D₁₃	C₂×C₄	1	2	yes	—
45	C₂²×Dic₁₃	C₂×C₄	1	2	yes	—
46	D₄×D₁₃	—	2	4	no	—
47	C₂×D₅₂	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄×D₁₃	—	4	2	no	—
49	C₂×C₄×D₁₃	—	4	2	no	—
50	C₂×C₄×D₁₃	C₂×C₄	1	1	yes	characteristic
51	C₂×C₄×D₁₃	C₂×C₄	1	2	yes	—
52	C₂×C₄×D₁₃	C₂×C₄	1	2	yes	—
53	C₂×C₄×D₁₃	C₂×C₄	1	1	yes	characteristic
54	C₂×C₄×D₁₃	C₂³	1	1	yes	characteristic
55	C₂×C₄×D₁₃	C₂³	1	2	yes	—
56	C₁₃×M₁₆	—	4	2	no	—
57	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
58	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
59	C₂×C₈×C₁₃	C₂³	1	2	yes	—
60	C₁₃×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
61	C₁₃×C₂²⋊C₄	C₂×C₄	1	2	yes	—
62	C₄²×C₁₃	C₂³	1	1	yes	characteristic
63	C₂²⋊Dic₁₃	C₂×C₄	1	2	yes	—
64	#6103	C₂×C₄	1	2	yes	—
65	#6103	C₂×C₄	1	2	yes	—
66	C₄⋊Dic₁₃	C₂×C₄	1	1	yes	characteristic
67	#6101	C₂×C₄	1	2	yes	—
68	C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
69	C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
70	C₄×Dic₁₃	C₂³	1	1	yes	characteristic
71	#6099	—	4	2	no	—
72	C₂×#1055	C₂³	1	1	yes	characteristic
73	C₂×#1055	C₂³	1	2	yes	—
74	C₂×#1055	C₂³	1	1	yes	characteristic
75	#6094	—	4	2	no	—
76	#6094	—	4	2	no	—
77	#2975	—	13	1	no	maximal, Sylow
78	C₂³×C₁₃	C₂²×C₄	1	2	yes	—
79	C₂²×D₁₃	—	2	4	no	—
80	C₂²×D₁₃	—	4	2	no	—
81	C₂²×D₁₃	—	4	2	no	—
82	C₂²×D₁₃	C₂²×C₄	1	2	yes	—
83	C₂²×D₁₃	C₂²×C₄	1	1	yes	characteristic
84	D₄×C₁₃	—	2	2	no	—
85	C₂×C₄×C₁₃	—	2	2	no	—
86	C₂×C₄×C₁₃	—	2	2	no	—
87	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
88	C₂×C₄×C₁₃	C₂²×C₄	1	2	yes	—
89	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
90	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
91	C₂²⋊D₁₃	—	2	4	no	—
92	C₂×Dic₁₃	—	4	2	no	—
93	C₂×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
94	C₂×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
95	C₂×Dic₁₃	C₂²×C₄	1	2	yes	—
96	C₂×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
97	D₅₂	—	2	2	no	—
98	C₄×D₁₃	—	2	2	no	—
99	C₄×D₁₃	—	2	2	no	—
100	C₄×D₁₃	—	2	1	no	—
101	C₄×D₁₃	—	2	1	no	—
102	C₄×D₁₃	—	4	2	no	—
103	C₄×D₁₃	—	4	2	no	—
104	C₄×D₁₃	—	4	1	no	—
105	C₄×D₁₃	M₁₆	1	4	yes	complemented
106	C₈×C₁₃	—	2	1	no	—
107	C₈×C₁₃	—	2	2	no	—
108	C₈×C₁₃	—	2	1	no	—
109	#1055	—	2	1	no	—
110	#1055	—	2	2	no	—
111	#1055	—	2	1	no	—
112	C₂×C₄×D₄	—	13	1	no	—
113	#436	—	13	2	no	—
114	#436	—	13	1	no	—
115	#436	—	13	1	no	—
116	#436	—	13	2	no	—
117	#436	—	13	1	no	—
118	#436	D₁₃	1	1	yes	characteristic, complemented
119	#431	—	13	1	no	—
120	#431	—	13	1	no	—
121	#407	—	13	2	no	—
122	#407	—	13	2	no	—
123	C₂²×C₁₃	—	2	2	no	—
124	C₂²×C₁₃	—	2	2	no	—
125	C₂²×C₁₃	—	2	2	no	—
126	C₂²×C₁₃	C₂³×C₄	1	1	yes	characteristic
127	C₂×D₁₃	—	2	1	no	—
128	C₂×D₁₃	—	2	1	no	—
129	C₂×D₁₃	—	2	2	no	—
130	C₂×D₁₃	—	2	2	no	—
131	C₂×D₁₃	—	2	2	no	—
132	C₂×D₁₃	—	4	8	no	—
133	C₂×D₁₃	C₂×M₁₆	1	2	yes	—
134	C₄×C₁₃	—	2	2	no	—
135	C₄×C₁₃	—	2	1	no	—
136	C₄×C₁₃	C₂×M₁₆	1	2	yes	—
137	C₄×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
138	C₄×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
139	Dic₁₃	—	2	1	no	—
140	Dic₁₃	—	2	2	no	—
141	Dic₁₃	—	2	1	no	—
142	Dic₁₃	C₂×M₁₆	1	2	yes	—
143	C₂²×D₄	—	13	1	no	—
144	C₂³×C₄	—	13	2	no	—
145	C₂×M₁₆	—	13	2	no	—
146	C₂×M₁₆	—	13	2	no	—
147	C₂×M₁₆	—	13	2	no	—
148	C₂×M₁₆	C₂×D₁₃	1	2	yes	complemented
149	C₄×D₄	—	13	1	no	—
150	C₄×D₄	—	13	2	no	—
151	C₄×D₄	—	13	1	no	—
152	C₄×D₄	—	13	2	no	—
153	C₄×D₄	—	13	1	no	—
154	C₄×D₄	C₂×D₁₃	1	1	yes	characteristic
155	C₂×C₄⋊C₄	—	13	1	no	—
156	C₂×C₂²⋊C₄	—	13	2	no	—
157	C₂×C₄²	—	13	1	no	—
158	C₄⋊C₈	—	13	1	no	—
159	C₄⋊C₈	—	13	1	no	—
160	C₄⋊C₈	—	13	1	no	—
161	C₄⋊C₈	C₂×D₁₃	1	1	yes	characteristic, complemented
162	C₂²⋊C₈	—	13	2	no	—
163	C₂²⋊C₈	—	13	2	no	—
164	C₂²⋊C₈	—	13	1	no	—
165	C₂²⋊C₈	—	13	1	no	—
166	C₂²⋊C₈	—	13	2	no	—
167	C₂²⋊C₈	—	13	2	no	—
168	C₂²⋊C₈	C₂×D₁₃	1	2	yes	complemented
169	C₄×C₈	—	13	2	no	—
170	C₄×C₈	—	13	1	no	—
171	C₄×C₈	C₂×D₁₃	1	1	yes	characteristic, complemented
172	C₂×C₁₃	—	4	2	no	—
173	C₂×C₁₃	#518	1	1	yes	characteristic
174	C₂×C₁₃	C₂²×M₁₆	1	1	yes	characteristic
175	C₂×C₁₃	#568	1	1	yes	characteristic
176	D₁₃	—	2	2	no	—
177	D₁₃	—	4	2	no	—
178	C₂⁴	—	13	2	no	—
179	C₂×D₄	—	13	2	no	—
180	C₂×D₄	—	13	1	no	—
181	C₂×D₄	—	26	4	no	—
182	C₂×D₄	C₄×D₁₃	1	1	yes	characteristic
183	C₂²×C₄	—	13	1	no	—
184	C₂²×C₄	—	13	2	no	—
185	C₂²×C₄	—	13	1	no	—
186	C₂²×C₄	—	13	2	no	—
187	C₂²×C₄	—	13	1	no	—
188	C₂²×C₄	—	13	2	no	—
189	C₂²×C₄	—	13	2	no	—
190	C₂²×C₄	—	52	2	no	—
191	C₂²×C₄	—	52	2	no	—
192	C₂²×C₄	C₂²×D₁₃	1	2	yes	—
193	M₁₆	—	4	2	no	—
194	M₁₆	—	52	2	no	—
195	M₁₆	—	52	2	no	—
196	M₁₆	—	52	2	no	—
197	C₂×C₈	—	13	1	no	—
198	C₂×C₈	—	13	2	no	—
199	C₂×C₈	—	13	1	no	—
200	C₂×C₈	C₂²×D₁₃	1	2	yes	—
201	C₂×C₈	C₂²×D₁₃	1	1	yes	characteristic
202	C₂×C₈	C₂²×D₁₃	1	1	yes	characteristic
203	C₄⋊C₄	—	13	2	no	—
204	C₄⋊C₄	—	13	1	no	—
205	C₄⋊C₄	C₄×D₁₃	1	1	yes	characteristic
206	C₂²⋊C₄	—	13	2	no	—
207	C₂²⋊C₄	—	13	2	no	—
208	C₂²⋊C₄	—	13	2	no	—
209	C₂²⋊C₄	C₄×D₁₃	1	2	yes	—
210	C₄²	—	13	1	no	—
211	C₄²	—	13	1	no	—
212	C₄²	—	13	1	no	—
213	C₄²	C₂²×D₁₃	1	1	yes	characteristic
214	C₁₃	#2975	1	1	yes	characteristic, complemented, Sylow
215	C₂³	—	13	1	no	—
216	C₂³	—	13	2	no	—
217	C₂³	—	26	4	no	—
218	C₂³	—	52	2	no	—
219	C₂³	—	52	2	no	—
220	C₂³	C₂×C₄×D₁₃	1	2	yes	—
221	D₄	—	2	2	no	—
222	D₄	—	26	4	no	—
223	D₄	—	26	2	no	—
224	C₂×C₄	—	2	2	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	13	1	no	—
227	C₂×C₄	—	13	4	no	—
228	C₂×C₄	—	13	1	no	—
229	C₂×C₄	—	13	1	no	—
230	C₂×C₄	—	13	2	no	—
231	C₂×C₄	—	26	2	no	—
232	C₂×C₄	—	26	1	no	—
233	C₂×C₄	—	26	2	no	—
234	C₂×C₄	—	26	1	no	—
235	C₂×C₄	—	52	1	no	—
236	C₂×C₄	—	52	2	no	—
237	C₂×C₄	—	52	2	no	—
238	C₂×C₄	—	52	2	no	—
239	C₂×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
240	C₂×C₄	C₂×C₄×D₁₃	1	2	yes	—
241	C₂×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
242	C₂×C₄	C₂³×D₁₃	1	1	yes	characteristic
243	C₈	—	2	1	no	—
244	C₈	—	2	2	no	—
245	C₈	—	2	1	no	—
246	C₈	—	26	1	no	—
247	C₈	—	26	2	no	—
248	C₈	—	26	1	no	—
249	C₂²	—	2	2	no	—
250	C₂²	—	2	2	no	—
251	C₂²	—	2	2	no	—
252	C₂²	—	13	2	no	—
253	C₂²	—	26	1	no	—
254	C₂²	—	26	1	no	—
255	C₂²	—	26	2	no	—
256	C₂²	—	26	2	no	—
257	C₂²	—	26	2	no	—
258	C₂²	—	52	8	no	—
259	C₂²	C₂²×C₄×D₁₃	1	1	yes	characteristic
260	C₄	—	2	1	no	—
261	C₄	—	2	2	no	—
262	C₄	—	13	2	no	—
263	C₄	—	26	1	no	—
264	C₄	—	26	2	no	—
265	C₄	—	26	1	no	—
266	C₄	M₁₆×D₁₃	1	2	yes	—
267	C₄	#88251	1	1	yes	characteristic
268	C₄	#88256	1	1	yes	characteristic
269	C₂	—	4	2	no	—
270	C₂	—	26	2	no	—
271	C₂	—	52	2	no	—
272	C₂	#11719250	1	1	yes	characteristic
273	C₂	C₂×M₁₆×D₁₃	1	1	yes	characteristic
274	C₂	#11719455	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₃
commutator, socle	C₂²×C₁₃	C₂³×C₄
Fitting	C₁₃×#436	C₂
prime radical	C₂×D₄×D₁₃	C₄
cosocle	C₂×C₄×C₁₃	C₂⁴
hypercenter	#436	D₁₃
hypocenter	C₁₃	#2975
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ #436

Complements

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

N	H ≅ G/N	type
2. C₁₃×#436	C₂	semidirect
3. #11718789	C₂	semidirect
4. #11718787	C₂	semidirect
5. #11718673	C₂	semidirect
6. #11718499	C₂	semidirect
7. #11718494	C₂	semidirect
8. #11718390	C₂	semidirect
9. #11718387	C₂	semidirect
10. #11718349	C₂	semidirect
11. #11718346	C₂	semidirect
14. C₂×C₁₃×M₁₆	C₂²	semidirect
18. C₂×#6099	C₂²	semidirect
19. C₂×#6094	C₂²	semidirect
20. C₂×#6094	C₂²	semidirect
27. C₁₃×C₄⋊C₈	C₂²	semidirect
28. C₁₃×C₂²⋊C₈	C₂²	semidirect
29. C₄×C₈×C₁₃	C₂²	semidirect
30. C₂²⋊#1055	C₂²	semidirect
34. #88060	C₂²	semidirect
35. #88060	C₂²	semidirect
36. #88054	C₂²	semidirect
37. #88054	C₂²	semidirect
38. C₈×Dic₁₃	C₂²	semidirect
39. C₄⋊#1055	C₂²	semidirect
40. C₄×#1055	C₂²	semidirect
105. C₄×D₁₃	M₁₆	semidirect
118. #436	D₁₃	semidirect
148. C₂×M₁₆	C₂×D₁₃	semidirect
161. C₄⋊C₈	C₂×D₁₃	semidirect
168. C₂²⋊C₈	C₂×D₁₃	semidirect
171. C₄×C₈	C₂×D₁₃	semidirect
214. C₁₃	#2975	semidirect

Maximal subgroups

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

subgroup	index
C₄×D₄×D₁₃	2
C₁₃×#436	2
#11718789	2
#11718787	2
#11718673	2
#11718499	2
#11718494	2
#11718390	2
#11718387	2
#11718349	2
#11718346	2
#2975	13