Subgroups — #15834950

Order 3328.

564 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13125419	C₂	1	1	yes	maximal, characteristic
2	#13123934	C₂	1	2	yes	maximal
3	#13113744	C₂	1	1	yes	maximal, characteristic, complemented
4	#13113684	C₂	1	2	yes	maximal, complemented
5	#13113683	C₂	1	2	yes	maximal, complemented
6	#13113656	C₂	1	1	yes	maximal, characteristic, complemented
7	#13113655	C₂	1	1	yes	maximal, characteristic, complemented
8	#13113581	C₂	1	1	yes	maximal, characteristic
9	#13113255	C₂	1	2	yes	maximal
10	#13113122	C₂	1	1	yes	maximal, characteristic
11	C₁₃×#1593	C₂	1	1	yes	maximal, characteristic, complemented
12	#11719283	C₂²	1	2	yes	—
13	C₁₃×#446	C₂²	1	2	yes	—
14	C₄×C₁₃×Q₁₆	C₂²	1	1	yes	characteristic
15	#11718849	C₄	1	1	yes	characteristic
16	#11718842	C₄	1	1	yes	characteristic
17	C₄×#6107	C₂²	1	2	yes	—
18	Q₈×#1055	C₂²	1	2	yes	—
19	#11718548	C₄	1	1	yes	characteristic
20	#11718535	C₄	1	1	yes	characteristic
21	#11718501	—	2	2	no	—
22	#11718494	C₂²	1	2	yes	—
23	#11718492	C₂²	1	2	yes	—
24	#11718491	—	2	2	no	—
25	#11718474	C₄	1	2	yes	—
26	#11718453	C₄	1	2	yes	—
27	#11718370	—	2	2	no	—
28	#11718362	—	2	2	no	—
29	C₄×Dic₅₂	C₂²	1	1	yes	characteristic
30	C₄×#6094	C₂²	1	1	yes	characteristic
31	#11718346	C₂²	1	1	yes	characteristic
32	C₄×C₈×D₁₃	C₂²	1	1	yes	characteristic
33	C₈×Dic₂₆	C₂²	1	2	yes	—
34	C₁₃×#334	C₂²	1	1	yes	characteristic
35	C₁₃×Q₈⋊C₈	C₂²	1	2	yes	—
36	C₁₃×#322	C₂²	1	1	yes	characteristic
37	Q₈⋊#1055	C₂²	1	2	yes	—
38	#11718076	C₂²	1	2	yes	—
39	#11718071	C₂²	1	2	yes	—
40	#11718049	C₂²	1	1	yes	characteristic
41	#11718048	C₂²	1	2	yes	—
42	#11718047	C₂²	1	1	yes	characteristic
43	#11718047	C₂²	1	1	yes	characteristic
44	#11718046	C₂²	1	1	yes	characteristic
45	C₂×C₁₃×Q₁₆	C₂×C₄	1	1	yes	characteristic
46	C₄×Q₈×C₁₃	C₂³	1	2	yes	—
47	#88200	C₂×C₄	1	2	yes	—
48	#88198	C₂×C₄	1	2	yes	—
49	C₂×#6107	C₂×C₄	1	2	yes	—
50	C₂×Dic₅₂	C₂×C₄	1	1	yes	characteristic
51	C₂×#6094	—	2	1	no	—
52	C₂×C₈×D₁₃	C₂×C₄	1	1	yes	characteristic
53	C₂×C₈×D₁₃	C₂×C₄	1	1	yes	characteristic
54	#88151	C₂×C₄	1	2	yes	—
55	#88150	—	2	2	no	—
56	#88143	—	2	2	no	—
57	#88142	C₂×C₄	1	2	yes	—
58	#88126	—	2	2	no	—
59	C₄²×D₁₃	C₂³	1	1	yes	characteristic
60	C₄×Dic₂₆	—	2	2	no	—
61	C₄×Dic₂₆	C₂³	1	2	yes	—
62	C₁₃×#107	C₂×C₄	1	1	yes	characteristic
63	C₁₃×C₄⋊C₈	—	2	2	no	—
64	C₁₃×C₄⋊C₈	C₂³	1	2	yes	—
65	C₁₃×#103	C₂×C₄	1	2	yes	—
66	C₁₃×#97	—	2	2	no	—
67	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
68	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
69	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
70	#88075	C₂×C₄	1	2	yes	—
71	#88060	—	2	2	no	—
72	#88060	D₄	1	2	yes	—
73	#88058	C₂×C₄	1	1	yes	characteristic
74	#88056	C₂×C₄	1	2	yes	—
75	#88055	—	2	1	no	—
76	#88055	—	2	2	no	—
77	#88054	—	2	2	no	—
78	#88054	—	2	2	no	—
79	#88054	D₄	1	2	yes	—
80	C₈×Dic₁₃	—	2	2	no	—
81	C₈×Dic₁₃	—	2	2	no	—
82	C₈×Dic₁₃	C₂×C₄	1	1	yes	characteristic
83	C₈×Dic₁₃	C₂×C₄	1	1	yes	characteristic
84	#88050	C₂×C₄	1	2	yes	—
85	#88047	C₂×C₄	1	2	yes	—
86	C₄⋊#1055	—	2	2	no	—
87	C₄⋊#1055	C₂³	1	2	yes	—
88	C₄×#1055	—	2	2	no	—
89	C₄×#1055	C₂³	1	1	yes	characteristic
90	C₄×#1055	C₂³	1	1	yes	characteristic
91	C₄×#1055	C₂³	1	1	yes	characteristic
92	#18090	—	13	1	no	maximal, Sylow
93	C₂×Q₈×C₁₃	C₂²×C₄	1	2	yes	—
94	C₂×C₄×D₁₃	C₂²×C₄	1	1	yes	characteristic
95	C₂×C₄×D₁₃	C₂×D₄	1	1	yes	characteristic
96	C₂×C₄×D₁₃	#41	1	1	yes	characteristic
97	C₂×Dic₂₆	—	2	2	no	—
98	C₂×Dic₂₆	C₂²×C₄	1	2	yes	—
99	C₁₃×Q₁₆	—	2	2	no	—
100	C₂×C₈×C₁₃	—	2	2	no	—
101	C₂×C₈×C₁₃	—	2	1	no	—
102	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
103	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
104	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
105	C₁₃×C₄⋊C₄	—	2	2	no	—
106	C₁₃×C₄⋊C₄	C₂²×C₄	1	2	yes	—
107	C₄²×C₁₃	—	2	2	no	—
108	C₄²×C₁₃	C₂⁴	1	1	yes	characteristic
109	#6107	—	4	2	no	—
110	#6103	—	2	2	no	—
111	#6103	—	2	2	no	—
112	C₄⋊Dic₁₃	—	2	2	no	—
113	C₄⋊Dic₁₃	C₂²×C₄	1	2	yes	—
114	#6101	—	2	2	no	—
115	#6101	—	2	2	no	—
116	#6101	—	2	2	no	—
117	#6101	—	2	2	no	—
118	#6101	—	2	2	no	—
119	C₄×Dic₁₃	—	2	2	no	—
120	C₄×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
121	C₄×Dic₁₃	C₂×D₄	1	1	yes	characteristic
122	C₄×Dic₁₃	#41	1	1	yes	characteristic
123	C₂×#1055	—	2	2	no	—
124	C₂×#1055	—	2	1	no	—
125	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
126	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
127	C₂×#1055	C₂×D₄	1	2	yes	—
128	Dic₅₂	—	2	2	no	—
129	#6094	—	4	2	no	—
130	C₈×D₁₃	—	2	2	no	—
131	C₈×D₁₃	—	4	1	no	—
132	#3170	—	13	1	no	—
133	#2986	—	13	2	no	—
134	#1593	—	13	1	no	—
135	#1593	D₁₃	1	1	yes	characteristic, complemented
136	#1584	—	13	2	no	—
137	#1584	—	13	2	no	—
138	#1579	—	13	1	no	—
139	#1579	—	13	1	no	—
140	#1564	—	13	1	no	—
141	#1485	—	13	2	no	—
142	#1451	—	13	1	no	—
143	C₂²×D₁₃	C₄×D₄	1	1	yes	characteristic
144	Q₈×C₁₃	—	2	2	no	—
145	Q₈×C₁₃	—	2	2	no	—
146	C₂×C₄×C₁₃	—	2	2	no	—
147	C₂×C₄×C₁₃	—	2	2	no	—
148	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
149	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
150	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
151	C₂×Dic₁₃	—	2	2	no	—
152	C₂×Dic₁₃	—	2	2	no	—
153	C₂×Dic₁₃	C₄×D₄	1	1	yes	characteristic
154	C₂×Dic₁₃	C₄×D₄	1	1	yes	characteristic
155	C₂×Dic₁₃	C₄×D₄	1	1	yes	characteristic
156	C₄×D₁₃	—	2	1	no	—
157	C₄×D₁₃	—	2	1	no	—
158	C₄×D₁₃	—	2	2	no	—
159	C₄×D₁₃	—	2	2	no	—
160	Dic₂₆	—	2	2	no	—
161	Dic₂₆	—	2	2	no	—
162	Dic₂₆	—	4	4	no	—
163	C₈×C₁₃	—	2	1	no	—
164	C₈×C₁₃	—	2	2	no	—
165	C₈×C₁₃	—	4	1	no	—
166	C₈×C₁₃	—	4	2	no	—
167	C₈×C₁₃	#131	1	2	yes	—
168	#1055	—	2	1	no	—
169	#1055	—	2	1	no	—
170	#1055	—	2	2	no	—
171	#1055	—	4	1	no	—
172	#1055	—	4	2	no	—
173	#533	—	13	2	no	—
174	#500	—	13	1	no	—
175	#494	—	13	1	no	—
176	#487	—	13	2	no	—
177	#467	—	13	1	no	—
178	#467	—	13	1	no	—
179	#467	—	13	2	no	—
180	#446	—	13	2	no	—
181	#446	—	26	2	no	—
182	#446	C₂×D₁₃	1	2	yes	—
183	C₈×Q₈	—	13	2	no	—
184	C₈×Q₈	—	13	2	no	—
185	C₈×Q₈	—	26	2	no	—
186	C₄×Q₁₆	—	13	2	no	—
187	C₄×Q₁₆	—	13	1	no	—
188	C₄×Q₁₆	C₂×D₁₃	1	1	yes	characteristic
189	#433	—	26	2	no	—
190	#433	—	26	2	no	—
191	#431	—	13	1	no	—
192	#431	—	13	2	no	—
193	C₄×M₁₆	—	13	1	no	—
194	C₂×C₄×C₈	—	13	1	no	—
195	#335	—	13	2	no	—
196	#334	—	13	1	no	—
197	#334	C₂×D₁₃	1	1	yes	characteristic
198	Q₈⋊C₈	—	13	2	no	—
199	Q₈⋊C₈	—	13	2	no	—
200	Q₈⋊C₈	—	13	2	no	—
201	Q₈⋊C₈	C₂×D₁₃	1	2	yes	—
202	#322	—	13	1	no	—
203	#322	—	13	1	no	—
204	#322	—	13	1	no	—
205	#322	C₂×D₁₃	1	1	yes	characteristic
206	C₂²×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
207	C₂×D₁₃	—	2	1	no	—
208	C₂×D₁₃	—	2	1	no	—
209	C₂×D₁₃	—	2	1	no	—
210	C₄×C₁₃	—	2	1	no	—
211	C₄×C₁₃	—	4	2	no	—
212	C₄×C₁₃	—	4	2	no	—
213	C₄×C₁₃	C₂×#131	1	1	yes	characteristic
214	C₄×C₁₃	#568	1	1	yes	characteristic
215	C₄×C₁₃	#578	1	2	yes	—
216	Dic₁₃	—	2	1	no	—
217	Dic₁₃	—	2	1	no	—
218	Dic₁₃	—	4	2	no	—
219	Dic₁₃	—	4	2	no	—
220	Dic₁₃	#444	1	2	yes	—
221	C₂×Q₁₆	—	13	2	no	—
222	C₂×Q₁₆	—	13	1	no	—
223	C₂×Q₁₆	C₄×D₁₃	1	1	yes	characteristic
224	C₂×M₁₆	—	26	1	no	—
225	C₂²×C₈	—	13	1	no	—
226	C₂²×C₈	—	13	1	no	—
227	C₄⋊Q₈	—	13	2	no	—
228	C₄⋊Q₈	—	13	2	no	—
229	#125	—	26	2	no	—
230	C₂²⋊Q₈	—	13	2	no	—
231	C₂²⋊Q₈	—	13	2	no	—
232	C₂²⋊Q₈	—	26	2	no	—
233	C₄×Q₈	—	13	2	no	—
234	C₄×Q₈	—	26	2	no	—
235	C₄×Q₈	C₂²×D₁₃	1	2	yes	—
236	#117	—	26	2	no	—
237	C₂×C₄²	—	13	1	no	—
238	#107	—	13	1	no	—
239	#107	—	13	2	no	—
240	#107	C₄×D₁₃	1	1	yes	characteristic
241	C₄⋊C₈	—	2	2	no	—
242	C₄⋊C₈	—	13	2	no	—
243	C₄⋊C₈	—	13	2	no	—
244	C₄⋊C₈	—	26	2	no	—
245	C₄⋊C₈	—	26	2	no	—
246	C₄⋊C₈	—	26	2	no	—
247	C₄⋊C₈	C₂²×D₁₃	1	2	yes	—
248	#103	—	13	2	no	—
249	#103	—	13	2	no	—
250	#103	—	13	2	no	—
251	#103	C₄×D₁₃	1	2	yes	—
252	C₂²⋊C₈	—	13	2	no	—
253	C₂²⋊C₈	—	26	2	no	—
254	#97	—	2	2	no	—
255	#97	—	26	1	no	—
256	#97	—	26	2	no	—
257	C₄×C₈	—	13	1	no	—
258	C₄×C₈	—	13	1	no	—
259	C₄×C₈	—	13	1	no	—
260	C₄×C₈	—	13	1	no	—
261	C₄×C₈	—	13	1	no	—
262	C₄×C₈	—	26	2	no	—
263	C₄×C₈	—	26	2	no	—
264	C₄×C₈	—	26	2	no	—
265	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
266	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
267	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
268	C₂×C₁₃	#2935	1	1	yes	characteristic
269	C₂×C₁₃	#2950	1	1	yes	characteristic
270	C₂×C₁₃	C₂×#444	1	1	yes	characteristic
271	D₁₃	—	4	1	no	—
272	C₂×Q₈	—	13	2	no	—
273	C₂×Q₈	—	26	2	no	—
274	C₂×Q₈	C₂×C₄×D₁₃	1	2	yes	—
275	C₂²×C₄	—	13	1	no	—
276	C₂²×C₄	—	13	1	no	—
277	C₂²×C₄	—	13	1	no	—
278	Q₁₆	—	2	2	no	—
279	Q₁₆	—	26	2	no	—
280	Q₁₆	—	52	2	no	—
281	M₁₆	—	52	2	no	—
282	C₂×C₈	—	2	2	no	—
283	C₂×C₈	—	2	1	no	—
284	C₂×C₈	—	13	1	no	—
285	C₂×C₈	—	13	1	no	—
286	C₂×C₈	—	13	2	no	—
287	C₂×C₈	—	26	2	no	—
288	C₂×C₈	—	26	2	no	—
289	C₂×C₈	—	26	1	no	—
290	C₂×C₈	—	52	1	no	—
291	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
292	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
293	C₂×C₈	D₄×D₁₃	1	2	yes	—
294	C₄⋊C₄	—	2	2	no	—
295	C₄⋊C₄	—	13	2	no	—
296	C₄⋊C₄	—	26	2	no	—
297	C₄⋊C₄	—	26	2	no	—
298	C₄⋊C₄	—	26	2	no	—
299	C₄⋊C₄	—	26	2	no	—
300	C₄⋊C₄	—	26	2	no	—
301	C₄⋊C₄	—	26	2	no	—
302	C₄⋊C₄	C₂×C₄×D₁₃	1	2	yes	—
303	C₂²⋊C₄	—	26	2	no	—
304	C₂²⋊C₄	—	26	2	no	—
305	C₄²	—	2	2	no	—
306	C₄²	—	13	1	no	—
307	C₄²	—	13	1	no	—
308	C₄²	—	13	1	no	—
309	C₄²	—	26	2	no	—
310	C₄²	C₂³×D₁₃	1	1	yes	characteristic
311	C₁₃	#18090	1	1	yes	characteristic, complemented, Sylow
312	C₂³	—	13	1	no	—
313	Q₈	—	2	2	no	—
314	Q₈	—	2	2	no	—
315	Q₈	—	26	2	no	—
316	Q₈	—	26	2	no	—
317	Q₈	—	52	4	no	—
318	C₂×C₄	—	2	2	no	—
319	C₂×C₄	—	2	2	no	—
320	C₂×C₄	—	13	1	no	—
321	C₂×C₄	—	13	1	no	—
322	C₂×C₄	—	13	1	no	—
323	C₂×C₄	—	26	2	no	—
324	C₂×C₄	—	26	1	no	—
325	C₂×C₄	—	26	1	no	—
326	C₂×C₄	—	26	2	no	—
327	C₂×C₄	—	26	2	no	—
328	C₂×C₄	—	26	2	no	—
329	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
330	C₂×C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
331	C₂×C₄	#41×D₁₃	1	1	yes	characteristic
332	C₈	—	2	1	no	—
333	C₈	—	2	2	no	—
334	C₈	—	4	1	no	—
335	C₈	—	4	2	no	—
336	C₈	—	26	1	no	—
337	C₈	—	26	1	no	—
338	C₈	—	26	2	no	—
339	C₈	—	52	1	no	—
340	C₈	—	52	2	no	—
341	C₈	#88161	1	2	yes	—
342	C₂²	—	26	1	no	—
343	C₂²	—	26	1	no	—
344	C₂²	—	26	1	no	—
345	C₂²	C₄×D₄×D₁₃	1	1	yes	characteristic
346	C₄	—	2	1	no	—
347	C₄	—	4	2	no	—
348	C₄	—	4	2	no	—
349	C₄	—	13	2	no	—
350	C₄	—	26	1	no	—
351	C₄	—	26	1	no	—
352	C₄	—	52	2	no	—
353	C₄	—	52	2	no	—
354	C₄	C₂×#88161	1	1	yes	characteristic
355	C₄	#11719455	1	1	yes	characteristic
356	C₄	#11719484	1	2	yes	—
357	C₂	—	52	1	no	—
358	C₂	#13123581	1	1	yes	characteristic
359	C₂	#13123701	1	1	yes	characteristic
360	C₂	D₁₃×#444	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₁₃
commutator	C₂×C₄×C₁₃	C₂³×C₄
Frattini	C₄²	C₂³×D₁₃
Fitting	C₁₃×#1593	C₂
socle	C₂²×C₁₃	C₂×C₄×D₄
prime radical	C₂²×D₁₃	C₄×D₄
cosocle	C₄²×C₁₃	C₂⁴
hypercenter	#1593	D₁₃
hypocenter	C₁₃	#18090
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #1593

Complements

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

N	H ≅ G/N	type
3. #13113744	C₂	semidirect
4. #13113684	C₂	semidirect
5. #13113683	C₂	semidirect
6. #13113656	C₂	semidirect
7. #13113655	C₂	semidirect
11. C₁₃×#1593	C₂	semidirect
135. #1593	D₁₃	semidirect
311. C₁₃	#18090	semidirect

Maximal subgroups

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

subgroup	index
#13125419	2
#13123934	2
#13113744	2
#13113684	2
#13113683	2
#13113656	2
#13113655	2
#13113581	2
#13113255	2
#13113122	2
C₁₃×#1593	2
#18090	13