Subgroups — #16016000

Order 3328.

682 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₈×D₁₃	C₂	1	1	yes	maximal, characteristic
2	#13116638	C₂	1	4	yes	maximal, complemented
3	#13116634	C₂	1	4	yes	maximal, complemented
4	#13116592	C₂	1	1	yes	maximal, characteristic, complemented
5	#13116446	C₂	1	2	yes	maximal
6	M₃₂×Dic₁₃	C₂	1	2	yes	maximal, complemented
7	C₁₃×#2150	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×C₈×D₁₃	C₄	1	1	yes	characteristic
9	C₂²×C₈×D₁₃	C₂²	1	1	yes	characteristic
10	C₂×C₄²×D₁₃	C₄	1	1	yes	characteristic
11	C₂×C₁₃×M₃₂	C₂²	1	2	yes	—
12	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
13	C₂×C₈×Dic₁₃	C₄	1	1	yes	characteristic
14	C₂×C₈×Dic₁₃	C₂²	1	1	yes	characteristic
15	C₂×#88052	C₂²	1	2	yes	—
16	C₂×C₄×#1055	C₄	1	1	yes	characteristic
17	C₄×C₈×D₁₃	C₄	1	2	yes	—
18	C₄×C₈×D₁₃	C₄	1	2	yes	—
19	C₄×C₈×D₁₃	C₂²	1	2	yes	—
20	C₄×C₈×D₁₃	C₂²	1	2	yes	—
21	C₁₃×C₄⋊C₁₆	C₂²	1	4	yes	complemented
22	#11718099	C₂²	1	8	yes	—
23	#11718094	C₂²	1	4	yes	complemented
24	C₁₆×Dic₁₃	C₂²	1	4	yes	complemented
25	C₄⋊#6090	C₂²	1	4	yes	complemented
26	C₂²×C₄×D₁₃	C₈	1	2	yes	—
27	C₂²×C₄×D₁₃	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
30	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄×Dic₁₃	C₈	1	2	yes	—
32	C₂×C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
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	—	2	4	no	—
38	C₂×C₈×D₁₃	C₂×C₄	1	2	yes	—
39	C₂×C₈×D₁₃	C₂×C₄	1	2	yes	—
40	C₂×C₈×D₁₃	C₂×C₄	1	2	yes	—
41	C₂×C₈×D₁₃	C₂³	1	2	yes	—
42	C₄²×D₁₃	C₈	1	2	yes	—
43	C₄²×D₁₃	C₈	1	2	yes	—
44	C₄²×D₁₃	C₂×C₄	1	2	yes	—
45	C₄²×D₁₃	C₂×C₄	1	2	yes	—
46	C₁₃×M₃₂	—	2	4	no	—
47	C₂×C₁₃×C₁₆	C₂³	1	4	yes	—
48	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
49	C₄×C₈×C₁₃	C₂³	1	2	yes	—
50	C₈×Dic₁₃	C₂×C₄	1	2	yes	—
51	C₈×Dic₁₃	C₂×C₄	1	2	yes	—
52	C₈×Dic₁₃	C₂×C₄	1	1	yes	characteristic
53	C₈×Dic₁₃	C₂×C₄	1	1	yes	characteristic
54	C₈×Dic₁₃	C₂³	1	1	yes	characteristic
55	C₈×Dic₁₃	C₂³	1	1	yes	characteristic
56	#88052	—	2	4	no	—
57	C₂×#6090	C₂³	1	4	yes	—
58	C₄×#1055	C₂×C₄	1	2	yes	—
59	C₄×#1055	C₂×C₄	1	2	yes	—
60	#33418	—	13	1	no	maximal, Sylow
61	C₂³×D₁₃	C₂×C₈	1	1	yes	characteristic
62	C₂²×C₄×C₁₃	C₂×C₈	1	2	yes	—
63	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
64	C₂²×Dic₁₃	C₂×C₈	1	1	yes	characteristic
65	C₂²×Dic₁₃	C₂×C₈	1	2	yes	—
66	C₂×C₄×D₁₃	—	2	2	no	—
67	C₂×C₄×D₁₃	—	2	2	no	—
68	C₂×C₄×D₁₃	—	2	8	no	—
69	C₂×C₄×D₁₃	C₂×C₈	1	2	yes	—
70	C₂×C₄×D₁₃	C₂×C₈	1	2	yes	—
71	C₂×C₄×D₁₃	C₂×C₈	1	2	yes	—
72	C₂×C₄×D₁₃	C₂×C₈	1	2	yes	—
73	C₂×C₄×D₁₃	C₂×C₈	1	2	yes	—
74	C₂×C₄×D₁₃	C₂²×C₄	1	2	yes	—
75	C₂×C₈×C₁₃	—	2	2	no	—
76	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
77	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₈×C₁₃	#41	1	2	yes	—
79	C₂×C₈×C₁₃	#41	1	2	yes	—
80	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
81	C₄²×C₁₃	C₂×C₈	1	1	yes	characteristic
82	C₄²×C₁₃	C₂×C₈	1	1	yes	characteristic
83	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
84	C₄×Dic₁₃	C₂×C₈	1	2	yes	—
85	C₄×Dic₁₃	C₂×C₈	1	2	yes	—
86	C₄×Dic₁₃	C₂×C₈	1	2	yes	—
87	C₄×Dic₁₃	C₂×C₈	1	2	yes	—
88	C₄×Dic₁₃	C₂×C₈	1	2	yes	—
89	C₄×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
90	C₄×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
91	C₂×#1055	—	2	2	no	—
92	C₂×#1055	M₁₆	1	4	yes	—
93	C₂×#1055	C₂²×C₄	1	2	yes	—
94	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
95	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
96	C₈×D₁₃	—	2	4	no	—
97	C₈×D₁₃	—	2	2	no	—
98	C₈×D₁₃	—	2	2	no	—
99	C₈×D₁₃	—	2	4	no	—
100	C₈×D₁₃	—	2	4	no	—
101	C₁₃×C₁₆	—	2	4	no	—
102	#6090	—	2	4	no	—
103	C₂²×C₄×C₈	—	13	1	no	—
104	#2163	—	13	4	no	—
105	#2162	—	13	4	no	—
106	#2150	—	13	1	no	—
107	#2150	D₁₃	1	1	yes	characteristic, complemented
108	#2112	—	13	2	no	—
109	C₄×M₃₂	—	13	2	no	—
110	C₂³×C₁₃	C₂²×C₈	1	1	yes	characteristic
111	C₂²×D₁₃	—	2	2	no	—
112	C₂²×D₁₃	—	2	4	no	—
113	C₂²×D₁₃	C₂²×C₈	1	2	yes	—
114	C₂×C₄×C₁₃	—	2	4	no	—
115	C₂×C₄×C₁₃	#117	1	2	yes	—
116	C₂×C₄×C₁₃	C₂²×C₈	1	2	yes	—
117	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
118	C₂×C₄×C₁₃	C₂²×C₈	1	2	yes	—
119	C₂×C₄×C₁₃	C₂×M₁₆	1	2	yes	—
120	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
121	C₂×Dic₁₃	—	2	2	no	—
122	C₂×Dic₁₃	—	2	4	no	—
123	C₂×Dic₁₃	C₂²×C₈	1	2	yes	—
124	C₂×Dic₁₃	C₂²×C₈	1	2	yes	—
125	C₂×Dic₁₃	C₂²×C₈	1	2	yes	—
126	C₄×D₁₃	—	2	4	no	—
127	C₄×D₁₃	—	2	2	no	—
128	C₄×D₁₃	—	2	2	no	—
129	C₄×D₁₃	—	2	4	no	—
130	C₄×D₁₃	—	2	4	no	—
131	C₄×D₁₃	—	2	4	no	—
132	C₄×D₁₃	M₃₂	1	8	yes	complemented
133	C₈×C₁₃	—	2	2	no	—
134	C₈×C₁₃	#117	1	2	yes	—
135	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
136	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
137	#1055	—	2	2	no	—
138	#1055	C₂×M₁₆	1	2	yes	—
139	#1055	C₂×M₁₆	1	2	yes	—
140	C₂³×C₈	—	13	1	no	—
141	C₂³×C₈	—	13	1	no	—
142	C₂²×C₄²	—	13	1	no	—
143	C₂×M₃₂	—	13	2	no	—
144	C₂×M₃₂	C₂×D₁₃	1	2	yes	—
145	C₂×C₄×C₈	—	13	1	no	—
146	C₂×C₄×C₈	—	13	1	no	—
147	C₂×C₄×C₈	—	13	1	no	—
148	C₂×C₄×C₈	—	13	2	no	—
149	C₂×C₄×C₈	—	13	2	no	—
150	C₂×C₄×C₈	—	13	2	no	—
151	C₂×C₄×C₈	—	13	2	no	—
152	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
153	C₄⋊C₁₆	—	13	4	no	—
154	C₄⋊C₁₆	C₂×D₁₃	1	4	yes	complemented
155	C₂²⋊C₁₆	—	13	8	no	—
156	#346	—	13	4	no	—
157	C₄×C₁₆	—	13	4	no	—
158	C₂²×C₁₃	—	2	2	no	—
159	C₂²×C₁₃	#431	1	2	yes	—
160	C₂²×C₁₃	C₂³×C₈	1	1	yes	characteristic
161	C₂×D₁₃	—	2	8	no	—
162	C₂×D₁₃	—	2	2	no	—
163	C₂×D₁₃	—	2	2	no	—
164	C₂×D₁₃	C₂×M₃₂	1	4	yes	—
165	C₄×C₁₃	—	2	2	no	—
166	C₄×C₁₃	#431	1	2	yes	—
167	C₄×C₁₃	C₂×M₃₂	1	4	yes	—
168	C₄×C₁₃	C₂×#117	1	1	yes	characteristic
169	C₄×C₁₃	C₂²×M₁₆	1	1	yes	characteristic
170	Dic₁₃	—	2	2	no	—
171	Dic₁₃	—	2	2	no	—
172	Dic₁₃	C₂×M₃₂	1	4	yes	—
173	C₂³×C₄	—	13	2	no	—
174	C₂³×C₄	—	13	1	no	—
175	C₂²×C₈	—	13	1	no	—
176	C₂²×C₈	—	13	1	no	—
177	C₂²×C₈	—	13	2	no	—
178	C₂²×C₈	—	13	2	no	—
179	C₂²×C₈	—	13	2	no	—
180	C₂²×C₈	—	13	2	no	—
181	C₂²×C₈	—	26	4	no	—
182	C₂²×C₈	—	26	2	no	—
183	C₂²×C₈	—	26	2	no	—
184	C₂²×C₈	C₄×D₁₃	1	1	yes	characteristic
185	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
186	C₂×C₄²	—	13	2	no	—
187	C₂×C₄²	—	13	1	no	—
188	C₂×C₄²	—	13	2	no	—
189	C₂×C₄²	—	13	2	no	—
190	C₂×C₄²	—	13	2	no	—
191	C₂×C₄²	—	13	2	no	—
192	C₂×C₄²	C₄×D₁₃	1	1	yes	characteristic
193	M₃₂	—	2	4	no	—
194	M₃₂	—	26	4	no	—
195	C₂×C₁₆	—	13	4	no	—
196	C₂×C₁₆	C₂²×D₁₃	1	4	yes	—
197	C₄×C₈	—	13	1	no	—
198	C₄×C₈	—	13	1	no	—
199	C₄×C₈	—	13	2	no	—
200	C₄×C₈	—	13	2	no	—
201	C₄×C₈	—	13	1	no	—
202	C₄×C₈	—	13	1	no	—
203	C₄×C₈	—	13	2	no	—
204	C₄×C₈	—	13	2	no	—
205	C₄×C₈	C₄×D₁₃	1	2	yes	—
206	C₄×C₈	C₂²×D₁₃	1	2	yes	—
207	C₂×C₁₃	—	2	2	no	—
208	C₂×C₁₃	C₂×#431	1	1	yes	characteristic
209	C₂×C₁₃	C₂²×M₃₂	1	2	yes	—
210	D₁₃	—	2	4	no	—
211	C₂⁴	—	13	1	no	—
212	C₂²×C₄	—	13	2	no	—
213	C₂²×C₄	—	13	2	no	—
214	C₂²×C₄	—	13	1	no	—
215	C₂²×C₄	—	13	2	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	8	no	—
221	C₂²×C₄	—	26	2	no	—
222	C₂²×C₄	—	26	2	no	—
223	C₂²×C₄	C₈×D₁₃	1	2	yes	—
224	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
225	C₂×C₈	—	2	2	no	—
226	C₂×C₈	—	13	4	no	—
227	C₂×C₈	—	13	1	no	—
228	C₂×C₈	—	13	1	no	—
229	C₂×C₈	—	13	2	no	—
230	C₂×C₈	—	26	4	no	—
231	C₂×C₈	—	26	4	no	—
232	C₂×C₈	—	26	2	no	—
233	C₂×C₈	—	26	4	no	—
234	C₂×C₈	—	26	2	no	—
235	C₂×C₈	—	26	2	no	—
236	C₂×C₈	C₂×C₄×D₁₃	1	2	yes	—
237	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
238	C₂×C₈	#6129	1	2	yes	—
239	C₂×C₈	#6131	1	2	yes	—
240	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
241	C₄²	—	13	2	no	—
242	C₄²	—	13	2	no	—
243	C₄²	—	13	2	no	—
244	C₄²	—	13	2	no	—
245	C₄²	—	13	2	no	—
246	C₄²	—	13	1	no	—
247	C₄²	—	13	1	no	—
248	C₄²	C₈×D₁₃	1	1	yes	characteristic
249	C₄²	C₈×D₁₃	1	1	yes	characteristic
250	C₄²	C₂×C₄×D₁₃	1	2	yes	—
251	C₁₆	—	2	4	no	—
252	C₁₆	—	26	4	no	—
253	C₁₃	#33418	1	1	yes	characteristic, complemented, Sylow
254	C₂³	—	13	2	no	—
255	C₂³	—	26	4	no	—
256	C₂³	—	26	2	no	—
257	C₂³	C₂×C₈×D₁₃	1	1	yes	characteristic
258	C₂×C₄	—	2	4	no	—
259	C₂×C₄	—	13	2	no	—
260	C₂×C₄	—	13	8	no	—
261	C₂×C₄	—	13	2	no	—
262	C₂×C₄	—	13	2	no	—
263	C₂×C₄	—	26	4	no	—
264	C₂×C₄	—	26	4	no	—
265	C₂×C₄	—	26	4	no	—
266	C₂×C₄	—	26	2	no	—
267	C₂×C₄	—	26	2	no	—
268	C₂×C₄	—	26	2	no	—
269	C₂×C₄	—	26	4	no	—
270	C₂×C₄	—	26	4	no	—
271	C₂×C₄	#88146	1	2	yes	—
272	C₂×C₄	C₂×C₈×D₁₃	1	2	yes	—
273	C₂×C₄	C₂×C₈×D₁₃	1	1	yes	characteristic
274	C₂×C₄	C₂×C₈×D₁₃	1	2	yes	—
275	C₂×C₄	M₁₆×D₁₃	1	2	yes	—
276	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
277	C₈	—	2	2	no	—
278	C₈	—	13	2	no	—
279	C₈	—	13	2	no	—
280	C₈	—	26	2	no	—
281	C₈	#88146	1	2	yes	—
282	C₈	C₂×#6129	1	1	yes	characteristic
283	C₈	C₂×#6131	1	1	yes	characteristic
284	C₂²	—	2	2	no	—
285	C₂²	—	13	4	no	—
286	C₂²	—	26	8	no	—
287	C₂²	—	26	2	no	—
288	C₂²	—	26	2	no	—
289	C₂²	#11718494	1	2	yes	—
290	C₂²	C₂²×C₈×D₁₃	1	1	yes	characteristic
291	C₄	—	2	2	no	—
292	C₄	—	13	4	no	—
293	C₄	—	26	2	no	—
294	C₄	—	26	2	no	—
295	C₄	#11718494	1	2	yes	—
296	C₄	D₁₃×M₃₂	1	4	yes	—
297	C₄	C₂×#88146	1	1	yes	characteristic
298	C₄	C₂×M₁₆×D₁₃	1	1	yes	characteristic
299	C₂	—	2	2	no	—
300	C₂	—	26	4	no	—
301	C₂	C₂×#11718494	1	1	yes	characteristic
302	C₂	C₂×D₁₃×M₃₂	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₁₃
commutator, socle	C₂²×C₁₃	C₂³×C₈
Fitting	C₁₃×#2150	C₂
prime radical	C₂³×D₁₃	C₂×C₈
cosocle	C₂×C₈×C₁₃	C₂⁴
hypercenter	#2150	D₁₃
hypocenter	C₁₃	#33418
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₈ ◁ #2150

Complements

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

N	H ≅ G/N	type
2. #13116638	C₂	semidirect
3. #13116634	C₂	semidirect
4. #13116592	C₂	semidirect
6. M₃₂×Dic₁₃	C₂	semidirect
7. C₁₃×#2150	C₂	semidirect
21. C₁₃×C₄⋊C₁₆	C₂²	semidirect
23. #11718094	C₂²	semidirect
24. C₁₆×Dic₁₃	C₂²	semidirect
25. C₄⋊#6090	C₂²	semidirect
107. #2150	D₁₃	semidirect
132. C₄×D₁₃	M₃₂	semidirect
154. C₄⋊C₁₆	C₂×D₁₃	semidirect
253. C₁₃	#33418	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×C₈×D₁₃	2
#13116638	2
#13116634	2
#13116592	2
#13116446	2
M₃₂×Dic₁₃	2
C₁₃×#2150	2
#33418	13