Subgroups — #13123717

Order 1664.

484 conjugacy classes of subgroups.

The subgroup lattice has 280 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#520	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#88204	C₂	1	1	yes	maximal, characteristic
3	#11719228	C₂	1	1	yes	maximal, characteristic, complemented
4	#11718893	C₂	1	1	yes	maximal, characteristic, complemented
5	#11718691	C₂	1	2	yes	maximal, complemented
6	C₄×#6106	C₂	1	2	yes	maximal, complemented
7	C₄×#6105	C₂	1	2	yes	maximal, complemented
8	#11718675	C₂	1	2	yes	maximal, complemented
9	#11718658	C₂	1	1	yes	maximal, characteristic, complemented
10	#11718627	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×#88048	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₁₃×#41	C₂²	1	1	yes	characteristic
13	C₂×#6127	C₂²	1	1	yes	characteristic
14	C₄×Q₈×C₁₃	C₂²	1	2	yes	—
15	C₄×D₄×C₁₃	—	2	2	no	—
16	C₄×D₄×C₁₃	C₂²	1	2	yes	—
17	C₁₃×#117	—	2	1	no	—
18	C₁₃×#117	C₂²	1	1	yes	characteristic
19	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
20	C₂×C₁₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
21	#88204	C₄	1	8	yes	complemented
22	C₂×#6107	C₂²	1	1	yes	characteristic
23	C₂×#6106	C₂²	1	1	yes	characteristic
24	C₂×#6105	C₂²	1	1	yes	characteristic
25	C₂×#6104	C₂²	1	1	yes	characteristic
26	C₂×C₄⋊Dic₁₃	C₂²	1	1	yes	characteristic
27	C₂×#6099	C₂²	1	1	yes	characteristic
28	C₂²×#1055	C₂²	1	1	yes	characteristic
29	#88146	—	2	1	no	—
30	C₄⋊C₄×D₁₃	—	2	1	no	—
31	#88135	—	2	2	no	—
32	C₄×D₅₂	C₂²	1	2	yes	—
33	C₄×Dic₂₆	C₂²	1	2	yes	—
34	#88075	C₂²	1	2	yes	—
35	#88072	C₂²	1	2	yes	—
36	#88050	C₂²	1	2	yes	—
37	#88049	C₂²	1	2	yes	—
38	#88048	C₂²	1	2	yes	—
39	#88048	C₂²	1	2	yes	—
40	#88043	C₂²	1	2	yes	—
41	C₄×#1055	C₂²	1	2	yes	—
42	C₁₃×#41	—	2	2	no	—
43	C₁₃×#41	C₂×C₄	1	4	yes	complemented
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	—	2	1	no	—
49	C₂²×C₄×C₁₃	—	2	2	no	—
50	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
51	C₂×C₂²⋊D₁₃	—	2	1	no	—
52	C₂²×Dic₁₃	—	2	1	no	—
53	#6127	—	2	2	no	—
54	#6127	C₂×C₄	1	4	yes	complemented
55	C₂×D₅₂	C₂³	1	1	yes	characteristic
56	C₂×C₄×D₁₃	—	2	1	no	—
57	C₂×C₄×D₁₃	—	2	2	no	—
58	C₂×Dic₂₆	C₂³	1	1	yes	characteristic
59	C₁₃×C₄⋊C₄	—	2	1	no	—
60	C₁₃×C₄⋊C₄	—	2	2	no	—
61	C₁₃×C₄⋊C₄	D₄	1	1	yes	characteristic
62	C₁₃×C₄⋊C₄	D₄	1	1	yes	characteristic
63	C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
64	C₁₃×C₂²⋊C₄	—	2	2	no	—
65	C₁₃×C₂²⋊C₄	D₄	1	2	yes	—
66	C₄²×C₁₃	—	2	2	no	—
67	C₄²×C₁₃	C₂³	1	2	yes	—
68	#6107	C₂×C₄	1	4	yes	complemented
69	#6106	C₂×C₄	1	4	yes	complemented
70	#6105	C₂×C₄	1	4	yes	complemented
71	#6104	C₂×C₄	1	4	yes	complemented
72	#6103	—	2	2	no	—
73	C₄⋊Dic₁₃	—	2	1	no	—
74	C₄⋊Dic₁₃	C₂³	1	2	yes	—
75	#6101	—	2	2	no	—
76	C₄×Dic₁₃	—	2	2	no	—
77	#6099	—	2	2	no	—
78	C₂×#1055	C₂×C₄	1	4	yes	complemented
79	C₂×#1055	C₂³	1	2	yes	—
80	C₂×#1055	C₂³	1	1	yes	characteristic
81	C₂×#1055	C₂³	1	1	yes	characteristic
82	#2952	—	13	1	no	maximal, Sylow
83	C₂³×C₁₃	—	2	1	no	—
84	C₂³×C₁₃	C₂×D₄	1	1	yes	characteristic
85	C₂²×D₁₃	—	2	1	no	—
86	Q₈×C₁₃	—	2	1	no	—
87	Q₈×C₁₃	C₂²×C₄	1	2	yes	—
88	D₄×C₁₃	—	2	4	no	—
89	D₄×C₁₃	—	2	1	no	—
90	D₄×C₁₃	C₂²×C₄	1	2	yes	—
91	C₂×C₄×C₁₃	—	2	1	no	—
92	C₂×C₄×C₁₃	—	2	4	no	—
93	C₂×C₄×C₁₃	—	2	2	no	—
94	C₂×C₄×C₁₃	—	8	2	no	—
95	C₂×C₄×C₁₃	—	8	1	no	—
96	C₂×C₄×C₁₃	C₂²×C₄	1	2	yes	—
97	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
98	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
99	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
100	C₂×C₄×C₁₃	#41	1	2	yes	—
101	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
102	C₂²⋊D₁₃	—	2	4	no	—
103	C₂×Dic₁₃	—	2	1	no	—
104	C₂×Dic₁₃	—	2	2	no	—
105	C₂×Dic₁₃	—	8	1	no	—
106	D₅₂	—	2	1	no	—
107	D₅₂	C₂²×C₄	1	2	yes	—
108	C₄×D₁₃	—	2	4	no	—
109	C₄×D₁₃	—	8	2	no	—
110	Dic₂₆	—	2	1	no	—
111	Dic₂₆	C₂²×C₄	1	2	yes	—
112	#1055	—	2	2	no	—
113	#1055	C₂²×C₄	1	2	yes	—
114	#1055	C₂²×C₄	1	2	yes	—
115	C₂×#135	—	13	1	no	—
116	#520	—	13	1	no	—
117	#520	D₁₃	1	1	yes	characteristic, complemented
118	#442	—	13	2	no	—
119	#441	—	13	2	no	—
120	C₄×SD₁₆	—	13	2	no	—
121	C₄×SD₁₆	—	13	2	no	—
122	C₂×#106	—	13	1	no	—
123	#417	—	13	1	no	—
124	#417	—	13	1	no	—
125	#405	—	13	1	no	—
126	C₂²×C₁₃	—	2	1	no	—
127	C₂²×C₁₃	—	2	1	no	—
128	C₂²×C₁₃	—	2	2	no	—
129	C₂²×C₁₃	—	4	1	no	—
130	C₂²×C₁₃	—	4	1	no	—
131	C₂²×C₁₃	C₄×D₄	1	2	yes	—
132	C₂²×C₁₃	C₂²×D₄	1	1	yes	characteristic
133	C₂×D₁₃	—	2	2	no	—
134	C₂×D₁₃	—	4	1	no	—
135	C₂×D₁₃	—	4	1	no	—
136	C₄×C₁₃	—	2	2	no	—
137	C₄×C₁₃	—	2	2	no	—
138	C₄×C₁₃	—	2	2	no	—
139	C₄×C₁₃	—	4	2	no	—
140	C₄×C₁₃	C₄×D₄	1	2	yes	—
141	C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
142	C₄×C₁₃	C₂×#41	1	1	yes	characteristic
143	Dic₁₃	—	2	2	no	—
144	Dic₁₃	—	4	2	no	—
145	C₂×#41	—	13	1	no	—
146	C₂×#41	C₂×D₁₃	1	1	yes	characteristic
147	#135	—	13	8	no	—
148	C₂×Q₁₆	—	13	1	no	—
149	C₂×SD₁₆	—	13	1	no	—
150	C₂×SD₁₆	—	13	1	no	—
151	C₂×D₈	—	13	1	no	—
152	C₂×M₁₆	—	13	1	no	—
153	C₂²×C₈	—	13	1	no	—
154	C₄×Q₈	—	13	2	no	—
155	C₄×Q₈	C₂×D₁₃	1	2	yes	—
156	C₄×D₄	—	2	2	no	—
157	C₄×D₄	—	13	2	no	—
158	C₄×D₄	—	26	2	no	—
159	C₄×D₄	C₂×D₁₃	1	2	yes	—
160	#117	—	2	1	no	—
161	#117	—	26	1	no	—
162	#117	C₂×D₁₃	1	1	yes	characteristic
163	C₂×C₄⋊C₄	—	2	1	no	—
164	C₂×C₄⋊C₄	—	13	1	no	—
165	C₂×C₄⋊C₄	—	26	1	no	—
166	C₂×C₄⋊C₄	C₂×D₁₃	1	1	yes	characteristic
167	#106	—	13	2	no	—
168	#106	—	13	2	no	—
169	#103	—	13	2	no	—
170	#103	—	13	2	no	—
171	#102	—	13	2	no	—
172	#102	—	13	2	no	—
173	#97	—	13	2	no	—
174	C₄×C₈	—	13	2	no	—
175	C₂×C₁₃	—	2	2	no	—
176	C₂×C₁₃	—	4	2	no	—
177	C₂×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
178	C₂×C₁₃	#577	1	1	yes	characteristic
179	C₂×C₁₃	#577	1	1	yes	characteristic
180	D₁₃	—	4	2	no	—
181	#41	—	2	2	no	—
182	#41	—	13	4	no	—
183	#41	—	26	2	no	—
184	#41	C₄×D₁₃	1	4	yes	complemented
185	C₂×Q₈	—	13	1	no	—
186	C₂×Q₈	C₂²×D₁₃	1	1	yes	characteristic
187	C₂×D₄	—	2	1	no	—
188	C₂×D₄	—	13	1	no	—
189	C₂×D₄	—	26	1	no	—
190	C₂×D₄	C₂²×D₁₃	1	1	yes	characteristic
191	C₂²×C₄	—	2	1	no	—
192	C₂²×C₄	—	2	1	no	—
193	C₂²×C₄	—	2	2	no	—
194	C₂²×C₄	—	26	1	no	—
195	C₂²×C₄	—	26	1	no	—
196	C₂²×C₄	—	26	2	no	—
197	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
198	Q₁₆	—	13	4	no	—
199	SD₁₆	—	13	4	no	—
200	SD₁₆	—	13	4	no	—
201	D₈	—	13	4	no	—
202	M₁₆	—	26	2	no	—
203	C₂×C₈	—	13	4	no	—
204	C₂×C₈	—	13	1	no	—
205	C₂×C₈	—	13	1	no	—
206	C₂×C₈	—	13	2	no	—
207	C₄⋊C₄	—	2	1	no	—
208	C₄⋊C₄	—	2	2	no	—
209	C₄⋊C₄	—	13	2	no	—
210	C₄⋊C₄	—	26	1	no	—
211	C₄⋊C₄	—	26	2	no	—
212	C₄⋊C₄	C₂²⋊D₁₃	1	1	yes	characteristic
213	C₄⋊C₄	C₂²⋊D₁₃	1	1	yes	characteristic
214	C₄⋊C₄	C₂²×D₁₃	1	2	yes	—
215	C₂²⋊C₄	—	2	2	no	—
216	C₂²⋊C₄	—	26	2	no	—
217	C₂²⋊C₄	C₂²⋊D₁₃	1	2	yes	—
218	C₄²	—	2	2	no	—
219	C₄²	—	26	2	no	—
220	C₄²	C₂²×D₁₃	1	2	yes	—
221	C₁₃	#2952	1	1	yes	characteristic, complemented, Sylow
222	C₂³	—	2	1	no	—
223	C₂³	—	26	1	no	—
224	C₂³	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
225	Q₈	—	2	1	no	—
226	Q₈	—	13	2	no	—
227	Q₈	—	26	1	no	—
228	Q₈	C₂×C₄×D₁₃	1	2	yes	—
229	D₄	—	2	4	no	—
230	D₄	—	2	1	no	—
231	D₄	—	13	2	no	—
232	D₄	—	26	4	no	—
233	D₄	—	26	1	no	—
234	D₄	C₂×C₄×D₁₃	1	2	yes	—
235	C₂×C₄	—	2	1	no	—
236	C₂×C₄	—	2	4	no	—
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	8	2	no	—
239	C₂×C₄	—	8	1	no	—
240	C₂×C₄	—	26	1	no	—
241	C₂×C₄	—	26	4	no	—
242	C₂×C₄	—	26	2	no	—
243	C₂×C₄	—	104	2	no	—
244	C₂×C₄	—	104	1	no	—
245	C₂×C₄	C₂×C₄×D₁₃	1	2	yes	—
246	C₂×C₄	#6127	1	2	yes	—
247	C₂×C₄	C₂×C₂²⋊D₁₃	1	2	yes	—
248	C₂×C₄	C₂×C₂²⋊D₁₃	1	2	yes	—
249	C₂×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
250	C₂×C₄	C₂³×D₁₃	1	1	yes	characteristic
251	C₈	—	13	2	no	—
252	C₈	—	13	2	no	—
253	C₈	—	26	2	no	—
254	C₂²	—	2	1	no	—
255	C₂²	—	2	1	no	—
256	C₂²	—	2	2	no	—
257	C₂²	—	4	1	no	—
258	C₂²	—	4	1	no	—
259	C₂²	—	26	2	no	—
260	C₂²	—	52	1	no	—
261	C₂²	—	52	1	no	—
262	C₂²	C₄×C₂²⋊D₁₃	1	2	yes	—
263	C₂²	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
264	C₄	—	2	2	no	—
265	C₄	—	2	2	no	—
266	C₄	—	2	2	no	—
267	C₄	—	4	2	no	—
268	C₄	—	26	2	no	—
269	C₄	—	52	2	no	—
270	C₄	C₄×C₂²⋊D₁₃	1	2	yes	—
271	C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
272	C₄	C₂×#6127	1	1	yes	characteristic
273	C₂	—	2	2	no	—
274	C₂	—	4	2	no	—
275	C₂	—	52	2	no	—
276	C₂	C₂×C₄×C₂²⋊D₁₃	1	1	yes	characteristic
277	C₂	#11719541	1	1	yes	characteristic
278	C₂	#11719542	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₁₃
commutator	C₄×C₁₃	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₃
Fitting	C₁₃×#520	C₂
socle	C₂²×C₁₃	C₂²×D₄
prime radical	C₂×#88204	C₂
cosocle	C₂×C₄×C₁₃	C₂⁴
hypercenter	#520	D₁₃
hypocenter	C₁₃	#2952
LCS	G ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ #117 ◁ #520

Complements

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

N	H ≅ G/N	type
1. C₁₃×#520	C₂	semidirect
3. #11719228	C₂	semidirect
4. #11718893	C₂	semidirect
5. #11718691	C₂	semidirect
6. C₄×#6106	C₂	semidirect
7. C₄×#6105	C₂	semidirect
8. #11718675	C₂	semidirect
9. #11718658	C₂	semidirect
10. #11718627	C₂	semidirect
11. C₂×#88048	C₂	semidirect
21. #88204	C₄	semidirect
43. C₁₃×#41	C₂×C₄	semidirect
54. #6127	C₂×C₄	semidirect
68. #6107	C₂×C₄	semidirect
69. #6106	C₂×C₄	semidirect
70. #6105	C₂×C₄	semidirect
71. #6104	C₂×C₄	semidirect
78. C₂×#1055	C₂×C₄	semidirect
117. #520	D₁₃	semidirect
184. #41	C₄×D₁₃	semidirect
221. C₁₃	#2952	semidirect

Maximal subgroups

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

subgroup	index
C₁₃×#520	2
C₂×#88204	2
#11719228	2
#11718893	2
#11718691	2
C₄×#6106	2
C₄×#6105	2
#11718675	2
#11718658	2
#11718627	2
C₂×#88048	2
#2952	13