Subgroups — #15751430

Order 3328.

390 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13114585	C₂	1	1	yes	maximal, characteristic
2	#13114582	C₂	1	1	yes	maximal, characteristic
3	#13112662	C₂	1	2	yes	maximal
4	#13112659	C₂	1	2	yes	maximal
5	C₁₃×#2863	C₂	1	1	yes	maximal, characteristic
6	C₁₃×#391	C₄	1	1	yes	characteristic
7	C₁₃×#391	C₄	1	1	yes	characteristic
8	C₁₃×#391	C₂²	1	1	yes	characteristic
9	C₁₃×#389	C₄	1	1	yes	characteristic
10	C₁₃×#389	C₄	1	1	yes	characteristic
11	C₁₃×#389	C₂²	1	1	yes	characteristic
12	C₁₃×#387	—	2	1	no	—
13	C₁₃×#387	—	2	1	no	—
14	C₁₃×#387	—	2	1	no	—
15	C₁₃×#383	—	2	1	no	—
16	C₄³×C₁₃	C₂²	1	1	yes	characteristic
17	C₂×#88043	—	2	1	no	—
18	C₂×C₄×#1055	—	2	1	no	—
19	#11718116	—	2	1	no	—
20	#11718116	—	2	1	no	—
21	#11718116	C₂²	1	2	yes	—
22	#11718116	C₂²	1	2	yes	—
23	C₂²×Q₈×C₁₃	C₂×C₄	1	1	yes	characteristic
24	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
25	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
26	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
27	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
28	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
29	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
33	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
34	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
35	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
36	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
37	C₂²×#1055	—	2	2	no	—
38	C₂²×#1055	—	2	2	no	—
39	C₁₃×#95	—	2	1	no	—
40	C₁₃×#95	—	2	1	no	—
41	C₁₃×#95	—	2	1	no	—
42	C₁₃×#95	—	2	1	no	—
43	C₁₃×#95	—	2	1	no	—
44	C₁₃×#95	—	2	1	no	—
45	C₁₃×#95	—	2	1	no	—
46	C₁₃×#95	—	2	1	no	—
47	C₁₃×#95	—	2	1	no	—
48	#88043	—	2	4	no	—
49	C₄×#1055	—	2	4	no	—
50	#7693	—	13	1	no	maximal, Sylow
51	C₂×Q₈×C₁₃	—	2	2	no	—
52	C₂×Q₈×C₁₃	—	4	1	no	—
53	C₂×Q₈×C₁₃	—	4	1	no	—
54	C₂²×C₄×C₁₃	—	2	1	no	—
55	C₂²×C₄×C₁₃	—	2	1	no	—
56	C₂²×C₄×C₁₃	—	2	1	no	—
57	C₂²×C₄×C₁₃	—	2	1	no	—
58	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
59	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
60	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
61	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
62	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
63	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
64	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
65	C₁₃×C₄⋊C₄	—	2	2	no	—
66	C₁₃×C₄⋊C₄	—	4	1	no	—
67	C₁₃×C₄⋊C₄	—	4	1	no	—
68	C₁₃×C₄⋊C₄	—	4	2	no	—
69	C₁₃×C₄⋊C₄	—	4	2	no	—
70	C₁₃×C₄⋊C₄	—	4	2	no	—
71	C₄²×C₁₃	—	2	2	no	—
72	C₄²×C₁₃	—	2	2	no	—
73	C₄²×C₁₃	—	2	2	no	—
74	C₄²×C₁₃	—	2	2	no	—
75	C₄²×C₁₃	—	2	2	no	—
76	C₄²×C₁₃	—	4	1	no	—
77	C₄²×C₁₃	—	4	1	no	—
78	C₂×#1055	—	2	4	no	—
79	C₂×#1055	—	2	4	no	—
80	C₂×#1055	—	4	2	no	—
81	C₂×#1055	—	4	2	no	—
82	C₂×#1055	—	4	2	no	—
83	C₂×#1055	—	4	2	no	—
84	#2863	D₁₃	1	1	yes	characteristic
85	#1745	—	13	1	no	—
86	#1744	—	13	1	no	—
87	#1272	—	13	2	no	—
88	#1271	—	13	2	no	—
89	C₂³×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
90	Q₈×C₁₃	—	8	2	no	—
91	C₂×C₄×C₁₃	—	2	1	no	—
92	C₂×C₄×C₁₃	—	2	1	no	—
93	C₂×C₄×C₁₃	—	2	1	no	—
94	C₂×C₄×C₁₃	—	2	1	no	—
95	C₂×C₄×C₁₃	—	2	1	no	—
96	C₂×C₄×C₁₃	—	2	1	no	—
97	C₂×C₄×C₁₃	—	2	1	no	—
98	C₂×C₄×C₁₃	—	2	1	no	—
99	C₂×C₄×C₁₃	—	2	1	no	—
100	C₂×C₄×C₁₃	—	2	1	no	—
101	C₂×C₄×C₁₃	—	2	1	no	—
102	C₂×C₄×C₁₃	—	2	1	no	—
103	C₂×C₄×C₁₃	—	2	1	no	—
104	C₂×C₄×C₁₃	—	2	1	no	—
105	C₂×C₄×C₁₃	—	2	1	no	—
106	C₂×C₄×C₁₃	—	2	1	no	—
107	C₂×C₄×C₁₃	—	4	1	no	—
108	C₂×C₄×C₁₃	—	4	1	no	—
109	C₂×C₄×C₁₃	—	4	1	no	—
110	C₂×C₄×C₁₃	—	4	1	no	—
111	C₂×C₄×C₁₃	—	4	1	no	—
112	C₂×C₄×C₁₃	—	4	1	no	—
113	C₂×C₄×C₁₃	—	4	1	no	—
114	C₂×C₄×C₁₃	—	4	1	no	—
115	C₂×C₄×C₁₃	—	4	1	no	—
116	C₂×C₄×C₁₃	—	4	1	no	—
117	C₂×C₄×C₁₃	—	4	1	no	—
118	C₂×C₄×C₁₃	—	4	1	no	—
119	C₂×C₄×C₁₃	#104	1	2	yes	—
120	C₂×C₄×C₁₃	#104	1	2	yes	—
121	C₂×C₄×C₁₃	#131	1	2	yes	—
122	C₂×C₄×C₁₃	#135	1	2	yes	—
123	C₂×C₄×C₁₃	#137	1	2	yes	—
124	#1055	—	4	4	no	—
125	#1055	—	4	4	no	—
126	C₂×#97	—	26	1	no	—
127	C₂×C₄×C₈	—	26	1	no	—
128	#391	Dic₁₃	1	1	yes	characteristic
129	#391	Dic₁₃	1	1	yes	characteristic
130	#391	C₂×D₁₃	1	1	yes	characteristic
131	#389	Dic₁₃	1	1	yes	characteristic
132	#389	Dic₁₃	1	1	yes	characteristic
133	#389	C₂×D₁₃	1	1	yes	characteristic
134	#387	—	2	1	no	—
135	#387	—	2	1	no	—
136	#387	—	2	1	no	—
137	#383	—	2	1	no	—
138	C₄³	C₂×D₁₃	1	1	yes	characteristic
139	#336	—	13	2	no	—
140	#336	—	13	2	no	—
141	#336	—	26	1	no	—
142	#336	—	26	1	no	—
143	C₂²×C₁₃	#408	1	1	yes	characteristic
144	C₂²×C₁₃	#410	1	1	yes	characteristic
145	C₂²×C₁₃	#413	1	1	yes	characteristic
146	C₂²×C₁₃	#416	1	1	yes	characteristic
147	C₂²×C₁₃	#419	1	1	yes	characteristic
148	C₂²×C₁₃	C₂×#104	1	1	yes	characteristic
149	C₂²×C₁₃	C₂×#104	1	1	yes	characteristic
150	C₄×C₁₃	—	2	2	no	—
151	C₄×C₁₃	—	2	2	no	—
152	C₄×C₁₃	—	2	2	no	—
153	C₄×C₁₃	—	2	2	no	—
154	C₄×C₁₃	—	2	2	no	—
155	C₄×C₁₃	—	4	1	no	—
156	C₄×C₁₃	—	4	1	no	—
157	C₄×C₁₃	—	8	1	no	—
158	C₄×C₁₃	—	8	1	no	—
159	C₄×C₁₃	—	8	1	no	—
160	C₄×C₁₃	—	8	1	no	—
161	C₂²×Q₈	C₂×Dic₁₃	1	1	yes	characteristic
162	C₂²×C₈	—	26	2	no	—
163	C₂²×C₈	—	26	2	no	—
164	C₂×C₄⋊C₄	—	2	1	no	—
165	C₂×C₄⋊C₄	—	2	1	no	—
166	C₂×C₄⋊C₄	—	2	1	no	—
167	C₂×C₄⋊C₄	C₂×Dic₁₃	1	1	yes	characteristic
168	C₂×C₄⋊C₄	C₂×Dic₁₃	1	1	yes	characteristic
169	C₂×C₄⋊C₄	C₂×Dic₁₃	1	1	yes	characteristic
170	C₂×C₄²	C₂×Dic₁₃	1	1	yes	characteristic
171	C₂×C₄²	C₂×Dic₁₃	1	1	yes	characteristic
172	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
173	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
174	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
175	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
176	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
177	#97	—	26	4	no	—
178	C₄×C₈	—	26	4	no	—
179	#95	—	2	1	no	—
180	#95	—	2	1	no	—
181	#95	—	2	1	no	—
182	#95	—	2	1	no	—
183	#95	—	2	1	no	—
184	#95	—	2	1	no	—
185	#95	—	2	1	no	—
186	#95	—	2	1	no	—
187	#95	—	2	1	no	—
188	C₂×C₁₃	#1473	1	1	yes	characteristic
189	C₂×C₁₃	#1485	1	1	yes	characteristic
190	C₂×C₁₃	#1494	1	1	yes	characteristic
191	C₂×C₁₃	#1500	1	1	yes	characteristic
192	C₂×C₁₃	#1506	1	1	yes	characteristic
193	C₂×C₁₃	#1524	1	1	yes	characteristic
194	C₂×C₁₃	#1554	1	1	yes	characteristic
195	C₂×Q₈	—	2	2	no	—
196	C₂×Q₈	—	4	1	no	—
197	C₂×Q₈	—	4	1	no	—
198	C₂²×C₄	—	2	1	no	—
199	C₂²×C₄	—	2	1	no	—
200	C₂²×C₄	—	2	1	no	—
201	C₂²×C₄	—	2	1	no	—
202	C₂²×C₄	C₂²⋊Dic₁₃	1	1	yes	characteristic
203	C₂²×C₄	C₂²⋊Dic₁₃	1	1	yes	characteristic
204	C₂²×C₄	C₂²⋊Dic₁₃	1	1	yes	characteristic
205	C₂²×C₄	C₂²⋊Dic₁₃	1	1	yes	characteristic
206	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
207	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
208	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
209	C₂×C₈	—	26	4	no	—
210	C₂×C₈	—	26	4	no	—
211	C₂×C₈	—	52	2	no	—
212	C₂×C₈	—	52	2	no	—
213	C₂×C₈	—	52	2	no	—
214	C₂×C₈	—	52	2	no	—
215	C₄⋊C₄	—	2	2	no	—
216	C₄⋊C₄	—	4	1	no	—
217	C₄⋊C₄	—	4	1	no	—
218	C₄⋊C₄	—	4	2	no	—
219	C₄⋊C₄	—	4	2	no	—
220	C₄⋊C₄	—	4	2	no	—
221	C₄²	—	2	2	no	—
222	C₄²	—	2	2	no	—
223	C₄²	—	2	2	no	—
224	C₄²	—	2	2	no	—
225	C₄²	—	2	2	no	—
226	C₄²	—	4	1	no	—
227	C₄²	—	4	1	no	—
228	C₁₃	#7693	1	1	yes	characteristic, complemented, Sylow
229	C₂³	C₂×C₂²⋊Dic₁₃	1	1	yes	characteristic
230	Q₈	—	8	2	no	—
231	C₂×C₄	—	2	1	no	—
232	C₂×C₄	—	2	1	no	—
233	C₂×C₄	—	2	1	no	—
234	C₂×C₄	—	2	1	no	—
235	C₂×C₄	—	2	1	no	—
236	C₂×C₄	—	2	1	no	—
237	C₂×C₄	—	2	1	no	—
238	C₂×C₄	—	2	1	no	—
239	C₂×C₄	—	2	1	no	—
240	C₂×C₄	—	2	1	no	—
241	C₂×C₄	—	2	1	no	—
242	C₂×C₄	—	2	1	no	—
243	C₂×C₄	—	2	1	no	—
244	C₂×C₄	—	2	1	no	—
245	C₂×C₄	—	2	1	no	—
246	C₂×C₄	—	2	1	no	—
247	C₂×C₄	—	4	1	no	—
248	C₂×C₄	—	4	1	no	—
249	C₂×C₄	—	4	1	no	—
250	C₂×C₄	—	4	1	no	—
251	C₂×C₄	—	4	1	no	—
252	C₂×C₄	—	4	1	no	—
253	C₂×C₄	—	4	1	no	—
254	C₂×C₄	—	4	1	no	—
255	C₂×C₄	—	4	1	no	—
256	C₂×C₄	—	4	1	no	—
257	C₂×C₄	—	4	1	no	—
258	C₂×C₄	—	4	1	no	—
259	C₂×C₄	#88077	1	2	yes	—
260	C₂×C₄	#88077	1	2	yes	—
261	C₂×C₄	#88196	1	2	yes	—
262	C₂×C₄	#88202	1	2	yes	—
263	C₂×C₄	#88204	1	2	yes	—
264	C₈	—	52	4	no	—
265	C₈	—	52	4	no	—
266	C₂²	#11718886	1	1	yes	characteristic
267	C₂²	#11718894	1	1	yes	characteristic
268	C₂²	#11718895	1	1	yes	characteristic
269	C₂²	C₂×#88077	1	1	yes	characteristic
270	C₂²	C₂×#88077	1	1	yes	characteristic
271	C₂²	#11718898	1	1	yes	characteristic
272	C₂²	#11718901	1	1	yes	characteristic
273	C₄	—	2	2	no	—
274	C₄	—	2	2	no	—
275	C₄	—	2	2	no	—
276	C₄	—	2	2	no	—
277	C₄	—	2	2	no	—
278	C₄	—	4	1	no	—
279	C₄	—	4	1	no	—
280	C₄	—	8	1	no	—
281	C₄	—	8	1	no	—
282	C₄	—	8	1	no	—
283	C₄	—	8	1	no	—
284	C₂	#13113205	1	1	yes	characteristic
285	C₂	#13113257	1	1	yes	characteristic
286	C₂	#13113296	1	1	yes	characteristic
287	C₂	#13113322	1	1	yes	characteristic
288	C₂	#13113348	1	1	yes	characteristic
289	C₂	#13113424	1	1	yes	characteristic
290	C₂	#13113546	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊Dic₁₃
commutator	C₂²×C₄×C₁₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₃
Fitting	C₁₃×#2863	C₂
socle, prime radical	C₂³×C₁₃	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₁₃	C₂³
hypercenter	#2863	D₁₃
hypocenter	C₁₃	#7693
LCS	G ▷ C₂²×C₄×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #2863

Complements

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

N	H ≅ G/N	type
228. C₁₃	#7693	semidirect

Maximal subgroups

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

subgroup	index
#13114585	2
#13114582	2
#13112662	2
#13112659	2
C₁₃×#2863	2
#7693	13