Subgroups — #15762167

Order 3328.

522 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×#11718749	C₂	1	1	yes	maximal, characteristic
2	#13115785	C₂	1	1	yes	maximal, characteristic, complemented
3	#13115762	C₂	1	1	yes	maximal, characteristic
4	#13114701	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#11718072	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#11718077	C₂	1	1	yes	maximal, characteristic
7	C₂×C₁₃×#333	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₁₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
9	C₂×#88128	C₂²	1	1	yes	characteristic
10	C₂×C₁₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
11	C₂×C₁₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
12	C₂×#88061	—	2	2	no	—
13	C₂×C₈×Dic₁₃	—	2	1	no	—
14	#11718749	—	2	2	no	—
15	#11718749	C₄	1	4	yes	complemented
16	C₂×#88049	—	2	1	no	—
17	C₂×#88049	—	2	1	no	—
18	C₄×C₄⋊Dic₁₃	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊#1055	C₂²	1	1	yes	characteristic
20	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
21	C₁₃×#333	—	2	2	no	—
22	#11718142	—	2	1	no	—
23	#11718121	—	2	2	no	—
24	#11718077	—	2	2	no	—
25	#11718072	—	2	2	no	—
26	C₂²×Q₈×C₁₃	—	2	1	no	—
27	C₂²×D₅₂	—	2	1	no	—
28	C₂²×D₅₂	—	2	1	no	—
29	C₂²×D₅₂	—	2	1	no	—
30	C₂²×C₈×C₁₃	—	2	1	no	—
31	C₂²×C₈×C₁₃	—	2	1	no	—
32	C₁₃×C₄⋊Q₈	—	2	1	no	—
33	C₁₃×C₄⋊Q₈	—	2	1	no	—
34	C₁₃×C₄⋊Q₈	—	2	1	no	—
35	C₁₃×C₄⋊Q₈	C₂×C₄	1	2	yes	—
36	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
37	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
38	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
39	C₂×C₄⋊Dic₁₃	D₄	1	2	yes	—
40	C₂×C₄×Dic₁₃	—	2	1	no	—
41	C₂²×#1055	—	2	1	no	—
42	C₂²×#1055	D₄	1	1	yes	characteristic
43	C₂²×#1055	D₄	1	1	yes	characteristic
44	#88128	—	2	1	no	—
45	#88128	—	2	1	no	—
46	#88128	—	2	1	no	—
47	#88128	C₂×C₄	1	2	yes	—
48	C₁₃×C₄⋊C₈	—	2	1	no	—
49	C₁₃×C₄⋊C₈	—	2	1	no	—
50	C₁₃×C₄⋊C₈	—	2	1	no	—
51	C₁₃×C₄⋊C₈	D₄	1	2	yes	—
52	#88071	—	2	1	no	—
53	#88061	—	4	4	no	—
54	C₈×Dic₁₃	—	4	2	no	—
55	#88049	—	2	4	no	—
56	#88049	—	2	4	no	—
57	C₄⋊#1055	—	2	1	no	—
58	C₄⋊#1055	—	2	1	no	—
59	C₄×#1055	C₂×C₄	1	2	yes	—
60	C₄×#1055	D₄	1	2	yes	—
61	#9735	—	13	1	no	maximal, Sylow
62	C₂³×D₁₃	—	4	1	no	—
63	C₂×Q₈×C₁₃	—	2	2	no	—
64	C₂×Q₈×C₁₃	—	4	1	no	—
65	C₂×Q₈×C₁₃	—	8	2	no	—
66	C₂²×C₄×C₁₃	—	4	1	no	—
67	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
70	C₂²×Dic₁₃	—	2	2	no	—
71	C₂×D₅₂	—	2	2	no	—
72	C₂×D₅₂	—	2	2	no	—
73	C₂×D₅₂	—	2	2	no	—
74	C₂×D₅₂	—	4	2	no	—
75	C₂×D₅₂	—	4	1	no	—
76	C₂×D₅₂	—	4	1	no	—
77	C₂×D₅₂	—	4	2	no	—
78	C₂×D₅₂	—	4	1	no	—
79	C₂×D₅₂	—	4	1	no	—
80	C₂×D₅₂	—	4	1	no	—
81	C₂×D₅₂	—	4	1	no	—
82	C₂×D₅₂	—	4	1	no	—
83	C₂×D₅₂	—	8	2	no	—
84	C₂×C₈×C₁₃	—	2	2	no	—
85	C₂×C₈×C₁₃	—	4	1	no	—
86	C₂×C₈×C₁₃	—	4	1	no	—
87	C₂×C₈×C₁₃	—	4	1	no	—
88	C₂×C₈×C₁₃	—	4	2	no	—
89	C₁₃×C₄⋊C₄	—	2	2	no	—
90	C₁₃×C₄⋊C₄	—	2	2	no	—
91	C₁₃×C₄⋊C₄	—	4	1	no	—
92	C₁₃×C₄⋊C₄	—	4	1	no	—
93	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
94	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
95	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
96	C₄²×C₁₃	#41	1	1	yes	characteristic
97	C₄⋊Dic₁₃	—	2	2	no	—
98	C₄⋊Dic₁₃	SD₁₆	1	4	yes	—
99	C₄×Dic₁₃	—	4	1	no	—
100	C₄×Dic₁₃	—	4	1	no	—
101	C₂×#1055	—	2	1	no	—
102	C₂×#1055	—	2	1	no	—
103	C₂×#1055	—	2	1	no	—
104	C₂×#1055	—	2	1	no	—
105	C₂×#1055	—	4	1	no	—
106	C₂×#1055	—	4	2	no	—
107	C₂×#1055	C₂²⋊C₄	1	2	yes	—
108	C₂×#1055	C₂²⋊C₄	1	2	yes	—
109	C₂×#486	—	13	1	no	—
110	#1987	—	13	1	no	—
111	#1982	—	13	1	no	—
112	C₈×C₄⋊C₄	—	13	1	no	—
113	C₂×#334	—	13	1	no	—
114	C₂×#333	D₁₃	1	1	yes	characteristic, complemented
115	C₂×#331	—	13	1	no	—
116	C₂³×C₁₃	#121	1	1	yes	characteristic
117	C₂²×D₁₃	—	4	2	no	—
118	C₂²×D₁₃	—	8	1	no	—
119	C₂²×D₁₃	—	8	1	no	—
120	C₂²×D₁₃	—	8	2	no	—
121	C₂²×D₁₃	—	8	2	no	—
122	Q₈×C₁₃	—	8	2	no	—
123	Q₈×C₁₃	—	8	2	no	—
124	C₂×C₄×C₁₃	—	2	2	no	—
125	C₂×C₄×C₁₃	—	4	2	no	—
126	C₂×C₄×C₁₃	—	8	2	no	—
127	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
128	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
129	C₂×C₄×C₁₃	C₄×D₄	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	#127	1	1	yes	characteristic
133	C₂×C₄×C₁₃	C₂×SD₁₆	1	2	yes	—
134	C₂×C₄×C₁₃	#135	1	1	yes	characteristic
135	C₂×C₄×C₁₃	#135	1	1	yes	characteristic
136	C₂×C₄×C₁₃	#135	1	2	yes	—
137	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
138	C₂×C₄×C₁₃	#137	1	1	yes	characteristic
139	C₂×Dic₁₃	—	2	4	no	—
140	C₂×Dic₁₃	—	4	2	no	—
141	C₂×Dic₁₃	—	4	2	no	—
142	D₅₂	—	4	2	no	—
143	D₅₂	—	4	2	no	—
144	D₅₂	—	4	4	no	—
145	D₅₂	—	8	2	no	—
146	D₅₂	—	8	2	no	—
147	D₅₂	—	8	1	no	—
148	D₅₂	—	8	1	no	—
149	D₅₂	—	8	1	no	—
150	D₅₂	—	8	1	no	—
151	C₈×C₁₃	—	4	2	no	—
152	C₈×C₁₃	—	8	1	no	—
153	#1055	—	2	2	no	—
154	#1055	—	2	2	no	—
155	#1055	—	8	1	no	—
156	C₂×C₄⋊Q₈	C₂×D₁₃	1	1	yes	characteristic
157	C₂×#127	—	13	1	no	—
158	#486	—	13	4	no	—
159	#486	—	26	2	no	—
160	C₂×C₄⋊C₈	—	13	1	no	—
161	C₂×C₄⋊C₈	C₂×D₁₃	1	1	yes	characteristic
162	C₂×C₄⋊C₈	C₂×D₁₃	1	1	yes	characteristic
163	C₂×#102	—	26	1	no	—
164	C₂×#102	—	26	1	no	—
165	C₂×#102	—	26	2	no	—
166	C₂×C₄×C₈	—	13	1	no	—
167	C₂×C₄×C₈	—	26	1	no	—
168	C₄×C₄⋊C₄	—	13	1	no	—
169	#340	—	26	2	no	—
170	#336	—	26	1	no	—
171	#334	—	26	2	no	—
172	#333	—	2	2	no	—
173	#331	—	26	2	no	—
174	C₂²×C₁₃	#390	1	1	yes	characteristic
175	C₂²×C₁₃	#463	1	1	yes	characteristic
176	C₂²×C₁₃	#464	1	1	yes	characteristic
177	C₂²×C₁₃	#465	1	2	yes	—
178	C₂²×C₁₃	#472	1	2	yes	—
179	C₂×D₁₃	—	8	2	no	—
180	C₂×D₁₃	—	8	2	no	—
181	C₂×D₁₃	—	8	2	no	—
182	C₂×D₁₃	—	16	2	no	—
183	C₂×D₁₃	—	16	2	no	—
184	C₄×C₁₃	—	2	1	no	—
185	C₄×C₁₃	—	2	1	no	—
186	C₄×C₁₃	—	8	2	no	—
187	C₄×C₁₃	#416	1	1	yes	characteristic
188	C₄×C₁₃	#417	1	1	yes	characteristic
189	C₄×C₁₃	C₄×SD₁₆	1	2	yes	—
190	C₄×C₁₃	#463	1	1	yes	characteristic
191	C₄×C₁₃	#464	1	1	yes	characteristic
192	C₄×C₁₃	#492	1	1	yes	characteristic
193	C₄×C₁₃	#495	1	1	yes	characteristic
194	Dic₁₃	—	4	4	no	—
195	C₂²×Q₈	—	2	1	no	—
196	C₂²×D₄	—	26	1	no	—
197	C₂²×D₄	—	26	1	no	—
198	C₂²×D₄	—	26	1	no	—
199	C₂²×C₈	—	2	1	no	—
200	C₂²×C₈	—	2	1	no	—
201	C₂²×C₈	—	13	1	no	—
202	C₂²×C₈	—	13	1	no	—
203	C₂²×C₈	—	26	1	no	—
204	C₄⋊Q₈	—	2	1	no	—
205	C₄⋊Q₈	—	2	1	no	—
206	C₄⋊Q₈	—	2	1	no	—
207	C₄⋊Q₈	C₄×D₁₃	1	2	yes	—
208	#127	—	13	2	no	—
209	#127	—	26	1	no	—
210	#127	—	26	1	no	—
211	#127	—	26	1	no	—
212	C₂×C₄⋊C₄	—	2	1	no	—
213	C₂×C₄⋊C₄	—	2	1	no	—
214	C₂×C₄⋊C₄	—	13	2	no	—
215	C₂×C₄²	—	26	1	no	—
216	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
217	C₄⋊C₈	—	2	1	no	—
218	C₄⋊C₈	—	2	1	no	—
219	C₄⋊C₈	—	2	1	no	—
220	C₄⋊C₈	—	26	1	no	—
221	C₄⋊C₈	—	26	1	no	—
222	C₄⋊C₈	C₂²⋊D₁₃	1	2	yes	—
223	#102	—	26	4	no	—
224	#102	—	26	4	no	—
225	#102	—	52	4	no	—
226	C₄×C₈	—	13	2	no	—
227	C₄×C₈	—	13	2	no	—
228	C₄×C₈	—	52	2	no	—
229	#95	—	26	1	no	—
230	C₂×C₁₃	#1677	1	2	yes	—
231	C₂×C₁₃	#1678	1	2	yes	—
232	C₂×C₁₃	#1968	1	1	yes	characteristic
233	C₂×C₁₃	#1978	1	1	yes	characteristic
234	C₂×C₁₃	#1983	1	1	yes	characteristic
235	D₁₃	—	16	2	no	—
236	C₂⁴	—	52	1	no	—
237	C₂×Q₈	—	2	2	no	—
238	C₂×Q₈	—	4	1	no	—
239	C₂×Q₈	—	8	2	no	—
240	C₂×D₄	—	26	2	no	—
241	C₂×D₄	—	26	2	no	—
242	C₂×D₄	—	26	2	no	—
243	C₂×D₄	—	52	1	no	—
244	C₂×D₄	—	52	2	no	—
245	C₂×D₄	—	52	1	no	—
246	C₂×D₄	—	52	1	no	—
247	C₂×D₄	—	52	1	no	—
248	C₂×D₄	—	52	2	no	—
249	C₂×D₄	—	52	1	no	—
250	C₂×D₄	—	52	1	no	—
251	C₂×D₄	—	52	1	no	—
252	C₂×D₄	—	104	2	no	—
253	C₂²×C₄	—	4	1	no	—
254	C₂²×C₄	—	26	2	no	—
255	C₂²×C₄	#6127	1	1	yes	characteristic
256	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
257	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
258	C₂×C₈	—	2	2	no	—
259	C₂×C₈	—	4	1	no	—
260	C₂×C₈	—	4	1	no	—
261	C₂×C₈	—	4	2	no	—
262	C₂×C₈	—	4	1	no	—
263	C₂×C₈	—	13	2	no	—
264	C₂×C₈	—	13	2	no	—
265	C₂×C₈	—	26	1	no	—
266	C₂×C₈	—	26	1	no	—
267	C₂×C₈	—	26	1	no	—
268	C₂×C₈	—	26	1	no	—
269	C₂×C₈	—	52	2	no	—
270	C₂×C₈	—	52	1	no	—
271	C₄⋊C₄	—	2	2	no	—
272	C₄⋊C₄	—	2	2	no	—
273	C₄⋊C₄	—	4	1	no	—
274	C₄⋊C₄	—	4	1	no	—
275	C₄⋊C₄	—	13	4	no	—
276	C₄⋊C₄	—	26	2	no	—
277	C₄²	—	52	1	no	—
278	C₄²	—	52	1	no	—
279	C₄²	C₂×C₄×D₁₃	1	1	yes	characteristic
280	C₄²	D₄×D₁₃	1	1	yes	characteristic
281	C₄²	#6129	1	1	yes	characteristic
282	C₄²	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
283	C₁₃	#9735	1	1	yes	characteristic, complemented, Sylow
284	C₂³	—	52	2	no	—
285	C₂³	—	104	2	no	—
286	C₂³	—	104	1	no	—
287	C₂³	—	104	2	no	—
288	C₂³	—	104	1	no	—
289	C₂³	#88138	1	1	yes	characteristic
290	Q₈	—	8	2	no	—
291	Q₈	—	8	2	no	—
292	D₄	—	52	4	no	—
293	D₄	—	52	2	no	—
294	D₄	—	52	2	no	—
295	D₄	—	104	2	no	—
296	D₄	—	104	2	no	—
297	D₄	—	104	1	no	—
298	D₄	—	104	1	no	—
299	D₄	—	104	1	no	—
300	D₄	—	104	1	no	—
301	C₂×C₄	—	2	2	no	—
302	C₂×C₄	—	4	2	no	—
303	C₂×C₄	—	8	2	no	—
304	C₂×C₄	—	26	4	no	—
305	C₂×C₄	—	52	2	no	—
306	C₂×C₄	—	52	2	no	—
307	C₂×C₄	C₂²⋊C₄×D₁₃	1	1	yes	characteristic
308	C₂×C₄	#88135	1	1	yes	characteristic
309	C₂×C₄	#88139	1	1	yes	characteristic
310	C₂×C₄	SD₁₆×D₁₃	1	2	yes	—
311	C₂×C₄	#88168	1	1	yes	characteristic
312	C₂×C₄	#88170	1	1	yes	characteristic
313	C₂×C₄	#88172	1	1	yes	characteristic
314	C₂×C₄	#88173	1	2	yes	—
315	C₂×C₄	#88173	1	1	yes	characteristic
316	C₂×C₄	C₄×C₂²⋊D₁₃	1	1	yes	characteristic
317	C₂×C₄	#88193	1	1	yes	characteristic
318	C₂×C₄	#88194	1	1	yes	characteristic
319	C₈	—	4	2	no	—
320	C₈	—	8	1	no	—
321	C₈	—	26	2	no	—
322	C₈	—	26	2	no	—
323	C₈	—	104	1	no	—
324	C₂²	—	104	2	no	—
325	C₂²	—	104	2	no	—
326	C₂²	—	104	2	no	—
327	C₂²	—	208	2	no	—
328	C₂²	—	208	2	no	—
329	C₂²	#11718410	1	2	yes	—
330	C₂²	#11718459	1	1	yes	characteristic
331	C₂²	#11718468	1	2	yes	—
332	C₂²	#11718480	1	1	yes	characteristic
333	C₂²	#11718618	1	1	yes	characteristic
334	C₄	—	2	1	no	—
335	C₄	—	2	1	no	—
336	C₄	—	8	2	no	—
337	C₄	—	52	4	no	—
338	C₄	#11718449	1	2	yes	—
339	C₄	#11718464	1	1	yes	characteristic
340	C₄	#11718465	1	1	yes	characteristic
341	C₄	#11718826	1	1	yes	characteristic
342	C₄	#11718836	1	1	yes	characteristic
343	C₄	#11718846	1	1	yes	characteristic
344	C₄	#11718852	1	1	yes	characteristic
345	C₂	—	208	2	no	—
346	C₂	#13114248	1	2	yes	—
347	C₂	#13114255	1	2	yes	—
348	C₂	#13115687	1	1	yes	characteristic
349	C₂	#13115743	1	1	yes	characteristic
350	C₂	#13115765	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#88138
commutator	C₄²×C₁₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₃
Fitting	C₂×C₁₃×#333	C₂
socle	C₂³×C₁₃	#121
prime radical	C₂×#88128	C₂²
cosocle	C₂×C₄²×C₁₃	C₂³
hypercenter	C₂×#333	D₁₃
hypocenter	C₁₃	#9735
LCS	G ▷ C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#333

Complements

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

N	H ≅ G/N	type
2. #13115785	C₂	semidirect
4. #13114701	C₂	semidirect
5. C₂×#11718072	C₂	semidirect
7. C₂×C₁₃×#333	C₂	semidirect
15. #11718749	C₄	semidirect
114. C₂×#333	D₁₃	semidirect
283. C₁₃	#9735	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11718749	2
#13115785	2
#13115762	2
#13114701	2
C₂×#11718072	2
C₂×#11718077	2
C₂×C₁₃×#333	2
#9735	13