Subgroups — #16017589

Order 3328.

574 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₁₃×#500	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄×#88166	C₂	1	1	yes	maximal, characteristic
3	#13116989	C₂	1	1	yes	maximal, characteristic, complemented
4	#13116985	C₂	1	1	yes	maximal, characteristic, complemented
5	#13116983	C₂	1	1	yes	maximal, characteristic, complemented
6	#13116977	C₂	1	2	yes	maximal, complemented
7	#13116971	C₂	1	2	yes	maximal, complemented
8	#13116623	C₂	1	2	yes	maximal, complemented
9	#13116590	C₂	1	1	yes	maximal, characteristic, complemented
10	#13116545	C₂	1	2	yes	maximal
11	C₁₃×#2226	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#88166	C₂²	1	1	yes	characteristic
13	D₁₃×C₄⋊Q₈	—	2	1	no	—
14	C₄×#6129	—	2	1	no	—
15	C₁₃×#500	C₂²	1	1	yes	characteristic
16	C₄×C₁₃×D₈	C₂²	1	1	yes	characteristic
17	D₈×Dic₁₃	C₂²	1	1	yes	characteristic
18	#11718739	—	2	1	no	—
19	C₄×#6105	—	2	1	no	—
20	C₂×#88038	C₂²	1	2	yes	—
21	#11718541	C₂²	1	1	yes	characteristic
22	D₁₃×#107	—	2	1	no	—
23	D₁₃×#107	C₂²	1	2	yes	—
24	#11718535	—	2	1	no	—
25	#11718535	C₂²	1	2	yes	—
26	#11718534	C₂²	1	1	yes	characteristic
27	#11718433	—	2	1	no	—
28	#11718419	—	2	1	no	—
29	C₄×Dic₅₂	C₂²	1	1	yes	characteristic
30	C₄×C₈×D₁₃	C₂²	1	1	yes	characteristic
31	#11718343	C₂²	1	1	yes	characteristic
32	C₁₃×#367	C₂²	1	2	yes	—
33	C₁₃×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
34	C₁₃×#357	C₂²	1	2	yes	—
35	#11718154	C₂²	1	2	yes	—
36	#11718096	C₂²	1	2	yes	—
37	#11718095	C₂²	1	2	yes	—
38	#11718093	C₂²	1	2	yes	—
39	#11718084	C₂²	1	2	yes	—
40	#11718082	C₂²	1	2	yes	—
41	#11718081	C₂²	1	2	yes	—
42	C₄⋊#6090	C₂²	1	1	yes	characteristic
43	C₂×Q₈×D₁₃	—	4	1	no	—
44	C₂×#6129	—	2	1	no	—
45	C₂×C₁₃×D₈	C₂³	1	1	yes	characteristic
46	C₁₃×C₄⋊Q₈	—	2	1	no	—
47	C₄×D₄×C₁₃	—	2	1	no	—
48	#88198	—	4	1	no	—
49	D₄×Dic₁₃	—	2	1	no	—
50	C₂×#6105	—	2	1	no	—
51	C₄×C₂²⋊D₁₃	—	4	1	no	—
52	C₂×C₄×Dic₁₃	—	4	1	no	—
53	#88166	—	2	2	no	—
54	#88166	Q₈	1	4	yes	complemented
55	C₂×Dic₅₂	C₂³	1	1	yes	characteristic
56	C₂×C₈×D₁₃	D₄	1	1	yes	characteristic
57	C₂×C₈×D₁₃	C₂³	1	1	yes	characteristic
58	#88146	—	2	1	no	—
59	C₄⋊C₄×D₁₃	—	2	2	no	—
60	C₄⋊C₄×D₁₃	—	4	1	no	—
61	#88142	—	2	2	no	—
62	#88142	—	4	1	no	—
63	#88141	—	2	1	no	—
64	#88135	—	4	1	no	—
65	#88131	—	4	1	no	—
66	C₄²×D₁₃	D₄	1	1	yes	characteristic
67	#88123	—	2	1	no	—
68	C₄×Dic₂₆	—	2	1	no	—
69	C₂×C₁₃×C₁₆	C₂³	1	2	yes	—
70	C₁₃×#107	—	2	1	no	—
71	C₁₃×#107	C₂³	1	2	yes	—
72	C₁₃×#107	C₂³	1	1	yes	characteristic
73	C₁₃×#102	—	2	1	no	—
74	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
75	#88072	—	2	1	no	—
76	#88058	—	2	1	no	—
77	#88058	C₂³	1	2	yes	—
78	#88058	C₂³	1	1	yes	characteristic
79	#88056	—	2	1	no	—
80	C₈×Dic₁₃	D₄	1	1	yes	characteristic
81	C₈×Dic₁₃	C₂³	1	1	yes	characteristic
82	C₂×#6090	C₂³	1	2	yes	—
83	#88050	—	2	1	no	—
84	#88048	—	2	1	no	—
85	#88047	—	2	2	no	—
86	#88047	—	2	2	no	—
87	C₄×#1055	D₄	1	1	yes	characteristic
88	#88038	—	2	4	no	—
89	#33629	—	13	1	no	maximal, Sylow
90	C₂×Q₈×C₁₃	—	4	1	no	—
91	C₂×D₄×C₁₃	—	2	1	no	—
92	C₂²×C₄×C₁₃	—	4	1	no	—
93	C₂×C₂²⋊D₁₃	—	4	1	no	—
94	C₂²×Dic₁₃	—	4	1	no	—
95	C₂²×Dic₁₃	—	4	1	no	—
96	Q₈×D₁₃	—	8	4	no	—
97	#6129	—	2	4	no	—
98	#6129	—	4	2	no	—
99	C₂×C₄×D₁₃	—	4	2	no	—
100	C₂×C₄×D₁₃	D₈	1	1	yes	characteristic
101	C₂×C₄×D₁₃	D₈	1	1	yes	characteristic
102	C₂×C₄×D₁₃	C₂×D₄	1	1	yes	characteristic
103	C₂×Dic₂₆	—	2	1	no	—
104	C₂×Dic₂₆	—	4	1	no	—
105	C₂×Dic₂₆	—	4	2	no	—
106	C₁₃×D₈	—	2	1	no	—
107	C₁₃×D₈	C₂×Q₈	1	2	yes	complemented
108	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
109	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
110	C₁₃×C₄⋊C₄	—	2	1	no	—
111	C₁₃×C₄⋊C₄	—	2	2	no	—
112	C₁₃×C₄⋊C₄	—	4	1	no	—
113	C₁₃×C₂²⋊C₄	—	4	1	no	—
114	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
115	C₂²⋊Dic₁₃	—	4	1	no	—
116	#6105	—	2	4	no	—
117	#6103	—	4	1	no	—
118	C₄⋊Dic₁₃	—	2	1	no	—
119	C₄⋊Dic₁₃	—	2	2	no	—
120	C₄⋊Dic₁₃	—	4	1	no	—
121	#6101	—	4	1	no	—
122	#6101	—	4	1	no	—
123	#6101	—	4	2	no	—
124	#6101	—	4	2	no	—
125	C₄×Dic₁₃	—	4	1	no	—
126	C₄×Dic₁₃	—	4	1	no	—
127	C₄×Dic₁₃	—	4	1	no	—
128	C₄×Dic₁₃	D₈	1	1	yes	characteristic
129	C₄×Dic₁₃	D₈	1	1	yes	characteristic
130	C₄×Dic₁₃	C₂×D₄	1	1	yes	characteristic
131	C₂×#1055	C₂×D₄	1	1	yes	characteristic
132	C₂×#1055	C₂×D₄	1	1	yes	characteristic
133	Dic₅₂	—	2	1	no	—
134	Dic₅₂	C₂×Q₈	1	2	yes	complemented
135	C₈×D₁₃	—	2	2	no	—
136	C₈×D₁₃	C₂×Q₈	1	2	yes	—
137	C₈×D₁₃	#41	1	2	yes	—
138	C₁₃×C₁₆	—	2	2	no	—
139	#6090	—	2	2	no	—
140	C₂×#500	—	13	1	no	—
141	C₄×#135	—	13	1	no	—
142	#2227	—	13	1	no	—
143	#2227	—	13	1	no	—
144	#2226	—	13	1	no	—
145	#2226	D₁₃	1	1	yes	characteristic, complemented
146	#2225	—	13	2	no	—
147	#2224	—	13	2	no	—
148	#2159	—	13	2	no	—
149	#2150	—	13	1	no	—
150	#2139	—	13	2	no	—
151	C₂³×C₁₃	—	4	1	no	—
152	C₂²×D₁₃	C₂×D₈	1	1	yes	characteristic
153	Q₈×C₁₃	—	8	2	no	—
154	D₄×C₁₃	—	2	2	no	—
155	D₄×C₁₃	—	4	1	no	—
156	C₂×C₄×C₁₃	—	4	1	no	—
157	C₂×C₄×C₁₃	—	4	2	no	—
158	C₂×C₄×C₁₃	—	8	2	no	—
159	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
160	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
161	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
162	C₂²⋊D₁₃	—	4	4	no	—
163	C₂×Dic₁₃	—	4	1	no	—
164	C₂×Dic₁₃	—	4	4	no	—
165	C₂×Dic₁₃	—	4	1	no	—
166	C₂×Dic₁₃	—	4	2	no	—
167	C₂×Dic₁₃	—	8	2	no	—
168	C₂×Dic₁₃	C₂×D₈	1	1	yes	characteristic
169	C₂×Dic₁₃	C₂×D₈	1	1	yes	characteristic
170	C₂×Dic₁₃	C₂×D₈	1	1	yes	characteristic
171	C₄×D₁₃	—	2	2	no	—
172	C₄×D₁₃	—	2	2	no	—
173	C₄×D₁₃	—	8	4	no	—
174	C₄×D₁₃	C₂²⋊Q₈	1	2	yes	—
175	C₄×D₁₃	#137	1	2	yes	—
176	Dic₂₆	—	2	2	no	—
177	Dic₂₆	—	4	1	no	—
178	Dic₂₆	—	8	2	no	—
179	Dic₂₆	—	8	4	no	—
180	C₈×C₁₃	—	2	1	no	—
181	C₈×C₁₃	C₂²×Q₈	1	1	yes	characteristic
182	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
183	#1055	—	2	1	no	—
184	#1055	C₂²⋊Q₈	1	2	yes	—
185	C₂×#135	—	13	1	no	—
186	C₂×C₄⋊Q₈	—	26	1	no	—
187	C₄×#41	—	26	1	no	—
188	C₂×M₃₂	—	13	2	no	—
189	#500	—	13	2	no	—
190	#500	—	13	1	no	—
191	#500	—	26	1	no	—
192	#500	—	26	1	no	—
193	#500	C₂×D₁₃	1	1	yes	characteristic
194	C₄×Q₁₆	—	13	1	no	—
195	C₄×Q₁₆	—	13	1	no	—
196	C₄×SD₁₆	—	26	1	no	—
197	C₄×SD₁₆	—	26	1	no	—
198	C₄×D₈	—	13	1	no	—
199	C₄×D₈	C₂×D₁₃	1	1	yes	characteristic
200	#427	—	13	1	no	—
201	C₂×#107	—	13	2	no	—
202	C₂×#107	—	26	1	no	—
203	#416	—	26	1	no	—
204	C₂×C₄×C₈	—	13	1	no	—
205	#367	—	13	2	no	—
206	#367	C₂×D₁₃	1	2	yes	—
207	#366	—	13	2	no	—
208	#366	—	13	2	no	—
209	C₄⋊C₁₆	—	13	2	no	—
210	C₄⋊C₁₆	—	13	1	no	—
211	C₄⋊C₁₆	C₂×D₁₃	1	1	yes	characteristic
212	#358	—	13	2	no	—
213	#358	—	13	2	no	—
214	#357	—	13	2	no	—
215	#357	C₂×D₁₃	1	2	yes	—
216	C₂²×C₁₃	—	4	2	no	—
217	C₂²×C₁₃	—	8	1	no	—
218	C₂²×C₁₃	—	8	1	no	—
219	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
220	C₂×D₁₃	—	2	1	no	—
221	C₂×D₁₃	—	2	1	no	—
222	C₂×D₁₃	#474	1	2	yes	—
223	C₄×C₁₃	—	2	1	no	—
224	C₄×C₁₃	—	8	2	no	—
225	C₄×C₁₃	—	8	1	no	—
226	C₄×C₁₃	#510	1	2	yes	—
227	C₄×C₁₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
228	C₄×C₁₃	C₂×#137	1	1	yes	characteristic
229	Dic₁₃	—	2	1	no	—
230	Dic₁₃	—	4	2	no	—
231	Dic₁₃	—	8	2	no	—
232	Dic₁₃	—	8	1	no	—
233	Dic₁₃	#474	1	2	yes	—
234	Dic₁₃	#509	1	2	yes	—
235	C₂×#41	—	26	1	no	—
236	C₂²×Q₈	—	52	1	no	—
237	#135	—	13	4	no	—
238	#135	—	26	2	no	—
239	C₂×Q₁₆	—	13	1	no	—
240	C₂×SD₁₆	—	26	1	no	—
241	C₂×D₈	C₂²×D₁₃	1	1	yes	characteristic
242	C₂²×C₈	—	13	1	no	—
243	C₂²×C₈	—	13	1	no	—
244	C₄⋊Q₈	—	2	1	no	—
245	C₄⋊Q₈	—	26	2	no	—
246	C₄⋊Q₈	—	26	1	no	—
247	C₄⋊Q₈	—	52	1	no	—
248	C₄⋊Q₈	—	52	1	no	—
249	C₄×Q₈	—	26	1	no	—
250	C₄×Q₈	—	26	1	no	—
251	C₄×D₄	—	2	1	no	—
252	C₄×D₄	—	26	1	no	—
253	C₄×D₄	—	52	1	no	—
254	C₄×D₄	—	52	1	no	—
255	#117	—	26	1	no	—
256	#117	—	52	1	no	—
257	C₂×C₄⋊C₄	—	26	2	no	—
258	C₂×C₄⋊C₄	—	52	1	no	—
259	C₂×C₄²	—	13	1	no	—
260	C₂×C₄²	—	52	1	no	—
261	M₃₂	—	26	4	no	—
262	C₂×C₁₆	—	13	2	no	—
263	C₂×C₁₆	C₂²×D₁₃	1	2	yes	—
264	#107	—	2	1	no	—
265	#107	—	13	2	no	—
266	#107	—	13	1	no	—
267	#107	—	26	2	no	—
268	#107	—	26	2	no	—
269	#107	—	26	1	no	—
270	#107	C₂²×D₁₃	1	2	yes	—
271	#107	C₂²×D₁₃	1	1	yes	characteristic
272	#106	—	26	1	no	—
273	#103	—	26	1	no	—
274	#103	—	26	1	no	—
275	#102	—	2	1	no	—
276	#102	—	26	1	no	—
277	C₄×C₈	—	13	1	no	—
278	C₄×C₈	—	13	1	no	—
279	C₄×C₈	—	13	1	no	—
280	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
281	C₂×C₁₃	—	8	2	no	—
282	C₂×C₁₃	C₂×#474	1	1	yes	characteristic
283	C₂×C₁₃	C₂×#509	1	1	yes	characteristic
284	C₂×C₁₃	C₂×#510	1	1	yes	characteristic
285	D₁₃	—	2	2	no	—
286	#41	—	26	4	no	—
287	#41	—	52	2	no	—
288	C₂×Q₈	—	4	1	no	—
289	C₂×Q₈	—	26	1	no	—
290	C₂×Q₈	—	52	2	no	—
291	C₂×Q₈	—	52	1	no	—
292	C₂×Q₈	—	104	4	no	—
293	C₂×D₄	—	2	1	no	—
294	C₂×D₄	—	52	1	no	—
295	C₂²×C₄	—	4	1	no	—
296	C₂²×C₄	—	13	1	no	—
297	C₂²×C₄	—	13	1	no	—
298	C₂²×C₄	—	13	1	no	—
299	C₂²×C₄	—	52	2	no	—
300	C₂²×C₄	—	52	1	no	—
301	C₂²×C₄	—	52	1	no	—
302	Q₁₆	—	13	2	no	—
303	Q₁₆	—	26	1	no	—
304	SD₁₆	—	26	4	no	—
305	D₈	—	2	1	no	—
306	D₈	Q₈×D₁₃	1	2	yes	complemented
307	C₂×C₈	—	13	1	no	—
308	C₂×C₈	—	13	2	no	—
309	C₂×C₈	—	13	2	no	—
310	C₂×C₈	—	13	1	no	—
311	C₂×C₈	—	26	2	no	—
312	C₂×C₈	D₄×D₁₃	1	1	yes	characteristic
313	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
314	C₄⋊C₄	—	2	1	no	—
315	C₄⋊C₄	—	2	2	no	—
316	C₄⋊C₄	—	4	1	no	—
317	C₄⋊C₄	—	26	1	no	—
318	C₄⋊C₄	—	26	2	no	—
319	C₄⋊C₄	—	52	2	no	—
320	C₄⋊C₄	—	52	2	no	—
321	C₄⋊C₄	—	52	1	no	—
322	C₄⋊C₄	—	52	1	no	—
323	C₄⋊C₄	—	52	1	no	—
324	C₂²⋊C₄	—	4	1	no	—
325	C₂²⋊C₄	—	52	1	no	—
326	C₂²⋊C₄	—	52	1	no	—
327	C₄²	—	13	1	no	—
328	C₄²	—	13	1	no	—
329	C₄²	—	13	1	no	—
330	C₄²	—	52	1	no	—
331	C₄²	—	52	1	no	—
332	C₄²	—	52	1	no	—
333	C₄²	D₄×D₁₃	1	1	yes	characteristic
334	C₁₆	—	2	2	no	—
335	C₁₆	—	26	2	no	—
336	C₁₃	#33629	1	1	yes	characteristic, complemented, Sylow
337	C₂³	—	4	1	no	—
338	C₂³	—	13	1	no	—
339	Q₈	—	8	2	no	—
340	Q₈	—	26	2	no	—
341	Q₈	—	52	1	no	—
342	Q₈	—	104	2	no	—
343	Q₈	—	104	4	no	—
344	D₄	—	2	2	no	—
345	D₄	—	4	1	no	—
346	D₄	—	52	4	no	—
347	C₂×C₄	—	4	1	no	—
348	C₂×C₄	—	4	2	no	—
349	C₂×C₄	—	8	2	no	—
350	C₂×C₄	—	13	1	no	—
351	C₂×C₄	—	13	1	no	—
352	C₂×C₄	—	13	2	no	—
353	C₂×C₄	—	13	2	no	—
354	C₂×C₄	—	13	1	no	—
355	C₂×C₄	—	26	2	no	—
356	C₂×C₄	—	26	2	no	—
357	C₂×C₄	—	52	4	no	—
358	C₂×C₄	—	52	1	no	—
359	C₂×C₄	—	52	1	no	—
360	C₂×C₄	—	52	2	no	—
361	C₂×C₄	—	104	4	no	—
362	C₂×C₄	—	104	2	no	—
363	C₂×C₄	D₈×D₁₃	1	1	yes	characteristic
364	C₂×C₄	D₈×D₁₃	1	1	yes	characteristic
365	C₂×C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
366	C₈	—	2	1	no	—
367	C₈	—	13	2	no	—
368	C₈	—	26	1	no	—
369	C₈	C₂×Q₈×D₁₃	1	1	yes	characteristic
370	C₈	#41×D₁₃	1	1	yes	characteristic
371	C₂²	—	4	2	no	—
372	C₂²	—	8	1	no	—
373	C₂²	—	8	1	no	—
374	C₂²	—	13	2	no	—
375	C₂²	—	26	1	no	—
376	C₂²	—	26	1	no	—
377	C₂²	C₂×D₈×D₁₃	1	1	yes	characteristic
378	C₄	—	2	1	no	—
379	C₄	—	8	2	no	—
380	C₄	—	8	1	no	—
381	C₄	—	13	2	no	—
382	C₄	—	13	2	no	—
383	C₄	—	26	1	no	—
384	C₄	—	52	2	no	—
385	C₄	—	104	1	no	—
386	C₄	—	104	2	no	—
387	C₄	#11718576	1	2	yes	—
388	C₄	D₁₃×C₂²⋊Q₈	1	1	yes	characteristic
389	C₄	D₁₃×#137	1	1	yes	characteristic
390	C₂	—	8	2	no	—
391	C₂	—	26	2	no	—
392	C₂	D₁₃×#474	1	1	yes	characteristic
393	C₂	D₁₃×#509	1	1	yes	characteristic
394	C₂	C₂×#11718576	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₈×D₁₃
commutator, cosocle	C₂×C₈×C₁₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₁₃
Fitting	C₁₃×#2226	C₂
socle	C₂²×C₁₃	C₂²×D₈
prime radical	C₂×#88166	C₂²
hypercenter	#2226	D₁₃
hypocenter	C₁₃	#33629
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2226

Complements

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

N	H ≅ G/N	type
1. D₁₃×#500	C₂	semidirect
3. #13116989	C₂	semidirect
4. #13116985	C₂	semidirect
5. #13116983	C₂	semidirect
6. #13116977	C₂	semidirect
7. #13116971	C₂	semidirect
8. #13116623	C₂	semidirect
9. #13116590	C₂	semidirect
11. C₁₃×#2226	C₂	semidirect
54. #88166	Q₈	semidirect
107. C₁₃×D₈	C₂×Q₈	semidirect
134. Dic₅₂	C₂×Q₈	semidirect
145. #2226	D₁₃	semidirect
306. D₈	Q₈×D₁₃	semidirect
336. C₁₃	#33629	semidirect

Maximal subgroups

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

subgroup	index
D₁₃×#500	2
C₄×#88166	2
#13116989	2
#13116985	2
#13116983	2
#13116977	2
#13116971	2
#13116623	2
#13116590	2
#13116545	2
C₁₃×#2226	2
#33629	13