Subgroups — #15833164

Order 3328.

658 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13124845	C₂	1	1	yes	maximal, characteristic
2	#13123770	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#11718349	C₂	1	1	yes	maximal, characteristic
4	#13113740	C₂	1	4	yes	maximal, complemented
5	#13113716	C₂	1	4	yes	maximal, complemented
6	#13113611	C₂	1	1	yes	maximal, characteristic, complemented
7	#13113262	C₂	1	1	yes	maximal, characteristic
8	#13113240	C₂	1	1	yes	maximal, characteristic, complemented
9	C₁₃×#1452	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×#6094	—	2	1	no	—
11	C₂×C₄×D₅₂	C₂²	1	1	yes	characteristic
12	#11719175	C₂²	1	1	yes	characteristic
13	C₁₃×#431	C₂²	1	1	yes	characteristic
14	C₁₃×#431	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
16	C₂×#88060	—	2	1	no	—
17	#11718763	—	2	1	no	—
18	#11718762	—	2	1	no	—
19	#11718585	C₂²	1	1	yes	characteristic
20	C₂×C₄⋊#1055	C₂²	1	1	yes	characteristic
21	#11718503	C₄	1	2	yes	—
22	#11718502	C₄	1	2	yes	—
23	#11718392	C₄	1	2	yes	—
24	#11718382	C₄	1	2	yes	—
25	C₄×#6095	C₂²	1	4	yes	—
26	#11718349	—	2	1	no	—
27	#11718349	—	2	1	no	—
28	#11718349	C₂²	1	2	yes	—
29	#11718349	C₂²	1	2	yes	—
30	#11718340	—	2	1	no	—
31	#11718340	C₂²	1	2	yes	complemented
32	C₈×Dic₂₆	—	2	1	no	—
33	C₈×Dic₂₆	C₂²	1	2	yes	complemented
34	C₁₃×#322	C₂²	1	2	yes	complemented
35	C₁₃×#322	C₂²	1	2	yes	complemented
36	#11718051	C₂²	1	4	yes	—
37	#11718050	C₂²	1	4	yes	complemented
38	#11718048	C₂²	1	4	yes	complemented
39	C₂²×D₅₂	C₂×C₄	1	1	yes	characteristic
40	C₂²×C₄×D₁₃	—	2	1	no	—
41	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
42	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
43	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
44	C₂×#6103	—	2	1	no	—
45	#88180	—	2	1	no	—
46	C₂×C₄⋊Dic₁₃	C₂×C₄	1	1	yes	characteristic
47	#88178	—	2	1	no	—
48	#88178	C₂×C₄	1	2	yes	—
49	C₂×#6099	—	2	1	no	—
50	C₂²×#1055	—	2	1	no	—
51	C₂×#6095	C₂×C₄	1	4	yes	—
52	C₂×#6094	—	2	2	no	—
53	C₂×#6094	—	2	2	no	—
54	C₂×#6094	—	4	4	no	—
55	C₄×D₅₂	—	2	1	no	—
56	C₄×D₅₂	—	2	1	no	—
57	C₄×D₅₂	C₂×C₄	1	2	yes	—
58	C₄×D₅₂	C₂³	1	2	yes	—
59	#88124	—	2	1	no	—
60	#88123	C₂×C₄	1	1	yes	characteristic
61	#88123	C₂×C₄	1	1	yes	characteristic
62	C₄×Dic₂₆	—	2	1	no	—
63	C₄×Dic₂₆	C₂³	1	2	yes	—
64	C₁₃×C₄⋊C₈	—	2	1	no	—
65	C₁₃×C₄⋊C₈	C₂×C₄	1	2	yes	—
66	C₁₃×C₂²⋊C₈	—	2	1	no	—
67	C₁₃×C₂²⋊C₈	C₂×C₄	1	2	yes	—
68	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
69	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
70	C₄×C₈×C₁₃	C₂³	1	2	yes	—
71	C₄×C₈×C₁₃	C₂³	1	2	yes	—
72	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
73	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
74	#88061	C₂×C₄	1	4	yes	—
75	#88060	—	2	2	no	—
76	#88060	—	2	2	no	—
77	#88057	C₂×C₄	1	4	yes	—
78	#88056	C₂×C₄	1	4	yes	—
79	#88055	—	2	2	no	—
80	#88054	—	2	2	no	—
81	#88054	—	2	2	no	—
82	C₈×Dic₁₃	—	2	2	no	—
83	C₄⋊#1055	—	2	1	no	—
84	C₄⋊#1055	—	2	1	no	—
85	C₄⋊#1055	C₂³	1	2	yes	—
86	C₄⋊#1055	C₂³	1	2	yes	—
87	#17933	—	13	1	no	maximal, Sylow
88	C₂³×D₁₃	—	2	1	no	—
89	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
90	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
91	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
92	C₂²×Dic₁₃	—	2	1	no	—
93	C₂×D₅₂	—	2	1	no	—
94	C₂×D₅₂	—	4	2	no	—
95	C₂×D₅₂	C₂²×C₄	1	2	yes	—
96	C₂×C₄×D₁₃	—	2	2	no	—
97	C₂×C₄×D₁₃	—	2	2	no	—
98	C₂×C₄×D₁₃	—	4	2	no	—
99	C₂×C₄×D₁₃	—	4	2	no	—
100	C₂×Dic₂₆	—	2	1	no	—
101	C₂×Dic₂₆	C₂²×C₄	1	2	yes	—
102	C₂×C₈×C₁₃	—	2	2	no	—
103	C₂×C₈×C₁₃	—	2	2	no	—
104	C₂×C₈×C₁₃	—	2	2	no	—
105	C₂×C₈×C₁₃	C₂²×C₄	1	4	yes	—
106	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
107	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
108	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
109	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
110	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
111	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
112	C₄²×C₁₃	#41	1	1	yes	characteristic
113	C₄²×C₁₃	C₂⁴	1	1	yes	characteristic
114	C₂²⋊Dic₁₃	—	2	2	no	—
115	#6103	—	2	2	no	—
116	#6103	—	2	2	no	—
117	C₄⋊Dic₁₃	—	2	1	no	—
118	C₄⋊Dic₁₃	—	2	1	no	—
119	C₄⋊Dic₁₃	—	2	1	no	—
120	C₄⋊Dic₁₃	—	2	1	no	—
121	C₄⋊Dic₁₃	C₂²×C₄	1	2	yes	—
122	C₄⋊Dic₁₃	C₂²×C₄	1	2	yes	—
123	#6101	—	2	2	no	—
124	#6101	—	2	2	no	—
125	C₄×Dic₁₃	—	2	2	no	—
126	#6099	—	4	2	no	—
127	C₂×#1055	—	2	2	no	—
128	C₂×#1055	—	2	2	no	—
129	C₂×#1055	—	4	2	no	—
130	#6095	—	4	4	no	—
131	#6094	—	4	4	no	—
132	#6094	—	4	4	no	—
133	#3101	—	13	1	no	—
134	#2960	—	13	1	no	—
135	C₂×#436	—	13	1	no	—
136	#1592	—	13	4	no	—
137	#1588	—	13	4	no	—
138	#1571	—	13	1	no	—
139	#1486	—	13	1	no	—
140	#1481	—	13	1	no	—
141	#1452	D₁₃	1	1	yes	characteristic, complemented
142	C₂³×C₁₃	C₄×D₄	1	1	yes	characteristic
143	C₂²×D₁₃	—	2	2	no	—
144	C₂²×D₁₃	—	4	2	no	—
145	C₂²×D₁₃	—	4	2	no	—
146	C₂²×D₁₃	—	4	2	no	—
147	C₂×C₄×C₁₃	—	2	2	no	—
148	C₂×C₄×C₁₃	—	4	1	no	—
149	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
150	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
151	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
152	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	2	yes	—
153	C₂×C₄×C₁₃	#137	1	2	yes	—
154	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
155	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
156	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
157	C₂×Dic₁₃	—	2	2	no	—
158	C₂×Dic₁₃	—	2	2	no	—
159	C₂×Dic₁₃	—	2	2	no	—
160	C₂×Dic₁₃	—	4	2	no	—
161	D₅₂	—	2	2	no	—
162	D₅₂	—	2	2	no	—
163	D₅₂	—	4	2	no	—
164	C₄×D₁₃	—	4	2	no	—
165	C₄×D₁₃	—	4	2	no	—
166	C₄×D₁₃	—	4	4	no	—
167	Dic₂₆	—	2	4	no	—
168	Dic₂₆	—	4	2	no	—
169	C₈×C₁₃	—	2	4	no	—
170	C₈×C₁₃	—	2	1	no	—
171	C₈×C₁₃	—	2	1	no	—
172	C₈×C₁₃	—	2	1	no	—
173	C₈×C₁₃	—	2	1	no	—
174	C₈×C₁₃	—	4	2	no	—
175	#1055	—	4	2	no	—
176	#1055	—	4	2	no	—
177	C₂²×M₁₆	—	26	1	no	—
178	#533	—	13	1	no	—
179	C₂×C₄×D₄	—	13	1	no	—
180	#483	—	13	2	no	—
181	#476	—	13	2	no	—
182	#461	—	13	2	no	—
183	#450	—	13	2	no	—
184	#446	—	13	2	no	—
185	#446	—	26	1	no	—
186	C₈×Q₈	—	13	2	no	—
187	C₈×Q₈	—	26	1	no	—
188	C₄×SD₁₆	—	13	4	no	—
189	#436	—	13	2	no	—
190	#436	—	13	2	no	—
191	#436	—	26	1	no	—
192	#436	—	26	1	no	—
193	#431	C₂×D₁₃	1	1	yes	characteristic
194	#431	C₂×D₁₃	1	1	yes	characteristic
195	#424	—	26	1	no	—
196	#423	—	13	1	no	—
197	C₂×C₄⋊C₈	—	13	1	no	—
198	C₂×C₂²⋊C₈	—	26	1	no	—
199	#405	—	26	1	no	—
200	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
201	#335	—	13	4	no	—
202	Q₈⋊C₈	—	13	4	no	—
203	D₄⋊C₈	—	13	4	no	—
204	#322	C₂×D₁₃	1	2	yes	complemented
205	#322	C₂×D₁₃	1	2	yes	complemented
206	C₂²×C₁₃	—	2	1	no	—
207	C₂²×C₁₃	—	2	1	no	—
208	C₂²×C₁₃	#440	1	2	yes	—
209	C₂²×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
210	C₂×D₁₃	—	4	2	no	—
211	C₂×D₁₃	—	4	2	no	—
212	C₂×D₁₃	—	4	2	no	—
213	C₂×D₁₃	—	8	4	no	—
214	C₄×C₁₃	—	2	1	no	—
215	C₄×C₁₃	—	2	1	no	—
216	C₄×C₁₃	—	2	1	no	—
217	C₄×C₁₃	#440	1	2	yes	—
218	C₄×C₁₃	#568	1	2	yes	—
219	C₄×C₁₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
220	C₄×C₁₃	C₂×#137	1	1	yes	characteristic
221	Dic₁₃	—	4	2	no	—
222	Dic₁₃	—	4	2	no	—
223	Dic₁₃	—	4	2	no	—
224	C₂²×D₄	—	13	1	no	—
225	C₂³×C₄	—	26	1	no	—
226	C₂×SD₁₆	—	13	4	no	—
227	C₂×M₁₆	—	26	1	no	—
228	C₂×M₁₆	—	26	2	no	—
229	C₂×M₁₆	—	26	2	no	—
230	C₂×M₁₆	—	52	4	no	—
231	C₂²×C₈	—	26	1	no	—
232	C₂²×C₈	D₅₂	1	1	yes	characteristic
233	C₂²×C₈	D₅₂	1	1	yes	characteristic
234	C₄⋊Q₈	—	13	1	no	—
235	C₄⋊Q₈	—	13	1	no	—
236	#125	—	26	1	no	—
237	C₂²⋊Q₈	—	13	2	no	—
238	C₂²⋊Q₈	—	26	1	no	—
239	C₄×Q₈	—	13	2	no	—
240	C₄×Q₈	—	26	1	no	—
241	C₄×D₄	—	13	2	no	—
242	C₄×D₄	—	13	2	no	—
243	C₄×D₄	—	26	1	no	—
244	C₄×D₄	—	26	1	no	—
245	#117	—	26	1	no	—
246	C₂×C₄⋊C₄	—	13	1	no	—
247	C₂×C₂²⋊C₄	—	26	1	no	—
248	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
249	#106	—	13	4	no	—
250	C₄⋊C₈	—	2	1	no	—
251	C₄⋊C₈	—	13	2	no	—
252	C₄⋊C₈	—	13	2	no	—
253	C₄⋊C₈	—	26	2	no	—
254	C₄⋊C₈	—	26	2	no	—
255	C₄⋊C₈	—	26	1	no	—
256	C₄⋊C₈	—	26	1	no	—
257	C₄⋊C₈	C₄×D₁₃	1	2	yes	—
258	#103	—	13	4	no	—
259	#102	—	13	4	no	—
260	C₂²⋊C₈	—	2	1	no	—
261	C₂²⋊C₈	—	26	2	no	—
262	C₂²⋊C₈	—	26	2	no	—
263	C₂²⋊C₈	C₄×D₁₃	1	2	yes	—
264	#97	—	26	2	no	—
265	C₄×C₈	—	26	2	no	—
266	C₄×C₈	D₅₂	1	1	yes	characteristic
267	C₄×C₈	D₅₂	1	1	yes	characteristic
268	C₄×C₈	C₂²×D₁₃	1	2	yes	—
269	C₄×C₈	C₂²×D₁₃	1	2	yes	—
270	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
271	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
272	C₂×C₁₃	—	2	2	no	—
273	C₂×C₁₃	#2930	1	1	yes	characteristic
274	C₂×C₁₃	C₂×#440	1	1	yes	characteristic
275	C₂×C₁₃	#2955	1	1	yes	characteristic
276	D₁₃	—	8	2	no	—
277	C₂⁴	—	26	1	no	—
278	C₂×Q₈	—	13	2	no	—
279	C₂×Q₈	—	26	1	no	—
280	C₂×D₄	—	13	2	no	—
281	C₂×D₄	—	26	1	no	—
282	C₂×D₄	—	52	2	no	—
283	C₂²×C₄	—	26	1	no	—
284	C₂²×C₄	—	26	2	no	—
285	C₂²×C₄	—	26	2	no	—
286	C₂²×C₄	—	52	2	no	—
287	C₂²×C₄	—	52	2	no	—
288	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
289	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
290	C₂²×C₄	#6127	1	1	yes	characteristic
291	SD₁₆	—	52	4	no	—
292	M₁₆	—	52	4	no	—
293	M₁₆	—	52	4	no	—
294	M₁₆	—	52	2	no	—
295	C₂×C₈	—	2	2	no	—
296	C₂×C₈	—	2	2	no	—
297	C₂×C₈	—	2	2	no	—
298	C₂×C₈	—	26	2	no	—
299	C₂×C₈	—	26	2	no	—
300	C₂×C₈	—	52	2	no	—
301	C₂×C₈	C₂×C₄×D₁₃	1	4	yes	—
302	C₂×C₈	C₂×D₅₂	1	1	yes	characteristic
303	C₂×C₈	C₂×D₅₂	1	1	yes	characteristic
304	C₂×C₈	C₂×D₅₂	1	1	yes	characteristic
305	C₂×C₈	C₂×D₅₂	1	1	yes	characteristic
306	C₄⋊C₄	—	13	2	no	—
307	C₄⋊C₄	—	13	2	no	—
308	C₄⋊C₄	—	26	1	no	—
309	C₄⋊C₄	—	26	1	no	—
310	C₄⋊C₄	—	26	1	no	—
311	C₄⋊C₄	—	26	2	no	—
312	C₄⋊C₄	—	26	1	no	—
313	C₄⋊C₄	—	26	2	no	—
314	C₂²⋊C₄	—	26	2	no	—
315	C₂²⋊C₄	—	26	2	no	—
316	C₂²⋊C₄	—	26	2	no	—
317	C₄²	—	26	2	no	—
318	C₄²	C₂×C₄×D₁₃	1	1	yes	characteristic
319	C₄²	C₂×D₅₂	1	1	yes	characteristic
320	C₄²	#6127	1	1	yes	characteristic
321	C₄²	C₂³×D₁₃	1	1	yes	characteristic
322	C₁₃	#17933	1	1	yes	characteristic, complemented, Sylow
323	C₂³	—	26	2	no	—
324	C₂³	—	52	2	no	—
325	C₂³	—	52	2	no	—
326	C₂³	—	52	2	no	—
327	C₂³	C₄×D₅₂	1	1	yes	characteristic
328	Q₈	—	26	4	no	—
329	Q₈	—	52	2	no	—
330	D₄	—	26	2	no	—
331	D₄	—	26	2	no	—
332	D₄	—	52	2	no	—
333	C₂×C₄	—	2	2	no	—
334	C₂×C₄	—	4	1	no	—
335	C₂×C₄	—	26	2	no	—
336	C₂×C₄	—	26	2	no	—
337	C₂×C₄	—	26	2	no	—
338	C₂×C₄	—	52	2	no	—
339	C₂×C₄	—	52	2	no	—
340	C₂×C₄	—	52	2	no	—
341	C₂×C₄	—	52	4	no	—
342	C₂×C₄	C₄×D₅₂	1	1	yes	characteristic
343	C₂×C₄	C₄×D₅₂	1	1	yes	characteristic
344	C₂×C₄	C₄×D₅₂	1	1	yes	characteristic
345	C₂×C₄	#88162	1	2	yes	—
346	C₂×C₄	#88163	1	2	yes	—
347	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
348	C₂×C₄	C₂²×D₅₂	1	1	yes	characteristic
349	C₂×C₄	C₂×#6127	1	1	yes	characteristic
350	C₈	—	2	1	no	—
351	C₈	—	2	1	no	—
352	C₈	—	2	1	no	—
353	C₈	—	2	1	no	—
354	C₈	—	2	4	no	—
355	C₈	—	4	2	no	—
356	C₈	—	52	2	no	—
357	C₈	—	52	2	no	—
358	C₂²	—	2	1	no	—
359	C₂²	—	2	1	no	—
360	C₂²	—	52	2	no	—
361	C₂²	—	52	2	no	—
362	C₂²	—	52	2	no	—
363	C₂²	—	104	4	no	—
364	C₂²	#11718371	1	2	yes	—
365	C₂²	C₂×C₄×D₅₂	1	1	yes	characteristic
366	C₄	—	2	1	no	—
367	C₄	—	2	1	no	—
368	C₄	—	2	1	no	—
369	C₄	—	52	2	no	—
370	C₄	—	52	2	no	—
371	C₄	—	52	2	no	—
372	C₄	#11718371	1	2	yes	—
373	C₄	C₂×#88162	1	1	yes	characteristic
374	C₄	C₂×#88163	1	1	yes	characteristic
375	C₄	#11719455	1	2	yes	—
376	C₂	—	2	2	no	—
377	C₂	—	104	2	no	—
378	C₂	#13123541	1	1	yes	characteristic
379	C₂	C₂×#11718371	1	1	yes	characteristic
380	C₂	#13123740	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₅₂
commutator	C₂×C₄×C₁₃	C₂³×C₄
Frattini	C₄²	C₂³×D₁₃
Fitting	C₁₃×#1452	C₂
socle	C₂²×C₁₃	C₂×C₄×D₄
prime radical	C₂²×D₅₂	C₂×C₄
cosocle	C₄²×C₁₃	C₂⁴
hypercenter	#1452	D₁₃
hypocenter	C₁₃	#17933
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ #1452

Complements

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

N	H ≅ G/N	type
2. #13123770	C₂	semidirect
4. #13113740	C₂	semidirect
5. #13113716	C₂	semidirect
6. #13113611	C₂	semidirect
8. #13113240	C₂	semidirect
9. C₁₃×#1452	C₂	semidirect
31. #11718340	C₂²	semidirect
33. C₈×Dic₂₆	C₂²	semidirect
34. C₁₃×#322	C₂²	semidirect
35. C₁₃×#322	C₂²	semidirect
37. #11718050	C₂²	semidirect
38. #11718048	C₂²	semidirect
141. #1452	D₁₃	semidirect
204. #322	C₂×D₁₃	semidirect
205. #322	C₂×D₁₃	semidirect
322. C₁₃	#17933	semidirect

Maximal subgroups

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

subgroup	index
#13124845	2
#13123770	2
C₂×#11718349	2
#13113740	2
#13113716	2
#13113611	2
#13113262	2
#13113240	2
C₁₃×#1452	2
#17933	13