Subgroups — #15828513

Order 3328.

890 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈×#6127	C₂	1	1	yes	maximal, characteristic
2	#13123792	C₂	1	1	yes	maximal, characteristic
3	C₄×#88161	C₂	1	1	yes	maximal, characteristic
4	#13113120	C₂	1	1	yes	maximal, characteristic, complemented
5	#13113118	C₂	1	2	yes	maximal, complemented
6	#13113115	C₂	1	2	yes	maximal, complemented
7	M₁₆×#1055	C₂	1	1	yes	maximal, characteristic, complemented
8	C₈×#6094	C₂	1	2	yes	maximal, complemented
9	C₂×#11718046	C₂	1	1	yes	maximal, characteristic, complemented
10	D₁₃×#322	C₂	1	2	yes	maximal, complemented
11	C₈×C₁₃×M₁₆	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#88161	C₄	1	2	yes	—
13	C₄×#6127	C₂²	1	1	yes	characteristic
14	C₁₃×#431	C₂²	1	1	yes	characteristic
15	C₄×C₁₃×M₁₆	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
17	#11718783	C₄	1	2	yes	—
18	C₈×C₂²⋊D₁₃	C₄	1	4	yes	—
19	#11718586	C₂²	1	1	yes	characteristic
20	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
21	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
22	#11718495	C₄	1	2	yes	—
23	#11718491	C₄	1	2	yes	—
24	#11718386	C₄	1	4	yes	—
25	#11718369	C₄	1	4	yes	—
26	C₄×#6094	C₂²	1	2	yes	—
27	C₈×D₅₂	C₄	1	2	yes	—
28	#11718346	C₂²	1	2	yes	—
29	#11718346	C₂²	1	2	yes	—
30	C₄×C₈×D₁₃	C₂²	1	2	yes	—
31	C₄×C₈×D₁₃	C₂²	1	2	yes	—
32	C₄×C₈×D₁₃	C₂²	1	2	yes	—
33	C₈×Dic₂₆	C₄	1	2	yes	—
34	C₁₃×#322	C₂²	1	2	yes	—
35	C₈²×C₁₃	C₂²	1	2	yes	—
36	#11718047	C₂²	1	2	yes	—
37	#11718046	C₂²	1	1	yes	characteristic
38	#11718046	C₂²	1	2	yes	—
39	#11718046	C₂²	1	1	yes	characteristic
40	#11718046	C₂²	1	1	yes	characteristic
41	#11718046	C₂²	1	1	yes	characteristic
42	C₈×#1055	C₂²	1	2	yes	—
43	C₈×#1055	C₂²	1	2	yes	—
44	C₂×#6127	C₂×C₄	1	1	yes	characteristic
45	C₂×C₁₃×M₁₆	C₂×C₄	1	2	yes	—
46	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
47	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
48	C₄×C₂²⋊D₁₃	C₂×C₄	1	2	yes	—
49	C₄×C₂²⋊D₁₃	C₂×C₄	1	2	yes	—
50	#88180	C₂×C₄	1	1	yes	characteristic
51	C₂²×#1055	C₂×C₄	1	2	yes	—
52	C₂²×#1055	C₂×C₄	1	2	yes	—
53	#88161	C₈	1	16	yes	complemented
54	C₂×#6094	C₂×C₄	1	4	yes	—
55	C₂×C₈×D₁₃	C₂×C₄	1	4	yes	—
56	C₂×C₈×D₁₃	C₂×C₄	1	4	yes	—
57	C₂×C₈×D₁₃	C₂×C₄	1	4	yes	—
58	C₄×D₅₂	C₂×C₄	1	1	yes	characteristic
59	C₄×D₅₂	C₂×C₄	1	1	yes	characteristic
60	#88126	C₂×C₄	1	2	yes	—
61	C₄²×D₁₃	C₂³	1	2	yes	—
62	C₄×Dic₂₆	C₂×C₄	1	1	yes	characteristic
63	C₄×Dic₂₆	C₂×C₄	1	1	yes	characteristic
64	C₁₃×C₄⋊C₈	C₂×C₄	1	2	yes	—
65	C₁₃×C₂²⋊C₈	C₂×C₄	1	2	yes	—
66	C₁₃×#97	C₂×C₄	1	2	yes	—
67	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
68	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
69	C₄×C₈×C₁₃	C₂³	1	2	yes	—
70	C₄×C₈×C₁₃	C₂³	1	2	yes	—
71	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
72	C₂²⋊#1055	C₂×C₄	1	2	yes	—
73	#88060	C₂×C₄	1	4	yes	—
74	#88060	C₂×C₄	1	4	yes	—
75	#88055	C₂×C₄	1	4	yes	—
76	#88054	C₂×C₄	1	4	yes	—
77	#88054	C₂×C₄	1	4	yes	—
78	C₈×Dic₁₃	C₂×C₄	1	4	yes	—
79	C₈×Dic₁₃	C₂×C₄	1	4	yes	—
80	C₈×Dic₁₃	C₂×C₄	1	4	yes	—
81	C₄⋊#1055	C₂×C₄	1	2	yes	—
82	C₄×#1055	C₂×C₄	1	2	yes	—
83	C₄×#1055	C₂×C₄	1	2	yes	—
84	C₄×#1055	C₂³	1	1	yes	characteristic
85	C₄×#1055	C₂³	1	1	yes	characteristic
86	C₄×#1055	C₂³	1	2	yes	—
87	C₄×#1055	C₂³	1	1	yes	characteristic
88	C₄×#1055	C₂³	1	1	yes	characteristic
89	#17316	—	13	1	no	maximal, Sylow
90	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
91	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
92	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
93	C₂×C₂²⋊D₁₃	C₄²	1	2	yes	—
94	#6127	C₂×C₈	1	8	yes	—
95	C₂×D₅₂	C₄²	1	1	yes	characteristic
96	C₂×C₄×D₁₃	C₂²×C₄	1	2	yes	—
97	C₂×C₄×D₁₃	C₂²×C₄	1	2	yes	—
98	C₂×C₄×D₁₃	C₂²×C₄	1	2	yes	—
99	C₂×Dic₂₆	C₄²	1	1	yes	characteristic
100	C₁₃×M₁₆	C₂×C₈	1	8	yes	complemented
101	C₂×C₈×C₁₃	—	4	2	no	—
102	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
103	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
104	C₂×C₈×C₁₃	C₂²×C₄	1	4	yes	—
105	C₂×C₈×C₁₃	C₂²×C₄	1	4	yes	—
106	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
107	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
108	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
109	C₄²×C₁₃	C₂⁴	1	1	yes	characteristic
110	C₂²⋊Dic₁₃	C₄²	1	2	yes	—
111	#6103	C₄²	1	2	yes	—
112	#6103	C₄²	1	2	yes	—
113	C₄⋊Dic₁₃	C₄²	1	1	yes	characteristic
114	C₄⋊Dic₁₃	C₄²	1	1	yes	characteristic
115	#6101	C₄²	1	2	yes	—
116	#6101	C₄²	1	2	yes	—
117	C₄×Dic₁₃	C₂²×C₄	1	2	yes	—
118	C₄×Dic₁₃	C₂²×C₄	1	2	yes	—
119	C₄×Dic₁₃	C₂²×C₄	1	2	yes	—
120	C₂×#1055	—	4	2	no	—
121	C₂×#1055	C₂×C₈	1	8	yes	complemented
122	C₂×#1055	C₂²×C₄	1	4	yes	—
123	C₂×#1055	C₂²×C₄	1	2	yes	—
124	C₂×#1055	C₂²×C₄	1	2	yes	—
125	C₂×#1055	C₂²×C₄	1	2	yes	—
126	C₂×#1055	C₂²×C₄	1	2	yes	—
127	#6094	C₂×C₈	1	16	yes	complemented
128	C₈×D₁₃	—	4	4	no	—
129	C₈×D₁₃	—	4	4	no	—
130	C₈×D₁₃	C₂×C₈	1	16	yes	complemented
131	C₈×#41	—	13	1	no	—
132	C₈×#41	—	13	1	no	—
133	C₄×#131	—	13	1	no	—
134	#1450	—	13	1	no	—
135	#1450	—	13	2	no	—
136	C₈×M₁₆	—	13	1	no	—
137	C₈×M₁₆	—	13	2	no	—
138	C₈×M₁₆	—	13	2	no	—
139	C₈×M₁₆	D₁₃	1	1	yes	characteristic, complemented
140	C₂×#322	—	13	1	no	—
141	C₂×#322	—	13	2	no	—
142	C₂³×C₁₃	C₂×C₄²	1	1	yes	characteristic
143	C₂²×D₁₃	C₂×C₄²	1	2	yes	—
144	C₂×C₄×C₁₃	—	2	2	no	—
145	C₂×C₄×C₁₃	—	2	2	no	—
146	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
147	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
148	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
149	C₂×C₄×C₁₃	C₂²×C₈	1	4	yes	—
150	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
151	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
152	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
153	C₂²⋊D₁₃	C₄×C₈	1	8	yes	—
154	C₂×Dic₁₃	C₂×C₄²	1	2	yes	—
155	C₂×Dic₁₃	C₂×C₄²	1	2	yes	—
156	C₂×Dic₁₃	C₂×C₄²	1	2	yes	—
157	D₅₂	C₄×C₈	1	4	yes	—
158	C₄×D₁₃	—	2	4	no	—
159	C₄×D₁₃	—	2	4	no	—
160	C₄×D₁₃	C₂²×C₈	1	8	yes	—
161	Dic₂₆	C₄×C₈	1	4	yes	—
162	C₈×C₁₃	—	2	2	no	—
163	C₈×C₁₃	—	2	4	no	—
164	C₈×C₁₃	—	2	2	no	—
165	C₈×C₁₃	C₂²×C₈	1	8	yes	—
166	#1055	—	2	2	no	—
167	#1055	—	2	4	no	—
168	#1055	—	2	2	no	—
169	#1055	C₂²×C₈	1	4	yes	—
170	#1055	C₂²×C₈	1	4	yes	—
171	C₂×#131	—	13	2	no	—
172	C₄×#41	—	13	1	no	—
173	C₈×Q₈	—	13	2	no	—
174	C₈×Q₈	—	13	2	no	—
175	C₈×D₄	—	13	4	no	—
176	C₈×D₄	—	13	2	no	—
177	C₈×D₄	—	13	4	no	—
178	C₈×D₄	—	13	2	no	—
179	#431	—	13	1	no	—
180	#431	—	13	2	no	—
181	#431	—	13	2	no	—
182	#431	C₂×D₁₃	1	1	yes	characteristic
183	#405	—	13	2	no	—
184	#405	—	13	4	no	—
185	C₄×M₁₆	—	13	2	no	—
186	C₄×M₁₆	C₂×D₁₃	1	1	yes	characteristic
187	C₂×C₄×C₈	—	13	1	no	—
188	C₂×C₄×C₈	—	13	1	no	—
189	C₂×C₄×C₈	—	13	2	no	—
190	C₂×C₄×C₈	—	13	2	no	—
191	C₂×C₄×C₈	—	13	2	no	—
192	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
193	#322	—	13	1	no	—
194	#322	—	13	1	no	—
195	#322	—	13	1	no	—
196	#322	—	13	2	no	—
197	#322	—	13	1	no	—
198	#322	—	13	2	no	—
199	#322	C₂×D₁₃	1	2	yes	—
200	C₈²	—	13	2	no	—
201	C₈²	—	13	2	no	—
202	C₈²	C₂×D₁₃	1	2	yes	—
203	C₂²×C₁₃	—	2	1	no	—
204	C₂²×C₁₃	—	2	1	no	—
205	C₂²×C₁₃	C₂×C₄×C₈	1	2	yes	—
206	C₂²×C₁₃	C₂²×C₄²	1	1	yes	characteristic
207	C₂×D₁₃	—	2	2	no	—
208	C₂×D₁₃	—	2	2	no	—
209	C₂×D₁₃	C₂×C₄×C₈	1	4	yes	—
210	C₄×C₁₃	—	2	1	no	—
211	C₄×C₁₃	—	2	1	no	—
212	C₄×C₁₃	C₂×C₄×C₈	1	2	yes	—
213	C₄×C₁₃	C₂³×C₈	1	2	yes	—
214	C₄×C₁₃	#568	1	2	yes	—
215	C₄×C₁₃	#568	1	2	yes	—
216	Dic₁₃	—	2	2	no	—
217	Dic₁₃	—	2	2	no	—
218	Dic₁₃	C₂×C₄×C₈	1	4	yes	—
219	C₂×#41	—	13	1	no	—
220	#131	—	13	16	no	—
221	C₂×M₁₆	—	13	4	no	—
222	C₂×M₁₆	C₄×D₁₃	1	2	yes	—
223	C₂²×C₈	—	13	2	no	—
224	C₂²×C₈	—	13	4	no	—
225	C₂²×C₈	—	13	2	no	—
226	C₂²×C₈	—	13	4	no	—
227	C₂²×C₈	—	13	4	no	—
228	C₂²×C₈	C₄×D₁₃	1	2	yes	—
229	C₄×Q₈	—	13	1	no	—
230	C₄×Q₈	—	13	1	no	—
231	C₄×D₄	—	13	2	no	—
232	C₄×D₄	—	13	2	no	—
233	C₄×D₄	—	13	1	no	—
234	C₄×D₄	—	13	1	no	—
235	#117	—	13	1	no	—
236	#117	—	13	2	no	—
237	C₂×C₄²	—	13	2	no	—
238	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
239	C₄⋊C₈	—	13	4	no	—
240	C₄⋊C₈	—	13	4	no	—
241	C₄⋊C₈	—	13	2	no	—
242	C₄⋊C₈	C₄×D₁₃	1	2	yes	—
243	C₂²⋊C₈	—	13	2	no	—
244	C₂²⋊C₈	—	13	4	no	—
245	C₂²⋊C₈	—	13	4	no	—
246	C₂²⋊C₈	C₄×D₁₃	1	2	yes	—
247	#97	—	13	4	no	—
248	#97	C₄×D₁₃	1	2	yes	—
249	C₄×C₈	—	13	4	no	—
250	C₄×C₈	—	13	4	no	—
251	C₄×C₈	—	13	4	no	—
252	C₄×C₈	—	13	2	no	—
253	C₄×C₈	—	13	2	no	—
254	C₄×C₈	—	13	1	no	—
255	C₄×C₈	—	13	1	no	—
256	C₄×C₈	—	13	2	no	—
257	C₄×C₈	—	13	1	no	—
258	C₄×C₈	—	13	1	no	—
259	C₄×C₈	C₄×D₁₃	1	2	yes	—
260	C₄×C₈	C₂²×D₁₃	1	2	yes	—
261	C₄×C₈	C₂²×D₁₃	1	2	yes	—
262	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
263	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
264	C₂×C₁₃	—	2	2	no	—
265	C₂×C₁₃	C₂²×C₄×C₈	1	1	yes	characteristic
266	C₂×C₁₃	#2875	1	1	yes	characteristic
267	C₂×C₁₃	#2875	1	1	yes	characteristic
268	D₁₃	—	2	4	no	—
269	#41	—	13	8	no	—
270	C₂×Q₈	—	13	1	no	—
271	C₂×D₄	—	13	2	no	—
272	C₂×D₄	—	13	1	no	—
273	C₂²×C₄	—	13	2	no	—
274	C₂²×C₄	—	13	2	no	—
275	C₂²×C₄	—	13	2	no	—
276	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
277	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
278	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
279	M₁₆	—	13	16	no	—
280	M₁₆	C₈×D₁₃	1	8	yes	complemented
281	C₂×C₈	—	4	2	no	—
282	C₂×C₈	—	13	8	no	—
283	C₂×C₈	—	13	2	no	—
284	C₂×C₈	—	13	2	no	—
285	C₂×C₈	—	13	16	no	—
286	C₂×C₈	—	13	2	no	—
287	C₂×C₈	—	13	4	no	—
288	C₂×C₈	—	13	2	no	—
289	C₂×C₈	—	52	4	no	—
290	C₂×C₈	—	52	4	no	—
291	C₂×C₈	—	52	2	no	—
292	C₂×C₈	C₂×C₄×D₁₃	1	2	yes	—
293	C₂×C₈	C₂×C₄×D₁₃	1	4	yes	—
294	C₂×C₈	C₂×C₄×D₁₃	1	2	yes	—
295	C₂×C₈	C₂×C₄×D₁₃	1	4	yes	—
296	C₄⋊C₄	—	13	2	no	—
297	C₄⋊C₄	—	13	2	no	—
298	C₄⋊C₄	—	13	1	no	—
299	C₄⋊C₄	—	13	1	no	—
300	C₂²⋊C₄	—	13	2	no	—
301	C₂²⋊C₄	—	13	2	no	—
302	C₂²⋊C₄	—	13	2	no	—
303	C₄²	—	13	2	no	—
304	C₄²	—	13	2	no	—
305	C₄²	—	13	2	no	—
306	C₄²	C₂×C₄×D₁₃	1	1	yes	characteristic
307	C₄²	C₂×C₄×D₁₃	1	1	yes	characteristic
308	C₄²	C₂×C₄×D₁₃	1	1	yes	characteristic
309	C₄²	C₂³×D₁₃	1	1	yes	characteristic
310	C₁₃	#17316	1	1	yes	characteristic, complemented, Sylow
311	C₂³	—	13	2	no	—
312	C₂³	C₄²×D₁₃	1	1	yes	characteristic
313	Q₈	—	13	4	no	—
314	D₄	—	13	8	no	—
315	D₄	—	13	4	no	—
316	C₂×C₄	—	2	2	no	—
317	C₂×C₄	—	2	2	no	—
318	C₂×C₄	—	13	2	no	—
319	C₂×C₄	—	13	8	no	—
320	C₂×C₄	—	13	2	no	—
321	C₂×C₄	—	13	2	no	—
322	C₂×C₄	—	26	4	no	—
323	C₂×C₄	—	26	4	no	—
324	C₂×C₄	C₄²×D₁₃	1	1	yes	characteristic
325	C₂×C₄	C₄²×D₁₃	1	1	yes	characteristic
326	C₂×C₄	C₄²×D₁₃	1	1	yes	characteristic
327	C₂×C₄	C₂×C₈×D₁₃	1	4	yes	—
328	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
329	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
330	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
331	C₈	—	2	2	no	—
332	C₈	—	2	2	no	—
333	C₈	—	2	4	no	—
334	C₈	—	13	4	no	—
335	C₈	—	13	4	no	—
336	C₈	—	26	4	no	—
337	C₈	—	26	2	no	—
338	C₈	—	26	2	no	—
339	C₈	C₂×C₈×D₁₃	1	8	yes	—
340	C₂²	—	2	1	no	—
341	C₂²	—	2	1	no	—
342	C₂²	—	13	4	no	—
343	C₂²	—	26	2	no	—
344	C₂²	—	26	2	no	—
345	C₂²	C₄×C₈×D₁₃	1	2	yes	—
346	C₂²	C₂×C₄²×D₁₃	1	1	yes	characteristic
347	C₄	—	2	1	no	—
348	C₄	—	2	1	no	—
349	C₄	—	13	4	no	—
350	C₄	—	26	2	no	—
351	C₄	—	26	2	no	—
352	C₄	C₄×C₈×D₁₃	1	2	yes	—
353	C₄	C₂²×C₈×D₁₃	1	2	yes	—
354	C₄	#11719450	1	2	yes	—
355	C₄	#11719455	1	2	yes	—
356	C₂	—	2	2	no	—
357	C₂	—	26	4	no	—
358	C₂	C₂×C₄×C₈×D₁₃	1	1	yes	characteristic
359	C₂	#13123263	1	1	yes	characteristic
360	C₂	#13123265	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₄²	C₂³×D₁₃
commutator	C₂×C₁₃	C₂²×C₄×C₈
Fitting	C₈×C₁₃×M₁₆	C₂
socle	C₂²×C₁₃	C₂²×C₄²
prime radical	C₂×#6127	C₂×C₄
cosocle	C₄²×C₁₃	C₂⁴
hypercenter	C₈×M₁₆	D₁₃
hypocenter	C₁₃	#17316
LCS	G ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₄² ◁ C₈×M₁₆

Complements

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

N	H ≅ G/N	type
4. #13113120	C₂	semidirect
5. #13113118	C₂	semidirect
6. #13113115	C₂	semidirect
7. M₁₆×#1055	C₂	semidirect
8. C₈×#6094	C₂	semidirect
9. C₂×#11718046	C₂	semidirect
10. D₁₃×#322	C₂	semidirect
11. C₈×C₁₃×M₁₆	C₂	semidirect
53. #88161	C₈	semidirect
100. C₁₃×M₁₆	C₂×C₈	semidirect
121. C₂×#1055	C₂×C₈	semidirect
127. #6094	C₂×C₈	semidirect
130. C₈×D₁₃	C₂×C₈	semidirect
139. C₈×M₁₆	D₁₃	semidirect
280. M₁₆	C₈×D₁₃	semidirect
310. C₁₃	#17316	semidirect

Maximal subgroups

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

subgroup	index
C₈×#6127	2
#13123792	2
C₄×#88161	2
#13113120	2
#13113118	2
#13113115	2
M₁₆×#1055	2
C₈×#6094	2
C₂×#11718046	2
D₁₃×#322	2
C₈×C₁₃×M₁₆	2
#17316	13