Subgroups — #15861649

Order 3328.

1122 conjugacy classes of subgroups.

The subgroup lattice has 286 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718586	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×C₄⋊Dic₁₃	C₂	1	1	yes	maximal, characteristic
3	#13114701	C₂	1	4	yes	maximal, complemented
4	C₈×C₄⋊Dic₁₃	C₂	1	4	yes	maximal, complemented
5	C₂²⋊C₈×Dic₁₃	C₂	1	4	yes	maximal
6	C₂×C₁₃×#431	C₂	1	1	yes	maximal, characteristic
7	C₂²×C₄²×C₁₃	C₂²	1	1	yes	characteristic
8	C₂²×C₄⋊Dic₁₃	C₄	1	2	yes	—
9	C₂²×C₄×Dic₁₃	C₂²	1	2	yes	—
10	C₁₃×#431	C₄	1	8	yes	complemented
11	C₂×C₁₃×C₄⋊C₈	C₂²	1	2	yes	—
12	C₂×C₁₃×C₂²⋊C₈	C₂²	1	2	yes	—
13	C₂×C₄×C₈×C₁₃	C₂²	1	2	yes	—
14	C₂×#88071	C₄	1	2	yes	—
15	C₂×C₂²⋊#1055	C₂²	1	2	yes	—
16	C₂×C₈×Dic₁₃	C₂²	1	8	yes	—
17	C₄×C₄⋊Dic₁₃	C₄	1	2	yes	—
18	C₄×C₄⋊Dic₁₃	C₄	1	2	yes	—
19	C₄×C₄⋊Dic₁₃	C₂²	1	2	yes	—
20	C₄×C₄⋊Dic₁₃	C₂²	1	2	yes	—
21	#11718586	C₄	1	8	yes	complemented
22	C₂×C₄⋊#1055	C₂²	1	2	yes	—
23	C₂×C₄×#1055	C₂²	1	2	yes	—
24	#11718142	C₂²	1	8	yes	—
25	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
26	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
27	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
28	C₂³×Dic₁₃	C₂×C₄	1	4	yes	—
29	C₂²×C₈×C₁₃	C₂³	1	4	yes	—
30	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄²×C₁₃	C₂×C₄	1	4	yes	—
33	C₂×C₄²×C₁₃	C₂×C₄	1	4	yes	—
34	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
35	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
36	C₂×C₄⋊Dic₁₃	C₈	1	16	yes	complemented
37	C₂×C₄⋊Dic₁₃	C₂×C₄	1	4	yes	—
38	C₂×C₄⋊Dic₁₃	C₂×C₄	1	4	yes	—
39	C₂×C₄×Dic₁₃	—	4	4	no	—
40	C₂×C₄×Dic₁₃	C₂×C₄	1	4	yes	—
41	C₂×C₄×Dic₁₃	C₂³	1	4	yes	—
42	C₂²×#1055	C₂³	1	4	yes	—
43	C₁₃×C₄⋊C₈	C₂×C₄	1	8	yes	complemented
44	C₁₃×C₂²⋊C₈	C₂×C₄	1	8	yes	—
45	C₄×C₈×C₁₃	C₂×C₄	1	8	yes	complemented
46	#88071	C₂×C₄	1	4	yes	—
47	#88071	C₂×C₄	1	4	yes	—
48	C₂²⋊#1055	C₂×C₄	1	8	yes	—
49	C₈×Dic₁₃	—	4	8	no	—
50	C₄⋊#1055	C₂×C₄	1	8	yes	complemented
51	C₄×#1055	C₂×C₄	1	8	yes	complemented
52	#20533	—	13	1	no	maximal, Sylow
53	C₂⁴×C₁₃	C₂²×C₄	1	1	yes	characteristic
54	C₂²×C₄×C₁₃	—	2	2	no	—
55	C₂²×C₄×C₁₃	—	2	2	no	—
56	C₂²×C₄×C₁₃	—	4	2	no	—
57	C₂²×C₄×C₁₃	C₄²	1	4	yes	—
58	C₂²×C₄×C₁₃	C₄²	1	4	yes	—
59	C₂²×C₄×C₁₃	C₂×C₈	1	8	yes	—
60	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
61	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
62	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
63	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
64	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
65	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
66	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
67	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
68	C₂²×Dic₁₃	—	4	4	no	—
69	C₂²×Dic₁₃	—	4	4	no	—
70	C₂²×Dic₁₃	C₂×C₈	1	16	yes	—
71	C₂²×Dic₁₃	C₂²×C₄	1	8	yes	—
72	C₂×C₈×C₁₃	—	2	4	no	—
73	C₂×C₈×C₁₃	—	2	4	no	—
74	C₂×C₈×C₁₃	C₂²×C₄	1	8	yes	—
75	C₄²×C₁₃	C₄²	1	4	yes	—
76	C₄²×C₁₃	C₄²	1	4	yes	—
77	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
78	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
79	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
80	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
81	C₄⋊Dic₁₃	C₂×C₈	1	16	yes	—
82	C₄⋊Dic₁₃	C₂×C₈	1	16	yes	—
83	C₄×Dic₁₃	—	2	8	no	—
84	C₄×Dic₁₃	—	4	4	no	—
85	C₂×#1055	—	2	4	no	—
86	C₂×#1055	—	2	4	no	—
87	C₂×#1055	C₂²×C₄	1	8	yes	—
88	C₂×#431	—	13	1	no	—
89	C₂×#431	D₁₃	1	1	yes	characteristic
90	C₂×C₄×C₄⋊C₄	—	13	1	no	—
91	C₈×C₄⋊C₄	—	13	4	no	—
92	C₈×C₄⋊C₄	—	13	4	no	—
93	C₄×C₂²⋊C₈	—	13	4	no	—
94	C₂³×C₁₃	—	2	1	no	—
95	C₂³×C₁₃	—	2	1	no	—
96	C₂³×C₁₃	—	2	2	no	—
97	C₂³×C₁₃	C₂×C₄²	1	2	yes	—
98	C₂³×C₁₃	C₂²×C₈	1	4	yes	—
99	C₂³×C₁₃	C₂³×C₄	1	1	yes	characteristic
100	C₂×C₄×C₁₃	—	2	1	no	—
101	C₂×C₄×C₁₃	—	2	1	no	—
102	C₂×C₄×C₁₃	—	2	2	no	—
103	C₂×C₄×C₁₃	—	2	2	no	—
104	C₂×C₄×C₁₃	—	2	4	no	—
105	C₂×C₄×C₁₃	—	2	4	no	—
106	C₂×C₄×C₁₃	—	4	4	no	—
107	C₂×C₄×C₁₃	C₄×C₈	1	16	yes	complemented
108	C₂×C₄×C₁₃	C₂×C₄²	1	2	yes	—
109	C₂×C₄×C₁₃	C₂×C₄²	1	2	yes	—
110	C₂×C₄×C₁₃	C₂×C₄²	1	2	yes	—
111	C₂×C₄×C₁₃	C₂×C₄²	1	2	yes	—
112	C₂×C₄×C₁₃	C₂×C₄²	1	2	yes	—
113	C₂×C₄×C₁₃	C₂²×C₈	1	4	yes	—
114	C₂×C₄×C₁₃	C₂²×C₈	1	4	yes	—
115	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
116	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
117	C₂×C₄×C₁₃	H(5,𝔽₂)	1	2	yes	—
118	C₂×C₄×C₁₃	2₋¹⁺⁴	1	2	yes	—
119	C₂×Dic₁₃	—	2	8	no	—
120	C₂×Dic₁₃	—	2	8	no	—
121	C₂×Dic₁₃	—	4	16	no	—
122	C₂×Dic₁₃	C₂²×C₈	1	16	yes	—
123	C₈×C₁₃	—	2	8	no	—
124	#1055	—	2	8	no	—
125	C₂²×C₄⋊C₄	—	13	2	no	—
126	C₂²×C₄²	—	13	2	no	—
127	C₂²×C₄²	C₂×D₁₃	1	1	yes	characteristic
128	#431	—	13	8	no	—
129	#431	Dic₁₃	1	8	yes	complemented
130	C₂×C₄⋊C₈	—	13	2	no	—
131	C₂×C₄⋊C₈	C₂×D₁₃	1	2	yes	—
132	C₂×C₂²⋊C₈	—	13	2	no	—
133	C₂×C₂²⋊C₈	C₂×D₁₃	1	2	yes	—
134	C₂×C₄×C₈	—	13	2	no	—
135	C₂×C₄×C₈	—	13	8	no	—
136	C₂×C₄×C₈	C₂×D₁₃	1	2	yes	—
137	C₄×C₄⋊C₄	—	13	2	no	—
138	C₄×C₄⋊C₄	—	13	2	no	—
139	C₄×C₄⋊C₄	—	13	2	no	—
140	C₄×C₄⋊C₄	—	13	2	no	—
141	C₂×#95	—	13	2	no	—
142	#336	—	13	8	no	—
143	C₂²×C₁₃	—	2	2	no	—
144	C₂²×C₁₃	—	2	4	no	—
145	C₂²×C₁₃	—	2	4	no	—
146	C₂²×C₁₃	—	2	2	no	—
147	C₂²×C₁₃	C₂×C₄×C₈	1	4	yes	—
148	C₂²×C₁₃	C₂²×C₄²	1	1	yes	characteristic
149	C₂²×C₁₃	#518	1	1	yes	characteristic
150	C₂²×C₁₃	#519	1	1	yes	characteristic
151	C₂²×C₁₃	C₂³×C₈	1	2	yes	—
152	C₂²×C₁₃	#568	1	2	yes	—
153	C₄×C₁₃	—	2	2	no	—
154	C₄×C₁₃	—	2	2	no	—
155	C₄×C₁₃	—	2	2	no	—
156	C₄×C₁₃	C₂×C₄×C₈	1	4	yes	—
157	C₄×C₁₃	C₂×C₄×C₈	1	4	yes	—
158	C₄×C₁₃	#520	1	2	yes	—
159	C₄×C₁₃	#520	1	2	yes	—
160	Dic₁₃	—	2	16	no	—
161	C₂³×C₄	—	13	4	no	—
162	C₂³×C₄	C₄×D₁₃	1	1	yes	characteristic
163	C₂³×C₄	C₄×D₁₃	1	1	yes	characteristic
164	C₂³×C₄	C₂²×D₁₃	1	1	yes	characteristic
165	C₂²×C₈	—	13	4	no	—
166	C₂²×C₈	C₂²×D₁₃	1	4	yes	—
167	C₂×C₄⋊C₄	—	13	16	no	—
168	C₂×C₄⋊C₄	—	13	4	no	—
169	C₂×C₄⋊C₄	—	13	4	no	—
170	C₂×C₄²	—	13	4	no	—
171	C₂×C₄²	—	13	4	no	—
172	C₂×C₄²	—	52	4	no	—
173	C₂×C₄²	C₄×D₁₃	1	1	yes	characteristic
174	C₂×C₄²	C₄×D₁₃	1	1	yes	characteristic
175	C₂×C₄²	C₄×D₁₃	1	4	yes	—
176	C₂×C₄²	C₂×Dic₁₃	1	4	yes	—
177	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
178	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
179	C₄⋊C₈	—	13	8	no	—
180	C₄⋊C₈	C₂×Dic₁₃	1	8	yes	complemented
181	C₂²⋊C₈	—	13	8	no	—
182	C₂²⋊C₈	C₂×Dic₁₃	1	8	yes	—
183	C₄×C₈	—	13	8	no	—
184	C₄×C₈	—	52	8	no	—
185	C₄×C₈	C₂×Dic₁₃	1	8	yes	complemented
186	#95	—	13	4	no	—
187	#95	—	13	4	no	—
188	C₂×C₁₃	—	2	4	no	—
189	C₂×C₁₃	#2274	1	1	yes	characteristic
190	C₂×C₁₃	C₂²×C₄×C₈	1	1	yes	characteristic
191	C₂×C₁₃	#2873	1	1	yes	characteristic
192	C₂×C₁₃	#2972	1	2	yes	—
193	C₂×C₁₃	#2982	1	2	yes	—
194	C₂⁴	C₂×C₄×D₁₃	1	1	yes	characteristic
195	C₂²×C₄	—	2	2	no	—
196	C₂²×C₄	—	2	2	no	—
197	C₂²×C₄	—	4	2	no	—
198	C₂²×C₄	—	13	8	no	—
199	C₂²×C₄	—	13	16	no	—
200	C₂²×C₄	—	52	4	no	—
201	C₂²×C₄	—	52	4	no	—
202	C₂²×C₄	C₈×D₁₃	1	8	yes	—
203	C₂²×C₄	C₄×Dic₁₃	1	4	yes	—
204	C₂²×C₄	C₄×Dic₁₃	1	4	yes	—
205	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
206	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
207	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
208	C₂²×C₄	C₂×C₄×D₁₃	1	2	yes	—
209	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
210	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
211	C₂²×C₄	C₂²×Dic₁₃	1	2	yes	—
212	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
213	C₂×C₈	—	2	4	no	—
214	C₂×C₈	—	2	4	no	—
215	C₂×C₈	—	13	8	no	—
216	C₂×C₈	—	26	4	no	—
217	C₂×C₈	—	26	4	no	—
218	C₂×C₈	C₂²×Dic₁₃	1	8	yes	—
219	C₄⋊C₄	—	13	16	no	—
220	C₄⋊C₄	—	13	16	no	—
221	C₄²	—	26	8	no	—
222	C₄²	—	52	4	no	—
223	C₄²	C₄×Dic₁₃	1	4	yes	—
224	C₄²	C₄×Dic₁₃	1	4	yes	—
225	C₄²	C₂×C₄×D₁₃	1	2	yes	—
226	C₄²	C₂×C₄×D₁₃	1	2	yes	—
227	C₄²	C₂²×Dic₁₃	1	2	yes	—
228	C₄²	C₂²×Dic₁₃	1	2	yes	—
229	C₁₃	#20533	1	1	yes	characteristic, complemented, Sylow
230	C₂³	—	2	2	no	—
231	C₂³	—	2	1	no	—
232	C₂³	—	2	1	no	—
233	C₂³	C₂×C₈×D₁₃	1	4	yes	—
234	C₂³	C₂×C₄×Dic₁₃	1	2	yes	—
235	C₂³	C₂²×C₄×D₁₃	1	1	yes	characteristic
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	2	1	no	—
239	C₂×C₄	—	2	1	no	—
240	C₂×C₄	—	2	4	no	—
241	C₂×C₄	—	2	4	no	—
242	C₂×C₄	—	4	4	no	—
243	C₂×C₄	—	13	16	no	—
244	C₂×C₄	—	26	8	no	—
245	C₂×C₄	—	26	8	no	—
246	C₂×C₄	—	52	16	no	—
247	C₂×C₄	C₈×Dic₁₃	1	16	yes	complemented
248	C₂×C₄	C₂×C₈×D₁₃	1	4	yes	—
249	C₂×C₄	C₂×C₈×D₁₃	1	4	yes	—
250	C₂×C₄	C₂×C₄×Dic₁₃	1	2	yes	—
251	C₂×C₄	C₂×C₄×Dic₁₃	1	2	yes	—
252	C₂×C₄	C₂×C₄×Dic₁₃	1	2	yes	—
253	C₂×C₄	C₂×C₄×Dic₁₃	1	2	yes	—
254	C₂×C₄	C₂×C₄×Dic₁₃	1	2	yes	—
255	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
256	C₂×C₄	#88256	1	2	yes	—
257	C₂×C₄	#88257	1	2	yes	—
258	C₂×C₄	C₂³×Dic₁₃	1	1	yes	characteristic
259	C₈	—	2	8	no	—
260	C₈	—	26	8	no	—
261	C₂²	—	2	2	no	—
262	C₂²	—	2	4	no	—
263	C₂²	—	2	4	no	—
264	C₂²	—	2	2	no	—
265	C₂²	C₂×C₈×Dic₁₃	1	4	yes	—
266	C₂²	#11719221	1	1	yes	characteristic
267	C₂²	#11719226	1	1	yes	characteristic
268	C₂²	C₂²×C₈×D₁₃	1	2	yes	—
269	C₂²	#11719450	1	2	yes	—
270	C₂²	C₂²×C₄×Dic₁₃	1	1	yes	characteristic
271	C₄	—	2	2	no	—
272	C₄	—	2	2	no	—
273	C₄	—	2	2	no	—
274	C₄	—	26	16	no	—
275	C₄	C₂×C₈×Dic₁₃	1	4	yes	—
276	C₄	C₂×C₈×Dic₁₃	1	4	yes	—
277	C₄	#11719228	1	2	yes	—
278	C₄	#11719532	1	2	yes	—
279	C₂	—	2	4	no	—
280	C₂	#13117192	1	1	yes	characteristic
281	C₂	C₂²×C₈×Dic₁₃	1	1	yes	characteristic
282	C₂	#13123254	1	1	yes	characteristic
283	C₂	#13123859	1	2	yes	—
284	C₂	#13123916	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₁₃
commutator	C₂×C₁₃	C₂²×C₄×C₈
Fitting	C₂×C₁₃×#431	C₂
socle	C₂³×C₁₃	C₂³×C₄
prime radical	C₂⁴×C₁₃	C₂²×C₄
cosocle	C₂²×C₄×C₁₃	C₂⁴
hypercenter	C₂×#431	D₁₃
hypocenter	C₁₃	#20533
LCS	G ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#431

Complements

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

N	H ≅ G/N	type
3. #13114701	C₂	semidirect
4. C₈×C₄⋊Dic₁₃	C₂	semidirect
10. C₁₃×#431	C₄	semidirect
21. #11718586	C₄	semidirect
36. C₂×C₄⋊Dic₁₃	C₈	semidirect
43. C₁₃×C₄⋊C₈	C₂×C₄	semidirect
45. C₄×C₈×C₁₃	C₂×C₄	semidirect
50. C₄⋊#1055	C₂×C₄	semidirect
51. C₄×#1055	C₂×C₄	semidirect
107. C₂×C₄×C₁₃	C₄×C₈	semidirect
129. #431	Dic₁₃	semidirect
180. C₄⋊C₈	C₂×Dic₁₃	semidirect
185. C₄×C₈	C₂×Dic₁₃	semidirect
229. C₁₃	#20533	semidirect
247. C₂×C₄	C₈×Dic₁₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11718586	2
C₂×C₄×C₄⋊Dic₁₃	2
#13114701	2
C₈×C₄⋊Dic₁₃	2
C₂²⋊C₈×Dic₁₃	2
C₂×C₁₃×#431	2
#20533	13