Subgroups — #15770937

Order 3328.

686 conjugacy classes of subgroups.

The subgroup lattice has 344 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13123400	C₂	1	1	yes	maximal, characteristic
2	#13114569	C₂	1	1	yes	maximal, characteristic
3	#13114555	C₂	1	2	yes	maximal
4	C₂×#11718123	C₂	1	2	yes	maximal
5	C₁₃×#2317	C₂	1	1	yes	maximal, characteristic
6	C₂×C₁₃×#117	C₂²	1	1	yes	characteristic
7	C₂×C₁₃×#117	C₂²	1	1	yes	characteristic
8	C₂²×C₁₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
9	C₂×#88180	C₂²	1	2	yes	—
10	C₂²×#6099	C₂²	1	2	yes	—
11	C₁₃×#389	—	2	2	no	—
12	C₁₃×#385	—	2	2	no	—
13	C₁₃×#384	C₄	1	2	yes	complemented
14	C₁₃×#382	C₄	1	2	yes	complemented
15	#11718660	—	2	4	no	—
16	#11718585	—	2	2	no	—
17	#11718123	—	2	4	no	—
18	#11718121	—	2	4	no	—
19	#11718118	—	2	2	no	—
20	C₂³×C₄×C₁₃	—	2	1	no	—
21	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
22	C₁₃×#117	—	2	2	no	—
23	C₁₃×#117	—	2	1	no	—
24	C₁₃×#117	—	2	1	no	—
25	C₁₃×#117	—	2	2	no	—
26	C₁₃×#117	D₄	1	2	yes	—
27	C₁₃×#117	Q₈	1	2	yes	—
28	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
29	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
30	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
31	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
32	C₂×C₁₃×C₄⋊C₄	—	2	4	no	—
33	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
34	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
35	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
36	C₂×C₁₃×C₂²⋊C₄	—	2	1	no	—
37	C₂×C₁₃×C₂²⋊C₄	D₄	1	2	yes	—
38	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
39	C₂×C₄²×C₁₃	D₄	1	2	yes	—
40	C₂×C₂²⋊Dic₁₃	—	2	2	no	—
41	#88180	—	2	2	no	—
42	#88180	—	2	2	no	—
43	#88180	—	2	2	no	—
44	#88180	Q₈	1	4	yes	—
45	C₂×C₄⋊Dic₁₃	—	2	2	no	—
46	C₂×C₄×Dic₁₃	—	2	2	no	—
47	C₂×#6099	—	2	2	no	—
48	C₂×#6099	—	2	2	no	—
49	C₂×#6099	—	2	2	no	—
50	C₂×#6099	—	2	2	no	—
51	C₂×#6099	—	2	2	no	—
52	C₂×#6099	—	2	2	no	—
53	C₂²×#1055	D₄	1	4	yes	—
54	C₁₃×#95	—	2	4	no	—
55	C₄⋊#1055	—	2	8	no	—
56	#11225	—	13	1	no	maximal, Sylow
57	C₂⁴×C₁₃	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄×C₁₃	—	2	2	no	—
59	C₂²×C₄×C₁₃	—	2	2	no	—
60	C₂²×C₄×C₁₃	—	4	2	no	—
61	C₂²×C₄×C₁₃	—	8	2	no	—
62	C₂²×C₄×C₁₃	C₂²⋊C₄	1	2	yes	—
63	C₂²×C₄×C₁₃	C₂²⋊C₄	1	2	yes	—
64	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
65	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
67	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₁₃	C₂×Q₈	1	1	yes	characteristic
69	C₂²×C₄×C₁₃	C₂×Q₈	1	2	yes	—
70	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
71	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
72	C₂²×C₄×C₁₃	#41	1	2	yes	—
73	C₂²×C₄×C₁₃	#41	1	2	yes	—
74	C₂²×Dic₁₃	—	4	2	no	—
75	C₁₃×C₄⋊C₄	—	2	2	no	—
76	C₁₃×C₄⋊C₄	—	2	2	no	—
77	C₁₃×C₄⋊C₄	—	2	2	no	—
78	C₁₃×C₄⋊C₄	—	2	2	no	—
79	C₁₃×C₄⋊C₄	—	2	4	no	—
80	C₁₃×C₄⋊C₄	—	4	1	no	—
81	C₁₃×C₄⋊C₄	—	4	1	no	—
82	C₁₃×C₄⋊C₄	—	8	4	no	—
83	C₁₃×C₂²⋊C₄	—	2	2	no	—
84	C₁₃×C₂²⋊C₄	—	2	4	no	—
85	C₁₃×C₂²⋊C₄	—	2	2	no	—
86	C₄²×C₁₃	—	2	2	no	—
87	C₄²×C₁₃	—	2	2	no	—
88	C₄²×C₁₃	—	2	2	no	—
89	C₄²×C₁₃	C₄⋊C₄	1	4	yes	—
90	C₂²⋊Dic₁₃	—	2	4	no	—
91	C₂²⋊Dic₁₃	—	4	2	no	—
92	C₄⋊Dic₁₃	—	2	4	no	—
93	C₄⋊Dic₁₃	—	4	2	no	—
94	C₄×Dic₁₃	—	2	4	no	—
95	C₄×Dic₁₃	—	4	2	no	—
96	#6099	—	4	2	no	—
97	#6099	—	4	2	no	—
98	#6099	—	4	2	no	—
99	#6099	—	4	2	no	—
100	C₂×#1055	—	2	4	no	—
101	C₂×#1055	—	2	4	no	—
102	C₂×#1055	—	2	4	no	—
103	#2905	—	13	1	no	—
104	#2317	D₁₃	1	1	yes	characteristic
105	#1740	—	13	1	no	—
106	#1736	—	13	2	no	—
107	C₂×H(3,ℤ/4)	—	13	2	no	—
108	C₂³×C₁₃	—	4	1	no	—
109	C₂³×C₁₃	—	4	1	no	—
110	C₂³×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
111	C₂³×C₁₃	#121	1	2	yes	—
112	C₂³×C₁₃	#121	1	1	yes	characteristic
113	C₂³×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
114	C₂³×C₁₃	C₂²⋊Q₈	1	2	yes	—
115	C₂×C₄×C₁₃	—	2	4	no	—
116	C₂×C₄×C₁₃	—	2	2	no	—
117	C₂×C₄×C₁₃	—	2	2	no	—
118	C₂×C₄×C₁₃	—	4	2	no	—
119	C₂×C₄×C₁₃	—	4	2	no	—
120	C₂×C₄×C₁₃	—	4	2	no	—
121	C₂×C₄×C₁₃	—	4	2	no	—
122	C₂×C₄×C₁₃	—	4	4	no	—
123	C₂×C₄×C₁₃	—	4	2	no	—
124	C₂×C₄×C₁₃	—	8	4	no	—
125	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
126	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
127	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
128	C₂×C₄×C₁₃	C₄×Q₈	1	2	yes	—
129	C₂×C₄×C₁₃	#120	1	1	yes	characteristic
130	C₂×C₄×C₁₃	#121	1	2	yes	—
131	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
132	C₂×C₄×C₁₃	C₂²⋊Q₈	1	2	yes	—
133	C₂×C₄×C₁₃	C₂²⋊Q₈	1	2	yes	—
134	C₂×C₄×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
135	C₂×C₄×C₁₃	#123	1	2	yes	—
136	C₂×C₄×C₁₃	#123	1	1	yes	characteristic
137	C₂×C₄×C₁₃	#124	1	1	yes	characteristic
138	C₂×C₄×C₁₃	#125	1	2	yes	—
139	C₂×C₄×C₁₃	#125	1	2	yes	—
140	C₂×C₄×C₁₃	#126	1	2	yes	—
141	C₂×C₄×C₁₃	C₄⋊Q₈	1	2	yes	—
142	C₂×C₄×C₁₃	C₄⋊Q₈	1	1	yes	characteristic
143	C₂×Dic₁₃	—	4	4	no	—
144	C₂×Dic₁₃	—	8	2	no	—
145	C₂×Dic₁₃	—	8	2	no	—
146	#1055	—	4	4	no	—
147	C₂²×M₁₆	—	13	2	no	—
148	C₂×#117	—	13	2	no	—
149	C₂×#117	C₂×D₁₃	1	1	yes	characteristic
150	C₂×#117	C₂×D₁₃	1	1	yes	characteristic
151	C₂²×C₄⋊C₄	C₂×D₁₃	1	1	yes	characteristic
152	#424	—	26	4	no	—
153	#423	—	26	2	no	—
154	#389	—	2	2	no	—
155	#385	—	2	2	no	—
156	#384	Dic₁₃	1	2	yes	complemented
157	#382	Dic₁₃	1	2	yes	complemented
158	#342	—	26	2	no	—
159	#340	—	26	4	no	—
160	H(3,ℤ/4)	—	26	4	no	—
161	C₂²×C₁₃	—	2	2	no	—
162	C₂²×C₁₃	—	2	1	no	—
163	C₂²×C₁₃	—	2	2	no	—
164	C₂²×C₁₃	—	2	1	no	—
165	C₂²×C₁₃	—	4	1	no	—
166	C₂²×C₁₃	—	4	1	no	—
167	C₂²×C₁₃	—	4	2	no	—
168	C₂²×C₁₃	#380	1	1	yes	characteristic
169	C₂²×C₁₃	#386	1	1	yes	characteristic
170	C₂²×C₁₃	#391	1	1	yes	characteristic
171	C₂²×C₁₃	#396	1	2	yes	—
172	C₂²×C₁₃	#398	1	2	yes	—
173	C₄×C₁₃	—	4	4	no	—
174	C₄×C₁₃	—	4	2	no	—
175	C₄×C₁₃	—	4	2	no	—
176	C₄×C₁₃	—	8	2	no	—
177	C₄×C₁₃	#387	1	2	yes	—
178	C₄×C₁₃	#389	1	2	yes	—
179	C₄×C₁₃	#394	1	1	yes	characteristic
180	C₄×C₁₃	#395	1	1	yes	characteristic
181	C₄×C₁₃	#398	1	1	yes	characteristic
182	C₄×C₁₃	#400	1	1	yes	characteristic
183	Dic₁₃	—	8	4	no	—
184	C₂³×C₄	—	2	1	no	—
185	C₂³×C₄	C₂²×D₁₃	1	1	yes	characteristic
186	C₂×M₁₆	—	26	2	no	—
187	C₂×M₁₆	—	26	2	no	—
188	C₂×M₁₆	—	26	2	no	—
189	C₂×M₁₆	—	26	2	no	—
190	C₂×M₁₆	—	26	2	no	—
191	C₂×M₁₆	—	26	2	no	—
192	C₂²×C₈	—	13	4	no	—
193	#117	—	2	1	no	—
194	#117	—	2	1	no	—
195	#117	—	2	2	no	—
196	#117	—	2	2	no	—
197	#117	—	13	4	no	—
198	#117	—	26	2	no	—
199	#117	—	26	2	no	—
200	#117	—	26	2	no	—
201	#117	Dic₂₆	1	2	yes	—
202	#117	D₅₂	1	2	yes	—
203	C₂×C₄⋊C₄	—	2	1	no	—
204	C₂×C₄⋊C₄	—	2	2	no	—
205	C₂×C₄⋊C₄	—	2	2	no	—
206	C₂×C₄⋊C₄	—	2	1	no	—
207	C₂×C₄⋊C₄	—	2	4	no	—
208	C₂×C₄⋊C₄	—	26	2	no	—
209	C₂×C₄⋊C₄	C₂×Dic₁₃	1	2	yes	—
210	C₂×C₄⋊C₄	C₂×Dic₁₃	1	1	yes	characteristic
211	C₂×C₄⋊C₄	C₂×Dic₁₃	1	1	yes	characteristic
212	C₂×C₂²⋊C₄	—	2	1	no	—
213	C₂×C₂²⋊C₄	—	26	2	no	—
214	C₂×C₂²⋊C₄	C₂²⋊D₁₃	1	2	yes	—
215	C₂×C₄²	—	26	2	no	—
216	C₂×C₄²	C₂×Dic₁₃	1	2	yes	—
217	C₂×C₄²	C₂²⋊D₁₃	1	2	yes	—
218	C₄⋊C₈	—	26	8	no	—
219	#95	—	2	4	no	—
220	C₂×C₁₃	—	2	2	no	—
221	C₂×C₁₃	—	2	1	no	—
222	C₂×C₁₃	—	2	1	no	—
223	C₂×C₁₃	—	2	2	no	—
224	C₂×C₁₃	#1441	1	1	yes	characteristic
225	C₂×C₁₃	#1909	1	1	yes	characteristic
226	C₂×C₁₃	#1910	1	1	yes	characteristic
227	C₂⁴	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
228	C₂²×C₄	—	2	2	no	—
229	C₂²×C₄	—	2	2	no	—
230	C₂²×C₄	—	4	2	no	—
231	C₂²×C₄	—	8	2	no	—
232	C₂²×C₄	—	52	2	no	—
233	C₂²×C₄	C₂²⋊Dic₁₃	1	2	yes	—
234	C₂²×C₄	C₂²⋊Dic₁₃	1	2	yes	—
235	C₂²×C₄	C₂×Dic₂₆	1	1	yes	characteristic
236	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
237	C₂²×C₄	#6127	1	1	yes	characteristic
238	C₂²×C₄	#6127	1	1	yes	characteristic
239	C₂²×C₄	D₄×D₁₃	1	2	yes	—
240	C₂²×C₄	#6129	1	2	yes	—
241	C₂²×C₄	Q₈×D₁₃	1	2	yes	—
242	C₂²×C₄	#6131	1	2	yes	—
243	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
244	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
245	M₁₆	—	52	2	no	—
246	M₁₆	—	52	2	no	—
247	M₁₆	—	52	2	no	—
248	M₁₆	—	52	2	no	—
249	C₂×C₈	—	26	4	no	—
250	C₂×C₈	—	26	4	no	—
251	C₂×C₈	—	26	4	no	—
252	C₄⋊C₄	—	2	2	no	—
253	C₄⋊C₄	—	2	2	no	—
254	C₄⋊C₄	—	2	2	no	—
255	C₄⋊C₄	—	2	2	no	—
256	C₄⋊C₄	—	2	4	no	—
257	C₄⋊C₄	—	4	1	no	—
258	C₄⋊C₄	—	4	1	no	—
259	C₄⋊C₄	—	8	4	no	—
260	C₄⋊C₄	—	26	4	no	—
261	C₄⋊C₄	—	52	2	no	—
262	C₂²⋊C₄	—	2	2	no	—
263	C₂²⋊C₄	—	2	2	no	—
264	C₂²⋊C₄	—	2	4	no	—
265	C₂²⋊C₄	—	26	4	no	—
266	C₂²⋊C₄	—	52	2	no	—
267	C₄²	—	2	2	no	—
268	C₄²	—	2	2	no	—
269	C₄²	—	2	2	no	—
270	C₄²	—	26	4	no	—
271	C₄²	—	52	2	no	—
272	C₄²	C₄⋊Dic₁₃	1	4	yes	—
273	C₁₃	#11225	1	1	yes	characteristic, complemented, Sylow
274	C₂³	—	4	1	no	—
275	C₂³	—	4	1	no	—
276	C₂³	#88178	1	1	yes	characteristic
277	C₂³	#88184	1	1	yes	characteristic
278	C₂³	#88193	1	2	yes	—
279	C₂³	#88200	1	2	yes	—
280	C₂³	C₂×C₂²⋊Dic₁₃	1	1	yes	characteristic
281	C₂×C₄	—	2	4	no	—
282	C₂×C₄	—	2	2	no	—
283	C₂×C₄	—	2	2	no	—
284	C₂×C₄	—	4	2	no	—
285	C₂×C₄	—	4	2	no	—
286	C₂×C₄	—	4	2	no	—
287	C₂×C₄	—	4	2	no	—
288	C₂×C₄	—	4	4	no	—
289	C₂×C₄	—	4	2	no	—
290	C₂×C₄	—	8	4	no	—
291	C₂×C₄	—	52	4	no	—
292	C₂×C₄	—	104	2	no	—
293	C₂×C₄	—	104	2	no	—
294	C₂×C₄	#88142	1	2	yes	—
295	C₂×C₄	#88143	1	2	yes	—
296	C₂×C₄	#88144	1	2	yes	—
297	C₂×C₄	#88148	1	2	yes	—
298	C₂×C₄	#88149	1	2	yes	—
299	C₂×C₄	#88150	1	2	yes	—
300	C₂×C₄	#88151	1	2	yes	—
301	C₂×C₄	#88152	1	2	yes	—
302	C₂×C₄	#88178	1	1	yes	characteristic
303	C₂×C₄	C₂×C₄⋊Dic₁₃	1	1	yes	characteristic
304	C₂×C₄	#88180	1	1	yes	characteristic
305	C₂×C₄	#88184	1	1	yes	characteristic
306	C₂×C₄	D₄×Dic₁₃	1	2	yes	—
307	C₂×C₄	#88189	1	1	yes	characteristic
308	C₂×C₄	#88191	1	1	yes	characteristic
309	C₂×C₄	#88198	1	1	yes	characteristic
310	C₂×C₄	Q₈×Dic₁₃	1	2	yes	—
311	C₂×C₄	#88201	1	1	yes	characteristic
312	C₈	—	52	4	no	—
313	C₂²	—	2	1	no	—
314	C₂²	—	2	2	no	—
315	C₂²	—	2	1	no	—
316	C₂²	—	2	2	no	—
317	C₂²	—	4	1	no	—
318	C₂²	—	4	1	no	—
319	C₂²	—	4	2	no	—
320	C₂²	#11718645	1	2	yes	—
321	C₂²	#11718655	1	2	yes	—
322	C₂²	#11718873	1	1	yes	characteristic
323	C₂²	#11718881	1	1	yes	characteristic
324	C₂²	#11718891	1	1	yes	characteristic
325	C₄	—	4	4	no	—
326	C₄	—	4	2	no	—
327	C₄	—	4	2	no	—
328	C₄	—	8	2	no	—
329	C₄	—	104	4	no	—
330	C₄	#11718640	1	1	yes	characteristic
331	C₄	#11718642	1	2	yes	—
332	C₄	#11718644	1	1	yes	characteristic
333	C₄	#11718646	1	2	yes	—
334	C₄	#11718647	1	1	yes	characteristic
335	C₄	#11718652	1	1	yes	characteristic
336	C₂	—	2	1	no	—
337	C₂	—	2	2	no	—
338	C₂	—	2	1	no	—
339	C₂	—	2	2	no	—
340	C₂	#13113089	1	1	yes	characteristic
341	C₂	#13115359	1	1	yes	characteristic
342	C₂	#13115364	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#11718881
commutator	C₂²×C₄×C₁₃	C₂²×C₄
Frattini	C₂³×C₄	C₂²×D₁₃
Fitting	C₁₃×#2317	C₂
socle	C₂²×C₁₃	#386
prime radical	C₂⁴×C₁₃	C₂×D₄
cosocle	C₂³×C₄×C₁₃	C₂³
hypercenter	#2317	D₁₃
hypocenter	C₁₃	#11225
LCS	G ▷ C₂²×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #2317

Complements

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

N	H ≅ G/N	type
13. C₁₃×#384	C₄	semidirect
14. C₁₃×#382	C₄	semidirect
156. #384	Dic₁₃	semidirect
157. #382	Dic₁₃	semidirect
273. C₁₃	#11225	semidirect

Maximal subgroups

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

subgroup	index
#13123400	2
#13114569	2
#13114555	2
C₂×#11718123	2
C₁₃×#2317	2
#11225	13