Subgroups — #15834698

Order 3328.

586 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13125422	C₂	1	1	yes	maximal, characteristic
2	#13123783	C₂	1	2	yes	maximal
3	#13113734	C₂	1	2	yes	maximal, complemented
4	#13113688	C₂	1	4	yes	maximal, complemented
5	C₈×Dic₅₂	C₂	1	2	yes	maximal, complemented
6	C₂×#11718050	C₂	1	1	yes	maximal, characteristic
7	#13113240	C₂	1	2	yes	maximal
8	C₈×C₁₃×M₁₆	C₂	1	1	yes	maximal, characteristic
9	#11719175	C₂²	1	2	yes	—
10	C₁₃×#431	C₂²	1	1	yes	characteristic
11	C₄×C₁₃×M₁₆	C₂²	1	1	yes	characteristic
12	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
13	#11718783	—	2	2	no	—
14	#11718780	—	2	2	no	—
15	#11718777	C₄	1	2	yes	—
16	#11718777	C₄	1	2	yes	—
17	C₂×C₄⋊#1055	C₂²	1	2	yes	—
18	#11718362	—	2	4	no	—
19	#11718360	C₄	1	1	yes	characteristic
20	C₄×Dic₅₂	C₂²	1	2	yes	—
21	C₄×Dic₅₂	C₂²	1	2	yes	—
22	#11718343	C₄	1	1	yes	characteristic
23	#11718342	C₄	1	2	yes	—
24	#11718340	C₂²	1	4	yes	—
25	C₈×Dic₂₆	C₂²	1	4	yes	—
26	C₁₃×#322	C₂²	1	2	yes	—
27	C₈²×C₁₃	C₂²	1	2	yes	—
28	#11718050	C₂²	1	1	yes	characteristic
29	#11718050	C₂²	1	2	yes	—
30	#11718050	C₂²	1	1	yes	characteristic
31	#11718048	C₂²	1	4	yes	—
32	#11718048	C₂²	1	4	yes	—
33	C₂×C₁₃×M₁₆	D₄	1	1	yes	characteristic
34	C₂×C₁₃×M₁₆	D₄	1	1	yes	characteristic
35	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
36	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
38	#88180	—	2	2	no	—
39	#88178	—	2	2	no	—
40	#88178	C₂×C₄	1	4	yes	—
41	C₂²×#1055	—	2	2	no	—
42	C₂×Dic₅₂	C₂×C₄	1	2	yes	—
43	C₂×Dic₅₂	C₂×C₄	1	2	yes	—
44	#88124	—	2	2	no	—
45	#88123	C₂×C₄	1	2	yes	—
46	#88123	C₂×C₄	1	2	yes	—
47	C₄×Dic₂₆	—	2	2	no	—
48	C₄×Dic₂₆	C₂³	1	4	yes	—
49	C₁₃×C₄⋊C₈	—	2	1	no	—
50	C₁₃×C₂²⋊C₈	—	2	1	no	—
51	C₁₃×#97	D₄	1	2	yes	—
52	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
53	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
54	C₄×C₈×C₁₃	C₂³	1	2	yes	—
55	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
56	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
57	C₄×C₈×C₁₃	C₂³	1	2	yes	—
58	#88058	C₂×C₄	1	2	yes	—
59	#88058	C₂×C₄	1	2	yes	—
60	#88056	C₂×C₄	1	4	yes	—
61	#88056	C₂×C₄	1	4	yes	—
62	#88055	—	2	4	no	—
63	#88054	—	2	4	no	—
64	#88054	—	2	4	no	—
65	C₈×Dic₁₃	—	2	4	no	—
66	C₄⋊#1055	—	2	2	no	—
67	C₄⋊#1055	C₂³	1	2	yes	—
68	C₄⋊#1055	C₂³	1	2	yes	—
69	#18068	—	13	1	no	maximal, Sylow
70	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
71	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
73	C₂×Dic₂₆	—	2	2	no	—
74	C₂×Dic₂₆	C₂²×C₄	1	4	yes	—
75	C₁₃×M₁₆	—	2	2	no	—
76	C₁₃×M₁₆	Q₁₆	1	4	yes	complemented
77	C₂×C₈×C₁₃	—	2	2	no	—
78	C₂×C₈×C₁₃	—	2	2	no	—
79	C₂×C₈×C₁₃	—	4	1	no	—
80	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
81	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
82	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
83	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
84	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
85	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
86	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
87	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
88	C₄²×C₁₃	#41	1	1	yes	characteristic
89	C₄²×C₁₃	C₂⁴	1	1	yes	characteristic
90	C₂²⋊Dic₁₃	—	2	4	no	—
91	C₄⋊Dic₁₃	—	2	2	no	—
92	C₄⋊Dic₁₃	—	2	2	no	—
93	C₄⋊Dic₁₃	—	2	2	no	—
94	C₄⋊Dic₁₃	C₂²×C₄	1	4	yes	—
95	#6101	—	2	4	no	—
96	#6101	—	2	4	no	—
97	C₄×Dic₁₃	—	2	4	no	—
98	C₂×#1055	—	2	2	no	—
99	C₂×#1055	—	2	2	no	—
100	C₂×#1055	—	4	4	no	—
101	Dic₅₂	—	2	4	no	—
102	Dic₅₂	—	4	2	no	—
103	#3170	—	13	1	no	—
104	#2962	—	13	2	no	—
105	#1591	—	13	2	no	—
106	#1584	—	13	4	no	—
107	C₈×Q₁₆	—	13	2	no	—
108	C₂×#335	—	13	1	no	—
109	#1481	—	13	2	no	—
110	C₈×M₁₆	D₁₃	1	1	yes	characteristic
111	C₂³×C₁₃	C₄×D₄	1	1	yes	characteristic
112	C₂×C₄×C₁₃	—	2	1	no	—
113	C₂×C₄×C₁₃	—	2	1	no	—
114	C₂×C₄×C₁₃	—	4	1	no	—
115	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
116	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
117	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
118	C₂×C₄×C₁₃	C₂×Q₁₆	1	2	yes	—
119	C₂×C₄×C₁₃	#135	1	2	yes	—
120	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
121	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
122	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
123	C₂×Dic₁₃	—	2	4	no	—
124	C₂×Dic₁₃	—	2	4	no	—
125	Dic₂₆	—	2	4	no	—
126	Dic₂₆	—	2	4	no	—
127	Dic₂₆	—	8	2	no	—
128	C₈×C₁₃	—	2	1	no	—
129	C₈×C₁₃	—	2	1	no	—
130	C₈×C₁₃	—	2	1	no	—
131	C₈×C₁₃	—	2	2	no	—
132	C₈×C₁₃	—	4	2	no	—
133	C₈×C₁₃	#131	1	2	yes	—
134	C₈×C₁₃	C₂×Q₁₆	1	4	yes	—
135	#1055	—	4	2	no	—
136	#1055	—	4	2	no	—
137	#533	—	13	2	no	—
138	#500	—	13	1	no	—
139	#494	—	13	1	no	—
140	#487	—	13	2	no	—
141	#467	—	13	2	no	—
142	#467	—	13	2	no	—
143	#446	—	13	4	no	—
144	#446	—	26	4	no	—
145	C₈×Q₈	—	13	4	no	—
146	C₄×Q₁₆	—	13	2	no	—
147	C₄×Q₁₆	—	13	2	no	—
148	#431	C₂×D₁₃	1	1	yes	characteristic
149	#424	—	26	2	no	—
150	C₂×C₄⋊C₈	—	13	2	no	—
151	#405	—	26	2	no	—
152	C₄×M₁₆	C₂×D₁₃	1	1	yes	characteristic
153	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
154	#335	—	13	1	no	—
155	#335	—	13	2	no	—
156	#335	—	13	1	no	—
157	Q₈⋊C₈	—	13	4	no	—
158	Q₈⋊C₈	—	13	4	no	—
159	#322	C₂×D₁₃	1	2	yes	—
160	C₈²	C₂×D₁₃	1	2	yes	—
161	C₂²×C₁₃	—	2	1	no	—
162	C₂²×C₁₃	—	2	1	no	—
163	C₂²×C₁₃	C₄×Q₁₆	1	2	yes	—
164	C₂²×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
165	C₄×C₁₃	—	2	1	no	—
166	C₄×C₁₃	—	2	1	no	—
167	C₄×C₁₃	—	2	1	no	—
168	C₄×C₁₃	C₄×Q₁₆	1	2	yes	—
169	C₄×C₁₃	C₂×#131	1	1	yes	characteristic
170	C₄×C₁₃	#568	1	1	yes	characteristic
171	C₄×C₁₃	C₂²×Q₁₆	1	1	yes	characteristic
172	C₄×C₁₃	C₂×#135	1	1	yes	characteristic
173	Dic₁₃	—	4	4	no	—
174	Dic₁₃	—	4	4	no	—
175	C₂×Q₁₆	—	13	2	no	—
176	C₂×Q₁₆	—	13	2	no	—
177	C₂×M₁₆	D₅₂	1	1	yes	characteristic
178	C₂×M₁₆	D₅₂	1	1	yes	characteristic
179	C₂²×C₈	—	26	2	no	—
180	C₂²×C₈	C₄×D₁₃	1	1	yes	characteristic
181	C₂²×C₈	C₄×D₁₃	1	1	yes	characteristic
182	C₄⋊Q₈	—	13	2	no	—
183	C₄⋊Q₈	—	13	2	no	—
184	#125	—	26	2	no	—
185	C₂²⋊Q₈	—	13	4	no	—
186	C₂²⋊Q₈	—	26	2	no	—
187	C₄×Q₈	—	13	4	no	—
188	C₄×Q₈	—	26	2	no	—
189	#117	—	26	2	no	—
190	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
191	#107	—	13	2	no	—
192	#107	—	13	2	no	—
193	C₄⋊C₈	—	2	1	no	—
194	C₄⋊C₈	—	13	2	no	—
195	C₄⋊C₈	—	13	2	no	—
196	C₄⋊C₈	—	26	4	no	—
197	C₄⋊C₈	—	26	4	no	—
198	C₄⋊C₈	—	26	2	no	—
199	#103	—	13	4	no	—
200	#103	—	13	4	no	—
201	C₂²⋊C₈	—	2	1	no	—
202	#97	—	26	4	no	—
203	#97	D₅₂	1	2	yes	—
204	C₄×C₈	—	26	4	no	—
205	C₄×C₈	C₄×D₁₃	1	1	yes	characteristic
206	C₄×C₈	C₄×D₁₃	1	1	yes	characteristic
207	C₄×C₈	C₂²×D₁₃	1	2	yes	—
208	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
209	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
210	C₄×C₈	C₂²×D₁₃	1	2	yes	—
211	C₂×C₁₃	—	2	2	no	—
212	C₂×C₁₃	#2935	1	1	yes	characteristic
213	C₂×C₁₃	C₂×C₄×Q₁₆	1	1	yes	characteristic
214	C₂×C₁₃	#2957	1	1	yes	characteristic
215	C₂×Q₈	—	13	4	no	—
216	C₂×Q₈	—	26	2	no	—
217	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
218	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
219	C₂²×C₄	#6127	1	1	yes	characteristic
220	Q₁₆	—	26	4	no	—
221	Q₁₆	—	52	2	no	—
222	M₁₆	—	2	2	no	—
223	M₁₆	Dic₅₂	1	4	yes	complemented
224	C₂×C₈	—	2	2	no	—
225	C₂×C₈	—	2	2	no	—
226	C₂×C₈	—	4	1	no	—
227	C₂×C₈	—	26	2	no	—
228	C₂×C₈	—	26	2	no	—
229	C₂×C₈	—	52	4	no	—
230	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
231	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
232	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
233	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
234	C₂×C₈	C₂×D₅₂	1	2	yes	—
235	C₂×C₈	C₂×D₅₂	1	2	yes	—
236	C₄⋊C₄	—	13	4	no	—
237	C₄⋊C₄	—	26	2	no	—
238	C₄⋊C₄	—	26	2	no	—
239	C₄⋊C₄	—	26	4	no	—
240	C₄⋊C₄	—	26	2	no	—
241	C₄⋊C₄	—	26	4	no	—
242	C₂²⋊C₄	—	26	4	no	—
243	C₄²	—	26	4	no	—
244	C₄²	C₂×C₄×D₁₃	1	1	yes	characteristic
245	C₄²	C₂×D₅₂	1	1	yes	characteristic
246	C₄²	#6127	1	1	yes	characteristic
247	C₄²	C₂³×D₁₃	1	1	yes	characteristic
248	C₁₃	#18068	1	1	yes	characteristic, complemented, Sylow
249	C₂³	C₄×D₅₂	1	1	yes	characteristic
250	Q₈	—	26	4	no	—
251	Q₈	—	26	4	no	—
252	Q₈	—	104	2	no	—
253	C₂×C₄	—	2	1	no	—
254	C₂×C₄	—	2	1	no	—
255	C₂×C₄	—	4	1	no	—
256	C₂×C₄	—	26	4	no	—
257	C₂×C₄	—	26	4	no	—
258	C₂×C₄	C₄×D₅₂	1	1	yes	characteristic
259	C₂×C₄	C₄×D₅₂	1	1	yes	characteristic
260	C₂×C₄	C₄×D₅₂	1	1	yes	characteristic
261	C₂×C₄	#88158	1	2	yes	—
262	C₂×C₄	C₂×Dic₅₂	1	2	yes	—
263	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
264	C₂×C₄	C₂²×D₅₂	1	1	yes	characteristic
265	C₂×C₄	C₂×#6127	1	1	yes	characteristic
266	C₈	—	2	2	no	—
267	C₈	—	2	1	no	—
268	C₈	—	2	1	no	—
269	C₈	—	2	1	no	—
270	C₈	—	4	2	no	—
271	C₈	—	52	2	no	—
272	C₈	—	52	2	no	—
273	C₈	C₂×Dic₅₂	1	4	yes	—
274	C₈	#88161	1	2	yes	—
275	C₂²	—	2	1	no	—
276	C₂²	—	2	1	no	—
277	C₂²	C₄×Dic₅₂	1	2	yes	—
278	C₂²	C₂×C₄×D₅₂	1	1	yes	characteristic
279	C₄	—	2	1	no	—
280	C₄	—	2	1	no	—
281	C₄	—	2	1	no	—
282	C₄	—	52	4	no	—
283	C₄	—	52	4	no	—
284	C₄	C₄×Dic₅₂	1	2	yes	—
285	C₄	C₂×#88158	1	1	yes	characteristic
286	C₄	C₂²×Dic₅₂	1	1	yes	characteristic
287	C₄	C₂×#88161	1	1	yes	characteristic
288	C₄	#11719450	1	1	yes	characteristic
289	C₂	—	2	2	no	—
290	C₂	#13123582	1	1	yes	characteristic
291	C₂	C₂×C₄×Dic₅₂	1	1	yes	characteristic
292	C₂	#13123754	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₅₂
commutator	C₂×C₄×C₁₃	C₂³×C₄
Frattini	C₄²	C₂³×D₁₃
Fitting	C₈×C₁₃×M₁₆	C₂
socle	C₂²×C₁₃	C₂×C₄×D₄
prime radical	C₂³×C₁₃	C₄×D₄
cosocle	C₄²×C₁₃	C₂⁴
hypercenter	C₈×M₁₆	D₁₃
hypocenter	C₁₃	#18068
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄×M₁₆ ◁ C₈×M₁₆

Complements

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

N	H ≅ G/N	type
3. #13113734	C₂	semidirect
4. #13113688	C₂	semidirect
5. C₈×Dic₅₂	C₂	semidirect
76. C₁₃×M₁₆	Q₁₆	semidirect
223. M₁₆	Dic₅₂	semidirect
248. C₁₃	#18068	semidirect

Maximal subgroups

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

subgroup	index
#13125422	2
#13123783	2
#13113734	2
#13113688	2
C₈×Dic₅₂	2
C₂×#11718050	2
#13113240	2
C₈×C₁₃×M₁₆	2
#18068	13