Subgroups — #15758564

Order 3328.

644 conjugacy classes of subgroups.

The subgroup lattice has 376 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×#11718078	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	#11718078	—	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	#9129	—	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	2	no	—
81	C₂×Dic₂₆	—	4	1	no	—
82	C₂×Dic₂₆	—	4	1	no	—
83	C₂×Dic₂₆	—	4	2	no	—
84	C₂×C₈×C₁₃	—	2	4	no	—
85	C₂×C₈×C₁₃	—	4	8	no	—
86	C₂×C₈×C₁₃	—	4	2	no	—
87	C₂×C₈×C₁₃	—	4	2	no	—
88	C₄²×C₁₃	—	2	2	no	—
89	C₄²×C₁₃	—	2	1	no	—
90	C₄²×C₁₃	—	2	2	no	—
91	C₄²×C₁₃	—	2	1	no	—
92	C₄²×C₁₃	C₂×D₄	1	2	yes	—
93	C₄²×C₁₃	#41	1	2	yes	—
94	C₂²⋊Dic₁₃	—	4	4	no	—
95	C₄⋊Dic₁₃	—	2	4	no	—
96	C₄⋊Dic₁₃	—	4	1	no	—
97	C₄⋊Dic₁₃	—	4	1	no	—
98	C₄⋊Dic₁₃	—	4	1	no	—
99	C₄⋊Dic₁₃	—	4	1	no	—
100	C₄⋊Dic₁₃	—	4	1	no	—
101	C₄⋊Dic₁₃	—	4	1	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	2	no	—
156	Dic₂₆	—	8	2	no	—
157	C₈×C₁₃	—	4	4	no	—
158	#1055	—	8	2	no	—
159	C₂³×C₈	—	2	1	no	—
160	C₂×C₄⋊Q₈	—	13	1	no	—
161	C₂×C₂²⋊Q₈	—	26	1	no	—
162	C₂²×C₄²	C₂×D₁₃	1	1	yes	characteristic
163	C₂×C₄⋊C₈	—	2	1	no	—
164	C₂×C₄⋊C₈	—	2	4	no	—
165	C₂×C₄⋊C₈	—	13	2	no	—
166	C₂×C₄⋊C₈	C₂×D₁₃	1	2	yes	—
167	C₂×C₂²⋊C₈	—	26	1	no	—
168	#391	—	26	1	no	—
169	#389	—	13	2	no	—
170	#389	—	26	1	no	—
171	#384	—	13	1	no	—
172	#380	—	13	2	no	—
173	#380	—	26	1	no	—
174	#336	—	26	1	no	—
175	#332	—	26	4	no	—
176	C₂²×C₁₃	—	2	2	no	—
177	C₂²×C₁₃	—	2	2	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	#332	1	4	yes	complemented
183	C₂²×C₁₃	#386	1	1	yes	characteristic
184	C₂²×C₁₃	C₂×#101	1	1	yes	characteristic
185	C₂²×C₁₃	#413	1	1	yes	characteristic
186	C₂²×C₁₃	C₂×#102	1	1	yes	characteristic
187	C₂²×C₁₃	C₂×#103	1	1	yes	characteristic
188	C₂²×C₁₃	#416	1	1	yes	characteristic
189	C₂²×C₁₃	#417	1	1	yes	characteristic
190	C₄×C₁₃	—	2	2	no	—
191	C₄×C₁₃	—	2	2	no	—
192	C₄×C₁₃	—	2	2	no	—
193	C₄×C₁₃	—	4	1	no	—
194	C₄×C₁₃	#450	1	1	yes	characteristic
195	C₄×C₁₃	#451	1	1	yes	characteristic
196	C₄×C₁₃	#465	1	2	yes	—
197	C₄×C₁₃	#466	1	2	yes	—
198	C₄×C₁₃	#483	1	1	yes	characteristic
199	C₄×C₁₃	#484	1	1	yes	characteristic
200	Dic₁₃	—	8	2	no	—
201	Dic₁₃	—	8	2	no	—
202	C₂²×Q₈	—	26	1	no	—
203	C₂³×C₄	C₄×D₁₃	1	1	yes	characteristic
204	C₂³×C₄	D₅₂	1	1	yes	characteristic
205	C₂³×C₄	C₂²⋊D₁₃	1	1	yes	characteristic
206	C₂²×C₈	—	2	4	no	—
207	C₂²×C₈	—	2	2	no	—
208	C₂²×C₈	—	4	2	no	—
209	C₂²×C₈	—	4	2	no	—
210	C₂²×C₈	—	26	2	no	—
211	C₄⋊Q₈	—	26	2	no	—
212	C₄⋊Q₈	—	26	2	no	—
213	C₂²⋊Q₈	—	52	2	no	—
214	C₂²⋊Q₈	—	52	2	no	—
215	C₂×C₄⋊C₄	—	13	2	no	—
216	C₂×C₄⋊C₄	—	13	2	no	—
217	C₂×C₄⋊C₄	—	26	2	no	—
218	C₂×C₄⋊C₄	—	26	1	no	—
219	C₂×C₄⋊C₄	—	26	1	no	—
220	C₂×C₄⋊C₄	—	26	1	no	—
221	C₂×C₂²⋊C₄	—	26	2	no	—
222	C₂×C₄²	—	2	2	no	—
223	C₂×C₄²	—	2	2	no	—
224	C₂×C₄²	C₄×D₁₃	1	1	yes	characteristic
225	C₂×C₄²	D₅₂	1	1	yes	characteristic
226	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
227	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
228	C₄⋊C₈	—	2	4	no	—
229	C₄⋊C₈	—	2	2	no	—
230	C₄⋊C₈	—	26	2	no	—
231	C₄⋊C₈	—	26	2	no	—
232	C₄⋊C₈	C₂²⋊D₁₃	1	4	yes	—
233	C₂²⋊C₈	—	52	2	no	—
234	#95	—	26	2	no	—
235	#95	—	26	2	no	—
236	C₂×C₁₃	—	2	4	no	—
237	C₂×C₁₃	C₂×#332	1	1	yes	characteristic
238	C₂×C₁₃	#1542	1	1	yes	characteristic
239	C₂×C₁₃	#1547	1	1	yes	characteristic
240	C₂×C₁₃	#1552	1	1	yes	characteristic
241	C₂×C₁₃	#1872	1	1	yes	characteristic
242	C₂×C₁₃	#1893	1	1	yes	characteristic
243	C₂×C₁₃	#1901	1	1	yes	characteristic
244	C₂⁴	#6103	1	1	yes	characteristic
245	C₂×Q₈	—	52	1	no	—
246	C₂×Q₈	—	52	1	no	—
247	C₂×Q₈	—	52	2	no	—
248	C₂×Q₈	—	52	2	no	—
249	C₂²×C₄	—	2	2	no	—
250	C₂²×C₄	—	2	1	no	—
251	C₂²×C₄	—	2	2	no	—
252	C₂²×C₄	—	2	1	no	—
253	C₂²×C₄	—	2	1	no	—
254	C₂²×C₄	—	2	1	no	—
255	C₂²×C₄	—	2	2	no	—
256	C₂²×C₄	—	2	2	no	—
257	C₂²×C₄	—	4	1	no	—
258	C₂²×C₄	—	26	2	no	—
259	C₂²×C₄	—	26	2	no	—
260	C₂²×C₄	#6095	1	2	yes	—
261	C₂²×C₄	D₁₀₄	1	2	yes	—
262	C₂²×C₄	#6103	1	1	yes	characteristic
263	C₂²×C₄	#6103	1	1	yes	characteristic
264	C₂²×C₄	#6103	1	1	yes	characteristic
265	C₂²×C₄	#6105	1	2	yes	—
266	C₂²×C₄	#6107	1	2	yes	—
267	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
268	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
269	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
270	C₂×C₈	—	2	4	no	—
271	C₂×C₈	—	4	8	no	—
272	C₂×C₈	—	4	2	no	—
273	C₂×C₈	—	4	2	no	—
274	C₂×C₈	—	52	2	no	—
275	C₂×C₈	—	52	2	no	—
276	C₂×C₈	—	52	2	no	—
277	C₄⋊C₄	—	26	4	no	—
278	C₄⋊C₄	—	52	2	no	—
279	C₄⋊C₄	—	52	1	no	—
280	C₄⋊C₄	—	52	1	no	—
281	C₄⋊C₄	—	52	1	no	—
282	C₄⋊C₄	—	52	1	no	—
283	C₄⋊C₄	—	52	1	no	—
284	C₄⋊C₄	—	52	1	no	—
285	C₄⋊C₄	—	52	4	no	—
286	C₂²⋊C₄	—	52	4	no	—
287	C₄²	—	2	2	no	—
288	C₄²	—	2	1	no	—
289	C₄²	—	2	2	no	—
290	C₄²	—	2	1	no	—
291	C₄²	#6127	1	2	yes	—
292	C₄²	C₂×C₂²⋊D₁₃	1	2	yes	—
293	C₁₃	#9129	1	1	yes	characteristic, complemented, Sylow
294	C₂³	—	2	1	no	—
295	C₂³	—	2	1	no	—
296	C₂³	—	2	1	no	—
297	C₂³	—	2	1	no	—
298	C₂³	#88050	1	2	yes	—
299	C₂³	#88061	1	2	yes	—
300	C₂³	#88064	1	2	yes	—
301	C₂³	C₂×#6103	1	1	yes	characteristic
302	Q₈	—	104	2	no	—
303	Q₈	—	104	2	no	—
304	C₂×C₄	—	2	1	no	—
305	C₂×C₄	—	2	1	no	—
306	C₂×C₄	—	2	1	no	—
307	C₂×C₄	—	2	1	no	—
308	C₂×C₄	—	2	1	no	—
309	C₂×C₄	—	2	1	no	—
310	C₂×C₄	—	2	4	no	—
311	C₂×C₄	—	2	4	no	—
312	C₂×C₄	—	2	2	no	—
313	C₂×C₄	—	2	2	no	—
314	C₂×C₄	—	2	2	no	—
315	C₂×C₄	—	2	2	no	—
316	C₂×C₄	—	2	1	no	—
317	C₂×C₄	—	2	1	no	—
318	C₂×C₄	—	2	1	no	—
319	C₂×C₄	—	4	4	no	—
320	C₂×C₄	—	52	2	no	—
321	C₂×C₄	—	52	2	no	—
322	C₂×C₄	—	52	2	no	—
323	C₂×C₄	—	52	2	no	—
324	C₂×C₄	—	52	2	no	—
325	C₂×C₄	—	52	2	no	—
326	C₂×C₄	#88050	1	2	yes	—
327	C₂×C₄	#88061	1	2	yes	—
328	C₂×C₄	C₂×#6095	1	1	yes	characteristic
329	C₂×C₄	C₂×D₁₀₄	1	1	yes	characteristic
330	C₂×C₄	#88158	1	2	yes	—
331	C₂×C₄	C₄×C₂²⋊D₁₃	1	2	yes	—
332	C₂×C₄	#88183	1	1	yes	characteristic
333	C₂×C₄	#88184	1	2	yes	—
334	C₂×C₄	#88186	1	1	yes	characteristic
335	C₂×C₄	C₂×#6105	1	1	yes	characteristic
336	C₂×C₄	#88196	1	1	yes	characteristic
337	C₂×C₄	C₂×#6107	1	1	yes	characteristic
338	C₂×C₄	#88207	1	1	yes	characteristic
339	C₈	—	4	4	no	—
340	C₈	—	104	2	no	—
341	C₂²	—	2	2	no	—
342	C₂²	—	2	2	no	—
343	C₂²	—	2	2	no	—
344	C₂²	—	2	2	no	—
345	C₂²	—	2	2	no	—
346	C₂²	—	2	2	no	—
347	C₂²	#11718078	1	4	yes	complemented
348	C₂²	#11718627	1	1	yes	characteristic
349	C₂²	C₂×#88050	1	1	yes	characteristic
350	C₂²	C₂×#88061	1	1	yes	characteristic
351	C₂²	#11718774	1	1	yes	characteristic
352	C₂²	#11718793	1	1	yes	characteristic
353	C₂²	C₂×#88064	1	1	yes	characteristic
354	C₂²	#11718874	1	1	yes	characteristic
355	C₄	—	2	2	no	—
356	C₄	—	2	2	no	—
357	C₄	—	2	2	no	—
358	C₄	—	4	1	no	—
359	C₄	—	104	2	no	—
360	C₄	—	104	2	no	—
361	C₄	#11718631	1	1	yes	characteristic
362	C₄	#11718632	1	1	yes	characteristic
363	C₄	#11718775	1	2	yes	—
364	C₄	#11718776	1	2	yes	—
365	C₄	#11718879	1	1	yes	characteristic
366	C₄	#11718889	1	1	yes	characteristic
367	C₂	—	2	4	no	—
368	C₂	C₂×#11718078	1	1	yes	characteristic
369	C₂	#13113494	1	1	yes	characteristic
370	C₂	#13113511	1	1	yes	characteristic
371	C₂	#13113534	1	1	yes	characteristic
372	C₂	#13115143	1	1	yes	characteristic
373	C₂	#13115264	1	1	yes	characteristic
374	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	#9129
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×#11718078	C₂	semidirect
4. #13112720	C₂	semidirect
182. C₂²×C₁₃	#332	semidirect
293. C₁₃	#9129	semidirect
347. C₂²	#11718078	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×#11718078	2
#13112720	2
C₂²×C₁₃×C₄⋊C₈	2
#9129	13