Subgroups — #16789966

Order 3328.

450 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13131168	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₄⋊#6092	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#11718270	C₂	1	2	yes	maximal
4	#13129891	C₂	1	2	yes	maximal, complemented
5	C₄×#88161	C₂	1	1	yes	maximal, characteristic
6	C₂×#88161	C₄	1	1	yes	characteristic
7	C₂×#88161	C₂²	1	1	yes	characteristic
8	C₄×#6127	C₄	1	1	yes	characteristic
9	C₂×#88106	C₂²	1	2	yes	—
10	C₂²×#6092	C₂²	1	2	yes	—
11	C₄×C₁₃×M₁₆	C₄	1	1	yes	characteristic
12	#11718783	C₄	1	1	yes	characteristic
13	#11718783	C₂²	1	1	yes	characteristic
14	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
15	#11718369	—	2	1	no	—
16	#11718369	—	2	1	no	—
17	C₄×#6094	—	2	1	no	—
18	C₄×C₈×D₁₃	—	2	1	no	—
19	#11718270	—	2	4	no	—
20	C₄⋊#6092	—	2	1	no	—
21	C₄⋊#6092	—	2	1	no	—
22	C₄⋊#6092	—	2	1	no	—
23	C₄⋊#6092	—	2	1	no	—
24	C₂×#6127	C₂×C₄	1	1	yes	characteristic
25	C₂×C₁₃×M₁₆	C₂×C₄	1	1	yes	characteristic
26	C₂×C₁₃×M₁₆	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
28	C₄×C₂²⋊D₁₃	—	2	1	no	—
29	C₄×C₂²⋊D₁₃	—	2	1	no	—
30	#88180	C₂×C₄	1	1	yes	characteristic
31	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
32	C₂²×#1055	C₂³	1	1	yes	characteristic
33	#88161	—	2	2	no	—
34	#88161	—	2	2	no	—
35	#88161	—	2	1	no	—
36	#88161	—	2	1	no	—
37	#88161	D₄	1	2	yes	—
38	#88161	Q₈	1	2	yes	—
39	C₂×#6094	—	2	1	no	—
40	C₂×#6094	—	2	1	no	—
41	C₂×C₈×D₁₃	—	2	1	no	—
42	C₂×C₈×D₁₃	—	2	1	no	—
43	C₄×D₅₂	C₈	1	1	yes	characteristic
44	C₄×D₅₂	C₈	1	1	yes	characteristic
45	#88126	—	2	1	no	—
46	C₄²×D₁₃	—	2	1	no	—
47	C₄×Dic₂₆	C₈	1	1	yes	characteristic
48	C₄×Dic₂₆	C₈	1	1	yes	characteristic
49	#88106	—	2	2	no	—
50	#88106	—	2	2	no	—
51	C₂×#6092	—	2	2	no	—
52	C₂×#6092	—	2	2	no	—
53	C₂×#6092	—	2	2	no	—
54	C₂×#6092	—	2	2	no	—
55	C₁₃×#97	—	2	1	no	—
56	C₄×C₈×C₁₃	—	2	1	no	—
57	#88055	—	2	1	no	—
58	#88055	—	2	1	no	—
59	C₈×Dic₁₃	—	2	1	no	—
60	C₈×Dic₁₃	—	2	1	no	—
61	C₄×#1055	D₄	1	1	yes	characteristic
62	C₄×#1055	D₄	1	1	yes	characteristic
63	C₄×#1055	D₄	1	1	yes	characteristic
64	C₄×#1055	D₄	1	1	yes	characteristic
65	#12787	—	13	1	no	maximal, Sylow
66	C₂²×C₄×C₁₃	M₁₆	1	1	yes	characteristic
67	C₂²×C₄×C₁₃	M₁₆	1	1	yes	characteristic
68	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
69	C₂×C₂²⋊D₁₃	—	2	1	no	—
70	#6127	—	2	1	no	—
71	#6127	—	2	1	no	—
72	#6127	C₄⋊C₄	1	2	yes	—
73	#6127	M₁₆	1	2	yes	—
74	C₂×D₅₂	C₂×C₈	1	1	yes	characteristic
75	C₂×C₄×D₁₃	—	2	1	no	—
76	C₂×C₄×D₁₃	—	2	1	no	—
77	C₂×C₄×D₁₃	—	2	1	no	—
78	C₂×Dic₂₆	C₂×C₈	1	1	yes	characteristic
79	C₁₃×M₁₆	—	2	1	no	—
80	C₁₃×M₁₆	—	2	1	no	—
81	C₁₃×M₁₆	C₄⋊C₄	1	2	yes	—
82	C₁₃×M₁₆	M₁₆	1	2	yes	—
83	C₂×C₈×C₁₃	—	2	1	no	—
84	C₂×C₈×C₁₃	—	2	1	no	—
85	C₄²×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
86	C₄²×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
87	C₄²×C₁₃	C₂×C₈	1	1	yes	characteristic
88	C₄²×C₁₃	C₂×C₈	1	1	yes	characteristic
89	C₂²⋊Dic₁₃	—	2	1	no	—
90	#6103	—	2	1	no	—
91	#6103	—	2	1	no	—
92	C₄⋊Dic₁₃	C₂×C₈	1	1	yes	characteristic
93	C₄⋊Dic₁₃	C₂×C₈	1	1	yes	characteristic
94	#6101	—	2	1	no	—
95	#6101	—	2	1	no	—
96	C₄×Dic₁₃	—	2	1	no	—
97	C₄×Dic₁₃	—	2	1	no	—
98	C₄×Dic₁₃	—	2	1	no	—
99	C₂×#1055	—	2	1	no	—
100	C₂×#1055	—	2	1	no	—
101	C₂×#1055	C₂²⋊C₄	1	1	yes	characteristic
102	C₂×#1055	C₂²⋊C₄	1	1	yes	characteristic
103	C₂×#1055	C₂×D₄	1	1	yes	characteristic
104	C₂×#1055	C₂×D₄	1	1	yes	characteristic
105	C₂×#1055	C₂×D₄	1	1	yes	characteristic
106	C₂×#1055	C₂×Q₈	1	1	yes	characteristic
107	C₂×#1055	#41	1	1	yes	characteristic
108	C₂×#1055	#41	1	1	yes	characteristic
109	#6094	—	2	4	no	—
110	#6094	—	2	2	no	—
111	#6094	—	2	2	no	—
112	C₈×D₁₃	—	2	4	no	—
113	C₈×D₁₃	—	2	2	no	—
114	C₈×D₁₃	—	2	2	no	—
115	#6092	—	4	2	no	—
116	#6092	—	4	2	no	—
117	C₄×#131	—	13	1	no	—
118	#2150	—	13	1	no	—
119	C₂×C₄⋊C₁₆	—	13	1	no	—
120	C₂×#350	—	13	2	no	—
121	#1379	—	13	2	no	—
122	C₂³×C₁₃	C₂×M₁₆	1	1	yes	characteristic
123	C₂²×D₁₃	—	2	1	no	—
124	C₂×C₄×C₁₃	—	2	2	no	—
125	C₂×C₄×C₁₃	—	4	1	no	—
126	C₂×C₄×C₁₃	C₂²⋊C₈	1	1	yes	characteristic
127	C₂×C₄×C₁₃	C₂²⋊C₈	1	1	yes	characteristic
128	C₂×C₄×C₁₃	C₂²⋊C₈	1	1	yes	characteristic
129	C₂×C₄×C₁₃	C₂²⋊C₈	1	1	yes	characteristic
130	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
131	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
132	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
133	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
134	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
135	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
136	C₂²⋊D₁₃	—	2	2	no	—
137	C₂²⋊D₁₃	—	2	2	no	—
138	C₂×Dic₁₃	—	2	1	no	—
139	C₂×Dic₁₃	—	2	1	no	—
140	C₂×Dic₁₃	—	2	1	no	—
141	D₅₂	—	2	1	no	—
142	D₅₂	C₄⋊C₈	1	2	yes	—
143	C₄×D₁₃	—	2	2	no	—
144	C₄×D₁₃	—	2	2	no	—
145	C₄×D₁₃	—	4	2	no	—
146	C₄×D₁₃	—	4	2	no	—
147	Dic₂₆	—	2	1	no	—
148	Dic₂₆	C₄⋊C₈	1	2	yes	—
149	C₈×C₁₃	—	2	2	no	—
150	C₈×C₁₃	—	2	2	no	—
151	#1055	—	2	1	no	—
152	#1055	—	2	1	no	—
153	#1055	#121	1	1	yes	characteristic
154	#1055	#121	1	1	yes	characteristic
155	#1055	C₂²⋊Q₈	1	1	yes	characteristic
156	#1055	C₂²⋊Q₈	1	1	yes	characteristic
157	C₂×#131	—	13	1	no	—
158	C₂×#131	—	13	1	no	—
159	C₄×#41	—	13	1	no	—
160	C₂×M₃₂	—	13	2	no	—
161	C₂²×C₁₆	—	13	2	no	—
162	#405	—	13	1	no	—
163	#405	—	13	1	no	—
164	#405	—	26	1	no	—
165	#405	—	26	1	no	—
166	C₄×M₁₆	—	26	1	no	—
167	C₄×M₁₆	#261	1	1	yes	characteristic
168	C₂×C₄×C₈	—	13	1	no	—
169	C₂×C₄×C₈	—	26	1	no	—
170	C₄⋊C₁₆	—	26	1	no	—
171	C₄⋊C₁₆	—	26	1	no	—
172	C₄⋊C₁₆	—	26	1	no	—
173	C₄⋊C₁₆	—	26	1	no	—
174	#350	—	26	4	no	—
175	C₂²×C₁₃	—	2	1	no	—
176	C₂²×C₁₃	—	2	1	no	—
177	C₂²×C₁₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
178	C₂²×C₁₃	#423	1	1	yes	characteristic
179	C₂²×C₁₃	#432	1	1	yes	characteristic
180	C₂×D₁₃	—	2	2	no	—
181	C₂×D₁₃	—	4	1	no	—
182	C₂×D₁₃	—	4	1	no	—
183	C₄×C₁₃	—	2	1	no	—
184	C₄×C₁₃	—	2	1	no	—
185	C₄×C₁₃	—	2	1	no	—
186	C₄×C₁₃	#350	1	2	yes	complemented
187	C₄×C₁₃	#380	1	1	yes	characteristic
188	C₄×C₁₃	#407	1	1	yes	characteristic
189	C₄×C₁₃	C₂×C₄⋊C₈	1	1	yes	characteristic
190	C₄×C₁₃	#431	1	1	yes	characteristic
191	Dic₁₃	—	2	2	no	—
192	Dic₁₃	—	4	1	no	—
193	Dic₁₃	—	4	1	no	—
194	C₂×#41	—	13	1	no	—
195	#131	—	13	2	no	—
196	#131	—	13	2	no	—
197	#131	—	26	1	no	—
198	#131	—	26	1	no	—
199	#131	—	26	2	no	—
200	#131	—	26	2	no	—
201	C₂×M₁₆	—	26	1	no	—
202	C₂×M₁₆	—	26	1	no	—
203	C₂×M₁₆	C₂×#261	1	1	yes	characteristic
204	C₂×M₁₆	C₂×#261	1	1	yes	characteristic
205	C₂²×C₈	—	13	1	no	—
206	C₂²×C₈	—	13	1	no	—
207	C₂²×C₈	—	26	1	no	—
208	C₂²×C₈	—	26	1	no	—
209	C₄×Q₈	—	13	1	no	—
210	C₄×Q₈	—	13	1	no	—
211	C₄×D₄	—	13	1	no	—
212	C₄×D₄	—	13	1	no	—
213	C₄×D₄	—	26	1	no	—
214	C₄×D₄	—	26	1	no	—
215	#117	—	13	1	no	—
216	#117	—	26	1	no	—
217	C₂×C₄²	—	26	1	no	—
218	C₂×C₄²	C₂×#261	1	1	yes	characteristic
219	M₃₂	—	26	2	no	—
220	M₃₂	—	26	2	no	—
221	C₂×C₁₆	—	26	2	no	—
222	C₂×C₁₆	—	26	2	no	—
223	C₂×C₁₆	—	26	2	no	—
224	C₂×C₁₆	—	26	2	no	—
225	#97	—	2	1	no	—
226	#97	—	26	1	no	—
227	#97	—	26	1	no	—
228	C₄×C₈	—	2	1	no	—
229	C₄×C₈	—	13	1	no	—
230	C₄×C₈	—	13	1	no	—
231	C₄×C₈	—	13	1	no	—
232	C₄×C₈	—	13	1	no	—
233	C₄×C₈	—	26	1	no	—
234	C₄×C₈	—	26	1	no	—
235	C₂×C₁₃	—	2	1	no	—
236	C₂×C₁₃	—	2	1	no	—
237	C₂×C₁₃	#1797	1	1	yes	characteristic
238	C₂×C₁₃	C₂×#350	1	1	yes	characteristic
239	C₂×C₁₃	#2117	1	1	yes	characteristic
240	D₁₃	—	4	2	no	—
241	#41	—	13	2	no	—
242	#41	—	13	2	no	—
243	#41	—	26	1	no	—
244	#41	—	26	1	no	—
245	C₂×Q₈	—	13	1	no	—
246	C₂×D₄	—	13	1	no	—
247	C₂×D₄	—	26	1	no	—
248	C₂²×C₄	—	26	1	no	—
249	C₂²×C₄	—	26	1	no	—
250	C₂²×C₄	—	26	1	no	—
251	C₂²×C₄	#6118	1	1	yes	characteristic
252	C₂²×C₄	#6118	1	1	yes	characteristic
253	C₂²×C₄	C₂²×#261	1	1	yes	characteristic
254	M₁₆	—	2	1	no	—
255	M₁₆	—	2	1	no	—
256	M₁₆	—	26	2	no	—
257	M₁₆	—	26	2	no	—
258	M₁₆	—	26	4	no	—
259	M₁₆	#6118	1	2	yes	—
260	M₁₆	C₄⋊#261	1	2	yes	—
261	C₂×C₈	—	2	1	no	—
262	C₂×C₈	—	2	1	no	—
263	C₂×C₈	—	13	1	no	—
264	C₂×C₈	—	13	1	no	—
265	C₂×C₈	—	13	1	no	—
266	C₂×C₈	—	13	1	no	—
267	C₂×C₈	—	13	1	no	—
268	C₂×C₈	—	13	1	no	—
269	C₂×C₈	—	13	1	no	—
270	C₂×C₈	—	13	1	no	—
271	C₂×C₈	—	26	2	no	—
272	C₂×C₈	—	26	2	no	—
273	C₂×C₈	—	26	4	no	—
274	C₂×C₈	—	26	1	no	—
275	C₂×C₈	—	26	1	no	—
276	C₄⋊C₄	—	13	1	no	—
277	C₄⋊C₄	—	13	1	no	—
278	C₄⋊C₄	—	26	1	no	—
279	C₄⋊C₄	—	26	1	no	—
280	C₂²⋊C₄	—	26	1	no	—
281	C₂²⋊C₄	—	26	1	no	—
282	C₂²⋊C₄	—	26	1	no	—
283	C₄²	—	26	1	no	—
284	C₄²	—	26	1	no	—
285	C₄²	—	26	1	no	—
286	C₄²	D₁₃⋊C₈	1	1	yes	characteristic
287	C₄²	D₁₃⋊C₈	1	1	yes	characteristic
288	C₄²	C₂²⋊#261	1	1	yes	characteristic
289	C₄²	C₂²⋊#261	1	1	yes	characteristic
290	C₁₆	—	52	2	no	—
291	C₁₆	—	52	2	no	—
292	C₁₃	#12787	1	1	yes	characteristic, complemented, Sylow
293	C₂³	—	26	1	no	—
294	C₂³	C₂×#6118	1	1	yes	characteristic
295	Q₈	—	13	2	no	—
296	Q₈	—	26	1	no	—
297	D₄	—	13	2	no	—
298	D₄	—	26	2	no	—
299	D₄	—	26	2	no	—
300	D₄	—	26	1	no	—
301	C₂×C₄	—	2	2	no	—
302	C₂×C₄	—	4	1	no	—
303	C₂×C₄	—	26	1	no	—
304	C₂×C₄	—	26	2	no	—
305	C₂×C₄	—	26	2	no	—
306	C₂×C₄	—	26	1	no	—
307	C₂×C₄	—	26	1	no	—
308	C₂×C₄	—	52	2	no	—
309	C₂×C₄	—	52	2	no	—
310	C₂×C₄	#88111	1	1	yes	characteristic
311	C₂×C₄	#88111	1	1	yes	characteristic
312	C₂×C₄	#88111	1	1	yes	characteristic
313	C₂×C₄	#88111	1	1	yes	characteristic
314	C₂×C₄	C₂×D₁₃⋊C₈	1	1	yes	characteristic
315	C₂×C₄	C₂×#6118	1	1	yes	characteristic
316	C₂×C₄	D₁₃⋊M₁₆	1	1	yes	characteristic
317	C₂×C₄	C₂×C₄⋊#261	1	1	yes	characteristic
318	C₂×C₄	#88237	1	1	yes	characteristic
319	C₂×C₄	C₂×C₂²⋊#261	1	1	yes	characteristic
320	C₈	—	2	2	no	—
321	C₈	—	2	2	no	—
322	C₈	—	13	1	no	—
323	C₈	—	13	1	no	—
324	C₈	—	13	1	no	—
325	C₈	—	13	1	no	—
326	C₈	—	26	1	no	—
327	C₈	—	26	1	no	—
328	C₂²	—	2	1	no	—
329	C₂²	—	2	1	no	—
330	C₂²	—	26	2	no	—
331	C₂²	—	52	1	no	—
332	C₂²	—	52	1	no	—
333	C₂²	#11719052	1	1	yes	characteristic
334	C₂²	#11719054	1	1	yes	characteristic
335	C₂²	C₂×#88111	1	1	yes	characteristic
336	C₄	—	2	1	no	—
337	C₄	—	2	1	no	—
338	C₄	—	2	1	no	—
339	C₄	—	26	2	no	—
340	C₄	—	52	1	no	—
341	C₄	—	52	1	no	—
342	C₄	#11718270	1	2	yes	complemented
343	C₄	D₁₃⋊(C₄⋊C₈)	1	1	yes	characteristic
344	C₄	#11719051	1	1	yes	characteristic
345	C₄	#11719127	1	1	yes	characteristic
346	C₄	#11719136	1	1	yes	characteristic
347	C₂	—	2	1	no	—
348	C₂	—	2	1	no	—
349	C₂	—	52	2	no	—
350	C₂	#13130691	1	1	yes	characteristic
351	C₂	C₂×#11718270	1	1	yes	characteristic
352	C₂	#13131080	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊#261
commutator	C₂×C₄×C₁₃	C₂²×C₈
Frattini	C₂²×C₄	C₂²×#261
Fitting	C₄×C₁₃×M₁₆	C₄
socle	C₂²×C₁₃	C₂×C₂²⋊C₈
prime radical	C₂×#6127	C₂×C₄
cosocle	C₂²×#1055	C₂³
hypercenter	C₄×M₁₆	#261
hypocenter	C₁₃	#12787
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×M₁₆

Complements

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

N	H ≅ G/N	type
1. #13131168	C₂	semidirect
2. C₂×C₄⋊#6092	C₂	semidirect
4. #13129891	C₂	semidirect
186. C₄×C₁₃	#350	semidirect
292. C₁₃	#12787	semidirect
342. C₄	#11718270	semidirect

Maximal subgroups

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

subgroup	index
#13131168	2
C₂×C₄⋊#6092	2
C₂×#11718270	2
#13129891	2
C₄×#88161	2
#12787	13