Subgroups — #15753727

Order 3328.

570 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×D₅₂	C₂	1	1	yes	maximal, characteristic
2	C₄×#88049	C₂	1	1	yes	maximal, characteristic
3	C₂×#11718075	C₂	1	1	yes	maximal, characteristic
4	C₂×#11718046	C₂	1	1	yes	maximal, characteristic, complemented
5	#13112677	C₂	1	1	yes	maximal, characteristic, complemented
6	#13112670	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₃×#1770	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×C₈×D₁₃	—	2	1	no	—
9	C₂×C₄×D₅₂	C₂²	1	1	yes	characteristic
10	C₂×C₁₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
11	C₂×C₁₃×#97	—	2	1	no	—
12	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
13	C₄×C₁₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
14	C₂×#88060	—	2	1	no	—
15	C₂×#88049	C₄	1	2	yes	—
16	C₂×C₄⋊#1055	C₂²	1	1	yes	characteristic
17	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
18	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
19	C₈×D₅₂	—	2	2	no	—
20	C₈×D₅₂	C₄	1	4	yes	complemented
21	C₁₃×#336	—	2	1	no	—
22	#11718121	C₄	1	2	yes	—
23	#11718075	C₄	1	4	yes	complemented
24	#11718046	—	2	2	no	—
25	C₂²×D₅₂	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₄×D₁₃	—	2	1	no	—
27	C₂²×C₈×C₁₃	—	2	1	no	—
28	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
29	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
30	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
31	C₂×C₄²×C₁₃	—	2	1	no	—
32	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
33	C₂×#6103	—	2	1	no	—
34	C₂×C₄⋊Dic₁₃	C₂×C₄	1	1	yes	characteristic
35	C₂²×#1055	—	2	1	no	—
36	C₂²×#1055	—	2	1	no	—
37	C₂²×#1055	C₂×C₄	1	2	yes	—
38	C₂×C₈×D₁₃	—	2	4	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	C₂×C₄	1	2	yes	—
45	C₁₃×C₄⋊C₈	—	2	1	no	—
46	C₁₃×C₄⋊C₈	C₂×C₄	1	2	yes	—
47	C₁₃×#97	—	4	2	no	—
48	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
49	C₄×C₈×C₁₃	D₄	1	2	yes	—
50	C₁₃×#95	—	2	1	no	—
51	#88060	—	2	4	no	—
52	#88049	—	2	4	no	—
53	C₄⋊#1055	—	2	1	no	—
54	C₄⋊#1055	C₂×C₄	1	2	yes	—
55	C₄×#1055	—	2	1	no	—
56	C₄×#1055	—	2	1	no	—
57	C₄×#1055	—	2	1	no	—
58	C₄×#1055	C₂×C₄	1	2	yes	—
59	#8204	—	13	1	no	maximal, Sylow
60	C₂³×D₁₃	—	2	1	no	—
61	C₂²×C₄×C₁₃	—	2	2	no	—
62	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
63	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
64	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
65	C₂²×Dic₁₃	—	2	1	no	—
66	C₂×D₅₂	—	2	1	no	—
67	C₂×D₅₂	—	2	2	no	—
68	C₂×D₅₂	C₄²	1	2	yes	—
69	C₂×D₅₂	M₁₆	1	4	yes	—
70	C₂×C₄×D₁₃	—	2	2	no	—
71	C₂×C₄×D₁₃	—	2	2	no	—
72	C₂×C₄×D₁₃	—	4	1	no	—
73	C₂×C₄×D₁₃	—	4	1	no	—
74	C₂×C₄×D₁₃	—	4	1	no	—
75	C₂×C₄×D₁₃	—	4	1	no	—
76	C₂×C₈×C₁₃	—	2	1	no	—
77	C₂×C₈×C₁₃	—	2	1	no	—
78	C₂×C₈×C₁₃	—	2	1	no	—
79	C₂×C₈×C₁₃	—	2	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	C₂²⋊C₄	1	2	yes	—
84	C₂×C₈×C₁₃	C₂²⋊C₄	1	2	yes	—
85	C₁₃×C₄⋊C₄	—	2	2	no	—
86	C₁₃×C₄⋊C₄	—	2	2	no	—
87	C₄²×C₁₃	—	4	1	no	—
88	C₄²×C₁₃	—	4	1	no	—
89	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
90	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
91	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
92	C₄²×C₁₃	#41	1	1	yes	characteristic
93	#6103	—	2	2	no	—
94	#6103	—	2	2	no	—
95	C₄⋊Dic₁₃	—	2	1	no	—
96	C₄⋊Dic₁₃	C₄²	1	2	yes	—
97	C₂×#1055	—	2	4	no	—
98	C₂×#1055	—	2	2	no	—
99	C₂×#1055	—	2	2	no	—
100	C₂×#1055	—	2	2	no	—
101	C₂×#1055	—	2	2	no	—
102	C₂×#1055	—	2	2	no	—
103	C₂×#1055	—	4	1	no	—
104	C₂×#1055	—	4	1	no	—
105	C₈×D₁₃	—	4	8	no	—
106	C₂×C₈×D₄	—	13	1	no	—
107	#1770	D₁₃	1	1	yes	characteristic, complemented
108	C₄×#102	—	13	1	no	—
109	C₂×D₄⋊C₈	—	13	1	no	—
110	C₂×#322	—	13	1	no	—
111	#1279	—	13	1	no	—
112	#1277	—	13	1	no	—
113	C₂³×C₁₃	C₄×D₄	1	1	yes	characteristic
114	C₂²×D₁₃	—	2	2	no	—
115	C₂²×D₁₃	—	2	4	no	—
116	C₂²×D₁₃	—	4	1	no	—
117	C₂²×D₁₃	—	4	1	no	—
118	C₂²×D₁₃	—	4	2	no	—
119	C₂×C₄×C₁₃	—	2	2	no	—
120	C₂×C₄×C₁₃	—	4	2	no	—
121	C₂×C₄×C₁₃	—	4	2	no	—
122	C₂×C₄×C₁₃	—	4	2	no	—
123	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
124	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
125	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
126	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
127	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
128	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
129	C₂×C₄×C₁₃	C₂×M₁₆	1	2	yes	—
130	C₂×C₄×C₁₃	#131	1	2	yes	—
131	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
132	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
133	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
134	C₂×C₄×C₁₃	#137	1	1	yes	characteristic
135	C₂×Dic₁₃	—	2	2	no	—
136	C₂×Dic₁₃	—	4	1	no	—
137	C₂×Dic₁₃	—	4	1	no	—
138	D₅₂	—	2	2	no	—
139	D₅₂	—	2	4	no	—
140	D₅₂	#97	1	4	yes	complemented
141	C₄×D₁₃	—	4	2	no	—
142	C₄×D₁₃	—	4	2	no	—
143	C₄×D₁₃	—	4	2	no	—
144	C₄×D₁₃	—	4	2	no	—
145	C₈×C₁₃	—	2	2	no	—
146	C₈×C₁₃	—	2	2	no	—
147	C₈×C₁₃	—	4	2	no	—
148	#1055	—	2	4	no	—
149	#1055	—	4	2	no	—
150	#1055	—	4	2	no	—
151	C₂³×C₈	—	26	1	no	—
152	C₂×C₄×D₄	—	13	1	no	—
153	C₈×D₄	—	13	4	no	—
154	C₈×D₄	—	26	2	no	—
155	C₂×C₄⋊C₈	—	13	1	no	—
156	C₂×C₄⋊C₈	C₂×D₁₃	1	1	yes	characteristic
157	C₂×#102	—	13	2	no	—
158	C₂×C₂²⋊C₈	—	26	1	no	—
159	C₂×#97	—	2	1	no	—
160	C₂×C₄×C₈	—	13	1	no	—
161	C₂×C₄×C₈	—	13	1	no	—
162	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
163	C₄×C₄⋊C₄	C₂×D₁₃	1	1	yes	characteristic
164	#340	—	13	2	no	—
165	#336	—	2	1	no	—
166	D₄⋊C₈	—	13	4	no	—
167	#322	—	26	2	no	—
168	C₂²×C₁₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
169	C₂²×C₁₃	#435	1	2	yes	—
170	C₂²×C₁₃	#440	1	1	yes	characteristic
171	C₂²×C₁₃	#442	1	1	yes	characteristic
172	C₂²×C₁₃	#443	1	2	yes	—
173	C₂×D₁₃	—	2	4	no	—
174	C₂×D₁₃	—	4	4	no	—
175	C₂×D₁₃	—	4	4	no	—
176	C₂×D₁₃	—	4	2	no	—
177	C₂×D₁₃	—	4	2	no	—
178	C₄×C₁₃	—	2	1	no	—
179	C₄×C₁₃	—	2	1	no	—
180	C₄×C₁₃	—	8	2	no	—
181	C₄×C₁₃	C₂×#97	1	1	yes	characteristic
182	C₄×C₁₃	#405	1	1	yes	characteristic
183	C₄×C₁₃	#417	1	1	yes	characteristic
184	C₄×C₁₃	#418	1	1	yes	characteristic
185	C₄×C₁₃	#435	1	2	yes	—
186	C₄×C₁₃	#440	1	1	yes	characteristic
187	C₄×C₁₃	#442	1	1	yes	characteristic
188	Dic₁₃	—	4	2	no	—
189	C₂²×D₄	—	13	1	no	—
190	C₂³×C₄	—	26	1	no	—
191	C₂²×C₈	—	2	1	no	—
192	C₂²×C₈	—	13	2	no	—
193	C₂²×C₈	—	26	1	no	—
194	C₂²×C₈	—	26	1	no	—
195	C₂²×C₈	—	26	4	no	—
196	C₂²×C₈	—	52	2	no	—
197	C₂²×C₈	—	52	2	no	—
198	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
199	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
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×C₄⋊C₄	C₄×D₁₃	1	2	yes	—
206	C₂×C₂²⋊C₄	—	26	1	no	—
207	C₂×C₄²	—	2	1	no	—
208	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
209	C₄⋊C₈	—	2	1	no	—
210	C₄⋊C₈	—	13	2	no	—
211	C₄⋊C₈	—	26	1	no	—
212	C₄⋊C₈	C₄×D₁₃	1	2	yes	—
213	#102	—	26	4	no	—
214	C₂²⋊C₈	—	26	4	no	—
215	#97	—	4	2	no	—
216	C₄×C₈	—	13	2	no	—
217	C₄×C₈	—	26	1	no	—
218	C₄×C₈	—	26	1	no	—
219	C₄×C₈	—	26	1	no	—
220	C₄×C₈	C₄×D₁₃	1	2	yes	—
221	C₄×C₈	D₅₂	1	2	yes	—
222	#95	—	2	1	no	—
223	C₂×C₁₃	#1585	1	2	yes	—
224	C₂×C₁₃	#1586	1	2	yes	—
225	C₂×C₁₃	#1752	1	1	yes	characteristic
226	C₂×C₁₃	#1762	1	1	yes	characteristic
227	C₂×C₁₃	#1764	1	1	yes	characteristic
228	D₁₃	—	4	4	no	—
229	C₂⁴	—	26	1	no	—
230	C₂×D₄	—	13	2	no	—
231	C₂×D₄	—	13	4	no	—
232	C₂×D₄	—	26	1	no	—
233	C₂×D₄	—	26	2	no	—
234	C₂²×C₄	—	2	2	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	C₂×C₄×D₁₃	1	1	yes	characteristic
243	C₂²×C₄	#6127	1	1	yes	characteristic
244	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
245	C₂×C₈	—	2	1	no	—
246	C₂×C₈	—	2	1	no	—
247	C₂×C₈	—	2	1	no	—
248	C₂×C₈	—	2	1	no	—
249	C₂×C₈	—	2	2	no	—
250	C₂×C₈	—	4	1	no	—
251	C₂×C₈	—	4	1	no	—
252	C₂×C₈	—	26	4	no	—
253	C₂×C₈	—	26	2	no	—
254	C₂×C₈	—	26	2	no	—
255	C₂×C₈	—	26	2	no	—
256	C₂×C₈	—	26	2	no	—
257	C₂×C₈	—	26	2	no	—
258	C₂×C₈	—	52	8	no	—
259	C₂×C₈	—	52	1	no	—
260	C₂×C₈	—	52	1	no	—
261	C₂×C₈	#6103	1	2	yes	—
262	C₂×C₈	#6103	1	2	yes	—
263	C₄⋊C₄	—	2	2	no	—
264	C₄⋊C₄	—	2	2	no	—
265	C₄⋊C₄	—	13	2	no	—
266	C₄⋊C₄	—	26	1	no	—
267	C₂²⋊C₄	—	26	2	no	—
268	C₂²⋊C₄	—	26	2	no	—
269	C₄²	—	4	1	no	—
270	C₄²	—	4	1	no	—
271	C₄²	C₂×C₄×D₁₃	1	1	yes	characteristic
272	C₄²	C₂×C₄×D₁₃	1	1	yes	characteristic
273	C₄²	C₂×D₅₂	1	1	yes	characteristic
274	C₄²	#6127	1	1	yes	characteristic
275	C₁₃	#8204	1	1	yes	characteristic, complemented, Sylow
276	C₂³	—	26	4	no	—
277	C₂³	—	26	2	no	—
278	C₂³	—	52	1	no	—
279	C₂³	—	52	1	no	—
280	C₂³	—	52	2	no	—
281	C₂³	C₄×C₂²⋊D₁₃	1	1	yes	characteristic
282	D₄	—	13	4	no	—
283	D₄	—	26	4	no	—
284	D₄	—	26	2	no	—
285	C₂×C₄	—	2	2	no	—
286	C₂×C₄	—	4	2	no	—
287	C₂×C₄	—	4	2	no	—
288	C₂×C₄	—	4	2	no	—
289	C₂×C₄	—	26	2	no	—
290	C₂×C₄	—	52	1	no	—
291	C₂×C₄	—	52	1	no	—
292	C₂×C₄	—	52	2	no	—
293	C₂×C₄	—	52	2	no	—
294	C₂×C₄	—	52	2	no	—
295	C₂×C₄	—	52	2	no	—
296	C₂×C₄	C₄²×D₁₃	1	1	yes	characteristic
297	C₂×C₄	#88126	1	1	yes	characteristic
298	C₂×C₄	C₄×D₅₂	1	1	yes	characteristic
299	C₂×C₄	C₄×D₅₂	1	1	yes	characteristic
300	C₂×C₄	M₁₆×D₁₃	1	2	yes	—
301	C₂×C₄	#88161	1	2	yes	—
302	C₂×C₄	C₂×#6103	1	1	yes	characteristic
303	C₂×C₄	C₄×C₂²⋊D₁₃	1	1	yes	characteristic
304	C₂×C₄	#88186	1	1	yes	characteristic
305	C₂×C₄	#88196	1	1	yes	characteristic
306	C₂×C₄	#88203	1	1	yes	characteristic
307	C₂×C₄	#88203	1	1	yes	characteristic
308	C₈	—	2	2	no	—
309	C₈	—	2	2	no	—
310	C₈	—	4	2	no	—
311	C₈	—	26	4	no	—
312	C₈	—	52	2	no	—
313	C₈	—	52	2	no	—
314	C₂²	—	26	4	no	—
315	C₂²	—	52	2	no	—
316	C₂²	—	52	2	no	—
317	C₂²	—	52	4	no	—
318	C₂²	—	52	4	no	—
319	C₂²	C₄×#6103	1	1	yes	characteristic
320	C₂²	#11718675	1	1	yes	characteristic
321	C₂²	#11718687	1	1	yes	characteristic
322	C₂²	#11718788	1	2	yes	—
323	C₂²	#11718805	1	2	yes	—
324	C₄	—	2	1	no	—
325	C₄	—	2	1	no	—
326	C₄	—	8	2	no	—
327	C₄	—	52	2	no	—
328	C₄	D₁₃×#97	1	1	yes	characteristic
329	C₄	#11718367	1	2	yes	—
330	C₄	#11718369	1	1	yes	characteristic
331	C₄	#11718627	1	1	yes	characteristic
332	C₄	#11718662	1	1	yes	characteristic
333	C₄	#11718675	1	1	yes	characteristic
334	C₄	#11718687	1	1	yes	characteristic
335	C₂	—	52	4	no	—
336	C₂	#13113691	1	2	yes	—
337	C₂	#13113698	1	2	yes	—
338	C₂	#13114613	1	1	yes	characteristic
339	C₂	#13114664	1	1	yes	characteristic
340	C₂	#13114675	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₁₃
commutator	C₂×C₄×C₁₃	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₁₃
Fitting	C₁₃×#1770	C₂
socle	C₂³×C₁₃	C₄×D₄
prime radical	C₂²×D₅₂	C₂×C₄
cosocle	C₂×C₄²×C₁₃	C₂³
hypercenter	#1770	D₁₃
hypocenter	C₁₃	#8204
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ #1770

Complements

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

N	H ≅ G/N	type
4. C₂×#11718046	C₂	semidirect
5. #13112677	C₂	semidirect
6. #13112670	C₂	semidirect
7. C₁₃×#1770	C₂	semidirect
20. C₈×D₅₂	C₄	semidirect
23. #11718075	C₄	semidirect
107. #1770	D₁₃	semidirect
140. D₅₂	#97	semidirect
275. C₁₃	#8204	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₈×D₅₂	2
C₄×#88049	2
C₂×#11718075	2
C₂×#11718046	2
#13112677	2
#13112670	2
C₁₃×#1770	2
#8204	13