Subgroups — #15768680

Order 3328.

686 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718146	C₂	1	2	yes	maximal
2	C₂×#11718144	C₂	1	1	yes	maximal, characteristic
3	#13114557	C₂	1	2	yes	maximal
4	C₂×#11718143	C₂	1	1	yes	maximal, characteristic
5	C₂×C₁₃×#405	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₁₃×M₁₆	C₂²	1	1	yes	characteristic
7	C₂³×C₈×C₁₃	C₂²	1	1	yes	characteristic
8	C₂×C₁₃×#117	C₂²	1	1	yes	characteristic
9	C₂²×C₄⋊Dic₁₃	C₂²	1	2	yes	—
10	C₂²×#6099	C₂²	1	2	yes	—
11	C₁₃×#405	—	2	2	no	—
12	C₁₃×#405	C₄	1	4	yes	complemented
13	C₂×C₁₃×#97	—	2	1	no	—
14	C₂×C₄×C₈×C₁₃	—	2	1	no	—
15	#11718146	C₄	1	8	yes	complemented
16	#11718144	—	2	2	no	—
17	#11718143	—	2	1	no	—
18	#11718143	—	2	1	no	—
19	#11718121	—	2	2	no	—
20	#11718121	—	2	2	no	—
21	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
22	C₂³×Dic₁₃	—	2	2	no	—
23	C₂×C₁₃×M₁₆	—	2	1	no	—
24	C₂×C₁₃×M₁₆	—	2	2	no	—
25	C₂×C₁₃×M₁₆	C₂×C₄	1	4	yes	—
26	C₂×C₁₃×M₁₆	C₂×C₄	1	2	yes	—
27	C₂²×C₈×C₁₃	—	2	1	no	—
28	C₂²×C₈×C₁₃	—	2	2	no	—
29	C₂²×C₈×C₁₃	—	2	2	no	—
30	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
31	C₂²×C₈×C₁₃	D₄	1	2	yes	—
32	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
33	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
34	C₁₃×#117	—	2	1	no	—
35	C₁₃×#117	—	2	1	no	—
36	C₁₃×#117	C₂×C₄	1	2	yes	—
37	C₁₃×#117	Q₈	1	2	yes	—
38	C₂×C₁₃×C₄⋊C₄	D₄	1	1	yes	characteristic
39	C₂×C₁₃×C₄⋊C₄	D₄	1	1	yes	characteristic
40	C₂×C₁₃×C₂²⋊C₄	—	2	1	no	—
41	C₂×C₄²×C₁₃	—	2	1	no	—
42	C₂×C₄⋊Dic₁₃	—	2	2	no	—
43	C₂×C₄⋊Dic₁₃	—	2	2	no	—
44	C₂×C₄⋊Dic₁₃	—	2	2	no	—
45	C₂×C₄⋊Dic₁₃	—	2	2	no	—
46	C₂×C₄⋊Dic₁₃	—	2	2	no	—
47	C₂×C₄⋊Dic₁₃	C₂×C₄	1	4	yes	—
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	C₂×#6099	C₂×C₄	1	4	yes	—
54	C₂²×#1055	—	2	2	no	—
55	C₁₃×#97	—	2	4	no	—
56	C₄×C₈×C₁₃	—	2	4	no	—
57	#10827	—	13	1	no	maximal, Sylow
58	C₂⁴×C₁₃	#41	1	1	yes	characteristic
59	C₂²×C₄×C₁₃	—	2	1	no	—
60	C₂²×C₄×C₁₃	—	2	1	no	—
61	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
62	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
63	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
64	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₁₃	C₂×Q₈	1	1	yes	characteristic
67	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
68	C₂²×C₄×C₁₃	#41	1	2	yes	—
69	C₂²×C₄×C₁₃	#41	1	2	yes	—
70	C₂²×C₄×C₁₃	#41	1	2	yes	—
71	C₂²×Dic₁₃	—	2	4	no	—
72	C₂²×Dic₁₃	—	4	2	no	—
73	C₂²×Dic₁₃	—	4	2	no	—
74	C₂²×Dic₁₃	—	8	2	no	—
75	C₂²×Dic₁₃	—	8	2	no	—
76	C₁₃×M₁₆	—	2	2	no	—
77	C₁₃×M₁₆	—	2	4	no	—
78	C₁₃×M₁₆	C₄²	1	4	yes	—
79	C₂×C₈×C₁₃	—	2	2	no	—
80	C₂×C₈×C₁₃	—	2	2	no	—
81	C₂×C₈×C₁₃	—	2	2	no	—
82	C₂×C₈×C₁₃	—	2	4	no	—
83	C₂×C₈×C₁₃	—	2	2	no	—
84	C₂×C₈×C₁₃	—	2	2	no	—
85	C₂×C₈×C₁₃	—	2	4	no	—
86	C₂×C₈×C₁₃	C₂²⋊C₄	1	2	yes	—
87	C₂×C₈×C₁₃	C₂²⋊C₄	1	2	yes	—
88	C₁₃×C₄⋊C₄	—	2	1	no	—
89	C₁₃×C₄⋊C₄	—	2	1	no	—
90	C₁₃×C₄⋊C₄	C₄⋊C₄	1	2	yes	—
91	C₁₃×C₄⋊C₄	C₄⋊C₄	1	2	yes	—
92	C₁₃×C₂²⋊C₄	—	2	2	no	—
93	C₁₃×C₂²⋊C₄	—	2	2	no	—
94	C₄²×C₁₃	—	2	2	no	—
95	C₄²×C₁₃	—	2	2	no	—
96	C₄⋊Dic₁₃	—	2	4	no	—
97	C₄⋊Dic₁₃	—	2	4	no	—
98	C₄⋊Dic₁₃	—	4	2	no	—
99	C₄⋊Dic₁₃	—	4	2	no	—
100	#6099	—	2	4	no	—
101	#6099	—	2	4	no	—
102	#6099	—	4	2	no	—
103	#6099	—	4	2	no	—
104	C₂×#1055	—	2	4	no	—
105	C₂×#1055	—	4	2	no	—
106	C₂×#1055	—	4	2	no	—
107	C₂×#405	D₁₃	1	1	yes	characteristic
108	C₂×#343	—	13	2	no	—
109	C₂×#341	—	13	1	no	—
110	#1736	—	13	2	no	—
111	C₂×#340	—	13	1	no	—
112	C₂³×C₁₃	—	2	2	no	—
113	C₂³×C₁₃	—	2	1	no	—
114	C₂³×C₁₃	—	2	1	no	—
115	C₂³×C₁₃	#117	1	1	yes	characteristic
116	C₂³×C₁₃	#117	1	2	yes	—
117	C₂³×C₁₃	#121	1	1	yes	characteristic
118	C₂³×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
119	C₂³×C₁₃	#126	1	2	yes	—
120	C₂×C₄×C₁₃	—	2	2	no	—
121	C₂×C₄×C₁₃	—	2	2	no	—
122	C₂×C₄×C₁₃	—	4	2	no	—
123	C₂×C₄×C₁₃	—	4	1	no	—
124	C₂×C₄×C₁₃	—	4	1	no	—
125	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
126	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
127	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
128	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
129	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
130	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
131	C₂×C₄×C₁₃	#117	1	2	yes	—
132	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
133	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
134	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
135	C₂×C₄×C₁₃	C₄×Q₈	1	2	yes	—
136	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
137	C₂×C₄×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
138	C₂×C₄×C₁₃	#123	1	2	yes	—
139	C₂×C₄×C₁₃	#123	1	2	yes	—
140	C₂×C₄×C₁₃	#124	1	2	yes	—
141	C₂×C₄×C₁₃	#125	1	2	yes	—
142	C₂×C₄×C₁₃	#126	1	2	yes	—
143	C₂×Dic₁₃	—	4	4	no	—
144	C₂×Dic₁₃	—	8	4	no	—
145	C₂×Dic₁₃	—	8	2	no	—
146	C₂×Dic₁₃	—	8	2	no	—
147	C₂×Dic₁₃	—	8	4	no	—
148	C₈×C₁₃	—	2	4	no	—
149	C₈×C₁₃	—	2	2	no	—
150	C₈×C₁₃	—	2	2	no	—
151	#1055	—	4	4	no	—
152	C₂²×M₁₆	—	13	2	no	—
153	C₂²×M₁₆	C₂×D₁₃	1	1	yes	characteristic
154	C₂³×C₈	C₂×D₁₃	1	1	yes	characteristic
155	C₂×#117	C₂×D₁₃	1	1	yes	characteristic
156	C₂²×C₄⋊C₄	—	13	2	no	—
157	#405	—	2	2	no	—
158	#405	Dic₁₃	1	4	yes	complemented
159	C₂×#97	—	2	1	no	—
160	C₂×C₄×C₈	—	2	1	no	—
161	#343	—	13	8	no	—
162	#341	—	26	2	no	—
163	#340	—	26	1	no	—
164	#340	—	26	1	no	—
165	#340	—	26	2	no	—
166	#340	—	26	2	no	—
167	C₂²×C₁₃	—	2	1	no	—
168	C₂²×C₁₃	—	2	1	no	—
169	C₂²×C₁₃	—	2	1	no	—
170	C₂²×C₁₃	—	2	1	no	—
171	C₂²×C₁₃	—	2	2	no	—
172	C₂²×C₁₃	—	2	2	no	—
173	C₂²×C₁₃	—	2	2	no	—
174	C₂²×C₁₃	—	2	2	no	—
175	C₂²×C₁₃	#376	1	1	yes	characteristic
176	C₂²×C₁₃	#380	1	1	yes	characteristic
177	C₂²×C₁₃	#383	1	1	yes	characteristic
178	C₂²×C₁₃	#387	1	2	yes	—
179	C₂²×C₁₃	#388	1	2	yes	—
180	C₂²×C₁₃	#471	1	2	yes	—
181	C₂²×C₁₃	#472	1	1	yes	characteristic
182	C₂²×C₁₃	#473	1	1	yes	characteristic
183	C₄×C₁₃	—	4	2	no	—
184	C₄×C₁₃	—	4	2	no	—
185	C₄×C₁₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
186	C₄×C₁₃	C₄×C₄⋊C₄	1	1	yes	characteristic
187	C₄×C₁₃	#381	1	1	yes	characteristic
188	C₄×C₁₃	#382	1	1	yes	characteristic
189	C₄×C₁₃	#387	1	2	yes	—
190	C₄×C₁₃	#388	1	2	yes	—
191	Dic₁₃	—	8	4	no	—
192	C₂³×C₄	—	26	2	no	—
193	C₂³×C₄	C₂²×D₁₃	1	1	yes	characteristic
194	C₂×M₁₆	—	2	2	no	—
195	C₂×M₁₆	—	2	1	no	—
196	C₂×M₁₆	—	13	4	no	—
197	C₂×M₁₆	—	26	2	no	—
198	C₂×M₁₆	—	26	2	no	—
199	C₂×M₁₆	—	26	2	no	—
200	C₂×M₁₆	—	26	2	no	—
201	C₂×M₁₆	—	26	2	no	—
202	C₂×M₁₆	C₄×D₁₃	1	4	yes	—
203	C₂×M₁₆	C₂×Dic₁₃	1	2	yes	—
204	C₂²×C₈	—	2	2	no	—
205	C₂²×C₈	—	2	2	no	—
206	C₂²×C₈	—	2	1	no	—
207	C₂²×C₈	—	26	2	no	—
208	C₂²×C₈	C₂×Dic₁₃	1	2	yes	—
209	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
210	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
211	C₂²×C₈	C₂²⋊D₁₃	1	2	yes	—
212	#117	—	2	1	no	—
213	#117	—	2	1	no	—
214	#117	Dic₂₆	1	2	yes	—
215	#117	C₂×Dic₁₃	1	2	yes	—
216	C₂×C₄⋊C₄	—	13	4	no	—
217	C₂×C₄⋊C₄	—	26	2	no	—
218	C₂×C₄⋊C₄	—	26	2	no	—
219	C₂×C₄⋊C₄	—	26	2	no	—
220	C₂×C₄⋊C₄	—	26	2	no	—
221	C₂×C₄⋊C₄	—	26	2	no	—
222	C₂×C₄⋊C₄	D₅₂	1	1	yes	characteristic
223	C₂×C₄⋊C₄	D₅₂	1	1	yes	characteristic
224	C₂×C₂²⋊C₄	—	2	1	no	—
225	C₂×C₄²	—	2	1	no	—
226	#97	—	2	4	no	—
227	C₄×C₈	—	2	4	no	—
228	C₂×C₁₃	—	2	1	no	—
229	C₂×C₁₃	—	2	1	no	—
230	C₂×C₁₃	—	2	2	no	—
231	C₂×C₁₃	#1434	1	1	yes	characteristic
232	C₂×C₁₃	#1815	1	1	yes	characteristic
233	C₂×C₁₃	#1816	1	1	yes	characteristic
234	C₂×C₁₃	#1930	1	2	yes	—
235	C₂×C₁₃	#1931	1	2	yes	—
236	C₂⁴	#6127	1	1	yes	characteristic
237	C₂²×C₄	—	2	1	no	—
238	C₂²×C₄	—	2	1	no	—
239	C₂²×C₄	—	26	4	no	—
240	C₂²×C₄	—	52	2	no	—
241	C₂²×C₄	—	52	2	no	—
242	C₂²×C₄	—	104	2	no	—
243	C₂²×C₄	—	104	2	no	—
244	C₂²×C₄	C₂×Dic₂₆	1	1	yes	characteristic
245	C₂²×C₄	C₂×C₄×D₁₃	1	2	yes	—
246	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
247	C₂²×C₄	#6127	1	2	yes	—
248	C₂²×C₄	#6127	1	2	yes	—
249	C₂²×C₄	#6127	1	2	yes	—
250	C₂²×C₄	#6127	1	1	yes	characteristic
251	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
252	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
253	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
254	M₁₆	—	2	2	no	—
255	M₁₆	—	2	4	no	—
256	M₁₆	—	26	4	no	—
257	M₁₆	—	26	4	no	—
258	M₁₆	—	52	2	no	—
259	M₁₆	—	52	2	no	—
260	M₁₆	C₄×Dic₁₃	1	4	yes	—
261	C₂×C₈	—	2	2	no	—
262	C₂×C₈	—	2	2	no	—
263	C₂×C₈	—	2	4	no	—
264	C₂×C₈	—	2	2	no	—
265	C₂×C₈	—	2	2	no	—
266	C₂×C₈	—	2	4	no	—
267	C₂×C₈	—	2	2	no	—
268	C₂×C₈	—	26	4	no	—
269	C₂×C₈	—	52	2	no	—
270	C₂×C₈	—	52	2	no	—
271	C₂×C₈	C₂²⋊Dic₁₃	1	2	yes	—
272	C₂×C₈	C₂²⋊Dic₁₃	1	2	yes	—
273	C₄⋊C₄	—	2	1	no	—
274	C₄⋊C₄	—	2	1	no	—
275	C₄⋊C₄	—	26	4	no	—
276	C₄⋊C₄	—	26	4	no	—
277	C₄⋊C₄	—	52	2	no	—
278	C₄⋊C₄	—	52	2	no	—
279	C₄⋊C₄	C₄⋊Dic₁₃	1	2	yes	—
280	C₄⋊C₄	C₄⋊Dic₁₃	1	2	yes	—
281	C₂²⋊C₄	—	2	2	no	—
282	C₂²⋊C₄	—	2	2	no	—
283	C₄²	—	2	2	no	—
284	C₄²	—	2	2	no	—
285	C₁₃	#10827	1	1	yes	characteristic, complemented, Sylow
286	C₂³	—	2	2	no	—
287	C₂³	—	2	1	no	—
288	C₂³	—	2	1	no	—
289	C₂³	#88126	1	2	yes	—
290	C₂³	#88130	1	2	yes	—
291	C₂³	#88178	1	1	yes	characteristic
292	C₂³	#88180	1	1	yes	characteristic
293	C₂³	#88184	1	1	yes	characteristic
294	C₂×C₄	—	2	2	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	4	2	no	—
297	C₂×C₄	—	4	1	no	—
298	C₂×C₄	—	4	1	no	—
299	C₂×C₄	—	52	4	no	—
300	C₂×C₄	—	104	4	no	—
301	C₂×C₄	—	104	2	no	—
302	C₂×C₄	—	104	2	no	—
303	C₂×C₄	—	104	4	no	—
304	C₂×C₄	C₄×Dic₂₆	1	2	yes	—
305	C₂×C₄	#88124	1	2	yes	—
306	C₂×C₄	#88126	1	2	yes	—
307	C₂×C₄	C₄×D₅₂	1	2	yes	—
308	C₂×C₄	#88129	1	2	yes	—
309	C₂×C₄	#88130	1	2	yes	—
310	C₂×C₄	C₂×C₄×Dic₁₃	1	1	yes	characteristic
311	C₂×C₄	#88178	1	1	yes	characteristic
312	C₂×C₄	C₂×C₄⋊Dic₁₃	1	1	yes	characteristic
313	C₂×C₄	#88180	1	1	yes	characteristic
314	C₂×C₄	#88180	1	1	yes	characteristic
315	C₂×C₄	#88180	1	1	yes	characteristic
316	C₂×C₄	C₄×C₂²⋊D₁₃	1	2	yes	—
317	C₂×C₄	C₄×C₂²⋊D₁₃	1	2	yes	—
318	C₂×C₄	#88183	1	2	yes	—
319	C₂×C₄	#88183	1	2	yes	—
320	C₂×C₄	#88184	1	1	yes	characteristic
321	C₂×C₄	C₂×C₂²⋊Dic₁₃	1	1	yes	characteristic
322	C₈	—	2	4	no	—
323	C₈	—	2	2	no	—
324	C₈	—	2	2	no	—
325	C₈	—	52	4	no	—
326	C₂²	—	2	1	no	—
327	C₂²	—	2	1	no	—
328	C₂²	—	2	1	no	—
329	C₂²	—	2	1	no	—
330	C₂²	—	2	2	no	—
331	C₂²	—	2	2	no	—
332	C₂²	—	2	2	no	—
333	C₂²	—	2	2	no	—
334	C₂²	#11718593	1	1	yes	characteristic
335	C₂²	#11718594	1	2	yes	—
336	C₂²	#11718598	1	1	yes	characteristic
337	C₂²	#11718601	1	2	yes	—
338	C₂²	#11718802	1	2	yes	—
339	C₂²	#11718803	1	1	yes	characteristic
340	C₂²	#11718804	1	1	yes	characteristic
341	C₂²	#11718873	1	1	yes	characteristic
342	C₄	—	4	2	no	—
343	C₄	—	4	2	no	—
344	C₄	—	104	4	no	—
345	C₄	#11718594	1	2	yes	—
346	C₄	C₄×C₄⋊Dic₁₃	1	1	yes	characteristic
347	C₄	#11718597	1	1	yes	characteristic
348	C₄	#11718601	1	2	yes	—
349	C₄	C₄×C₂²⋊Dic₁₃	1	1	yes	characteristic
350	C₄	#11718872	1	1	yes	characteristic
351	C₂	—	2	1	no	—
352	C₂	—	2	2	no	—
353	C₂	—	2	1	no	—
354	C₂	#13113063	1	1	yes	characteristic
355	C₂	#13114895	1	1	yes	characteristic
356	C₂	#13114900	1	1	yes	characteristic
357	C₂	#13115491	1	2	yes	—
358	C₂	#13115498	1	2	yes	—

	subgroup	G/H
center	C₂³	#88184
commutator	C₂×C₄×C₁₃	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₁₃
Fitting	C₂×C₁₃×#405	C₂
socle	C₂³×C₁₃	#121
prime radical	C₂⁴×C₁₃	#41
cosocle	C₂³×C₄×C₁₃	C₂³
hypercenter	C₂×#405	D₁₃
hypocenter	C₁₃	#10827
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂×#405

Complements

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

N	H ≅ G/N	type
12. C₁₃×#405	C₄	semidirect
15. #11718146	C₄	semidirect
158. #405	Dic₁₃	semidirect
285. C₁₃	#10827	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11718146	2
C₂×#11718144	2
#13114557	2
C₂×#11718143	2
C₂×C₁₃×#405	2
#10827	13