Subgroups — #15828632

Order 3328.

806 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13123467	C₂	1	2	yes	maximal
2	C₄×M₁₆×D₁₃	C₂	1	1	yes	maximal, characteristic
3	#13113139	C₂	1	2	yes	maximal, complemented
4	#13113139	C₂	1	1	yes	maximal, characteristic, complemented
5	#13113134	C₂	1	1	yes	maximal, characteristic, complemented
6	#13113132	C₂	1	2	yes	maximal, complemented
7	#13113130	C₂	1	1	yes	maximal, characteristic, complemented
8	#13113127	C₂	1	2	yes	maximal, complemented
9	#13113121	C₂	1	1	yes	maximal, characteristic, complemented
10	#13113118	C₂	1	1	yes	maximal, characteristic, complemented
11	C₁₃×#1455	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×M₁₆×D₁₃	C₄	1	2	yes	—
13	C₂×C₄²×D₁₃	C₂²	1	1	yes	characteristic
14	C₄×C₁₃×M₁₆	C₂²	1	2	yes	—
15	C₄×C₁₃×M₁₆	C₂²	1	1	yes	characteristic
16	#11718787	C₄	1	4	yes	—
17	#11718779	C₄	1	4	yes	—
18	M₁₆×Dic₁₃	C₄	1	2	yes	—
19	C₄×#6099	C₂²	1	2	yes	—
20	C₄×#6099	C₂²	1	1	yes	characteristic
21	#11718368	C₄	1	2	yes	—
22	#11718366	C₄	1	4	yes	—
23	#11718366	C₄	1	4	yes	—
24	D₁₃×#97	C₄	1	2	yes	—
25	C₄×#6094	C₂²	1	1	yes	characteristic
26	C₄×#6094	C₂²	1	1	yes	characteristic
27	#11718346	C₂²	1	2	yes	—
28	#11718346	C₂²	1	2	yes	—
29	#11718346	C₂²	1	2	yes	—
30	#11718346	C₂²	1	2	yes	—
31	C₄×C₈×D₁₃	C₂²	1	1	yes	characteristic
32	C₄×C₈×D₁₃	C₂²	1	1	yes	characteristic
33	C₁₃×#322	C₂²	1	1	yes	characteristic, complemented
34	C₁₃×#322	C₂²	1	2	yes	complemented
35	C₈²×C₁₃	C₂²	1	1	yes	characteristic, complemented
36	#11718047	C₂²	1	1	yes	characteristic, complemented
37	#11718047	C₂²	1	2	yes	complemented
38	#11718046	C₂²	1	2	yes	complemented
39	#11718046	C₂²	1	2	yes	complemented
40	#11718046	C₂²	1	2	yes	complemented
41	C₈×#1055	C₂²	1	1	yes	characteristic, complemented
42	C₈×#1055	C₂²	1	2	yes	complemented
43	C₂²×C₄×D₁₃	C₂×C₄	1	1	yes	characteristic
44	C₂²×C₄×D₁₃	C₂×C₄	1	2	yes	—
45	C₂×C₁₃×M₁₆	C₂×C₄	1	2	yes	—
46	C₂×C₁₃×M₁₆	C₂×C₄	1	4	yes	—
47	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
48	C₂×C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
49	C₂×C₄×Dic₁₃	C₂×C₄	1	2	yes	—
50	C₂×#6099	C₂×C₄	1	2	yes	—
51	C₂×#6099	C₂×C₄	1	4	yes	—
52	M₁₆×D₁₃	—	2	8	no	—
53	C₂×#6094	C₂×C₄	1	2	yes	—
54	C₂×#6094	C₂×C₄	1	2	yes	—
55	C₂×C₈×D₁₃	C₂×C₄	1	2	yes	—
56	C₂×C₈×D₁₃	C₂×C₄	1	2	yes	—
57	C₄²×D₁₃	C₂×C₄	1	1	yes	characteristic
58	C₄²×D₁₃	C₂×C₄	1	2	yes	—
59	C₄²×D₁₃	C₂×C₄	1	2	yes	—
60	C₄²×D₁₃	C₂×C₄	1	1	yes	characteristic
61	C₄²×D₁₃	C₂³	1	1	yes	characteristic
62	C₄²×D₁₃	C₂³	1	1	yes	characteristic
63	C₁₃×#97	C₂×C₄	1	2	yes	—
64	C₁₃×#97	C₂×C₄	1	4	yes	—
65	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
66	C₄×C₈×C₁₃	C₂³	1	2	yes	—
67	C₄×C₈×C₁₃	C₂³	1	2	yes	—
68	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
69	#88060	C₂×C₄	1	4	yes	—
70	#88060	C₂×C₄	1	4	yes	—
71	#88060	C₂×C₄	1	4	yes	—
72	#88060	C₂×C₄	1	4	yes	—
73	#88055	C₂×C₄	1	2	yes	—
74	#88055	C₂×C₄	1	2	yes	—
75	#88054	C₂×C₄	1	4	yes	—
76	#88054	C₂×C₄	1	4	yes	—
77	#88054	C₂×C₄	1	4	yes	—
78	#88054	C₂×C₄	1	4	yes	—
79	C₈×Dic₁₃	C₂×C₄	1	2	yes	—
80	C₈×Dic₁₃	C₂×C₄	1	2	yes	—
81	#88043	C₂×C₄	1	2	yes	—
82	#88043	C₂×C₄	1	4	yes	—
83	C₄×#1055	C₂³	1	1	yes	characteristic
84	C₄×#1055	C₂³	1	2	yes	—
85	C₄×#1055	C₂³	1	2	yes	—
86	C₄×#1055	C₂³	1	1	yes	characteristic
87	#17334	—	13	1	no	maximal, Sylow
88	C₂³×D₁₃	C₄²	1	1	yes	characteristic
89	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
90	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
91	C₂²×Dic₁₃	C₄²	1	1	yes	characteristic
92	C₂²×Dic₁₃	C₄²	1	2	yes	—
93	C₂×C₄×D₁₃	—	2	4	no	—
94	C₂×C₄×D₁₃	—	4	4	no	—
95	C₂×C₄×D₁₃	C₄²	1	1	yes	characteristic
96	C₂×C₄×D₁₃	C₄²	1	1	yes	characteristic
97	C₂×C₄×D₁₃	C₄²	1	2	yes	—
98	C₂×C₄×D₁₃	C₄²	1	2	yes	—
99	C₂×C₄×D₁₃	C₂²×C₄	1	1	yes	characteristic
100	C₂×C₄×D₁₃	C₂²×C₄	1	2	yes	—
101	C₂×C₄×D₁₃	C₂²×C₄	1	2	yes	—
102	C₂×C₄×D₁₃	C₂²×C₄	1	1	yes	characteristic
103	C₁₃×M₁₆	—	2	4	no	—
104	C₁₃×M₁₆	—	4	4	no	—
105	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
106	C₂×C₈×C₁₃	C₂²×C₄	1	4	yes	—
107	C₂×C₈×C₁₃	C₂²×C₄	1	4	yes	—
108	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
109	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
110	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
111	C₄²×C₁₃	C₂⁴	1	1	yes	characteristic
112	C₄×Dic₁₃	C₄²	1	2	yes	—
113	C₄×Dic₁₃	C₄²	1	2	yes	—
114	C₄×Dic₁₃	C₄²	1	2	yes	—
115	C₄×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
116	C₄×Dic₁₃	C₂²×C₄	1	2	yes	—
117	C₄×Dic₁₃	C₂²×C₄	1	2	yes	—
118	C₄×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
119	#6099	—	2	4	no	—
120	#6099	—	4	4	no	—
121	C₂×#1055	C₂²×C₄	1	2	yes	—
122	C₂×#1055	C₂²×C₄	1	4	yes	—
123	C₂×#1055	C₂²×C₄	1	4	yes	—
124	C₂×#1055	C₂²×C₄	1	2	yes	—
125	#6094	—	2	4	no	—
126	#6094	—	2	4	no	—
127	C₈×D₁₃	—	2	4	no	—
128	C₈×D₁₃	M₁₆	1	8	yes	complemented
129	#2920	—	13	2	no	—
130	C₂×C₄×M₁₆	—	13	1	no	—
131	#1455	—	13	2	no	—
132	#1455	—	13	1	no	—
133	#1455	D₁₃	1	1	yes	characteristic, complemented
134	#1453	—	13	2	no	—
135	#1453	—	13	1	no	—
136	#1453	—	13	1	no	—
137	#1452	—	13	2	no	—
138	#1450	—	13	1	no	—
139	#1450	—	13	1	no	—
140	C₂³×C₁₃	C₂×C₄²	1	1	yes	characteristic
141	C₂²×D₁₃	—	2	2	no	—
142	C₂²×D₁₃	—	4	2	no	—
143	C₂²×D₁₃	C₂×C₄²	1	1	yes	characteristic
144	C₂²×D₁₃	C₂×C₄²	1	1	yes	characteristic
145	C₂×C₄×C₁₃	—	2	2	no	—
146	C₂×C₄×C₁₃	—	2	4	no	—
147	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
148	C₂×C₄×C₁₃	C₂×C₄²	1	2	yes	—
149	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
150	C₂×C₄×C₁₃	C₂³×C₄	1	2	yes	—
151	C₂×Dic₁₃	—	2	2	no	—
152	C₂×Dic₁₃	—	4	2	no	—
153	C₂×Dic₁₃	C₂×C₄²	1	1	yes	characteristic
154	C₂×Dic₁₃	C₂×C₄²	1	2	yes	—
155	C₂×Dic₁₃	C₂×C₄²	1	2	yes	—
156	C₂×Dic₁₃	C₂×C₄²	1	1	yes	characteristic
157	C₄×D₁₃	—	2	2	no	—
158	C₄×D₁₃	—	2	2	no	—
159	C₄×D₁₃	—	2	4	no	—
160	C₄×D₁₃	—	2	4	no	—
161	C₄×D₁₃	—	2	2	no	—
162	C₄×D₁₃	—	4	2	no	—
163	C₄×D₁₃	—	4	2	no	—
164	C₄×D₁₃	C₂×M₁₆	1	4	yes	—
165	C₈×C₁₃	—	2	2	no	—
166	C₈×C₁₃	—	2	4	no	—
167	C₈×C₁₃	—	2	4	no	—
168	C₈×C₁₃	C₂×M₁₆	1	4	yes	—
169	#1055	—	2	2	no	—
170	#1055	—	2	4	no	—
171	#1055	—	2	4	no	—
172	#1055	C₂×M₁₆	1	4	yes	—
173	C₂²×M₁₆	—	13	2	no	—
174	C₂²×C₄²	—	13	1	no	—
175	#432	—	13	4	no	—
176	#432	—	13	4	no	—
177	#431	—	13	2	no	—
178	#431	—	13	2	no	—
179	#431	—	13	2	no	—
180	#431	—	13	2	no	—
181	#423	—	13	4	no	—
182	#407	—	13	4	no	—
183	C₄×M₁₆	—	13	2	no	—
184	C₄×M₁₆	—	13	1	no	—
185	C₄×M₁₆	—	13	2	no	—
186	C₄×M₁₆	—	13	2	no	—
187	C₄×M₁₆	—	13	1	no	—
188	C₄×M₁₆	—	13	1	no	—
189	C₄×M₁₆	C₂×D₁₃	1	1	yes	characteristic
190	C₄×M₁₆	C₂×D₁₃	1	2	yes	—
191	C₂×#97	—	13	2	no	—
192	C₂×C₄×C₈	—	13	1	no	—
193	C₂×C₄×C₈	—	13	1	no	—
194	#322	—	13	2	no	—
195	#322	—	13	2	no	—
196	#322	—	13	2	no	—
197	#322	—	13	1	no	—
198	#322	—	13	2	no	—
199	#322	C₂×D₁₃	1	1	yes	characteristic, complemented
200	#322	C₂×D₁₃	1	2	yes	complemented
201	C₈²	—	13	2	no	—
202	C₈²	—	13	1	no	—
203	C₈²	C₂×D₁₃	1	1	yes	characteristic, complemented
204	C₂²×C₁₃	—	2	1	no	—
205	C₂²×C₁₃	—	2	2	no	—
206	C₂²×C₁₃	C₂²×C₄²	1	1	yes	characteristic
207	C₂×D₁₃	—	2	2	no	—
208	C₂×D₁₃	—	2	1	no	—
209	C₂×D₁₃	—	2	2	no	—
210	C₂×D₁₃	—	4	4	no	—
211	C₂×D₁₃	C₄×M₁₆	1	2	yes	—
212	C₄×C₁₃	—	2	1	no	—
213	C₄×C₁₃	—	2	2	no	—
214	C₄×C₁₃	C₂²×M₁₆	1	2	yes	—
215	C₄×C₁₃	#568	1	4	yes	—
216	Dic₁₃	—	2	1	no	—
217	Dic₁₃	—	2	2	no	—
218	Dic₁₃	—	2	2	no	—
219	Dic₁₃	C₄×M₁₆	1	2	yes	—
220	C₂³×C₄	—	13	1	no	—
221	C₂³×C₄	—	13	2	no	—
222	C₂×M₁₆	—	13	2	no	—
223	C₂×M₁₆	—	13	4	no	—
224	C₂×M₁₆	—	13	2	no	—
225	C₂×M₁₆	—	13	2	no	—
226	C₂×M₁₆	—	26	8	no	—
227	C₂×M₁₆	C₄×D₁₃	1	2	yes	—
228	C₂×M₁₆	C₄×D₁₃	1	4	yes	—
229	C₂²×C₈	—	13	2	no	—
230	C₂²×C₈	—	13	2	no	—
231	C₂×C₄²	—	13	1	no	—
232	C₂×C₄²	—	13	2	no	—
233	C₂×C₄²	—	13	1	no	—
234	C₂×C₄²	—	13	1	no	—
235	C₂×C₄²	—	13	2	no	—
236	C₂×C₄²	—	13	2	no	—
237	C₂×C₄²	—	13	1	no	—
238	C₂×C₄²	—	13	1	no	—
239	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
240	C₄⋊C₈	—	13	4	no	—
241	C₄⋊C₈	—	13	4	no	—
242	C₄⋊C₈	—	13	4	no	—
243	C₄⋊C₈	—	13	4	no	—
244	C₂²⋊C₈	—	13	4	no	—
245	C₂²⋊C₈	—	13	4	no	—
246	C₂²⋊C₈	—	13	4	no	—
247	C₂²⋊C₈	—	13	4	no	—
248	#97	—	13	2	no	—
249	#97	—	13	2	no	—
250	#97	—	13	2	no	—
251	#97	—	13	4	no	—
252	#97	C₄×D₁₃	1	2	yes	—
253	#97	C₄×D₁₃	1	4	yes	—
254	C₄×C₈	—	13	2	no	—
255	C₄×C₈	—	13	2	no	—
256	C₄×C₈	—	13	1	no	—
257	C₄×C₈	—	13	2	no	—
258	C₄×C₈	—	13	2	no	—
259	C₄×C₈	—	13	1	no	—
260	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
261	C₄×C₈	C₂²×D₁₃	1	2	yes	—
262	C₄×C₈	C₂²×D₁₃	1	2	yes	—
263	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
264	C₂×C₁₃	—	4	1	no	—
265	C₂×C₁₃	C₂×C₄×M₁₆	1	1	yes	characteristic
266	C₂×C₁₃	#2873	1	2	yes	—
267	D₁₃	—	2	2	no	—
268	D₁₃	—	4	1	no	—
269	C₂⁴	—	13	1	no	—
270	C₂²×C₄	—	13	1	no	—
271	C₂²×C₄	—	13	1	no	—
272	C₂²×C₄	—	13	2	no	—
273	C₂²×C₄	—	13	1	no	—
274	C₂²×C₄	—	13	1	no	—
275	C₂²×C₄	—	13	2	no	—
276	C₂²×C₄	—	13	2	no	—
277	C₂²×C₄	—	13	1	no	—
278	C₂²×C₄	—	13	2	no	—
279	C₂²×C₄	—	13	2	no	—
280	C₂²×C₄	—	26	4	no	—
281	C₂²×C₄	—	52	4	no	—
282	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
283	C₂²×C₄	C₂×C₄×D₁₃	1	2	yes	—
284	M₁₆	—	2	4	no	—
285	M₁₆	—	4	4	no	—
286	M₁₆	—	26	4	no	—
287	M₁₆	—	26	4	no	—
288	M₁₆	—	26	4	no	—
289	M₁₆	—	52	4	no	—
290	C₂×C₈	—	13	2	no	—
291	C₂×C₈	—	13	8	no	—
292	C₂×C₈	—	13	2	no	—
293	C₂×C₈	—	13	4	no	—
294	C₂×C₈	—	13	4	no	—
295	C₂×C₈	—	26	4	no	—
296	C₂×C₈	C₂×C₄×D₁₃	1	2	yes	—
297	C₂×C₈	C₂×C₄×D₁₃	1	4	yes	—
298	C₂×C₈	C₂×C₄×D₁₃	1	4	yes	—
299	C₂×C₈	C₂×C₄×D₁₃	1	2	yes	—
300	C₄²	—	13	2	no	—
301	C₄²	—	13	2	no	—
302	C₄²	—	13	2	no	—
303	C₄²	—	13	1	no	—
304	C₄²	—	13	1	no	—
305	C₄²	—	13	2	no	—
306	C₄²	—	13	2	no	—
307	C₄²	C₂×C₄×D₁₃	1	1	yes	characteristic
308	C₄²	C₂×C₄×D₁₃	1	2	yes	—
309	C₄²	C₂³×D₁₃	1	1	yes	characteristic
310	C₁₃	#17334	1	1	yes	characteristic, complemented, Sylow
311	C₂³	—	13	1	no	—
312	C₂³	—	13	1	no	—
313	C₂³	—	26	2	no	—
314	C₂³	—	52	2	no	—
315	C₂³	C₄²×D₁₃	1	1	yes	characteristic
316	C₂×C₄	—	2	2	no	—
317	C₂×C₄	—	2	4	no	—
318	C₂×C₄	—	13	1	no	—
319	C₂×C₄	—	13	4	no	—
320	C₂×C₄	—	13	1	no	—
321	C₂×C₄	—	13	2	no	—
322	C₂×C₄	—	13	2	no	—
323	C₂×C₄	—	26	2	no	—
324	C₂×C₄	—	26	2	no	—
325	C₂×C₄	—	26	2	no	—
326	C₂×C₄	—	26	2	no	—
327	C₂×C₄	—	26	4	no	—
328	C₂×C₄	—	26	4	no	—
329	C₂×C₄	—	52	2	no	—
330	C₂×C₄	—	52	2	no	—
331	C₂×C₄	—	52	2	no	—
332	C₂×C₄	C₄²×D₁₃	1	1	yes	characteristic
333	C₂×C₄	C₄²×D₁₃	1	2	yes	—
334	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
335	C₂×C₄	C₂²×C₄×D₁₃	1	2	yes	—
336	C₈	—	2	2	no	—
337	C₈	—	2	4	no	—
338	C₈	—	2	4	no	—
339	C₈	—	13	4	no	—
340	C₈	—	26	2	no	—
341	C₈	—	26	4	no	—
342	C₈	—	26	4	no	—
343	C₈	M₁₆×D₁₃	1	4	yes	—
344	C₂²	—	2	1	no	—
345	C₂²	—	2	2	no	—
346	C₂²	—	13	2	no	—
347	C₂²	—	26	2	no	—
348	C₂²	—	26	1	no	—
349	C₂²	—	26	2	no	—
350	C₂²	—	52	4	no	—
351	C₂²	C₂×C₄²×D₁₃	1	1	yes	characteristic
352	C₄	—	2	1	no	—
353	C₄	—	2	2	no	—
354	C₄	—	13	2	no	—
355	C₄	—	26	1	no	—
356	C₄	—	26	2	no	—
357	C₄	—	26	2	no	—
358	C₄	C₂×M₁₆×D₁₃	1	2	yes	—
359	C₄	#11719450	1	4	yes	—
360	C₂	—	4	1	no	—
361	C₂	—	26	2	no	—
362	C₂	—	52	1	no	—
363	C₂	C₄×M₁₆×D₁₃	1	1	yes	characteristic
364	C₂	#13123253	1	2	yes	—

	subgroup	G/H
center, Frattini	C₄²	C₂³×D₁₃
commutator, socle	C₂²×C₁₃	C₂²×C₄²
Fitting	C₁₃×#1455	C₂
prime radical	C₂³×D₁₃	C₄²
cosocle	C₄²×C₁₃	C₂⁴
hypercenter	#1455	D₁₃
hypocenter	C₁₃	#17334
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₄² ◁ #1455

Complements

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

N	H ≅ G/N	type
3. #13113139	C₂	semidirect
4. #13113139	C₂	semidirect
5. #13113134	C₂	semidirect
6. #13113132	C₂	semidirect
7. #13113130	C₂	semidirect
8. #13113127	C₂	semidirect
9. #13113121	C₂	semidirect
10. #13113118	C₂	semidirect
11. C₁₃×#1455	C₂	semidirect
33. C₁₃×#322	C₂²	semidirect
34. C₁₃×#322	C₂²	semidirect
35. C₈²×C₁₃	C₂²	semidirect
36. #11718047	C₂²	semidirect
37. #11718047	C₂²	semidirect
38. #11718046	C₂²	semidirect
39. #11718046	C₂²	semidirect
40. #11718046	C₂²	semidirect
41. C₈×#1055	C₂²	semidirect
42. C₈×#1055	C₂²	semidirect
128. C₈×D₁₃	M₁₆	semidirect
133. #1455	D₁₃	semidirect
199. #322	C₂×D₁₃	semidirect
200. #322	C₂×D₁₃	semidirect
203. C₈²	C₂×D₁₃	semidirect
310. C₁₃	#17334	semidirect

Maximal subgroups

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

subgroup	index
#13123467	2
C₄×M₁₆×D₁₃	2
#13113139	2
#13113134	2
#13113132	2
#13113130	2
#13113127	2
#13113121	2
#13113118	2
C₁₃×#1455	2
#17334	13