Subgroups — #15751450

Order 3328.

378 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13114953	C₂	1	1	yes	maximal, characteristic
2	#13114689	C₂	1	1	yes	maximal, characteristic
3	#13112659	C₂	1	2	yes	maximal
4	#13112659	C₂	1	2	yes	maximal
5	C₁₃×#2864	C₂	1	1	yes	maximal, characteristic
6	C₁₃×#389	C₄	1	1	yes	characteristic
7	C₁₃×#389	C₄	1	1	yes	characteristic
8	C₁₃×#389	C₂²	1	1	yes	characteristic
9	C₁₃×#389	C₂²	1	1	yes	characteristic
10	C₁₃×#387	—	2	1	no	—
11	C₁₃×#387	—	2	1	no	—
12	C₁₃×#387	—	2	1	no	—
13	C₁₃×#387	—	2	1	no	—
14	C₁₃×#382	C₄	1	1	yes	characteristic
15	C₁₃×#382	C₄	1	1	yes	characteristic
16	C₄³×C₁₃	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊#1055	—	2	1	no	—
18	C₂×C₄⋊#1055	—	2	1	no	—
19	C₂×#88043	—	2	1	no	—
20	#11718116	—	2	1	no	—
21	#11718116	C₂²	1	2	yes	—
22	#11718116	C₂²	1	2	yes	—
23	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
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×C₁₃×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄²×C₁₃	C₂×C₄	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	D₄	1	1	yes	characteristic
37	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
38	C₂²×#1055	—	2	2	no	—
39	C₂²×#1055	—	2	2	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	C₄⋊#1055	—	2	4	no	—
49	C₄⋊#1055	—	4	2	no	—
50	#88043	—	4	2	no	—
51	#7697	—	13	1	no	maximal, Sylow
52	C₂²×C₄×C₁₃	—	2	1	no	—
53	C₂²×C₄×C₁₃	—	2	1	no	—
54	C₂²×C₄×C₁₃	—	2	1	no	—
55	C₂²×C₄×C₁₃	—	2	1	no	—
56	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
57	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
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	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
63	C₁₃×C₄⋊C₄	—	2	2	no	—
64	C₁₃×C₄⋊C₄	—	2	2	no	—
65	C₁₃×C₄⋊C₄	—	4	1	no	—
66	C₁₃×C₄⋊C₄	—	4	1	no	—
67	C₁₃×C₄⋊C₄	—	4	2	no	—
68	C₁₃×C₄⋊C₄	—	4	2	no	—
69	C₁₃×C₄⋊C₄	—	4	1	no	—
70	C₁₃×C₄⋊C₄	—	4	1	no	—
71	C₁₃×C₄⋊C₄	—	4	1	no	—
72	C₁₃×C₄⋊C₄	—	4	1	no	—
73	C₄²×C₁₃	—	2	2	no	—
74	C₄²×C₁₃	—	2	2	no	—
75	C₄²×C₁₃	—	2	2	no	—
76	C₄²×C₁₃	—	2	1	no	—
77	C₄²×C₁₃	—	2	1	no	—
78	C₄²×C₁₃	—	2	1	no	—
79	C₄²×C₁₃	—	2	1	no	—
80	C₄²×C₁₃	—	4	1	no	—
81	C₄²×C₁₃	—	4	1	no	—
82	C₂×#1055	—	2	4	no	—
83	C₂×#1055	—	4	2	no	—
84	C₂×#1055	—	4	2	no	—
85	C₂×#1055	—	4	2	no	—
86	C₂×#1055	—	4	2	no	—
87	C₂×#1055	—	4	2	no	—
88	#2864	D₁₃	1	1	yes	characteristic
89	#1831	—	13	1	no	—
90	#1767	—	13	1	no	—
91	#1271	—	13	2	no	—
92	#1271	—	13	2	no	—
93	C₂³×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
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	—	2	1	no	—
108	C₂×C₄×C₁₃	—	2	1	no	—
109	C₂×C₄×C₁₃	—	2	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	—	4	1	no	—
120	C₂×C₄×C₁₃	—	4	1	no	—
121	C₂×C₄×C₁₃	—	4	1	no	—
122	C₂×C₄×C₁₃	#101	1	2	yes	—
123	C₂×C₄×C₁₃	#104	1	2	yes	—
124	C₂×C₄×C₁₃	#131	1	2	yes	—
125	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
126	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
127	C₂×C₄×C₁₃	#137	1	1	yes	characteristic
128	C₂×C₄×C₁₃	#137	1	1	yes	characteristic
129	#1055	—	4	4	no	—
130	#1055	—	8	2	no	—
131	C₂×C₄⋊C₈	—	26	1	no	—
132	C₂×C₄⋊C₈	—	26	1	no	—
133	C₂×#97	—	26	1	no	—
134	#389	Dic₁₃	1	1	yes	characteristic
135	#389	Dic₁₃	1	1	yes	characteristic
136	#389	C₂×D₁₃	1	1	yes	characteristic
137	#389	C₂×D₁₃	1	1	yes	characteristic
138	#387	—	2	1	no	—
139	#387	—	2	1	no	—
140	#387	—	2	1	no	—
141	#387	—	2	1	no	—
142	#382	Dic₁₃	1	1	yes	characteristic
143	#382	Dic₁₃	1	1	yes	characteristic
144	C₄³	C₂×D₁₃	1	1	yes	characteristic
145	#336	—	13	2	no	—
146	#336	—	13	2	no	—
147	#336	—	26	1	no	—
148	C₂²×C₁₃	#408	1	1	yes	characteristic
149	C₂²×C₁₃	#410	1	1	yes	characteristic
150	C₂²×C₁₃	C₂×#101	1	1	yes	characteristic
151	C₂²×C₁₃	#417	1	1	yes	characteristic
152	C₂²×C₁₃	#417	1	1	yes	characteristic
153	C₂²×C₁₃	C₂×#104	1	1	yes	characteristic
154	C₂²×C₁₃	#421	1	1	yes	characteristic
155	C₄×C₁₃	—	2	2	no	—
156	C₄×C₁₃	—	2	2	no	—
157	C₄×C₁₃	—	2	1	no	—
158	C₄×C₁₃	—	2	1	no	—
159	C₄×C₁₃	—	2	1	no	—
160	C₄×C₁₃	—	2	1	no	—
161	C₄×C₁₃	—	2	2	no	—
162	C₄×C₁₃	—	4	1	no	—
163	C₄×C₁₃	—	4	1	no	—
164	C₄×C₁₃	—	8	1	no	—
165	C₄×C₁₃	—	8	1	no	—
166	C₄×C₁₃	—	8	1	no	—
167	C₄×C₁₃	—	8	1	no	—
168	C₂²×C₈	—	26	2	no	—
169	C₂²×C₈	—	26	2	no	—
170	C₂×C₄⋊C₄	—	2	1	no	—
171	C₂×C₄⋊C₄	—	2	1	no	—
172	C₂×C₄⋊C₄	—	2	1	no	—
173	C₂×C₄⋊C₄	—	2	1	no	—
174	C₂×C₄⋊C₄	C₂×Dic₁₃	1	1	yes	characteristic
175	C₂×C₄⋊C₄	C₂×Dic₁₃	1	1	yes	characteristic
176	C₂×C₄⋊C₄	C₂×Dic₁₃	1	1	yes	characteristic
177	C₂×C₄⋊C₄	C₂×Dic₁₃	1	1	yes	characteristic
178	C₂×C₄²	C₂×Dic₁₃	1	1	yes	characteristic
179	C₂×C₄²	C₂×Dic₁₃	1	1	yes	characteristic
180	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
181	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
182	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
183	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
184	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
185	C₄⋊C₈	—	26	4	no	—
186	C₄⋊C₈	—	52	2	no	—
187	#97	—	52	2	no	—
188	#95	—	2	1	no	—
189	#95	—	2	1	no	—
190	#95	—	2	1	no	—
191	#95	—	2	1	no	—
192	#95	—	2	1	no	—
193	#95	—	2	1	no	—
194	#95	—	2	1	no	—
195	#95	—	2	1	no	—
196	C₂×C₁₃	#1470	1	1	yes	characteristic
197	C₂×C₁₃	#1483	1	1	yes	characteristic
198	C₂×C₁₃	#1495	1	1	yes	characteristic
199	C₂×C₁₃	#1509	1	1	yes	characteristic
200	C₂×C₁₃	#1512	1	1	yes	characteristic
201	C₂×C₁₃	#1531	1	1	yes	characteristic
202	C₂×C₁₃	#1561	1	1	yes	characteristic
203	C₂²×C₄	—	2	1	no	—
204	C₂²×C₄	—	2	1	no	—
205	C₂²×C₄	—	2	1	no	—
206	C₂²×C₄	—	2	1	no	—
207	C₂²×C₄	C₂²⋊Dic₁₃	1	1	yes	characteristic
208	C₂²×C₄	C₂²⋊Dic₁₃	1	1	yes	characteristic
209	C₂²×C₄	C₂²⋊Dic₁₃	1	1	yes	characteristic
210	C₂²×C₄	C₂²⋊Dic₁₃	1	1	yes	characteristic
211	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
212	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
213	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
214	C₂×C₈	—	26	4	no	—
215	C₂×C₈	—	52	2	no	—
216	C₂×C₈	—	52	2	no	—
217	C₂×C₈	—	52	2	no	—
218	C₂×C₈	—	52	2	no	—
219	C₂×C₈	—	52	2	no	—
220	C₄⋊C₄	—	2	2	no	—
221	C₄⋊C₄	—	2	2	no	—
222	C₄⋊C₄	—	4	1	no	—
223	C₄⋊C₄	—	4	1	no	—
224	C₄⋊C₄	—	4	2	no	—
225	C₄⋊C₄	—	4	2	no	—
226	C₄⋊C₄	—	4	1	no	—
227	C₄⋊C₄	—	4	1	no	—
228	C₄⋊C₄	—	4	1	no	—
229	C₄⋊C₄	—	4	1	no	—
230	C₄²	—	2	2	no	—
231	C₄²	—	2	2	no	—
232	C₄²	—	2	2	no	—
233	C₄²	—	2	1	no	—
234	C₄²	—	2	1	no	—
235	C₄²	—	2	1	no	—
236	C₄²	—	2	1	no	—
237	C₄²	—	4	1	no	—
238	C₄²	—	4	1	no	—
239	C₁₃	#7697	1	1	yes	characteristic, complemented, Sylow
240	C₂³	C₂×C₂²⋊Dic₁₃	1	1	yes	characteristic
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	—	2	1	no	—
248	C₂×C₄	—	2	1	no	—
249	C₂×C₄	—	2	1	no	—
250	C₂×C₄	—	2	1	no	—
251	C₂×C₄	—	2	1	no	—
252	C₂×C₄	—	2	1	no	—
253	C₂×C₄	—	2	1	no	—
254	C₂×C₄	—	2	1	no	—
255	C₂×C₄	—	2	1	no	—
256	C₂×C₄	—	2	1	no	—
257	C₂×C₄	—	4	1	no	—
258	C₂×C₄	—	4	1	no	—
259	C₂×C₄	—	4	1	no	—
260	C₂×C₄	—	4	1	no	—
261	C₂×C₄	—	4	1	no	—
262	C₂×C₄	—	4	1	no	—
263	C₂×C₄	—	4	1	no	—
264	C₂×C₄	—	4	1	no	—
265	C₂×C₄	—	4	1	no	—
266	C₂×C₄	—	4	1	no	—
267	C₂×C₄	—	4	1	no	—
268	C₂×C₄	—	4	1	no	—
269	C₂×C₄	#88076	1	2	yes	—
270	C₂×C₄	#88077	1	2	yes	—
271	C₂×C₄	#88202	1	2	yes	—
272	C₂×C₄	#88203	1	1	yes	characteristic
273	C₂×C₄	#88203	1	1	yes	characteristic
274	C₂×C₄	#88205	1	1	yes	characteristic
275	C₂×C₄	#88205	1	1	yes	characteristic
276	C₈	—	52	4	no	—
277	C₈	—	104	2	no	—
278	C₂²	C₂×#88076	1	1	yes	characteristic
279	C₂²	#11718893	1	1	yes	characteristic
280	C₂²	#11718893	1	1	yes	characteristic
281	C₂²	#11718895	1	1	yes	characteristic
282	C₂²	C₂×#88077	1	1	yes	characteristic
283	C₂²	#11718897	1	1	yes	characteristic
284	C₂²	#11718901	1	1	yes	characteristic
285	C₄	—	2	1	no	—
286	C₄	—	2	1	no	—
287	C₄	—	2	1	no	—
288	C₄	—	2	1	no	—
289	C₄	—	2	2	no	—
290	C₄	—	2	2	no	—
291	C₄	—	2	2	no	—
292	C₄	—	4	1	no	—
293	C₄	—	4	1	no	—
294	C₄	—	8	1	no	—
295	C₄	—	8	1	no	—
296	C₄	—	8	1	no	—
297	C₄	—	8	1	no	—
298	C₂	#13113192	1	1	yes	characteristic
299	C₂	#13113249	1	1	yes	characteristic
300	C₂	#13113301	1	1	yes	characteristic
301	C₂	#13113361	1	1	yes	characteristic
302	C₂	#13113374	1	1	yes	characteristic
303	C₂	#13113451	1	1	yes	characteristic
304	C₂	#13113574	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×#2864	C₂
socle, prime radical	C₂³×C₁₃	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₁₃	C₂³
hypercenter	#2864	D₁₃
hypocenter	C₁₃	#7697
LCS	G ▷ C₂²×C₄×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #2864

Complements

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

N	H ≅ G/N	type
239. C₁₃	#7697	semidirect

Maximal subgroups

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

subgroup	index
#13114953	2
#13114689	2
#13112659	2
C₁₃×#2864	2
#7697	13