Subgroups — #16789318

Order 3328.

482 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#131×#261	C₂	1	1	yes	maximal, characteristic
2	D₁₃⋊#350	C₂	1	1	yes	maximal, characteristic
3	M₃₂×#261	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#11718216	C₂	1	1	yes	maximal, characteristic, complemented
5	#13129892	C₂	1	1	yes	maximal, characteristic, complemented
6	#13129889	C₂	1	1	yes	maximal, characteristic, complemented
7	D₁₃×#350	C₂	1	1	yes	maximal, characteristic
8	#41×#261	C₄	1	1	yes	characteristic
9	D₁₃⋊#131	C₄	1	1	yes	characteristic
10	D₁₃×#131	C₂²	1	1	yes	characteristic
11	#11719099	—	2	1	no	—
12	#11719099	C₄	1	1	yes	characteristic
13	M₁₆×#261	—	2	1	no	—
14	M₁₆×#261	C₄	1	1	yes	characteristic
15	#11719089	—	2	1	no	—
16	#11719089	C₂²	1	1	yes	characteristic
17	C₂×C₈×#261	—	2	1	no	—
18	C₂×C₈×#261	C₂²	1	1	yes	characteristic
19	D₁₃⋊M₃₂	C₂²	1	1	yes	characteristic
20	C₂×D₁₃⋊C₁₆	C₂²	1	1	yes	characteristic
21	D₁₃×M₃₂	C₂²	1	1	yes	characteristic
22	C₂×C₁₆×D₁₃	C₂²	1	1	yes	characteristic
23	#11718304	C₄	1	2	yes	complemented
24	#11718270	C₄	1	2	yes	complemented
25	#11718216	—	2	1	no	—
26	#11718216	—	2	1	no	—
27	#11718216	—	2	1	no	—
28	C₁₆×#261	—	2	1	no	—
29	C₁₃×#350	C₄	1	1	yes	characteristic, complemented
30	#11718153	C₄	1	1	yes	characteristic, complemented
31	#11718107	C₄	1	1	yes	characteristic, complemented
32	#11718100	C₄	1	1	yes	characteristic, complemented
33	#41×D₁₃	C₂×C₄	1	1	yes	characteristic
34	Q₈×#261	C₈	1	2	yes	—
35	#88240	C₈	1	2	yes	—
36	D₄×#261	—	2	2	no	—
37	D₄×#261	C₈	1	2	yes	—
38	#88238	—	2	2	no	—
39	#88238	C₈	1	2	yes	—
40	#88237	—	2	1	no	—
41	#88237	C₂×C₄	1	1	yes	characteristic
42	C₂×C₄×#261	—	2	1	no	—
43	C₂×C₄×#261	C₂×C₄	1	1	yes	characteristic
44	D₁₃⋊M₁₆	—	2	1	no	—
45	D₁₃⋊M₁₆	C₂×C₄	1	1	yes	characteristic
46	C₂×D₁₃⋊C₈	—	2	1	no	—
47	C₂×D₁₃⋊C₈	C₂×C₄	1	1	yes	characteristic
48	C₁₃×#131	C₂×C₄	1	1	yes	characteristic
49	#88202	C₂×C₄	1	1	yes	characteristic
50	#88161	—	2	1	no	—
51	#88161	C₂×C₄	1	1	yes	characteristic
52	M₁₆×D₁₃	—	2	1	no	—
53	M₁₆×D₁₃	C₂×C₄	1	1	yes	characteristic
54	#88155	—	2	1	no	—
55	#88155	C₂×C₄	1	1	yes	characteristic
56	C₂×C₈×D₁₃	—	2	1	no	—
57	C₂×C₈×D₁₃	C₂³	1	1	yes	characteristic
58	#88106	C₂×C₄	1	2	yes	complemented
59	C₂×#6092	C₂×C₄	1	2	yes	complemented
60	#88100	—	2	1	no	—
61	#88100	—	2	1	no	—
62	#88100	—	2	1	no	—
63	C₈×#261	—	2	1	no	—
64	C₈×#261	—	2	1	no	—
65	C₈×#261	—	2	1	no	—
66	C₈×#261	D₄	1	1	yes	characteristic
67	C₈×#261	D₄	1	1	yes	characteristic
68	C₈×#261	D₄	1	1	yes	characteristic
69	C₈×#261	D₄	1	1	yes	characteristic
70	#88098	—	2	1	no	—
71	D₁₃⋊C₁₆	—	2	1	no	—
72	D₁₃⋊C₁₆	—	2	1	no	—
73	D₁₃⋊C₁₆	—	2	1	no	—
74	C₁₃×M₃₂	C₂×C₄	1	1	yes	characteristic, complemented
75	C₂×C₁₃×C₁₆	C₂×C₄	1	1	yes	characteristic, complemented
76	#88052	C₂×C₄	1	1	yes	characteristic, complemented
77	C₂×#6090	C₂×C₄	1	1	yes	characteristic, complemented
78	#88038	—	2	1	no	—
79	C₁₆×D₁₃	—	2	1	no	—
80	C₁₆×D₁₃	—	2	1	no	—
81	C₁₆×D₁₃	—	2	1	no	—
82	#12604	—	13	1	no	maximal, Sylow
83	C₂²×#261	—	2	2	no	—
84	C₂²×#261	M₁₆	1	2	yes	—
85	C₁₃×#41	C₄²	1	1	yes	characteristic
86	#6131	C₂×C₈	1	1	yes	characteristic
87	Q₈×D₁₃	C₂×C₈	1	1	yes	characteristic
88	#6129	—	2	1	no	—
89	#6129	C₂×C₈	1	1	yes	characteristic
90	D₄×D₁₃	—	2	1	no	—
91	D₄×D₁₃	C₂×C₈	1	1	yes	characteristic
92	#6127	—	2	1	no	—
93	#6127	C₄²	1	1	yes	characteristic
94	C₂×C₄×D₁₃	—	2	1	no	—
95	C₂×C₄×D₁₃	C₂²×C₄	1	1	yes	characteristic
96	C₂²⋊#261	—	2	1	no	—
97	C₂²⋊#261	—	2	2	no	—
98	#6122	—	2	1	no	—
99	#6122	—	2	2	no	—
100	C₂×#1057	—	2	2	no	—
101	C₂×#1057	M₁₆	1	2	yes	—
102	C₄⋊#261	—	2	2	no	—
103	C₄⋊#261	C₂×C₈	1	2	yes	—
104	C₄×#261	—	2	2	no	—
105	C₄×#261	—	2	1	no	—
106	C₄×#261	C₂²⋊C₄	1	1	yes	characteristic
107	C₄×#261	C₂²⋊C₄	1	1	yes	characteristic
108	C₄×#261	C₂×C₈	1	2	yes	—
109	#6118	—	2	2	no	—
110	#6118	C₂×C₈	1	2	yes	—
111	D₁₃⋊C₈	—	2	1	no	—
112	D₁₃⋊C₈	—	2	2	no	—
113	D₁₃⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
114	D₁₃⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
115	D₁₃⋊C₈	C₂×C₈	1	2	yes	—
116	C₁₃×M₁₆	—	2	1	no	—
117	C₁₃×M₁₆	C₄²	1	1	yes	characteristic
118	C₂×C₈×C₁₃	—	2	1	no	—
119	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
120	#6099	—	2	1	no	—
121	#6099	C₄²	1	1	yes	characteristic
122	C₂×#1055	—	2	1	no	—
123	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
124	#6094	—	2	1	no	—
125	#6094	—	2	1	no	—
126	#6094	—	2	1	no	—
127	C₈×D₁₃	—	2	1	no	—
128	C₈×D₁₃	—	2	1	no	—
129	C₈×D₁₃	—	2	1	no	—
130	C₈×D₁₃	C₂×D₄	1	1	yes	characteristic
131	C₈×D₁₃	C₂×D₄	1	1	yes	characteristic
132	C₈×D₁₃	#41	1	1	yes	characteristic
133	C₈×D₁₃	#41	1	1	yes	characteristic
134	#6092	—	2	2	no	—
135	#6092	—	2	2	no	—
136	C₁₃×C₁₆	—	2	1	no	—
137	C₁₃×C₁₆	—	2	1	no	—
138	#6090	—	2	1	no	—
139	#6090	—	2	1	no	—
140	C₄×#131	—	13	1	no	—
141	C₂×#350	—	13	1	no	—
142	C₂×#350	—	13	1	no	—
143	C₄×M₃₂	—	13	1	no	—
144	C₂×#346	—	13	1	no	—
145	#1379	—	13	1	no	—
146	#1379	—	13	1	no	—
147	C₂²×D₁₃	—	2	1	no	—
148	C₂²×D₁₃	C₂×M₁₆	1	1	yes	characteristic
149	C₂×#261	—	2	2	no	—
150	C₂×#261	—	4	2	no	—
151	C₂×#261	—	4	4	no	—
152	C₂×#261	C₂²⋊C₈	1	2	yes	—
153	C₂×#261	C₂²⋊C₈	1	2	yes	—
154	Q₈×C₁₃	C₄×C₈	1	1	yes	characteristic
155	D₄×C₁₃	—	2	1	no	—
156	D₄×C₁₃	C₄×C₈	1	1	yes	characteristic
157	C₂×C₄×C₁₃	—	2	1	no	—
158	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
159	C₂²⋊D₁₃	—	2	1	no	—
160	C₂²⋊D₁₃	—	2	1	no	—
161	C₂²⋊D₁₃	—	2	1	no	—
162	C₂×Dic₁₃	—	2	1	no	—
163	C₂×Dic₁₃	C₂×M₁₆	1	1	yes	characteristic
164	D₅₂	—	2	1	no	—
165	D₅₂	C₄×C₈	1	1	yes	characteristic
166	C₄×D₁₃	—	2	1	no	—
167	C₄×D₁₃	—	2	1	no	—
168	C₄×D₁₃	—	2	1	no	—
169	C₄×D₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
170	C₄×D₁₃	#117	1	1	yes	characteristic
171	C₄×D₁₃	C₂²×C₈	1	1	yes	characteristic
172	C₄×D₁₃	C₂²×C₈	1	1	yes	characteristic
173	Dic₂₆	—	2	1	no	—
174	Dic₂₆	C₄×C₈	1	1	yes	characteristic
175	#1057	—	2	2	no	—
176	#1057	C₂²⋊C₈	1	2	yes	—
177	#1057	C₂²⋊C₈	1	2	yes	—
178	C₈×C₁₃	—	2	1	no	—
179	C₈×C₁₃	C₄×D₄	1	1	yes	characteristic
180	C₈×C₁₃	C₄×D₄	1	1	yes	characteristic
181	#1055	—	2	1	no	—
182	#1055	C₄×D₄	1	1	yes	characteristic
183	#1055	C₄×D₄	1	1	yes	characteristic
184	C₂×#131	—	13	1	no	—
185	C₂×#131	—	13	1	no	—
186	C₄×#41	—	13	1	no	—
187	C₂×M₃₂	—	13	1	no	—
188	C₂×M₃₂	—	13	1	no	—
189	C₂²×C₁₆	—	13	1	no	—
190	C₂²×C₁₆	—	13	1	no	—
191	#405	—	13	1	no	—
192	#405	—	13	1	no	—
193	#405	—	26	1	no	—
194	#405	—	26	1	no	—
195	C₄×M₁₆	—	13	1	no	—
196	C₄×M₁₆	—	26	1	no	—
197	C₂×C₄×C₈	—	13	1	no	—
198	C₂×C₄×C₈	—	26	1	no	—
199	#350	—	13	2	no	—
200	#350	—	13	2	no	—
201	#350	—	13	1	no	—
202	#350	—	13	1	no	—
203	#350	—	13	1	no	—
204	#350	#261	1	1	yes	characteristic, complemented
205	#346	—	26	1	no	—
206	#346	—	26	1	no	—
207	#346	—	26	1	no	—
208	C₄×C₁₆	—	26	1	no	—
209	C₂²×C₁₃	—	2	1	no	—
210	C₂²×C₁₃	C₄×M₁₆	1	1	yes	characteristic
211	C₂×D₁₃	—	2	2	no	—
212	C₂×D₁₃	—	2	1	no	—
213	C₂×D₁₃	—	4	2	no	—
214	C₂×D₁₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
215	C₂×D₁₃	#431	1	1	yes	characteristic
216	#261	—	2	2	no	—
217	#261	—	2	2	no	—
218	#261	—	4	2	no	—
219	C₄×C₁₃	—	2	1	no	—
220	C₄×C₁₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
221	C₄×C₁₃	C₂×C₄×C₈	1	1	yes	characteristic
222	Dic₁₃	—	2	1	no	—
223	Dic₁₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
224	Dic₁₃	#431	1	1	yes	characteristic
225	C₂×#41	—	13	1	no	—
226	#131	—	13	2	no	—
227	#131	—	13	2	no	—
228	#131	—	13	1	no	—
229	#131	—	13	1	no	—
230	#131	—	13	1	no	—
231	#131	—	26	1	no	—
232	#131	—	26	1	no	—
233	#131	—	26	2	no	—
234	#131	C₂×#261	1	1	yes	characteristic
235	C₂×M₁₆	—	13	1	no	—
236	C₂×M₁₆	—	13	1	no	—
237	C₂×M₁₆	—	26	1	no	—
238	C₂×M₁₆	—	26	1	no	—
239	C₂²×C₈	—	13	1	no	—
240	C₂²×C₈	—	13	1	no	—
241	C₂²×C₈	—	26	1	no	—
242	C₂²×C₈	—	26	1	no	—
243	C₄×Q₈	—	13	2	no	—
244	C₄×D₄	—	13	2	no	—
245	C₄×D₄	—	26	2	no	—
246	#117	—	13	1	no	—
247	#117	—	26	1	no	—
248	C₂×C₄²	—	13	1	no	—
249	C₂×C₄²	—	26	1	no	—
250	M₃₂	—	13	2	no	—
251	M₃₂	—	13	1	no	—
252	M₃₂	—	26	1	no	—
253	M₃₂	—	26	1	no	—
254	M₃₂	C₂×#261	1	1	yes	characteristic, complemented
255	C₂×C₁₆	—	13	2	no	—
256	C₂×C₁₆	—	13	1	no	—
257	C₂×C₁₆	—	26	1	no	—
258	C₂×C₁₆	—	26	1	no	—
259	C₂×C₁₆	—	26	1	no	—
260	C₂×C₁₆	—	26	1	no	—
261	C₂×C₁₆	—	26	1	no	—
262	C₂×C₁₆	—	26	1	no	—
263	C₂×C₁₆	C₂×#261	1	1	yes	characteristic, complemented
264	#97	—	26	1	no	—
265	#97	—	26	1	no	—
266	#97	—	26	1	no	—
267	C₄×C₈	—	13	1	no	—
268	C₄×C₈	—	13	1	no	—
269	C₄×C₈	—	13	1	no	—
270	C₄×C₈	—	13	1	no	—
271	C₄×C₈	—	26	1	no	—
272	C₄×C₈	—	26	1	no	—
273	C₄×C₈	—	26	1	no	—
274	C₂×C₁₃	—	2	1	no	—
275	C₂×C₁₃	—	4	1	no	—
276	C₂×C₁₃	C₄×C₂²⋊C₈	1	1	yes	characteristic
277	D₁₃	—	2	1	no	—
278	D₁₃	—	4	1	no	—
279	D₁₃	#2117	1	2	yes	—
280	#41	—	13	1	no	—
281	#41	—	13	1	no	—
282	#41	—	13	1	no	—
283	#41	—	26	1	no	—
284	#41	—	26	1	no	—
285	#41	C₄×#261	1	1	yes	characteristic
286	C₂×Q₈	—	13	1	no	—
287	C₂×D₄	—	13	1	no	—
288	C₂×D₄	—	26	1	no	—
289	C₂²×C₄	—	13	1	no	—
290	C₂²×C₄	—	13	2	no	—
291	C₂²×C₄	—	26	1	no	—
292	C₂²×C₄	—	26	2	no	—
293	M₁₆	—	2	1	no	—
294	M₁₆	—	13	2	no	—
295	M₁₆	—	13	1	no	—
296	M₁₆	—	26	1	no	—
297	M₁₆	—	26	1	no	—
298	M₁₆	—	26	1	no	—
299	M₁₆	—	26	2	no	—
300	M₁₆	—	26	2	no	—
301	M₁₆	—	26	1	no	—
302	M₁₆	—	26	1	no	—
303	M₁₆	C₄×#261	1	1	yes	characteristic
304	C₂×C₈	—	2	1	no	—
305	C₂×C₈	—	13	1	no	—
306	C₂×C₈	—	13	1	no	—
307	C₂×C₈	—	13	2	no	—
308	C₂×C₈	—	13	2	no	—
309	C₂×C₈	—	13	1	no	—
310	C₂×C₈	—	13	1	no	—
311	C₂×C₈	—	13	1	no	—
312	C₂×C₈	—	13	1	no	—
313	C₂×C₈	—	13	1	no	—
314	C₂×C₈	—	26	1	no	—
315	C₂×C₈	—	26	1	no	—
316	C₂×C₈	—	26	1	no	—
317	C₂×C₈	—	26	2	no	—
318	C₂×C₈	—	26	2	no	—
319	C₂×C₈	—	26	1	no	—
320	C₂×C₈	—	26	1	no	—
321	C₂×C₈	C₂²×#261	1	1	yes	characteristic
322	C₄⋊C₄	—	13	2	no	—
323	C₄⋊C₄	—	26	2	no	—
324	C₂²⋊C₄	—	26	2	no	—
325	C₂²⋊C₄	—	26	1	no	—
326	C₄²	—	13	1	no	—
327	C₄²	—	13	1	no	—
328	C₄²	—	13	2	no	—
329	C₄²	—	26	1	no	—
330	C₄²	—	26	2	no	—
331	C₁₆	—	2	1	no	—
332	C₁₆	—	2	1	no	—
333	C₁₆	—	26	2	no	—
334	C₁₆	—	26	2	no	—
335	C₁₆	—	26	1	no	—
336	C₁₆	—	26	1	no	—
337	C₁₃	#12604	1	1	yes	characteristic, complemented, Sylow
338	C₂³	—	13	1	no	—
339	C₂³	—	26	1	no	—
340	Q₈	—	13	1	no	—
341	Q₈	—	26	1	no	—
342	Q₈	C₈×#261	1	1	yes	characteristic
343	D₄	—	2	1	no	—
344	D₄	—	13	1	no	—
345	D₄	—	26	1	no	—
346	D₄	—	26	1	no	—
347	D₄	—	26	1	no	—
348	D₄	—	26	1	no	—
349	D₄	C₈×#261	1	1	yes	characteristic
350	C₂×C₄	—	2	1	no	—
351	C₂×C₄	—	13	1	no	—
352	C₂×C₄	—	13	1	no	—
353	C₂×C₄	—	13	1	no	—
354	C₂×C₄	—	13	1	no	—
355	C₂×C₄	—	13	1	no	—
356	C₂×C₄	—	13	2	no	—
357	C₂×C₄	—	13	2	no	—
358	C₂×C₄	—	26	1	no	—
359	C₂×C₄	—	26	1	no	—
360	C₂×C₄	—	26	1	no	—
361	C₂×C₄	—	26	1	no	—
362	C₂×C₄	—	26	2	no	—
363	C₂×C₄	—	52	2	no	—
364	C₂×C₄	—	52	4	no	—
365	C₂×C₄	C₂×C₄×#261	1	1	yes	characteristic
366	C₈	—	2	1	no	—
367	C₈	—	13	2	no	—
368	C₈	—	13	2	no	—
369	C₈	—	13	1	no	—
370	C₈	—	13	1	no	—
371	C₈	—	26	2	no	—
372	C₈	—	26	1	no	—
373	C₈	D₄×#261	1	1	yes	characteristic
374	C₈	D₄×#261	1	1	yes	characteristic
375	C₂²	—	2	1	no	—
376	C₂²	—	13	1	no	—
377	C₂²	—	13	1	no	—
378	C₂²	—	26	2	no	—
379	C₂²	—	26	1	no	—
380	C₂²	—	52	2	no	—
381	C₂²	M₁₆×#261	1	1	yes	characteristic
382	C₄	—	2	1	no	—
383	C₄	—	13	1	no	—
384	C₄	—	13	1	no	—
385	C₄	—	26	2	no	—
386	C₄	—	26	2	no	—
387	C₄	—	26	1	no	—
388	C₄	—	52	2	no	—
389	C₄	C₂²⋊C₄×#261	1	1	yes	characteristic
390	C₄	C₂×C₈×#261	1	1	yes	characteristic
391	C₂	—	2	1	no	—
392	C₂	—	4	1	no	—
393	C₂	—	13	2	no	—
394	C₂	—	26	1	no	—
395	C₂	—	52	1	no	—
396	C₂	C₂²⋊C₈×#261	1	1	yes	characteristic

	subgroup	G/H
center	C₈	D₄×#261
commutator	C₄×C₁₃	C₂×C₄×C₈
Frattini	C₂×C₈	C₂²×#261
Fitting	C₁₃×#350	C₄
socle	C₂×C₁₃	C₄×C₂²⋊C₈
prime radical	#41×D₁₃	C₂×C₄
cosocle	C₂×C₈×D₁₃	C₂³
hypercenter	#350	#261
hypocenter	C₁₃	#12604
LCS	G ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₈ ◁ C₂×C₈ ◁ #350

Complements

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

N	H ≅ G/N	type
3. M₃₂×#261	C₂	semidirect
4. C₂×#11718216	C₂	semidirect
5. #13129892	C₂	semidirect
6. #13129889	C₂	semidirect
23. #11718304	C₄	semidirect
24. #11718270	C₄	semidirect
29. C₁₃×#350	C₄	semidirect
30. #11718153	C₄	semidirect
31. #11718107	C₄	semidirect
32. #11718100	C₄	semidirect
58. #88106	C₂×C₄	semidirect
59. C₂×#6092	C₂×C₄	semidirect
74. C₁₃×M₃₂	C₂×C₄	semidirect
75. C₂×C₁₃×C₁₆	C₂×C₄	semidirect
76. #88052	C₂×C₄	semidirect
77. C₂×#6090	C₂×C₄	semidirect
204. #350	#261	semidirect
254. M₃₂	C₂×#261	semidirect
263. C₂×C₁₆	C₂×#261	semidirect
337. C₁₃	#12604	semidirect

Maximal subgroups

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

subgroup	index
#131×#261	2
D₁₃⋊#350	2
M₃₂×#261	2
C₂×#11718216	2
#13129892	2
#13129889	2
D₁₃×#350	2
#12604	13