Subgroups — #15756052

Order 3328.

466 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718495	C₂	1	1	yes	maximal, characteristic
2	#13114973	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×C₈×D₁₃	—	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×#88060	—	2	1	no	—
14	#11718586	C₂²	1	1	yes	characteristic
15	#11718585	C₂²	1	1	yes	characteristic
16	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
17	#11718495	—	2	1	no	—
18	#11718495	—	2	1	no	—
19	#11718495	C₄	1	2	yes	—
20	#11718495	C₄	1	2	yes	—
21	#11718150	—	2	1	no	—
22	#11718150	—	2	1	no	—
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×C₈×D₁₃	—	2	2	no	—
38	C₂×C₈×D₁₃	—	2	2	no	—
39	C₂×C₈×D₁₃	—	4	2	no	—
40	C₂×C₈×D₁₃	—	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	#88060	—	2	2	no	—
54	#88060	—	2	2	no	—
55	C₄⋊#1055	—	2	1	no	—
56	C₄⋊#1055	—	2	1	no	—
57	C₄⋊#1055	—	2	1	no	—
58	C₄×#1055	—	2	1	no	—
59	C₄×#1055	C₂×C₄	1	1	yes	characteristic
60	C₄×#1055	C₂×C₄	1	1	yes	characteristic
61	C₄×#1055	D₄	1	1	yes	characteristic
62	C₄×#1055	D₄	1	1	yes	characteristic
63	#8660	—	13	1	no	maximal, Sylow
64	C₂³×D₁₃	—	2	1	no	—
65	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₁₃	C₂×Q₈	1	1	yes	characteristic
67	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
68	C₂²×Dic₁₃	—	2	1	no	—
69	C₂×D₅₂	—	2	1	no	—
70	C₂×D₅₂	—	2	4	no	—
71	C₂×D₅₂	C₄⋊C₄	1	2	yes	—
72	C₂×C₄×D₁₃	—	2	2	no	—
73	C₂×C₄×D₁₃	—	2	2	no	—
74	C₂×C₄×D₁₃	—	4	1	no	—
75	C₂×C₄×D₁₃	—	4	1	no	—
76	C₂×C₄×D₁₃	—	4	1	no	—
77	C₂×C₄×D₁₃	—	4	1	no	—
78	C₁₃×M₁₆	—	4	1	no	—
79	C₁₃×M₁₆	—	4	1	no	—
80	C₂×C₈×C₁₃	—	2	2	no	—
81	C₂×C₈×C₁₃	—	4	1	no	—
82	C₂×C₈×C₁₃	—	4	1	no	—
83	C₂×C₈×C₁₃	—	4	1	no	—
84	C₂×C₈×C₁₃	—	4	1	no	—
85	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
86	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
87	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
88	C₄²×C₁₃	#41	1	1	yes	characteristic
89	#6103	—	2	2	no	—
90	#6103	—	2	2	no	—
91	C₄⋊Dic₁₃	—	2	1	no	—
92	C₄⋊Dic₁₃	C₄⋊C₄	1	2	yes	—
93	#6099	—	4	1	no	—
94	#6099	—	4	1	no	—
95	C₂×#1055	—	2	1	no	—
96	C₂×#1055	—	2	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	C₈×D₁₃	—	4	4	no	—
106	C₈×D₁₃	—	4	4	no	—
107	C₂×C₈×D₄	—	13	1	no	—
108	#1837	—	13	1	no	—
109	#1564	—	13	1	no	—
110	#1486	—	13	1	no	—
111	#1486	—	13	1	no	—
112	#1301	—	13	1	no	—
113	#1301	D₁₃	1	1	yes	characteristic, complemented
114	C₂³×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
115	C₂²×D₁₃	—	2	4	no	—
116	C₂²×D₁₃	—	2	2	no	—
117	C₂²×D₁₃	—	4	2	no	—
118	C₂²×D₁₃	—	4	1	no	—
119	C₂²×D₁₃	—	4	1	no	—
120	C₂×C₄×C₁₃	—	2	2	no	—
121	C₂×C₄×C₁₃	—	4	1	no	—
122	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
123	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
124	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
125	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
126	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
127	C₂×C₄×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
128	C₂×C₄×C₁₃	#135	1	1	yes	characteristic
129	C₂×C₄×C₁₃	#135	1	1	yes	characteristic
130	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
131	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
132	C₂×Dic₁₃	—	2	2	no	—
133	C₂×Dic₁₃	—	4	1	no	—
134	C₂×Dic₁₃	—	4	1	no	—
135	D₅₂	—	2	4	no	—
136	D₅₂	—	2	2	no	—
137	D₅₂	#108	1	4	yes	—
138	C₄×D₁₃	—	4	2	no	—
139	C₄×D₁₃	—	4	2	no	—
140	C₄×D₁₃	—	4	2	no	—
141	C₄×D₁₃	—	4	2	no	—
142	C₈×C₁₃	—	4	2	no	—
143	C₈×C₁₃	—	8	1	no	—
144	C₈×C₁₃	—	8	1	no	—
145	#1055	—	2	1	no	—
146	#1055	—	2	1	no	—
147	#1055	—	2	1	no	—
148	#1055	—	2	1	no	—
149	#1055	—	8	1	no	—
150	#1055	—	8	1	no	—
151	C₂³×C₈	—	26	1	no	—
152	C₂×C₄×D₄	—	13	1	no	—
153	C₈×D₄	—	13	2	no	—
154	C₈×D₄	—	13	2	no	—
155	C₈×D₄	—	26	1	no	—
156	C₈×D₄	—	26	1	no	—
157	#431	—	13	1	no	—
158	#431	C₂×D₁₃	1	1	yes	characteristic
159	#423	—	13	1	no	—
160	#423	C₂×D₁₃	1	1	yes	characteristic
161	C₂×C₄⋊C₈	C₂×D₁₃	1	1	yes	characteristic
162	C₂×C₂²⋊C₈	—	26	1	no	—
163	C₂×C₄×C₈	—	13	1	no	—
164	#341	—	26	1	no	—
165	#341	—	26	1	no	—
166	#335	—	26	1	no	—
167	#334	—	26	1	no	—
168	D₄⋊C₈	—	26	2	no	—
169	D₄⋊C₈	—	26	2	no	—
170	C₂²×C₁₃	—	2	1	no	—
171	C₂²×C₁₃	—	2	1	no	—
172	C₂²×C₁₃	#380	1	1	yes	characteristic
173	C₂²×C₁₃	#478	1	1	yes	characteristic
174	C₂²×C₁₃	#478	1	1	yes	characteristic
175	C₂×D₁₃	—	2	4	no	—
176	C₂×D₁₃	—	4	4	no	—
177	C₂×D₁₃	—	4	4	no	—
178	C₂×D₁₃	—	4	2	no	—
179	C₂×D₁₃	—	4	2	no	—
180	C₄×C₁₃	—	2	1	no	—
181	C₄×C₁₃	—	2	1	no	—
182	C₄×C₁₃	—	2	1	no	—
183	C₄×C₁₃	#416	1	1	yes	characteristic
184	C₄×C₁₃	#418	1	1	yes	characteristic
185	C₄×C₁₃	C₂×#108	1	2	yes	—
186	C₄×C₁₃	#463	1	1	yes	characteristic
187	C₄×C₁₃	#463	1	1	yes	characteristic
188	Dic₁₃	—	4	2	no	—
189	C₂²×D₄	—	13	1	no	—
190	C₂³×C₄	—	26	1	no	—
191	C₂×M₁₆	—	2	1	no	—
192	C₂×M₁₆	—	26	1	no	—
193	C₂²×C₈	—	2	1	no	—
194	C₂²×C₈	—	13	1	no	—
195	C₂²×C₈	—	13	1	no	—
196	C₂²×C₈	—	26	2	no	—
197	C₂²×C₈	—	26	2	no	—
198	C₂²×C₈	—	52	2	no	—
199	C₂²×C₈	—	52	2	no	—
200	C₄×D₄	—	13	2	no	—
201	C₄×D₄	—	13	2	no	—
202	C₄×D₄	—	26	1	no	—
203	C₄×D₄	—	26	1	no	—
204	C₂×C₄⋊C₄	—	13	1	no	—
205	C₂×C₂²⋊C₄	—	26	1	no	—
206	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
207	C₄⋊C₈	—	2	1	no	—
208	C₄⋊C₈	—	2	1	no	—
209	C₄⋊C₈	—	2	1	no	—
210	C₄⋊C₈	—	2	1	no	—
211	C₄⋊C₈	—	26	1	no	—
212	C₄⋊C₈	—	26	1	no	—
213	C₄⋊C₈	—	26	1	no	—
214	C₄⋊C₈	C₄×D₁₃	1	2	yes	—
215	C₂²⋊C₈	—	2	1	no	—
216	C₂²⋊C₈	—	26	2	no	—
217	C₂²⋊C₈	—	26	2	no	—
218	C₂²⋊C₈	—	26	1	no	—
219	C₄×C₈	—	2	1	no	—
220	C₄×C₈	—	13	1	no	—
221	C₄×C₈	—	13	1	no	—
222	C₄×C₈	—	13	1	no	—
223	C₄×C₈	—	13	1	no	—
224	C₄×C₈	—	26	1	no	—
225	C₂×C₁₃	—	2	1	no	—
226	C₂×C₁₃	—	2	1	no	—
227	C₂×C₁₃	#1804	1	1	yes	characteristic
228	C₂×C₁₃	#1810	1	1	yes	characteristic
229	C₂×C₁₃	#1814	1	1	yes	characteristic
230	D₁₃	—	4	4	no	—
231	C₂⁴	—	26	1	no	—
232	C₂×D₄	—	13	2	no	—
233	C₂×D₄	—	26	4	no	—
234	C₂×D₄	—	26	1	no	—
235	C₂²×C₄	—	26	1	no	—
236	C₂²×C₄	—	26	2	no	—
237	C₂²×C₄	—	26	2	no	—
238	C₂²×C₄	—	52	1	no	—
239	C₂²×C₄	—	52	1	no	—
240	C₂²×C₄	—	52	1	no	—
241	C₂²×C₄	—	52	1	no	—
242	C₂²×C₄	#6127	1	1	yes	characteristic
243	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
244	C₂²×C₄	Q₈×D₁₃	1	1	yes	characteristic
245	M₁₆	—	4	1	no	—
246	M₁₆	—	4	1	no	—
247	M₁₆	—	52	1	no	—
248	M₁₆	—	52	1	no	—
249	C₂×C₈	—	2	2	no	—
250	C₂×C₈	—	4	1	no	—
251	C₂×C₈	—	4	1	no	—
252	C₂×C₈	—	4	1	no	—
253	C₂×C₈	—	4	1	no	—
254	C₂×C₈	—	13	1	no	—
255	C₂×C₈	—	13	1	no	—
256	C₂×C₈	—	13	1	no	—
257	C₂×C₈	—	13	1	no	—
258	C₂×C₈	—	26	1	no	—
259	C₂×C₈	—	26	1	no	—
260	C₂×C₈	—	26	1	no	—
261	C₂×C₈	—	26	1	no	—
262	C₂×C₈	—	52	4	no	—
263	C₂×C₈	—	52	4	no	—
264	C₂×C₈	—	52	1	no	—
265	C₂×C₈	—	52	1	no	—
266	C₄⋊C₄	—	13	2	no	—
267	C₄⋊C₄	—	26	1	no	—
268	C₂²⋊C₄	—	26	2	no	—
269	C₂²⋊C₄	—	26	2	no	—
270	C₄²	C₂×C₄×D₁₃	1	1	yes	characteristic
271	C₄²	#6127	1	1	yes	characteristic
272	C₄²	D₄×D₁₃	1	1	yes	characteristic
273	C₄²	D₄×D₁₃	1	1	yes	characteristic
274	C₁₃	#8660	1	1	yes	characteristic, complemented, Sylow
275	C₂³	—	26	4	no	—
276	C₂³	—	26	2	no	—
277	C₂³	—	52	2	no	—
278	C₂³	—	52	1	no	—
279	C₂³	—	52	1	no	—
280	C₂³	#88150	1	1	yes	characteristic
281	D₄	—	13	4	no	—
282	D₄	—	26	4	no	—
283	D₄	—	26	2	no	—
284	C₂×C₄	—	2	2	no	—
285	C₂×C₄	—	4	1	no	—
286	C₂×C₄	—	26	2	no	—
287	C₂×C₄	—	52	1	no	—
288	C₂×C₄	—	52	1	no	—
289	C₂×C₄	—	52	2	no	—
290	C₂×C₄	—	52	2	no	—
291	C₂×C₄	—	52	2	no	—
292	C₂×C₄	—	52	2	no	—
293	C₂×C₄	#88126	1	1	yes	characteristic
294	C₂×C₄	C₂²⋊C₄×D₁₃	1	1	yes	characteristic
295	C₂×C₄	#88138	1	1	yes	characteristic
296	C₂×C₄	#88138	1	1	yes	characteristic
297	C₂×C₄	C₄⋊C₄×D₁₃	1	1	yes	characteristic
298	C₂×C₄	#88150	1	1	yes	characteristic
299	C₂×C₄	#88165	1	1	yes	characteristic
300	C₂×C₄	#88168	1	1	yes	characteristic
301	C₂×C₄	#88170	1	1	yes	characteristic
302	C₂×C₄	#88173	1	1	yes	characteristic
303	C₈	—	4	2	no	—
304	C₈	—	8	1	no	—
305	C₈	—	8	1	no	—
306	C₈	—	26	1	no	—
307	C₈	—	26	1	no	—
308	C₈	—	26	1	no	—
309	C₈	—	26	1	no	—
310	C₈	—	104	1	no	—
311	C₈	—	104	1	no	—
312	C₂²	—	2	1	no	—
313	C₂²	—	2	1	no	—
314	C₂²	—	26	4	no	—
315	C₂²	—	52	4	no	—
316	C₂²	—	52	4	no	—
317	C₂²	—	52	2	no	—
318	C₂²	—	52	2	no	—
319	C₂²	#11718328	1	1	yes	characteristic
320	C₂²	#11718532	1	1	yes	characteristic
321	C₂²	#11718552	1	1	yes	characteristic
322	C₄	—	2	1	no	—
323	C₄	—	2	1	no	—
324	C₄	—	2	1	no	—
325	C₄	—	52	2	no	—
326	C₄	#11718431	1	1	yes	characteristic
327	C₄	#11718448	1	1	yes	characteristic
328	C₄	#11718465	1	1	yes	characteristic
329	C₄	#11718480	1	1	yes	characteristic
330	C₄	D₁₃×#108	1	2	yes	—
331	C₂	—	2	1	no	—
332	C₂	—	2	1	no	—
333	C₂	—	52	4	no	—
334	C₂	#13114845	1	1	yes	characteristic
335	C₂	#13114873	1	1	yes	characteristic
336	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	#8660
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. #13114973	C₂	semidirect
3. #13113582	C₂	semidirect
6. #13112733	C₂	semidirect
7. C₁₃×#1301	C₂	semidirect
113. #1301	D₁₃	semidirect
274. C₁₃	#8660	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11718495	2
#13114973	2
#13113582	2
#13113262	2
#13113260	2
#13112733	2
C₁₃×#1301	2
#8660	13