Subgroups — #15834432

Order 3328.

786 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×D₁₀₄	C₂	1	1	yes	maximal, characteristic
2	#13123571	C₂	1	2	yes	maximal, complemented
3	#13113749	C₂	1	2	yes	maximal, complemented
4	#13113699	C₂	1	4	yes	maximal, complemented
5	#13113659	C₂	1	2	yes	maximal, complemented
6	#13113260	C₂	1	2	yes	maximal, complemented
7	#13113125	C₂	1	1	yes	maximal, characteristic, complemented
8	C₁₃×#1566	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×M₁₆×D₁₃	—	2	2	no	—
10	C₂²×D₁₀₄	C₄	1	1	yes	characteristic
11	C₂×C₄×D₅₂	C₂²	1	2	yes	—
12	C₁₃×#423	C₂²	1	2	yes	—
13	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
14	#11718787	—	2	2	no	—
15	C₂×#88061	C₄	1	2	yes	—
16	C₂×#88058	C₄	1	1	yes	characteristic
17	#11718586	C₂²	1	1	yes	characteristic
18	C₄×#6099	C₂²	1	1	yes	characteristic
19	#11718499	C₂²	1	4	yes	complemented
20	#11718497	—	2	4	no	—
21	#11718495	C₂²	1	4	yes	complemented
22	C₄×D₁₀₄	C₄	1	2	yes	—
23	C₄×D₁₀₄	C₄	1	1	yes	characteristic
24	C₄×D₁₀₄	C₄	1	1	yes	characteristic
25	C₄×D₁₀₄	C₂²	1	1	yes	characteristic
26	C₄×D₁₀₄	C₂²	1	2	yes	—
27	C₄×D₁₀₄	C₂²	1	1	yes	characteristic
28	C₁₃×#334	C₂²	1	2	yes	complemented
29	C₁₃×#334	C₂²	1	2	yes	complemented
30	#11718075	C₂²	1	4	yes	complemented
31	#11718075	C₂²	1	4	yes	complemented
32	#11718047	C₂²	1	2	yes	complemented
33	#11718046	C₂²	1	2	yes	complemented
34	C₂²×D₅₂	C₂×C₄	1	2	yes	—
35	C₂²×C₄×D₁₃	—	2	2	no	—
36	C₂×C₁₃×M₁₆	—	2	2	no	—
37	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
38	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
40	C₂×#6103	—	2	2	no	—
41	C₂×C₄⋊Dic₁₃	C₂×C₄	1	2	yes	—
42	C₂×#6099	D₄	1	2	yes	—
43	M₁₆×D₁₃	—	4	8	no	—
44	C₂×D₁₀₄	—	2	4	no	—
45	C₂×D₁₀₄	C₂×C₄	1	2	yes	—
46	C₂×D₁₀₄	C₂×C₄	1	1	yes	characteristic
47	C₂×D₁₀₄	C₂×C₄	1	1	yes	characteristic
48	C₂×#6094	—	2	4	no	—
49	C₂×C₈×D₁₃	—	2	4	no	—
50	C₄×D₅₂	—	2	2	no	—
51	C₄×D₅₂	—	2	2	no	—
52	C₄×D₅₂	C₂×C₄	1	2	yes	—
53	C₄×D₅₂	C₂×C₄	1	2	yes	—
54	C₄×D₅₂	C₂³	1	2	yes	—
55	C₄×D₅₂	C₂³	1	2	yes	—
56	C₁₃×C₄⋊C₈	—	2	2	no	—
57	C₁₃×C₄⋊C₈	C₂³	1	4	yes	—
58	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
59	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
60	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
61	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
62	C₂²⋊#1055	—	2	1	no	—
63	#88061	C₂×C₄	1	2	yes	—
64	#88061	C₂×C₄	1	2	yes	—
65	#88061	C₂×C₄	1	2	yes	—
66	#88061	C₂×C₄	1	2	yes	—
67	#88060	—	2	4	no	—
68	#88060	—	2	4	no	—
69	#88058	C₂×C₄	1	2	yes	—
70	#88058	C₂×C₄	1	1	yes	characteristic
71	#88058	C₂×C₄	1	1	yes	characteristic
72	C₄⋊#1055	—	2	1	no	—
73	#88043	D₄	1	2	yes	—
74	C₄×#1055	C₂³	1	2	yes	—
75	C₄×#1055	C₂³	1	2	yes	—
76	#18044	—	13	1	no	maximal, Sylow
77	C₂³×D₁₃	—	2	2	no	—
78	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
79	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
80	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
81	C₂²×Dic₁₃	—	2	2	no	—
82	C₂×D₅₂	—	2	4	no	—
83	C₂×D₅₂	—	2	2	no	—
84	C₂×D₅₂	—	4	2	no	—
85	C₂×D₅₂	C₂²×C₄	1	2	yes	—
86	C₂×D₅₂	C₂²×C₄	1	2	yes	—
87	C₂×C₄×D₁₃	—	2	2	no	—
88	C₂×C₄×D₁₃	—	2	2	no	—
89	C₂×C₄×D₁₃	—	2	2	no	—
90	C₂×C₄×D₁₃	—	2	2	no	—
91	C₂×C₄×D₁₃	—	4	4	no	—
92	C₂×C₄×D₁₃	—	4	4	no	—
93	C₁₃×M₁₆	—	4	4	no	—
94	C₂×C₈×C₁₃	—	2	2	no	—
95	C₂×C₈×C₁₃	—	2	4	no	—
96	C₂×C₈×C₁₃	—	4	1	no	—
97	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
98	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
99	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
100	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
101	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
102	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
103	C₄²×C₁₃	#41	1	1	yes	characteristic
104	C₄²×C₁₃	C₂⁴	1	1	yes	characteristic
105	#6103	—	2	2	no	—
106	#6103	—	2	2	no	—
107	#6103	—	2	2	no	—
108	#6103	—	2	2	no	—
109	C₄⋊Dic₁₃	—	2	2	no	—
110	C₄⋊Dic₁₃	C₂²×C₄	1	2	yes	—
111	C₄⋊Dic₁₃	C₂²×C₄	1	2	yes	—
112	#6099	—	2	4	no	—
113	C₂×#1055	—	2	2	no	—
114	C₂×#1055	C₂×D₄	1	4	yes	—
115	D₁₀₄	—	2	4	no	—
116	D₁₀₄	—	2	2	no	—
117	D₁₀₄	M₁₆	1	4	yes	complemented
118	#6094	—	4	8	no	—
119	C₈×D₁₃	—	4	8	no	—
120	C₂×C₄×D₈	—	13	1	no	—
121	D₄×M₁₆	—	13	2	no	—
122	#1594	—	13	2	no	—
123	#1586	—	13	4	no	—
124	#1580	—	13	2	no	—
125	#1566	D₁₃	1	1	yes	characteristic, complemented
126	#1486	—	13	2	no	—
127	#1451	—	13	1	no	—
128	C₂³×C₁₃	C₄×D₄	1	1	yes	characteristic
129	C₂²×D₁₃	—	2	2	no	—
130	C₂²×D₁₃	—	2	2	no	—
131	C₂²×D₁₃	—	4	4	no	—
132	C₂²×D₁₃	—	4	4	no	—
133	C₂²×D₁₃	—	8	2	no	—
134	C₂×C₄×C₁₃	—	2	1	no	—
135	C₂×C₄×C₁₃	—	2	1	no	—
136	C₂×C₄×C₁₃	—	4	1	no	—
137	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
138	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
139	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
140	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	4	yes	—
141	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
142	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
143	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
144	C₂×Dic₁₃	—	2	2	no	—
145	C₂×Dic₁₃	—	2	2	no	—
146	C₂×Dic₁₃	—	4	4	no	—
147	D₅₂	—	2	2	no	—
148	D₅₂	—	2	2	no	—
149	D₅₂	—	2	2	no	—
150	D₅₂	—	4	4	no	—
151	D₅₂	C₂×M₁₆	1	4	yes	complemented
152	C₄×D₁₃	—	4	2	no	—
153	C₄×D₁₃	—	4	2	no	—
154	C₄×D₁₃	—	4	2	no	—
155	C₄×D₁₃	—	4	2	no	—
156	C₄×D₁₃	—	4	4	no	—
157	C₄×D₁₃	—	4	4	no	—
158	C₈×C₁₃	—	2	1	no	—
159	C₈×C₁₃	—	2	1	no	—
160	C₈×C₁₃	—	2	1	no	—
161	C₈×C₁₃	—	4	4	no	—
162	C₈×C₁₃	C₂×M₁₆	1	2	yes	complemented
163	#1055	—	2	4	no	—
164	#1055	—	4	2	no	—
165	C₂²×D₈	—	13	1	no	—
166	C₂²×M₁₆	—	26	2	no	—
167	C₂×C₄×D₄	—	13	2	no	—
168	C₄×D₈	—	13	1	no	—
169	C₄×D₈	—	13	2	no	—
170	C₄×D₈	—	13	1	no	—
171	C₄×D₈	—	13	1	no	—
172	C₄×D₈	—	13	2	no	—
173	C₄×D₈	—	13	1	no	—
174	#436	—	13	4	no	—
175	#435	—	26	4	no	—
176	C₈×D₄	—	13	4	no	—
177	#431	—	13	1	no	—
178	C₂×#107	—	13	1	no	—
179	#423	C₂×D₁₃	1	2	yes	—
180	C₂×#102	—	13	2	no	—
181	#407	—	26	2	no	—
182	C₄×M₁₆	—	13	1	no	—
183	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
184	#334	C₂×D₁₃	1	2	yes	complemented
185	#334	C₂×D₁₃	1	2	yes	complemented
186	D₄⋊C₈	—	13	4	no	—
187	D₄⋊C₈	—	13	4	no	—
188	#322	—	13	2	no	—
189	#322	—	13	2	no	—
190	C₂²×C₁₃	—	2	1	no	—
191	C₂²×C₁₃	—	2	1	no	—
192	C₂²×C₁₃	#442	1	2	yes	—
193	C₂²×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
194	C₂×D₁₃	—	2	4	no	—
195	C₂×D₁₃	—	4	2	no	—
196	C₂×D₁₃	—	4	2	no	—
197	C₂×D₁₃	—	4	2	no	—
198	C₂×D₁₃	—	4	2	no	—
199	C₂×D₁₃	—	4	2	no	—
200	C₂×D₁₃	—	8	8	no	—
201	C₄×C₁₃	—	2	1	no	—
202	C₄×C₁₃	—	2	1	no	—
203	C₄×C₁₃	—	2	1	no	—
204	C₄×C₁₃	#442	1	2	yes	—
205	C₄×C₁₃	C₂²×M₁₆	1	1	yes	characteristic
206	C₄×C₁₃	#568	1	1	yes	characteristic
207	C₄×C₁₃	C₂×AGL(1,ℤ/8)	1	2	yes	—
208	Dic₁₃	—	4	2	no	—
209	Dic₁₃	—	4	2	no	—
210	C₂²×D₄	—	13	2	no	—
211	C₂³×C₄	—	26	2	no	—
212	C₂×D₈	—	13	1	no	—
213	C₂×D₈	—	13	2	no	—
214	C₂×D₈	—	13	1	no	—
215	C₂×D₈	—	26	4	no	—
216	C₂×M₁₆	—	2	2	no	—
217	C₂×M₁₆	—	13	2	no	—
218	C₂×M₁₆	—	26	4	no	—
219	C₂×M₁₆	—	52	8	no	—
220	C₂²×C₈	—	26	4	no	—
221	C₂²×C₈	C₄×D₁₃	1	1	yes	characteristic
222	C₂²×C₈	C₄×D₁₃	1	1	yes	characteristic
223	C₄×D₄	—	13	2	no	—
224	C₄×D₄	—	13	2	no	—
225	C₄×D₄	—	13	2	no	—
226	C₄×D₄	—	13	2	no	—
227	C₄×D₄	—	26	2	no	—
228	C₄×D₄	—	26	2	no	—
229	C₂×C₄⋊C₄	—	13	2	no	—
230	C₂×C₂²⋊C₄	—	26	2	no	—
231	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
232	#107	—	13	2	no	—
233	#107	—	13	1	no	—
234	#107	—	13	1	no	—
235	C₄⋊C₈	—	2	2	no	—
236	C₄⋊C₈	—	26	1	no	—
237	C₄⋊C₈	C₂²×D₁₃	1	4	yes	—
238	#102	—	13	2	no	—
239	#102	—	13	2	no	—
240	#102	—	13	2	no	—
241	#102	—	13	2	no	—
242	C₂²⋊C₈	—	26	4	no	—
243	C₂²⋊C₈	—	26	4	no	—
244	C₂²⋊C₈	—	26	1	no	—
245	#97	—	13	2	no	—
246	C₄×C₈	—	13	2	no	—
247	C₄×C₈	—	13	2	no	—
248	C₄×C₈	C₄×D₁₃	1	1	yes	characteristic
249	C₄×C₈	C₄×D₁₃	1	1	yes	characteristic
250	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
251	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
252	C₂×C₁₃	—	2	2	no	—
253	C₂×C₁₃	D₄×M₁₆	1	1	yes	characteristic
254	C₂×C₁₃	C₂×#442	1	1	yes	characteristic
255	C₂×C₁₃	#2957	1	1	yes	characteristic
256	D₁₃	—	4	4	no	—
257	D₁₃	—	8	2	no	—
258	C₂⁴	—	26	2	no	—
259	C₂×D₄	—	13	2	no	—
260	C₂×D₄	—	13	2	no	—
261	C₂×D₄	—	26	4	no	—
262	C₂×D₄	—	26	2	no	—
263	C₂×D₄	—	52	2	no	—
264	C₂²×C₄	—	26	2	no	—
265	C₂²×C₄	—	26	2	no	—
266	C₂²×C₄	—	26	2	no	—
267	C₂²×C₄	—	26	2	no	—
268	C₂²×C₄	—	26	2	no	—
269	C₂²×C₄	—	52	4	no	—
270	C₂²×C₄	—	52	4	no	—
271	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
272	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
273	C₂²×C₄	#6131	1	1	yes	characteristic
274	D₈	—	13	4	no	—
275	D₈	—	26	4	no	—
276	D₈	—	26	2	no	—
277	M₁₆	—	4	4	no	—
278	M₁₆	—	26	4	no	—
279	M₁₆	—	52	8	no	—
280	C₂×C₈	—	2	2	no	—
281	C₂×C₈	—	2	4	no	—
282	C₂×C₈	—	4	1	no	—
283	C₂×C₈	—	13	4	no	—
284	C₂×C₈	—	26	2	no	—
285	C₂×C₈	—	52	8	no	—
286	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
287	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
288	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
289	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
290	C₄⋊C₄	—	13	2	no	—
291	C₄⋊C₄	—	13	2	no	—
292	C₄⋊C₄	—	26	2	no	—
293	C₂²⋊C₄	—	26	2	no	—
294	C₂²⋊C₄	—	26	2	no	—
295	C₂²⋊C₄	—	26	2	no	—
296	C₂²⋊C₄	—	26	2	no	—
297	C₄²	C₂×C₄×D₁₃	1	1	yes	characteristic
298	C₄²	D₄×D₁₃	1	1	yes	characteristic
299	C₄²	#6131	1	1	yes	characteristic
300	C₄²	C₂³×D₁₃	1	1	yes	characteristic
301	C₁₃	#18044	1	1	yes	characteristic, complemented, Sylow
302	C₂³	—	26	2	no	—
303	C₂³	—	26	2	no	—
304	C₂³	—	52	4	no	—
305	C₂³	—	52	4	no	—
306	C₂³	—	104	2	no	—
307	C₂³	#88147	1	1	yes	characteristic
308	D₄	—	13	4	no	—
309	D₄	—	26	2	no	—
310	D₄	—	26	2	no	—
311	D₄	—	26	2	no	—
312	D₄	—	52	4	no	—
313	C₂×C₄	—	2	1	no	—
314	C₂×C₄	—	2	1	no	—
315	C₂×C₄	—	4	1	no	—
316	C₂×C₄	—	26	2	no	—
317	C₂×C₄	—	26	2	no	—
318	C₂×C₄	—	52	4	no	—
319	C₂×C₄	—	52	2	no	—
320	C₂×C₄	—	52	2	no	—
321	C₂×C₄	—	52	2	no	—
322	C₂×C₄	—	52	2	no	—
323	C₂×C₄	—	52	4	no	—
324	C₂×C₄	—	52	4	no	—
325	C₂×C₄	#88147	1	1	yes	characteristic
326	C₂×C₄	#88147	1	1	yes	characteristic
327	C₂×C₄	#88147	1	1	yes	characteristic
328	C₂×C₄	#88168	1	4	yes	—
329	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
330	C₂×C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
331	C₂×C₄	C₂×#6131	1	1	yes	characteristic
332	C₈	—	2	1	no	—
333	C₈	—	2	1	no	—
334	C₈	—	2	1	no	—
335	C₈	—	4	4	no	—
336	C₈	—	26	4	no	—
337	C₈	—	52	2	no	—
338	C₈	M₁₆×D₁₃	1	2	yes	complemented
339	C₂²	—	2	1	no	—
340	C₂²	—	2	1	no	—
341	C₂²	—	26	4	no	—
342	C₂²	—	52	2	no	—
343	C₂²	—	52	2	no	—
344	C₂²	—	52	2	no	—
345	C₂²	—	52	2	no	—
346	C₂²	—	52	2	no	—
347	C₂²	—	104	8	no	—
348	C₂²	#11718530	1	2	yes	—
349	C₂²	C₂×#88147	1	1	yes	characteristic
350	C₄	—	2	1	no	—
351	C₄	—	2	1	no	—
352	C₄	—	2	1	no	—
353	C₄	—	52	2	no	—
354	C₄	—	52	2	no	—
355	C₄	#11718530	1	2	yes	—
356	C₄	C₂×M₁₆×D₁₃	1	1	yes	characteristic
357	C₄	#11719450	1	1	yes	characteristic
358	C₄	C₂×#88168	1	2	yes	—
359	C₂	—	2	2	no	—
360	C₂	—	52	4	no	—
361	C₂	—	104	2	no	—
362	C₂	#13123571	1	1	yes	characteristic
363	C₂	C₂×#11718530	1	1	yes	characteristic
364	C₂	#13123752	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₁₃
commutator	C₂×C₄×C₁₃	C₂³×C₄
Frattini	C₄²	C₂³×D₁₃
Fitting	C₁₃×#1566	C₂
socle	C₂²×C₁₃	C₂×C₄×D₄
prime radical	C₂²×D₁₀₄	C₄
cosocle	C₄²×C₁₃	C₂⁴
hypercenter	#1566	D₁₃
hypocenter	C₁₃	#18044
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1566

Complements

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

N	H ≅ G/N	type
2. #13123571	C₂	semidirect
3. #13113749	C₂	semidirect
4. #13113699	C₂	semidirect
5. #13113659	C₂	semidirect
6. #13113260	C₂	semidirect
7. #13113125	C₂	semidirect
8. C₁₃×#1566	C₂	semidirect
19. #11718499	C₂²	semidirect
21. #11718495	C₂²	semidirect
28. C₁₃×#334	C₂²	semidirect
29. C₁₃×#334	C₂²	semidirect
30. #11718075	C₂²	semidirect
31. #11718075	C₂²	semidirect
32. #11718047	C₂²	semidirect
33. #11718046	C₂²	semidirect
117. D₁₀₄	M₁₆	semidirect
125. #1566	D₁₃	semidirect
151. D₅₂	C₂×M₁₆	semidirect
162. C₈×C₁₃	C₂×M₁₆	semidirect
184. #334	C₂×D₁₃	semidirect
185. #334	C₂×D₁₃	semidirect
301. C₁₃	#18044	semidirect
338. C₈	M₁₆×D₁₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×D₁₀₄	2
#13123571	2
#13113749	2
#13113699	2
#13113659	2
#13113260	2
#13113125	2
C₁₃×#1566	2
#18044	13