Subgroups — #15753448

Order 3328.

506 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718350	C₂	1	1	yes	maximal, characteristic
2	#13113148	C₂	1	2	yes	maximal
3	#13113119	C₂	1	1	yes	maximal, characteristic, complemented
4	#13112680	C₂	1	2	yes	maximal, complemented
5	C₁₃×#1450	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×#6094	C₄	1	1	yes	characteristic
7	C₂²×C₈×D₁₃	C₄	1	1	yes	characteristic
8	C₂×#88126	C₂²	1	1	yes	characteristic
9	C₁₃×#431	C₂²	1	2	yes	—
10	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
11	C₂×#88055	C₄	1	1	yes	characteristic
12	C₂×C₈×Dic₁₃	C₄	1	1	yes	characteristic
13	#11718586	C₂²	1	2	yes	—
14	C₂×#88043	C₂²	1	1	yes	characteristic
15	#11718350	—	2	1	no	—
16	#11718350	—	2	1	no	—
17	#11718350	—	2	1	no	—
18	#11718350	—	2	1	no	—
19	C₁₃×#322	—	2	1	no	—
20	C₈²×C₁₃	—	2	1	no	—
21	#11718061	C₄	1	4	yes	—
22	#11718060	C₄	1	4	yes	—
23	#11718047	—	2	1	no	—
24	C₈×#1055	—	2	1	no	—
25	C₂²×C₄×D₁₃	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
29	C₂×#6103	C₂×C₄	1	2	yes	—
30	C₂×#6101	C₂×C₄	1	2	yes	—
31	C₂×C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
32	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
33	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
34	C₂×#6094	—	2	1	no	—
35	C₂×#6094	—	2	1	no	—
36	C₂×#6094	—	2	2	no	—
37	C₂×#6094	C₈	1	4	yes	complemented
38	C₂×C₈×D₁₃	—	2	1	no	—
39	C₂×C₈×D₁₃	—	2	1	no	—
40	C₂×C₈×D₁₃	—	2	2	no	—
41	C₂×C₈×D₁₃	C₈	1	4	yes	complemented
42	#88126	—	2	1	no	—
43	#88126	—	2	1	no	—
44	#88126	—	2	2	no	—
45	C₁₃×C₄⋊C₈	—	2	2	no	—
46	C₁₃×C₂²⋊C₈	C₂×C₄	1	4	yes	—
47	C₄×C₈×C₁₃	—	2	2	no	—
48	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
49	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
50	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
51	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
52	C₂²⋊#1055	C₂×C₄	1	4	yes	—
53	#88055	—	2	1	no	—
54	#88055	—	2	1	no	—
55	C₈×Dic₁₃	—	2	1	no	—
56	C₈×Dic₁₃	—	2	1	no	—
57	C₄⋊#1055	—	2	2	no	—
58	#88043	—	2	1	no	—
59	#88043	—	2	1	no	—
60	C₄×#1055	—	2	2	no	—
61	#8156	—	13	1	no	maximal, Sylow
62	C₂³×D₁₃	C₄²	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×Dic₁₃	C₄²	1	1	yes	characteristic
66	C₂²×Dic₁₃	C₄²	1	2	yes	—
67	C₂×C₄×D₁₃	—	2	1	no	—
68	C₂×C₄×D₁₃	—	2	1	no	—
69	C₂×C₄×D₁₃	—	2	1	no	—
70	C₂×C₄×D₁₃	—	2	1	no	—
71	C₂×C₄×D₁₃	C₂×C₈	1	2	yes	—
72	C₂×C₄×D₁₃	C₂×C₈	1	2	yes	—
73	C₂×C₈×C₁₃	—	2	4	no	—
74	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
75	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
76	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
77	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
78	C₂×C₈×C₁₃	C₂×C₈	1	2	yes	—
79	C₂×C₈×C₁₃	C₂×C₈	1	2	yes	—
80	C₂×C₈×C₁₃	M₁₆	1	2	yes	—
81	C₂×C₈×C₁₃	M₁₆	1	2	yes	—
82	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
83	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
84	C₄²×C₁₃	#41	1	2	yes	—
85	#6103	—	2	2	no	—
86	#6103	—	2	2	no	—
87	#6101	—	2	2	no	—
88	#6101	—	2	2	no	—
89	C₄×Dic₁₃	—	2	1	no	—
90	C₄×Dic₁₃	—	2	1	no	—
91	C₂×#1055	—	2	1	no	—
92	C₂×#1055	—	2	1	no	—
93	C₂×#1055	—	2	1	no	—
94	C₂×#1055	—	2	1	no	—
95	C₂×#1055	—	2	4	no	—
96	C₂×#1055	C₂×C₈	1	2	yes	—
97	C₂×#1055	C₂×C₈	1	2	yes	—
98	#6094	—	2	4	no	—
99	#6094	—	2	4	no	—
100	C₈×D₁₃	—	2	4	no	—
101	C₈×D₁₃	—	2	4	no	—
102	C₂×#405	—	13	1	no	—
103	#1458	—	13	2	no	—
104	#1450	—	13	1	no	—
105	#1450	D₁₃	1	1	yes	characteristic, complemented
106	#1281	—	13	2	no	—
107	C₂³×C₁₃	C₂×C₄²	1	1	yes	characteristic
108	C₂²×D₁₃	—	2	1	no	—
109	C₂²×D₁₃	—	2	1	no	—
110	C₂²×D₁₃	—	4	2	no	—
111	C₂²×D₁₃	C₄×C₈	1	2	yes	—
112	C₂×C₄×C₁₃	—	4	2	no	—
113	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
114	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
115	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
116	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
117	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
118	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
119	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
120	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
121	C₂×Dic₁₃	—	2	1	no	—
122	C₂×Dic₁₃	—	2	1	no	—
123	C₂×Dic₁₃	—	2	2	no	—
124	C₂×Dic₁₃	—	4	2	no	—
125	C₂×Dic₁₃	C₄×C₈	1	2	yes	—
126	C₄×D₁₃	—	2	2	no	—
127	C₄×D₁₃	—	2	2	no	—
128	C₄×D₁₃	—	2	2	no	—
129	C₄×D₁₃	—	2	2	no	—
130	C₈×C₁₃	—	4	4	no	—
131	C₈×C₁₃	C₂²⋊C₈	1	2	yes	—
132	C₈×C₁₃	C₂²⋊C₈	1	2	yes	—
133	C₈×C₁₃	C₂²⋊C₈	1	2	yes	—
134	C₈×C₁₃	C₂²⋊C₈	1	2	yes	—
135	#1055	—	2	2	no	—
136	#1055	—	2	2	no	—
137	#1055	—	4	4	no	—
138	C₂²×M₁₆	—	13	1	no	—
139	C₂³×C₈	—	13	1	no	—
140	C₂×#117	—	13	1	no	—
141	#431	—	13	2	no	—
142	#431	C₂×D₁₃	1	2	yes	—
143	#405	—	26	1	no	—
144	#405	—	26	1	no	—
145	#405	—	26	1	no	—
146	#405	—	26	1	no	—
147	C₂×#97	—	13	1	no	—
148	C₂×#97	—	13	1	no	—
149	C₂×C₄×C₈	—	13	1	no	—
150	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
151	#324	—	13	4	no	—
152	#323	—	13	4	no	—
153	#322	—	2	1	no	—
154	#322	—	26	1	no	—
155	C₈²	—	2	1	no	—
156	C₈²	—	26	1	no	—
157	C₂²×C₁₃	—	2	2	no	—
158	C₂²×C₁₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
159	C₂²×C₁₃	C₂×C₄×C₈	1	1	yes	characteristic
160	C₂²×C₁₃	C₄×M₁₆	1	1	yes	characteristic
161	C₂×D₁₃	—	2	2	no	—
162	C₂×D₁₃	—	4	2	no	—
163	C₂×D₁₃	—	4	2	no	—
164	C₂×D₁₃	—	4	2	no	—
165	C₄×C₁₃	—	2	2	no	—
166	C₄×C₁₃	—	2	2	no	—
167	C₄×C₁₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
168	C₄×C₁₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
169	C₄×C₁₃	#431	1	1	yes	characteristic
170	C₄×C₁₃	#431	1	1	yes	characteristic
171	Dic₁₃	—	2	2	no	—
172	Dic₁₃	—	4	2	no	—
173	C₂³×C₄	—	13	1	no	—
174	C₂×M₁₆	—	13	4	no	—
175	C₂×M₁₆	—	26	2	no	—
176	C₂×M₁₆	—	26	1	no	—
177	C₂×M₁₆	—	26	1	no	—
178	C₂²×C₈	—	13	1	no	—
179	C₂²×C₈	—	13	1	no	—
180	C₂²×C₈	—	13	4	no	—
181	C₂²×C₈	—	26	2	no	—
182	C₂²×C₈	—	26	1	no	—
183	C₂²×C₈	—	26	1	no	—
184	C₂²×C₈	C₄×D₁₃	1	1	yes	characteristic
185	C₂²×C₈	C₄×D₁₃	1	1	yes	characteristic
186	#117	—	26	1	no	—
187	#117	—	26	2	no	—
188	#117	—	26	1	no	—
189	C₂×C₄⋊C₄	—	13	2	no	—
190	C₂×C₂²⋊C₄	—	13	2	no	—
191	C₂×C₄²	—	13	1	no	—
192	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
193	C₄⋊C₈	—	2	2	no	—
194	C₄⋊C₈	—	26	2	no	—
195	C₂²⋊C₈	—	13	4	no	—
196	C₂²⋊C₈	C₄×D₁₃	1	4	yes	—
197	#97	—	26	1	no	—
198	#97	—	26	1	no	—
199	#97	—	26	1	no	—
200	#97	—	26	1	no	—
201	C₄×C₈	—	2	2	no	—
202	C₄×C₈	—	26	1	no	—
203	C₄×C₈	—	26	1	no	—
204	C₄×C₈	—	26	2	no	—
205	C₄×C₈	D₅₂	1	1	yes	characteristic
206	C₄×C₈	D₅₂	1	1	yes	characteristic
207	C₄×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
208	C₄×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
209	C₂×C₁₃	—	2	1	no	—
210	C₂×C₁₃	—	2	1	no	—
211	C₂×C₁₃	C₄×C₂²⋊C₈	1	1	yes	characteristic
212	C₂×C₁₃	#1757	1	2	yes	—
213	D₁₃	—	4	2	no	—
214	C₂⁴	—	13	1	no	—
215	C₂²×C₄	—	13	1	no	—
216	C₂²×C₄	—	13	2	no	—
217	C₂²×C₄	—	13	2	no	—
218	C₂²×C₄	—	13	2	no	—
219	C₂²×C₄	—	26	1	no	—
220	C₂²×C₄	—	26	1	no	—
221	C₂²×C₄	—	26	1	no	—
222	C₂²×C₄	—	26	1	no	—
223	C₂²×C₄	C₂×C₄×D₁₃	1	2	yes	—
224	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
225	M₁₆	—	26	4	no	—
226	M₁₆	—	26	4	no	—
227	C₂×C₈	—	2	4	no	—
228	C₂×C₈	—	13	2	no	—
229	C₂×C₈	—	13	2	no	—
230	C₂×C₈	—	26	1	no	—
231	C₂×C₈	—	26	1	no	—
232	C₂×C₈	—	26	1	no	—
233	C₂×C₈	—	26	1	no	—
234	C₂×C₈	—	26	4	no	—
235	C₂×C₈	—	26	4	no	—
236	C₂×C₈	—	26	4	no	—
237	C₂×C₈	C₈×D₁₃	1	2	yes	—
238	C₂×C₈	C₈×D₁₃	1	2	yes	—
239	C₂×C₈	#6094	1	2	yes	—
240	C₂×C₈	#6094	1	2	yes	—
241	C₂×C₈	#6103	1	1	yes	characteristic
242	C₂×C₈	#6103	1	1	yes	characteristic
243	C₂×C₈	#6103	1	1	yes	characteristic
244	C₂×C₈	#6103	1	1	yes	characteristic
245	C₄⋊C₄	—	26	2	no	—
246	C₄⋊C₄	—	26	2	no	—
247	C₂²⋊C₄	—	26	2	no	—
248	C₂²⋊C₄	—	26	2	no	—
249	C₄²	—	26	1	no	—
250	C₄²	—	26	1	no	—
251	C₄²	C₂×D₅₂	1	1	yes	characteristic
252	C₄²	#6127	1	2	yes	—
253	C₄²	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
254	C₁₃	#8156	1	1	yes	characteristic, complemented, Sylow
255	C₂³	—	13	2	no	—
256	C₂³	—	26	1	no	—
257	C₂³	—	26	1	no	—
258	C₂³	—	52	2	no	—
259	C₂³	C₄²×D₁₃	1	1	yes	characteristic
260	C₂×C₄	—	4	2	no	—
261	C₂×C₄	—	13	2	no	—
262	C₂×C₄	—	26	1	no	—
263	C₂×C₄	—	26	1	no	—
264	C₂×C₄	—	26	2	no	—
265	C₂×C₄	—	26	2	no	—
266	C₂×C₄	—	26	2	no	—
267	C₂×C₄	—	26	2	no	—
268	C₂×C₄	—	26	2	no	—
269	C₂×C₄	—	52	2	no	—
270	C₂×C₄	#88126	1	1	yes	characteristic
271	C₂×C₄	C₄×D₅₂	1	2	yes	—
272	C₂×C₄	C₂×C₈×D₁₃	1	1	yes	characteristic
273	C₂×C₄	C₂×C₈×D₁₃	1	1	yes	characteristic
274	C₂×C₄	C₂×#6094	1	1	yes	characteristic
275	C₂×C₄	C₂×#6094	1	1	yes	characteristic
276	C₂×C₄	C₂×#6103	1	1	yes	characteristic
277	C₂×C₄	C₄×C₂²⋊D₁₃	1	2	yes	—
278	C₈	—	4	4	no	—
279	C₈	—	26	2	no	—
280	C₈	—	26	2	no	—
281	C₈	—	52	4	no	—
282	C₈	#88060	1	2	yes	—
283	C₈	#88060	1	2	yes	—
284	C₈	#88060	1	2	yes	—
285	C₈	#88060	1	2	yes	—
286	C₂²	—	2	2	no	—
287	C₂²	—	26	2	no	—
288	C₂²	—	52	2	no	—
289	C₂²	—	52	2	no	—
290	C₂²	—	52	2	no	—
291	C₂²	C₄×C₈×D₁₃	1	1	yes	characteristic
292	C₂²	C₄×#6094	1	1	yes	characteristic
293	C₂²	C₄×#6103	1	1	yes	characteristic
294	C₄	—	2	2	no	—
295	C₄	—	2	2	no	—
296	C₄	—	26	2	no	—
297	C₄	—	52	2	no	—
298	C₄	#11718346	1	1	yes	characteristic
299	C₄	#11718346	1	1	yes	characteristic
300	C₄	C₂×#88060	1	1	yes	characteristic
301	C₄	C₂×#88060	1	1	yes	characteristic
302	C₂	—	2	1	no	—
303	C₂	—	2	1	no	—
304	C₂	—	52	2	no	—
305	C₂	C₄×#88060	1	1	yes	characteristic
306	C₂	#13114637	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	#6103
commutator	C₂²×C₁₃	C₂×C₄×C₈
Frattini	C₂×C₄²	C₂²×D₁₃
Fitting	C₁₃×#1450	C₂
socle	C₂²×C₁₃	C₄×C₂²⋊C₄
prime radical	C₂³×D₁₃	C₄²
cosocle	C₂×C₄²×C₁₃	C₂³
hypercenter	#1450	D₁₃
hypocenter	C₁₃	#8156
LCS	G ▷ C₂²×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ #1450

Complements

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

N	H ≅ G/N	type
3. #13113119	C₂	semidirect
4. #13112680	C₂	semidirect
5. C₁₃×#1450	C₂	semidirect
37. C₂×#6094	C₈	semidirect
41. C₂×C₈×D₁₃	C₈	semidirect
105. #1450	D₁₃	semidirect
254. C₁₃	#8156	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11718350	2
#13113148	2
#13113119	2
#13112680	2
C₁₃×#1450	2
#8156	13