Subgroups — #15833481

Order 3328.

734 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13125411	C₂	1	1	yes	maximal, characteristic
2	#13123548	C₂	1	1	yes	maximal, characteristic, complemented
3	#13123547	C₂	1	1	yes	maximal, characteristic, complemented
4	#13113747	C₂	1	4	yes	maximal, complemented
5	#13113657	C₂	1	4	yes	maximal, complemented
6	#13113591	C₂	1	1	yes	maximal, characteristic, complemented
7	#13113288	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#11718046	C₂	1	1	yes	maximal, characteristic, complemented
9	C₁₃×#1492	C₂	1	1	yes	maximal, characteristic, complemented
10	C₁₃×#532	C₂²	1	1	yes	characteristic
11	C₂×#88202	—	2	1	no	—
12	C₂×#88161	—	2	1	no	—
13	#11719182	C₂²	1	1	yes	characteristic
14	C₁₃×#423	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
16	#11718895	—	2	1	no	—
17	#11718790	—	2	1	no	—
18	#11718749	C₄	1	1	yes	characteristic
19	#11718739	C₄	1	1	yes	characteristic
20	#11718734	C₄	1	1	yes	characteristic
21	#11718731	C₄	1	1	yes	characteristic
22	#11718699	C₄	1	4	yes	—
23	C₄×#6104	C₂²	1	4	yes	—
24	#11718673	—	2	1	no	—
25	#11718673	C₂²	1	2	yes	—
26	D₄×#1055	—	2	1	no	—
27	D₄×#1055	C₂²	1	2	yes	—
28	#11718585	C₂²	1	1	yes	characteristic
29	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
30	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
31	#11718499	—	2	1	no	—
32	#11718499	C₂²	1	2	yes	—
33	#11718495	—	2	1	no	—
34	#11718495	C₂²	1	2	yes	—
35	C₁₃×D₄⋊C₈	C₂²	1	4	yes	—
36	#11718072	C₂²	1	4	yes	—
37	#11718051	C₂²	1	4	yes	—
38	#11718046	C₂²	1	2	yes	—
39	#11718046	C₂²	1	2	yes	—
40	C₂×C₁₃×#41	—	2	1	no	—
41	C₂×#6127	—	2	1	no	—
42	C₂×C₁₃×M₁₆	—	2	1	no	—
43	C₂²×C₈×C₁₃	—	2	1	no	—
44	C₁₃×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
45	C₁₃×#127	C₂×C₄	1	1	yes	characteristic
46	C₁₃×#124	—	2	1	no	—
47	C₁₃×#121	—	2	1	no	—
48	C₁₃×#121	C₂×C₄	1	2	yes	—
49	C₄×D₄×C₁₃	—	2	1	no	—
50	C₄×D₄×C₁₃	C₂³	1	2	yes	—
51	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
52	#88202	—	4	4	no	—
53	C₂×#6104	C₂×C₄	1	4	yes	—
54	#88184	—	2	1	no	—
55	#88184	C₂×C₄	1	2	yes	—
56	C₂×#6099	—	2	1	no	—
57	C₂×#6099	—	2	2	no	—
58	C₂²×#1055	—	2	2	no	—
59	C₂²×#1055	C₂×C₄	1	2	yes	—
60	C₂²×#1055	D₄	1	1	yes	characteristic
61	C₂²×#1055	D₄	1	1	yes	characteristic
62	#88161	—	4	4	no	—
63	C₂×#6094	—	2	2	no	—
64	C₂×C₈×D₁₃	—	2	2	no	—
65	#88129	—	2	1	no	—
66	#88128	C₂×C₄	1	1	yes	characteristic
67	C₄×D₅₂	—	2	1	no	—
68	C₄×D₅₂	C₂³	1	2	yes	—
69	#88123	C₂×C₄	1	1	yes	characteristic
70	C₁₃×C₄⋊C₈	—	2	1	no	—
71	C₁₃×C₄⋊C₈	C₂³	1	2	yes	—
72	C₄×C₈×C₁₃	—	2	1	no	—
73	C₄×C₈×C₁₃	C₂³	1	2	yes	—
74	#88072	C₂×C₄	1	4	yes	—
75	C₂²⋊#1055	—	2	2	no	—
76	C₂²⋊#1055	—	2	2	no	—
77	#88060	—	2	2	no	—
78	#88060	—	2	2	no	—
79	#88049	C₂×C₄	1	4	yes	—
80	#88047	C₂×C₄	1	4	yes	—
81	C₄⋊#1055	—	2	1	no	—
82	C₄⋊#1055	C₂³	1	2	yes	—
83	C₄×#1055	C₂×C₄	1	1	yes	characteristic
84	C₄×#1055	C₂×C₄	1	1	yes	characteristic
85	C₄×#1055	D₄	1	1	yes	characteristic
86	C₄×#1055	D₄	1	1	yes	characteristic
87	C₄×#1055	C₂³	1	2	yes	—
88	C₄×#1055	C₂³	1	1	yes	characteristic
89	C₄×#1055	C₂³	1	1	yes	characteristic
90	#17962	—	13	1	no	maximal, Sylow
91	C₁₃×#41	—	4	2	no	—
92	C₁₃×#41	—	4	2	no	—
93	C₂×Q₈×C₁₃	—	2	1	no	—
94	C₂×D₄×C₁₃	—	2	2	no	—
95	C₂×D₄×C₁₃	—	2	1	no	—
96	C₂×D₄×C₁₃	—	2	1	no	—
97	C₂×D₄×C₁₃	C₂²×C₄	1	2	yes	—
98	C₂²×C₄×C₁₃	—	2	2	no	—
99	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
100	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
101	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
102	C₂×C₂²⋊D₁₃	—	2	2	no	—
103	#6127	—	4	2	no	—
104	#6127	—	4	2	no	—
105	C₂×D₅₂	—	2	1	no	—
106	C₂×D₅₂	—	2	1	no	—
107	C₂×D₅₂	C₂²×C₄	1	2	yes	—
108	C₂×C₄×D₁₃	—	2	2	no	—
109	C₂×Dic₂₆	—	2	1	no	—
110	C₁₃×M₁₆	—	4	2	no	—
111	C₂×C₈×C₁₃	—	2	2	no	—
112	C₂×C₈×C₁₃	—	2	2	no	—
113	C₂×C₈×C₁₃	—	4	2	no	—
114	C₁₃×C₄⋊C₄	—	2	1	no	—
115	C₁₃×C₄⋊C₄	C₂²×C₄	1	2	yes	—
116	C₁₃×C₂²⋊C₄	—	2	2	no	—
117	C₁₃×C₂²⋊C₄	—	2	2	no	—
118	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
119	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
120	C₄²×C₁₃	#41	1	1	yes	characteristic
121	C₄²×C₁₃	C₂⁴	1	1	yes	characteristic
122	#6104	—	2	8	no	—
123	#6103	—	2	2	no	—
124	#6103	—	2	2	no	—
125	C₄⋊Dic₁₃	—	2	1	no	—
126	C₄⋊Dic₁₃	C₂²×C₄	1	2	yes	—
127	#6099	—	4	4	no	—
128	#6099	—	4	2	no	—
129	C₂×#1055	—	2	2	no	—
130	C₂×#1055	—	2	2	no	—
131	C₂×#1055	—	2	4	no	—
132	C₂×#1055	—	2	2	no	—
133	C₂×#1055	—	4	4	no	—
134	C₂×#1055	C₂²×C₄	1	2	yes	—
135	C₂×#1055	C₂²×C₄	1	2	yes	—
136	C₂×#1055	C₂×D₄	1	1	yes	characteristic
137	C₂×#1055	C₂×D₄	1	1	yes	characteristic
138	C₂×#1055	C₂×D₄	1	1	yes	characteristic
139	C₂×#1055	C₂×D₄	1	1	yes	characteristic
140	#6094	—	4	4	no	—
141	C₈×D₁₃	—	4	4	no	—
142	#3169	—	13	1	no	—
143	#2931	—	13	1	no	—
144	#2931	—	13	1	no	—
145	#1594	—	13	4	no	—
146	#1580	—	13	4	no	—
147	#1566	—	13	1	no	—
148	#1492	—	13	1	no	—
149	#1492	D₁₃	1	1	yes	characteristic, complemented
150	C₂×#322	—	13	1	no	—
151	C₂³×C₁₃	—	2	2	no	—
152	C₂³×C₁₃	C₄×D₄	1	1	yes	characteristic
153	C₂²×D₁₃	—	2	2	no	—
154	Q₈×C₁₃	—	4	2	no	—
155	D₄×C₁₃	—	2	4	no	—
156	D₄×C₁₃	—	4	4	no	—
157	D₄×C₁₃	—	4	2	no	—
158	D₄×C₁₃	—	4	2	no	—
159	C₂×C₄×C₁₃	—	2	2	no	—
160	C₂×C₄×C₁₃	—	2	2	no	—
161	C₂×C₄×C₁₃	—	2	2	no	—
162	C₂×C₄×C₁₃	—	4	2	no	—
163	C₂×C₄×C₁₃	—	4	2	no	—
164	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
165	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
166	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
167	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	2	yes	—
168	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	2	yes	—
169	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
170	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
171	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
172	C₂²⋊D₁₃	—	4	4	no	—
173	C₂×Dic₁₃	—	2	2	no	—
174	D₅₂	—	2	4	no	—
175	D₅₂	—	4	2	no	—
176	D₅₂	—	4	2	no	—
177	C₄×D₁₃	—	4	2	no	—
178	C₄×D₁₃	—	4	2	no	—
179	Dic₂₆	—	4	2	no	—
180	C₈×C₁₃	—	4	2	no	—
181	C₈×C₁₃	—	4	2	no	—
182	#1055	—	2	2	no	—
183	#1055	—	2	1	no	—
184	#1055	—	2	1	no	—
185	#1055	—	2	1	no	—
186	#1055	—	2	1	no	—
187	#1055	—	4	2	no	—
188	#1055	#131	1	4	yes	complemented
189	C₂×#131	—	26	1	no	—
190	C₂×#131	—	26	1	no	—
191	#532	—	13	1	no	—
192	#532	C₂×D₁₃	1	1	yes	characteristic
193	#500	—	13	1	no	—
194	#493	—	13	1	no	—
195	#486	—	13	1	no	—
196	#486	—	13	1	no	—
197	#466	—	13	4	no	—
198	C₄×D₈	—	13	4	no	—
199	#436	—	13	2	no	—
200	#436	—	13	2	no	—
201	#436	—	26	1	no	—
202	#436	—	26	1	no	—
203	C₈×D₄	—	13	2	no	—
204	C₈×D₄	—	13	2	no	—
205	C₈×D₄	—	26	1	no	—
206	C₈×D₄	—	26	1	no	—
207	#423	—	13	1	no	—
208	#423	C₂×D₁₃	1	1	yes	characteristic
209	#408	—	26	1	no	—
210	#408	—	26	1	no	—
211	C₂×C₄×C₈	—	13	1	no	—
212	C₂×C₄×C₈	—	13	1	no	—
213	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
214	#334	—	13	4	no	—
215	D₄⋊C₈	—	13	4	no	—
216	D₄⋊C₈	C₂×D₁₃	1	4	yes	—
217	#322	—	13	2	no	—
218	#322	—	13	2	no	—
219	C₂²×C₁₃	—	2	1	no	—
220	C₂²×C₁₃	—	2	1	no	—
221	C₂²×C₁₃	—	4	2	no	—
222	C₂²×C₁₃	—	4	2	no	—
223	C₂²×C₁₃	—	4	2	no	—
224	C₂²×C₁₃	#442	1	2	yes	—
225	C₂²×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
226	C₂×D₁₃	—	4	2	no	—
227	C₂×D₁₃	—	4	2	no	—
228	C₂×D₁₃	—	4	2	no	—
229	C₄×C₁₃	—	2	1	no	—
230	C₄×C₁₃	—	2	1	no	—
231	C₄×C₁₃	—	4	2	no	—
232	C₄×C₁₃	#442	1	2	yes	—
233	C₄×C₁₃	#444	1	2	yes	—
234	C₄×C₁₃	C₂×#131	1	2	yes	—
235	C₄×C₁₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
236	C₄×C₁₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
237	Dic₁₃	—	4	2	no	—
238	C₂×#41	—	2	1	no	—
239	C₂×#41	—	26	1	no	—
240	C₂×D₈	—	13	4	no	—
241	#131	—	52	4	no	—
242	#131	—	52	4	no	—
243	C₂×M₁₆	—	2	1	no	—
244	C₂×M₁₆	—	26	2	no	—
245	C₂×M₁₆	—	26	2	no	—
246	C₂×M₁₆	—	26	1	no	—
247	C₂²×C₈	—	2	1	no	—
248	C₂²×C₈	—	13	2	no	—
249	C₂²×C₈	—	13	1	no	—
250	C₂²×C₈	—	13	1	no	—
251	C₂²×C₈	—	26	2	no	—
252	C₂²×C₈	—	26	2	no	—
253	C₄⋊Q₈	—	13	1	no	—
254	C₄⋊Q₈	C₄×D₁₃	1	1	yes	characteristic
255	#127	—	13	1	no	—
256	#127	C₄×D₁₃	1	1	yes	characteristic
257	#124	—	2	1	no	—
258	#124	—	26	1	no	—
259	#121	—	2	1	no	—
260	#121	—	13	2	no	—
261	#121	—	26	1	no	—
262	#121	C₄×D₁₃	1	2	yes	—
263	C₄×D₄	—	2	1	no	—
264	C₄×D₄	—	13	2	no	—
265	C₄×D₄	—	26	1	no	—
266	C₄×D₄	C₂²×D₁₃	1	2	yes	—
267	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
268	#107	—	13	4	no	—
269	C₄⋊C₈	—	2	1	no	—
270	C₄⋊C₈	—	13	2	no	—
271	C₄⋊C₈	—	26	1	no	—
272	C₄⋊C₈	C₂²×D₁₃	1	2	yes	—
273	#102	—	13	4	no	—
274	#102	—	13	4	no	—
275	C₂²⋊C₈	—	26	2	no	—
276	C₂²⋊C₈	—	26	2	no	—
277	C₂²⋊C₈	—	26	2	no	—
278	C₂²⋊C₈	—	26	2	no	—
279	C₄×C₈	—	2	1	no	—
280	C₄×C₈	—	13	1	no	—
281	C₄×C₈	—	13	1	no	—
282	C₄×C₈	—	13	1	no	—
283	C₄×C₈	—	13	1	no	—
284	C₄×C₈	—	13	2	no	—
285	C₄×C₈	—	13	1	no	—
286	C₄×C₈	—	13	1	no	—
287	C₄×C₈	C₂²×D₁₃	1	2	yes	—
288	C₂×C₁₃	—	2	2	no	—
289	C₂×C₁₃	—	8	2	no	—
290	C₂×C₁₃	#2931	1	1	yes	characteristic
291	C₂×C₁₃	C₂×#442	1	1	yes	characteristic
292	C₂×C₁₃	C₂×#444	1	1	yes	characteristic
293	D₁₃	—	8	2	no	—
294	#41	—	4	2	no	—
295	#41	—	4	2	no	—
296	#41	—	52	2	no	—
297	#41	—	52	2	no	—
298	C₂×Q₈	—	2	1	no	—
299	C₂×Q₈	—	26	1	no	—
300	C₂×D₄	—	2	2	no	—
301	C₂×D₄	—	2	1	no	—
302	C₂×D₄	—	2	1	no	—
303	C₂×D₄	—	13	2	no	—
304	C₂×D₄	—	26	2	no	—
305	C₂×D₄	—	26	1	no	—
306	C₂×D₄	—	26	1	no	—
307	C₂×D₄	C₂×C₄×D₁₃	1	2	yes	—
308	C₂²×C₄	—	2	2	no	—
309	C₂²×C₄	—	26	2	no	—
310	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
311	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
312	C₂²×C₄	#6129	1	1	yes	characteristic
313	D₈	—	26	8	no	—
314	M₁₆	—	4	2	no	—
315	M₁₆	—	52	4	no	—
316	M₁₆	—	52	4	no	—
317	M₁₆	—	52	2	no	—
318	C₂×C₈	—	2	2	no	—
319	C₂×C₈	—	2	2	no	—
320	C₂×C₈	—	4	2	no	—
321	C₂×C₈	—	13	2	no	—
322	C₂×C₈	—	13	2	no	—
323	C₂×C₈	—	13	1	no	—
324	C₂×C₈	—	13	1	no	—
325	C₂×C₈	—	13	1	no	—
326	C₂×C₈	—	13	1	no	—
327	C₂×C₈	—	26	4	no	—
328	C₂×C₈	—	26	2	no	—
329	C₂×C₈	—	26	2	no	—
330	C₂×C₈	—	26	2	no	—
331	C₂×C₈	—	52	4	no	—
332	C₂×C₈	—	52	4	no	—
333	C₄⋊C₄	—	2	1	no	—
334	C₄⋊C₄	—	13	2	no	—
335	C₄⋊C₄	—	26	1	no	—
336	C₄⋊C₄	C₂×C₄×D₁₃	1	2	yes	—
337	C₂²⋊C₄	—	2	2	no	—
338	C₂²⋊C₄	—	2	2	no	—
339	C₂²⋊C₄	—	26	2	no	—
340	C₂²⋊C₄	—	26	2	no	—
341	C₄²	C₂×C₄×D₁₃	1	1	yes	characteristic
342	C₄²	D₄×D₁₃	1	1	yes	characteristic
343	C₄²	#6129	1	1	yes	characteristic
344	C₄²	C₂³×D₁₃	1	1	yes	characteristic
345	C₁₃	#17962	1	1	yes	characteristic, complemented, Sylow
346	C₂³	—	2	2	no	—
347	C₂³	—	26	2	no	—
348	C₂³	#88135	1	1	yes	characteristic
349	Q₈	—	4	2	no	—
350	Q₈	—	52	2	no	—
351	D₄	—	2	4	no	—
352	D₄	—	4	4	no	—
353	D₄	—	4	2	no	—
354	D₄	—	4	2	no	—
355	D₄	—	26	4	no	—
356	D₄	—	52	4	no	—
357	D₄	—	52	2	no	—
358	D₄	—	52	2	no	—
359	C₂×C₄	—	2	2	no	—
360	C₂×C₄	—	2	2	no	—
361	C₂×C₄	—	2	2	no	—
362	C₂×C₄	—	4	2	no	—
363	C₂×C₄	—	4	2	no	—
364	C₂×C₄	—	26	2	no	—
365	C₂×C₄	—	52	2	no	—
366	C₂×C₄	—	52	2	no	—
367	C₂×C₄	#88135	1	1	yes	characteristic
368	C₂×C₄	#88135	1	1	yes	characteristic
369	C₂×C₄	#88135	1	1	yes	characteristic
370	C₂×C₄	#88165	1	2	yes	—
371	C₂×C₄	#88168	1	2	yes	—
372	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
373	C₂×C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
374	C₂×C₄	C₂×#6129	1	1	yes	characteristic
375	C₈	—	4	2	no	—
376	C₈	—	4	2	no	—
377	C₈	—	13	4	no	—
378	C₈	—	26	2	no	—
379	C₈	—	26	1	no	—
380	C₈	—	26	1	no	—
381	C₈	—	26	1	no	—
382	C₈	—	26	1	no	—
383	C₈	—	52	2	no	—
384	C₂²	—	2	1	no	—
385	C₂²	—	2	1	no	—
386	C₂²	—	4	2	no	—
387	C₂²	—	4	2	no	—
388	C₂²	—	4	2	no	—
389	C₂²	—	52	2	no	—
390	C₂²	—	52	2	no	—
391	C₂²	—	52	2	no	—
392	C₂²	#11718446	1	2	yes	—
393	C₂²	C₂×#88135	1	1	yes	characteristic
394	C₄	—	2	1	no	—
395	C₄	—	2	1	no	—
396	C₄	—	4	2	no	—
397	C₄	—	52	2	no	—
398	C₄	#11718446	1	2	yes	—
399	C₄	#11718484	1	2	yes	—
400	C₄	D₁₃×#131	1	2	yes	—
401	C₄	C₂×#88165	1	1	yes	characteristic
402	C₄	C₂×#88168	1	1	yes	characteristic
403	C₂	—	2	2	no	—
404	C₂	—	8	2	no	—
405	C₂	—	104	2	no	—
406	C₂	#13123544	1	1	yes	characteristic
407	C₂	C₂×#11718446	1	1	yes	characteristic
408	C₂	C₂×#11718484	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×#1492	C₂
socle	C₂²×C₁₃	C₂×C₄×D₄
prime radical	#13125411	C₂
cosocle	C₄²×C₁₃	C₂⁴
hypercenter	#1492	D₁₃
hypocenter	C₁₃	#17962
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1492

Complements

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

N	H ≅ G/N	type
2. #13123548	C₂	semidirect
3. #13123547	C₂	semidirect
4. #13113747	C₂	semidirect
5. #13113657	C₂	semidirect
6. #13113591	C₂	semidirect
7. #13113288	C₂	semidirect
8. C₂×#11718046	C₂	semidirect
9. C₁₃×#1492	C₂	semidirect
149. #1492	D₁₃	semidirect
188. #1055	#131	semidirect
345. C₁₃	#17962	semidirect

Maximal subgroups

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

subgroup	index
#13125411	2
#13123548	2
#13123547	2
#13113747	2
#13113657	2
#13113591	2
#13113288	2
C₂×#11718046	2
C₁₃×#1492	2
#17962	13