Subgroups — #15752780

Order 3328.

562 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13123469	C₂	1	1	yes	maximal, characteristic
2	#13117234	C₂	1	1	yes	maximal, characteristic
3	#13112702	C₂	1	4	yes	maximal, complemented
4	C₁₃×#2920	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₄²×C₁₃	C₂²	1	1	yes	characteristic
6	C₁₃×#432	C₄	1	2	yes	complemented
7	C₁₃×#432	C₂²	1	2	yes	—
8	C₁₃×#431	—	2	1	no	—
9	C₁₃×#431	—	2	1	no	—
10	C₁₃×#423	—	2	1	no	—
11	C₁₃×#407	—	2	1	no	—
12	C₄×C₁₃×M₁₆	C₄	1	2	yes	complemented
13	#11718873	C₄	1	2	yes	—
14	#11718872	—	2	1	no	—
15	C₄×C₂²⋊Dic₁₃	—	2	1	no	—
16	#11718868	—	2	1	no	—
17	#11718598	C₂²	1	2	yes	—
18	#11718597	—	2	1	no	—
19	C₄×C₄⋊Dic₁₃	—	2	1	no	—
20	#11718593	C₄	1	2	yes	—
21	#11718587	—	2	1	no	—
22	#11718587	—	2	1	no	—
23	#11718586	C₄	1	2	yes	complemented
24	#11718586	C₄	1	2	yes	complemented
25	#11718585	—	2	1	no	—
26	C₄×#6099	C₂²	1	2	yes	—
27	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
28	C₂³×C₄×C₁₃	D₄	1	1	yes	characteristic
29	C₂³×C₄×C₁₃	D₄	1	1	yes	characteristic
30	C₂×C₁₃×M₁₆	—	2	2	no	—
31	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄²×C₁₃	D₄	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	Q₈	1	1	yes	characteristic
41	C₂×C₄²×C₁₃	Q₈	1	1	yes	characteristic
42	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
43	C₂×C₂²⋊Dic₁₃	—	2	2	no	—
44	C₂×C₄⋊Dic₁₃	—	2	2	no	—
45	C₂×C₄×Dic₁₃	—	2	2	no	—
46	C₂×#6099	—	2	2	no	—
47	C₁₃×C₄⋊C₈	—	2	2	no	—
48	C₁₃×C₄⋊C₈	—	2	2	no	—
49	C₁₃×C₂²⋊C₈	—	2	2	no	—
50	C₁₃×C₂²⋊C₈	—	2	2	no	—
51	C₁₃×#97	C₂×C₄	1	4	yes	—
52	C₄×C₈×C₁₃	—	2	2	no	—
53	#88071	—	2	2	no	—
54	#88071	—	2	2	no	—
55	#88071	—	2	2	no	—
56	#88071	C₂×C₄	1	4	yes	—
57	C₂²⋊#1055	—	2	2	no	—
58	C₂²⋊#1055	—	2	2	no	—
59	C₄⋊#1055	—	2	2	no	—
60	C₄⋊#1055	—	2	2	no	—
61	#88043	—	2	2	no	—
62	C₄×#1055	C₂×C₄	1	4	yes	—
63	#7976	—	13	1	no	maximal, Sylow
64	C₂⁴×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
65	C₂²×C₄×C₁₃	—	4	2	no	—
66	C₂²×C₄×C₁₃	—	4	2	no	—
67	C₂²×C₄×C₁₃	—	4	2	no	—
68	C₂²×C₄×C₁₃	C₄²	1	1	yes	characteristic
69	C₂²×C₄×C₁₃	C₄²	1	1	yes	characteristic
70	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
71	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
72	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
73	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
74	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
75	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
76	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
77	C₂²×C₄×C₁₃	C₄⋊C₄	1	1	yes	characteristic
78	C₂²×C₄×C₁₃	C₄⋊C₄	1	1	yes	characteristic
79	C₂²×C₄×C₁₃	C₄⋊C₄	1	1	yes	characteristic
80	C₂²×C₄×C₁₃	C₄⋊C₄	1	1	yes	characteristic
81	C₂²×C₄×C₁₃	C₄⋊C₄	1	1	yes	characteristic
82	C₂²×C₄×C₁₃	C₄⋊C₄	1	1	yes	characteristic
83	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
84	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
85	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
86	C₂²×Dic₁₃	—	2	4	no	—
87	C₁₃×M₁₆	—	4	4	no	—
88	C₂×C₈×C₁₃	—	2	4	no	—
89	C₄²×C₁₃	C₄²	1	1	yes	characteristic
90	C₄²×C₁₃	C₄²	1	1	yes	characteristic
91	C₄²×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
92	C₄²×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
93	C₄²×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
94	C₄²×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
95	C₄²×C₁₃	C₄⋊C₄	1	1	yes	characteristic
96	C₄²×C₁₃	C₄⋊C₄	1	1	yes	characteristic
97	C₄²×C₁₃	C₄⋊C₄	1	1	yes	characteristic
98	C₄²×C₁₃	C₄⋊C₄	1	1	yes	characteristic
99	C₄²×C₁₃	C₄⋊C₄	1	1	yes	characteristic
100	C₄²×C₁₃	C₄⋊C₄	1	1	yes	characteristic
101	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
102	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
103	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
104	C₄²×C₁₃	C₂×Q₈	1	1	yes	characteristic
105	C₂²⋊Dic₁₃	—	4	4	no	—
106	C₄⋊Dic₁₃	—	4	4	no	—
107	C₄×Dic₁₃	—	4	4	no	—
108	#6099	—	4	4	no	—
109	C₂×#1055	—	2	4	no	—
110	#2920	—	13	1	no	—
111	#2920	D₁₃	1	1	yes	characteristic
112	#2283	—	13	1	no	—
113	#1288	—	13	4	no	—
114	C₂³×C₁₃	—	2	2	no	—
115	C₂³×C₁₃	—	4	1	no	—
116	C₂³×C₁₃	—	4	1	no	—
117	C₂³×C₁₃	#95	1	1	yes	characteristic
118	C₂³×C₁₃	#95	1	1	yes	characteristic
119	C₂³×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
120	C₂×C₄×C₁₃	—	2	2	no	—
121	C₂×C₄×C₁₃	—	2	2	no	—
122	C₂×C₄×C₁₃	—	2	2	no	—
123	C₂×C₄×C₁₃	—	2	2	no	—
124	C₂×C₄×C₁₃	—	2	2	no	—
125	C₂×C₄×C₁₃	—	2	2	no	—
126	C₂×C₄×C₁₃	—	2	2	no	—
127	C₂×C₄×C₁₃	—	2	2	no	—
128	C₂×C₄×C₁₃	—	4	1	no	—
129	C₂×C₄×C₁₃	—	4	1	no	—
130	C₂×C₄×C₁₃	—	4	1	no	—
131	C₂×C₄×C₁₃	—	4	1	no	—
132	C₂×C₄×C₁₃	—	4	1	no	—
133	C₂×C₄×C₁₃	—	4	1	no	—
134	C₂×C₄×C₁₃	—	4	1	no	—
135	C₂×C₄×C₁₃	—	4	1	no	—
136	C₂×C₄×C₁₃	—	4	1	no	—
137	C₂×C₄×C₁₃	—	4	1	no	—
138	C₂×C₄×C₁₃	#95	1	1	yes	characteristic
139	C₂×C₄×C₁₃	#95	1	1	yes	characteristic
140	C₂×C₄×C₁₃	#95	1	1	yes	characteristic
141	C₂×C₄×C₁₃	#95	1	1	yes	characteristic
142	C₂×C₄×C₁₃	#95	1	1	yes	characteristic
143	C₂×C₄×C₁₃	#95	1	1	yes	characteristic
144	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
145	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
146	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
147	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
148	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
149	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
150	C₂×Dic₁₃	—	4	4	no	—
151	C₂×Dic₁₃	—	4	4	no	—
152	C₂×Dic₁₃	—	4	4	no	—
153	C₈×C₁₃	—	4	4	no	—
154	#1055	—	4	4	no	—
155	C₂²×C₄²	C₂×D₁₃	1	1	yes	characteristic
156	#432	—	26	1	no	—
157	#432	—	26	1	no	—
158	#432	Dic₁₃	1	2	yes	complemented
159	#432	C₂×D₁₃	1	2	yes	—
160	#431	—	2	1	no	—
161	#431	—	2	1	no	—
162	#431	—	13	2	no	—
163	#431	—	13	2	no	—
164	#423	—	2	1	no	—
165	#423	—	26	1	no	—
166	#407	—	2	1	no	—
167	#407	—	26	1	no	—
168	C₄×M₁₆	—	13	2	no	—
169	C₄×M₁₆	Dic₁₃	1	2	yes	complemented
170	#383	—	13	2	no	—
171	#382	—	26	1	no	—
172	#381	—	26	1	no	—
173	#380	—	13	2	no	—
174	C₄×C₄⋊C₄	—	26	1	no	—
175	C₄×C₂²⋊C₄	—	26	1	no	—
176	#376	—	13	2	no	—
177	C₂²×C₁₃	—	2	1	no	—
178	C₂²×C₁₃	—	2	1	no	—
179	C₂²×C₁₃	—	2	1	no	—
180	C₂²×C₁₃	—	2	1	no	—
181	C₂²×C₁₃	—	2	1	no	—
182	C₂²×C₁₃	—	2	1	no	—
183	C₂²×C₁₃	—	2	1	no	—
184	C₂²×C₁₃	—	2	1	no	—
185	C₂²×C₁₃	—	4	4	no	—
186	C₂²×C₁₃	C₂×#95	1	1	yes	characteristic
187	C₂²×C₁₃	#421	1	2	yes	—
188	C₄×C₁₃	—	2	1	no	—
189	C₄×C₁₃	—	2	1	no	—
190	C₄×C₁₃	—	2	1	no	—
191	C₄×C₁₃	—	2	1	no	—
192	C₄×C₁₃	—	2	1	no	—
193	C₄×C₁₃	—	2	1	no	—
194	C₄×C₁₃	—	2	1	no	—
195	C₄×C₁₃	—	2	1	no	—
196	C₄×C₁₃	—	2	1	no	—
197	C₄×C₁₃	—	2	1	no	—
198	C₄×C₁₃	H(3,ℤ/4)	1	2	yes	—
199	C₄×C₁₃	#344	1	2	yes	—
200	Dic₁₃	—	8	4	no	—
201	C₂³×C₄	C₄×D₁₃	1	1	yes	characteristic
202	C₂³×C₄	D₅₂	1	1	yes	characteristic
203	C₂³×C₄	C₂²⋊D₁₃	1	1	yes	characteristic
204	C₂×M₁₆	—	2	2	no	—
205	C₂×M₁₆	—	26	2	no	—
206	C₂×C₄⋊C₄	—	26	2	no	—
207	C₂×C₂²⋊C₄	—	26	2	no	—
208	C₂×C₄²	—	26	2	no	—
209	C₂×C₄²	Dic₂₆	1	1	yes	characteristic
210	C₂×C₄²	Dic₂₆	1	1	yes	characteristic
211	C₂×C₄²	C₄×D₁₃	1	1	yes	characteristic
212	C₂×C₄²	C₄×D₁₃	1	1	yes	characteristic
213	C₂×C₄²	C₄×D₁₃	1	1	yes	characteristic
214	C₂×C₄²	D₅₂	1	1	yes	characteristic
215	C₂×C₄²	C₂×Dic₁₃	1	1	yes	characteristic
216	C₂×C₄²	C₂×Dic₁₃	1	1	yes	characteristic
217	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
218	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
219	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
220	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
221	C₄⋊C₈	—	2	2	no	—
222	C₄⋊C₈	—	2	2	no	—
223	C₄⋊C₈	—	26	2	no	—
224	C₄⋊C₈	—	26	2	no	—
225	C₂²⋊C₈	—	2	2	no	—
226	C₂²⋊C₈	—	2	2	no	—
227	C₂²⋊C₈	—	26	2	no	—
228	C₂²⋊C₈	—	26	2	no	—
229	#97	—	26	2	no	—
230	#97	C₂×Dic₁₃	1	4	yes	—
231	C₄×C₈	—	2	2	no	—
232	C₄×C₈	—	13	4	no	—
233	#95	—	13	4	no	—
234	#95	—	26	2	no	—
235	#95	—	26	2	no	—
236	#95	—	26	2	no	—
237	C₂×C₁₃	—	2	2	no	—
238	C₂×C₁₃	—	4	1	no	—
239	C₂×C₁₃	—	4	1	no	—
240	C₂×C₁₃	C₂×H(3,ℤ/4)	1	1	yes	characteristic
241	C₂×C₁₃	#1733	1	1	yes	characteristic
242	C₂×C₁₃	C₂×#344	1	1	yes	characteristic
243	C₂⁴	#6103	1	1	yes	characteristic
244	C₂²×C₄	—	4	2	no	—
245	C₂²×C₄	—	4	2	no	—
246	C₂²×C₄	—	4	2	no	—
247	C₂²×C₄	—	26	4	no	—
248	C₂²×C₄	C₄×Dic₁₃	1	1	yes	characteristic
249	C₂²×C₄	C₄×Dic₁₃	1	1	yes	characteristic
250	C₂²×C₄	#6101	1	1	yes	characteristic
251	C₂²×C₄	#6101	1	1	yes	characteristic
252	C₂²×C₄	#6101	1	1	yes	characteristic
253	C₂²×C₄	#6101	1	1	yes	characteristic
254	C₂²×C₄	C₄⋊Dic₁₃	1	1	yes	characteristic
255	C₂²×C₄	C₄⋊Dic₁₃	1	1	yes	characteristic
256	C₂²×C₄	#6103	1	1	yes	characteristic
257	C₂²×C₄	#6103	1	1	yes	characteristic
258	C₂²×C₄	#6103	1	1	yes	characteristic
259	C₂²×C₄	#6103	1	1	yes	characteristic
260	C₂²×C₄	#6103	1	1	yes	characteristic
261	C₂²×C₄	C₂²⋊Dic₁₃	1	1	yes	characteristic
262	C₂²×C₄	C₂²⋊Dic₁₃	1	1	yes	characteristic
263	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
264	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
265	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
266	M₁₆	—	4	4	no	—
267	M₁₆	—	52	4	no	—
268	C₂×C₈	—	2	4	no	—
269	C₂×C₈	—	26	4	no	—
270	C₄⋊C₄	—	52	4	no	—
271	C₂²⋊C₄	—	52	4	no	—
272	C₄²	—	52	4	no	—
273	C₄²	C₄×Dic₁₃	1	1	yes	characteristic
274	C₄²	C₄×Dic₁₃	1	1	yes	characteristic
275	C₄²	#6101	1	1	yes	characteristic
276	C₄²	#6101	1	1	yes	characteristic
277	C₄²	#6101	1	1	yes	characteristic
278	C₄²	#6101	1	1	yes	characteristic
279	C₄²	C₄⋊Dic₁₃	1	1	yes	characteristic
280	C₄²	C₄⋊Dic₁₃	1	1	yes	characteristic
281	C₄²	#6103	1	1	yes	characteristic
282	C₄²	#6103	1	1	yes	characteristic
283	C₄²	C₂²⋊Dic₁₃	1	1	yes	characteristic
284	C₄²	C₂²⋊Dic₁₃	1	1	yes	characteristic
285	C₄²	C₂×Dic₂₆	1	1	yes	characteristic
286	C₄²	C₂×C₄×D₁₃	1	1	yes	characteristic
287	C₄²	C₂²×Dic₁₃	1	1	yes	characteristic
288	C₄²	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
289	C₁₃	#7976	1	1	yes	characteristic, complemented, Sylow
290	C₂³	—	2	2	no	—
291	C₂³	—	4	1	no	—
292	C₂³	—	4	1	no	—
293	C₂³	#88071	1	1	yes	characteristic
294	C₂³	#88071	1	1	yes	characteristic
295	C₂³	C₂×#6103	1	1	yes	characteristic
296	C₂×C₄	—	2	2	no	—
297	C₂×C₄	—	2	2	no	—
298	C₂×C₄	—	2	2	no	—
299	C₂×C₄	—	2	2	no	—
300	C₂×C₄	—	2	2	no	—
301	C₂×C₄	—	2	2	no	—
302	C₂×C₄	—	2	2	no	—
303	C₂×C₄	—	2	2	no	—
304	C₂×C₄	—	4	1	no	—
305	C₂×C₄	—	4	1	no	—
306	C₂×C₄	—	4	1	no	—
307	C₂×C₄	—	4	1	no	—
308	C₂×C₄	—	4	1	no	—
309	C₂×C₄	—	4	1	no	—
310	C₂×C₄	—	4	1	no	—
311	C₂×C₄	—	4	1	no	—
312	C₂×C₄	—	4	1	no	—
313	C₂×C₄	—	4	1	no	—
314	C₂×C₄	—	52	4	no	—
315	C₂×C₄	—	52	4	no	—
316	C₂×C₄	—	52	4	no	—
317	C₂×C₄	#88071	1	1	yes	characteristic
318	C₂×C₄	#88071	1	1	yes	characteristic
319	C₂×C₄	#88071	1	1	yes	characteristic
320	C₂×C₄	#88071	1	1	yes	characteristic
321	C₂×C₄	#88071	1	1	yes	characteristic
322	C₂×C₄	#88071	1	1	yes	characteristic
323	C₂×C₄	C₂×C₄×Dic₁₃	1	1	yes	characteristic
324	C₂×C₄	C₂×#6101	1	1	yes	characteristic
325	C₂×C₄	C₂×#6101	1	1	yes	characteristic
326	C₂×C₄	C₂×C₄⋊Dic₁₃	1	1	yes	characteristic
327	C₂×C₄	C₂×#6103	1	1	yes	characteristic
328	C₂×C₄	C₂×C₂²⋊Dic₁₃	1	1	yes	characteristic
329	C₈	—	4	4	no	—
330	C₈	—	52	4	no	—
331	C₂²	—	2	1	no	—
332	C₂²	—	2	1	no	—
333	C₂²	—	2	1	no	—
334	C₂²	—	2	1	no	—
335	C₂²	—	2	1	no	—
336	C₂²	—	2	1	no	—
337	C₂²	—	2	1	no	—
338	C₂²	—	2	1	no	—
339	C₂²	—	4	4	no	—
340	C₂²	#11718800	1	2	yes	—
341	C₂²	C₂×#88071	1	1	yes	characteristic
342	C₄	—	2	1	no	—
343	C₄	—	2	1	no	—
344	C₄	—	2	1	no	—
345	C₄	—	2	1	no	—
346	C₄	—	2	1	no	—
347	C₄	—	2	1	no	—
348	C₄	—	2	1	no	—
349	C₄	—	2	1	no	—
350	C₄	—	2	1	no	—
351	C₄	—	2	1	no	—
352	C₄	—	104	4	no	—
353	C₄	#11718123	1	2	yes	—
354	C₄	#11718151	1	2	yes	—
355	C₂	—	2	2	no	—
356	C₂	—	4	1	no	—
357	C₂	—	4	1	no	—
358	C₂	C₂×#11718123	1	1	yes	characteristic
359	C₂	#13114547	1	1	yes	characteristic
360	C₂	C₂×#11718151	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#88071
commutator	C₂×C₄×C₁₃	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₁₃
Fitting	C₁₃×#2920	C₂
socle	C₂²×C₁₃	C₂×#95
prime radical	C₂⁴×C₁₃	C₂²⋊C₄
cosocle	C₂×C₄²×C₁₃	C₂³
hypercenter	#2920	D₁₃
hypocenter	C₁₃	#7976
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ #2920

Complements

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

N	H ≅ G/N	type
3. #13112702	C₂	semidirect
6. C₁₃×#432	C₄	semidirect
12. C₄×C₁₃×M₁₆	C₄	semidirect
23. #11718586	C₄	semidirect
24. #11718586	C₄	semidirect
158. #432	Dic₁₃	semidirect
169. C₄×M₁₆	Dic₁₃	semidirect
289. C₁₃	#7976	semidirect

Maximal subgroups

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

subgroup	index
#13123469	2
#13117234	2
#13112702	2
C₁₃×#2920	2
#7976	13