Subgroups — #16785720

Order 3328.

806 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#88103	C₂	1	2	yes	maximal
2	C₂×#11718268	C₂	1	2	yes	maximal
3	#13130500	C₂	1	2	yes	maximal
4	C₂×C₄×C₈×D₁₃	C₂	1	1	yes	maximal, characteristic
5	C₂²×#6118	C₂²	1	4	yes	—
6	C₂²×C₈×D₁₃	C₂²	1	1	yes	characteristic
7	C₂²×C₈×D₁₃	C₂²	1	1	yes	characteristic
8	C₂×C₄²×D₁₃	C₂²	1	1	yes	characteristic
9	C₂×#88103	—	2	4	no	—
10	C₂×#88103	—	2	4	no	—
11	C₂×#88103	C₄	1	8	yes	complemented
12	C₂×C₄×C₈×C₁₃	C₄	1	1	yes	characteristic
13	C₂×C₈×Dic₁₃	C₄	1	1	yes	characteristic
14	C₂×C₈×Dic₁₃	C₄	1	1	yes	characteristic
15	C₂×C₄×#1055	C₄	1	1	yes	characteristic
16	C₄×C₈×D₁₃	—	2	4	no	—
17	#11718290	—	2	4	no	—
18	#11718268	—	2	4	no	—
19	C₂²×C₄×D₁₃	Q₈	1	2	yes	—
20	C₂²×C₄×D₁₃	C₂³	1	1	yes	characteristic
21	C₂²×#1057	—	2	4	no	—
22	C₂×#6118	—	2	4	no	—
23	C₂×#6118	—	2	4	no	—
24	C₂×#6118	—	2	4	no	—
25	C₂×#6118	—	2	4	no	—
26	C₂×#6118	—	2	4	no	—
27	C₂×#6118	C₂×C₄	1	8	yes	—
28	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄×Dic₁₃	D₄	1	2	yes	—
33	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
34	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
35	C₂×C₈×D₁₃	—	2	2	no	—
36	C₂×C₈×D₁₃	—	2	2	no	—
37	C₂×C₈×D₁₃	C₂×C₄	1	4	yes	—
38	C₂×C₈×D₁₃	D₄	1	2	yes	—
39	C₂×C₈×D₁₃	D₄	1	2	yes	—
40	C₂×C₈×D₁₃	D₄	1	4	yes	—
41	C₂×C₈×D₁₃	Q₈	1	2	yes	—
42	C₂×C₈×D₁₃	Q₈	1	2	yes	—
43	C₄²×D₁₃	—	2	2	no	—
44	C₄²×D₁₃	—	2	2	no	—
45	#88103	—	2	16	no	—
46	C₄×C₈×C₁₃	—	2	2	no	—
47	C₈×Dic₁₃	—	2	2	no	—
48	C₈×Dic₁₃	—	2	2	no	—
49	C₄×#1055	—	2	2	no	—
50	#10864	—	13	1	no	maximal, Sylow
51	C₂³×D₁₃	C₂×Q₈	1	1	yes	characteristic
52	C₂²×C₄×C₁₃	C₄⋊C₄	1	2	yes	—
53	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
54	C₂²×Dic₁₃	C₄⋊C₄	1	2	yes	—
55	C₂²×Dic₁₃	C₂×D₄	1	1	yes	characteristic
56	C₂×C₄×D₁₃	—	2	4	no	—
57	C₂×C₄×D₁₃	—	2	4	no	—
58	C₂×C₄×D₁₃	—	2	4	no	—
59	C₂×C₄×D₁₃	C₂²×C₄	1	2	yes	—
60	C₂×C₄×D₁₃	C₂×D₄	1	1	yes	characteristic
61	C₂×C₄×D₁₃	C₂×D₄	1	2	yes	—
62	C₂×C₄×D₁₃	C₂×D₄	1	1	yes	characteristic
63	C₂×C₄×D₁₃	C₂×Q₈	1	1	yes	characteristic
64	C₂×C₄×D₁₃	C₂×Q₈	1	1	yes	characteristic
65	C₂×C₄×D₁₃	#41	1	2	yes	—
66	C₂×C₄×D₁₃	#41	1	2	yes	—
67	C₂×#1057	—	2	8	no	—
68	C₂×#1057	—	4	4	no	—
69	C₂×#1057	—	4	4	no	—
70	#6118	—	2	8	no	—
71	#6118	—	2	8	no	—
72	#6118	—	4	4	no	—
73	#6118	—	4	4	no	—
74	C₂×C₈×C₁₃	—	2	1	no	—
75	C₂×C₈×C₁₃	—	2	1	no	—
76	C₂×C₈×C₁₃	C₄²	1	2	yes	—
77	C₂×C₈×C₁₃	C₂²⋊C₄	1	2	yes	—
78	C₂×C₈×C₁₃	C₄⋊C₄	1	2	yes	—
79	C₂×C₈×C₁₃	C₄⋊C₄	1	2	yes	—
80	C₄²×C₁₃	—	2	1	no	—
81	C₄²×C₁₃	—	2	1	no	—
82	C₄×Dic₁₃	—	2	1	no	—
83	C₄×Dic₁₃	—	2	1	no	—
84	C₄×Dic₁₃	—	2	4	no	—
85	C₂×#1055	—	2	1	no	—
86	C₂×#1055	—	2	1	no	—
87	C₂×#1055	C₄²	1	2	yes	—
88	C₂×#1055	C₂²⋊C₄	1	2	yes	—
89	C₂×#1055	C₄⋊C₄	1	2	yes	—
90	C₂×#1055	C₄⋊C₄	1	2	yes	—
91	C₈×D₁₃	—	2	4	no	—
92	C₈×D₁₃	—	2	4	no	—
93	C₈×D₁₃	C₂²⋊C₄	1	8	yes	—
94	C₈×D₁₃	C₄⋊C₄	1	8	yes	—
95	C₂²×#108	—	13	2	no	—
96	C₂²×C₄×C₈	—	13	1	no	—
97	C₂×#341	—	13	2	no	—
98	#1728	—	13	2	no	—
99	C₂³×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
100	C₂²×D₁₃	—	2	2	no	—
101	C₂²×D₁₃	—	2	2	no	—
102	C₂²×D₁₃	C₄×Q₈	1	2	yes	—
103	C₂²×D₁₃	C₂²⋊Q₈	1	2	yes	—
104	C₂²×D₁₃	C₄⋊Q₈	1	1	yes	characteristic
105	C₂²×D₁₃	C₄⋊Q₈	1	1	yes	characteristic
106	C₂×C₄×C₁₃	—	2	2	no	—
107	C₂×C₄×C₁₃	—	2	2	no	—
108	C₂×C₄×C₁₃	#108	1	4	yes	complemented
109	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
110	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
111	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
112	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
113	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
114	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
115	C₂×Dic₁₃	—	2	2	no	—
116	C₂×Dic₁₃	—	2	2	no	—
117	C₂×Dic₁₃	#108	1	4	yes	complemented
118	C₂×Dic₁₃	C₄×D₄	1	2	yes	—
119	C₂×Dic₁₃	#121	1	2	yes	—
120	C₂×Dic₁₃	#127	1	1	yes	characteristic
121	C₂×Dic₁₃	C₄⋊Q₈	1	1	yes	characteristic
122	C₄×D₁₃	—	2	16	no	—
123	C₄×D₁₃	C₂×C₂²⋊C₄	1	2	yes	—
124	C₄×D₁₃	C₂×C₄⋊C₄	1	2	yes	—
125	C₄×D₁₃	C₄×D₄	1	2	yes	—
126	C₄×D₁₃	C₄×Q₈	1	2	yes	—
127	C₄×D₁₃	#121	1	2	yes	—
128	C₄×D₁₃	C₂²⋊Q₈	1	2	yes	—
129	C₄×D₁₃	#124	1	2	yes	—
130	C₄×D₁₃	#125	1	2	yes	—
131	#1057	—	4	8	no	—
132	C₈×C₁₃	—	2	2	no	—
133	C₈×C₁₃	#95	1	4	yes	—
134	#1055	—	2	2	no	—
135	#1055	#95	1	4	yes	—
136	C₂²×M₁₆	—	13	4	no	—
137	C₂³×C₈	—	13	1	no	—
138	C₂³×C₈	—	13	1	no	—
139	C₂²×C₄²	—	13	1	no	—
140	C₂×#108	—	13	8	no	—
141	C₂×#108	—	26	4	no	—
142	C₂×#108	—	26	4	no	—
143	C₂×C₄×C₈	—	13	1	no	—
144	C₂×C₄×C₈	—	13	1	no	—
145	C₂×C₄×C₈	—	13	1	no	—
146	C₂×C₄×C₈	—	26	4	no	—
147	C₂×C₄×C₈	#261	1	1	yes	characteristic
148	#341	—	26	4	no	—
149	#336	—	26	4	no	—
150	C₂²×C₁₃	C₄×C₄⋊C₄	1	1	yes	characteristic
151	C₂²×C₁₃	#380	1	1	yes	characteristic
152	C₂²×C₁₃	#384	1	1	yes	characteristic
153	C₂²×C₁₃	C₂×#108	1	2	yes	—
154	C₂²×C₁₃	C₂×#108	1	2	yes	—
155	C₂×D₁₃	—	2	4	no	—
156	C₂×D₁₃	—	2	1	no	—
157	C₂×D₁₃	—	2	1	no	—
158	C₂×D₁₃	—	2	1	no	—
159	C₂×D₁₃	—	2	1	no	—
160	C₂×D₁₃	—	2	4	no	—
161	C₂×D₁₃	#389	1	2	yes	—
162	C₂×D₁₃	#391	1	2	yes	—
163	C₄×C₁₃	—	2	4	no	—
164	C₄×C₁₃	C₂×#95	1	1	yes	characteristic
165	C₄×C₁₃	C₄×C₄⋊C₄	1	1	yes	characteristic
166	C₄×C₁₃	#380	1	1	yes	characteristic
167	C₄×C₁₃	#382	1	1	yes	characteristic
168	Dic₁₃	—	2	4	no	—
169	Dic₁₃	#389	1	2	yes	—
170	Dic₁₃	#390	1	2	yes	—
171	C₂³×C₄	—	13	1	no	—
172	C₂³×C₄	—	13	2	no	—
173	C₂×M₁₆	—	13	8	no	—
174	C₂×M₁₆	—	26	4	no	—
175	C₂×M₁₆	—	26	4	no	—
176	C₂×M₁₆	—	26	4	no	—
177	C₂×M₁₆	—	26	4	no	—
178	C₂×M₁₆	—	26	4	no	—
179	C₂²×C₈	—	13	1	no	—
180	C₂²×C₈	—	13	4	no	—
181	C₂²×C₈	—	13	2	no	—
182	C₂²×C₈	—	13	2	no	—
183	C₂²×C₈	—	13	4	no	—
184	C₂²×C₈	—	13	1	no	—
185	C₂²×C₈	—	13	2	no	—
186	C₂²×C₈	—	13	2	no	—
187	C₂²×C₈	—	26	2	no	—
188	C₂²×C₈	—	26	2	no	—
189	C₂²×C₈	—	26	4	no	—
190	C₂²×C₈	C₂×#261	1	1	yes	characteristic
191	C₂²×C₈	C₂×#261	1	1	yes	characteristic
192	C₂×C₄²	—	13	2	no	—
193	C₂×C₄²	—	13	1	no	—
194	C₂×C₄²	—	26	2	no	—
195	C₂×C₄²	—	26	2	no	—
196	C₂×C₄²	C₂×#261	1	1	yes	characteristic
197	#108	—	26	16	no	—
198	C₄×C₈	—	2	2	no	—
199	C₄×C₈	—	26	2	no	—
200	C₄×C₈	—	26	2	no	—
201	C₄×C₈	—	26	2	no	—
202	C₂×C₁₃	C₄⋊#95	1	1	yes	characteristic
203	C₂×C₁₃	C₄×#108	1	2	yes	—
204	C₂×C₁₃	#1810	1	2	yes	—
205	C₂×C₁₃	#1848	1	1	yes	characteristic
206	C₂×C₁₃	#1848	1	1	yes	characteristic
207	D₁₃	—	2	2	no	—
208	D₁₃	—	2	2	no	—
209	C₂⁴	—	13	1	no	—
210	C₂²×C₄	—	13	1	no	—
211	C₂²×C₄	—	13	2	no	—
212	C₂²×C₄	—	13	1	no	—
213	C₂²×C₄	—	13	1	no	—
214	C₂²×C₄	—	13	1	no	—
215	C₂²×C₄	—	13	1	no	—
216	C₂²×C₄	—	13	2	no	—
217	C₂²×C₄	—	13	2	no	—
218	C₂²×C₄	—	13	2	no	—
219	C₂²×C₄	—	13	2	no	—
220	C₂²×C₄	—	26	4	no	—
221	C₂²×C₄	—	26	4	no	—
222	C₂²×C₄	—	26	4	no	—
223	C₂²×C₄	C₄⋊#261	1	2	yes	—
224	C₂²×C₄	C₂²×#261	1	1	yes	characteristic
225	M₁₆	—	26	8	no	—
226	M₁₆	—	26	8	no	—
227	M₁₆	—	52	4	no	—
228	M₁₆	—	52	4	no	—
229	C₂×C₈	—	2	1	no	—
230	C₂×C₈	—	2	1	no	—
231	C₂×C₈	—	13	2	no	—
232	C₂×C₈	—	13	2	no	—
233	C₂×C₈	—	13	2	no	—
234	C₂×C₈	—	13	8	no	—
235	C₂×C₈	—	13	8	no	—
236	C₂×C₈	—	13	2	no	—
237	C₂×C₈	—	26	4	no	—
238	C₂×C₈	—	26	4	no	—
239	C₂×C₈	—	26	1	no	—
240	C₂×C₈	—	26	1	no	—
241	C₂×C₈	—	26	8	no	—
242	C₂×C₈	—	52	4	no	—
243	C₂×C₈	—	52	4	no	—
244	C₂×C₈	C₄×#261	1	2	yes	—
245	C₂×C₈	C₄⋊#261	1	2	yes	—
246	C₂×C₈	C₄⋊#261	1	2	yes	—
247	C₂×C₈	C₂²⋊#261	1	2	yes	—
248	C₄²	—	2	1	no	—
249	C₄²	—	2	1	no	—
250	C₄²	—	26	4	no	—
251	C₄²	—	26	1	no	—
252	C₄²	—	26	1	no	—
253	C₁₃	#10864	1	1	yes	characteristic, complemented, Sylow
254	C₂³	—	13	2	no	—
255	C₂³	—	13	1	no	—
256	C₂³	—	13	1	no	—
257	C₂³	—	13	2	no	—
258	C₂³	—	26	2	no	—
259	C₂³	—	26	2	no	—
260	C₂³	C₂×C₄⋊#261	1	1	yes	characteristic
261	C₂×C₄	—	2	2	no	—
262	C₂×C₄	—	2	2	no	—
263	C₂×C₄	—	13	2	no	—
264	C₂×C₄	—	13	1	no	—
265	C₂×C₄	—	13	1	no	—
266	C₂×C₄	—	13	2	no	—
267	C₂×C₄	—	13	2	no	—
268	C₂×C₄	—	13	2	no	—
269	C₂×C₄	—	13	2	no	—
270	C₂×C₄	—	13	2	no	—
271	C₂×C₄	—	13	2	no	—
272	C₂×C₄	—	13	2	no	—
273	C₂×C₄	—	13	2	no	—
274	C₂×C₄	—	13	2	no	—
275	C₂×C₄	—	13	4	no	—
276	C₂×C₄	—	26	16	no	—
277	C₂×C₄	—	26	2	no	—
278	C₂×C₄	—	26	2	no	—
279	C₂×C₄	#88103	1	4	yes	complemented
280	C₂×C₄	C₂×C₄×#261	1	1	yes	characteristic
281	C₂×C₄	C₂×C₄⋊#261	1	1	yes	characteristic
282	C₂×C₄	C₂×C₄⋊#261	1	1	yes	characteristic
283	C₂×C₄	#88237	1	1	yes	characteristic
284	C₂×C₄	#88237	1	1	yes	characteristic
285	C₂×C₄	C₂×C₂²⋊#261	1	1	yes	characteristic
286	C₈	—	2	2	no	—
287	C₈	—	13	4	no	—
288	C₈	—	26	2	no	—
289	C₈	—	52	8	no	—
290	C₈	(C₂×#261)⋊C₄	1	4	yes	—
291	C₂²	—	13	2	no	—
292	C₂²	—	13	2	no	—
293	C₂²	—	26	1	no	—
294	C₂²	—	26	1	no	—
295	C₂²	—	26	1	no	—
296	C₂²	—	26	1	no	—
297	C₂²	—	26	4	no	—
298	C₂²	—	26	4	no	—
299	C₂²	C₄×C₄⋊#261	1	1	yes	characteristic
300	C₂²	#11719059	1	1	yes	characteristic
301	C₂²	C₂×#88103	1	2	yes	—
302	C₂²	D₁₃⋊#108	1	2	yes	—
303	C₂²	#11719136	1	1	yes	characteristic
304	C₄	—	2	4	no	—
305	C₄	—	13	2	no	—
306	C₄	—	13	2	no	—
307	C₄	—	26	4	no	—
308	C₄	C₄×C₄⋊#261	1	1	yes	characteristic
309	C₄	#11719060	1	1	yes	characteristic
310	C₄	C₂×(C₂×#261)⋊C₄	1	1	yes	characteristic
311	C₄	#11719136	1	1	yes	characteristic
312	C₂	—	26	2	no	—
313	C₂	—	26	2	no	—
314	C₂	#13130014	1	1	yes	characteristic
315	C₂	C₄×#88103	1	2	yes	—
316	C₂	#13130715	1	2	yes	—
317	C₂	#13130775	1	1	yes	characteristic
318	C₂	#13130776	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×C₄⋊#261
commutator	C₂×C₄×C₁₃	C₂×C₄²
Frattini	C₂²×C₄	C₂²×#261
Fitting	C₂×C₄×C₈×C₁₃	C₄
socle	C₂³×C₁₃	C₂×C₄⋊C₄
prime radical	C₂³×D₁₃	C₂×Q₈
cosocle	C₂²×C₄×D₁₃	C₂³
hypercenter	C₂×C₄×C₈	#261
hypocenter	C₁₃	#10864
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
11. C₂×#88103	C₄	semidirect
108. C₂×C₄×C₁₃	#108	semidirect
117. C₂×Dic₁₃	#108	semidirect
253. C₁₃	#10864	semidirect
279. C₂×C₄	#88103	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#88103	2
C₂×#11718268	2
#13130500	2
C₂×C₄×C₈×D₁₃	2
#10864	13