Subgroups — #15758574

Order 3328.

644 conjugacy classes of subgroups.

The subgroup lattice has 380 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13123188	C₂	1	1	yes	maximal, characteristic
2	#13114816	C₂	1	1	yes	maximal, characteristic
3	C₂×#11718074	C₂	1	2	yes	maximal, complemented
4	#13112720	C₂	1	2	yes	maximal, complemented
5	C₂²×C₁₃×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
6	C₂³×C₈×C₁₃	—	2	1	no	—
7	C₂²×C₄²×C₁₃	C₂²	1	1	yes	characteristic
8	C₂×#88178	—	2	1	no	—
9	C₂×#88123	C₂²	1	1	yes	characteristic
10	C₂×C₁₃×C₄⋊C₈	—	2	1	no	—
11	C₂×C₁₃×C₄⋊C₈	—	2	4	no	—
12	C₂×C₁₃×C₄⋊C₈	C₂²	1	2	yes	—
13	#11718873	—	2	1	no	—
14	#11718873	C₄	1	2	yes	—
15	C₂×C₂²⋊#1055	—	2	1	no	—
16	#11718596	C₂²	1	1	yes	characteristic
17	#11718590	—	2	1	no	—
18	#11718589	—	2	1	no	—
19	#11718589	C₄	1	2	yes	—
20	C₂×C₄⋊#1055	C₂²	1	2	yes	—
21	#11718116	—	2	1	no	—
22	#11718074	—	2	4	no	—
23	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
24	C₂³×C₄×C₁₃	D₄	1	1	yes	characteristic
25	C₂³×C₄×C₁₃	D₄	1	1	yes	characteristic
26	C₂²×Dic₂₆	—	2	1	no	—
27	C₂²×C₈×C₁₃	—	2	4	no	—
28	C₂²×C₈×C₁₃	—	2	2	no	—
29	C₂²×C₈×C₁₃	—	4	2	no	—
30	C₂²×C₈×C₁₃	—	4	2	no	—
31	C₂×C₄²×C₁₃	—	2	2	no	—
32	C₂×C₄²×C₁₃	—	2	2	no	—
33	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
35	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
36	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
37	C₂×C₂²⋊Dic₁₃	—	2	2	no	—
38	C₂×C₄⋊Dic₁₃	—	2	1	no	—
39	C₂×C₄⋊Dic₁₃	—	2	1	no	—
40	C₂×C₄⋊Dic₁₃	—	2	1	no	—
41	C₂×C₄⋊Dic₁₃	C₂×C₄	1	2	yes	—
42	C₂×C₄⋊Dic₁₃	C₂×C₄	1	2	yes	—
43	#88178	—	4	2	no	—
44	#88178	—	4	2	no	—
45	C₂×#6101	—	2	2	no	—
46	C₂²×#1055	—	2	2	no	—
47	#88123	—	2	2	no	—
48	#88123	—	2	2	no	—
49	C₁₃×C₄⋊C₈	—	2	4	no	—
50	C₁₃×C₄⋊C₈	—	2	2	no	—
51	C₁₃×C₄⋊C₈	D₄	1	4	yes	—
52	#88071	—	2	2	no	—
53	#88071	—	2	2	no	—
54	C₂²⋊#1055	—	4	2	no	—
55	C₄⋊#1055	—	2	2	no	—
56	C₄⋊#1055	—	2	2	no	—
57	#9131	—	13	1	no	maximal, Sylow
58	C₂⁴×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
59	C₂²×C₄×C₁₃	—	2	1	no	—
60	C₂²×C₄×C₁₃	—	2	2	no	—
61	C₂²×C₄×C₁₃	—	2	1	no	—
62	C₂²×C₄×C₁₃	—	2	2	no	—
63	C₂²×C₄×C₁₃	—	2	1	no	—
64	C₂²×C₄×C₁₃	—	2	1	no	—
65	C₂²×C₄×C₁₃	—	2	2	no	—
66	C₂²×C₄×C₁₃	—	2	2	no	—
67	C₂²×C₄×C₁₃	—	4	1	no	—
68	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
69	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
70	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
71	C₂²×C₄×C₁₃	D₈	1	2	yes	—
72	C₂²×C₄×C₁₃	SD₁₆	1	2	yes	—
73	C₂²×C₄×C₁₃	SD₁₆	1	2	yes	—
74	C₂²×C₄×C₁₃	Q₁₆	1	2	yes	—
75	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
76	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
78	C₂²×Dic₁₃	—	2	2	no	—
79	C₂²×Dic₁₃	—	2	2	no	—
80	C₂×Dic₂₆	—	4	1	no	—
81	C₂×Dic₂₆	—	4	1	no	—
82	C₂×Dic₂₆	—	4	1	no	—
83	C₂×Dic₂₆	—	4	1	no	—
84	C₂×Dic₂₆	—	4	1	no	—
85	C₂×Dic₂₆	—	4	1	no	—
86	C₂×C₈×C₁₃	—	2	4	no	—
87	C₂×C₈×C₁₃	—	4	8	no	—
88	C₂×C₈×C₁₃	—	4	2	no	—
89	C₂×C₈×C₁₃	—	4	2	no	—
90	C₄²×C₁₃	—	2	2	no	—
91	C₄²×C₁₃	—	2	1	no	—
92	C₄²×C₁₃	—	2	2	no	—
93	C₄²×C₁₃	—	2	1	no	—
94	C₄²×C₁₃	C₂×D₄	1	2	yes	—
95	C₄²×C₁₃	#41	1	2	yes	—
96	C₂²⋊Dic₁₃	—	4	4	no	—
97	C₄⋊Dic₁₃	—	2	4	no	—
98	C₄⋊Dic₁₃	—	4	2	no	—
99	C₄⋊Dic₁₃	—	4	1	no	—
100	C₄⋊Dic₁₃	—	4	1	no	—
101	C₄⋊Dic₁₃	—	4	2	no	—
102	C₄⋊Dic₁₃	—	4	2	no	—
103	#6101	—	4	4	no	—
104	C₂×#1055	—	4	2	no	—
105	C₂×#1055	—	4	2	no	—
106	C₂×#1055	—	4	2	no	—
107	C₂²×C₄⋊C₈	D₁₃	1	1	yes	characteristic
108	#2857	—	13	1	no	—
109	#1797	—	13	1	no	—
110	C₂×#332	—	13	2	no	—
111	#1296	—	13	2	no	—
112	C₂³×C₁₃	—	2	1	no	—
113	C₂³×C₁₃	—	2	1	no	—
114	C₂³×C₁₃	—	2	1	no	—
115	C₂³×C₁₃	—	2	1	no	—
116	C₂³×C₁₃	#101	1	2	yes	—
117	C₂³×C₁₃	#102	1	2	yes	—
118	C₂³×C₁₃	#103	1	2	yes	—
119	C₂³×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
120	C₂×C₄×C₁₃	—	2	1	no	—
121	C₂×C₄×C₁₃	—	2	1	no	—
122	C₂×C₄×C₁₃	—	2	1	no	—
123	C₂×C₄×C₁₃	—	2	1	no	—
124	C₂×C₄×C₁₃	—	2	1	no	—
125	C₂×C₄×C₁₃	—	2	1	no	—
126	C₂×C₄×C₁₃	—	2	2	no	—
127	C₂×C₄×C₁₃	—	2	2	no	—
128	C₂×C₄×C₁₃	—	2	4	no	—
129	C₂×C₄×C₁₃	—	2	4	no	—
130	C₂×C₄×C₁₃	—	2	2	no	—
131	C₂×C₄×C₁₃	—	2	2	no	—
132	C₂×C₄×C₁₃	—	2	1	no	—
133	C₂×C₄×C₁₃	—	2	1	no	—
134	C₂×C₄×C₁₃	—	2	1	no	—
135	C₂×C₄×C₁₃	—	4	4	no	—
136	C₂×C₄×C₁₃	#102	1	2	yes	—
137	C₂×C₄×C₁₃	#103	1	2	yes	—
138	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
139	C₂×C₄×C₁₃	#120	1	1	yes	characteristic
140	C₂×C₄×C₁₃	#121	1	2	yes	—
141	C₂×C₄×C₁₃	#123	1	1	yes	characteristic
142	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
143	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
144	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
145	C₂×C₄×C₁₃	C₂×Q₁₆	1	1	yes	characteristic
146	C₂×C₄×C₁₃	#135	1	2	yes	—
147	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
148	C₂×C₄×C₁₃	#137	1	1	yes	characteristic
149	C₂×Dic₁₃	—	4	2	no	—
150	C₂×Dic₁₃	—	4	2	no	—
151	C₂×Dic₁₃	—	4	2	no	—
152	C₂×Dic₁₃	—	4	2	no	—
153	C₂×Dic₁₃	—	4	2	no	—
154	C₂×Dic₁₃	—	4	2	no	—
155	Dic₂₆	—	8	1	no	—
156	Dic₂₆	—	8	1	no	—
157	Dic₂₆	—	8	1	no	—
158	Dic₂₆	—	8	1	no	—
159	C₈×C₁₃	—	4	4	no	—
160	#1055	—	8	2	no	—
161	C₂³×C₈	—	2	1	no	—
162	C₂×C₄⋊Q₈	—	13	1	no	—
163	C₂×C₂²⋊Q₈	—	26	1	no	—
164	C₂²×C₄²	C₂×D₁₃	1	1	yes	characteristic
165	C₂×C₄⋊C₈	—	2	1	no	—
166	C₂×C₄⋊C₈	—	2	4	no	—
167	C₂×C₄⋊C₈	—	13	2	no	—
168	C₂×C₄⋊C₈	C₂×D₁₃	1	2	yes	—
169	C₂×C₂²⋊C₈	—	26	1	no	—
170	#391	—	26	1	no	—
171	#389	—	13	2	no	—
172	#389	—	26	1	no	—
173	#384	—	13	1	no	—
174	#380	—	13	2	no	—
175	#380	—	26	1	no	—
176	#336	—	26	1	no	—
177	#332	—	26	4	no	—
178	C₂²×C₁₃	—	2	2	no	—
179	C₂²×C₁₃	—	2	2	no	—
180	C₂²×C₁₃	—	2	2	no	—
181	C₂²×C₁₃	—	2	2	no	—
182	C₂²×C₁₃	—	2	2	no	—
183	C₂²×C₁₃	—	2	2	no	—
184	C₂²×C₁₃	#332	1	4	yes	complemented
185	C₂²×C₁₃	#386	1	1	yes	characteristic
186	C₂²×C₁₃	C₂×#101	1	1	yes	characteristic
187	C₂²×C₁₃	#413	1	1	yes	characteristic
188	C₂²×C₁₃	C₂×#102	1	1	yes	characteristic
189	C₂²×C₁₃	C₂×#103	1	1	yes	characteristic
190	C₂²×C₁₃	#416	1	1	yes	characteristic
191	C₂²×C₁₃	#417	1	1	yes	characteristic
192	C₄×C₁₃	—	2	2	no	—
193	C₄×C₁₃	—	2	2	no	—
194	C₄×C₁₃	—	2	2	no	—
195	C₄×C₁₃	—	4	1	no	—
196	C₄×C₁₃	#447	1	1	yes	characteristic
197	C₄×C₁₃	#448	1	1	yes	characteristic
198	C₄×C₁₃	#465	1	2	yes	—
199	C₄×C₁₃	#467	1	2	yes	—
200	C₄×C₁₃	#480	1	1	yes	characteristic
201	C₄×C₁₃	#481	1	1	yes	characteristic
202	Dic₁₃	—	8	2	no	—
203	Dic₁₃	—	8	2	no	—
204	C₂²×Q₈	—	26	1	no	—
205	C₂³×C₄	C₄×D₁₃	1	1	yes	characteristic
206	C₂³×C₄	D₅₂	1	1	yes	characteristic
207	C₂³×C₄	C₂²⋊D₁₃	1	1	yes	characteristic
208	C₂²×C₈	—	2	4	no	—
209	C₂²×C₈	—	2	2	no	—
210	C₂²×C₈	—	4	2	no	—
211	C₂²×C₈	—	4	2	no	—
212	C₂²×C₈	—	26	2	no	—
213	C₄⋊Q₈	—	26	2	no	—
214	C₄⋊Q₈	—	26	2	no	—
215	C₂²⋊Q₈	—	52	2	no	—
216	C₂²⋊Q₈	—	52	2	no	—
217	C₂×C₄⋊C₄	—	13	2	no	—
218	C₂×C₄⋊C₄	—	13	2	no	—
219	C₂×C₄⋊C₄	—	26	2	no	—
220	C₂×C₄⋊C₄	—	26	1	no	—
221	C₂×C₄⋊C₄	—	26	1	no	—
222	C₂×C₄⋊C₄	—	26	1	no	—
223	C₂×C₂²⋊C₄	—	26	2	no	—
224	C₂×C₄²	—	2	2	no	—
225	C₂×C₄²	—	2	2	no	—
226	C₂×C₄²	C₄×D₁₃	1	1	yes	characteristic
227	C₂×C₄²	D₅₂	1	1	yes	characteristic
228	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
229	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
230	C₄⋊C₈	—	2	4	no	—
231	C₄⋊C₈	—	2	2	no	—
232	C₄⋊C₈	—	26	2	no	—
233	C₄⋊C₈	—	26	2	no	—
234	C₄⋊C₈	C₂²⋊D₁₃	1	4	yes	—
235	C₂²⋊C₈	—	52	2	no	—
236	#95	—	26	2	no	—
237	#95	—	26	2	no	—
238	C₂×C₁₃	—	2	4	no	—
239	C₂×C₁₃	C₂×#332	1	1	yes	characteristic
240	C₂×C₁₃	#1540	1	1	yes	characteristic
241	C₂×C₁₃	#1550	1	1	yes	characteristic
242	C₂×C₁₃	#1552	1	1	yes	characteristic
243	C₂×C₁₃	#1874	1	1	yes	characteristic
244	C₂×C₁₃	#1892	1	1	yes	characteristic
245	C₂×C₁₃	#1901	1	1	yes	characteristic
246	C₂⁴	#6103	1	1	yes	characteristic
247	C₂×Q₈	—	52	1	no	—
248	C₂×Q₈	—	52	1	no	—
249	C₂×Q₈	—	52	1	no	—
250	C₂×Q₈	—	52	1	no	—
251	C₂×Q₈	—	52	1	no	—
252	C₂×Q₈	—	52	1	no	—
253	C₂²×C₄	—	2	2	no	—
254	C₂²×C₄	—	2	1	no	—
255	C₂²×C₄	—	2	2	no	—
256	C₂²×C₄	—	2	1	no	—
257	C₂²×C₄	—	2	1	no	—
258	C₂²×C₄	—	2	1	no	—
259	C₂²×C₄	—	2	2	no	—
260	C₂²×C₄	—	2	2	no	—
261	C₂²×C₄	—	4	1	no	—
262	C₂²×C₄	—	26	2	no	—
263	C₂²×C₄	—	26	2	no	—
264	C₂²×C₄	#6095	1	2	yes	—
265	C₂²×C₄	Dic₅₂	1	2	yes	—
266	C₂²×C₄	#6103	1	1	yes	characteristic
267	C₂²×C₄	#6103	1	1	yes	characteristic
268	C₂²×C₄	#6103	1	1	yes	characteristic
269	C₂²×C₄	#6104	1	2	yes	—
270	C₂²×C₄	#6106	1	2	yes	—
271	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
272	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
273	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
274	C₂×C₈	—	2	4	no	—
275	C₂×C₈	—	4	8	no	—
276	C₂×C₈	—	4	2	no	—
277	C₂×C₈	—	4	2	no	—
278	C₂×C₈	—	52	2	no	—
279	C₂×C₈	—	52	2	no	—
280	C₂×C₈	—	52	2	no	—
281	C₄⋊C₄	—	26	4	no	—
282	C₄⋊C₄	—	52	2	no	—
283	C₄⋊C₄	—	52	2	no	—
284	C₄⋊C₄	—	52	2	no	—
285	C₄⋊C₄	—	52	1	no	—
286	C₄⋊C₄	—	52	1	no	—
287	C₄⋊C₄	—	52	4	no	—
288	C₂²⋊C₄	—	52	4	no	—
289	C₄²	—	2	2	no	—
290	C₄²	—	2	1	no	—
291	C₄²	—	2	2	no	—
292	C₄²	—	2	1	no	—
293	C₄²	#6127	1	2	yes	—
294	C₄²	C₂×C₂²⋊D₁₃	1	2	yes	—
295	C₁₃	#9131	1	1	yes	characteristic, complemented, Sylow
296	C₂³	—	2	1	no	—
297	C₂³	—	2	1	no	—
298	C₂³	—	2	1	no	—
299	C₂³	—	2	1	no	—
300	C₂³	#88049	1	2	yes	—
301	C₂³	#88056	1	2	yes	—
302	C₂³	#88064	1	2	yes	—
303	C₂³	C₂×#6103	1	1	yes	characteristic
304	Q₈	—	104	1	no	—
305	Q₈	—	104	1	no	—
306	Q₈	—	104	1	no	—
307	Q₈	—	104	1	no	—
308	C₂×C₄	—	2	1	no	—
309	C₂×C₄	—	2	1	no	—
310	C₂×C₄	—	2	1	no	—
311	C₂×C₄	—	2	1	no	—
312	C₂×C₄	—	2	1	no	—
313	C₂×C₄	—	2	1	no	—
314	C₂×C₄	—	2	4	no	—
315	C₂×C₄	—	2	4	no	—
316	C₂×C₄	—	2	2	no	—
317	C₂×C₄	—	2	2	no	—
318	C₂×C₄	—	2	2	no	—
319	C₂×C₄	—	2	2	no	—
320	C₂×C₄	—	2	1	no	—
321	C₂×C₄	—	2	1	no	—
322	C₂×C₄	—	2	1	no	—
323	C₂×C₄	—	4	4	no	—
324	C₂×C₄	—	52	2	no	—
325	C₂×C₄	—	52	2	no	—
326	C₂×C₄	—	52	2	no	—
327	C₂×C₄	—	52	2	no	—
328	C₂×C₄	—	52	2	no	—
329	C₂×C₄	—	52	2	no	—
330	C₂×C₄	#88049	1	2	yes	—
331	C₂×C₄	#88056	1	2	yes	—
332	C₂×C₄	C₂×#6095	1	1	yes	characteristic
333	C₂×C₄	#88158	1	2	yes	—
334	C₂×C₄	C₂×Dic₅₂	1	1	yes	characteristic
335	C₂×C₄	C₄×C₂²⋊D₁₃	1	2	yes	—
336	C₂×C₄	#88183	1	1	yes	characteristic
337	C₂×C₄	#88184	1	2	yes	—
338	C₂×C₄	C₂×#6104	1	1	yes	characteristic
339	C₂×C₄	#88186	1	1	yes	characteristic
340	C₂×C₄	C₂×#6106	1	1	yes	characteristic
341	C₂×C₄	#88196	1	1	yes	characteristic
342	C₂×C₄	#88207	1	1	yes	characteristic
343	C₈	—	4	4	no	—
344	C₈	—	104	2	no	—
345	C₂²	—	2	2	no	—
346	C₂²	—	2	2	no	—
347	C₂²	—	2	2	no	—
348	C₂²	—	2	2	no	—
349	C₂²	—	2	2	no	—
350	C₂²	—	2	2	no	—
351	C₂²	#11718074	1	4	yes	complemented
352	C₂²	C₂×#88049	1	1	yes	characteristic
353	C₂²	#11718627	1	1	yes	characteristic
354	C₂²	C₂×#88056	1	1	yes	characteristic
355	C₂²	#11718774	1	1	yes	characteristic
356	C₂²	#11718793	1	1	yes	characteristic
357	C₂²	C₂×#88064	1	1	yes	characteristic
358	C₂²	#11718874	1	1	yes	characteristic
359	C₄	—	2	2	no	—
360	C₄	—	2	2	no	—
361	C₄	—	2	2	no	—
362	C₄	—	4	1	no	—
363	C₄	—	104	2	no	—
364	C₄	—	104	2	no	—
365	C₄	#11718628	1	1	yes	characteristic
366	C₄	#11718629	1	1	yes	characteristic
367	C₄	#11718775	1	2	yes	—
368	C₄	#11718777	1	2	yes	—
369	C₄	#11718878	1	1	yes	characteristic
370	C₄	#11718888	1	1	yes	characteristic
371	C₂	—	2	4	no	—
372	C₂	C₂×#11718074	1	1	yes	characteristic
373	C₂	#13113486	1	1	yes	characteristic
374	C₂	#13113524	1	1	yes	characteristic
375	C₂	#13113534	1	1	yes	characteristic
376	C₂	#13115153	1	1	yes	characteristic
377	C₂	#13115259	1	1	yes	characteristic
378	C₂	#13115304	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#6103
commutator	C₂²×C₄×C₁₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₃
Fitting	C₂²×C₁₃×C₄⋊C₈	C₂
socle	C₂³×C₁₃	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₁₃	C₂²⋊C₄
cosocle	C₂×C₄²×C₁₃	C₂³
hypercenter	C₂²×C₄⋊C₈	D₁₃
hypocenter	C₁₃	#9131
LCS	G ▷ C₂²×C₄×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂²×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
3. C₂×#11718074	C₂	semidirect
4. #13112720	C₂	semidirect
184. C₂²×C₁₃	#332	semidirect
295. C₁₃	#9131	semidirect
351. C₂²	#11718074	semidirect

Maximal subgroups

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

subgroup	index
#13123188	2
#13114816	2
C₂×#11718074	2
#13112720	2
C₂²×C₁₃×C₄⋊C₈	2
#9131	13