Subgroups — #16792639

Order 3328.

874 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13131513	C₂	1	2	yes	maximal
2	#13130049	C₂	1	2	yes	maximal, complemented
3	#13130046	C₂	1	4	yes	maximal, complemented
4	#13130039	C₂	1	2	yes	maximal, complemented
5	C₈×#6118	C₂	1	4	yes	maximal, complemented
6	C₂×C₄×C₈×D₁₃	C₂	1	1	yes	maximal, characteristic
7	C₂²×C₈×D₁₃	C₄	1	1	yes	characteristic
8	C₂²×C₈×D₁₃	C₄	1	1	yes	characteristic
9	C₂×C₄²×D₁₃	C₂²	1	1	yes	characteristic
10	#11719130	C₄	1	4	yes	—
11	#11719127	C₄	1	4	yes	—
12	#11719119	C₂²	1	4	yes	—
13	#11719054	C₄	1	4	yes	—
14	#11719052	C₄	1	4	yes	—
15	#11719051	C₂²	1	4	yes	—
16	C₄×#6118	C₂²	1	4	yes	—
17	C₂×C₄×C₈×C₁₃	C₄	1	1	yes	characteristic
18	C₂×C₈×Dic₁₃	C₂²	1	1	yes	characteristic
19	C₂×C₈×Dic₁₃	C₂²	1	1	yes	characteristic
20	C₂×C₄×#1055	C₄	1	1	yes	characteristic
21	C₄×C₈×D₁₃	C₄	1	2	yes	—
22	C₄×C₈×D₁₃	C₄	1	2	yes	—
23	C₄×C₈×D₁₃	C₂²	1	2	yes	—
24	C₄×C₈×D₁₃	C₂²	1	2	yes	—
25	#1057⋊C₈	C₂²	1	8	yes	complemented
26	#11718251	C₂²	1	4	yes	complemented
27	C₈×#1057	C₂²	1	4	yes	complemented
28	C₂²×C₄×D₁₃	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₄×D₁₃	C₂×C₄	1	2	yes	—
30	C₂×#6118	C₂×C₄	1	8	yes	—
31	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄×Dic₁₃	C₂×C₄	1	2	yes	—
35	C₂×C₄×Dic₁₃	C₂³	1	1	yes	characteristic
36	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
37	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
38	C₂×C₈×D₁₃	—	2	4	no	—
39	C₂×C₈×D₁₃	—	2	4	no	—
40	C₂×C₈×D₁₃	C₂×C₄	1	2	yes	—
41	C₂×C₈×D₁₃	C₂×C₄	1	2	yes	—
42	C₂×C₈×D₁₃	C₂×C₄	1	2	yes	—
43	C₂×C₈×D₁₃	C₂×C₄	1	2	yes	—
44	C₄²×D₁₃	C₂×C₄	1	2	yes	—
45	C₄²×D₁₃	C₂×C₄	1	2	yes	—
46	C₄²×D₁₃	C₂×C₄	1	2	yes	—
47	C₄²×D₁₃	C₂³	1	2	yes	—
48	C₂²⋊#1057	C₂×C₄	1	8	yes	—
49	#88111	C₂×C₄	1	8	yes	—
50	#88110	C₂×C₄	1	8	yes	—
51	C₄⋊#1057	C₂×C₄	1	8	yes	—
52	C₄⋊#1057	C₂×C₄	1	8	yes	—
53	C₄×#1057	C₂³	1	8	yes	—
54	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
55	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
56	C₈×Dic₁₃	C₂×C₄	1	2	yes	—
57	C₈×Dic₁₃	C₂×C₄	1	1	yes	characteristic
58	C₈×Dic₁₃	C₂×C₄	1	1	yes	characteristic
59	C₈×Dic₁₃	C₂³	1	1	yes	characteristic
60	C₈×Dic₁₃	C₂³	1	1	yes	characteristic
61	C₈×Dic₁₃	C₂³	1	2	yes	—
62	C₄×#1055	C₂×C₄	1	2	yes	—
63	C₄×#1055	C₂×C₄	1	2	yes	—
64	#17323	—	13	1	no	maximal, Sylow
65	C₂³×D₁₃	C₄²	1	1	yes	characteristic
66	C₂²×C₄×C₁₃	C₄²	1	2	yes	—
67	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
68	C₂²×Dic₁₃	C₄²	1	1	yes	characteristic
69	C₂²×Dic₁₃	C₂²×C₄	1	2	yes	—
70	C₂×C₄×D₁₃	—	2	2	no	—
71	C₂×C₄×D₁₃	—	2	2	no	—
72	C₂×C₄×D₁₃	—	2	8	no	—
73	C₂×C₄×D₁₃	C₄²	1	2	yes	—
74	C₂×C₄×D₁₃	C₄²	1	2	yes	—
75	C₂×C₄×D₁₃	C₄²	1	2	yes	—
76	C₂×C₄×D₁₃	C₂²×C₄	1	2	yes	—
77	C₂×C₄×D₁₃	C₂²×C₄	1	2	yes	—
78	C₂×C₄×D₁₃	C₂²×C₄	1	2	yes	—
79	C₂×#1057	C₂²×C₄	1	16	yes	—
80	#6118	—	2	16	no	—
81	C₂×C₈×C₁₃	—	2	2	no	—
82	C₂×C₈×C₁₃	M₁₆	1	4	yes	complemented
83	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
84	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
86	C₄²×C₁₃	C₄²	1	1	yes	characteristic
87	C₄²×C₁₃	C₄²	1	1	yes	characteristic
88	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
89	C₄×Dic₁₃	C₄²	1	2	yes	—
90	C₄×Dic₁₃	C₄²	1	2	yes	—
91	C₄×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
92	C₄×Dic₁₃	C₂²×C₄	1	2	yes	—
93	C₄×Dic₁₃	C₂²×C₄	1	2	yes	—
94	C₄×Dic₁₃	C₂²×C₄	1	2	yes	—
95	C₄×Dic₁₃	C₂⁴	1	1	yes	characteristic
96	C₂×#1055	—	2	2	no	—
97	C₂×#1055	M₁₆	1	4	yes	complemented
98	C₂×#1055	C₂²×C₄	1	2	yes	—
99	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
100	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
101	C₈×D₁₃	—	2	4	no	—
102	C₈×D₁₃	—	2	4	no	—
103	C₈×D₁₃	—	2	4	no	—
104	C₈×D₁₃	—	2	4	no	—
105	#2920	—	13	2	no	—
106	C₂²×C₄×C₈	—	13	1	no	—
107	#1452	—	13	2	no	—
108	#1451	—	13	4	no	—
109	#1450	—	13	2	no	—
110	C₈×M₁₆	—	13	4	no	—
111	C₂³×C₁₃	C₂×C₄²	1	1	yes	characteristic
112	C₂²×D₁₃	—	2	2	no	—
113	C₂²×D₁₃	—	2	4	no	—
114	C₂²×D₁₃	C₂×C₄²	1	2	yes	—
115	C₂×C₄×C₁₃	—	2	4	no	—
116	C₂×C₄×C₁₃	C₂×C₄²	1	2	yes	—
117	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
118	C₂×C₄×C₁₃	C₂×C₄²	1	2	yes	—
119	C₂×C₄×C₁₃	C₂×M₁₆	1	2	yes	—
120	C₂×C₄×C₁₃	C₂×M₁₆	1	2	yes	—
121	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
122	C₂×Dic₁₃	—	2	2	no	—
123	C₂×Dic₁₃	—	2	4	no	—
124	C₂×Dic₁₃	C₂×C₄²	1	2	yes	—
125	C₂×Dic₁₃	C₂×C₄²	1	2	yes	—
126	C₂×Dic₁₃	C₂³×C₄	1	2	yes	—
127	C₄×D₁₃	—	2	4	no	—
128	C₄×D₁₃	—	2	2	no	—
129	C₄×D₁₃	—	2	2	no	—
130	C₄×D₁₃	—	2	4	no	—
131	C₄×D₁₃	—	2	4	no	—
132	C₄×D₁₃	—	2	4	no	—
133	C₄×D₁₃	#131	1	8	yes	—
134	#1057	—	2	16	no	—
135	C₈×C₁₃	—	2	2	no	—
136	C₈×C₁₃	C₂×M₁₆	1	2	yes	—
137	C₈×C₁₃	C₂×M₁₆	1	2	yes	—
138	#1055	—	2	2	no	—
139	#1055	C₂×M₁₆	1	2	yes	—
140	#1055	C₂×M₁₆	1	2	yes	—
141	C₂³×C₈	—	13	1	no	—
142	C₂³×C₈	—	13	1	no	—
143	C₂²×C₄²	—	13	1	no	—
144	#432	—	13	4	no	—
145	#432	—	13	4	no	—
146	#431	—	13	4	no	—
147	#431	—	13	4	no	—
148	#423	—	13	4	no	—
149	#407	—	13	4	no	—
150	C₄×M₁₆	—	13	4	no	—
151	C₂×C₄×C₈	—	13	1	no	—
152	C₂×C₄×C₈	—	13	1	no	—
153	C₂×C₄×C₈	—	13	1	no	—
154	C₂×C₄×C₈	—	13	2	no	—
155	C₂×C₄×C₈	—	13	2	no	—
156	C₂×C₄×C₈	—	13	2	no	—
157	C₂×C₄×C₈	—	13	2	no	—
158	C₂×C₄×C₈	#261	1	1	yes	characteristic
159	#322	—	13	4	no	—
160	#322	—	13	8	no	—
161	C₈²	—	13	4	no	—
162	C₂²×C₁₃	—	2	2	no	—
163	C₂²×C₁₃	C₄×M₁₆	1	2	yes	—
164	C₂²×C₁₃	C₂²×C₄²	1	1	yes	characteristic
165	C₂×D₁₃	—	2	8	no	—
166	C₂×D₁₃	—	2	2	no	—
167	C₂×D₁₃	—	2	2	no	—
168	C₂×D₁₃	#405	1	4	yes	—
169	C₄×C₁₃	—	2	2	no	—
170	C₄×C₁₃	C₄×M₁₆	1	2	yes	—
171	C₄×C₁₃	#405	1	4	yes	—
172	C₄×C₁₃	C₂²×M₁₆	1	1	yes	characteristic
173	C₄×C₁₃	C₂²×M₁₆	1	1	yes	characteristic
174	Dic₁₃	—	2	2	no	—
175	Dic₁₃	—	2	2	no	—
176	Dic₁₃	C₂×#131	1	4	yes	—
177	C₂³×C₄	—	13	2	no	—
178	C₂³×C₄	—	13	1	no	—
179	C₂×M₁₆	—	13	8	no	—
180	C₂²×C₈	—	13	1	no	—
181	C₂²×C₈	—	13	1	no	—
182	C₂²×C₈	—	13	2	no	—
183	C₂²×C₈	—	13	2	no	—
184	C₂²×C₈	—	13	2	no	—
185	C₂²×C₈	—	13	2	no	—
186	C₂²×C₈	—	26	4	no	—
187	C₂²×C₈	—	26	4	no	—
188	C₂²×C₈	C₂×#261	1	1	yes	characteristic
189	C₂²×C₈	C₂×#261	1	1	yes	characteristic
190	C₂×C₄²	—	13	2	no	—
191	C₂×C₄²	—	13	1	no	—
192	C₂×C₄²	—	13	2	no	—
193	C₂×C₄²	—	13	2	no	—
194	C₂×C₄²	—	13	2	no	—
195	C₂×C₄²	—	13	2	no	—
196	C₂×C₄²	C₂×#261	1	1	yes	characteristic
197	C₄⋊C₈	—	13	8	no	—
198	C₄⋊C₈	—	13	8	no	—
199	C₂²⋊C₈	—	13	8	no	—
200	C₂²⋊C₈	—	13	8	no	—
201	#97	—	13	8	no	—
202	C₄×C₈	—	13	8	no	—
203	C₄×C₈	—	13	1	no	—
204	C₄×C₈	—	13	2	no	—
205	C₄×C₈	—	13	1	no	—
206	C₄×C₈	—	13	2	no	—
207	C₄×C₈	—	13	1	no	—
208	C₄×C₈	—	13	1	no	—
209	C₄×C₈	—	13	2	no	—
210	C₄×C₈	—	13	2	no	—
211	C₄×C₈	C₂×#261	1	2	yes	—
212	C₄×C₈	C₂×#261	1	2	yes	—
213	C₂×C₁₃	—	2	2	no	—
214	C₂×C₁₃	C₂×C₄×M₁₆	1	1	yes	characteristic
215	C₂×C₁₃	C₂×#405	1	2	yes	—
216	D₁₃	—	2	4	no	—
217	C₂⁴	—	13	1	no	—
218	C₂²×C₄	—	13	2	no	—
219	C₂²×C₄	—	13	2	no	—
220	C₂²×C₄	—	13	1	no	—
221	C₂²×C₄	—	13	2	no	—
222	C₂²×C₄	—	13	2	no	—
223	C₂²×C₄	—	13	2	no	—
224	C₂²×C₄	—	13	2	no	—
225	C₂²×C₄	—	13	2	no	—
226	C₂²×C₄	—	26	8	no	—
227	C₂²×C₄	—	26	2	no	—
228	C₂²×C₄	—	26	2	no	—
229	C₂²×C₄	C₄×#261	1	2	yes	—
230	C₂²×C₄	C₂²×#261	1	1	yes	characteristic
231	M₁₆	—	26	16	no	—
232	C₂×C₈	—	2	2	no	—
233	C₂×C₈	—	13	4	no	—
234	C₂×C₈	—	13	1	no	—
235	C₂×C₈	—	13	1	no	—
236	C₂×C₈	—	13	16	no	—
237	C₂×C₈	—	13	2	no	—
238	C₂×C₈	—	26	4	no	—
239	C₂×C₈	—	26	4	no	—
240	C₂×C₈	—	26	2	no	—
241	C₂×C₈	—	26	4	no	—
242	C₂×C₈	—	26	4	no	—
243	C₂×C₈	#6118	1	4	yes	complemented
244	C₂×C₈	C₂²×#261	1	2	yes	—
245	C₂×C₈	C₂²×#261	1	1	yes	characteristic
246	C₂×C₈	C₂²×#261	1	1	yes	characteristic
247	C₄²	—	13	2	no	—
248	C₄²	—	13	2	no	—
249	C₄²	—	13	2	no	—
250	C₄²	—	13	2	no	—
251	C₄²	—	13	2	no	—
252	C₄²	—	13	1	no	—
253	C₄²	—	13	1	no	—
254	C₄²	C₄×#261	1	1	yes	characteristic
255	C₄²	C₄×#261	1	1	yes	characteristic
256	C₄²	C₂²×#261	1	2	yes	—
257	C₁₃	#17323	1	1	yes	characteristic, complemented, Sylow
258	C₂³	—	13	2	no	—
259	C₂³	—	26	4	no	—
260	C₂³	—	26	2	no	—
261	C₂³	C₂×C₄×#261	1	1	yes	characteristic
262	C₂×C₄	—	2	4	no	—
263	C₂×C₄	—	13	2	no	—
264	C₂×C₄	—	13	8	no	—
265	C₂×C₄	—	13	2	no	—
266	C₂×C₄	—	13	2	no	—
267	C₂×C₄	—	26	4	no	—
268	C₂×C₄	—	26	4	no	—
269	C₂×C₄	—	26	4	no	—
270	C₂×C₄	—	26	2	no	—
271	C₂×C₄	—	26	4	no	—
272	C₂×C₄	—	26	4	no	—
273	C₂×C₄	—	26	2	no	—
274	C₂×C₄	—	26	2	no	—
275	C₂×C₄	C₂×#6118	1	2	yes	—
276	C₂×C₄	D₁₃⋊M₁₆	1	2	yes	—
277	C₂×C₄	C₂×C₄×#261	1	2	yes	—
278	C₂×C₄	C₂×C₄×#261	1	1	yes	characteristic
279	C₂×C₄	C₂×C₄×#261	1	2	yes	—
280	C₂×C₄	C₂³×#261	1	1	yes	characteristic
281	C₈	—	2	2	no	—
282	C₈	—	13	2	no	—
283	C₈	—	13	2	no	—
284	C₈	—	26	16	no	—
285	C₈	—	26	2	no	—
286	C₈	C₂×#6118	1	2	yes	—
287	C₈	C₂×#6118	1	2	yes	—
288	C₂²	—	2	2	no	—
289	C₂²	—	13	4	no	—
290	C₂²	—	26	8	no	—
291	C₂²	—	26	2	no	—
292	C₂²	—	26	2	no	—
293	C₂²	C₄×#6118	1	2	yes	—
294	C₂²	C₂²×C₄×#261	1	1	yes	characteristic
295	C₄	—	2	2	no	—
296	C₄	—	13	4	no	—
297	C₄	—	26	2	no	—
298	C₄	—	26	2	no	—
299	C₄	C₄×#6118	1	2	yes	—
300	C₄	#11719089	1	4	yes	—
301	C₄	C₂²×#6118	1	1	yes	characteristic
302	C₄	C₂×D₁₃⋊M₁₆	1	1	yes	characteristic
303	C₂	—	2	2	no	—
304	C₂	—	26	4	no	—
305	C₂	C₂×C₄×#6118	1	1	yes	characteristic
306	C₂	C₂×#11719089	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	C₂²×#261
commutator, socle	C₂²×C₁₃	C₂²×C₄²
Frattini	C₂×C₄	C₂³×#261
Fitting	C₂×C₄×C₈×C₁₃	C₄
prime radical	C₂³×D₁₃	C₄²
cosocle	C₄×Dic₁₃	C₂⁴
hypercenter	C₂×C₄×C₈	#261
hypocenter	C₁₃	#17323
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
2. #13130049	C₂	semidirect
3. #13130046	C₂	semidirect
4. #13130039	C₂	semidirect
5. C₈×#6118	C₂	semidirect
25. #1057⋊C₈	C₂²	semidirect
26. #11718251	C₂²	semidirect
27. C₈×#1057	C₂²	semidirect
82. C₂×C₈×C₁₃	M₁₆	semidirect
97. C₂×#1055	M₁₆	semidirect
243. C₂×C₈	#6118	semidirect
257. C₁₃	#17323	semidirect

Maximal subgroups

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

subgroup	index
#13131513	2
#13130049	2
#13130046	2
#13130039	2
C₈×#6118	2
C₂×C₄×C₈×D₁₃	2
#17323	13