Subgroups — #16781881

Order 3328.

462 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13130383	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#11718253	C₂	1	1	yes	maximal, characteristic, complemented
3	#13130131	C₂	1	2	yes	maximal
4	#13129682	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	C₂²	1	1	yes	characteristic
8	C₂×#88202	C₄	1	1	yes	characteristic
9	#11719119	C₂²	1	2	yes	—
10	C₂×C₄⋊#1057	C₂²	1	2	yes	—
11	#11718783	—	2	1	no	—
12	#11718783	C₄	1	1	yes	characteristic
13	M₁₆×Dic₁₃	—	2	1	no	—
14	M₁₆×Dic₁₃	C₂²	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	Q₈⋊#1057	—	2	2	no	—
20	D₄⋊#1057	—	2	2	no	—
21	#11718254	—	2	1	no	—
22	#11718253	—	2	1	no	—
23	#11718253	—	2	1	no	—
24	#11718253	—	2	1	no	—
25	C₂×C₁₃×#41	C₂×C₄	1	1	yes	characteristic
26	C₂²×#1057	—	2	2	no	—
27	C₁₃×#131	—	2	2	no	—
28	C₁₃×#131	C₈	1	4	yes	complemented
29	C₂×C₁₃×M₁₆	—	2	1	no	—
30	C₂×C₁₃×M₁₆	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₈×C₁₃	—	2	1	no	—
32	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
33	#88202	—	2	2	no	—
34	#88202	—	2	2	no	—
35	Q₈×Dic₁₃	D₄	1	1	yes	characteristic
36	Q₈×Dic₁₃	Q₈	1	1	yes	characteristic
37	D₄×Dic₁₃	—	2	1	no	—
38	D₄×Dic₁₃	—	2	1	no	—
39	D₄×Dic₁₃	D₄	1	1	yes	characteristic
40	D₄×Dic₁₃	Q₈	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	C₂³	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	C₂×C₄	1	1	yes	characteristic
49	C₂²⋊#1057	—	2	2	no	—
50	C₄⋊#1057	—	2	2	no	—
51	C₄⋊#1057	—	2	2	no	—
52	C₄⋊#1057	—	2	2	no	—
53	C₄×#1057	—	2	2	no	—
54	#88055	—	2	1	no	—
55	#88055	—	2	1	no	—
56	#88055	—	2	1	no	—
57	C₈×Dic₁₃	—	2	1	no	—
58	C₈×Dic₁₃	—	2	1	no	—
59	C₈×Dic₁₃	—	2	1	no	—
60	C₈×Dic₁₃	D₄	1	1	yes	characteristic
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	#8604	—	13	1	no	maximal, Sylow
65	C₁₃×#41	—	2	1	no	—
66	C₁₃×#41	—	2	1	no	—
67	C₁₃×#41	C₂×C₈	1	2	yes	—
68	C₁₃×#41	M₁₆	1	2	yes	—
69	C₂×Q₈×C₁₃	C₄⋊C₄	1	1	yes	characteristic
70	C₂×D₄×C₁₃	—	2	1	no	—
71	C₂×D₄×C₁₃	C₄⋊C₄	1	1	yes	characteristic
72	C₂²×C₄×C₁₃	—	2	1	no	—
73	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
74	C₂²×Dic₁₃	—	2	1	no	—
75	C₂²×Dic₁₃	—	2	1	no	—
76	C₂²×Dic₁₃	#41	1	1	yes	characteristic
77	C₂²×Dic₁₃	#41	1	1	yes	characteristic
78	C₂×#1057	—	4	2	no	—
79	C₂×#1057	—	4	2	no	—
80	C₂×#1057	—	8	2	no	—
81	C₁₃×M₁₆	—	2	1	no	—
82	C₁₃×M₁₆	—	2	4	no	—
83	C₁₃×M₁₆	C₂×C₈	1	2	yes	—
84	C₂×C₈×C₁₃	—	2	1	no	—
85	C₂×C₈×C₁₃	—	2	4	no	—
86	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
87	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
88	C₂×C₈×C₁₃	C₂×C₈	1	2	yes	—
89	C₂×C₈×C₁₃	M₁₆	1	2	yes	—
90	C₂²⋊Dic₁₃	—	2	1	no	—
91	C₂²⋊Dic₁₃	—	2	1	no	—
92	C₂²⋊Dic₁₃	—	2	1	no	—
93	C₄⋊Dic₁₃	—	2	1	no	—
94	C₄⋊Dic₁₃	—	2	1	no	—
95	C₄⋊Dic₁₃	C₄⋊C₄	1	1	yes	characteristic
96	C₄⋊Dic₁₃	C₄⋊C₄	1	1	yes	characteristic
97	C₄×Dic₁₃	—	2	1	no	—
98	C₄×Dic₁₃	—	2	1	no	—
99	C₄×Dic₁₃	—	2	1	no	—
100	C₄×Dic₁₃	C₂×D₄	1	1	yes	characteristic
101	C₄×Dic₁₃	C₂×D₄	1	1	yes	characteristic
102	C₄×Dic₁₃	C₂×D₄	1	1	yes	characteristic
103	C₄×Dic₁₃	C₂×Q₈	1	1	yes	characteristic
104	#6099	—	2	4	no	—
105	#6099	—	2	1	no	—
106	#6099	M₁₆	1	2	yes	—
107	C₂×#1055	—	2	4	no	—
108	C₂×#1055	—	2	1	no	—
109	C₂×#1055	—	2	1	no	—
110	C₂×#1055	—	2	1	no	—
111	C₂×#1055	C₂²⋊C₄	1	1	yes	characteristic
112	C₂×#1055	C₂²⋊C₄	1	1	yes	characteristic
113	C₄×#131	—	13	1	no	—
114	#1564	—	13	1	no	—
115	C₂×#334	—	13	1	no	—
116	#1477	—	13	2	no	—
117	#1283	—	13	2	no	—
118	C₂³×C₁₃	—	2	1	no	—
119	C₂³×C₁₃	#117	1	1	yes	characteristic
120	Q₈×C₁₃	—	2	1	no	—
121	Q₈×C₁₃	C₄⋊C₈	1	2	yes	—
122	D₄×C₁₃	—	2	2	no	—
123	D₄×C₁₃	—	2	2	no	—
124	D₄×C₁₃	—	2	1	no	—
125	D₄×C₁₃	C₄⋊C₈	1	2	yes	—
126	C₂×C₄×C₁₃	—	2	1	no	—
127	C₂×C₄×C₁₃	—	2	2	no	—
128	C₂×C₄×C₁₃	—	2	2	no	—
129	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
130	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
131	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
132	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
133	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
134	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
135	C₂×Dic₁₃	—	2	2	no	—
136	C₂×Dic₁₃	—	2	1	no	—
137	C₂×Dic₁₃	—	2	1	no	—
138	C₂×Dic₁₃	—	4	1	no	—
139	C₂×Dic₁₃	—	4	2	no	—
140	C₂×Dic₁₃	—	4	2	no	—
141	C₂×Dic₁₃	#121	1	1	yes	characteristic
142	C₂×Dic₁₃	#121	1	1	yes	characteristic
143	C₂×Dic₁₃	C₂²⋊Q₈	1	1	yes	characteristic
144	C₂×Dic₁₃	C₂²⋊Q₈	1	1	yes	characteristic
145	#1057	—	8	2	no	—
146	#1057	—	8	2	no	—
147	C₈×C₁₃	—	2	2	no	—
148	C₈×C₁₃	C₂²⋊C₈	1	2	yes	—
149	C₈×C₁₃	C₂²⋊C₈	1	2	yes	—
150	#1055	—	2	1	no	—
151	#1055	—	2	2	no	—
152	#1055	—	2	1	no	—
153	C₂×#131	—	13	1	no	—
154	C₂×#131	#261	1	1	yes	characteristic
155	C₄×#41	—	13	1	no	—
156	#431	—	13	2	no	—
157	C₂×C₄⋊C₈	—	13	2	no	—
158	#405	—	13	1	no	—
159	#405	—	13	1	no	—
160	#405	—	26	1	no	—
161	#405	—	26	1	no	—
162	C₄×M₁₆	—	13	1	no	—
163	C₄×M₁₆	—	26	1	no	—
164	C₂×C₄×C₈	—	13	1	no	—
165	C₂×C₄×C₈	—	26	1	no	—
166	#335	—	26	1	no	—
167	#334	—	26	1	no	—
168	#334	—	26	1	no	—
169	#334	—	26	1	no	—
170	Q₈⋊C₈	—	26	2	no	—
171	D₄⋊C₈	—	26	2	no	—
172	C₂²×C₁₃	—	2	1	no	—
173	C₂²×C₁₃	—	2	1	no	—
174	C₂²×C₁₃	—	2	2	no	—
175	C₂²×C₁₃	—	4	1	no	—
176	C₂²×C₁₃	—	4	1	no	—
177	C₂²×C₁₃	#380	1	1	yes	characteristic
178	C₂²×C₁₃	#431	1	1	yes	characteristic
179	C₂²×C₁₃	#432	1	1	yes	characteristic
180	C₄×C₁₃	—	2	2	no	—
181	C₄×C₁₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
182	C₄×C₁₃	#407	1	1	yes	characteristic
183	C₄×C₁₃	C₂×C₄⋊C₈	1	1	yes	characteristic
184	C₄×C₁₃	#423	1	1	yes	characteristic
185	Dic₁₃	—	2	1	no	—
186	Dic₁₃	—	2	1	no	—
187	Dic₁₃	—	2	1	no	—
188	Dic₁₃	—	4	1	no	—
189	Dic₁₃	—	4	1	no	—
190	Dic₁₃	#471	1	2	yes	—
191	C₂×#41	C₂×#261	1	1	yes	characteristic
192	#131	—	2	2	no	—
193	#131	—	26	2	no	—
194	#131	—	26	2	no	—
195	#131	#1057	1	4	yes	complemented
196	C₂×M₁₆	—	2	1	no	—
197	C₂×M₁₆	—	13	1	no	—
198	C₂×M₁₆	—	26	1	no	—
199	C₂×M₁₆	C₂×#261	1	1	yes	characteristic
200	C₂²×C₈	—	2	1	no	—
201	C₂²×C₈	—	13	1	no	—
202	C₂²×C₈	—	26	1	no	—
203	C₂²×C₈	—	26	2	no	—
204	C₂²×C₈	C₂×#261	1	1	yes	characteristic
205	C₄×Q₈	—	13	1	no	—
206	C₄×Q₈	—	13	1	no	—
207	C₄×D₄	—	13	1	no	—
208	C₄×D₄	—	13	1	no	—
209	C₄×D₄	—	26	1	no	—
210	C₄×D₄	—	26	1	no	—
211	#117	—	13	1	no	—
212	#117	—	26	1	no	—
213	C₂×C₄²	—	13	1	no	—
214	C₂×C₄²	—	26	1	no	—
215	C₄⋊C₈	—	26	2	no	—
216	C₄⋊C₈	—	26	2	no	—
217	C₄⋊C₈	—	26	2	no	—
218	C₂²⋊C₈	—	26	2	no	—
219	#97	—	26	1	no	—
220	#97	—	26	1	no	—
221	#97	—	26	1	no	—
222	C₄×C₈	—	13	1	no	—
223	C₄×C₈	—	13	1	no	—
224	C₄×C₈	—	13	1	no	—
225	C₄×C₈	—	13	1	no	—
226	C₄×C₈	—	26	1	no	—
227	C₄×C₈	—	26	2	no	—
228	C₄×C₈	—	26	1	no	—
229	C₄×C₈	—	26	1	no	—
230	C₂×C₁₃	—	2	1	no	—
231	C₂×C₁₃	—	2	1	no	—
232	C₂×C₁₃	—	4	2	no	—
233	C₂×C₁₃	#1797	1	1	yes	characteristic
234	C₂×C₁₃	#1814	1	1	yes	characteristic
235	C₂×C₁₃	#1815	1	1	yes	characteristic
236	#41	—	2	1	no	—
237	#41	—	2	1	no	—
238	#41	#6118	1	2	yes	—
239	#41	C₂×#1057	1	2	yes	—
240	C₂×Q₈	C₄⋊#261	1	1	yes	characteristic
241	C₂×D₄	—	2	1	no	—
242	C₂×D₄	C₄⋊#261	1	1	yes	characteristic
243	C₂²×C₄	—	2	1	no	—
244	C₂²×C₄	—	13	1	no	—
245	C₂²×C₄	—	13	1	no	—
246	C₂²×C₄	—	26	1	no	—
247	C₂²×C₄	—	26	1	no	—
248	C₂²×C₄	C₂²×#261	1	1	yes	characteristic
249	M₁₆	—	2	1	no	—
250	M₁₆	—	2	4	no	—
251	M₁₆	—	13	2	no	—
252	M₁₆	—	26	4	no	—
253	M₁₆	—	26	1	no	—
254	M₁₆	C₂×#1057	1	2	yes	—
255	C₂×C₈	—	2	1	no	—
256	C₂×C₈	—	2	4	no	—
257	C₂×C₈	—	13	1	no	—
258	C₂×C₈	—	13	1	no	—
259	C₂×C₈	—	26	4	no	—
260	C₂×C₈	—	26	1	no	—
261	C₂×C₈	—	26	1	no	—
262	C₂×C₈	—	26	1	no	—
263	C₂×C₈	—	52	2	no	—
264	C₂×C₈	—	52	2	no	—
265	C₂×C₈	—	104	2	no	—
266	C₂×C₈	C₂×#1057	1	2	yes	—
267	C₂×C₈	#6122	1	2	yes	—
268	C₂×C₈	C₂²⋊#261	1	1	yes	characteristic
269	C₂×C₈	C₂²⋊#261	1	1	yes	characteristic
270	C₄⋊C₄	—	13	1	no	—
271	C₄⋊C₄	—	13	1	no	—
272	C₄⋊C₄	—	26	1	no	—
273	C₄⋊C₄	—	26	1	no	—
274	C₂²⋊C₄	—	26	1	no	—
275	C₂²⋊C₄	—	26	1	no	—
276	C₂²⋊C₄	—	26	1	no	—
277	C₄²	—	13	1	no	—
278	C₄²	—	13	1	no	—
279	C₄²	—	13	1	no	—
280	C₄²	—	13	1	no	—
281	C₄²	—	26	1	no	—
282	C₄²	—	26	1	no	—
283	C₄²	—	26	1	no	—
284	C₁₃	#8604	1	1	yes	characteristic, complemented, Sylow
285	C₂³	—	2	1	no	—
286	C₂³	#88237	1	1	yes	characteristic
287	Q₈	—	2	1	no	—
288	Q₈	C₄⋊#1057	1	2	yes	—
289	D₄	—	2	2	no	—
290	D₄	—	2	2	no	—
291	D₄	—	2	1	no	—
292	D₄	C₄⋊#1057	1	2	yes	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	2	2	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	13	1	no	—
297	C₂×C₄	—	13	1	no	—
298	C₂×C₄	—	13	1	no	—
299	C₂×C₄	—	13	1	no	—
300	C₂×C₄	—	26	2	no	—
301	C₂×C₄	—	26	1	no	—
302	C₂×C₄	—	26	1	no	—
303	C₂×C₄	—	52	2	no	—
304	C₂×C₄	—	52	1	no	—
305	C₂×C₄	—	52	2	no	—
306	C₂×C₄	C₂×#6118	1	1	yes	characteristic
307	C₂×C₄	D₁₃⋊M₁₆	1	1	yes	characteristic
308	C₂×C₄	C₂×C₄⋊#261	1	1	yes	characteristic
309	C₂×C₄	C₂²×#1057	1	1	yes	characteristic
310	C₂×C₄	C₂×#6122	1	1	yes	characteristic
311	C₂×C₄	C₂×C₂²⋊#261	1	1	yes	characteristic
312	C₈	—	2	2	no	—
313	C₈	—	26	1	no	—
314	C₈	—	26	1	no	—
315	C₈	—	26	2	no	—
316	C₈	—	104	2	no	—
317	C₈	—	104	2	no	—
318	C₈	C₂²⋊#1057	1	2	yes	—
319	C₈	C₂²⋊#1057	1	2	yes	—
320	C₂²	—	2	1	no	—
321	C₂²	—	2	1	no	—
322	C₂²	—	2	2	no	—
323	C₂²	—	4	1	no	—
324	C₂²	—	4	1	no	—
325	C₂²	#11719119	1	1	yes	characteristic
326	C₂²	#11719130	1	1	yes	characteristic
327	C₂²	#11719136	1	1	yes	characteristic
328	C₄	—	2	2	no	—
329	C₄	—	13	2	no	—
330	C₄	—	26	1	no	—
331	C₄	—	26	1	no	—
332	C₄	—	26	1	no	—
333	C₄	—	52	1	no	—
334	C₄	—	52	1	no	—
335	C₄	C₂×C₄⋊#1057	1	1	yes	characteristic
336	C₄	#11719127	1	1	yes	characteristic
337	C₄	#11719129	1	1	yes	characteristic
338	C₄	C₂×C₂²⋊#1057	1	1	yes	characteristic
339	C₂	—	2	1	no	—
340	C₂	—	2	1	no	—
341	C₂	—	4	2	no	—
342	C₂	#13130690	1	1	yes	characteristic
343	C₂	#13130722	1	1	yes	characteristic
344	C₂	#13130724	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#11719136
commutator	C₂×C₄×C₁₃	C₂²×C₈
Frattini	C₂²×C₄	C₂²×#261
Fitting	C₂×C₁₃×#131	C₄
socle	C₂²×C₁₃	#380
prime radical	C₂×C₁₃×#41	C₂×C₄
cosocle	C₂×C₄×Dic₁₃	C₂³
hypercenter	C₂×#131	#261
hypocenter	C₁₃	#8604
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#131

Complements

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

N	H ≅ G/N	type
1. #13130383	C₂	semidirect
2. C₂×#11718253	C₂	semidirect
4. #13129682	C₂	semidirect
28. C₁₃×#131	C₈	semidirect
195. #131	#1057	semidirect
284. C₁₃	#8604	semidirect

Maximal subgroups

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

subgroup	index
#13130383	2
C₂×#11718253	2
#13130131	2
#13129682	2
#131×Dic₁₃	2
#8604	13