Subgroups — #15850624

Order 3328.

706 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13125422	C₂	1	1	yes	maximal, characteristic
2	#13125418	C₂	1	2	yes	maximal
3	#13125346	C₂	1	2	yes	maximal
4	#13114453	C₂	1	1	yes	maximal, characteristic, complemented
5	#13114452	C₂	1	1	yes	maximal, characteristic, complemented
6	#13114444	C₂	1	2	yes	maximal, complemented
7	#13114434	C₂	1	1	yes	maximal, characteristic, complemented
8	#13114432	C₂	1	2	yes	maximal, complemented
9	#13114430	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₈×#1055	C₂	1	1	yes	maximal, characteristic
11	C₁₃×#3170	C₂	1	1	yes	maximal, characteristic
12	C₁₃×#533	C₂²	1	2	yes	—
13	#11719175	C₂²	1	2	yes	—
14	C₁₃×#500	C₂²	1	1	yes	characteristic
15	C₁₃×#494	C₂²	1	1	yes	characteristic
16	C₁₃×#487	C₂²	1	2	yes	—
17	C₁₃×#467	—	2	2	no	—
18	C₄×C₁₃×Q₁₆	—	2	2	no	—
19	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
20	#11718777	—	2	2	no	—
21	#11718754	C₂²	1	2	yes	—
22	#11718753	C₂²	1	2	yes	—
23	#11718742	C₂²	1	2	yes	—
24	#11718740	C₂²	1	2	yes	—
25	#11718740	C₂²	1	2	yes	—
26	#11718739	C₂²	1	2	yes	—
27	#11718739	C₂²	1	2	yes	—
28	#11718739	C₂²	1	2	yes	—
29	#11718725	—	2	4	no	—
30	#11718712	—	2	4	no	—
31	C₄×#6107	—	2	4	no	—
32	#11718659	—	2	4	no	—
33	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
34	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
35	#11718360	C₂²	1	1	yes	characteristic
36	C₄×Dic₅₂	—	2	2	no	—
37	#11718343	C₂²	1	1	yes	characteristic
38	#11718342	C₂²	1	2	yes	—
39	C₈×#1055	C₂²	1	1	yes	characteristic
40	C₈×#1055	C₂²	1	1	yes	characteristic
41	C₈×#1055	C₂²	1	1	yes	characteristic
42	C₈×#1055	C₂²	1	1	yes	characteristic
43	C₂×C₁₃×Q₁₆	—	2	2	no	—
44	C₂²×C₈×C₁₃	D₄	1	2	yes	—
45	C₁₃×C₄⋊Q₈	C₂³	1	2	yes	—
46	C₁₃×C₄⋊Q₈	C₂³	1	2	yes	—
47	C₁₃×#125	—	2	2	no	—
48	C₁₃×C₂²⋊Q₈	—	2	2	no	—
49	C₁₃×C₂²⋊Q₈	—	2	2	no	—
50	C₄×Q₈×C₁₃	—	2	2	no	—
51	C₄×Q₈×C₁₃	—	2	2	no	—
52	C₁₃×#117	—	2	2	no	—
53	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
54	C₂×#6107	—	2	4	no	—
55	#88180	—	2	2	no	—
56	#88178	—	2	2	no	—
57	#88178	—	2	2	no	—
58	C₂²×#1055	D₄	1	2	yes	—
59	C₂²×#1055	D₄	1	2	yes	—
60	C₂×Dic₅₂	—	2	2	no	—
61	#88124	—	2	2	no	—
62	#88123	C₂³	1	2	yes	—
63	#88123	C₂³	1	2	yes	—
64	C₄×Dic₂₆	—	2	2	no	—
65	C₄×Dic₂₆	—	2	2	no	—
66	C₁₃×#107	—	2	2	no	—
67	C₁₃×#103	—	2	4	no	—
68	C₄×C₈×C₁₃	D₄	1	2	yes	—
69	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
70	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
71	#88075	—	2	4	no	—
72	#88058	—	2	2	no	—
73	#88056	—	2	4	no	—
74	#88050	—	2	4	no	—
75	#88048	—	2	4	no	—
76	#88048	—	2	4	no	—
77	#88047	—	2	4	no	—
78	#88047	—	2	4	no	—
79	#88047	—	2	4	no	—
80	C₄×#1055	D₄	1	2	yes	—
81	C₄×#1055	D₄	1	2	yes	—
82	C₄×#1055	C₂³	1	1	yes	characteristic
83	C₄×#1055	C₂³	1	1	yes	characteristic
84	C₄×#1055	C₂³	1	1	yes	characteristic
85	C₄×#1055	C₂³	1	1	yes	characteristic
86	#19604	—	13	1	no	maximal, Sylow
87	C₂×Q₈×C₁₃	—	2	2	no	—
88	C₂×Q₈×C₁₃	—	2	2	no	—
89	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
90	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
91	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
92	C₂×Dic₂₆	—	2	2	no	—
93	C₂×Dic₂₆	—	2	2	no	—
94	C₁₃×Q₁₆	—	4	4	no	—
95	C₂×C₈×C₁₃	—	2	4	no	—
96	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
97	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
98	C₁₃×C₄⋊C₄	—	2	2	no	—
99	C₁₃×C₄⋊C₄	—	2	2	no	—
100	C₁₃×C₄⋊C₄	—	2	2	no	—
101	C₁₃×C₄⋊C₄	—	2	2	no	—
102	C₁₃×C₄⋊C₄	—	4	2	no	—
103	C₁₃×C₄⋊C₄	—	4	2	no	—
104	C₁₃×C₂²⋊C₄	—	4	2	no	—
105	C₄²×C₁₃	—	4	2	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	C₂⁴	1	1	yes	characteristic
110	C₂²⋊Dic₁₃	—	4	2	no	—
111	#6107	—	4	8	no	—
112	C₄⋊Dic₁₃	—	2	2	no	—
113	C₄⋊Dic₁₃	—	2	2	no	—
114	C₄⋊Dic₁₃	—	2	2	no	—
115	C₄⋊Dic₁₃	—	2	2	no	—
116	#6101	—	4	2	no	—
117	#6101	—	4	2	no	—
118	C₄×Dic₁₃	—	4	2	no	—
119	C₂×#1055	—	2	4	no	—
120	C₂×#1055	Q₁₆	1	8	yes	complemented
121	C₂×#1055	C₂×D₄	1	2	yes	—
122	C₂×#1055	C₂×D₄	1	2	yes	—
123	C₂×#1055	C₂×D₄	1	2	yes	—
124	C₂×#1055	C₂×D₄	1	2	yes	—
125	Dic₅₂	—	4	4	no	—
126	#3170	—	13	1	no	—
127	#3170	—	13	2	no	—
128	#3170	D₁₃	1	1	yes	characteristic
129	#3160	—	13	2	no	—
130	#1713	—	13	1	no	—
131	#1713	—	13	1	no	—
132	#1710	—	13	2	no	—
133	#1707	—	13	1	no	—
134	#1707	—	13	1	no	—
135	#1707	—	13	2	no	—
136	C₂×C₈²	—	13	1	no	—
137	C₂³×C₁₃	#127	1	1	yes	characteristic
138	Q₈×C₁₃	—	4	4	no	—
139	Q₈×C₁₃	—	4	4	no	—
140	C₂×C₄×C₁₃	—	2	2	no	—
141	C₂×C₄×C₁₃	—	2	2	no	—
142	C₂×C₄×C₁₃	—	4	2	no	—
143	C₂×C₄×C₁₃	—	4	2	no	—
144	C₂×C₄×C₁₃	#127	1	1	yes	characteristic
145	C₂×C₄×C₁₃	#127	1	1	yes	characteristic
146	C₂×C₄×C₁₃	#127	1	1	yes	characteristic
147	C₂×C₄×C₁₃	C₂×Q₁₆	1	4	yes	—
148	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
149	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
150	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
151	C₂×Dic₁₃	—	4	2	no	—
152	C₂×Dic₁₃	—	4	2	no	—
153	Dic₂₆	—	4	4	no	—
154	Dic₂₆	—	4	4	no	—
155	C₈×C₁₃	—	2	2	no	—
156	C₈×C₁₃	#135	1	4	yes	—
157	#1055	—	2	2	no	—
158	#1055	C₂×Q₁₆	1	4	yes	—
159	#1055	C₂×Q₁₆	1	4	yes	—
160	#1055	#135	1	4	yes	—
161	#533	—	13	2	no	—
162	#533	C₂×D₁₃	1	2	yes	—
163	#500	—	13	2	no	—
164	#500	—	13	2	no	—
165	#500	—	13	2	no	—
166	#500	—	13	1	no	—
167	#500	C₂×D₁₃	1	1	yes	characteristic
168	#499	—	26	4	no	—
169	#498	—	13	2	no	—
170	#498	—	13	2	no	—
171	#494	—	13	2	no	—
172	#494	—	13	1	no	—
173	#494	C₂×D₁₃	1	1	yes	characteristic
174	#487	—	13	2	no	—
175	#487	—	13	2	no	—
176	#487	—	13	2	no	—
177	#487	C₂×D₁₃	1	2	yes	—
178	#467	—	2	2	no	—
179	#467	—	26	4	no	—
180	#467	—	26	2	no	—
181	C₄×Q₁₆	—	2	2	no	—
182	C₄×Q₁₆	—	26	4	no	—
183	C₄×Q₁₆	—	26	2	no	—
184	#427	—	26	4	no	—
185	C₂×C₄×C₈	—	13	1	no	—
186	C₂×C₄×C₈	—	13	1	no	—
187	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
188	C₈²	—	13	1	no	—
189	C₈²	—	13	1	no	—
190	C₈²	—	13	1	no	—
191	C₈²	—	13	1	no	—
192	C₂²×C₁₃	—	2	1	no	—
193	C₂²×C₁₃	—	2	1	no	—
194	C₂²×C₁₃	#494	1	2	yes	—
195	C₂²×C₁₃	C₂×#127	1	1	yes	characteristic
196	C₄×C₁₃	—	2	1	no	—
197	C₄×C₁₃	—	2	1	no	—
198	C₄×C₁₃	—	8	2	no	—
199	C₄×C₁₃	—	8	2	no	—
200	C₄×C₁₃	#494	1	2	yes	—
201	C₄×C₁₃	C₂²×Q₁₆	1	2	yes	—
202	C₄×C₁₃	C₂×#135	1	2	yes	—
203	C₄×C₁₃	C₂×#135	1	2	yes	—
204	Dic₁₃	—	8	2	no	—
205	Dic₁₃	—	8	2	no	—
206	C₂×Q₁₆	—	2	2	no	—
207	C₂×Q₁₆	—	26	4	no	—
208	C₂×Q₁₆	—	26	2	no	—
209	C₂²×C₈	—	13	2	no	—
210	C₂²×C₈	—	13	2	no	—
211	C₂²×C₈	C₂²⋊D₁₃	1	2	yes	—
212	C₄⋊Q₈	—	13	2	no	—
213	C₄⋊Q₈	—	13	2	no	—
214	C₄⋊Q₈	C₂²×D₁₃	1	2	yes	—
215	C₄⋊Q₈	C₂²×D₁₃	1	2	yes	—
216	#125	—	2	2	no	—
217	#125	—	26	2	no	—
218	C₂²⋊Q₈	—	2	2	no	—
219	C₂²⋊Q₈	—	2	2	no	—
220	C₂²⋊Q₈	—	26	2	no	—
221	C₂²⋊Q₈	—	26	2	no	—
222	C₄×Q₈	—	2	2	no	—
223	C₄×Q₈	—	2	2	no	—
224	C₄×Q₈	—	26	2	no	—
225	C₄×Q₈	—	26	2	no	—
226	#117	—	2	2	no	—
227	#117	—	26	2	no	—
228	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
229	#107	—	2	2	no	—
230	#107	—	26	4	no	—
231	#107	—	26	4	no	—
232	#107	—	26	4	no	—
233	#107	—	26	2	no	—
234	#106	—	26	4	no	—
235	#106	—	26	4	no	—
236	#103	—	2	4	no	—
237	#103	—	26	4	no	—
238	#103	—	26	4	no	—
239	#103	—	26	4	no	—
240	C₄×C₈	—	13	2	no	—
241	C₄×C₈	—	13	2	no	—
242	C₄×C₈	—	13	1	no	—
243	C₄×C₈	—	13	1	no	—
244	C₄×C₈	—	13	1	no	—
245	C₄×C₈	—	13	1	no	—
246	C₄×C₈	C₂²⋊D₁₃	1	2	yes	—
247	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
248	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
249	C₂×C₁₃	—	2	2	no	—
250	C₂×C₁₃	C₂×#494	1	1	yes	characteristic
251	C₂×C₁₃	#3147	1	1	yes	characteristic
252	C₂×C₁₃	#3147	1	1	yes	characteristic
253	C₂×Q₈	—	2	2	no	—
254	C₂×Q₈	—	2	2	no	—
255	C₂×Q₈	—	26	2	no	—
256	C₂×Q₈	—	26	2	no	—
257	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
258	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
259	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
260	Q₁₆	—	4	4	no	—
261	Q₁₆	—	52	8	no	—
262	Q₁₆	—	52	4	no	—
263	C₂×C₈	—	2	4	no	—
264	C₂×C₈	—	13	8	no	—
265	C₂×C₈	—	13	2	no	—
266	C₂×C₈	—	13	2	no	—
267	C₂×C₈	—	13	2	no	—
268	C₂×C₈	—	13	2	no	—
269	C₂×C₈	—	26	4	no	—
270	C₂×C₈	C₂×C₂²⋊D₁₃	1	2	yes	—
271	C₂×C₈	C₂×C₂²⋊D₁₃	1	2	yes	—
272	C₄⋊C₄	—	2	2	no	—
273	C₄⋊C₄	—	2	2	no	—
274	C₄⋊C₄	—	2	2	no	—
275	C₄⋊C₄	—	2	2	no	—
276	C₄⋊C₄	—	4	2	no	—
277	C₄⋊C₄	—	4	2	no	—
278	C₄⋊C₄	—	26	2	no	—
279	C₄⋊C₄	—	26	2	no	—
280	C₄⋊C₄	—	26	2	no	—
281	C₄⋊C₄	—	26	2	no	—
282	C₄⋊C₄	—	52	2	no	—
283	C₄⋊C₄	—	52	2	no	—
284	C₂²⋊C₄	—	4	2	no	—
285	C₂²⋊C₄	—	52	2	no	—
286	C₄²	—	4	2	no	—
287	C₄²	—	52	2	no	—
288	C₄²	D₄×D₁₃	1	1	yes	characteristic
289	C₄²	D₄×D₁₃	1	1	yes	characteristic
290	C₄²	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
291	C₄²	C₂³×D₁₃	1	1	yes	characteristic
292	C₁₃	#19604	1	1	yes	characteristic, complemented, Sylow
293	C₂³	#88194	1	1	yes	characteristic
294	Q₈	—	4	4	no	—
295	Q₈	—	4	4	no	—
296	Q₈	—	52	4	no	—
297	Q₈	—	52	4	no	—
298	C₂×C₄	—	2	2	no	—
299	C₂×C₄	—	2	2	no	—
300	C₂×C₄	—	4	2	no	—
301	C₂×C₄	—	4	2	no	—
302	C₂×C₄	—	52	2	no	—
303	C₂×C₄	—	52	2	no	—
304	C₂×C₄	Q₁₆×D₁₃	1	4	yes	—
305	C₂×C₄	#88194	1	1	yes	characteristic
306	C₂×C₄	#88194	1	1	yes	characteristic
307	C₂×C₄	#88194	1	1	yes	characteristic
308	C₂×C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
309	C₂×C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
310	C₂×C₄	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
311	C₈	—	2	2	no	—
312	C₈	—	13	4	no	—
313	C₈	—	13	4	no	—
314	C₈	—	13	4	no	—
315	C₈	—	26	2	no	—
316	C₈	#88204	1	4	yes	—
317	C₂²	—	2	1	no	—
318	C₂²	—	2	1	no	—
319	C₂²	#11718842	1	2	yes	—
320	C₂²	C₂×#88194	1	1	yes	characteristic
321	C₄	—	2	1	no	—
322	C₄	—	2	1	no	—
323	C₄	—	8	2	no	—
324	C₄	—	8	2	no	—
325	C₄	—	104	2	no	—
326	C₄	—	104	2	no	—
327	C₄	#11718842	1	2	yes	—
328	C₄	C₂×Q₁₆×D₁₃	1	2	yes	—
329	C₄	D₁₃×#135	1	2	yes	—
330	C₄	C₂×#88204	1	2	yes	—
331	C₂	—	2	2	no	—
332	C₂	C₂×#11718842	1	1	yes	characteristic
333	C₂	#13125260	1	1	yes	characteristic
334	C₂	#13125262	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#88194
commutator, cosocle	C₄²×C₁₃	C₂⁴
Frattini	C₄²	C₂³×D₁₃
Fitting	C₁₃×#3170	C₂
socle	C₂²×C₁₃	C₂×#127
prime radical	C₂³×C₁₃	#127
hypercenter	#3170	D₁₃
hypocenter	C₁₃	#19604
LCS	G ▷ C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3170

Complements

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

N	H ≅ G/N	type
4. #13114453	C₂	semidirect
5. #13114452	C₂	semidirect
6. #13114444	C₂	semidirect
7. #13114434	C₂	semidirect
8. #13114432	C₂	semidirect
9. #13114430	C₂	semidirect
120. C₂×#1055	Q₁₆	semidirect
292. C₁₃	#19604	semidirect

Maximal subgroups

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

subgroup	index
#13125422	2
#13125418	2
#13125346	2
#13114453	2
#13114452	2
#13114444	2
#13114434	2
#13114432	2
#13114430	2
C₂×C₈×#1055	2
C₁₃×#3170	2
#19604	13