Subgroups — #13131399

Order 1664.

564 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#88237	C₂	1	1	yes	maximal, characteristic
2	C₂×D₁₃⋊M₁₆	C₂	1	1	yes	maximal, characteristic
3	C₂×M₁₆×D₁₃	C₂	1	1	yes	maximal, characteristic
4	#11719099	C₂	1	4	yes	maximal, complemented
5	M₁₆×#261	C₂	1	4	yes	maximal, complemented
6	#11719089	C₂	1	4	yes	maximal, complemented
7	C₂²×C₄×D₁₃	C₂²	1	1	yes	characteristic
8	C₂×C₂²⋊#261	C₄	1	2	yes	—
9	C₂×#6122	C₄	1	2	yes	complemented
10	#88237	C₄	1	2	yes	—
11	#88237	C₄	1	2	yes	—
12	#88237	C₂²	1	2	yes	—
13	#88237	C₂²	1	2	yes	—
14	C₂×C₄⋊#261	C₄	1	2	yes	—
15	C₂×C₄×#261	C₂²	1	2	yes	—
16	D₁₃⋊M₁₆	C₄	1	2	yes	complemented
17	D₁₃⋊M₁₆	C₄	1	2	yes	complemented
18	D₁₃⋊M₁₆	C₂²	1	2	yes	—
19	D₁₃⋊M₁₆	C₂²	1	2	yes	—
20	C₂×#6118	C₄	1	2	yes	complemented
21	C₂×D₁₃⋊C₈	C₂²	1	2	yes	—
22	C₂×C₁₃×M₁₆	C₄	1	1	yes	characteristic, complemented
23	C₂×#6099	C₄	1	1	yes	characteristic, complemented
24	M₁₆×D₁₃	C₄	1	2	yes	complemented
25	M₁₆×D₁₃	C₄	1	2	yes	complemented
26	M₁₆×D₁₃	C₂²	1	2	yes	—
27	M₁₆×D₁₃	C₂²	1	2	yes	—
28	C₂×#6094	C₄	1	2	yes	complemented
29	C₂×C₈×D₁₃	C₂²	1	2	yes	—
30	#88100	C₂²	1	8	yes	complemented
31	C₈×#261	C₂²	1	8	yes	complemented
32	C₂³×D₁₃	C₂×C₄	1	1	yes	characteristic
33	C₂²×#261	C₂×C₄	1	4	yes	—
34	C₂²×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
35	C₂²×Dic₁₃	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄×D₁₃	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄×D₁₃	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄×D₁₃	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄×D₁₃	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄×D₁₃	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄×D₁₃	C₂×C₄	1	1	yes	characteristic
42	C₂×C₄×D₁₃	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄×D₁₃	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄×D₁₃	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄×D₁₃	C₂³	1	1	yes	characteristic
46	C₂×C₄×D₁₃	C₂³	1	1	yes	characteristic
47	C₂×C₄×D₁₃	C₂³	1	1	yes	characteristic
48	C₂²⋊#261	C₂×C₄	1	4	yes	—
49	C₂²⋊#261	C₂×C₄	1	4	yes	—
50	#6122	C₂×C₄	1	4	yes	—
51	#6122	C₂×C₄	1	4	yes	complemented
52	C₂×#1057	C₂×C₄	1	4	yes	complemented
53	C₄⋊#261	C₂×C₄	1	4	yes	—
54	C₄⋊#261	C₂×C₄	1	4	yes	—
55	C₄×#261	C₂×C₄	1	4	yes	—
56	C₄×#261	C₂³	1	4	yes	—
57	#6118	C₂×C₄	1	4	yes	—
58	#6118	C₂×C₄	1	4	yes	complemented
59	D₁₃⋊C₈	C₂×C₄	1	4	yes	complemented
60	D₁₃⋊C₈	C₂³	1	4	yes	—
61	C₁₃×M₁₆	C₂×C₄	1	2	yes	—
62	C₁₃×M₁₆	C₂×C₄	1	2	yes	complemented
63	C₂×C₈×C₁₃	C₂×C₄	1	2	yes	complemented
64	#6099	C₂×C₄	1	2	yes	—
65	#6099	C₂×C₄	1	2	yes	complemented
66	C₂×#1055	C₂×C₄	1	2	yes	complemented
67	#6094	C₂×C₄	1	4	yes	—
68	#6094	C₂×C₄	1	4	yes	complemented
69	C₈×D₁₃	C₂×C₄	1	4	yes	complemented
70	C₈×D₁₃	C₂³	1	4	yes	—
71	#2873	—	13	1	no	maximal, Sylow
72	C₂³×C₁₃	C₄²	1	1	yes	characteristic
73	C₂²×D₁₃	—	4	2	no	—
74	C₂²×D₁₃	C₄²	1	1	yes	characteristic
75	C₂²×D₁₃	C₄²	1	1	yes	characteristic
76	C₂²×D₁₃	C₄²	1	1	yes	characteristic
77	C₂²×D₁₃	C₂²×C₄	1	1	yes	characteristic
78	C₂²×D₁₃	C₂²×C₄	1	1	yes	characteristic
79	C₂²×D₁₃	C₂²×C₄	1	1	yes	characteristic
80	C₂×#261	—	4	4	no	—
81	C₂×#261	C₂²×C₄	1	8	yes	—
82	C₂×C₄×C₁₃	C₄²	1	1	yes	characteristic
83	C₂×C₄×C₁₃	C₄²	1	1	yes	characteristic
84	C₂×C₄×C₁₃	C₄²	1	1	yes	characteristic
85	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
86	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
87	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
88	C₂×Dic₁₃	C₄²	1	1	yes	characteristic
89	C₂×Dic₁₃	C₄²	1	1	yes	characteristic
90	C₂×Dic₁₃	C₄²	1	1	yes	characteristic
91	C₂×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
92	C₂×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
93	C₂×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
94	C₄×D₁₃	C₄²	1	1	yes	characteristic
95	C₄×D₁₃	C₄²	1	1	yes	characteristic
96	C₄×D₁₃	C₄²	1	1	yes	characteristic
97	C₄×D₁₃	C₄²	1	1	yes	characteristic
98	C₄×D₁₃	C₄²	1	1	yes	characteristic
99	C₄×D₁₃	C₄²	1	1	yes	characteristic
100	C₄×D₁₃	C₂²×C₄	1	1	yes	characteristic
101	C₄×D₁₃	C₂²×C₄	1	1	yes	characteristic
102	C₄×D₁₃	C₂²×C₄	1	1	yes	characteristic
103	C₄×D₁₃	C₂²×C₄	1	1	yes	characteristic
104	C₄×D₁₃	C₂²×C₄	1	1	yes	characteristic
105	C₄×D₁₃	C₂²×C₄	1	1	yes	characteristic
106	C₄×D₁₃	C₂²×C₄	1	1	yes	characteristic
107	C₄×D₁₃	C₂²×C₄	1	1	yes	characteristic
108	C₄×D₁₃	C₂²×C₄	1	1	yes	characteristic
109	C₄×D₁₃	C₂⁴	1	1	yes	characteristic
110	#1057	C₂²×C₄	1	8	yes	—
111	C₈×C₁₃	C₂²×C₄	1	4	yes	—
112	#1055	C₂²×C₄	1	4	yes	—
113	C₂²×M₁₆	—	13	1	no	—
114	C₂²×M₁₆	—	13	1	no	—
115	C₂×#117	—	13	1	no	—
116	#405	—	13	4	no	—
117	#405	—	13	4	no	—
118	C₄×M₁₆	—	13	4	no	—
119	C₂²×C₁₃	—	4	1	no	—
120	C₂²×C₁₃	C₂×C₄²	1	1	yes	characteristic
121	C₂²×C₁₃	C₂×C₄²	1	1	yes	characteristic
122	C₂²×C₁₃	C₂×C₄²	1	1	yes	characteristic
123	C₂×D₁₃	—	2	2	no	—
124	C₂×D₁₃	—	2	2	no	—
125	C₂×D₁₃	—	2	2	no	—
126	C₂×D₁₃	—	4	1	no	—
127	C₂×D₁₃	—	4	1	no	—
128	C₂×D₁₃	—	4	1	no	—
129	C₂×D₁₃	C₂×C₄²	1	1	yes	characteristic
130	C₂×D₁₃	C₂×C₄²	1	1	yes	characteristic
131	C₂×D₁₃	C₂×C₄²	1	1	yes	characteristic
132	C₂×D₁₃	C₂³×C₄	1	1	yes	characteristic
133	#261	—	2	8	no	—
134	C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
135	C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
136	C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
137	C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
138	Dic₁₃	C₂×C₄²	1	1	yes	characteristic
139	Dic₁₃	C₂×C₄²	1	1	yes	characteristic
140	Dic₁₃	C₂×C₄²	1	1	yes	characteristic
141	Dic₁₃	C₂³×C₄	1	1	yes	characteristic
142	C₂³×C₄	—	13	1	no	—
143	C₂×M₁₆	—	13	2	no	—
144	C₂×M₁₆	—	13	2	no	—
145	C₂×M₁₆	—	13	2	no	—
146	C₂×M₁₆	—	13	2	no	—
147	C₂×M₁₆	—	13	2	no	—
148	C₂×M₁₆	—	13	2	no	—
149	C₂×M₁₆	—	13	1	no	—
150	C₂×M₁₆	—	13	2	no	—
151	C₂×M₁₆	—	13	2	no	—
152	C₂×M₁₆	—	13	2	no	—
153	C₂×M₁₆	—	13	2	no	—
154	C₂×M₁₆	—	13	2	no	—
155	C₂×M₁₆	#261	1	1	yes	characteristic, complemented
156	C₂²×C₈	—	13	2	no	—
157	C₂²×C₈	—	13	2	no	—
158	#117	—	13	2	no	—
159	#117	—	13	2	no	—
160	#117	—	13	2	no	—
161	#117	—	13	2	no	—
162	C₂×C₄⋊C₄	—	13	2	no	—
163	C₂×C₂²⋊C₄	—	13	2	no	—
164	C₂×C₄²	—	13	2	no	—
165	#97	—	13	8	no	—
166	C₄×C₈	—	13	8	no	—
167	C₂×C₁₃	—	2	1	no	—
168	C₂×C₁₃	—	2	1	no	—
169	C₂×C₁₃	—	2	1	no	—
170	C₂×C₁₃	C₂²×C₄²	1	1	yes	characteristic
171	D₁₃	—	2	1	no	—
172	D₁₃	—	2	1	no	—
173	D₁₃	—	2	1	no	—
174	D₁₃	#568	1	2	yes	—
175	C₂⁴	—	13	1	no	—
176	C₂²×C₄	—	13	1	no	—
177	C₂²×C₄	—	13	1	no	—
178	C₂²×C₄	—	13	1	no	—
179	C₂²×C₄	—	13	1	no	—
180	C₂²×C₄	—	13	1	no	—
181	C₂²×C₄	—	13	1	no	—
182	C₂²×C₄	—	13	1	no	—
183	C₂²×C₄	—	13	1	no	—
184	C₂²×C₄	—	13	1	no	—
185	C₂²×C₄	—	13	1	no	—
186	C₂²×C₄	—	13	1	no	—
187	C₂²×C₄	—	13	1	no	—
188	C₂²×C₄	—	13	1	no	—
189	C₂²×C₄	—	13	4	no	—
190	C₂²×C₄	C₂×#261	1	1	yes	characteristic
191	M₁₆	—	13	4	no	—
192	M₁₆	—	13	4	no	—
193	M₁₆	—	13	4	no	—
194	M₁₆	—	13	4	no	—
195	M₁₆	—	13	2	no	—
196	M₁₆	—	13	2	no	—
197	M₁₆	—	13	4	no	—
198	M₁₆	—	13	4	no	—
199	M₁₆	C₂×#261	1	2	yes	—
200	M₁₆	C₂×#261	1	2	yes	complemented
201	C₂×C₈	—	13	4	no	—
202	C₂×C₈	—	13	4	no	—
203	C₂×C₈	—	13	4	no	—
204	C₂×C₈	—	13	2	no	—
205	C₂×C₈	—	13	4	no	—
206	C₂×C₈	—	13	4	no	—
207	C₂×C₈	C₂×#261	1	2	yes	complemented
208	C₄⋊C₄	—	13	4	no	—
209	C₄⋊C₄	—	13	4	no	—
210	C₂²⋊C₄	—	13	4	no	—
211	C₂²⋊C₄	—	13	4	no	—
212	C₄²	—	13	4	no	—
213	C₄²	—	13	4	no	—
214	C₁₃	#2873	1	1	yes	characteristic, complemented, Sylow
215	C₂³	—	13	1	no	—
216	C₂³	—	13	1	no	—
217	C₂³	—	13	1	no	—
218	C₂³	—	13	1	no	—
219	C₂³	—	13	1	no	—
220	C₂³	—	13	1	no	—
221	C₂³	—	52	2	no	—
222	C₂³	C₄×#261	1	1	yes	characteristic
223	C₂×C₄	—	13	1	no	—
224	C₂×C₄	—	13	1	no	—
225	C₂×C₄	—	13	1	no	—
226	C₂×C₄	—	13	1	no	—
227	C₂×C₄	—	13	1	no	—
228	C₂×C₄	—	13	1	no	—
229	C₂×C₄	—	13	1	no	—
230	C₂×C₄	—	13	1	no	—
231	C₂×C₄	—	13	1	no	—
232	C₂×C₄	—	13	1	no	—
233	C₂×C₄	—	13	1	no	—
234	C₂×C₄	—	13	1	no	—
235	C₂×C₄	—	13	1	no	—
236	C₂×C₄	—	13	1	no	—
237	C₂×C₄	—	13	1	no	—
238	C₂×C₄	—	13	1	no	—
239	C₂×C₄	—	13	1	no	—
240	C₂×C₄	—	13	1	no	—
241	C₂×C₄	—	13	1	no	—
242	C₂×C₄	—	13	1	no	—
243	C₂×C₄	—	13	1	no	—
244	C₂×C₄	—	13	1	no	—
245	C₂×C₄	—	13	8	no	—
246	C₂×C₄	—	52	4	no	—
247	C₂×C₄	C₄×#261	1	1	yes	characteristic
248	C₂×C₄	C₄×#261	1	1	yes	characteristic
249	C₂×C₄	C₄×#261	1	1	yes	characteristic
250	C₂×C₄	C₂²×#261	1	1	yes	characteristic
251	C₂×C₄	C₂²×#261	1	1	yes	characteristic
252	C₂×C₄	C₂²×#261	1	1	yes	characteristic
253	C₈	—	13	8	no	—
254	C₈	—	13	4	no	—
255	C₈	C₂²×#261	1	4	yes	—
256	C₂²	—	4	1	no	—
257	C₂²	—	13	1	no	—
258	C₂²	—	13	1	no	—
259	C₂²	—	13	1	no	—
260	C₂²	—	13	1	no	—
261	C₂²	—	26	2	no	—
262	C₂²	—	26	2	no	—
263	C₂²	—	26	2	no	—
264	C₂²	—	52	1	no	—
265	C₂²	—	52	1	no	—
266	C₂²	—	52	1	no	—
267	C₂²	C₂×C₄×#261	1	1	yes	characteristic
268	C₂²	C₂×C₄×#261	1	1	yes	characteristic
269	C₂²	C₂×C₄×#261	1	1	yes	characteristic
270	C₄	—	13	1	no	—
271	C₄	—	13	1	no	—
272	C₄	—	13	1	no	—
273	C₄	—	13	1	no	—
274	C₄	—	26	8	no	—
275	C₄	C₂×C₄×#261	1	1	yes	characteristic
276	C₄	C₂×C₄×#261	1	1	yes	characteristic
277	C₄	C₂×C₄×#261	1	1	yes	characteristic
278	C₄	C₂³×#261	1	1	yes	characteristic
279	C₂	—	2	1	no	—
280	C₂	—	2	1	no	—
281	C₂	—	2	1	no	—
282	C₂	—	13	2	no	—
283	C₂	—	26	1	no	—
284	C₂	—	26	1	no	—
285	C₂	—	26	1	no	—
286	C₂	C₂²×C₄×#261	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₄	C₂³×#261
commutator, socle	C₂×C₁₃	C₂²×C₄²
Fitting	C₂×C₁₃×M₁₆	C₄
prime radical	C₂³×D₁₃	C₂×C₄
cosocle	C₄×D₁₃	C₂⁴
hypercenter	C₂×M₁₆	#261
hypocenter	C₁₃	#2873
LCS	G ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₄ ◁ C₂×M₁₆

Complements

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

N	H ≅ G/N	type
4. #11719099	C₂	semidirect
5. M₁₆×#261	C₂	semidirect
6. #11719089	C₂	semidirect
9. C₂×#6122	C₄	semidirect
16. D₁₃⋊M₁₆	C₄	semidirect
17. D₁₃⋊M₁₆	C₄	semidirect
20. C₂×#6118	C₄	semidirect
22. C₂×C₁₃×M₁₆	C₄	semidirect
23. C₂×#6099	C₄	semidirect
24. M₁₆×D₁₃	C₄	semidirect
25. M₁₆×D₁₃	C₄	semidirect
28. C₂×#6094	C₄	semidirect
30. #88100	C₂²	semidirect
31. C₈×#261	C₂²	semidirect
51. #6122	C₂×C₄	semidirect
52. C₂×#1057	C₂×C₄	semidirect
58. #6118	C₂×C₄	semidirect
59. D₁₃⋊C₈	C₂×C₄	semidirect
62. C₁₃×M₁₆	C₂×C₄	semidirect
63. C₂×C₈×C₁₃	C₂×C₄	semidirect
65. #6099	C₂×C₄	semidirect
66. C₂×#1055	C₂×C₄	semidirect
68. #6094	C₂×C₄	semidirect
69. C₈×D₁₃	C₂×C₄	semidirect
155. C₂×M₁₆	#261	semidirect
200. M₁₆	C₂×#261	semidirect
207. C₂×C₈	C₂×#261	semidirect
214. C₁₃	#2873	semidirect

Maximal subgroups

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

subgroup	index
C₂×#88237	2
C₂×D₁₃⋊M₁₆	2
C₂×M₁₆×D₁₃	2
#11719099	2
M₁₆×#261	2
#11719089	2
#2873	13