Subgroups — #15753854

Order 3328.

488 conjugacy classes of subgroups.

The subgroup lattice has 316 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	C₄×#88043	C₂	1	1	yes	maximal, characteristic
3	C₂×#11718135	C₂	1	2	yes	maximal
4	#13112717	C₂	1	2	yes	maximal
5	C₄×C₁₃×#125	C₂	1	1	yes	maximal, characteristic
6	C₂×C₁₃×#125	C₂²	1	1	yes	characteristic
7	C₁₃×#389	C₄	1	1	yes	characteristic
8	C₁₃×#389	C₄	1	1	yes	characteristic
9	C₁₃×#387	—	2	1	no	—
10	C₁₃×#387	—	2	1	no	—
11	C₁₃×#382	C₂²	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×#88043	—	2	1	no	—
18	C₂×#88043	C₂²	1	2	yes	—
19	C₂×#88043	C₂²	1	2	yes	—
20	C₂×C₄×#1055	—	2	1	no	—
21	#11718135	C₄	1	8	yes	complemented
22	#11718116	—	2	1	no	—
23	#11718116	—	2	1	no	—
24	C₁₃×#125	—	2	2	no	—
25	C₁₃×#125	C₂×C₄	1	4	yes	—
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	—	2	1	no	—
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×C₁₃×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄²×C₁₃	—	2	1	no	—
35	C₂×C₄²×C₁₃	—	2	1	no	—
36	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄²×C₁₃	C₂×C₄	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	D₄	1	1	yes	characteristic
42	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
43	C₂²×#1055	—	2	2	no	—
44	C₂²×#1055	—	2	2	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	C₁₃×#95	—	2	1	no	—
51	#88043	—	2	4	no	—
52	#88043	—	2	2	no	—
53	#88043	—	2	2	no	—
54	#88043	C₂×C₄	1	4	yes	—
55	#88043	C₂×C₄	1	4	yes	—
56	C₄×#1055	—	2	4	no	—
57	#8227	—	13	1	no	maximal, Sylow
58	C₂²×C₄×C₁₃	—	2	1	no	—
59	C₂²×C₄×C₁₃	—	2	1	no	—
60	C₂²×C₄×C₁₃	—	2	1	no	—
61	C₂²×C₄×C₁₃	—	2	1	no	—
62	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
63	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
64	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
65	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
66	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
67	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
69	C₁₃×C₄⋊C₄	—	2	1	no	—
70	C₁₃×C₄⋊C₄	—	2	1	no	—
71	C₁₃×C₄⋊C₄	—	2	4	no	—
72	C₁₃×C₄⋊C₄	—	2	4	no	—
73	C₁₃×C₄⋊C₄	—	4	1	no	—
74	C₁₃×C₄⋊C₄	—	4	1	no	—
75	C₁₃×C₄⋊C₄	—	4	2	no	—
76	C₁₃×C₄⋊C₄	—	4	2	no	—
77	C₁₃×C₄⋊C₄	C₄²	1	2	yes	—
78	C₁₃×C₄⋊C₄	C₄²	1	2	yes	—
79	C₄²×C₁₃	—	2	2	no	—
80	C₄²×C₁₃	—	2	2	no	—
81	C₄²×C₁₃	—	2	2	no	—
82	C₄²×C₁₃	—	4	1	no	—
83	C₄²×C₁₃	—	4	1	no	—
84	C₄²×C₁₃	—	4	1	no	—
85	C₄²×C₁₃	—	4	2	no	—
86	C₄²×C₁₃	—	4	2	no	—
87	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
88	C₄²×C₁₃	#41	1	2	yes	—
89	C₂×#1055	—	2	4	no	—
90	C₂×#1055	—	2	4	no	—
91	C₂×#1055	—	2	4	no	—
92	C₂×#1055	—	2	4	no	—
93	C₂×#1055	—	4	2	no	—
94	C₂×#1055	—	4	2	no	—
95	C₄×#125	D₁₃	1	1	yes	characteristic
96	#1745	—	13	1	no	—
97	C₄×#97	—	13	1	no	—
98	C₂×#330	—	13	2	no	—
99	#1295	—	13	2	no	—
100	C₂³×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
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	1	no	—
113	C₂×C₄×C₁₃	—	2	2	no	—
114	C₂×C₄×C₁₃	—	2	2	no	—
115	C₂×C₄×C₁₃	—	2	1	no	—
116	C₂×C₄×C₁₃	—	2	1	no	—
117	C₂×C₄×C₁₃	—	2	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	—	4	1	no	—
125	C₂×C₄×C₁₃	—	4	1	no	—
126	C₂×C₄×C₁₃	—	4	1	no	—
127	C₂×C₄×C₁₃	—	4	1	no	—
128	C₂×C₄×C₁₃	#101	1	2	yes	—
129	C₂×C₄×C₁₃	#104	1	2	yes	—
130	C₂×C₄×C₁₃	#104	1	2	yes	—
131	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
132	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
133	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
134	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
135	#1055	—	2	8	no	—
136	#1055	—	4	4	no	—
137	C₂×#125	C₂×D₁₃	1	1	yes	characteristic
138	C₂×#97	—	13	2	no	—
139	C₂×#97	—	13	2	no	—
140	C₂×#97	—	26	1	no	—
141	C₂×C₄×C₈	—	26	1	no	—
142	#389	Dic₁₃	1	1	yes	characteristic
143	#389	Dic₁₃	1	1	yes	characteristic
144	#387	—	2	1	no	—
145	#387	—	2	1	no	—
146	#382	C₂×D₁₃	1	1	yes	characteristic
147	C₄×C₄⋊C₄	—	2	1	no	—
148	C₄×C₄⋊C₄	—	2	1	no	—
149	C₄×C₄⋊C₄	Dic₁₃	1	1	yes	characteristic
150	C₄×C₄⋊C₄	Dic₁₃	1	1	yes	characteristic
151	C₄³	C₂×D₁₃	1	1	yes	characteristic
152	#336	—	26	1	no	—
153	#336	—	26	1	no	—
154	#330	—	13	8	no	—
155	C₂²×C₁₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
156	C₂²×C₁₃	C₂×#101	1	1	yes	characteristic
157	C₂²×C₁₃	#413	1	1	yes	characteristic
158	C₂²×C₁₃	C₂×#104	1	1	yes	characteristic
159	C₂²×C₁₃	C₂×#104	1	1	yes	characteristic
160	C₂²×C₁₃	#421	1	1	yes	characteristic
161	C₂²×C₁₃	#421	1	1	yes	characteristic
162	C₄×C₁₃	—	2	1	no	—
163	C₄×C₁₃	—	2	1	no	—
164	C₄×C₁₃	—	2	2	no	—
165	C₄×C₁₃	—	2	2	no	—
166	C₄×C₁₃	—	2	2	no	—
167	C₄×C₁₃	—	2	2	no	—
168	C₄×C₁₃	—	2	2	no	—
169	C₄×C₁₃	—	4	2	no	—
170	C₄×C₁₃	—	4	2	no	—
171	C₄×C₁₃	—	4	1	no	—
172	C₄×C₁₃	—	8	1	no	—
173	C₄×C₁₃	—	8	1	no	—
174	C₂²×C₈	—	26	2	no	—
175	C₂²×C₈	—	26	2	no	—
176	#125	—	2	2	no	—
177	#125	C₄×D₁₃	1	4	yes	—
178	C₂×C₄⋊C₄	—	2	1	no	—
179	C₂×C₄⋊C₄	—	2	1	no	—
180	C₂×C₄⋊C₄	—	2	1	no	—
181	C₂×C₄⋊C₄	—	2	1	no	—
182	C₂×C₄⋊C₄	C₂×Dic₁₃	1	1	yes	characteristic
183	C₂×C₄⋊C₄	C₂×Dic₁₃	1	1	yes	characteristic
184	C₂×C₄⋊C₄	C₂×Dic₁₃	1	1	yes	characteristic
185	C₂×C₄⋊C₄	C₂×Dic₁₃	1	1	yes	characteristic
186	C₂×C₄²	—	2	1	no	—
187	C₂×C₄²	—	2	1	no	—
188	C₂×C₄²	C₂×Dic₁₃	1	1	yes	characteristic
189	C₂×C₄²	C₂×Dic₁₃	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	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
194	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
195	#97	—	13	4	no	—
196	#97	—	13	4	no	—
197	#97	—	26	4	no	—
198	#97	—	26	2	no	—
199	#97	—	26	2	no	—
200	C₄×C₈	—	26	4	no	—
201	#95	—	2	1	no	—
202	#95	—	2	1	no	—
203	#95	—	2	1	no	—
204	#95	—	2	1	no	—
205	#95	—	2	1	no	—
206	#95	—	2	1	no	—
207	C₂×C₁₃	#1528	1	1	yes	characteristic
208	C₂×C₁₃	#1529	1	1	yes	characteristic
209	C₂×C₁₃	#1531	1	1	yes	characteristic
210	C₂×C₁₃	#1532	1	1	yes	characteristic
211	C₂×C₁₃	C₄×#101	1	1	yes	characteristic
212	C₂×C₁₃	C₄×#104	1	1	yes	characteristic
213	C₂×C₁₃	#1759	1	1	yes	characteristic
214	C₂²×C₄	—	2	1	no	—
215	C₂²×C₄	—	2	1	no	—
216	C₂²×C₄	—	2	1	no	—
217	C₂²×C₄	—	2	1	no	—
218	C₂²×C₄	C₂²⋊Dic₁₃	1	1	yes	characteristic
219	C₂²×C₄	C₂²⋊Dic₁₃	1	1	yes	characteristic
220	C₂²×C₄	C₂²⋊Dic₁₃	1	1	yes	characteristic
221	C₂²×C₄	C₂²⋊Dic₁₃	1	1	yes	characteristic
222	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
223	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
224	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
225	C₂×C₈	—	26	4	no	—
226	C₂×C₈	—	26	4	no	—
227	C₂×C₈	—	26	4	no	—
228	C₂×C₈	—	26	4	no	—
229	C₂×C₈	—	52	2	no	—
230	C₂×C₈	—	52	2	no	—
231	C₄⋊C₄	—	2	1	no	—
232	C₄⋊C₄	—	2	1	no	—
233	C₄⋊C₄	—	2	4	no	—
234	C₄⋊C₄	—	2	4	no	—
235	C₄⋊C₄	—	4	1	no	—
236	C₄⋊C₄	—	4	1	no	—
237	C₄⋊C₄	—	4	2	no	—
238	C₄⋊C₄	—	4	2	no	—
239	C₄⋊C₄	C₄×Dic₁₃	1	2	yes	—
240	C₄⋊C₄	C₄×Dic₁₃	1	2	yes	—
241	C₄²	—	2	2	no	—
242	C₄²	—	2	2	no	—
243	C₄²	—	2	2	no	—
244	C₄²	—	4	1	no	—
245	C₄²	—	4	1	no	—
246	C₄²	—	4	1	no	—
247	C₄²	—	4	2	no	—
248	C₄²	—	4	2	no	—
249	C₄²	C₂×C₄×D₁₃	1	2	yes	—
250	C₄²	#6127	1	2	yes	—
251	C₁₃	#8227	1	1	yes	characteristic, complemented, Sylow
252	C₂³	C₂×C₂²⋊Dic₁₃	1	1	yes	characteristic
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	1	no	—
264	C₂×C₄	—	2	1	no	—
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	—	2	1	no	—
268	C₂×C₄	—	2	1	no	—
269	C₂×C₄	—	2	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	—	4	1	no	—
276	C₂×C₄	—	4	1	no	—
277	C₂×C₄	—	4	1	no	—
278	C₂×C₄	—	4	1	no	—
279	C₂×C₄	—	4	1	no	—
280	C₂×C₄	#88076	1	2	yes	—
281	C₂×C₄	#88077	1	2	yes	—
282	C₂×C₄	#88077	1	2	yes	—
283	C₂×C₄	C₂×C₄×Dic₁₃	1	1	yes	characteristic
284	C₂×C₄	#88180	1	1	yes	characteristic
285	C₂×C₄	C₄×C₂²⋊D₁₃	1	2	yes	—
286	C₂×C₄	C₄×C₂²⋊D₁₃	1	2	yes	—
287	C₈	—	26	8	no	—
288	C₈	—	52	4	no	—
289	C₂²	C₄×C₂²⋊Dic₁₃	1	1	yes	characteristic
290	C₂²	C₂×#88076	1	1	yes	characteristic
291	C₂²	C₂×#88077	1	1	yes	characteristic
292	C₂²	C₂×#88077	1	1	yes	characteristic
293	C₂²	#11718897	1	1	yes	characteristic
294	C₂²	#11718897	1	1	yes	characteristic
295	C₂²	#11718898	1	1	yes	characteristic
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	—	2	2	no	—
303	C₄	—	4	2	no	—
304	C₄	—	4	2	no	—
305	C₄	—	4	1	no	—
306	C₄	—	8	1	no	—
307	C₄	—	8	1	no	—
308	C₂	#13113438	1	1	yes	characteristic
309	C₂	#13113443	1	1	yes	characteristic
310	C₂	#13113451	1	1	yes	characteristic
311	C₂	#13113456	1	1	yes	characteristic
312	C₂	C₄×#88076	1	1	yes	characteristic
313	C₂	C₄×#88077	1	1	yes	characteristic
314	C₂	#13114646	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊Dic₁₃
commutator	C₂×C₄×C₁₃	C₂×C₄²
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	#8227
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
21. #11718135	C₄	semidirect
251. C₁₃	#8227	semidirect

Maximal subgroups

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

subgroup	index
#13114585	2
C₄×#88043	2
C₂×#11718135	2
#13112717	2
C₄×C₁₃×#125	2
#8227	13