Subgroups — #13119248

Order 1664.

412 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#389	C₂	1	1	yes	maximal, characteristic, complemented
2	#11718654	C₂	1	2	yes	maximal
3	#11718648	C₂	1	1	yes	maximal, characteristic
4	#11718648	C₂	1	1	yes	maximal, characteristic
5	#11718647	C₂	1	2	yes	maximal, complemented
6	#11718646	C₂	1	1	yes	maximal, characteristic, complemented
7	#11718638	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×#6103	C₂	1	1	yes	maximal, characteristic
9	C₄×C₄⋊Dic₁₃	C₂	1	1	yes	maximal, characteristic, complemented
10	#11718331	C₂	1	2	yes	maximal
11	#11718312	C₂	1	2	yes	maximal, complemented
12	C₂²×C₄×D₁₃	C₂²	1	1	yes	characteristic
13	C₂×C₁₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
14	C₂×C₁₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
15	C₂×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
16	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
17	C₂×#6103	C₂²	1	1	yes	characteristic
18	C₂×#6103	C₂²	1	2	yes	—
19	C₂×#6103	C₂²	1	1	yes	characteristic
20	C₂×#6103	C₂²	1	2	yes	—
21	C₂×C₄⋊Dic₁₃	C₂²	1	1	yes	characteristic
22	C₂×C₄⋊Dic₁₃	C₂²	1	1	yes	characteristic
23	C₂×C₄⋊Dic₁₃	C₂²	1	2	yes	—
24	C₂×C₄×Dic₁₃	C₂²	1	2	yes	—
25	C₂×C₄×Dic₁₃	C₂²	1	1	yes	characteristic
26	C₁₃×#95	C₂²	1	2	yes	—
27	#88071	C₂²	1	1	yes	characteristic
28	#88071	C₂²	1	2	yes	—
29	#88071	C₂²	1	2	yes	—
30	#88071	C₂²	1	2	yes	—
31	#88071	C₂²	1	2	yes	—
32	#88071	C₂²	1	2	yes	—
33	#88071	C₂²	1	2	yes	—
34	#88071	C₂²	1	1	yes	characteristic
35	C₂³×D₁₃	C₂³	1	1	yes	characteristic
36	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
37	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
38	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
39	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
40	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
41	C₂²×Dic₁₃	C₂³	1	1	yes	characteristic
42	C₂²×Dic₁₃	C₂³	1	1	yes	characteristic
43	C₂²×Dic₁₃	C₂³	1	2	yes	—
44	C₂²×Dic₁₃	C₂³	1	2	yes	—
45	C₂²×Dic₁₃	C₂³	1	1	yes	characteristic
46	C₂×C₄×D₁₃	—	2	2	no	—
47	C₂×C₄×D₁₃	—	2	2	no	—
48	C₂×C₄×D₁₃	—	2	2	no	—
49	C₁₃×C₄⋊C₄	—	2	2	no	—
50	C₁₃×C₄⋊C₄	—	2	2	no	—
51	C₁₃×C₄⋊C₄	—	4	2	no	—
52	C₄²×C₁₃	—	2	2	no	—
53	#6103	—	2	4	no	—
54	#6103	—	2	2	no	—
55	#6103	—	2	2	no	—
56	#6103	—	4	2	no	—
57	C₄⋊Dic₁₃	—	2	2	no	—
58	C₄⋊Dic₁₃	—	2	2	no	—
59	C₄⋊Dic₁₃	—	4	2	no	—
60	C₄×Dic₁₃	—	2	4	no	—
61	C₄×Dic₁₃	—	2	2	no	—
62	#2495	—	13	1	no	maximal, Sylow
63	C₂³×C₁₃	C₂⁴	1	1	yes	characteristic
64	C₂²×D₁₃	—	2	1	no	—
65	C₂²×D₁₃	—	2	1	no	—
66	C₂²×D₁₃	—	2	1	no	—
67	C₂²×D₁₃	—	2	1	no	—
68	C₂²×D₁₃	—	2	1	no	—
69	C₂²×D₁₃	—	2	1	no	—
70	C₂²×D₁₃	#41	1	2	yes	—
71	C₂×C₄×C₁₃	—	2	1	no	—
72	C₂×C₄×C₁₃	—	2	1	no	—
73	C₂×C₄×C₁₃	—	2	1	no	—
74	C₂×C₄×C₁₃	—	2	2	no	—
75	C₂×C₄×C₁₃	—	2	1	no	—
76	C₂×C₄×C₁₃	—	2	2	no	—
77	C₂×C₄×C₁₃	—	2	2	no	—
78	C₂×C₄×C₁₃	—	2	2	no	—
79	C₂×C₄×C₁₃	—	2	2	no	—
80	C₂×C₄×C₁₃	—	2	1	no	—
81	C₂×C₄×C₁₃	—	2	2	no	—
82	C₂×C₄×C₁₃	#41	1	2	yes	—
83	C₂×C₄×C₁₃	#41	1	2	yes	—
84	C₂×C₄×C₁₃	#41	1	2	yes	—
85	C₂×C₄×C₁₃	#41	1	2	yes	—
86	C₂×Dic₁₃	—	2	2	no	—
87	C₂×Dic₁₃	—	2	1	no	—
88	C₂×Dic₁₃	—	2	1	no	—
89	C₂×Dic₁₃	—	2	2	no	—
90	C₂×Dic₁₃	—	2	2	no	—
91	C₂×Dic₁₃	—	2	2	no	—
92	C₂×Dic₁₃	—	2	2	no	—
93	C₂×Dic₁₃	—	2	1	no	—
94	C₂×Dic₁₃	—	2	1	no	—
95	C₂×Dic₁₃	—	2	1	no	—
96	C₂×Dic₁₃	—	2	1	no	—
97	C₂×Dic₁₃	—	2	1	no	—
98	C₂×Dic₁₃	—	2	1	no	—
99	C₂×Dic₁₃	—	2	2	no	—
100	C₂×Dic₁₃	—	2	1	no	—
101	C₄×D₁₃	—	4	4	no	—
102	#400	—	13	2	no	—
103	#397	—	13	2	no	—
104	#389	—	13	1	no	—
105	#389	D₁₃	1	1	yes	characteristic, complemented
106	#388	—	13	2	no	—
107	#383	—	13	2	no	—
108	#382	—	13	1	no	—
109	#380	—	13	1	no	—
110	#380	—	13	1	no	—
111	C₄×C₄⋊C₄	—	13	1	no	—
112	C₄×C₂²⋊C₄	—	13	1	no	—
113	C₂²×C₁₃	C₂×#41	1	1	yes	characteristic
114	C₂²×C₁₃	C₂×#41	1	1	yes	characteristic
115	C₂²×C₁₃	C₂×#41	1	1	yes	characteristic
116	C₂²×C₁₃	C₂×#41	1	1	yes	characteristic
117	C₂²×C₁₃	C₂×#41	1	1	yes	characteristic
118	C₂²×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
119	C₂²×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
120	C₂×D₁₃	—	2	2	no	—
121	C₂×D₁₃	—	2	2	no	—
122	C₂×D₁₃	—	2	2	no	—
123	C₂×D₁₃	—	4	1	no	—
124	C₂×D₁₃	—	4	1	no	—
125	C₂×D₁₃	—	4	1	no	—
126	C₂×D₁₃	—	4	1	no	—
127	C₄×C₁₃	—	2	2	no	—
128	C₄×C₁₃	—	4	2	no	—
129	C₄×C₁₃	—	4	1	no	—
130	C₄×C₁₃	—	4	2	no	—
131	C₄×C₁₃	#126	1	4	yes	—
132	Dic₁₃	—	4	1	no	—
133	Dic₁₃	—	4	2	no	—
134	Dic₁₃	—	4	1	no	—
135	Dic₁₃	—	4	2	no	—
136	Dic₁₃	—	4	1	no	—
137	C₂³×C₄	—	13	1	no	—
138	C₂×C₄⋊C₄	—	13	1	no	—
139	C₂×C₄⋊C₄	—	13	1	no	—
140	C₂×C₄⋊C₄	—	13	2	no	—
141	C₂×C₄⋊C₄	C₂×D₁₃	1	1	yes	characteristic
142	C₂×C₄⋊C₄	C₂×D₁₃	1	1	yes	characteristic
143	C₂×C₄⋊C₄	C₂×D₁₃	1	2	yes	—
144	C₂×C₂²⋊C₄	—	13	1	no	—
145	C₂×C₂²⋊C₄	—	13	1	no	—
146	C₂×C₂²⋊C₄	—	13	2	no	—
147	C₂×C₂²⋊C₄	—	13	2	no	—
148	C₂×C₄²	—	13	1	no	—
149	C₂×C₄²	—	13	2	no	—
150	C₂×C₄²	C₂×D₁₃	1	1	yes	characteristic
151	#95	—	13	1	no	—
152	#95	—	13	1	no	—
153	#95	—	13	2	no	—
154	#95	—	13	2	no	—
155	#95	—	13	2	no	—
156	#95	—	13	2	no	—
157	#95	—	13	2	no	—
158	#95	—	13	2	no	—
159	#95	C₂×D₁₃	1	2	yes	—
160	C₂×C₁₃	C₂×#126	1	1	yes	characteristic
161	C₂×C₁₃	#529	1	1	yes	characteristic
162	C₂×C₁₃	#542	1	1	yes	characteristic
163	C₂×C₁₃	#552	1	1	yes	characteristic
164	C₂×C₁₃	#552	1	1	yes	characteristic
165	C₂×C₁₃	#553	1	1	yes	characteristic
166	C₂×C₁₃	#555	1	1	yes	characteristic
167	D₁₃	—	4	2	no	—
168	C₂⁴	—	13	1	no	—
169	C₂²×C₄	—	13	1	no	—
170	C₂²×C₄	—	13	1	no	—
171	C₂²×C₄	—	13	2	no	—
172	C₂²×C₄	—	13	2	no	—
173	C₂²×C₄	—	13	1	no	—
174	C₂²×C₄	—	26	2	no	—
175	C₂²×C₄	—	26	2	no	—
176	C₂²×C₄	—	26	2	no	—
177	C₂²×C₄	C₂²×D₁₃	1	2	yes	—
178	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
179	C₂²×C₄	C₂²×D₁₃	1	2	yes	—
180	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
181	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
182	C₄⋊C₄	—	2	2	no	—
183	C₄⋊C₄	—	2	2	no	—
184	C₄⋊C₄	—	4	2	no	—
185	C₄⋊C₄	—	26	2	no	—
186	C₄⋊C₄	—	26	2	no	—
187	C₄⋊C₄	—	52	2	no	—
188	C₂²⋊C₄	—	26	2	no	—
189	C₂²⋊C₄	—	26	4	no	—
190	C₂²⋊C₄	—	26	2	no	—
191	C₂²⋊C₄	—	52	2	no	—
192	C₄²	—	2	2	no	—
193	C₄²	—	26	4	no	—
194	C₄²	—	26	2	no	—
195	C₁₃	#2495	1	1	yes	characteristic, complemented, Sylow
196	C₂³	—	13	2	no	—
197	C₂³	—	26	1	no	—
198	C₂³	—	26	1	no	—
199	C₂³	—	26	1	no	—
200	C₂³	—	26	1	no	—
201	C₂³	—	26	1	no	—
202	C₂³	—	26	1	no	—
203	C₂³	C₂³×D₁₃	1	1	yes	characteristic
204	C₂×C₄	—	2	1	no	—
205	C₂×C₄	—	2	1	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	2	1	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	2	1	no	—
211	C₂×C₄	—	2	2	no	—
212	C₂×C₄	—	2	1	no	—
213	C₂×C₄	—	2	2	no	—
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	—	26	2	no	—
216	C₂×C₄	—	26	1	no	—
217	C₂×C₄	—	26	1	no	—
218	C₂×C₄	—	26	2	no	—
219	C₂×C₄	—	26	1	no	—
220	C₂×C₄	—	26	1	no	—
221	C₂×C₄	—	26	2	no	—
222	C₂×C₄	—	26	2	no	—
223	C₂×C₄	—	26	1	no	—
224	C₂×C₄	—	26	1	no	—
225	C₂×C₄	—	26	2	no	—
226	C₂×C₄	—	26	1	no	—
227	C₂×C₄	—	26	2	no	—
228	C₂×C₄	—	26	1	no	—
229	C₂×C₄	—	26	1	no	—
230	C₂×C₄	—	52	4	no	—
231	C₂×C₄	#6127	1	2	yes	—
232	C₂×C₄	#6129	1	2	yes	—
233	C₂×C₄	#6131	1	2	yes	—
234	C₂×C₄	#6131	1	2	yes	—
235	C₂²	—	26	2	no	—
236	C₂²	—	26	2	no	—
237	C₂²	—	26	2	no	—
238	C₂²	—	52	1	no	—
239	C₂²	—	52	1	no	—
240	C₂²	—	52	1	no	—
241	C₂²	—	52	1	no	—
242	C₂²	C₂×#6127	1	1	yes	characteristic
243	C₂²	C₂×#6129	1	1	yes	characteristic
244	C₂²	#88251	1	1	yes	characteristic
245	C₂²	C₂×#6131	1	1	yes	characteristic
246	C₂²	C₂×#6131	1	1	yes	characteristic
247	C₂²	#41×D₁₃	1	1	yes	characteristic
248	C₂²	#88257	1	1	yes	characteristic
249	C₄	—	2	2	no	—
250	C₄	—	4	2	no	—
251	C₄	—	4	1	no	—
252	C₄	—	4	2	no	—
253	C₄	—	52	2	no	—
254	C₄	—	52	1	no	—
255	C₄	—	52	1	no	—
256	C₄	—	52	2	no	—
257	C₄	—	52	1	no	—
258	C₄	#88152	1	4	yes	—
259	C₂	—	52	2	no	—
260	C₂	C₂×#88152	1	1	yes	characteristic
261	C₂	#11719266	1	1	yes	characteristic
262	C₂	#11719285	1	1	yes	characteristic
263	C₂	#11719319	1	1	yes	characteristic
264	C₂	#11719347	1	1	yes	characteristic
265	C₂	#11719399	1	1	yes	characteristic
266	C₂	#11719439	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₃
commutator, socle, cosocle	C₂³×C₁₃	C₂⁴
Fitting	C₁₃×#389	C₂
prime radical	C₂³×D₁₃	C₂³
hypercenter	#389	D₁₃
hypocenter	C₁₃	#2495
LCS	G ▷ C₂³×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ #389

Complements

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

N	H ≅ G/N	type
1. C₁₃×#389	C₂	semidirect
5. #11718647	C₂	semidirect
6. #11718646	C₂	semidirect
7. #11718638	C₂	semidirect
9. C₄×C₄⋊Dic₁₃	C₂	semidirect
11. #11718312	C₂	semidirect
105. #389	D₁₃	semidirect
195. C₁₃	#2495	semidirect

Maximal subgroups

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

subgroup	index
C₁₃×#389	2
#11718654	2
#11718648	2
#11718647	2
#11718646	2
#11718638	2
C₄×#6103	2
C₄×C₄⋊Dic₁₃	2
#11718331	2
#11718312	2
#2495	13