Subgroups — #16016327

Order 3328.

652 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13127969	C₂	1	1	yes	maximal, characteristic
2	D₁₃×#443	C₂	1	2	yes	maximal, complemented
3	#13116722	C₂	1	1	yes	maximal, characteristic, complemented
4	#13116721	C₂	1	1	yes	maximal, characteristic, complemented
5	#13116719	C₂	1	2	yes	maximal, complemented
6	#13116718	C₂	1	1	yes	maximal, characteristic, complemented
7	#13116715	C₂	1	2	yes	maximal, complemented
8	#13116629	C₂	1	1	yes	maximal, characteristic, complemented
9	#13116562	C₂	1	2	yes	maximal, complemented
10	C₄×#88038	C₂	1	1	yes	maximal, characteristic, complemented
11	C₁₃×#2178	C₂	1	1	yes	maximal, characteristic, complemented
12	D₁₃×#135	C₂²	1	2	yes	—
13	D₁₃×#131	—	2	2	no	—
14	C₁₃×#508	C₂²	1	1	yes	characteristic
15	C₁₃×#443	C₂²	1	2	yes	complemented
16	#11718857	C₂²	1	2	yes	complemented
17	#11718855	C₂²	1	2	yes	—
18	#11718805	—	2	2	no	—
19	#11718579	C₄	1	2	yes	complemented
20	#11718576	C₄	1	2	yes	complemented
21	#11718575	C₄	1	2	yes	complemented
22	#11718572	C₄	1	2	yes	complemented
23	#11718565	C₂²	1	1	yes	characteristic
24	C₂×#88038	C₂²	1	1	yes	characteristic
25	C₂×#88038	C₂²	1	1	yes	characteristic
26	#11718556	C₂²	1	2	yes	complemented
27	D₁₃×#108	C₂²	1	2	yes	—
28	#11718485	—	2	2	no	—
29	D₁₃×#104	—	2	2	no	—
30	#11718361	C₂²	1	2	yes	complemented
31	C₄×C₈×D₁₃	C₂²	1	1	yes	characteristic
32	C₁₃×#368	C₂²	1	1	yes	characteristic, complemented
33	C₁₃×#359	C₂²	1	2	yes	complemented
34	C₄×C₁₃×C₁₆	C₂²	1	1	yes	characteristic, complemented
35	#11718159	C₂²	1	2	yes	complemented
36	#11718102	C₂²	1	2	yes	complemented
37	#11718098	C₂²	1	1	yes	characteristic, complemented
38	#11718094	C₂²	1	1	yes	characteristic, complemented
39	#11718094	C₂²	1	1	yes	characteristic, complemented
40	#11718089	C₂²	1	2	yes	complemented
41	#11718085	C₂²	1	2	yes	complemented
42	C₄×#6090	C₂²	1	1	yes	characteristic, complemented
43	#41×D₁₃	—	2	2	no	—
44	C₁₃×#135	C₂³	1	2	yes	—
45	C₁₃×#131	—	2	2	no	—
46	#88204	—	2	2	no	—
47	#88202	—	2	2	no	—
48	#88173	C₂×C₄	1	2	yes	—
49	Q₁₆×D₁₃	C₂×C₄	1	2	yes	—
50	#88170	—	2	2	no	—
51	SD₁₆×D₁₃	—	2	2	no	—
52	#88166	C₂×C₄	1	2	yes	—
53	D₈×D₁₃	C₂×C₄	1	2	yes	—
54	#88161	—	2	2	no	—
55	#88161	—	4	2	no	—
56	M₁₆×D₁₃	—	2	2	no	—
57	M₁₆×D₁₃	—	4	2	no	—
58	#88158	C₂³	1	2	yes	—
59	#88155	—	2	2	no	—
60	#88155	—	4	2	no	—
61	C₂×C₈×D₁₃	—	4	2	no	—
62	C₂×C₈×D₁₃	D₄	1	1	yes	characteristic
63	C₂×C₈×D₁₃	C₂³	1	1	yes	characteristic
64	C₄²×D₁₃	D₄	1	1	yes	characteristic
65	C₁₃×Q₃₂	C₂×C₄	1	1	yes	characteristic, complemented
66	C₁₃×SD₃₂	C₂×C₄	1	2	yes	complemented
67	C₁₃×D₁₆	C₂×C₄	1	1	yes	characteristic, complemented
68	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
69	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
70	C₁₃×#108	C₂³	1	2	yes	—
71	C₁₃×#104	—	2	2	no	—
72	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
73	#88077	—	2	2	no	—
74	#88069	C₂×C₄	1	2	yes	complemented
75	#88068	C₂×C₄	1	2	yes	complemented
76	#88067	C₂×C₄	1	2	yes	complemented
77	#88066	C₂×C₄	1	2	yes	complemented
78	#88065	—	2	2	no	—
79	#88062	—	2	2	no	—
80	#88059	C₂³	1	2	yes	—
81	C₈×Dic₁₃	D₄	1	1	yes	characteristic
82	C₈×Dic₁₃	C₂³	1	1	yes	characteristic
83	C₂×#6090	C₂³	1	1	yes	characteristic
84	C₂×#6090	C₂³	1	1	yes	characteristic
85	#88045	—	2	2	no	—
86	C₄×#1055	D₄	1	1	yes	characteristic
87	Dic₁₀₄	C₂×C₄	1	1	yes	characteristic, complemented
88	#88040	C₂×C₄	1	2	yes	complemented
89	D₂₀₈	C₂×C₄	1	1	yes	characteristic, complemented
90	#88038	—	2	2	no	—
91	#88038	C₂×C₄	1	2	yes	—
92	#88038	C₂×C₄	1	2	yes	—
93	#33463	—	13	1	no	maximal, Sylow
94	C₁₃×#41	—	2	2	no	—
95	#6131	—	2	2	no	—
96	Q₈×D₁₃	—	2	2	no	—
97	#6129	—	2	2	no	—
98	#6129	—	4	2	no	—
99	D₄×D₁₃	—	2	2	no	—
100	D₄×D₁₃	—	4	2	no	—
101	#6127	—	2	2	no	—
102	#6127	—	4	2	no	—
103	C₂×C₄×D₁₃	—	2	1	no	—
104	C₂×C₄×D₁₃	—	4	2	no	—
105	C₂×C₄×D₁₃	C₂×D₄	1	1	yes	characteristic
106	C₁₃×Q₁₆	C₂²×C₄	1	2	yes	—
107	C₁₃×SD₁₆	—	2	2	no	—
108	C₁₃×D₈	C₂²×C₄	1	2	yes	—
109	C₁₃×M₁₆	—	2	2	no	—
110	C₁₃×M₁₆	—	4	2	no	—
111	C₂×C₈×C₁₃	—	4	2	no	—
112	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
113	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
114	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
115	#6107	—	2	2	no	—
116	#6106	—	2	2	no	—
117	#6105	—	2	2	no	—
118	#6104	—	2	2	no	—
119	C₄×Dic₁₃	—	2	1	no	—
120	C₄×Dic₁₃	C₂×D₄	1	1	yes	characteristic
121	#6099	—	2	2	no	—
122	#6099	—	4	2	no	—
123	C₂×#1055	—	4	2	no	—
124	C₂×#1055	C₂×D₄	1	1	yes	characteristic
125	C₂×#1055	C₂×D₄	1	1	yes	characteristic
126	Dic₅₂	C₂²×C₄	1	2	yes	—
127	D₁₀₄	C₂²×C₄	1	2	yes	—
128	#6095	—	2	2	no	—
129	#6094	—	4	2	no	—
130	#6094	—	4	2	no	—
131	#6094	—	4	2	no	—
132	C₈×D₁₃	—	4	2	no	—
133	C₈×D₁₃	—	4	2	no	—
134	C₈×D₁₃	—	4	2	no	—
135	C₈×D₁₃	D₈	1	2	yes	—
136	C₈×D₁₃	D₈	1	2	yes	—
137	C₈×D₁₃	C₂²×C₄	1	2	yes	—
138	C₈×D₁₃	#41	1	2	yes	—
139	C₁₃×C₁₆	—	2	1	no	—
140	C₁₃×C₁₆	C₂²×C₄	1	1	yes	characteristic
141	C₁₃×C₁₆	C₂²×C₄	1	1	yes	characteristic
142	#6090	—	2	1	no	—
143	#6090	C₂²×C₄	1	2	yes	—
144	#3414	—	13	1	no	—
145	C₂×#443	—	13	2	no	—
146	#2179	—	13	1	no	—
147	#2179	—	13	1	no	—
148	#2179	—	13	2	no	—
149	#2178	—	13	2	no	—
150	#2178	—	13	1	no	—
151	#2178	D₁₃	1	1	yes	characteristic, complemented
152	#2161	—	13	1	no	—
153	#2143	—	13	2	no	—
154	C₄×M₃₂	—	13	1	no	—
155	C₂²×D₁₃	—	4	2	no	—
156	C₂²×D₁₃	#135	1	1	yes	characteristic
157	Q₈×C₁₃	—	2	2	no	—
158	D₄×C₁₃	—	2	2	no	—
159	D₄×C₁₃	—	4	2	no	—
160	C₂×C₄×C₁₃	—	2	1	no	—
161	C₂×C₄×C₁₃	—	4	2	no	—
162	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
163	C₂²⋊D₁₃	—	4	2	no	—
164	C₂²⋊D₁₃	—	4	2	no	—
165	C₂²⋊D₁₃	—	4	2	no	—
166	C₂×Dic₁₃	—	2	1	no	—
167	C₂×Dic₁₃	—	4	2	no	—
168	C₂×Dic₁₃	#135	1	1	yes	characteristic
169	D₅₂	—	2	2	no	—
170	D₅₂	—	4	2	no	—
171	C₄×D₁₃	—	4	2	no	—
172	C₄×D₁₃	—	4	2	no	—
173	C₄×D₁₃	—	4	2	no	—
174	C₄×D₁₃	—	4	2	no	—
175	C₄×D₁₃	C₄×D₄	1	2	yes	—
176	C₄×D₁₃	C₂×D₈	1	2	yes	—
177	Dic₂₆	—	2	2	no	—
178	Dic₂₆	—	4	2	no	—
179	C₈×C₁₃	—	4	2	no	—
180	C₈×C₁₃	C₂×D₈	1	1	yes	characteristic
181	C₈×C₁₃	C₂×D₈	1	1	yes	characteristic
182	C₈×C₁₃	C₂³×C₄	1	1	yes	characteristic
183	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
184	#1055	—	4	2	no	—
185	#1055	C₄×D₄	1	2	yes	—
186	#1055	C₂×D₈	1	2	yes	—
187	C₂×#135	—	13	2	no	—
188	C₂×#131	—	26	2	no	—
189	#510	—	13	2	no	—
190	#510	—	13	2	no	—
191	#509	—	13	2	no	—
192	#509	—	13	2	no	—
193	#508	—	13	2	no	—
194	#508	—	13	1	no	—
195	#508	C₂×D₁₃	1	1	yes	characteristic
196	C₂×M₃₂	—	13	1	no	—
197	C₂×M₃₂	—	13	1	no	—
198	#443	—	13	2	no	—
199	#443	—	13	2	no	—
200	#443	—	13	2	no	—
201	#443	—	26	2	no	—
202	#443	—	26	2	no	—
203	#443	C₂×D₁₃	1	2	yes	complemented
204	C₂×#108	—	13	2	no	—
205	C₂×#104	—	26	2	no	—
206	C₂×C₄×C₈	—	13	1	no	—
207	#368	—	13	1	no	—
208	#368	—	13	2	no	—
209	#368	C₂×D₁₃	1	1	yes	characteristic, complemented
210	#359	—	13	2	no	—
211	#359	—	13	2	no	—
212	#359	—	13	2	no	—
213	#359	C₂×D₁₃	1	2	yes	complemented
214	#346	—	13	1	no	—
215	#346	—	13	1	no	—
216	C₄×C₁₆	—	13	1	no	—
217	C₄×C₁₆	C₂×D₁₃	1	1	yes	characteristic, complemented
218	C₂²×C₁₃	—	4	2	no	—
219	C₂²×C₁₃	C₂×#135	1	1	yes	characteristic
220	C₂×D₁₃	—	4	1	no	—
221	C₂×D₁₃	—	4	2	no	—
222	C₂×D₁₃	—	8	4	no	—
223	C₂×D₁₃	C₄×D₈	1	2	yes	—
224	C₄×C₁₃	—	2	2	no	—
225	C₄×C₁₃	—	4	2	no	—
226	C₄×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
227	C₄×C₁₃	C₂²×D₈	1	1	yes	characteristic
228	Dic₁₃	—	4	1	no	—
229	Dic₁₃	—	4	2	no	—
230	Dic₁₃	C₄×D₈	1	2	yes	—
231	C₂×#41	—	26	2	no	—
232	#135	—	13	2	no	—
233	#135	—	13	2	no	—
234	#135	—	13	2	no	—
235	#135	—	26	2	no	—
236	#135	—	26	2	no	—
237	#135	C₂²×D₁₃	1	2	yes	—
238	C₂×Q₁₆	—	13	2	no	—
239	C₂×SD₁₆	—	26	2	no	—
240	C₂×D₈	—	13	2	no	—
241	#131	—	2	2	no	—
242	#131	—	26	2	no	—
243	#131	—	26	2	no	—
244	#131	—	26	2	no	—
245	#131	—	52	2	no	—
246	#131	—	52	2	no	—
247	C₂×M₁₆	—	26	2	no	—
248	C₂×M₁₆	—	52	2	no	—
249	C₂²×C₈	—	13	1	no	—
250	C₂²×C₈	—	13	1	no	—
251	C₂²×C₈	—	52	2	no	—
252	C₂×C₄²	—	13	1	no	—
253	Q₃₂	—	13	2	no	—
254	Q₃₂	—	13	1	no	—
255	Q₃₂	C₄×D₁₃	1	1	yes	characteristic, complemented
256	SD₃₂	—	13	2	no	—
257	SD₃₂	—	13	2	no	—
258	SD₃₂	—	13	2	no	—
259	SD₃₂	C₄×D₁₃	1	2	yes	complemented
260	D₁₆	—	13	2	no	—
261	D₁₆	—	13	1	no	—
262	D₁₆	C₄×D₁₃	1	1	yes	characteristic, complemented
263	M₃₂	—	13	2	no	—
264	M₃₂	—	13	2	no	—
265	M₃₂	—	26	2	no	—
266	C₂×C₁₆	—	13	1	no	—
267	C₂×C₁₆	—	13	1	no	—
268	C₂×C₁₆	C₂²×D₁₃	1	1	yes	characteristic
269	C₂×C₁₆	C₂²×D₁₃	1	1	yes	characteristic
270	#108	—	13	2	no	—
271	#108	—	26	2	no	—
272	#108	C₂²×D₁₃	1	2	yes	—
273	#104	—	2	2	no	—
274	#104	—	26	2	no	—
275	#104	—	26	2	no	—
276	#104	—	26	2	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	—	2	1	no	—
282	C₂×C₁₃	—	8	2	no	—
283	C₂×C₁₃	C₂×C₄×D₈	1	1	yes	characteristic
284	D₁₃	—	2	2	no	—
285	D₁₃	—	8	2	no	—
286	#41	—	2	2	no	—
287	#41	—	26	2	no	—
288	#41	—	26	2	no	—
289	#41	—	26	2	no	—
290	#41	—	52	2	no	—
291	#41	—	52	2	no	—
292	C₂×Q₈	—	26	2	no	—
293	C₂×D₄	—	26	2	no	—
294	C₂×D₄	—	52	2	no	—
295	C₂²×C₄	—	13	1	no	—
296	C₂²×C₄	—	26	1	no	—
297	C₂²×C₄	—	52	2	no	—
298	Q₁₆	—	13	2	no	—
299	Q₁₆	—	26	2	no	—
300	Q₁₆	C₂×C₄×D₁₃	1	2	yes	—
301	SD₁₆	—	2	2	no	—
302	SD₁₆	—	26	2	no	—
303	SD₁₆	—	26	2	no	—
304	SD₁₆	—	26	2	no	—
305	D₈	—	13	2	no	—
306	D₈	—	26	2	no	—
307	D₈	C₂×C₄×D₁₃	1	2	yes	—
308	M₁₆	—	2	2	no	—
309	M₁₆	—	4	2	no	—
310	M₁₆	—	26	2	no	—
311	M₁₆	—	52	2	no	—
312	M₁₆	—	52	2	no	—
313	M₁₆	—	52	2	no	—
314	M₁₆	—	52	2	no	—
315	C₂×C₈	—	4	2	no	—
316	C₂×C₈	—	13	1	no	—
317	C₂×C₈	—	13	1	no	—
318	C₂×C₈	—	13	2	no	—
319	C₂×C₈	—	13	2	no	—
320	C₂×C₈	—	13	2	no	—
321	C₂×C₈	—	13	2	no	—
322	C₂×C₈	—	52	2	no	—
323	C₂×C₈	—	52	2	no	—
324	C₂×C₈	—	52	2	no	—
325	C₂×C₈	—	52	2	no	—
326	C₂×C₈	D₄×D₁₃	1	1	yes	characteristic
327	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
328	C₄²	—	13	1	no	—
329	C₄²	—	26	1	no	—
330	C₄²	D₄×D₁₃	1	1	yes	characteristic
331	C₁₆	—	2	1	no	—
332	C₁₆	—	13	2	no	—
333	C₁₆	—	26	1	no	—
334	C₁₆	C₂×C₄×D₁₃	1	1	yes	characteristic
335	C₁₆	C₂×C₄×D₁₃	1	1	yes	characteristic
336	C₁₃	#33463	1	1	yes	characteristic, complemented, Sylow
337	C₂³	—	13	1	no	—
338	C₂³	—	52	2	no	—
339	Q₈	—	2	2	no	—
340	Q₈	—	26	2	no	—
341	Q₈	—	52	2	no	—
342	D₄	—	2	2	no	—
343	D₄	—	4	2	no	—
344	D₄	—	26	2	no	—
345	D₄	—	52	2	no	—
346	D₄	—	52	2	no	—
347	D₄	—	52	2	no	—
348	D₄	—	52	2	no	—
349	C₂×C₄	—	2	1	no	—
350	C₂×C₄	—	4	2	no	—
351	C₂×C₄	—	13	1	no	—
352	C₂×C₄	—	13	2	no	—
353	C₂×C₄	—	13	2	no	—
354	C₂×C₄	—	26	1	no	—
355	C₂×C₄	—	52	2	no	—
356	C₂×C₄	—	52	2	no	—
357	C₂×C₄	—	52	2	no	—
358	C₂×C₄	—	52	2	no	—
359	C₂×C₄	—	52	2	no	—
360	C₂×C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
361	C₈	—	4	2	no	—
362	C₈	—	13	2	no	—
363	C₈	—	13	2	no	—
364	C₈	—	52	2	no	—
365	C₈	D₈×D₁₃	1	1	yes	characteristic
366	C₈	D₈×D₁₃	1	1	yes	characteristic
367	C₈	C₂²×C₄×D₁₃	1	1	yes	characteristic
368	C₈	#41×D₁₃	1	1	yes	characteristic
369	C₂²	—	4	2	no	—
370	C₂²	—	13	2	no	—
371	C₂²	—	52	1	no	—
372	C₂²	—	52	2	no	—
373	C₂²	—	104	4	no	—
374	C₂²	D₁₃×#135	1	1	yes	characteristic
375	C₄	—	2	2	no	—
376	C₄	—	4	2	no	—
377	C₄	—	13	2	no	—
378	C₄	—	52	1	no	—
379	C₄	—	52	2	no	—
380	C₄	C₄×D₄×D₁₃	1	1	yes	characteristic
381	C₄	C₂×D₈×D₁₃	1	1	yes	characteristic
382	C₂	—	2	1	no	—
383	C₂	—	8	2	no	—
384	C₂	—	26	2	no	—
385	C₂	—	104	2	no	—
386	C₂	C₄×D₈×D₁₃	1	1	yes	characteristic

	subgroup	G/H
center	C₈	D₈×D₁₃
commutator	C₈×C₁₃	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₁₃
Fitting	C₁₃×#2178	C₂
socle	C₂×C₁₃	C₂×C₄×D₈
prime radical	#13127969	C₂
cosocle	C₂×C₈×C₁₃	C₂⁴
hypercenter	#2178	D₁₃
hypocenter	C₁₃	#33463
LCS	G ▷ C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₈ ◁ C₂×C₈ ◁ C₄×C₈ ◁ #2178

Complements

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

N	H ≅ G/N	type
2. D₁₃×#443	C₂	semidirect
3. #13116722	C₂	semidirect
4. #13116721	C₂	semidirect
5. #13116719	C₂	semidirect
6. #13116718	C₂	semidirect
7. #13116715	C₂	semidirect
8. #13116629	C₂	semidirect
9. #13116562	C₂	semidirect
10. C₄×#88038	C₂	semidirect
11. C₁₃×#2178	C₂	semidirect
15. C₁₃×#443	C₂²	semidirect
16. #11718857	C₂²	semidirect
19. #11718579	C₄	semidirect
20. #11718576	C₄	semidirect
21. #11718575	C₄	semidirect
22. #11718572	C₄	semidirect
26. #11718556	C₂²	semidirect
30. #11718361	C₂²	semidirect
32. C₁₃×#368	C₂²	semidirect
33. C₁₃×#359	C₂²	semidirect
34. C₄×C₁₃×C₁₆	C₂²	semidirect
35. #11718159	C₂²	semidirect
36. #11718102	C₂²	semidirect
37. #11718098	C₂²	semidirect
38. #11718094	C₂²	semidirect
39. #11718094	C₂²	semidirect
40. #11718089	C₂²	semidirect
41. #11718085	C₂²	semidirect
42. C₄×#6090	C₂²	semidirect
65. C₁₃×Q₃₂	C₂×C₄	semidirect
66. C₁₃×SD₃₂	C₂×C₄	semidirect
67. C₁₃×D₁₆	C₂×C₄	semidirect
74. #88069	C₂×C₄	semidirect
75. #88068	C₂×C₄	semidirect
76. #88067	C₂×C₄	semidirect
77. #88066	C₂×C₄	semidirect
87. Dic₁₀₄	C₂×C₄	semidirect
88. #88040	C₂×C₄	semidirect
89. D₂₀₈	C₂×C₄	semidirect
151. #2178	D₁₃	semidirect
203. #443	C₂×D₁₃	semidirect
209. #368	C₂×D₁₃	semidirect
213. #359	C₂×D₁₃	semidirect
217. C₄×C₁₆	C₂×D₁₃	semidirect
255. Q₃₂	C₄×D₁₃	semidirect
259. SD₃₂	C₄×D₁₃	semidirect
262. D₁₆	C₄×D₁₃	semidirect
336. C₁₃	#33463	semidirect

Maximal subgroups

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

subgroup	index
#13127969	2
D₁₃×#443	2
#13116722	2
#13116721	2
#13116719	2
#13116718	2
#13116715	2
#13116629	2
#13116562	2
C₄×#88038	2
C₁₃×#2178	2
#33463	13