Subgroups — #15834948

Order 3328.

564 conjugacy classes of subgroups.

The subgroup lattice has 280 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	#13123937	C₂	1	2	yes	maximal
3	#13113744	C₂	1	2	yes	maximal, complemented
4	#13113685	C₂	1	4	yes	maximal, complemented
5	#13113654	C₂	1	2	yes	maximal, complemented
6	#13113256	C₂	1	2	yes	maximal
7	#13113125	C₂	1	1	yes	maximal, characteristic
8	C₁₃×#1564	C₂	1	1	yes	maximal, characteristic
9	#11719175	C₂²	1	2	yes	—
10	C₁₃×#431	C₂²	1	2	yes	—
11	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
12	#11718777	C₄	1	2	yes	—
13	#11718777	C₄	1	2	yes	—
14	#11718586	C₂²	1	1	yes	characteristic
15	C₄×#6099	C₂²	1	1	yes	characteristic
16	#11718492	C₂²	1	4	yes	—
17	#11718491	C₂²	1	4	yes	—
18	#11718381	—	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	—	2	2	no	—
25	C₈×Dic₂₆	—	2	2	no	—
26	C₁₃×#335	C₂²	1	2	yes	—
27	C₁₃×#334	C₂²	1	1	yes	characteristic
28	C₁₃×#334	C₂²	1	1	yes	characteristic
29	#11718076	C₂²	1	4	yes	—
30	#11718076	C₂²	1	4	yes	—
31	#11718047	C₂²	1	2	yes	—
32	#11718046	C₂²	1	2	yes	—
33	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
36	#88180	—	2	2	no	—
37	#88178	—	2	2	no	—
38	#88178	C₂×C₄	1	4	yes	—
39	C₂×#6099	—	2	1	no	—
40	C₂×Dic₅₂	C₂×C₄	1	2	yes	—
41	C₂×Dic₅₂	C₂×C₄	1	2	yes	—
42	#88124	—	2	2	no	—
43	#88123	C₂×C₄	1	2	yes	—
44	#88123	C₂×C₄	1	2	yes	—
45	C₄×Dic₂₆	—	2	2	no	—
46	C₄×Dic₂₆	C₂³	1	4	yes	—
47	C₁₃×C₄⋊C₈	C₂³	1	2	yes	—
48	C₁₃×C₄⋊C₈	C₂³	1	2	yes	—
49	C₁₃×C₂²⋊C₈	—	2	2	no	—
50	C₄×C₈×C₁₃	—	2	2	no	—
51	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
52	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
53	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
54	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
55	C₂²⋊#1055	D₄	1	2	yes	—
56	#88058	C₂×C₄	1	2	yes	—
57	#88058	C₂×C₄	1	2	yes	—
58	#88056	C₂×C₄	1	4	yes	—
59	#88056	C₂×C₄	1	4	yes	—
60	#88055	—	2	4	no	—
61	#88054	—	2	4	no	—
62	#88054	—	2	4	no	—
63	C₈×Dic₁₃	—	2	4	no	—
64	C₄⋊#1055	D₄	1	2	yes	—
65	#88043	—	2	1	no	—
66	C₄×#1055	C₂³	1	2	yes	—
67	C₄×#1055	C₂³	1	2	yes	—
68	#18090	—	13	1	no	maximal, Sylow
69	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
70	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
72	C₂×Dic₂₆	—	2	2	no	—
73	C₂×Dic₂₆	C₂²×C₄	1	4	yes	—
74	C₂×C₈×C₁₃	—	2	2	no	—
75	C₂×C₈×C₁₃	—	2	2	no	—
76	C₂×C₈×C₁₃	—	2	2	no	—
77	C₂×C₈×C₁₃	—	4	1	no	—
78	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
81	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
82	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
83	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
84	C₄²×C₁₃	#41	1	1	yes	characteristic
85	C₄²×C₁₃	C₂⁴	1	1	yes	characteristic
86	C₂²⋊Dic₁₃	—	2	4	no	—
87	C₄⋊Dic₁₃	—	2	2	no	—
88	C₄⋊Dic₁₃	—	2	2	no	—
89	C₄⋊Dic₁₃	—	2	2	no	—
90	C₄⋊Dic₁₃	C₂²×C₄	1	4	yes	—
91	#6101	—	2	4	no	—
92	#6101	—	2	4	no	—
93	C₄×Dic₁₃	—	2	4	no	—
94	#6099	—	4	2	no	—
95	C₂×#1055	—	2	2	no	—
96	C₂×#1055	C₂×D₄	1	4	yes	—
97	Dic₅₂	—	2	4	no	—
98	Dic₅₂	—	4	2	no	—
99	#3170	—	13	1	no	—
100	#2986	—	13	2	no	—
101	#1593	—	13	2	no	—
102	#1584	—	13	4	no	—
103	#1579	—	13	2	no	—
104	#1564	D₁₃	1	1	yes	characteristic
105	#1485	—	13	2	no	—
106	#1451	—	13	1	no	—
107	C₂³×C₁₃	C₄×D₄	1	1	yes	characteristic
108	C₂×C₄×C₁₃	—	2	1	no	—
109	C₂×C₄×C₁₃	—	2	1	no	—
110	C₂×C₄×C₁₃	—	2	2	no	—
111	C₂×C₄×C₁₃	—	2	2	no	—
112	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
113	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
114	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
115	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
116	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
117	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
118	C₂×Dic₁₃	—	2	4	no	—
119	C₂×Dic₁₃	—	2	4	no	—
120	Dic₂₆	—	2	4	no	—
121	Dic₂₆	—	2	4	no	—
122	Dic₂₆	—	4	4	no	—
123	C₈×C₁₃	—	2	1	no	—
124	C₈×C₁₃	—	2	1	no	—
125	C₈×C₁₃	—	2	1	no	—
126	C₈×C₁₃	—	4	2	no	—
127	C₈×C₁₃	—	4	2	no	—
128	C₈×C₁₃	#131	1	2	yes	—
129	#1055	—	2	4	no	—
130	#1055	—	4	2	no	—
131	#533	—	13	2	no	—
132	#500	—	13	1	no	—
133	#494	—	13	1	no	—
134	#487	—	13	2	no	—
135	#467	—	13	2	no	—
136	#467	—	13	2	no	—
137	#446	—	13	4	no	—
138	#446	—	26	2	no	—
139	C₈×Q₈	—	13	4	no	—
140	C₈×Q₈	—	26	2	no	—
141	C₄×Q₁₆	—	13	2	no	—
142	C₄×Q₁₆	—	13	2	no	—
143	#433	—	26	4	no	—
144	#431	—	13	1	no	—
145	#431	C₂×D₁₃	1	2	yes	—
146	C₄×M₁₆	—	13	1	no	—
147	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
148	#335	C₂×D₁₃	1	2	yes	—
149	#334	C₂×D₁₃	1	1	yes	characteristic
150	#334	C₂×D₁₃	1	1	yes	characteristic
151	Q₈⋊C₈	—	13	4	no	—
152	Q₈⋊C₈	—	13	4	no	—
153	#322	—	13	2	no	—
154	#322	—	13	2	no	—
155	C₂²×C₁₃	—	2	1	no	—
156	C₂²×C₁₃	—	2	1	no	—
157	C₂²×C₁₃	—	2	1	no	—
158	C₂²×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
159	C₄×C₁₃	—	2	1	no	—
160	C₄×C₁₃	—	2	1	no	—
161	C₄×C₁₃	—	2	1	no	—
162	C₄×C₁₃	#444	1	2	yes	—
163	C₄×C₁₃	C₂×#131	1	1	yes	characteristic
164	C₄×C₁₃	#568	1	1	yes	characteristic
165	C₄×C₁₃	#578	1	2	yes	—
166	Dic₁₃	—	4	4	no	—
167	Dic₁₃	—	4	4	no	—
168	C₂×Q₁₆	—	13	2	no	—
169	C₂×Q₁₆	—	13	2	no	—
170	C₂×M₁₆	—	26	1	no	—
171	C₂²×C₈	C₄×D₁₃	1	1	yes	characteristic
172	C₂²×C₈	C₄×D₁₃	1	1	yes	characteristic
173	C₄⋊Q₈	—	13	2	no	—
174	C₄⋊Q₈	—	13	2	no	—
175	#125	—	26	2	no	—
176	C₂²⋊Q₈	—	13	4	no	—
177	C₂²⋊Q₈	—	26	2	no	—
178	C₄×Q₈	—	13	4	no	—
179	C₄×Q₈	—	26	2	no	—
180	#117	—	26	2	no	—
181	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
182	#107	—	13	2	no	—
183	#107	—	13	2	no	—
184	C₄⋊C₈	—	13	2	no	—
185	C₄⋊C₈	—	26	4	no	—
186	C₄⋊C₈	—	26	4	no	—
187	C₄⋊C₈	C₂²×D₁₃	1	2	yes	—
188	C₄⋊C₈	C₂²×D₁₃	1	2	yes	—
189	#103	—	13	4	no	—
190	#103	—	13	4	no	—
191	C₂²⋊C₈	—	2	2	no	—
192	C₂²⋊C₈	—	13	2	no	—
193	#97	—	26	4	no	—
194	#97	—	26	1	no	—
195	C₄×C₈	—	2	2	no	—
196	C₄×C₈	—	13	2	no	—
197	C₄×C₈	—	13	2	no	—
198	C₄×C₈	—	26	4	no	—
199	C₄×C₈	C₄×D₁₃	1	1	yes	characteristic
200	C₄×C₈	C₄×D₁₃	1	1	yes	characteristic
201	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
202	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
203	C₂×C₁₃	—	4	1	no	—
204	C₂×C₁₃	#2935	1	1	yes	characteristic
205	C₂×C₁₃	#2950	1	1	yes	characteristic
206	C₂×C₁₃	C₂×#444	1	1	yes	characteristic
207	C₂×Q₈	—	13	4	no	—
208	C₂×Q₈	—	26	2	no	—
209	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
210	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
211	C₂²×C₄	#6131	1	1	yes	characteristic
212	Q₁₆	—	26	4	no	—
213	Q₁₆	—	52	2	no	—
214	M₁₆	—	52	2	no	—
215	C₂×C₈	—	2	2	no	—
216	C₂×C₈	—	2	2	no	—
217	C₂×C₈	—	2	2	no	—
218	C₂×C₈	—	4	1	no	—
219	C₂×C₈	—	13	4	no	—
220	C₂×C₈	—	26	2	no	—
221	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
222	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
223	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
224	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
225	C₄⋊C₄	—	13	4	no	—
226	C₄⋊C₄	—	26	2	no	—
227	C₄⋊C₄	—	26	2	no	—
228	C₄⋊C₄	—	26	4	no	—
229	C₄⋊C₄	—	26	2	no	—
230	C₄⋊C₄	—	26	4	no	—
231	C₂²⋊C₄	—	26	4	no	—
232	C₄²	—	26	4	no	—
233	C₄²	C₂×C₄×D₁₃	1	1	yes	characteristic
234	C₄²	D₄×D₁₃	1	1	yes	characteristic
235	C₄²	#6131	1	1	yes	characteristic
236	C₄²	C₂³×D₁₃	1	1	yes	characteristic
237	C₁₃	#18090	1	1	yes	characteristic, complemented, Sylow
238	C₂³	#88147	1	1	yes	characteristic
239	Q₈	—	26	4	no	—
240	Q₈	—	26	4	no	—
241	Q₈	—	52	4	no	—
242	C₂×C₄	—	2	1	no	—
243	C₂×C₄	—	2	1	no	—
244	C₂×C₄	—	2	2	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	26	4	no	—
247	C₂×C₄	—	26	4	no	—
248	C₂×C₄	#88147	1	1	yes	characteristic
249	C₂×C₄	#88147	1	1	yes	characteristic
250	C₂×C₄	#88147	1	1	yes	characteristic
251	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
252	C₂×C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
253	C₂×C₄	C₂×#6131	1	1	yes	characteristic
254	C₈	—	2	1	no	—
255	C₈	—	2	1	no	—
256	C₈	—	2	1	no	—
257	C₈	—	4	2	no	—
258	C₈	—	4	2	no	—
259	C₈	—	26	4	no	—
260	C₈	—	52	2	no	—
261	C₈	#88155	1	2	yes	—
262	C₂²	—	2	1	no	—
263	C₂²	—	2	1	no	—
264	C₂²	—	2	1	no	—
265	C₂²	C₂×#88147	1	1	yes	characteristic
266	C₄	—	2	1	no	—
267	C₄	—	2	1	no	—
268	C₄	—	2	1	no	—
269	C₄	—	52	4	no	—
270	C₄	—	52	4	no	—
271	C₄	#11718555	1	2	yes	—
272	C₄	C₂×#88155	1	1	yes	characteristic
273	C₄	#11719450	1	1	yes	characteristic
274	C₄	#11719476	1	2	yes	—
275	C₂	—	4	1	no	—
276	C₂	#13123579	1	1	yes	characteristic
277	C₂	#13123703	1	1	yes	characteristic
278	C₂	C₂×#11718555	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₁₃
commutator	C₂×C₄×C₁₃	C₂³×C₄
Frattini	C₄²	C₂³×D₁₃
Fitting	C₁₃×#1564	C₂
socle	C₂²×C₁₃	C₂×C₄×D₄
prime radical	C₂³×C₁₃	C₄×D₄
cosocle	C₄²×C₁₃	C₂⁴
hypercenter	#1564	D₁₃
hypocenter	C₁₃	#18090
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1564

Complements

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

N	H ≅ G/N	type
3. #13113744	C₂	semidirect
4. #13113685	C₂	semidirect
5. #13113654	C₂	semidirect
237. C₁₃	#18090	semidirect

Maximal subgroups

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

subgroup	index
#13125422	2
#13123937	2
#13113744	2
#13113685	2
#13113654	2
#13113256	2
#13113125	2
C₁₃×#1564	2
#18090	13