Subgroups — #15750966

Order 3328.

426 conjugacy classes of subgroups.

The subgroup lattice has 314 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13114979	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊#1055	C₂	1	1	yes	maximal, characteristic
3	#13112659	C₂	1	2	yes	maximal
4	#13112659	C₂	1	2	yes	maximal
5	C₄×C₁₃×#125	C₂	1	1	yes	maximal, characteristic
6	C₂×C₁₃×#125	C₄	1	1	yes	characteristic
7	C₁₃×#389	C₂²	1	1	yes	characteristic
8	C₁₃×#389	C₂²	1	1	yes	characteristic
9	C₁₃×#387	—	2	1	no	—
10	C₁₃×#387	—	2	1	no	—
11	C₁₃×#382	C₄	1	1	yes	characteristic
12	C₄×C₁₃×C₄⋊C₄	—	2	1	no	—
13	C₄×C₁₃×C₄⋊C₄	—	2	1	no	—
14	C₄×C₁₃×C₄⋊C₄	C₄	1	1	yes	characteristic
15	C₄×C₁₃×C₄⋊C₄	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	#11718116	—	2	1	no	—
20	#11718116	—	2	1	no	—
21	#11718116	C₂²	1	2	yes	—
22	#11718116	C₂²	1	2	yes	—
23	C₁₃×#125	—	2	2	no	—
24	C₁₃×#125	C₈	1	4	yes	complemented
25	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
26	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
27	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
28	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
29	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
30	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
31	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
32	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄²×C₁₃	—	2	1	no	—
34	C₂×C₄²×C₁₃	—	2	1	no	—
35	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
38	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
39	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
40	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
41	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
42	C₂²×#1055	—	2	2	no	—
43	C₂²×#1055	—	2	2	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	C₁₃×#95	—	2	1	no	—
49	C₁₃×#95	—	2	1	no	—
50	C₄⋊#1055	—	2	4	no	—
51	C₄×#1055	—	4	2	no	—
52	#7585	—	13	1	no	maximal, Sylow
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	—	2	1	no	—
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	C₂²×C₄	1	1	yes	characteristic
62	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
64	C₁₃×C₄⋊C₄	—	2	4	no	—
65	C₁₃×C₄⋊C₄	—	2	4	no	—
66	C₁₃×C₄⋊C₄	—	2	1	no	—
67	C₁₃×C₄⋊C₄	—	2	1	no	—
68	C₁₃×C₄⋊C₄	—	4	1	no	—
69	C₁₃×C₄⋊C₄	—	4	1	no	—
70	C₁₃×C₄⋊C₄	—	4	2	no	—
71	C₁₃×C₄⋊C₄	—	4	1	no	—
72	C₁₃×C₄⋊C₄	—	4	1	no	—
73	C₁₃×C₄⋊C₄	C₂×C₈	1	2	yes	—
74	C₁₃×C₄⋊C₄	C₂×C₈	1	2	yes	—
75	C₄²×C₁₃	—	2	2	no	—
76	C₄²×C₁₃	—	2	2	no	—
77	C₄²×C₁₃	—	2	2	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	C₄²×C₁₃	—	4	2	no	—
83	C₄²×C₁₃	—	4	2	no	—
84	C₄²×C₁₃	C₂×C₈	1	2	yes	—
85	C₄²×C₁₃	M₁₆	1	2	yes	—
86	C₂×#1055	—	2	4	no	—
87	C₂×#1055	—	4	2	no	—
88	C₂×#1055	—	4	2	no	—
89	C₂×#1055	—	4	2	no	—
90	C₂×#1055	—	4	2	no	—
91	C₂×#1055	—	4	2	no	—
92	C₄×#125	D₁₃	1	1	yes	characteristic
93	#1839	—	13	1	no	—
94	C₄×C₄⋊C₈	—	13	1	no	—
95	#1271	—	13	2	no	—
96	#1271	—	13	2	no	—
97	C₂³×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
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	—	2	1	no	—
111	C₂×C₄×C₁₃	—	2	1	no	—
112	C₂×C₄×C₁₃	—	2	2	no	—
113	C₂×C₄×C₁₃	—	2	2	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	—	4	1	no	—
123	C₂×C₄×C₁₃	—	4	1	no	—
124	C₂×C₄×C₁₃	C₂²⋊C₈	1	2	yes	—
125	C₂×C₄×C₁₃	C₂²⋊C₈	1	2	yes	—
126	C₂×C₄×C₁₃	#101	1	2	yes	—
127	C₂×C₄×C₁₃	#104	1	2	yes	—
128	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
129	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
130	C₂×C₄×C₁₃	#135	1	2	yes	—
131	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
132	C₂×C₄×C₁₃	#137	1	1	yes	characteristic
133	#1055	—	4	4	no	—
134	#1055	—	8	2	no	—
135	C₂×#125	Dic₁₃	1	1	yes	characteristic
136	C₂×C₄⋊C₈	—	26	1	no	—
137	C₂×C₄×C₈	—	26	1	no	—
138	#389	C₂×D₁₃	1	1	yes	characteristic
139	#389	C₂×D₁₃	1	1	yes	characteristic
140	#387	—	2	1	no	—
141	#387	—	2	1	no	—
142	#382	Dic₁₃	1	1	yes	characteristic
143	C₄×C₄⋊C₄	—	2	1	no	—
144	C₄×C₄⋊C₄	—	2	1	no	—
145	C₄×C₄⋊C₄	Dic₁₃	1	1	yes	characteristic
146	C₄×C₄⋊C₄	Dic₁₃	1	1	yes	characteristic
147	C₄³	C₂×D₁₃	1	1	yes	characteristic
148	#336	—	13	2	no	—
149	#336	—	13	2	no	—
150	#336	—	26	1	no	—
151	#336	—	26	1	no	—
152	C₂²×C₁₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
153	C₂²×C₁₃	#410	1	1	yes	characteristic
154	C₂²×C₁₃	C₂×#101	1	1	yes	characteristic
155	C₂²×C₁₃	#416	1	1	yes	characteristic
156	C₂²×C₁₃	#417	1	1	yes	characteristic
157	C₂²×C₁₃	C₂×#104	1	1	yes	characteristic
158	C₂²×C₁₃	#421	1	1	yes	characteristic
159	C₄×C₁₃	—	2	2	no	—
160	C₄×C₁₃	—	2	1	no	—
161	C₄×C₁₃	—	2	1	no	—
162	C₄×C₁₃	—	2	2	no	—
163	C₄×C₁₃	—	2	2	no	—
164	C₄×C₁₃	—	2	1	no	—
165	C₄×C₁₃	—	2	1	no	—
166	C₄×C₁₃	—	2	2	no	—
167	C₄×C₁₃	—	2	2	no	—
168	C₄×C₁₃	—	4	2	no	—
169	C₄×C₁₃	—	4	2	no	—
170	C₄×C₁₃	—	8	1	no	—
171	C₄×C₁₃	—	8	1	no	—
172	C₂²×C₈	—	26	2	no	—
173	C₂²×C₈	—	26	2	no	—
174	#125	—	2	2	no	—
175	#125	#1055	1	4	yes	complemented
176	C₂×C₄⋊C₄	—	2	1	no	—
177	C₂×C₄⋊C₄	—	2	1	no	—
178	C₂×C₄⋊C₄	—	2	1	no	—
179	C₂×C₄⋊C₄	—	2	1	no	—
180	C₂×C₄⋊C₄	C₂×Dic₁₃	1	1	yes	characteristic
181	C₂×C₄⋊C₄	C₂×Dic₁₃	1	1	yes	characteristic
182	C₂×C₄⋊C₄	C₂×Dic₁₃	1	1	yes	characteristic
183	C₂×C₄⋊C₄	C₂×Dic₁₃	1	1	yes	characteristic
184	C₂×C₄²	—	2	1	no	—
185	C₂×C₄²	—	2	1	no	—
186	C₂×C₄²	C₂×Dic₁₃	1	1	yes	characteristic
187	C₂×C₄²	C₂×Dic₁₃	1	1	yes	characteristic
188	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
189	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
190	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
191	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
192	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
193	C₄⋊C₈	—	26	4	no	—
194	C₄×C₈	—	52	2	no	—
195	#95	—	2	1	no	—
196	#95	—	2	1	no	—
197	#95	—	2	1	no	—
198	#95	—	2	1	no	—
199	#95	—	2	1	no	—
200	#95	—	2	1	no	—
201	C₂×C₁₃	#1461	1	1	yes	characteristic
202	C₂×C₁₃	#1476	1	1	yes	characteristic
203	C₂×C₁₃	#1477	1	1	yes	characteristic
204	C₂×C₁₃	#1509	1	1	yes	characteristic
205	C₂×C₁₃	#1521	1	1	yes	characteristic
206	C₂×C₁₃	#1528	1	1	yes	characteristic
207	C₂×C₁₃	#1552	1	1	yes	characteristic
208	C₂²×C₄	—	2	1	no	—
209	C₂²×C₄	—	2	1	no	—
210	C₂²×C₄	—	2	1	no	—
211	C₂²×C₄	—	2	1	no	—
212	C₂²×C₄	C₂²⋊Dic₁₃	1	1	yes	characteristic
213	C₂²×C₄	C₂²⋊Dic₁₃	1	1	yes	characteristic
214	C₂²×C₄	C₂²⋊Dic₁₃	1	1	yes	characteristic
215	C₂²×C₄	C₂²⋊Dic₁₃	1	1	yes	characteristic
216	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
217	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
218	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
219	C₂×C₈	—	26	4	no	—
220	C₂×C₈	—	52	2	no	—
221	C₂×C₈	—	52	2	no	—
222	C₂×C₈	—	52	2	no	—
223	C₂×C₈	—	52	2	no	—
224	C₂×C₈	—	52	2	no	—
225	C₄⋊C₄	—	2	4	no	—
226	C₄⋊C₄	—	2	4	no	—
227	C₄⋊C₄	—	2	1	no	—
228	C₄⋊C₄	—	2	1	no	—
229	C₄⋊C₄	—	4	1	no	—
230	C₄⋊C₄	—	4	1	no	—
231	C₄⋊C₄	—	4	1	no	—
232	C₄⋊C₄	—	4	2	no	—
233	C₄⋊C₄	—	4	1	no	—
234	C₄⋊C₄	C₂×#1055	1	2	yes	—
235	C₄⋊C₄	C₂×#1055	1	2	yes	—
236	C₄²	—	2	2	no	—
237	C₄²	—	2	2	no	—
238	C₄²	—	2	1	no	—
239	C₄²	—	2	2	no	—
240	C₄²	—	2	1	no	—
241	C₄²	—	4	1	no	—
242	C₄²	—	4	1	no	—
243	C₄²	—	4	2	no	—
244	C₄²	—	4	2	no	—
245	C₄²	C₂×#1055	1	2	yes	—
246	C₄²	#6099	1	2	yes	—
247	C₁₃	#7585	1	1	yes	characteristic, complemented, Sylow
248	C₂³	C₂×C₂²⋊Dic₁₃	1	1	yes	characteristic
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	—	2	1	no	—
258	C₂×C₄	—	2	1	no	—
259	C₂×C₄	—	2	1	no	—
260	C₂×C₄	—	2	1	no	—
261	C₂×C₄	—	2	1	no	—
262	C₂×C₄	—	2	1	no	—
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	—	2	2	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	—	4	1	no	—
270	C₂×C₄	—	4	1	no	—
271	C₂×C₄	—	4	1	no	—
272	C₂×C₄	—	4	1	no	—
273	C₂×C₄	—	4	1	no	—
274	C₂×C₄	—	4	1	no	—
275	C₂×C₄	C₂²⋊#1055	1	2	yes	—
276	C₂×C₄	C₂²⋊#1055	1	2	yes	—
277	C₂×C₄	#88076	1	2	yes	—
278	C₂×C₄	#88077	1	2	yes	—
279	C₂×C₄	C₂²×#1055	1	1	yes	characteristic
280	C₂×C₄	C₂×#6099	1	1	yes	characteristic
281	C₂×C₄	#88203	1	1	yes	characteristic
282	C₂×C₄	#88204	1	2	yes	—
283	C₂×C₄	#88205	1	1	yes	characteristic
284	C₈	—	52	4	no	—
285	C₈	—	104	2	no	—
286	C₂²	C₂×C₂²⋊#1055	1	1	yes	characteristic
287	C₂²	C₂×#88076	1	1	yes	characteristic
288	C₂²	#11718893	1	1	yes	characteristic
289	C₂²	#11718894	1	1	yes	characteristic
290	C₂²	C₂×#88077	1	1	yes	characteristic
291	C₂²	#11718897	1	1	yes	characteristic
292	C₂²	#11718901	1	1	yes	characteristic
293	C₄	—	2	1	no	—
294	C₄	—	2	1	no	—
295	C₄	—	2	2	no	—
296	C₄	—	2	1	no	—
297	C₄	—	2	1	no	—
298	C₄	—	2	2	no	—
299	C₄	—	2	2	no	—
300	C₄	—	2	2	no	—
301	C₄	—	2	2	no	—
302	C₄	—	4	2	no	—
303	C₄	—	4	2	no	—
304	C₄	—	8	1	no	—
305	C₄	—	8	1	no	—
306	C₂	#13113160	1	1	yes	characteristic
307	C₂	#13113218	1	1	yes	characteristic
308	C₂	#13113223	1	1	yes	characteristic
309	C₂	#13113361	1	1	yes	characteristic
310	C₂	#13113413	1	1	yes	characteristic
311	C₂	#13113438	1	1	yes	characteristic
312	C₂	#13113536	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊Dic₁₃
commutator	C₂×C₄×C₁₃	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₁₃
Fitting	C₄×C₁₃×#125	C₂
socle, prime radical	C₂³×C₁₃	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₁₃	C₂³
hypercenter	C₄×#125	D₁₃
hypocenter	C₁₃	#7585
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄×#125

Complements

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

N	H ≅ G/N	type
24. C₁₃×#125	C₈	semidirect
175. #125	#1055	semidirect
247. C₁₃	#7585	semidirect

Maximal subgroups

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

subgroup	index
#13114979	2
C₄×C₄⋊#1055	2
#13112659	2
C₄×C₁₃×#125	2
#7585	13