Subgroups — #15753873

Order 3328.

684 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	C₄×#88128	C₂	1	1	yes	maximal, characteristic
2	#13114689	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#11718077	C₂	1	2	yes	maximal
4	#13112720	C₂	1	2	yes	maximal, complemented
5	C₁₃×#1767	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#88128	C₂²	1	1	yes	characteristic
7	C₂×C₄×D₅₂	—	2	1	no	—
8	C₂×C₄×D₅₂	—	2	1	no	—
9	C₂×C₄×D₅₂	C₄	1	2	yes	—
10	C₂×C₁₃×C₄⋊C₈	C₂²	1	2	yes	—
11	C₂×C₁₃×#97	—	2	1	no	—
12	C₄³×C₁₃	C₂²	1	1	yes	characteristic
13	#11718600	—	2	1	no	—
14	#11718600	—	2	1	no	—
15	#11718600	C₄	1	2	yes	—
16	#11718596	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊#1055	C₂²	1	2	yes	—
18	C₂×#88043	—	2	1	no	—
19	C₁₃×#336	—	2	1	no	—
20	#11718116	—	2	1	no	—
21	#11718077	C₄	1	8	yes	complemented
22	C₂²×D₅₂	—	2	1	no	—
23	C₂²×D₅₂	—	2	1	no	—
24	C₂²×D₅₂	C₂×C₄	1	2	yes	—
25	C₂²×C₄×D₁₃	—	2	2	no	—
26	C₂²×C₈×C₁₃	—	2	2	no	—
27	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
30	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
31	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
32	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
33	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
34	C₂×#6103	—	2	2	no	—
35	C₂×#6103	—	2	2	no	—
36	C₂×#6103	—	2	2	no	—
37	C₂×C₄⋊Dic₁₃	—	2	1	no	—
38	C₂×C₄⋊Dic₁₃	—	2	1	no	—
39	C₂×C₄⋊Dic₁₃	C₂×C₄	1	2	yes	—
40	C₂²×#1055	—	2	2	no	—
41	#88128	—	2	2	no	—
42	#88128	C₂×C₄	1	4	yes	—
43	C₄×D₅₂	—	4	4	no	—
44	C₄×D₅₂	—	4	4	no	—
45	C₄×D₅₂	—	4	4	no	—
46	C₁₃×C₄⋊C₈	—	2	2	no	—
47	C₁₃×C₄⋊C₈	C₂×C₄	1	4	yes	complemented
48	C₁₃×#97	—	4	2	no	—
49	C₄⋊#1055	—	2	2	no	—
50	C₄⋊#1055	C₂×C₄	1	4	yes	complemented
51	#88043	—	4	2	no	—
52	#8231	—	13	1	no	maximal, Sylow
53	C₂³×D₁₃	—	2	2	no	—
54	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
55	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
56	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
57	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
58	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
59	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
61	C₂²×Dic₁₃	—	2	2	no	—
62	C₂×D₅₂	—	2	2	no	—
63	C₂×D₅₂	—	2	4	no	—
64	C₂×D₅₂	—	2	4	no	—
65	C₂×D₅₂	—	4	2	no	—
66	C₂×D₅₂	—	4	2	no	—
67	C₂×D₅₂	—	4	2	no	—
68	C₂×D₅₂	—	4	2	no	—
69	C₂×D₅₂	—	4	1	no	—
70	C₂×D₅₂	—	4	1	no	—
71	C₂×D₅₂	—	4	1	no	—
72	C₂×D₅₂	—	4	1	no	—
73	C₂×D₅₂	C₄²	1	4	yes	—
74	C₂×C₄×D₁₃	—	4	4	no	—
75	C₂×C₄×D₁₃	—	4	4	no	—
76	C₂×C₄×D₁₃	—	4	4	no	—
77	C₂×C₈×C₁₃	—	2	4	no	—
78	C₂×C₈×C₁₃	—	4	2	no	—
79	C₂×C₈×C₁₃	—	4	2	no	—
80	C₄²×C₁₃	—	2	2	no	—
81	C₄²×C₁₃	—	2	1	no	—
82	C₄²×C₁₃	—	2	2	no	—
83	C₄²×C₁₃	—	2	2	no	—
84	C₄²×C₁₃	—	2	1	no	—
85	C₄²×C₁₃	—	2	1	no	—
86	C₄²×C₁₃	—	2	1	no	—
87	C₄²×C₁₃	—	4	1	no	—
88	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
89	C₄²×C₁₃	#41	1	2	yes	—
90	#6103	—	4	4	no	—
91	#6103	—	4	4	no	—
92	#6103	—	4	4	no	—
93	C₄⋊Dic₁₃	—	4	2	no	—
94	C₄⋊Dic₁₃	—	4	2	no	—
95	C₄⋊Dic₁₃	—	4	2	no	—
96	C₂×#1055	—	2	4	no	—
97	C₂×#1055	—	4	2	no	—
98	C₂×#1055	—	4	2	no	—
99	C₄×#127	—	13	1	no	—
100	#1767	—	13	1	no	—
101	#1767	D₁₃	1	1	yes	characteristic, complemented
102	C₂×#331	—	13	2	no	—
103	#1296	—	13	2	no	—
104	C₂³×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
105	C₂²×D₁₃	—	2	4	no	—
106	C₂²×D₁₃	—	4	2	no	—
107	C₂²×D₁₃	—	4	2	no	—
108	C₂²×D₁₃	—	4	2	no	—
109	C₂²×D₁₃	—	4	2	no	—
110	C₂²×D₁₃	—	4	2	no	—
111	C₂²×D₁₃	—	4	2	no	—
112	C₂×C₄×C₁₃	—	2	1	no	—
113	C₂×C₄×C₁₃	—	2	1	no	—
114	C₂×C₄×C₁₃	—	2	1	no	—
115	C₂×C₄×C₁₃	—	2	1	no	—
116	C₂×C₄×C₁₃	—	2	1	no	—
117	C₂×C₄×C₁₃	—	2	1	no	—
118	C₂×C₄×C₁₃	—	2	1	no	—
119	C₂×C₄×C₁₃	—	2	1	no	—
120	C₂×C₄×C₁₃	—	2	1	no	—
121	C₂×C₄×C₁₃	—	2	1	no	—
122	C₂×C₄×C₁₃	—	2	1	no	—
123	C₂×C₄×C₁₃	—	2	1	no	—
124	C₂×C₄×C₁₃	—	2	1	no	—
125	C₂×C₄×C₁₃	#100	1	2	yes	—
126	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
127	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
128	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
129	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
130	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	2	yes	—
131	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	2	yes	—
132	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
133	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
134	C₂×C₄×C₁₃	#137	1	1	yes	characteristic
135	C₂×C₄×C₁₃	#137	1	1	yes	characteristic
136	C₂×Dic₁₃	—	4	2	no	—
137	C₂×Dic₁₃	—	4	2	no	—
138	C₂×Dic₁₃	—	4	2	no	—
139	D₅₂	—	4	4	no	—
140	D₅₂	—	4	4	no	—
141	D₅₂	—	4	4	no	—
142	D₅₂	—	4	4	no	—
143	D₅₂	—	4	4	no	—
144	D₅₂	—	4	4	no	—
145	C₄×D₁₃	—	8	8	no	—
146	C₈×C₁₃	—	4	4	no	—
147	#1055	—	4	4	no	—
148	C₂×#127	—	13	1	no	—
149	C₂×C₄×D₄	—	13	2	no	—
150	C₂×C₄×D₄	—	26	1	no	—
151	C₂×C₄×D₄	—	26	1	no	—
152	C₂×C₄⋊C₈	—	13	2	no	—
153	C₂×C₄⋊C₈	C₂×D₁₃	1	2	yes	—
154	C₂×#97	—	2	1	no	—
155	C₂×#97	—	26	1	no	—
156	#390	—	13	2	no	—
157	#390	—	26	1	no	—
158	#390	—	26	1	no	—
159	#384	—	13	1	no	—
160	C₄³	C₂×D₁₃	1	1	yes	characteristic
161	#336	—	2	1	no	—
162	#336	—	26	1	no	—
163	#331	—	13	8	no	—
164	C₂²×C₁₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
165	C₂²×C₁₃	C₂×#100	1	1	yes	characteristic
166	C₂²×C₁₃	#413	1	1	yes	characteristic
167	C₂²×C₁₃	#417	1	1	yes	characteristic
168	C₂²×C₁₃	#417	1	1	yes	characteristic
169	C₂²×C₁₃	#418	1	1	yes	characteristic
170	C₂²×C₁₃	#418	1	1	yes	characteristic
171	C₂×D₁₃	—	4	4	no	—
172	C₂×D₁₃	—	4	4	no	—
173	C₂×D₁₃	—	4	4	no	—
174	C₂×D₁₃	—	8	2	no	—
175	C₂×D₁₃	—	8	2	no	—
176	C₂×D₁₃	—	8	2	no	—
177	C₂×D₁₃	—	8	2	no	—
178	C₄×C₁₃	—	2	2	no	—
179	C₄×C₁₃	—	2	1	no	—
180	C₄×C₁₃	—	2	1	no	—
181	C₄×C₁₃	—	4	1	no	—
182	C₄×C₁₃	—	4	1	no	—
183	C₄×C₁₃	—	4	1	no	—
184	C₄×C₁₃	#440	1	2	yes	—
185	C₄×C₁₃	#440	1	2	yes	—
186	C₄×C₁₃	#442	1	2	yes	—
187	C₄×C₁₃	#442	1	2	yes	—
188	Dic₁₃	—	8	2	no	—
189	C₂²×D₄	—	13	2	no	—
190	C₂²×D₄	—	26	1	no	—
191	C₂²×D₄	—	26	1	no	—
192	C₂³×C₄	—	26	2	no	—
193	C₂²×C₈	—	2	2	no	—
194	C₂²×C₈	—	26	2	no	—
195	#127	—	13	4	no	—
196	#127	—	26	2	no	—
197	C₄×D₄	—	52	4	no	—
198	C₄×D₄	—	52	4	no	—
199	C₄×D₄	—	52	4	no	—
200	C₂×C₄⋊C₄	—	13	2	no	—
201	C₂×C₄⋊C₄	—	26	1	no	—
202	C₂×C₄⋊C₄	—	26	1	no	—
203	C₂×C₂²⋊C₄	—	26	2	no	—
204	C₂×C₂²⋊C₄	—	26	2	no	—
205	C₂×C₂²⋊C₄	—	26	2	no	—
206	C₂×C₄²	C₄×D₁₃	1	1	yes	characteristic
207	C₂×C₄²	C₄×D₁₃	1	1	yes	characteristic
208	C₂×C₄²	D₅₂	1	1	yes	characteristic
209	C₂×C₄²	D₅₂	1	1	yes	characteristic
210	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
211	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
212	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
213	C₄⋊C₈	—	2	2	no	—
214	C₄⋊C₈	—	13	4	no	—
215	C₄⋊C₈	—	26	2	no	—
216	C₄⋊C₈	C₄×D₁₃	1	4	yes	complemented
217	#97	—	4	2	no	—
218	#97	—	52	2	no	—
219	C₂×C₁₃	#1555	1	1	yes	characteristic
220	C₂×C₁₃	#1557	1	1	yes	characteristic
221	C₂×C₁₃	#1557	1	1	yes	characteristic
222	C₂×C₁₃	#1560	1	1	yes	characteristic
223	C₂×C₁₃	C₄×#100	1	1	yes	characteristic
224	C₂×C₁₃	#1762	1	1	yes	characteristic
225	C₂×C₁₃	#1762	1	1	yes	characteristic
226	D₁₃	—	8	4	no	—
227	C₂⁴	—	26	2	no	—
228	C₂×D₄	—	13	4	no	—
229	C₂×D₄	—	26	2	no	—
230	C₂×D₄	—	26	4	no	—
231	C₂×D₄	—	26	4	no	—
232	C₂×D₄	—	52	2	no	—
233	C₂×D₄	—	52	2	no	—
234	C₂×D₄	—	52	2	no	—
235	C₂×D₄	—	52	1	no	—
236	C₂×D₄	—	52	1	no	—
237	C₂×D₄	—	52	1	no	—
238	C₂×D₄	—	52	1	no	—
239	C₂×D₄	—	52	2	no	—
240	C₂²×C₄	—	26	2	no	—
241	C₂²×C₄	—	52	4	no	—
242	C₂²×C₄	—	52	4	no	—
243	C₂²×C₄	—	52	4	no	—
244	C₂²×C₄	#6103	1	1	yes	characteristic
245	C₂²×C₄	#6103	1	1	yes	characteristic
246	C₂²×C₄	#6103	1	1	yes	characteristic
247	C₂²×C₄	#6103	1	1	yes	characteristic
248	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
249	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
250	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
251	C₂×C₈	—	2	4	no	—
252	C₂×C₈	—	4	2	no	—
253	C₂×C₈	—	4	2	no	—
254	C₂×C₈	—	26	4	no	—
255	C₂×C₈	—	52	2	no	—
256	C₂×C₈	—	52	2	no	—
257	C₄⋊C₄	—	52	2	no	—
258	C₄⋊C₄	—	52	2	no	—
259	C₄⋊C₄	—	52	2	no	—
260	C₂²⋊C₄	—	52	4	no	—
261	C₂²⋊C₄	—	52	4	no	—
262	C₂²⋊C₄	—	52	4	no	—
263	C₄²	—	2	1	no	—
264	C₄²	—	2	2	no	—
265	C₄²	—	2	1	no	—
266	C₄²	—	2	1	no	—
267	C₄²	—	2	1	no	—
268	C₄²	—	2	2	no	—
269	C₄²	—	2	2	no	—
270	C₄²	—	4	1	no	—
271	C₄²	C₂×C₄×D₁₃	1	2	yes	—
272	C₄²	#6127	1	2	yes	—
273	C₁₃	#8231	1	1	yes	characteristic, complemented, Sylow
274	C₂³	—	26	4	no	—
275	C₂³	—	52	2	no	—
276	C₂³	—	52	2	no	—
277	C₂³	—	52	2	no	—
278	C₂³	—	52	2	no	—
279	C₂³	—	52	2	no	—
280	C₂³	—	52	2	no	—
281	C₂³	C₂×#6103	1	1	yes	characteristic
282	D₄	—	52	4	no	—
283	D₄	—	52	4	no	—
284	D₄	—	52	4	no	—
285	D₄	—	52	4	no	—
286	D₄	—	52	4	no	—
287	D₄	—	52	4	no	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	2	1	no	—
296	C₂×C₄	—	2	1	no	—
297	C₂×C₄	—	2	1	no	—
298	C₂×C₄	—	2	1	no	—
299	C₂×C₄	—	2	1	no	—
300	C₂×C₄	—	2	1	no	—
301	C₂×C₄	—	52	2	no	—
302	C₂×C₄	—	52	2	no	—
303	C₂×C₄	—	52	2	no	—
304	C₂×C₄	—	104	8	no	—
305	C₂×C₄	#88063	1	2	yes	—
306	C₂×C₄	C₄²×D₁₃	1	1	yes	characteristic
307	C₂×C₄	#88126	1	1	yes	characteristic
308	C₂×C₄	C₄×D₅₂	1	2	yes	—
309	C₂×C₄	#88162	1	2	yes	—
310	C₂×C₄	#88162	1	1	yes	characteristic
311	C₂×C₄	#88163	1	1	yes	characteristic
312	C₂×C₄	C₄×C₂²⋊D₁₃	1	2	yes	—
313	C₂×C₄	#88186	1	1	yes	characteristic
314	C₂×C₄	#88196	1	1	yes	characteristic
315	C₂×C₄	#88203	1	2	yes	—
316	C₈	—	4	4	no	—
317	C₈	—	52	4	no	—
318	C₂²	—	52	4	no	—
319	C₂²	—	52	4	no	—
320	C₂²	—	52	4	no	—
321	C₂²	—	104	2	no	—
322	C₂²	—	104	2	no	—
323	C₂²	—	104	2	no	—
324	C₂²	—	104	2	no	—
325	C₂²	C₄×#6103	1	1	yes	characteristic
326	C₂²	#11718627	1	1	yes	characteristic
327	C₂²	#11718662	1	1	yes	characteristic
328	C₂²	C₂×#88063	1	1	yes	characteristic
329	C₂²	#11718792	1	1	yes	characteristic
330	C₂²	#11718793	1	1	yes	characteristic
331	C₂²	#11718794	1	1	yes	characteristic
332	C₄	—	2	1	no	—
333	C₄	—	2	1	no	—
334	C₄	—	2	2	no	—
335	C₄	—	4	1	no	—
336	C₄	—	4	1	no	—
337	C₄	—	4	1	no	—
338	C₄	—	104	2	no	—
339	C₄	#11718371	1	2	yes	—
340	C₄	#11718372	1	2	yes	—
341	C₄	#11718675	1	2	yes	—
342	C₄	#11718687	1	2	yes	—
343	C₂	—	104	4	no	—
344	C₂	#13113549	1	1	yes	characteristic
345	C₂	#13113556	1	1	yes	characteristic
346	C₂	#13113559	1	1	yes	characteristic
347	C₂	#13113569	1	1	yes	characteristic
348	C₂	C₄×#88063	1	1	yes	characteristic
349	C₂	#13114662	1	1	yes	characteristic
350	C₂	#13114664	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#6103
commutator	C₂×C₄×C₁₃	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₁₃
Fitting	C₁₃×#1767	C₂
socle	C₂³×C₁₃	C₂×C₂²⋊C₄
prime radical	C₂×#88128	C₂²
cosocle	C₂×C₄²×C₁₃	C₂³
hypercenter	#1767	D₁₃
hypocenter	C₁₃	#8231
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #1767

Complements

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

N	H ≅ G/N	type
2. #13114689	C₂	semidirect
4. #13112720	C₂	semidirect
5. C₁₃×#1767	C₂	semidirect
21. #11718077	C₄	semidirect
47. C₁₃×C₄⋊C₈	C₂×C₄	semidirect
50. C₄⋊#1055	C₂×C₄	semidirect
101. #1767	D₁₃	semidirect
216. C₄⋊C₈	C₄×D₁₃	semidirect
273. C₁₃	#8231	semidirect

Maximal subgroups

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

subgroup	index
C₄×#88128	2
#13114689	2
C₂×#11718077	2
#13112720	2
C₁₃×#1767	2
#8231	13