Subgroups — #15755798

Order 3328.

462 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13113584	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#11718050	C₂	1	1	yes	maximal, characteristic, complemented
3	#13113223	C₂	1	2	yes	maximal
4	#13112687	C₂	1	2	yes	maximal, complemented
5	C₄×C₁₃×#131	C₂	1	1	yes	maximal, characteristic
6	C₂×C₁₃×#131	C₄	1	1	yes	characteristic
7	C₂×C₁₃×#131	C₄	1	1	yes	characteristic
8	C₄×C₁₃×#41	C₂²	1	1	yes	characteristic
9	C₁₃×#405	—	2	1	no	—
10	C₁₃×#405	—	2	1	no	—
11	C₁₃×#405	C₄	1	1	yes	characteristic
12	C₁₃×#405	C₄	1	1	yes	characteristic
13	C₄×C₁₃×M₁₆	—	2	1	no	—
14	C₄×C₁₃×M₁₆	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₈×C₁₃	—	2	1	no	—
16	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
17	#11718586	C₂²	1	2	yes	—
18	C₂×C₄⋊#1055	C₂²	1	2	yes	—
19	Q₈⋊#1055	—	2	2	no	—
20	D₄⋊#1055	—	2	2	no	—
21	#11718050	—	2	1	no	—
22	#11718050	—	2	1	no	—
23	#11718050	—	2	1	no	—
24	#11718049	—	2	1	no	—
25	C₂×C₁₃×#41	C₂×C₄	1	1	yes	characteristic
26	C₁₃×#131	—	2	2	no	—
27	C₁₃×#131	—	2	2	no	—
28	C₁₃×#131	—	2	2	no	—
29	C₁₃×#131	C₈	1	4	yes	complemented
30	C₂×C₁₃×M₁₆	—	2	1	no	—
31	C₂×C₁₃×M₁₆	—	2	1	no	—
32	C₂×C₁₃×M₁₆	C₂×C₄	1	1	yes	characteristic
33	C₂×C₁₃×M₁₆	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₈×C₁₃	—	2	1	no	—
35	C₂²×C₈×C₁₃	—	2	1	no	—
36	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
38	C₄×Q₈×C₁₃	D₄	1	1	yes	characteristic
39	C₄×Q₈×C₁₃	Q₈	1	1	yes	characteristic
40	C₄×D₄×C₁₃	—	2	1	no	—
41	C₄×D₄×C₁₃	—	2	1	no	—
42	C₄×D₄×C₁₃	D₄	1	1	yes	characteristic
43	C₄×D₄×C₁₃	Q₈	1	1	yes	characteristic
44	C₁₃×#117	—	2	1	no	—
45	C₁₃×#117	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄²×C₁₃	—	2	1	no	—
47	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
48	C₂²×#1055	—	2	2	no	—
49	C₁₃×#97	—	2	1	no	—
50	C₁₃×#97	—	2	1	no	—
51	C₁₃×#97	—	2	1	no	—
52	C₄×C₈×C₁₃	—	2	1	no	—
53	C₄×C₈×C₁₃	—	2	1	no	—
54	C₄×C₈×C₁₃	—	2	1	no	—
55	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
56	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
57	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
58	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
59	C₂²⋊#1055	—	2	2	no	—
60	C₄⋊#1055	—	2	2	no	—
61	C₄⋊#1055	—	2	2	no	—
62	C₄⋊#1055	—	2	2	no	—
63	C₄×#1055	—	2	2	no	—
64	#8605	—	13	1	no	maximal, Sylow
65	C₁₃×#41	—	2	1	no	—
66	C₁₃×#41	—	2	1	no	—
67	C₁₃×#41	C₂×C₈	1	2	yes	—
68	C₁₃×#41	M₁₆	1	2	yes	—
69	C₂×Q₈×C₁₃	C₄⋊C₄	1	1	yes	characteristic
70	C₂×D₄×C₁₃	—	2	1	no	—
71	C₂×D₄×C₁₃	C₄⋊C₄	1	1	yes	characteristic
72	C₂²×C₄×C₁₃	—	2	1	no	—
73	C₂²×C₄×C₁₃	—	2	1	no	—
74	C₂²×C₄×C₁₃	—	2	1	no	—
75	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
76	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
77	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
78	C₁₃×M₁₆	—	2	4	no	—
79	C₁₃×M₁₆	—	2	1	no	—
80	C₁₃×M₁₆	—	2	1	no	—
81	C₁₃×M₁₆	—	2	4	no	—
82	C₁₃×M₁₆	C₂×C₈	1	2	yes	—
83	C₁₃×M₁₆	M₁₆	1	2	yes	—
84	C₂×C₈×C₁₃	—	2	4	no	—
85	C₂×C₈×C₁₃	—	2	1	no	—
86	C₂×C₈×C₁₃	—	2	1	no	—
87	C₂×C₈×C₁₃	—	2	1	no	—
88	C₂×C₈×C₁₃	—	2	1	no	—
89	C₂×C₈×C₁₃	—	2	4	no	—
90	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
91	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
92	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
93	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
94	C₂×C₈×C₁₃	C₂×C₈	1	2	yes	—
95	C₂×C₈×C₁₃	M₁₆	1	2	yes	—
96	C₁₃×C₄⋊C₄	—	2	1	no	—
97	C₁₃×C₄⋊C₄	—	2	1	no	—
98	C₁₃×C₄⋊C₄	C₄⋊C₄	1	1	yes	characteristic
99	C₁₃×C₄⋊C₄	C₄⋊C₄	1	1	yes	characteristic
100	C₁₃×C₂²⋊C₄	—	2	1	no	—
101	C₁₃×C₂²⋊C₄	—	2	1	no	—
102	C₁₃×C₂²⋊C₄	—	2	1	no	—
103	C₄²×C₁₃	—	2	1	no	—
104	C₄²×C₁₃	—	2	1	no	—
105	C₄²×C₁₃	—	2	1	no	—
106	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
107	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
108	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
109	C₄²×C₁₃	C₂×Q₈	1	1	yes	characteristic
110	C₂×#1055	—	4	2	no	—
111	C₂×#1055	—	4	2	no	—
112	C₂×#1055	—	8	2	no	—
113	C₄×#131	D₁₃	1	1	yes	characteristic
114	#1564	—	13	1	no	—
115	C₂×#335	—	13	1	no	—
116	#1477	—	13	2	no	—
117	#1283	—	13	2	no	—
118	C₂³×C₁₃	—	2	1	no	—
119	C₂³×C₁₃	#117	1	1	yes	characteristic
120	Q₈×C₁₃	—	2	1	no	—
121	Q₈×C₁₃	C₄⋊C₈	1	2	yes	—
122	D₄×C₁₃	—	2	2	no	—
123	D₄×C₁₃	—	2	2	no	—
124	D₄×C₁₃	—	2	1	no	—
125	D₄×C₁₃	C₄⋊C₈	1	2	yes	—
126	C₂×C₄×C₁₃	—	2	1	no	—
127	C₂×C₄×C₁₃	—	2	2	no	—
128	C₂×C₄×C₁₃	—	2	2	no	—
129	C₂×C₄×C₁₃	—	2	1	no	—
130	C₂×C₄×C₁₃	—	2	1	no	—
131	C₂×C₄×C₁₃	—	2	1	no	—
132	C₂×C₄×C₁₃	—	2	1	no	—
133	C₂×C₄×C₁₃	—	4	1	no	—
134	C₂×C₄×C₁₃	—	4	2	no	—
135	C₂×C₄×C₁₃	—	4	1	no	—
136	C₂×C₄×C₁₃	—	4	1	no	—
137	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
138	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
139	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
140	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
141	C₂×C₄×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
142	C₂×C₄×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
143	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
144	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
145	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
146	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
147	C₈×C₁₃	—	2	2	no	—
148	C₈×C₁₃	—	2	1	no	—
149	C₈×C₁₃	—	2	2	no	—
150	C₈×C₁₃	—	2	1	no	—
151	C₈×C₁₃	C₂²⋊C₈	1	2	yes	—
152	C₈×C₁₃	C₂²⋊C₈	1	2	yes	—
153	#1055	—	8	2	no	—
154	#1055	—	8	2	no	—
155	C₂×#131	Dic₁₃	1	1	yes	characteristic
156	C₂×#131	Dic₁₃	1	1	yes	characteristic
157	C₄×#41	C₂×D₁₃	1	1	yes	characteristic
158	#431	—	13	2	no	—
159	C₂×C₄⋊C₈	—	13	2	no	—
160	#405	—	2	1	no	—
161	#405	—	2	1	no	—
162	#405	Dic₁₃	1	1	yes	characteristic
163	#405	Dic₁₃	1	1	yes	characteristic
164	C₄×M₁₆	—	2	1	no	—
165	C₄×M₁₆	C₂×D₁₃	1	1	yes	characteristic
166	C₂×C₄×C₈	—	2	1	no	—
167	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
168	#335	—	26	1	no	—
169	#335	—	26	1	no	—
170	#335	—	26	1	no	—
171	#334	—	26	1	no	—
172	Q₈⋊C₈	—	26	2	no	—
173	D₄⋊C₈	—	26	2	no	—
174	C₂²×C₁₃	—	2	1	no	—
175	C₂²×C₁₃	—	2	1	no	—
176	C₂²×C₁₃	—	2	2	no	—
177	C₂²×C₁₃	—	4	1	no	—
178	C₂²×C₁₃	—	4	1	no	—
179	C₂²×C₁₃	#380	1	1	yes	characteristic
180	C₂²×C₁₃	#431	1	1	yes	characteristic
181	C₂²×C₁₃	#432	1	1	yes	characteristic
182	C₄×C₁₃	—	2	2	no	—
183	C₄×C₁₃	—	2	1	no	—
184	C₄×C₁₃	—	2	1	no	—
185	C₄×C₁₃	—	2	1	no	—
186	C₄×C₁₃	—	4	1	no	—
187	C₄×C₁₃	—	4	1	no	—
188	C₄×C₁₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
189	C₄×C₁₃	#407	1	1	yes	characteristic
190	C₄×C₁₃	C₂×C₄⋊C₈	1	1	yes	characteristic
191	C₄×C₁₃	#423	1	1	yes	characteristic
192	C₄×C₁₃	#472	1	1	yes	characteristic
193	C₄×C₁₃	#473	1	1	yes	characteristic
194	C₂×#41	C₂×Dic₁₃	1	1	yes	characteristic
195	#131	—	2	2	no	—
196	#131	—	2	2	no	—
197	#131	—	2	2	no	—
198	#131	#1055	1	4	yes	complemented
199	C₂×M₁₆	—	2	1	no	—
200	C₂×M₁₆	—	2	1	no	—
201	C₂×M₁₆	C₂×Dic₁₃	1	1	yes	characteristic
202	C₂×M₁₆	C₂×Dic₁₃	1	1	yes	characteristic
203	C₂²×C₈	—	2	1	no	—
204	C₂²×C₈	—	2	1	no	—
205	C₂²×C₈	—	26	2	no	—
206	C₂²×C₈	C₂×Dic₁₃	1	1	yes	characteristic
207	C₂²×C₈	C₂×Dic₁₃	1	1	yes	characteristic
208	C₄×Q₈	Dic₂₆	1	1	yes	characteristic
209	C₄×Q₈	D₅₂	1	1	yes	characteristic
210	C₄×D₄	—	2	1	no	—
211	C₄×D₄	—	2	1	no	—
212	C₄×D₄	Dic₂₆	1	1	yes	characteristic
213	C₄×D₄	D₅₂	1	1	yes	characteristic
214	#117	—	2	1	no	—
215	#117	C₂×Dic₁₃	1	1	yes	characteristic
216	C₂×C₄²	—	2	1	no	—
217	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
218	C₄⋊C₈	—	26	2	no	—
219	C₄⋊C₈	—	26	2	no	—
220	C₄⋊C₈	—	26	2	no	—
221	C₂²⋊C₈	—	26	2	no	—
222	#97	—	2	1	no	—
223	#97	—	2	1	no	—
224	#97	—	2	1	no	—
225	C₄×C₈	—	2	1	no	—
226	C₄×C₈	—	2	1	no	—
227	C₄×C₈	—	2	1	no	—
228	C₄×C₈	—	26	2	no	—
229	C₄×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
230	C₄×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
231	C₄×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
232	C₄×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
233	C₂×C₁₃	—	2	1	no	—
234	C₂×C₁₃	—	2	1	no	—
235	C₂×C₁₃	—	4	2	no	—
236	C₂×C₁₃	#1797	1	1	yes	characteristic
237	C₂×C₁₃	#1814	1	1	yes	characteristic
238	C₂×C₁₃	#1816	1	1	yes	characteristic
239	#41	—	2	1	no	—
240	#41	—	2	1	no	—
241	#41	C₂×#1055	1	2	yes	—
242	#41	#6099	1	2	yes	—
243	C₂×Q₈	C₄⋊Dic₁₃	1	1	yes	characteristic
244	C₂×D₄	—	2	1	no	—
245	C₂×D₄	C₄⋊Dic₁₃	1	1	yes	characteristic
246	C₂²×C₄	—	2	1	no	—
247	C₂²×C₄	—	2	1	no	—
248	C₂²×C₄	—	2	1	no	—
249	C₂²×C₄	#6127	1	1	yes	characteristic
250	C₂²×C₄	#6127	1	1	yes	characteristic
251	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
252	M₁₆	—	2	4	no	—
253	M₁₆	—	2	1	no	—
254	M₁₆	—	2	1	no	—
255	M₁₆	—	2	4	no	—
256	M₁₆	C₂×#1055	1	2	yes	—
257	M₁₆	#6099	1	2	yes	—
258	C₂×C₈	—	2	4	no	—
259	C₂×C₈	—	2	1	no	—
260	C₂×C₈	—	2	1	no	—
261	C₂×C₈	—	2	1	no	—
262	C₂×C₈	—	2	1	no	—
263	C₂×C₈	—	2	4	no	—
264	C₂×C₈	—	52	2	no	—
265	C₂×C₈	—	52	2	no	—
266	C₂×C₈	—	104	2	no	—
267	C₂×C₈	C₂×#1055	1	2	yes	—
268	C₂×C₈	#6099	1	2	yes	—
269	C₂×C₈	C₂²⋊Dic₁₃	1	1	yes	characteristic
270	C₂×C₈	C₂²⋊Dic₁₃	1	1	yes	characteristic
271	C₂×C₈	C₂²⋊Dic₁₃	1	1	yes	characteristic
272	C₂×C₈	C₂²⋊Dic₁₃	1	1	yes	characteristic
273	C₄⋊C₄	—	2	1	no	—
274	C₄⋊C₄	—	2	1	no	—
275	C₄⋊C₄	C₄⋊Dic₁₃	1	1	yes	characteristic
276	C₄⋊C₄	C₄⋊Dic₁₃	1	1	yes	characteristic
277	C₂²⋊C₄	—	2	1	no	—
278	C₂²⋊C₄	—	2	1	no	—
279	C₂²⋊C₄	—	2	1	no	—
280	C₄²	—	2	1	no	—
281	C₄²	—	2	1	no	—
282	C₄²	—	2	1	no	—
283	C₄²	C₂×Dic₂₆	1	1	yes	characteristic
284	C₄²	C₂×D₅₂	1	1	yes	characteristic
285	C₄²	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
286	C₄²	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
287	C₁₃	#8605	1	1	yes	characteristic, complemented, Sylow
288	C₂³	—	2	1	no	—
289	C₂³	#88180	1	1	yes	characteristic
290	Q₈	—	2	1	no	—
291	Q₈	C₄⋊#1055	1	2	yes	—
292	D₄	—	2	2	no	—
293	D₄	—	2	2	no	—
294	D₄	—	2	1	no	—
295	D₄	C₄⋊#1055	1	2	yes	—
296	C₂×C₄	—	2	1	no	—
297	C₂×C₄	—	2	1	no	—
298	C₂×C₄	—	2	1	no	—
299	C₂×C₄	—	2	2	no	—
300	C₂×C₄	—	2	2	no	—
301	C₂×C₄	—	2	1	no	—
302	C₂×C₄	—	2	1	no	—
303	C₂×C₄	—	4	1	no	—
304	C₂×C₄	—	4	2	no	—
305	C₂×C₄	—	4	1	no	—
306	C₂×C₄	—	4	1	no	—
307	C₂×C₄	C₂²×#1055	1	1	yes	characteristic
308	C₂×C₄	C₂×#6099	1	1	yes	characteristic
309	C₂×C₄	C₂×#6099	1	1	yes	characteristic
310	C₂×C₄	C₂×#6099	1	1	yes	characteristic
311	C₂×C₄	#88178	1	1	yes	characteristic
312	C₂×C₄	#88178	1	1	yes	characteristic
313	C₂×C₄	C₂×C₄⋊Dic₁₃	1	1	yes	characteristic
314	C₂×C₄	#88184	1	1	yes	characteristic
315	C₂×C₄	#88184	1	1	yes	characteristic
316	C₂×C₄	C₂×C₂²⋊Dic₁₃	1	1	yes	characteristic
317	C₈	—	2	2	no	—
318	C₈	—	2	1	no	—
319	C₈	—	2	2	no	—
320	C₈	—	2	1	no	—
321	C₈	—	104	2	no	—
322	C₈	—	104	2	no	—
323	C₈	C₂²⋊#1055	1	2	yes	—
324	C₈	C₂²⋊#1055	1	2	yes	—
325	C₂²	—	2	1	no	—
326	C₂²	—	2	1	no	—
327	C₂²	—	2	2	no	—
328	C₂²	—	4	1	no	—
329	C₂²	—	4	1	no	—
330	C₂²	#11718586	1	1	yes	characteristic
331	C₂²	#11718587	1	1	yes	characteristic
332	C₂²	#11718873	1	1	yes	characteristic
333	C₄	—	2	1	no	—
334	C₄	—	2	1	no	—
335	C₄	—	2	1	no	—
336	C₄	—	2	2	no	—
337	C₄	—	4	1	no	—
338	C₄	—	4	1	no	—
339	C₄	C₂×C₄⋊#1055	1	1	yes	characteristic
340	C₄	#11718585	1	1	yes	characteristic
341	C₄	#11718803	1	1	yes	characteristic
342	C₄	#11718804	1	1	yes	characteristic
343	C₄	C₂×C₂²⋊#1055	1	1	yes	characteristic
344	C₄	#11718868	1	1	yes	characteristic
345	C₂	—	2	1	no	—
346	C₂	—	2	1	no	—
347	C₂	—	4	2	no	—
348	C₂	#13114816	1	1	yes	characteristic
349	C₂	#13114890	1	1	yes	characteristic
350	C₂	#13114900	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#88184
commutator	C₂×C₄×C₁₃	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₁₃
Fitting	C₄×C₁₃×#131	C₂
socle	C₂²×C₁₃	#380
prime radical	C₂×C₁₃×#41	C₂×C₄
cosocle	C₂×C₄²×C₁₃	C₂³
hypercenter	C₄×#131	D₁₃
hypocenter	C₁₃	#8605
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×#131

Complements

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

N	H ≅ G/N	type
1. #13113584	C₂	semidirect
2. C₂×#11718050	C₂	semidirect
4. #13112687	C₂	semidirect
29. C₁₃×#131	C₈	semidirect
198. #131	#1055	semidirect
287. C₁₃	#8605	semidirect

Maximal subgroups

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

subgroup	index
#13113584	2
C₂×#11718050	2
#13113223	2
#13112687	2
C₄×C₁₃×#131	2
#8605	13