Subgroups — #15762719

Order 3328.

578 conjugacy classes of subgroups.

The subgroup lattice has 354 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	#13115797	C₂	1	1	yes	maximal, characteristic, complemented
3	#13115642	C₂	1	1	yes	maximal, characteristic
4	#13115642	C₂	1	1	yes	maximal, characteristic
5	#13115638	C₂	1	1	yes	maximal, characteristic, complemented
6	#13114996	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₁₃×#322	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	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
12	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
13	C₂×#88061	—	2	2	no	—
14	C₂×#88061	—	2	2	no	—
15	C₂×#88061	—	2	2	no	—
16	C₂×#88058	—	2	2	no	—
17	C₂×#88056	—	2	2	no	—
18	#11718596	C₂²	1	1	yes	characteristic
19	#11718596	C₂²	1	1	yes	characteristic
20	#11718355	—	2	2	no	—
21	#11718355	C₄	1	4	yes	complemented
22	C₁₃×#322	—	2	2	no	—
23	#11718143	—	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	C₂²×C₈×C₁₃	D₄	1	2	yes	—
30	C₂²×C₈×C₁₃	D₄	1	2	yes	—
31	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
32	C₂×C₄⋊Dic₁₃	—	2	1	no	—
33	C₂×C₄⋊Dic₁₃	—	2	1	no	—
34	C₂×C₄⋊Dic₁₃	—	2	1	no	—
35	C₂×C₄⋊Dic₁₃	—	2	1	no	—
36	C₂×C₄⋊Dic₁₃	—	2	1	no	—
37	C₂×C₄⋊Dic₁₃	—	2	1	no	—
38	C₂×C₄⋊Dic₁₃	—	2	1	no	—
39	C₂×C₄⋊Dic₁₃	—	2	1	no	—
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	—	2	1	no	—
49	C₄×C₈×C₁₃	—	2	1	no	—
50	C₄×C₈×C₁₃	—	2	1	no	—
51	C₄×C₈×C₁₃	—	2	1	no	—
52	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
53	C₄×C₈×C₁₃	D₄	1	2	yes	—
54	#88061	—	2	8	no	—
55	#88061	—	4	4	no	—
56	#88061	—	4	4	no	—
57	#88058	—	4	4	no	—
58	#88056	—	4	4	no	—
59	#9830	—	13	1	no	maximal, Sylow
60	C₂³×D₁₃	—	4	1	no	—
61	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
64	C₂²×Dic₁₃	—	4	1	no	—
65	C₂²×Dic₁₃	—	4	1	no	—
66	C₂²×Dic₁₃	—	4	1	no	—
67	C₂×D₅₂	—	2	2	no	—
68	C₂×D₅₂	—	2	2	no	—
69	C₂×D₅₂	—	2	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	2	no	—
76	C₂×D₅₂	—	4	1	no	—
77	C₂×D₅₂	—	4	1	no	—
78	C₂×D₅₂	—	4	1	no	—
79	C₂×Dic₂₆	—	2	2	no	—
80	C₂×Dic₂₆	—	4	2	no	—
81	C₂×Dic₂₆	—	4	2	no	—
82	C₂×Dic₂₆	—	4	1	no	—
83	C₂×C₈×C₁₃	—	2	2	no	—
84	C₂×C₈×C₁₃	—	2	2	no	—
85	C₂×C₈×C₁₃	—	2	2	no	—
86	C₂×C₈×C₁₃	—	2	2	no	—
87	C₂×C₈×C₁₃	—	2	2	no	—
88	C₂×C₈×C₁₃	—	2	2	no	—
89	C₂×C₈×C₁₃	—	2	2	no	—
90	C₂×C₈×C₁₃	C₂²⋊C₄	1	4	yes	—
91	C₂×C₈×C₁₃	D₈	1	4	yes	—
92	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
93	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
94	C₄²×C₁₃	#41	1	1	yes	characteristic
95	C₄²×C₁₃	#41	1	1	yes	characteristic
96	C₄⋊Dic₁₃	—	2	2	no	—
97	C₄⋊Dic₁₃	—	2	2	no	—
98	C₄⋊Dic₁₃	—	4	1	no	—
99	C₄⋊Dic₁₃	—	4	1	no	—
100	C₄⋊Dic₁₃	—	4	2	no	—
101	C₄⋊Dic₁₃	—	4	2	no	—
102	C₄⋊Dic₁₃	—	4	2	no	—
103	C₄⋊Dic₁₃	—	4	2	no	—
104	C₄⋊Dic₁₃	—	4	2	no	—
105	C₄⋊Dic₁₃	—	4	2	no	—
106	C₂×#486	—	13	1	no	—
107	#1990	—	13	1	no	—
108	#1957	—	13	1	no	—
109	#1957	—	13	1	no	—
110	#1956	—	13	1	no	—
111	#1845	—	13	1	no	—
112	C₂×#322	D₁₃	1	1	yes	characteristic, complemented
113	C₂³×C₁₃	#127	1	1	yes	characteristic
114	C₂²×D₁₃	—	4	2	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	C₂²×D₁₃	—	8	1	no	—
121	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
122	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
123	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
124	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
125	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
126	C₂×C₄×C₁₃	#124	1	1	yes	characteristic
127	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
128	C₂×C₄×C₁₃	#135	1	2	yes	—
129	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	2	yes	—
130	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	2	yes	—
131	C₂×C₄×C₁₃	#137	1	2	yes	—
132	C₂×C₄×C₁₃	#137	1	2	yes	—
133	C₂×Dic₁₃	—	4	2	no	—
134	C₂×Dic₁₃	—	8	1	no	—
135	C₂×Dic₁₃	—	8	1	no	—
136	C₂×Dic₁₃	—	8	1	no	—
137	C₂×Dic₁₃	—	8	1	no	—
138	C₂×Dic₁₃	—	8	1	no	—
139	C₂×Dic₁₃	—	8	1	no	—
140	C₂×Dic₁₃	—	8	1	no	—
141	C₂×Dic₁₃	—	8	1	no	—
142	D₅₂	—	4	4	no	—
143	D₅₂	—	4	4	no	—
144	D₅₂	—	4	4	no	—
145	D₅₂	—	8	2	no	—
146	D₅₂	—	8	2	no	—
147	D₅₂	—	8	2	no	—
148	Dic₂₆	—	4	4	no	—
149	Dic₂₆	—	8	2	no	—
150	C₈×C₁₃	—	2	4	no	—
151	C₈×C₁₃	—	2	4	no	—
152	C₈×C₁₃	—	4	2	no	—
153	C₂×C₄⋊Q₈	—	13	1	no	—
154	C₂×#127	—	13	1	no	—
155	#486	—	13	4	no	—
156	#486	—	26	2	no	—
157	C₂×#107	—	26	2	no	—
158	C₂×#103	—	26	2	no	—
159	C₂×#102	—	26	2	no	—
160	C₂×#102	—	26	2	no	—
161	C₂×#102	—	26	2	no	—
162	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
163	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
164	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
165	#384	—	13	1	no	—
166	#384	—	13	1	no	—
167	#340	—	26	2	no	—
168	#322	—	2	2	no	—
169	C₂²×C₁₃	#390	1	1	yes	characteristic
170	C₂²×C₁₃	#493	1	1	yes	characteristic
171	C₂²×C₁₃	#495	1	1	yes	characteristic
172	C₂²×C₁₃	#496	1	1	yes	characteristic
173	C₂²×C₁₃	#496	1	1	yes	characteristic
174	C₂²×C₁₃	#497	1	1	yes	characteristic
175	C₂²×C₁₃	#497	1	1	yes	characteristic
176	C₂×D₁₃	—	8	2	no	—
177	C₂×D₁₃	—	8	2	no	—
178	C₂×D₁₃	—	8	2	no	—
179	C₂×D₁₃	—	16	1	no	—
180	C₂×D₁₃	—	16	1	no	—
181	C₂×D₁₃	—	16	1	no	—
182	C₂×D₁₃	—	16	1	no	—
183	C₄×C₁₃	#417	1	2	yes	—
184	C₄×C₁₃	C₄×D₈	1	2	yes	—
185	C₄×C₁₃	#440	1	2	yes	—
186	C₄×C₁₃	#466	1	2	yes	—
187	C₄×C₁₃	#468	1	2	yes	—
188	C₄×C₁₃	#489	1	2	yes	—
189	Dic₁₃	—	8	2	no	—
190	Dic₁₃	—	16	1	no	—
191	Dic₁₃	—	16	1	no	—
192	C₂²×Q₈	—	26	1	no	—
193	C₂²×D₄	—	26	1	no	—
194	C₂²×D₄	—	26	1	no	—
195	C₂²×D₄	—	26	1	no	—
196	C₂²×C₈	D₅₂	1	2	yes	—
197	C₂²×C₈	D₅₂	1	2	yes	—
198	C₂²×C₈	D₅₂	1	2	yes	—
199	C₄⋊Q₈	—	13	2	no	—
200	C₄⋊Q₈	—	26	1	no	—
201	C₄⋊Q₈	—	26	1	no	—
202	C₄⋊Q₈	—	26	1	no	—
203	#127	—	13	2	no	—
204	#127	—	26	1	no	—
205	#127	—	26	1	no	—
206	#127	—	26	1	no	—
207	C₂×C₄⋊C₄	—	26	1	no	—
208	C₂×C₄⋊C₄	—	26	1	no	—
209	C₂×C₄⋊C₄	—	26	1	no	—
210	C₂×C₄⋊C₄	—	26	1	no	—
211	C₂×C₄⋊C₄	—	26	1	no	—
212	C₂×C₄⋊C₄	—	26	1	no	—
213	C₂×C₄⋊C₄	—	26	1	no	—
214	C₂×C₄⋊C₄	—	26	1	no	—
215	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
216	#107	—	52	4	no	—
217	#103	—	52	4	no	—
218	#102	—	26	8	no	—
219	#102	—	52	4	no	—
220	#102	—	52	4	no	—
221	C₄×C₈	—	2	1	no	—
222	C₄×C₈	—	2	1	no	—
223	C₄×C₈	—	2	1	no	—
224	C₄×C₈	—	2	1	no	—
225	C₄×C₈	C₄×D₁₃	1	2	yes	—
226	C₄×C₈	C₂²⋊D₁₃	1	2	yes	—
227	C₂×C₁₃	#1719	1	1	yes	characteristic
228	C₂×C₁₃	#1720	1	1	yes	characteristic
229	C₂×C₁₃	#1721	1	1	yes	characteristic
230	C₂×C₁₃	#1722	1	1	yes	characteristic
231	C₂×C₁₃	#1959	1	1	yes	characteristic
232	C₂×C₁₃	#1970	1	1	yes	characteristic
233	C₂×C₁₃	#1973	1	1	yes	characteristic
234	D₁₃	—	16	2	no	—
235	C₂⁴	—	52	1	no	—
236	C₂×Q₈	—	26	2	no	—
237	C₂×Q₈	—	52	1	no	—
238	C₂×Q₈	—	52	2	no	—
239	C₂×Q₈	—	52	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	2	no	—
246	C₂×D₄	—	52	2	no	—
247	C₂×D₄	—	52	2	no	—
248	C₂×D₄	—	52	2	no	—
249	C₂×D₄	—	52	2	no	—
250	C₂×D₄	—	52	1	no	—
251	C₂×D₄	—	52	1	no	—
252	C₂²×C₄	—	52	1	no	—
253	C₂²×C₄	—	52	1	no	—
254	C₂²×C₄	—	52	1	no	—
255	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
256	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
257	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
258	C₂×C₈	—	2	2	no	—
259	C₂×C₈	—	2	2	no	—
260	C₂×C₈	—	2	2	no	—
261	C₂×C₈	—	2	2	no	—
262	C₂×C₈	—	2	2	no	—
263	C₂×C₈	—	2	2	no	—
264	C₂×C₈	—	2	2	no	—
265	C₂×C₈	D₁₀₄	1	4	yes	—
266	C₂×C₈	#6103	1	4	yes	—
267	C₄⋊C₄	—	26	2	no	—
268	C₄⋊C₄	—	26	2	no	—
269	C₄⋊C₄	—	52	1	no	—
270	C₄⋊C₄	—	52	1	no	—
271	C₄⋊C₄	—	52	2	no	—
272	C₄⋊C₄	—	52	2	no	—
273	C₄⋊C₄	—	52	2	no	—
274	C₄⋊C₄	—	52	2	no	—
275	C₄⋊C₄	—	52	2	no	—
276	C₄⋊C₄	—	52	2	no	—
277	C₄²	C₂×C₄×D₁₃	1	1	yes	characteristic
278	C₄²	#6127	1	1	yes	characteristic
279	C₄²	#6127	1	1	yes	characteristic
280	C₄²	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
281	C₁₃	#9830	1	1	yes	characteristic, complemented, Sylow
282	C₂³	—	52	2	no	—
283	C₂³	—	104	1	no	—
284	C₂³	—	104	1	no	—
285	C₂³	—	104	1	no	—
286	C₂³	—	104	1	no	—
287	C₂³	—	104	1	no	—
288	C₂³	—	104	1	no	—
289	C₂³	#88128	1	1	yes	characteristic
290	Q₈	—	52	4	no	—
291	Q₈	—	104	2	no	—
292	D₄	—	52	4	no	—
293	D₄	—	52	4	no	—
294	D₄	—	52	4	no	—
295	D₄	—	104	2	no	—
296	D₄	—	104	2	no	—
297	D₄	—	104	2	no	—
298	C₂×C₄	—	52	2	no	—
299	C₂×C₄	—	104	1	no	—
300	C₂×C₄	—	104	1	no	—
301	C₂×C₄	—	104	1	no	—
302	C₂×C₄	—	104	1	no	—
303	C₂×C₄	—	104	1	no	—
304	C₂×C₄	—	104	1	no	—
305	C₂×C₄	—	104	1	no	—
306	C₂×C₄	—	104	1	no	—
307	C₂×C₄	C₄×D₅₂	1	1	yes	characteristic
308	C₂×C₄	C₄×D₅₂	1	1	yes	characteristic
309	C₂×C₄	#88129	1	1	yes	characteristic
310	C₂×C₄	C₂×D₁₀₄	1	2	yes	—
311	C₂×C₄	#88158	1	2	yes	—
312	C₂×C₄	#88162	1	2	yes	—
313	C₂×C₄	#88162	1	2	yes	—
314	C₂×C₄	#88163	1	2	yes	—
315	C₂×C₄	#88163	1	2	yes	—
316	C₂×C₄	C₂×#6103	1	1	yes	characteristic
317	C₂×C₄	#88184	1	1	yes	characteristic
318	C₂×C₄	#88184	1	1	yes	characteristic
319	C₈	—	2	4	no	—
320	C₈	—	2	4	no	—
321	C₈	—	4	2	no	—
322	C₂²	—	104	2	no	—
323	C₂²	—	104	2	no	—
324	C₂²	—	104	2	no	—
325	C₂²	—	208	1	no	—
326	C₂²	—	208	1	no	—
327	C₂²	—	208	1	no	—
328	C₂²	—	208	1	no	—
329	C₂²	#11718356	1	1	yes	characteristic
330	C₂²	#11718357	1	1	yes	characteristic
331	C₂²	#11718373	1	1	yes	characteristic
332	C₂²	#11718373	1	1	yes	characteristic
333	C₂²	#11718376	1	1	yes	characteristic
334	C₂²	#11718376	1	1	yes	characteristic
335	C₂²	#11718600	1	1	yes	characteristic
336	C₄	—	104	2	no	—
337	C₄	—	208	1	no	—
338	C₄	—	208	1	no	—
339	C₄	C₄×D₁₀₄	1	2	yes	—
340	C₄	#11718371	1	2	yes	—
341	C₄	#11718375	1	2	yes	—
342	C₄	#11718776	1	2	yes	—
343	C₄	#11718794	1	2	yes	—
344	C₄	#11718795	1	2	yes	—
345	C₂	—	208	2	no	—
346	C₂	#13114489	1	1	yes	characteristic
347	C₂	#13114496	1	1	yes	characteristic
348	C₂	#13114502	1	1	yes	characteristic
349	C₂	#13114507	1	1	yes	characteristic
350	C₂	#13115652	1	1	yes	characteristic
351	C₂	#13115702	1	1	yes	characteristic
352	C₂	#13115717	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#88128
commutator	C₄²×C₁₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₃
Fitting	C₂×C₁₃×#322	C₂
socle	C₂³×C₁₃	#127
prime radical	C₂×#88128	C₂²
cosocle	C₂×C₄²×C₁₃	C₂³
hypercenter	C₂×#322	D₁₃
hypocenter	C₁₃	#9830
LCS	G ▷ C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#322

Complements

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

N	H ≅ G/N	type
2. #13115797	C₂	semidirect
5. #13115638	C₂	semidirect
6. #13114996	C₂	semidirect
7. C₂×C₁₃×#322	C₂	semidirect
21. #11718355	C₄	semidirect
112. C₂×#322	D₁₃	semidirect
281. C₁₃	#9830	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
#13115797	2
#13115642	2
#13115638	2
#13114996	2
C₂×C₁₃×#322	2
#9830	13