Subgroups — #15762669

Order 3328.

578 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	C₂×#11718355	C₂	1	1	yes	maximal, characteristic
2	#13115796	C₂	1	1	yes	maximal, characteristic, complemented
3	#13115641	C₂	1	1	yes	maximal, characteristic
4	#13115641	C₂	1	1	yes	maximal, characteristic
5	#13115637	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₈×#1055	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₃×#1834	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#88128	C₂²	1	1	yes	characteristic
9	C₂×#88123	C₂²	1	1	yes	characteristic
10	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
11	C₁₃×#384	C₂²	1	1	yes	characteristic
12	C₁₃×#384	C₂²	1	1	yes	characteristic
13	C₂×#88061	—	2	2	no	—
14	C₂×#88050	—	2	2	no	—
15	C₂×#88049	—	2	2	no	—
16	C₂×#88049	—	2	2	no	—
17	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
18	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
19	#11718355	—	2	2	no	—
20	#11718355	C₄	1	4	yes	complemented
21	C₁₃×#340	—	2	2	no	—
22	#11718121	—	2	2	no	—
23	C₈×#1055	—	2	2	no	—
24	C₂²×D₅₂	—	2	1	no	—
25	C₂²×D₅₂	—	2	1	no	—
26	C₂²×D₅₂	—	2	1	no	—
27	C₂²×Dic₂₆	—	2	1	no	—
28	C₂²×C₈×C₁₃	D₄	1	2	yes	—
29	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
30	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
31	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
32	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
33	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
34	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
35	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
36	C₂×C₄⋊Dic₁₃	—	2	1	no	—
37	C₂×C₄⋊Dic₁₃	—	2	1	no	—
38	C₂²×#1055	D₄	1	2	yes	—
39	C₂²×#1055	D₄	1	2	yes	—
40	#88128	—	2	1	no	—
41	#88128	—	2	1	no	—
42	#88128	—	2	1	no	—
43	#88128	C₂×C₄	1	2	yes	—
44	#88123	—	2	1	no	—
45	#88123	—	2	1	no	—
46	#88123	—	2	1	no	—
47	#88123	C₂×C₄	1	2	yes	—
48	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
49	C₄×C₈×C₁₃	D₄	1	2	yes	—
50	#88061	—	2	8	no	—
51	#88050	—	4	4	no	—
52	#88049	—	4	4	no	—
53	#88049	—	4	4	no	—
54	C₄×#1055	—	2	1	no	—
55	C₄×#1055	—	2	1	no	—
56	C₄×#1055	—	2	1	no	—
57	C₄×#1055	—	2	1	no	—
58	#9822	—	13	1	no	maximal, Sylow
59	C₂³×D₁₃	—	4	1	no	—
60	C₂²×C₄×C₁₃	—	4	1	no	—
61	C₂²×C₄×C₁₃	—	4	1	no	—
62	C₂²×C₄×C₁₃	C₂×D₄	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×Dic₁₃	—	4	1	no	—
66	C₂×D₅₂	—	2	2	no	—
67	C₂×D₅₂	—	2	2	no	—
68	C₂×D₅₂	—	2	2	no	—
69	C₂×D₅₂	—	4	2	no	—
70	C₂×D₅₂	—	4	2	no	—
71	C₂×D₅₂	—	4	2	no	—
72	C₂×D₅₂	—	4	2	no	—
73	C₂×D₅₂	—	4	2	no	—
74	C₂×D₅₂	—	4	2	no	—
75	C₂×D₅₂	—	4	1	no	—
76	C₂×D₅₂	—	4	1	no	—
77	C₂×D₅₂	—	4	1	no	—
78	C₂×Dic₂₆	—	2	2	no	—
79	C₂×Dic₂₆	—	4	2	no	—
80	C₂×Dic₂₆	—	4	2	no	—
81	C₂×Dic₂₆	—	4	1	no	—
82	C₂×C₈×C₁₃	—	2	2	no	—
83	C₂×C₈×C₁₃	—	2	2	no	—
84	C₂×C₈×C₁₃	C₂²⋊C₄	1	4	yes	—
85	C₁₃×C₄⋊C₄	—	4	2	no	—
86	C₁₃×C₄⋊C₄	—	4	2	no	—
87	C₁₃×C₄⋊C₄	—	4	2	no	—
88	C₁₃×C₄⋊C₄	—	4	2	no	—
89	C₁₃×C₄⋊C₄	—	4	2	no	—
90	C₁₃×C₄⋊C₄	—	4	2	no	—
91	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
92	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
93	C₄²×C₁₃	#41	1	1	yes	characteristic
94	C₄²×C₁₃	#41	1	1	yes	characteristic
95	C₄⋊Dic₁₃	—	2	2	no	—
96	C₄⋊Dic₁₃	—	2	2	no	—
97	C₄⋊Dic₁₃	—	4	1	no	—
98	C₄⋊Dic₁₃	—	4	1	no	—
99	C₂×#1055	—	2	2	no	—
100	C₂×#1055	—	2	2	no	—
101	C₂×#1055	—	2	2	no	—
102	C₂×#1055	—	2	2	no	—
103	C₂×#1055	D₈	1	4	yes	—
104	C₂×#1055	SD₁₆	1	4	yes	—
105	C₂×#486	—	13	1	no	—
106	#1990	—	13	1	no	—
107	#1957	—	13	1	no	—
108	#1957	—	13	1	no	—
109	#1956	—	13	1	no	—
110	#1834	D₁₃	1	1	yes	characteristic, complemented
111	C₂×C₈²	—	13	1	no	—
112	C₂³×C₁₃	#127	1	1	yes	characteristic
113	C₂²×D₁₃	—	4	2	no	—
114	C₂²×D₁₃	—	8	1	no	—
115	C₂²×D₁₃	—	8	1	no	—
116	C₂²×D₁₃	—	8	1	no	—
117	C₂²×D₁₃	—	8	1	no	—
118	C₂²×D₁₃	—	8	1	no	—
119	C₂²×D₁₃	—	8	1	no	—
120	C₂×C₄×C₁₃	—	8	1	no	—
121	C₂×C₄×C₁₃	—	8	1	no	—
122	C₂×C₄×C₁₃	—	8	1	no	—
123	C₂×C₄×C₁₃	—	8	1	no	—
124	C₂×C₄×C₁₃	—	8	1	no	—
125	C₂×C₄×C₁₃	—	8	1	no	—
126	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
127	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
128	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
129	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
130	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
131	C₂×C₄×C₁₃	#124	1	1	yes	characteristic
132	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
133	C₂×C₄×C₁₃	C₂×SD₁₆	1	2	yes	—
134	C₂×C₄×C₁₃	#135	1	2	yes	—
135	C₂×C₄×C₁₃	#135	1	2	yes	—
136	C₂×C₄×C₁₃	#135	1	2	yes	—
137	C₂×C₄×C₁₃	#135	1	2	yes	—
138	C₂×Dic₁₃	—	4	2	no	—
139	C₂×Dic₁₃	—	8	1	no	—
140	C₂×Dic₁₃	—	8	1	no	—
141	D₅₂	—	4	4	no	—
142	D₅₂	—	4	4	no	—
143	D₅₂	—	4	4	no	—
144	D₅₂	—	8	2	no	—
145	D₅₂	—	8	2	no	—
146	D₅₂	—	8	2	no	—
147	Dic₂₆	—	4	4	no	—
148	Dic₂₆	—	8	2	no	—
149	C₈×C₁₃	—	2	4	no	—
150	#1055	—	2	4	no	—
151	#1055	—	2	4	no	—
152	C₂×C₄⋊Q₈	—	13	1	no	—
153	C₂×#127	—	13	1	no	—
154	#486	—	13	4	no	—
155	#486	—	26	2	no	—
156	C₂×#103	—	26	2	no	—
157	C₂×#102	—	26	2	no	—
158	C₂×#102	—	26	2	no	—
159	C₂×#102	—	26	2	no	—
160	C₂×C₄×C₈	—	13	1	no	—
161	C₂×C₄×C₈	—	13	1	no	—
162	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
163	#384	C₂×D₁₃	1	1	yes	characteristic
164	#384	C₂×D₁₃	1	1	yes	characteristic
165	#340	—	2	2	no	—
166	#340	—	26	2	no	—
167	C₈²	—	26	2	no	—
168	C₂²×C₁₃	#390	1	1	yes	characteristic
169	C₂²×C₁₃	#492	1	1	yes	characteristic
170	C₂²×C₁₃	#493	1	1	yes	characteristic
171	C₂²×C₁₃	#495	1	1	yes	characteristic
172	C₂²×C₁₃	#495	1	1	yes	characteristic
173	C₂²×C₁₃	#495	1	1	yes	characteristic
174	C₂²×C₁₃	#495	1	1	yes	characteristic
175	C₂×D₁₃	—	8	2	no	—
176	C₂×D₁₃	—	8	2	no	—
177	C₂×D₁₃	—	8	2	no	—
178	C₂×D₁₃	—	16	1	no	—
179	C₂×D₁₃	—	16	1	no	—
180	C₂×D₁₃	—	16	1	no	—
181	C₂×D₁₃	—	16	1	no	—
182	C₄×C₁₃	—	16	1	no	—
183	C₄×C₁₃	—	16	1	no	—
184	C₄×C₁₃	#416	1	2	yes	—
185	C₄×C₁₃	C₄×D₈	1	2	yes	—
186	C₄×C₁₃	C₄×SD₁₆	1	2	yes	—
187	C₄×C₁₃	#465	1	2	yes	—
188	C₄×C₁₃	#466	1	2	yes	—
189	C₄×C₁₃	#488	1	2	yes	—
190	Dic₁₃	—	8	2	no	—
191	C₂²×Q₈	—	26	1	no	—
192	C₂²×D₄	—	26	1	no	—
193	C₂²×D₄	—	26	1	no	—
194	C₂²×D₄	—	26	1	no	—
195	C₂²×C₈	—	13	2	no	—
196	C₂²×C₈	—	13	2	no	—
197	C₂²×C₈	C₂²⋊D₁₃	1	2	yes	—
198	C₄⋊Q₈	—	13	2	no	—
199	C₄⋊Q₈	—	26	1	no	—
200	C₄⋊Q₈	—	26	1	no	—
201	C₄⋊Q₈	—	26	1	no	—
202	#127	—	13	2	no	—
203	#127	—	26	1	no	—
204	#127	—	26	1	no	—
205	#127	—	26	1	no	—
206	C₂×C₄⋊C₄	—	2	1	no	—
207	C₂×C₄⋊C₄	—	2	1	no	—
208	C₂×C₄⋊C₄	—	2	1	no	—
209	C₂×C₄⋊C₄	—	2	1	no	—
210	C₂×C₄⋊C₄	—	2	1	no	—
211	C₂×C₄⋊C₄	—	2	1	no	—
212	C₂×C₄⋊C₄	—	26	1	no	—
213	C₂×C₄⋊C₄	—	26	1	no	—
214	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
215	#103	—	52	4	no	—
216	#102	—	26	8	no	—
217	#102	—	52	4	no	—
218	#102	—	52	4	no	—
219	C₄×C₈	—	26	1	no	—
220	C₄×C₈	—	26	1	no	—
221	C₄×C₈	—	26	1	no	—
222	C₄×C₈	—	26	1	no	—
223	C₄×C₈	C₄×D₁₃	1	2	yes	—
224	C₄×C₈	D₅₂	1	2	yes	—
225	C₂×C₁₃	#1706	1	1	yes	characteristic
226	C₂×C₁₃	#1708	1	1	yes	characteristic
227	C₂×C₁₃	#1709	1	1	yes	characteristic
228	C₂×C₁₃	#1711	1	1	yes	characteristic
229	C₂×C₁₃	#1958	1	1	yes	characteristic
230	C₂×C₁₃	#1968	1	1	yes	characteristic
231	C₂×C₁₃	#1970	1	1	yes	characteristic
232	D₁₃	—	16	2	no	—
233	C₂⁴	—	52	1	no	—
234	C₂×Q₈	—	26	2	no	—
235	C₂×Q₈	—	52	1	no	—
236	C₂×Q₈	—	52	2	no	—
237	C₂×Q₈	—	52	2	no	—
238	C₂×D₄	—	26	2	no	—
239	C₂×D₄	—	26	2	no	—
240	C₂×D₄	—	26	2	no	—
241	C₂×D₄	—	52	1	no	—
242	C₂×D₄	—	52	1	no	—
243	C₂×D₄	—	52	2	no	—
244	C₂×D₄	—	52	2	no	—
245	C₂×D₄	—	52	2	no	—
246	C₂×D₄	—	52	2	no	—
247	C₂×D₄	—	52	1	no	—
248	C₂×D₄	—	52	2	no	—
249	C₂×D₄	—	52	2	no	—
250	C₂²×C₄	—	4	1	no	—
251	C₂²×C₄	—	4	1	no	—
252	C₂²×C₄	—	52	1	no	—
253	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
254	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
255	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
256	C₂×C₈	—	2	2	no	—
257	C₂×C₈	—	2	2	no	—
258	C₂×C₈	—	13	4	no	—
259	C₂×C₈	—	13	4	no	—
260	C₂×C₈	—	26	2	no	—
261	C₂×C₈	—	26	2	no	—
262	C₂×C₈	—	26	2	no	—
263	C₂×C₈	—	26	2	no	—
264	C₂×C₈	#6103	1	4	yes	—
265	C₄⋊C₄	—	4	2	no	—
266	C₄⋊C₄	—	4	2	no	—
267	C₄⋊C₄	—	4	2	no	—
268	C₄⋊C₄	—	4	2	no	—
269	C₄⋊C₄	—	4	2	no	—
270	C₄⋊C₄	—	4	2	no	—
271	C₄⋊C₄	—	26	2	no	—
272	C₄⋊C₄	—	26	2	no	—
273	C₄⋊C₄	—	52	1	no	—
274	C₄⋊C₄	—	52	1	no	—
275	C₄²	C₂×C₄×D₁₃	1	1	yes	characteristic
276	C₄²	C₂×D₅₂	1	1	yes	characteristic
277	C₄²	#6131	1	1	yes	characteristic
278	C₄²	#6131	1	1	yes	characteristic
279	C₁₃	#9822	1	1	yes	characteristic, complemented, Sylow
280	C₂³	—	52	2	no	—
281	C₂³	—	104	1	no	—
282	C₂³	—	104	1	no	—
283	C₂³	—	104	1	no	—
284	C₂³	—	104	1	no	—
285	C₂³	—	104	1	no	—
286	C₂³	—	104	1	no	—
287	C₂³	#88194	1	1	yes	characteristic
288	Q₈	—	52	4	no	—
289	Q₈	—	104	2	no	—
290	D₄	—	52	4	no	—
291	D₄	—	52	4	no	—
292	D₄	—	52	4	no	—
293	D₄	—	104	2	no	—
294	D₄	—	104	2	no	—
295	D₄	—	104	2	no	—
296	C₂×C₄	—	8	1	no	—
297	C₂×C₄	—	8	1	no	—
298	C₂×C₄	—	8	1	no	—
299	C₂×C₄	—	8	1	no	—
300	C₂×C₄	—	8	1	no	—
301	C₂×C₄	—	8	1	no	—
302	C₂×C₄	—	52	2	no	—
303	C₂×C₄	—	104	1	no	—
304	C₂×C₄	—	104	1	no	—
305	C₂×C₄	#88147	1	1	yes	characteristic
306	C₂×C₄	#88147	1	1	yes	characteristic
307	C₂×C₄	#88149	1	1	yes	characteristic
308	C₂×C₄	#88149	1	1	yes	characteristic
309	C₂×C₄	D₈×D₁₃	1	2	yes	—
310	C₂×C₄	SD₁₆×D₁₃	1	2	yes	—
311	C₂×C₄	#88170	1	2	yes	—
312	C₂×C₄	#88173	1	2	yes	—
313	C₂×C₄	C₂×#6103	1	1	yes	characteristic
314	C₂×C₄	#88201	1	1	yes	characteristic
315	C₂×C₄	#88204	1	2	yes	—
316	C₂×C₄	#88204	1	2	yes	—
317	C₈	—	2	4	no	—
318	C₈	—	26	4	no	—
319	C₈	—	26	4	no	—
320	C₂²	—	104	2	no	—
321	C₂²	—	104	2	no	—
322	C₂²	—	104	2	no	—
323	C₂²	—	208	1	no	—
324	C₂²	—	208	1	no	—
325	C₂²	—	208	1	no	—
326	C₂²	—	208	1	no	—
327	C₂²	#11718649	1	1	yes	characteristic
328	C₂²	#11718720	1	1	yes	characteristic
329	C₂²	#11718720	1	1	yes	characteristic
330	C₂²	#11718812	1	1	yes	characteristic
331	C₂²	#11718829	1	1	yes	characteristic
332	C₂²	#11718836	1	1	yes	characteristic
333	C₂²	#11718852	1	1	yes	characteristic
334	C₄	—	16	1	no	—
335	C₄	—	16	1	no	—
336	C₄	—	104	2	no	—
337	C₄	#11718513	1	2	yes	—
338	C₄	#11718525	1	2	yes	—
339	C₄	#11718533	1	2	yes	—
340	C₄	#11718544	1	2	yes	—
341	C₄	#11718663	1	2	yes	—
342	C₄	#11718729	1	2	yes	—
343	C₂	—	208	2	no	—
344	C₂	#13114427	1	1	yes	characteristic
345	C₂	#13114439	1	1	yes	characteristic
346	C₂	#13114440	1	1	yes	characteristic
347	C₂	#13114449	1	1	yes	characteristic
348	C₂	#13115645	1	1	yes	characteristic
349	C₂	#13115688	1	1	yes	characteristic
350	C₂	#13115700	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#88194
commutator	C₄²×C₁₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₃
Fitting	C₁₃×#1834	C₂
socle	C₂³×C₁₃	#127
prime radical	C₂×#88128	C₂²
cosocle	C₂×C₄²×C₁₃	C₂³
hypercenter	#1834	D₁₃
hypocenter	C₁₃	#9822
LCS	G ▷ C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1834

Complements

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

N	H ≅ G/N	type
2. #13115796	C₂	semidirect
5. #13115637	C₂	semidirect
6. C₂×C₈×#1055	C₂	semidirect
7. C₁₃×#1834	C₂	semidirect
20. #11718355	C₄	semidirect
110. #1834	D₁₃	semidirect
279. C₁₃	#9822	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11718355	2
#13115796	2
#13115641	2
#13115637	2
C₂×C₈×#1055	2
C₁₃×#1834	2
#9822	13