Subgroups — #16789967

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	#13131170	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#11718260	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#11718304	C₂	1	2	yes	maximal
4	#13129891	C₂	1	2	yes	maximal, complemented
5	#131×Dic₁₃	C₂	1	1	yes	maximal, characteristic
6	C₂×C₁₃×#131	C₄	1	1	yes	characteristic
7	#41×Dic₁₃	C₄	1	1	yes	characteristic
8	C₂×#88202	C₂²	1	1	yes	characteristic
9	C₂×#88106	C₂²	1	2	yes	—
10	C₂²×#6092	C₂²	1	2	yes	—
11	#11718783	—	2	1	no	—
12	#11718783	C₂²	1	1	yes	characteristic
13	M₁₆×Dic₁₃	—	2	1	no	—
14	M₁₆×Dic₁₃	C₄	1	1	yes	characteristic
15	#11718762	—	2	1	no	—
16	#11718762	C₄	1	1	yes	characteristic
17	C₂×C₈×Dic₁₃	—	2	1	no	—
18	C₂×C₈×Dic₁₃	C₂²	1	1	yes	characteristic
19	#11718304	—	2	4	no	—
20	#11718260	—	2	1	no	—
21	#11718260	—	2	1	no	—
22	#11718260	—	2	1	no	—
23	#11718260	—	2	1	no	—
24	C₂×C₁₃×#41	C₂×C₄	1	1	yes	characteristic
25	C₁₃×#131	—	2	2	no	—
26	C₁₃×#131	—	2	2	no	—
27	C₂×C₁₃×M₁₆	—	2	1	no	—
28	C₂×C₁₃×M₁₆	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₈×C₁₃	—	2	1	no	—
30	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
31	#88202	—	2	1	no	—
32	#88202	—	2	1	no	—
33	#88202	D₄	1	2	yes	—
34	#88202	Q₈	1	2	yes	—
35	Q₈×Dic₁₃	C₈	1	1	yes	characteristic
36	Q₈×Dic₁₃	C₈	1	1	yes	characteristic
37	D₄×Dic₁₃	—	2	1	no	—
38	D₄×Dic₁₃	—	2	1	no	—
39	D₄×Dic₁₃	C₈	1	1	yes	characteristic
40	D₄×Dic₁₃	C₈	1	1	yes	characteristic
41	#88180	—	2	1	no	—
42	#88180	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄×Dic₁₃	—	2	1	no	—
44	C₂×C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
45	C₂×#6099	—	2	1	no	—
46	C₂×#6099	C₂×C₄	1	1	yes	characteristic
47	C₂²×#1055	—	2	1	no	—
48	C₂²×#1055	C₂³	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	#88055	—	2	1	no	—
56	#88055	—	2	1	no	—
57	#88055	—	2	1	no	—
58	C₈×Dic₁₃	—	2	1	no	—
59	C₈×Dic₁₃	—	2	1	no	—
60	C₈×Dic₁₃	—	2	1	no	—
61	C₈×Dic₁₃	D₄	1	1	yes	characteristic
62	C₈×Dic₁₃	D₄	1	1	yes	characteristic
63	C₈×Dic₁₃	D₄	1	1	yes	characteristic
64	C₈×Dic₁₃	D₄	1	1	yes	characteristic
65	#12787	—	13	1	no	maximal, Sylow
66	C₁₃×#41	—	2	1	no	—
67	C₁₃×#41	—	2	1	no	—
68	C₁₃×#41	C₄⋊C₄	1	2	yes	—
69	C₁₃×#41	M₁₆	1	2	yes	—
70	C₂×Q₈×C₁₃	C₂×C₈	1	1	yes	characteristic
71	C₂×D₄×C₁₃	—	2	1	no	—
72	C₂×D₄×C₁₃	C₂×C₈	1	1	yes	characteristic
73	C₂²×C₄×C₁₃	—	2	1	no	—
74	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
75	C₂²×Dic₁₃	—	2	1	no	—
76	C₂²×Dic₁₃	—	2	1	no	—
77	C₂²×Dic₁₃	M₁₆	1	1	yes	characteristic
78	C₂²×Dic₁₃	M₁₆	1	1	yes	characteristic
79	C₁₃×M₁₆	—	2	4	no	—
80	C₁₃×M₁₆	—	2	1	no	—
81	C₁₃×M₁₆	M₁₆	1	2	yes	—
82	C₂×C₈×C₁₃	—	2	4	no	—
83	C₂×C₈×C₁₃	—	2	1	no	—
84	C₂×C₈×C₁₃	—	2	1	no	—
85	C₂×C₈×C₁₃	—	2	1	no	—
86	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
87	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
88	C₂²⋊Dic₁₃	—	2	1	no	—
89	C₂²⋊Dic₁₃	—	2	1	no	—
90	C₂²⋊Dic₁₃	—	2	1	no	—
91	C₄⋊Dic₁₃	—	2	1	no	—
92	C₄⋊Dic₁₃	—	2	1	no	—
93	C₄⋊Dic₁₃	C₂×C₈	1	1	yes	characteristic
94	C₄⋊Dic₁₃	C₂×C₈	1	1	yes	characteristic
95	C₄×Dic₁₃	—	2	1	no	—
96	C₄×Dic₁₃	—	2	1	no	—
97	C₄×Dic₁₃	—	2	1	no	—
98	C₄×Dic₁₃	C₂²⋊C₄	1	1	yes	characteristic
99	C₄×Dic₁₃	C₂²⋊C₄	1	1	yes	characteristic
100	C₄×Dic₁₃	C₂×C₈	1	1	yes	characteristic
101	C₄×Dic₁₃	C₂×C₈	1	1	yes	characteristic
102	#6099	—	2	1	no	—
103	#6099	—	2	2	no	—
104	#6099	—	2	2	no	—
105	#6099	C₄⋊C₄	1	2	yes	—
106	C₂×#1055	—	2	1	no	—
107	C₂×#1055	—	2	2	no	—
108	C₂×#1055	—	2	2	no	—
109	C₂×#1055	C₂×D₄	1	1	yes	characteristic
110	C₂×#1055	C₂×D₄	1	1	yes	characteristic
111	C₂×#1055	C₂×D₄	1	1	yes	characteristic
112	C₂×#1055	C₂×Q₈	1	1	yes	characteristic
113	C₂×#1055	#41	1	1	yes	characteristic
114	C₂×#1055	#41	1	1	yes	characteristic
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	—	2	1	no	—
123	C₂³×C₁₃	C₂×M₁₆	1	1	yes	characteristic
124	Q₈×C₁₃	—	2	1	no	—
125	Q₈×C₁₃	C₄⋊C₈	1	2	yes	—
126	D₄×C₁₃	—	2	2	no	—
127	D₄×C₁₃	—	2	2	no	—
128	D₄×C₁₃	—	2	1	no	—
129	D₄×C₁₃	C₄⋊C₈	1	2	yes	—
130	C₂×C₄×C₁₃	—	2	1	no	—
131	C₂×C₄×C₁₃	—	2	2	no	—
132	C₂×C₄×C₁₃	—	2	2	no	—
133	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
134	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
135	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
136	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
137	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
138	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
139	C₂×Dic₁₃	—	2	2	no	—
140	C₂×Dic₁₃	—	2	1	no	—
141	C₂×Dic₁₃	—	2	1	no	—
142	C₂×Dic₁₃	—	4	1	no	—
143	C₂×Dic₁₃	—	4	2	no	—
144	C₂×Dic₁₃	—	4	2	no	—
145	C₂×Dic₁₃	C₂²⋊C₈	1	1	yes	characteristic
146	C₂×Dic₁₃	C₂²⋊C₈	1	1	yes	characteristic
147	C₂×Dic₁₃	C₂²⋊C₈	1	1	yes	characteristic
148	C₂×Dic₁₃	C₂²⋊C₈	1	1	yes	characteristic
149	C₈×C₁₃	—	2	1	no	—
150	C₈×C₁₃	—	2	2	no	—
151	C₈×C₁₃	—	2	1	no	—
152	#1055	—	2	2	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	#261	1	1	yes	characteristic
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	—	13	1	no	—
167	C₄×M₁₆	—	26	1	no	—
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	—	2	2	no	—
178	C₂²×C₁₃	—	4	1	no	—
179	C₂²×C₁₃	—	4	1	no	—
180	C₂²×C₁₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
181	C₂²×C₁₃	#423	1	1	yes	characteristic
182	C₂²×C₁₃	#432	1	1	yes	characteristic
183	C₄×C₁₃	—	2	2	no	—
184	C₄×C₁₃	#380	1	1	yes	characteristic
185	C₄×C₁₃	#407	1	1	yes	characteristic
186	C₄×C₁₃	C₂×C₄⋊C₈	1	1	yes	characteristic
187	C₄×C₁₃	#431	1	1	yes	characteristic
188	Dic₁₃	—	2	1	no	—
189	Dic₁₃	—	2	1	no	—
190	Dic₁₃	—	2	1	no	—
191	Dic₁₃	—	4	1	no	—
192	Dic₁₃	—	4	1	no	—
193	Dic₁₃	#350	1	2	yes	complemented
194	C₂×#41	C₂×#261	1	1	yes	characteristic
195	#131	—	2	2	no	—
196	#131	—	2	2	no	—
197	#131	—	13	2	no	—
198	#131	—	13	2	no	—
199	#131	—	26	1	no	—
200	#131	—	26	1	no	—
201	C₂×M₁₆	—	2	1	no	—
202	C₂×M₁₆	—	13	1	no	—
203	C₂×M₁₆	—	26	1	no	—
204	C₂×M₁₆	C₂×#261	1	1	yes	characteristic
205	C₂²×C₈	—	2	1	no	—
206	C₂²×C₈	—	13	1	no	—
207	C₂²×C₈	—	26	1	no	—
208	C₂²×C₈	C₂×#261	1	1	yes	characteristic
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	—	13	1	no	—
218	C₂×C₄²	—	26	1	no	—
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	—	26	1	no	—
226	#97	—	26	1	no	—
227	#97	—	26	1	no	—
228	C₄×C₈	—	13	1	no	—
229	C₄×C₈	—	13	1	no	—
230	C₄×C₈	—	13	1	no	—
231	C₄×C₈	—	13	1	no	—
232	C₄×C₈	—	26	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	—	4	2	no	—
238	C₂×C₁₃	#1797	1	1	yes	characteristic
239	C₂×C₁₃	C₂×#350	1	1	yes	characteristic
240	C₂×C₁₃	#2117	1	1	yes	characteristic
241	#41	—	2	1	no	—
242	#41	—	2	1	no	—
243	#41	C₄⋊#261	1	2	yes	—
244	#41	#6122	1	2	yes	—
245	C₂×Q₈	D₁₃⋊C₈	1	1	yes	characteristic
246	C₂×D₄	—	2	1	no	—
247	C₂×D₄	D₁₃⋊C₈	1	1	yes	characteristic
248	C₂²×C₄	—	2	1	no	—
249	C₂²×C₄	—	13	1	no	—
250	C₂²×C₄	—	13	1	no	—
251	C₂²×C₄	—	26	1	no	—
252	C₂²×C₄	—	26	1	no	—
253	C₂²×C₄	C₂²×#261	1	1	yes	characteristic
254	M₁₆	—	2	4	no	—
255	M₁₆	—	2	1	no	—
256	M₁₆	—	13	2	no	—
257	M₁₆	—	26	2	no	—
258	M₁₆	—	26	2	no	—
259	M₁₆	—	26	1	no	—
260	M₁₆	#6122	1	2	yes	—
261	C₂×C₈	—	2	4	no	—
262	C₂×C₈	—	2	1	no	—
263	C₂×C₈	—	2	1	no	—
264	C₂×C₈	—	2	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	1	no	—
274	C₂×C₈	C₂²⋊#261	1	1	yes	characteristic
275	C₂×C₈	C₂²⋊#261	1	1	yes	characteristic
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	—	13	1	no	—
284	C₄²	—	13	1	no	—
285	C₄²	—	13	1	no	—
286	C₄²	—	13	1	no	—
287	C₄²	—	26	1	no	—
288	C₄²	—	26	1	no	—
289	C₄²	—	26	1	no	—
290	C₁₆	—	52	2	no	—
291	C₁₆	—	52	2	no	—
292	C₁₃	#12787	1	1	yes	characteristic, complemented, Sylow
293	C₂³	—	2	1	no	—
294	C₂³	D₁₃⋊M₁₆	1	1	yes	characteristic
295	Q₈	—	2	1	no	—
296	Q₈	#88112	1	2	yes	—
297	D₄	—	2	2	no	—
298	D₄	—	2	2	no	—
299	D₄	—	2	1	no	—
300	D₄	#88112	1	2	yes	—
301	C₂×C₄	—	2	1	no	—
302	C₂×C₄	—	2	2	no	—
303	C₂×C₄	—	2	2	no	—
304	C₂×C₄	—	13	1	no	—
305	C₂×C₄	—	13	1	no	—
306	C₂×C₄	—	13	1	no	—
307	C₂×C₄	—	13	1	no	—
308	C₂×C₄	—	26	2	no	—
309	C₂×C₄	—	26	1	no	—
310	C₂×C₄	—	26	1	no	—
311	C₂×C₄	—	52	2	no	—
312	C₂×C₄	—	52	1	no	—
313	C₂×C₄	—	52	2	no	—
314	C₂×C₄	C₂×D₁₃⋊C₈	1	1	yes	characteristic
315	C₂×C₄	C₂×C₄⋊#261	1	1	yes	characteristic
316	C₂×C₄	#88237	1	1	yes	characteristic
317	C₂×C₄	C₂×#6122	1	1	yes	characteristic
318	C₂×C₄	C₂×#6122	1	1	yes	characteristic
319	C₂×C₄	C₂×C₂²⋊#261	1	1	yes	characteristic
320	C₈	—	2	1	no	—
321	C₈	—	2	2	no	—
322	C₈	—	2	1	no	—
323	C₈	—	13	1	no	—
324	C₈	—	13	1	no	—
325	C₈	—	13	1	no	—
326	C₈	—	13	1	no	—
327	C₈	—	26	2	no	—
328	C₂²	—	2	1	no	—
329	C₂²	—	2	1	no	—
330	C₂²	—	2	2	no	—
331	C₂²	—	4	1	no	—
332	C₂²	—	4	1	no	—
333	C₂²	D₁₃⋊(C₂²⋊C₈)	1	1	yes	characteristic
334	C₂²	#11719128	1	1	yes	characteristic
335	C₂²	#11719129	1	1	yes	characteristic
336	C₄	—	2	2	no	—
337	C₄	—	13	2	no	—
338	C₄	—	26	1	no	—
339	C₄	—	26	1	no	—
340	C₄	—	26	1	no	—
341	C₄	—	52	1	no	—
342	C₄	—	52	1	no	—
343	C₄	C₂×#88112	1	1	yes	characteristic
344	C₄	#11719125	1	1	yes	characteristic
345	C₄	#11719136	1	1	yes	characteristic
346	C₄	#11719169	1	1	yes	characteristic
347	C₂	—	2	1	no	—
348	C₂	—	2	1	no	—
349	C₂	—	4	2	no	—
350	C₂	#13130692	1	1	yes	characteristic
351	C₂	D₁₃⋊#350	1	1	yes	characteristic
352	C₂	#13131079	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	C₂×C₁₃×#131	C₄
socle	C₂²×C₁₃	C₂×C₂²⋊C₈
prime radical	C₂×C₁₃×#41	C₂×C₄
cosocle	C₂²×#1055	C₂³
hypercenter	C₂×#131	#261
hypocenter	C₁₃	#12787
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ C₂×#131

Complements

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

N	H ≅ G/N	type
1. #13131170	C₂	semidirect
2. C₂×#11718260	C₂	semidirect
4. #13129891	C₂	semidirect
193. Dic₁₃	#350	semidirect
292. C₁₃	#12787	semidirect

Maximal subgroups

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

subgroup	index
#13131170	2
C₂×#11718260	2
C₂×#11718304	2
#13129891	2
#131×Dic₁₃	2
#12787	13