Subgroups — #15756062

Order 3328.

450 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718499	C₂	1	1	yes	maximal, characteristic
2	#13115007	C₂	1	1	yes	maximal, characteristic, complemented
3	#13113582	C₂	1	1	yes	maximal, characteristic, complemented
4	#13113262	C₂	1	1	yes	maximal, characteristic
5	#13113260	C₂	1	1	yes	maximal, characteristic
6	#13112733	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₃×#1301	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×#6094	—	2	1	no	—
9	C₂×C₄×D₅₂	C₂²	1	1	yes	characteristic
10	C₁₃×#431	C₂²	1	1	yes	characteristic
11	C₁₃×#423	C₂²	1	1	yes	characteristic
12	C₂×C₁₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
13	C₂×#88062	—	2	1	no	—
14	C₂×#88062	—	2	1	no	—
15	C₂×#88060	—	2	1	no	—
16	#11718586	C₂²	1	1	yes	characteristic
17	#11718585	C₂²	1	1	yes	characteristic
18	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
19	#11718499	—	2	1	no	—
20	#11718499	—	2	1	no	—
21	#11718499	C₄	1	2	yes	—
22	#11718499	C₄	1	2	yes	—
23	#11718075	—	2	2	no	—
24	#11718072	—	2	1	no	—
25	#11718071	—	2	1	no	—
26	#11718051	—	2	2	no	—
27	C₂²×D₅₂	Q₈	1	1	yes	characteristic
28	C₂²×C₄×D₁₃	—	2	1	no	—
29	C₂×C₁₃×M₁₆	—	2	1	no	—
30	C₂²×C₈×C₁₃	—	2	1	no	—
31	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
32	C₂×#6103	—	2	1	no	—
33	C₂×C₄⋊Dic₁₃	Q₈	1	1	yes	characteristic
34	C₂×#6099	—	2	1	no	—
35	C₂²×#1055	D₄	1	1	yes	characteristic
36	C₂²×#1055	D₄	1	1	yes	characteristic
37	C₂×#6094	—	2	2	no	—
38	C₂×#6094	—	2	2	no	—
39	C₂×#6094	—	4	2	no	—
40	C₂×#6094	—	4	2	no	—
41	C₄×D₅₂	—	2	1	no	—
42	C₄×D₅₂	—	2	1	no	—
43	C₄×D₅₂	C₂×C₄	1	2	yes	—
44	C₄×D₅₂	D₄	1	2	yes	—
45	C₁₃×C₄⋊C₈	—	2	1	no	—
46	C₁₃×C₄⋊C₈	—	2	1	no	—
47	C₁₃×C₄⋊C₈	—	2	1	no	—
48	C₁₃×C₄⋊C₈	—	2	1	no	—
49	C₁₃×C₄⋊C₈	C₂×C₄	1	2	yes	—
50	C₁₃×C₂²⋊C₈	—	2	1	no	—
51	C₄×C₈×C₁₃	—	2	1	no	—
52	C₂²⋊#1055	—	2	1	no	—
53	#88062	—	4	2	no	—
54	#88062	—	4	2	no	—
55	#88060	—	2	2	no	—
56	#88060	—	2	2	no	—
57	C₄⋊#1055	—	2	1	no	—
58	C₄⋊#1055	—	2	1	no	—
59	C₄⋊#1055	—	2	1	no	—
60	C₄×#1055	—	2	1	no	—
61	C₄×#1055	C₂×C₄	1	1	yes	characteristic
62	C₄×#1055	C₂×C₄	1	1	yes	characteristic
63	C₄×#1055	D₄	1	1	yes	characteristic
64	C₄×#1055	D₄	1	1	yes	characteristic
65	#8662	—	13	1	no	maximal, Sylow
66	C₂³×D₁₃	—	2	1	no	—
67	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₁₃	C₂×Q₈	1	1	yes	characteristic
69	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
70	C₂²×Dic₁₃	—	2	1	no	—
71	C₂×D₅₂	—	2	1	no	—
72	C₂×D₅₂	—	4	2	no	—
73	C₂×D₅₂	C₄⋊C₄	1	2	yes	—
74	C₂×C₄×D₁₃	—	2	2	no	—
75	C₂×C₄×D₁₃	—	2	2	no	—
76	C₂×C₄×D₁₃	—	4	1	no	—
77	C₂×C₄×D₁₃	—	4	1	no	—
78	C₂×C₄×D₁₃	—	4	1	no	—
79	C₂×C₄×D₁₃	—	4	1	no	—
80	C₁₃×M₁₆	—	4	1	no	—
81	C₁₃×M₁₆	—	4	1	no	—
82	C₂×C₈×C₁₃	—	2	2	no	—
83	C₂×C₈×C₁₃	—	4	1	no	—
84	C₂×C₈×C₁₃	—	4	1	no	—
85	C₂×C₈×C₁₃	—	4	1	no	—
86	C₂×C₈×C₁₃	—	4	1	no	—
87	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
88	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
89	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
90	C₄²×C₁₃	#41	1	1	yes	characteristic
91	#6103	—	2	2	no	—
92	#6103	—	2	2	no	—
93	C₄⋊Dic₁₃	—	2	1	no	—
94	C₄⋊Dic₁₃	C₄⋊C₄	1	2	yes	—
95	#6099	—	4	1	no	—
96	#6099	—	4	1	no	—
97	C₂×#1055	—	2	1	no	—
98	C₂×#1055	—	2	1	no	—
99	C₂×#1055	—	4	1	no	—
100	C₂×#1055	—	4	1	no	—
101	C₂×#1055	C₂²⋊C₄	1	1	yes	characteristic
102	C₂×#1055	C₂²⋊C₄	1	1	yes	characteristic
103	C₂×#1055	C₂²⋊C₄	1	1	yes	characteristic
104	C₂×#1055	C₂²⋊C₄	1	1	yes	characteristic
105	C₂×#1055	D₈	1	2	yes	—
106	C₂×#1055	SD₁₆	1	2	yes	—
107	#6094	—	4	4	no	—
108	#6094	—	4	4	no	—
109	C₂×#436	—	13	1	no	—
110	#1848	—	13	1	no	—
111	#1564	—	13	1	no	—
112	#1486	—	13	1	no	—
113	#1486	—	13	1	no	—
114	#1301	—	13	1	no	—
115	#1301	D₁₃	1	1	yes	characteristic, complemented
116	C₂³×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
117	C₂²×D₁₃	—	2	2	no	—
118	C₂²×D₁₃	—	4	2	no	—
119	C₂²×D₁₃	—	4	2	no	—
120	C₂²×D₁₃	—	4	1	no	—
121	C₂²×D₁₃	—	4	1	no	—
122	C₂×C₄×C₁₃	—	2	2	no	—
123	C₂×C₄×C₁₃	—	4	1	no	—
124	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
125	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
126	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
127	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
128	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
129	C₂×C₄×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
130	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
131	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
132	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
133	C₂×C₄×C₁₃	#137	1	1	yes	characteristic
134	C₂×Dic₁₃	—	2	2	no	—
135	C₂×Dic₁₃	—	4	1	no	—
136	C₂×Dic₁₃	—	4	1	no	—
137	D₅₂	—	2	4	no	—
138	D₅₂	—	4	2	no	—
139	C₄×D₁₃	—	4	2	no	—
140	C₄×D₁₃	—	4	2	no	—
141	C₄×D₁₃	—	4	2	no	—
142	C₄×D₁₃	—	4	2	no	—
143	C₈×C₁₃	—	4	2	no	—
144	C₈×C₁₃	—	8	1	no	—
145	C₈×C₁₃	—	8	1	no	—
146	#1055	—	2	1	no	—
147	#1055	—	2	1	no	—
148	#1055	—	8	1	no	—
149	#1055	—	8	1	no	—
150	#1055	#102	1	2	yes	—
151	#1055	#102	1	2	yes	—
152	C₂²×M₁₆	—	26	1	no	—
153	C₂×C₄×D₄	—	13	1	no	—
154	#436	—	13	2	no	—
155	#436	—	13	2	no	—
156	#436	—	26	1	no	—
157	#436	—	26	1	no	—
158	#431	—	13	1	no	—
159	#431	C₂×D₁₃	1	1	yes	characteristic
160	C₂×#108	—	26	1	no	—
161	C₂×#108	—	26	1	no	—
162	#423	—	13	1	no	—
163	#423	C₂×D₁₃	1	1	yes	characteristic
164	C₂×C₄⋊C₈	C₂×D₁₃	1	1	yes	characteristic
165	C₂×C₂²⋊C₈	—	26	1	no	—
166	C₂×C₄×C₈	—	13	1	no	—
167	#335	—	26	1	no	—
168	#334	—	26	1	no	—
169	D₄⋊C₈	—	26	2	no	—
170	D₄⋊C₈	—	26	2	no	—
171	C₂²×C₁₃	—	2	1	no	—
172	C₂²×C₁₃	—	2	1	no	—
173	C₂²×C₁₃	#380	1	1	yes	characteristic
174	C₂²×C₁₃	#474	1	1	yes	characteristic
175	C₂²×C₁₃	#476	1	1	yes	characteristic
176	C₂×D₁₃	—	4	2	no	—
177	C₂×D₁₃	—	4	2	no	—
178	C₂×D₁₃	—	4	2	no	—
179	C₂×D₁₃	—	8	2	no	—
180	C₂×D₁₃	—	8	2	no	—
181	C₄×C₁₃	—	2	1	no	—
182	C₄×C₁₃	—	2	1	no	—
183	C₄×C₁₃	—	2	1	no	—
184	C₄×C₁₃	C₂×#102	1	1	yes	characteristic
185	C₄×C₁₃	#417	1	1	yes	characteristic
186	C₄×C₁₃	#430	1	2	yes	—
187	C₄×C₁₃	#459	1	1	yes	characteristic
188	C₄×C₁₃	#461	1	1	yes	characteristic
189	Dic₁₃	—	4	2	no	—
190	C₂²×D₄	—	13	1	no	—
191	C₂³×C₄	—	26	1	no	—
192	C₂×M₁₆	—	2	1	no	—
193	C₂×M₁₆	—	26	2	no	—
194	C₂×M₁₆	—	26	2	no	—
195	C₂×M₁₆	—	26	1	no	—
196	C₂×M₁₆	—	52	2	no	—
197	C₂×M₁₆	—	52	2	no	—
198	C₂²×C₈	—	2	1	no	—
199	C₂²×C₈	—	13	1	no	—
200	C₂²×C₈	—	13	1	no	—
201	C₄×D₄	—	13	2	no	—
202	C₄×D₄	—	13	2	no	—
203	C₄×D₄	—	26	1	no	—
204	C₄×D₄	—	26	1	no	—
205	C₂×C₄⋊C₄	—	13	1	no	—
206	C₂×C₂²⋊C₄	—	26	1	no	—
207	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
208	#108	—	52	2	no	—
209	#108	—	52	2	no	—
210	C₄⋊C₈	—	2	1	no	—
211	C₄⋊C₈	—	2	1	no	—
212	C₄⋊C₈	—	2	1	no	—
213	C₄⋊C₈	—	2	1	no	—
214	C₄⋊C₈	—	26	1	no	—
215	C₄⋊C₈	—	26	1	no	—
216	C₄⋊C₈	—	26	1	no	—
217	C₄⋊C₈	C₄×D₁₃	1	2	yes	—
218	C₂²⋊C₈	—	2	1	no	—
219	C₂²⋊C₈	—	26	2	no	—
220	C₂²⋊C₈	—	26	2	no	—
221	C₂²⋊C₈	—	26	1	no	—
222	C₄×C₈	—	2	1	no	—
223	C₄×C₈	—	13	1	no	—
224	C₄×C₈	—	13	1	no	—
225	C₄×C₈	—	13	1	no	—
226	C₄×C₈	—	13	1	no	—
227	C₄×C₈	—	26	1	no	—
228	C₂×C₁₃	—	2	1	no	—
229	C₂×C₁₃	—	2	1	no	—
230	C₂×C₁₃	#1802	1	1	yes	characteristic
231	C₂×C₁₃	#1811	1	1	yes	characteristic
232	C₂×C₁₃	#1814	1	1	yes	characteristic
233	D₁₃	—	8	2	no	—
234	C₂⁴	—	26	1	no	—
235	C₂×D₄	—	13	2	no	—
236	C₂×D₄	—	26	1	no	—
237	C₂×D₄	—	52	2	no	—
238	C₂²×C₄	—	26	1	no	—
239	C₂²×C₄	—	26	2	no	—
240	C₂²×C₄	—	26	2	no	—
241	C₂²×C₄	—	52	1	no	—
242	C₂²×C₄	—	52	1	no	—
243	C₂²×C₄	—	52	1	no	—
244	C₂²×C₄	—	52	1	no	—
245	C₂²×C₄	#6127	1	1	yes	characteristic
246	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
247	C₂²×C₄	Q₈×D₁₃	1	1	yes	characteristic
248	M₁₆	—	4	1	no	—
249	M₁₆	—	4	1	no	—
250	M₁₆	—	52	4	no	—
251	M₁₆	—	52	4	no	—
252	M₁₆	—	52	1	no	—
253	M₁₆	—	52	1	no	—
254	C₂×C₈	—	2	2	no	—
255	C₂×C₈	—	4	1	no	—
256	C₂×C₈	—	4	1	no	—
257	C₂×C₈	—	4	1	no	—
258	C₂×C₈	—	4	1	no	—
259	C₂×C₈	—	13	1	no	—
260	C₂×C₈	—	13	1	no	—
261	C₂×C₈	—	13	2	no	—
262	C₂×C₈	—	13	2	no	—
263	C₂×C₈	—	13	1	no	—
264	C₂×C₈	—	13	1	no	—
265	C₂×C₈	—	26	1	no	—
266	C₂×C₈	—	26	1	no	—
267	C₂×C₈	—	52	1	no	—
268	C₂×C₈	—	52	1	no	—
269	C₄⋊C₄	—	13	2	no	—
270	C₄⋊C₄	—	26	1	no	—
271	C₂²⋊C₄	—	26	2	no	—
272	C₂²⋊C₄	—	26	2	no	—
273	C₄²	C₂×C₄×D₁₃	1	1	yes	characteristic
274	C₄²	#6127	1	1	yes	characteristic
275	C₄²	D₄×D₁₃	1	1	yes	characteristic
276	C₄²	D₄×D₁₃	1	1	yes	characteristic
277	C₁₃	#8662	1	1	yes	characteristic, complemented, Sylow
278	C₂³	—	26	2	no	—
279	C₂³	—	52	2	no	—
280	C₂³	—	52	2	no	—
281	C₂³	—	52	1	no	—
282	C₂³	—	52	1	no	—
283	C₂³	#88150	1	1	yes	characteristic
284	D₄	—	26	4	no	—
285	D₄	—	52	2	no	—
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	4	1	no	—
288	C₂×C₄	—	26	2	no	—
289	C₂×C₄	—	52	1	no	—
290	C₂×C₄	—	52	1	no	—
291	C₂×C₄	—	52	2	no	—
292	C₂×C₄	—	52	2	no	—
293	C₂×C₄	—	52	2	no	—
294	C₂×C₄	—	52	2	no	—
295	C₂×C₄	#88126	1	1	yes	characteristic
296	C₂×C₄	C₂²⋊C₄×D₁₃	1	1	yes	characteristic
297	C₂×C₄	#88138	1	1	yes	characteristic
298	C₂×C₄	#88138	1	1	yes	characteristic
299	C₂×C₄	C₄⋊C₄×D₁₃	1	1	yes	characteristic
300	C₂×C₄	#88150	1	1	yes	characteristic
301	C₂×C₄	D₈×D₁₃	1	1	yes	characteristic
302	C₂×C₄	SD₁₆×D₁₃	1	1	yes	characteristic
303	C₂×C₄	#88168	1	1	yes	characteristic
304	C₂×C₄	#88172	1	1	yes	characteristic
305	C₈	—	4	2	no	—
306	C₈	—	8	1	no	—
307	C₈	—	8	1	no	—
308	C₈	—	13	2	no	—
309	C₈	—	13	2	no	—
310	C₈	—	26	1	no	—
311	C₈	—	26	1	no	—
312	C₈	—	104	1	no	—
313	C₈	—	104	1	no	—
314	C₂²	—	2	1	no	—
315	C₂²	—	2	1	no	—
316	C₂²	—	52	2	no	—
317	C₂²	—	52	2	no	—
318	C₂²	—	52	2	no	—
319	C₂²	—	104	2	no	—
320	C₂²	—	104	2	no	—
321	C₂²	#11718328	1	1	yes	characteristic
322	C₂²	#11718531	1	1	yes	characteristic
323	C₂²	#11718551	1	1	yes	characteristic
324	C₄	—	2	1	no	—
325	C₄	—	2	1	no	—
326	C₄	—	2	1	no	—
327	C₄	—	52	2	no	—
328	C₄	D₁₃×#102	1	1	yes	characteristic
329	C₄	#11718447	1	1	yes	characteristic
330	C₄	#11718464	1	1	yes	characteristic
331	C₄	#11718479	1	1	yes	characteristic
332	C₄	#11718554	1	2	yes	—
333	C₂	—	2	1	no	—
334	C₂	—	2	1	no	—
335	C₂	—	104	2	no	—
336	C₂	#13114835	1	1	yes	characteristic
337	C₂	#13114877	1	1	yes	characteristic
338	C₂	#13114889	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#88138
commutator	C₄²×C₁₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₃
Fitting	C₁₃×#1301	C₂
socle	C₂²×C₁₃	#380
prime radical	C₂²×D₅₂	Q₈
cosocle	C₂×C₄²×C₁₃	C₂³
hypercenter	#1301	D₁₃
hypocenter	C₁₃	#8662
LCS	G ▷ C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1301

Complements

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

N	H ≅ G/N	type
2. #13115007	C₂	semidirect
3. #13113582	C₂	semidirect
6. #13112733	C₂	semidirect
7. C₁₃×#1301	C₂	semidirect
115. #1301	D₁₃	semidirect
277. C₁₃	#8662	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11718499	2
#13115007	2
#13113582	2
#13113262	2
#13113260	2
#13112733	2
C₁₃×#1301	2
#8662	13