Subgroups — #16016278

Order 3328.

718 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₁₃×D₁₆	C₂	1	1	yes	maximal, characteristic
2	#13123683	C₂	1	2	yes	maximal, complemented
3	#13116713	C₂	1	1	yes	maximal, characteristic, complemented
4	#13116712	C₂	1	1	yes	maximal, characteristic, complemented
5	#13116711	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄×#88066	C₂	1	2	yes	maximal, complemented
7	#13116680	C₂	1	2	yes	maximal, complemented
8	#13116619	C₂	1	1	yes	maximal, characteristic, complemented
9	#13116550	C₂	1	2	yes	maximal, complemented
10	#13116434	C₂	1	1	yes	maximal, characteristic, complemented
11	C₁₃×#2177	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×D₈×D₁₃	C₂²	1	2	yes	—
13	#11719250	—	2	2	no	—
14	C₂×C₁₃×D₁₆	C₂²	1	1	yes	characteristic
15	C₄×C₁₃×D₈	C₂²	1	2	yes	complemented
16	D₈×Dic₁₃	C₂²	1	2	yes	complemented
17	C₂×#88066	C₂²	1	2	yes	—
18	#11718675	—	2	2	no	—
19	D₁₃×D₁₆	C₄	1	8	yes	complemented
20	C₂×D₂₀₈	C₂²	1	1	yes	characteristic
21	C₂×#88038	C₂²	1	1	yes	characteristic
22	C₂×C₁₆×D₁₃	C₂²	1	1	yes	characteristic
23	#11718541	C₂²	1	2	yes	—
24	#11718533	C₂²	1	2	yes	complemented
25	#11718446	—	2	2	no	—
26	#11718431	—	2	2	no	—
27	C₄×D₁₀₄	C₂²	1	2	yes	complemented
28	#11718350	C₂²	1	1	yes	characteristic
29	C₁₃×#367	C₂²	1	1	yes	characteristic, complemented
30	C₁₃×#357	C₂²	1	2	yes	complemented
31	C₁₃×#346	C₂²	1	1	yes	characteristic, complemented
32	#11718154	C₂²	1	2	yes	complemented
33	#11718101	C₂²	1	2	yes	complemented
34	#11718097	C₂²	1	1	yes	characteristic, complemented
35	#11718094	C₂²	1	1	yes	characteristic, complemented
36	C₁₆×Dic₁₃	C₂²	1	1	yes	characteristic, complemented
37	#11718083	C₂²	1	2	yes	complemented
38	#11718081	C₂²	1	2	yes	complemented
39	C₄×#6090	C₂²	1	1	yes	characteristic, complemented
40	C₂×D₄×D₁₃	—	2	2	no	—
41	C₂×C₁₃×D₈	C₂³	1	2	yes	—
42	C₄×D₄×C₁₃	—	2	2	no	—
43	D₄×Dic₁₃	—	2	2	no	—
44	C₂×#6104	—	2	2	no	—
45	C₄×C₂²⋊D₁₃	—	4	2	no	—
46	C₂×#6103	—	4	2	no	—
47	D₈×D₁₃	—	2	4	no	—
48	D₈×D₁₃	C₂×C₄	1	8	yes	complemented
49	C₂×D₁₀₄	C₂³	1	2	yes	—
50	C₂×#6094	D₄	1	1	yes	characteristic
51	C₂×C₈×D₁₃	C₂³	1	1	yes	characteristic
52	#88147	—	2	2	no	—
53	#88146	—	2	2	no	—
54	#88135	—	4	2	no	—
55	C₂²⋊C₄×D₁₃	—	4	2	no	—
56	C₄×D₅₂	—	2	2	no	—
57	#88126	D₄	1	1	yes	characteristic
58	C₁₃×D₁₆	C₂×C₄	1	4	yes	complemented
59	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
60	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
61	C₁₃×#107	C₂³	1	2	yes	—
62	C₁₃×#102	—	2	2	no	—
63	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
64	#88072	—	2	2	no	—
65	#88066	C₂×C₄	1	8	yes	complemented
66	#88061	—	2	2	no	—
67	#88058	C₂³	1	2	yes	—
68	#88055	D₄	1	1	yes	characteristic
69	C₈×Dic₁₃	C₂³	1	1	yes	characteristic
70	C₂×#6090	C₂³	1	1	yes	characteristic
71	C₂×#6090	C₂³	1	1	yes	characteristic
72	#88049	—	2	2	no	—
73	#88048	—	2	2	no	—
74	#88043	D₄	1	1	yes	characteristic
75	D₂₀₈	C₂×C₄	1	4	yes	complemented
76	#88038	—	2	2	no	—
77	C₁₆×D₁₃	C₂×C₄	1	4	yes	complemented
78	#33456	—	13	1	no	maximal, Sylow
79	C₂³×D₁₃	—	4	2	no	—
80	C₂×D₄×C₁₃	—	2	2	no	—
81	C₂²×C₄×C₁₃	—	4	2	no	—
82	C₂×C₂²⋊D₁₃	—	4	2	no	—
83	C₂²×Dic₁₃	—	4	2	no	—
84	D₄×D₁₃	—	2	8	no	—
85	D₄×D₁₃	—	4	4	no	—
86	C₂×D₅₂	—	2	2	no	—
87	C₂×C₄×D₁₃	—	4	2	no	—
88	C₂×C₄×D₁₃	—	4	2	no	—
89	C₂×C₄×D₁₃	C₂×D₄	1	1	yes	characteristic
90	C₁₃×D₈	—	2	2	no	—
91	C₁₃×D₈	C₂²×C₄	1	4	yes	—
92	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
93	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
94	C₁₃×C₄⋊C₄	—	2	2	no	—
95	C₁₃×C₂²⋊C₄	—	4	2	no	—
96	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
97	C₂²⋊Dic₁₃	—	4	2	no	—
98	#6104	—	2	8	no	—
99	#6103	—	4	2	no	—
100	#6103	—	4	2	no	—
101	#6103	—	4	2	no	—
102	#6103	D₈	1	2	yes	—
103	C₄⋊Dic₁₃	—	2	2	no	—
104	#6101	D₈	1	2	yes	—
105	C₄×Dic₁₃	—	4	2	no	—
106	C₄×Dic₁₃	C₂×D₄	1	1	yes	characteristic
107	C₂×#1055	C₂×D₄	1	1	yes	characteristic
108	C₂×#1055	C₂×D₄	1	1	yes	characteristic
109	D₁₀₄	—	2	2	no	—
110	D₁₀₄	C₂²×C₄	1	4	yes	—
111	#6094	—	2	2	no	—
112	C₈×D₁₃	C₂²×C₄	1	2	yes	—
113	C₈×D₁₃	#41	1	2	yes	—
114	C₁₃×C₁₆	—	2	1	no	—
115	C₁₃×C₁₆	C₂²×C₄	1	2	yes	—
116	#6090	—	2	1	no	—
117	#6090	C₂²×C₄	1	2	yes	—
118	C₂²×D₁₆	—	13	1	no	—
119	#2948	—	13	2	no	—
120	#2177	—	13	1	no	—
121	#2177	—	13	1	no	—
122	#2177	—	13	1	no	—
123	#2177	D₁₃	1	1	yes	characteristic, complemented
124	C₄×D₁₆	—	13	2	no	—
125	C₄×D₁₆	—	13	2	no	—
126	#2158	—	13	1	no	—
127	#2140	—	13	2	no	—
128	#2108	—	13	1	no	—
129	C₂³×C₁₃	—	4	2	no	—
130	C₂²×D₁₃	—	4	2	no	—
131	C₂²×D₁₃	—	4	8	no	—
132	C₂²×D₁₃	—	16	2	no	—
133	C₂²×D₁₃	—	16	2	no	—
134	C₂²×D₁₃	C₂×D₈	1	1	yes	characteristic
135	D₄×C₁₃	—	2	4	no	—
136	D₄×C₁₃	—	4	2	no	—
137	C₂×C₄×C₁₃	—	4	2	no	—
138	C₂×C₄×C₁₃	—	16	2	no	—
139	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
140	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
141	C₂²⋊D₁₃	—	4	8	no	—
142	C₂×Dic₁₃	—	4	2	no	—
143	C₂×Dic₁₃	—	16	2	no	—
144	C₂×Dic₁₃	C₂×D₈	1	2	yes	—
145	C₂×Dic₁₃	C₂×D₈	1	1	yes	characteristic
146	D₅₂	—	2	4	no	—
147	D₅₂	—	4	2	no	—
148	C₄×D₁₃	—	16	2	no	—
149	C₄×D₁₃	—	16	2	no	—
150	C₄×D₁₃	C₄×D₄	1	2	yes	—
151	C₄×D₁₃	#135	1	2	yes	—
152	C₈×C₁₃	—	2	1	no	—
153	C₈×C₁₃	C₂³×C₄	1	1	yes	characteristic
154	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
155	#1055	—	2	1	no	—
156	#1055	C₄×D₄	1	2	yes	—
157	C₂²×D₈	—	13	2	no	—
158	#518	—	26	2	no	—
159	C₂×D₁₆	—	13	2	no	—
160	C₂×D₁₆	—	13	8	no	—
161	C₂×D₁₆	—	13	1	no	—
162	C₂×D₁₆	C₂×D₁₃	1	1	yes	characteristic
163	C₂×M₃₂	—	13	1	no	—
164	C₂²×C₁₆	—	13	1	no	—
165	#442	—	26	2	no	—
166	#442	—	26	2	no	—
167	C₄×D₈	—	13	2	no	—
168	C₄×D₈	—	13	2	no	—
169	C₄×D₈	—	13	2	no	—
170	C₄×D₈	C₂×D₁₃	1	2	yes	complemented
171	#427	—	13	2	no	—
172	#418	—	26	2	no	—
173	#405	—	13	1	no	—
174	#367	—	13	1	no	—
175	#367	C₂×D₁₃	1	1	yes	characteristic, complemented
176	#366	—	13	2	no	—
177	#357	—	13	2	no	—
178	#357	—	13	2	no	—
179	#357	—	13	2	no	—
180	#357	C₂×D₁₃	1	2	yes	complemented
181	#346	—	13	1	no	—
182	#346	C₂×D₁₃	1	1	yes	characteristic, complemented
183	C₄×C₁₆	—	13	1	no	—
184	C₄×C₁₆	—	13	1	no	—
185	C₂²×C₁₃	—	4	4	no	—
186	C₂²×C₁₃	—	8	2	no	—
187	C₂²×C₁₃	—	8	2	no	—
188	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
189	C₂×D₁₃	—	2	2	no	—
190	C₂×D₁₃	—	4	4	no	—
191	C₂×D₁₃	—	8	2	no	—
192	C₂×D₁₃	—	8	2	no	—
193	C₂×D₁₃	—	16	8	no	—
194	C₂×D₁₃	C₄×D₈	1	2	yes	—
195	C₄×C₁₃	—	2	2	no	—
196	C₄×C₁₃	—	8	2	no	—
197	C₄×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
198	C₄×C₁₃	C₂×#135	1	1	yes	characteristic
199	Dic₁₃	—	2	2	no	—
200	Dic₁₃	—	8	2	no	—
201	Dic₁₃	C₄×D₈	1	2	yes	—
202	C₂²×D₄	—	26	2	no	—
203	C₂×D₈	—	13	8	no	—
204	C₂×D₈	—	13	2	no	—
205	C₂×D₈	—	26	4	no	—
206	C₂×D₈	—	26	2	no	—
207	C₂×D₈	C₂²×D₁₃	1	2	yes	—
208	C₂×M₁₆	—	13	1	no	—
209	C₂²×C₈	—	13	1	no	—
210	C₄×D₄	—	2	2	no	—
211	C₄×D₄	—	26	2	no	—
212	C₄×D₄	—	26	2	no	—
213	C₄×D₄	—	26	2	no	—
214	C₄×D₄	—	52	2	no	—
215	C₄×D₄	—	52	2	no	—
216	#117	—	13	1	no	—
217	#117	—	26	2	no	—
218	C₂×C₂²⋊C₄	—	52	2	no	—
219	C₂×C₂²⋊C₄	—	52	2	no	—
220	D₁₆	—	13	8	no	—
221	D₁₆	—	13	4	no	—
222	D₁₆	C₄×D₁₃	1	4	yes	complemented
223	M₃₂	—	26	2	no	—
224	C₂×C₁₆	—	13	1	no	—
225	C₂×C₁₆	—	13	4	no	—
226	C₂×C₁₆	—	13	1	no	—
227	C₂×C₁₆	C₂²×D₁₃	1	1	yes	characteristic
228	C₂×C₁₆	C₂²×D₁₃	1	1	yes	characteristic
229	#107	—	13	2	no	—
230	#107	C₂²×D₁₃	1	2	yes	—
231	#106	—	26	2	no	—
232	#102	—	2	2	no	—
233	#102	—	26	2	no	—
234	#102	—	26	2	no	—
235	#102	—	26	2	no	—
236	#97	—	13	1	no	—
237	#97	—	13	1	no	—
238	C₄×C₈	—	13	1	no	—
239	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
240	C₂×C₁₃	—	8	4	no	—
241	C₂×C₁₃	C₂×C₄×D₈	1	1	yes	characteristic
242	C₂×C₁₃	#3415	1	2	yes	—
243	D₁₃	—	2	2	no	—
244	D₁₃	—	8	4	no	—
245	C₂⁴	—	52	2	no	—
246	C₂×D₄	—	2	2	no	—
247	C₂×D₄	—	26	8	no	—
248	C₂×D₄	—	26	2	no	—
249	C₂×D₄	—	52	4	no	—
250	C₂×D₄	—	52	2	no	—
251	C₂²×C₄	—	4	2	no	—
252	C₂²×C₄	—	13	1	no	—
253	C₂²×C₄	—	52	2	no	—
254	C₂²×C₄	—	52	2	no	—
255	C₂²×C₄	—	52	2	no	—
256	D₈	—	2	2	no	—
257	D₈	—	13	4	no	—
258	D₈	—	26	8	no	—
259	D₈	—	26	2	no	—
260	D₈	C₂×C₄×D₁₃	1	4	yes	—
261	M₁₆	—	26	2	no	—
262	C₂×C₈	—	13	1	no	—
263	C₂×C₈	—	13	2	no	—
264	C₂×C₈	—	13	2	no	—
265	C₂×C₈	—	13	1	no	—
266	C₂×C₈	D₄×D₁₃	1	1	yes	characteristic
267	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
268	C₄⋊C₄	—	2	2	no	—
269	C₄⋊C₄	—	13	2	no	—
270	C₄⋊C₄	—	26	2	no	—
271	C₂²⋊C₄	—	4	2	no	—
272	C₂²⋊C₄	—	13	2	no	—
273	C₂²⋊C₄	—	52	2	no	—
274	C₂²⋊C₄	—	52	2	no	—
275	C₂²⋊C₄	—	52	2	no	—
276	C₂²⋊C₄	—	52	2	no	—
277	C₄²	—	13	1	no	—
278	C₄²	—	52	2	no	—
279	C₄²	D₄×D₁₃	1	1	yes	characteristic
280	C₁₆	—	2	1	no	—
281	C₁₆	—	13	2	no	—
282	C₁₆	—	26	1	no	—
283	C₁₆	C₂×C₄×D₁₃	1	2	yes	—
284	C₁₃	#33456	1	1	yes	characteristic, complemented, Sylow
285	C₂³	—	4	2	no	—
286	C₂³	—	13	1	no	—
287	C₂³	—	52	8	no	—
288	C₂³	—	52	2	no	—
289	C₂³	—	208	2	no	—
290	C₂³	—	208	2	no	—
291	D₄	—	2	4	no	—
292	D₄	—	4	2	no	—
293	D₄	—	26	4	no	—
294	D₄	—	52	8	no	—
295	D₄	—	52	2	no	—
296	C₂×C₄	—	4	2	no	—
297	C₂×C₄	—	13	1	no	—
298	C₂×C₄	—	13	2	no	—
299	C₂×C₄	—	13	2	no	—
300	C₂×C₄	—	13	2	no	—
301	C₂×C₄	—	16	2	no	—
302	C₂×C₄	—	52	2	no	—
303	C₂×C₄	—	208	2	no	—
304	C₂×C₄	—	208	2	no	—
305	C₂×C₄	—	208	2	no	—
306	C₂×C₄	D₈×D₁₃	1	2	yes	—
307	C₂×C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
308	C₈	—	2	1	no	—
309	C₈	—	13	2	no	—
310	C₈	—	26	1	no	—
311	C₈	C₂²×C₄×D₁₃	1	1	yes	characteristic
312	C₈	#41×D₁₃	1	1	yes	characteristic
313	C₂²	—	4	4	no	—
314	C₂²	—	8	2	no	—
315	C₂²	—	8	2	no	—
316	C₂²	—	13	2	no	—
317	C₂²	—	26	2	no	—
318	C₂²	—	52	4	no	—
319	C₂²	—	104	2	no	—
320	C₂²	—	104	2	no	—
321	C₂²	—	208	8	no	—
322	C₂²	C₂×D₈×D₁₃	1	1	yes	characteristic
323	C₄	—	2	2	no	—
324	C₄	—	8	2	no	—
325	C₄	—	13	2	no	—
326	C₄	—	26	2	no	—
327	C₄	—	104	2	no	—
328	C₄	C₄×D₄×D₁₃	1	1	yes	characteristic
329	C₄	D₁₃×#135	1	1	yes	characteristic
330	C₂	—	8	4	no	—
331	C₂	—	26	2	no	—
332	C₂	—	104	4	no	—
333	C₂	C₄×D₈×D₁₃	1	1	yes	characteristic
334	C₂	#13127976	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₈×D₁₃
commutator	C₈×C₁₃	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₁₃
Fitting	C₁₃×#2177	C₂
socle	C₂²×C₁₃	C₂²×D₈
prime radical	C₂×D₁₃×D₁₆	C₂
cosocle	C₂×C₈×C₁₃	C₂⁴
hypercenter	#2177	D₁₃
hypocenter	C₁₃	#33456
LCS	G ▷ C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2177

Complements

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

N	H ≅ G/N	type
2. #13123683	C₂	semidirect
3. #13116713	C₂	semidirect
4. #13116712	C₂	semidirect
5. #13116711	C₂	semidirect
6. C₄×#88066	C₂	semidirect
7. #13116680	C₂	semidirect
8. #13116619	C₂	semidirect
9. #13116550	C₂	semidirect
10. #13116434	C₂	semidirect
11. C₁₃×#2177	C₂	semidirect
15. C₄×C₁₃×D₈	C₂²	semidirect
16. D₈×Dic₁₃	C₂²	semidirect
19. D₁₃×D₁₆	C₄	semidirect
24. #11718533	C₂²	semidirect
27. C₄×D₁₀₄	C₂²	semidirect
29. C₁₃×#367	C₂²	semidirect
30. C₁₃×#357	C₂²	semidirect
31. C₁₃×#346	C₂²	semidirect
32. #11718154	C₂²	semidirect
33. #11718101	C₂²	semidirect
34. #11718097	C₂²	semidirect
35. #11718094	C₂²	semidirect
36. C₁₆×Dic₁₃	C₂²	semidirect
37. #11718083	C₂²	semidirect
38. #11718081	C₂²	semidirect
39. C₄×#6090	C₂²	semidirect
48. D₈×D₁₃	C₂×C₄	semidirect
58. C₁₃×D₁₆	C₂×C₄	semidirect
65. #88066	C₂×C₄	semidirect
75. D₂₀₈	C₂×C₄	semidirect
77. C₁₆×D₁₃	C₂×C₄	semidirect
123. #2177	D₁₃	semidirect
170. C₄×D₈	C₂×D₁₃	semidirect
175. #367	C₂×D₁₃	semidirect
180. #357	C₂×D₁₃	semidirect
182. #346	C₂×D₁₃	semidirect
222. D₁₆	C₄×D₁₃	semidirect
284. C₁₃	#33456	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₁₃×D₁₆	2
#13123683	2
#13116713	2
#13116712	2
#13116711	2
C₄×#88066	2
#13116680	2
#13116619	2
#13116550	2
#13116434	2
C₁₃×#2177	2
#33456	13