Subgroups — #15871123

Order 3328.

730 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	#13125181	C₂	1	1	yes	maximal, characteristic
2	#13123770	C₂	1	1	yes	maximal, characteristic, complemented
3	#13115816	C₂	1	1	yes	maximal, characteristic, complemented
4	#13115669	C₂	1	1	yes	maximal, characteristic
5	#13115543	C₂	1	4	yes	maximal, complemented
6	#13114976	C₂	1	1	yes	maximal, characteristic, complemented
7	#13114678	C₂	1	4	yes	maximal, complemented
8	#13114522	C₂	1	1	yes	maximal, characteristic, complemented
9	C₁₃×#2931	C₂	1	1	yes	maximal, characteristic
10	C₂×C₁₃×#131	C₂²	1	2	yes	—
11	C₁₃×#532	C₂²	1	1	yes	characteristic
12	#11719175	C₂²	1	1	yes	characteristic
13	C₁₃×#436	—	2	2	no	—
14	C₈×D₄×C₁₃	—	2	2	no	—
15	C₁₃×#423	C₂²	1	1	yes	characteristic
16	C₁₃×#408	C₂²	1	2	yes	—
17	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
18	C₂×#88077	C₂²	1	4	yes	—
19	#11718894	C₂²	1	4	yes	—
20	#11718763	C₂²	1	2	yes	—
21	#11718762	C₂²	1	2	yes	—
22	C₂×#88055	C₂²	1	2	yes	—
23	C₂×C₈×Dic₁₃	C₂²	1	2	yes	—
24	#11718743	C₄	1	2	yes	complemented
25	#11718732	C₄	1	2	yes	complemented
26	#11718716	C₄	1	4	yes	complemented
27	#11718593	C₂²	1	1	yes	characteristic
28	#11718585	C₂²	1	1	yes	characteristic
29	C₂×#88043	C₂²	1	1	yes	characteristic
30	#11718340	—	2	2	no	—
31	C₈×Dic₂₆	—	2	2	no	—
32	#11718150	C₂²	1	4	yes	—
33	#11718147	C₂²	1	4	yes	—
34	C₂×C₁₃×#41	C₂³	1	2	yes	—
35	C₁₃×#131	—	2	8	no	—
36	C₂×C₁₃×M₁₆	—	2	2	no	—
37	C₂×C₁₃×M₁₆	C₂³	1	2	yes	—
38	C₂²×C₈×C₁₃	—	2	2	no	—
39	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
40	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
41	C₁₃×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
42	C₁₃×#127	C₂×C₄	1	1	yes	characteristic
43	C₁₃×#124	C₂×C₄	1	2	yes	—
44	C₁₃×#121	—	2	2	no	—
45	C₄×D₄×C₁₃	—	2	2	no	—
46	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
47	#88180	C₂³	1	2	yes	—
48	#88178	—	2	2	no	—
49	C₂×C₄×Dic₁₃	C₂³	1	2	yes	—
50	C₂×#6099	C₂³	1	2	yes	—
51	C₂²×#1055	C₂³	1	2	yes	—
52	#88124	C₂×C₄	1	2	yes	—
53	#88123	C₂×C₄	1	2	yes	—
54	C₄×Dic₂₆	—	2	2	no	—
55	C₁₃×C₄⋊C₈	—	2	2	no	—
56	C₁₃×C₂²⋊C₈	—	2	4	no	—
57	C₄×C₈×C₁₃	D₄	1	2	yes	—
58	C₄×C₈×C₁₃	D₄	1	2	yes	—
59	#88077	—	2	8	no	—
60	#88075	C₂×C₄	1	8	yes	—
61	#88072	C₂×C₄	1	8	yes	—
62	#88071	—	2	2	no	—
63	#88055	—	2	2	no	—
64	#88055	—	2	4	no	—
65	#88054	—	2	4	no	—
66	C₈×Dic₁₃	—	2	2	no	—
67	C₈×Dic₁₃	—	2	4	no	—
68	C₄⋊#1055	—	2	2	no	—
69	#88043	C₂×C₄	1	2	yes	—
70	#88043	C₂×C₄	1	2	yes	—
71	#21663	—	13	1	no	maximal, Sylow
72	C₁₃×#41	—	2	4	no	—
73	C₁₃×#41	—	2	4	no	—
74	C₂×Q₈×C₁₃	C₂²×C₄	1	2	yes	—
75	C₂×D₄×C₁₃	—	2	2	no	—
76	C₂×D₄×C₁₃	—	2	2	no	—
77	C₂×D₄×C₁₃	C₂²×C₄	1	2	yes	—
78	C₂²×C₄×C₁₃	—	2	2	no	—
79	C₂²×C₄×C₁₃	#41	1	2	yes	—
80	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
81	C₂²×Dic₁₃	—	2	2	no	—
82	C₂×Dic₂₆	—	2	2	no	—
83	C₁₃×M₁₆	—	2	4	no	—
84	C₁₃×M₁₆	—	4	4	no	—
85	C₂×C₈×C₁₃	—	2	2	no	—
86	C₂×C₈×C₁₃	—	2	2	no	—
87	C₂×C₈×C₁₃	—	4	4	no	—
88	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
89	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
90	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
91	C₂×C₈×C₁₃	#41	1	4	yes	—
92	C₁₃×C₄⋊C₄	—	2	2	no	—
93	C₁₃×C₂²⋊C₄	—	2	4	no	—
94	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
95	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
96	C₄²×C₁₃	C₂×D₄	1	2	yes	—
97	C₂²⋊Dic₁₃	—	2	2	no	—
98	C₄⋊Dic₁₃	—	2	2	no	—
99	C₄⋊Dic₁₃	C₂²×C₄	1	4	yes	—
100	#6101	—	2	4	no	—
101	C₄×Dic₁₃	—	2	2	no	—
102	C₄×Dic₁₃	—	2	2	no	—
103	C₄×Dic₁₃	#41	1	4	yes	—
104	#6099	—	2	4	no	—
105	C₂×#1055	—	2	2	no	—
106	C₂×#1055	—	2	2	no	—
107	C₂×#1055	—	2	2	no	—
108	C₂×#1055	C₂²×C₄	1	4	yes	—
109	#3135	—	13	1	no	—
110	#2960	—	13	1	no	—
111	#2931	D₁₃	1	1	yes	characteristic
112	#1995	—	13	1	no	—
113	#1963	—	13	1	no	—
114	#1937	—	13	4	no	—
115	#1838	—	13	1	no	—
116	#1765	—	13	4	no	—
117	#1726	—	13	1	no	—
118	C₂³×C₁₃	—	2	2	no	—
119	C₂³×C₁₃	C₂×#41	1	1	yes	characteristic
120	Q₈×C₁₃	—	2	4	no	—
121	D₄×C₁₃	—	2	4	no	—
122	D₄×C₁₃	—	4	4	no	—
123	D₄×C₁₃	—	4	4	no	—
124	C₂×C₄×C₁₃	—	2	2	no	—
125	C₂×C₄×C₁₃	—	4	2	no	—
126	C₂×C₄×C₁₃	—	4	2	no	—
127	C₂×C₄×C₁₃	—	4	2	no	—
128	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
129	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
130	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
131	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
132	C₂×C₄×C₁₃	C₂×#41	1	2	yes	—
133	C₂×C₄×C₁₃	C₂×#41	1	2	yes	—
134	C₂×Dic₁₃	—	2	4	no	—
135	C₂×Dic₁₃	—	2	4	no	—
136	C₂×Dic₁₃	—	4	4	no	—
137	Dic₂₆	—	4	4	no	—
138	C₈×C₁₃	—	2	2	no	—
139	C₈×C₁₃	—	4	2	no	—
140	C₈×C₁₃	#124	1	2	yes	—
141	C₈×C₁₃	#124	1	2	yes	—
142	#1055	—	2	4	no	—
143	#1055	—	4	2	no	—
144	C₂×#131	C₂×D₁₃	1	2	yes	—
145	#533	—	13	1	no	—
146	#532	C₂×D₁₃	1	1	yes	characteristic
147	#491	—	13	2	no	—
148	#490	—	13	2	no	—
149	#489	—	13	4	no	—
150	#446	—	26	2	no	—
151	C₈×Q₈	—	26	2	no	—
152	#436	—	2	2	no	—
153	C₈×D₄	—	2	2	no	—
154	#424	—	13	2	no	—
155	#423	—	13	1	no	—
156	#423	C₂×D₁₃	1	1	yes	characteristic
157	C₂×#104	—	13	4	no	—
158	#416	—	13	4	no	—
159	#408	C₂×D₁₃	1	2	yes	—
160	#405	—	13	2	no	—
161	C₂×#97	—	13	1	no	—
162	C₂×#97	—	13	2	no	—
163	C₂×C₄×C₈	—	13	2	no	—
164	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
165	#376	—	13	1	no	—
166	#341	—	13	4	no	—
167	#339	—	13	4	no	—
168	C₂²×C₁₃	—	2	2	no	—
169	C₂²×C₁₃	—	4	4	no	—
170	C₂²×C₁₃	—	4	2	no	—
171	C₂²×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
172	C₂²×C₁₃	#529	1	2	yes	—
173	C₄×C₁₃	—	2	2	no	—
174	C₄×C₁₃	—	2	2	no	—
175	C₄×C₁₃	—	4	2	no	—
176	C₄×C₁₃	C₄×#41	1	2	yes	—
177	C₄×C₁₃	C₂×#124	1	1	yes	characteristic
178	C₄×C₁₃	#532	1	1	yes	characteristic
179	Dic₁₃	—	4	4	no	—
180	Dic₁₃	—	4	4	no	—
181	C₂×#41	C₂²×D₁₃	1	2	yes	—
182	#131	—	2	8	no	—
183	C₂×M₁₆	—	2	2	no	—
184	C₂×M₁₆	—	13	2	no	—
185	C₂×M₁₆	C₂²×D₁₃	1	2	yes	—
186	C₂²×C₈	—	2	2	no	—
187	C₂²×C₈	—	13	2	no	—
188	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
189	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
190	C₄⋊Q₈	—	13	2	no	—
191	C₄⋊Q₈	C₄×D₁₃	1	1	yes	characteristic
192	#127	C₄×D₁₃	1	1	yes	characteristic
193	#125	—	13	2	no	—
194	#124	C₄×D₁₃	1	2	yes	—
195	C₂²⋊Q₈	—	26	2	no	—
196	#121	—	2	2	no	—
197	C₄×Q₈	—	26	2	no	—
198	C₄×D₄	—	2	2	no	—
199	#117	—	13	2	no	—
200	C₂×C₄²	—	13	2	no	—
201	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
202	C₄⋊C₈	—	2	2	no	—
203	C₄⋊C₈	—	26	4	no	—
204	C₄⋊C₈	—	26	2	no	—
205	#104	—	26	8	no	—
206	#103	—	13	8	no	—
207	#102	—	13	8	no	—
208	C₂²⋊C₈	—	2	4	no	—
209	#97	—	13	2	no	—
210	#97	—	13	2	no	—
211	#97	—	26	4	no	—
212	#97	—	26	2	no	—
213	C₄×C₈	—	26	4	no	—
214	C₄×C₈	—	26	2	no	—
215	C₄×C₈	C₂²⋊D₁₃	1	2	yes	—
216	C₄×C₈	C₂²⋊D₁₃	1	2	yes	—
217	#95	—	26	2	no	—
218	C₂×C₁₃	—	2	2	no	—
219	C₂×C₁₃	—	8	2	no	—
220	C₂×C₁₃	C₄×#124	1	1	yes	characteristic
221	C₂×C₁₃	#2955	1	2	yes	—
222	#41	—	2	4	no	—
223	#41	—	2	4	no	—
224	C₂×Q₈	—	26	2	no	—
225	C₂×Q₈	C₂×C₄×D₁₃	1	2	yes	—
226	C₂×D₄	—	2	2	no	—
227	C₂×D₄	—	2	2	no	—
228	C₂×D₄	C₂×C₄×D₁₃	1	2	yes	—
229	C₂²×C₄	—	2	2	no	—
230	C₂²×C₄	—	26	2	no	—
231	C₂²×C₄	#6127	1	2	yes	—
232	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
233	M₁₆	—	2	4	no	—
234	M₁₆	—	4	4	no	—
235	M₁₆	—	26	4	no	—
236	C₂×C₈	—	2	2	no	—
237	C₂×C₈	—	2	2	no	—
238	C₂×C₈	—	4	4	no	—
239	C₂×C₈	—	13	4	no	—
240	C₂×C₈	—	26	2	no	—
241	C₂×C₈	—	26	2	no	—
242	C₂×C₈	—	26	2	no	—
243	C₂×C₈	#6129	1	4	yes	—
244	C₂×C₈	C₂×C₂²⋊D₁₃	1	2	yes	—
245	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
246	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
247	C₄⋊C₄	—	2	2	no	—
248	C₄⋊C₄	—	13	4	no	—
249	C₄⋊C₄	—	26	4	no	—
250	C₄⋊C₄	—	26	2	no	—
251	C₂²⋊C₄	—	2	4	no	—
252	C₂²⋊C₄	—	26	2	no	—
253	C₄²	—	13	4	no	—
254	C₄²	—	26	2	no	—
255	C₄²	—	26	2	no	—
256	C₄²	C₂×C₄×D₁₃	1	1	yes	characteristic
257	C₄²	C₂×C₄×D₁₃	1	1	yes	characteristic
258	C₄²	C₂×C₂²⋊D₁₃	1	2	yes	—
259	C₁₃	#21663	1	1	yes	characteristic, complemented, Sylow
260	C₂³	—	2	2	no	—
261	C₂³	C₂×#6127	1	1	yes	characteristic
262	Q₈	—	2	4	no	—
263	Q₈	—	52	4	no	—
264	D₄	—	2	4	no	—
265	D₄	—	4	4	no	—
266	D₄	—	4	4	no	—
267	C₂×C₄	—	2	2	no	—
268	C₂×C₄	—	4	2	no	—
269	C₂×C₄	—	4	2	no	—
270	C₂×C₄	—	4	2	no	—
271	C₂×C₄	—	26	4	no	—
272	C₂×C₄	—	26	4	no	—
273	C₂×C₄	—	52	4	no	—
274	C₂×C₄	C₄×C₂²⋊D₁₃	1	2	yes	—
275	C₂×C₄	C₄×C₂²⋊D₁₃	1	2	yes	—
276	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
277	C₂×C₄	C₂×#6129	1	2	yes	—
278	C₂×C₄	#41×D₁₃	1	2	yes	—
279	C₂×C₄	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
280	C₈	—	2	2	no	—
281	C₈	—	4	2	no	—
282	C₈	—	26	4	no	—
283	C₈	—	52	2	no	—
284	C₈	#88190	1	2	yes	—
285	C₈	#88190	1	2	yes	—
286	C₂²	—	2	2	no	—
287	C₂²	—	4	4	no	—
288	C₂²	—	4	2	no	—
289	C₂²	#11719262	1	2	yes	—
290	C₂²	C₂×C₄×C₂²⋊D₁₃	1	1	yes	characteristic
291	C₄	—	2	2	no	—
292	C₄	—	2	2	no	—
293	C₄	—	4	2	no	—
294	C₄	—	52	4	no	—
295	C₄	—	52	4	no	—
296	C₄	C₄×#6129	1	2	yes	—
297	C₄	C₂×#88190	1	1	yes	characteristic
298	C₄	#11719537	1	1	yes	characteristic
299	C₂	—	2	2	no	—
300	C₂	—	8	2	no	—
301	C₂	C₄×#88190	1	1	yes	characteristic
302	C₂	#13123737	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂²×C₂²⋊D₁₃
commutator	C₂×C₄×C₁₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₁₃
Fitting	C₁₃×#2931	C₂
socle	C₂²×C₁₃	C₂×C₄×D₄
prime radical	C₁₃×#532	C₂²
cosocle	C₂²×C₄×C₁₃	C₂⁴
hypercenter	#2931	D₁₃
hypocenter	C₁₃	#21663
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ #2931

Complements

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

N	H ≅ G/N	type
2. #13123770	C₂	semidirect
3. #13115816	C₂	semidirect
5. #13115543	C₂	semidirect
6. #13114976	C₂	semidirect
7. #13114678	C₂	semidirect
8. #13114522	C₂	semidirect
24. #11718743	C₄	semidirect
25. #11718732	C₄	semidirect
26. #11718716	C₄	semidirect
259. C₁₃	#21663	semidirect

Maximal subgroups

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

subgroup	index
#13125181	2
#13123770	2
#13115816	2
#13115669	2
#13115543	2
#13114976	2
#13114678	2
#13114522	2
C₁₃×#2931	2
#21663	13