Subgroups — #16796556

Order 3328.

600 conjugacy classes of subgroups.

The subgroup lattice has 394 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13131668	C₂	1	2	yes	maximal, complemented
2	#13130476	C₂	1	1	yes	maximal, characteristic, complemented
3	#13130469	C₂	1	1	yes	maximal, characteristic, complemented
4	#13130445	C₂	1	2	yes	maximal, complemented
5	#13130441	C₂	1	2	yes	maximal, complemented
6	Q₁₆×#1057	C₂	1	1	yes	maximal, characteristic, complemented
7	D₈×#1057	C₂	1	1	yes	maximal, characteristic, complemented
8	#13130386	C₂	1	1	yes	maximal, characteristic, complemented
9	#13130148	C₂	1	2	yes	maximal, complemented
10	C₈×#6118	C₂	1	1	yes	maximal, characteristic, complemented
11	#13125403	C₂	1	1	yes	maximal, characteristic
12	#11719385	C₂²	1	2	yes	—
13	#11719158	C₂²	1	2	yes	—
14	Q₈×#1057	C₂²	1	2	yes	—
15	#11719145	C₂²	1	2	yes	—
16	D₄×#1057	C₂²	1	2	yes	—
17	#11719068	—	2	2	no	—
18	#11719066	—	2	2	no	—
19	#11719064	—	2	2	no	—
20	#11719062	—	2	2	no	—
21	#11719051	C₂²	1	2	yes	—
22	#11719051	C₂²	1	1	yes	characteristic
23	C₄×#6118	C₂²	1	1	yes	characteristic
24	C₁₃×#495	C₄	1	1	yes	characteristic
25	#11718849	C₄	1	1	yes	characteristic
26	Q₁₆×Dic₁₃	C₂²	1	1	yes	characteristic
27	#11718832	C₄	1	2	yes	—
28	SD₁₆×Dic₁₃	C₂²	1	2	yes	—
29	#11718818	C₄	1	1	yes	characteristic
30	D₈×Dic₁₃	C₂²	1	1	yes	characteristic
31	#11718717	C₄	1	2	yes	—
32	C₄×C₈×D₁₃	C₂²	1	1	yes	characteristic
33	#11718344	C₄	1	1	yes	characteristic
34	Q₈⋊#1057	C₂²	1	2	yes	—
35	Q₈⋊#1057	C₂²	1	2	yes	—
36	D₄⋊#1057	C₂²	1	2	yes	—
37	D₄⋊#1057	C₂²	1	2	yes	—
38	#11718255	C₂²	1	2	yes	—
39	#11718254	C₂²	1	1	yes	characteristic
40	#11718254	C₂²	1	1	yes	characteristic
41	#1057⋊C₈	C₂²	1	1	yes	characteristic
42	#1057⋊C₈	C₂²	1	1	yes	characteristic
43	C₈×#1057	C₂²	1	1	yes	characteristic
44	C₈×#1057	C₂²	1	1	yes	characteristic
45	C₂×#6122	—	2	2	no	—
46	C₂²×#1057	—	2	2	no	—
47	C₂×#6118	D₄	1	1	yes	characteristic
48	C₂×#6118	D₄	1	1	yes	characteristic
49	C₂×C₁₃×Q₁₆	C₂×C₄	1	1	yes	characteristic
50	C₂×C₁₃×SD₁₆	C₂×C₄	1	2	yes	—
51	C₂×C₁₃×D₈	C₂×C₄	1	1	yes	characteristic
52	C₁₃×#124	C₂×C₄	1	2	yes	—
53	#88200	C₂×C₄	1	2	yes	—
54	Q₈×Dic₁₃	C₂³	1	2	yes	—
55	#88192	C₂×C₄	1	2	yes	—
56	D₄×Dic₁₃	C₂³	1	2	yes	—
57	C₂×C₈×D₁₃	C₂×C₄	1	1	yes	characteristic
58	C₂×C₈×D₁₃	C₂×C₄	1	1	yes	characteristic
59	#88137	—	2	2	no	—
60	#88135	—	2	2	no	—
61	#88133	—	2	2	no	—
62	#88131	—	2	2	no	—
63	C₄²×D₁₃	C₂³	1	1	yes	characteristic
64	#88124	C₂×C₄	1	2	yes	—
65	C₂²⋊#1057	—	2	2	no	—
66	C₂²⋊#1057	—	2	2	no	—
67	#88112	—	2	2	no	—
68	#88111	—	2	2	no	—
69	#88111	—	2	1	no	—
70	#88110	—	2	2	no	—
71	#88110	D₄	1	1	yes	characteristic
72	#88110	D₄	1	1	yes	characteristic
73	C₄⋊#1057	—	2	2	no	—
74	C₄⋊#1057	—	2	1	no	—
75	C₄⋊#1057	C₂³	1	2	yes	—
76	C₄⋊#1057	C₂³	1	2	yes	—
77	C₄×#1057	—	2	2	no	—
78	C₄×#1057	—	2	2	no	—
79	C₄×#1057	C₂³	1	1	yes	characteristic
80	C₄×#1057	C₂³	1	1	yes	characteristic
81	C₄×#1057	C₂³	1	1	yes	characteristic
82	C₄×#1057	C₂³	1	1	yes	characteristic
83	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
84	#88075	C₂×C₄	1	2	yes	—
85	#88075	C₂×C₄	1	2	yes	—
86	#88072	C₂×C₄	1	2	yes	—
87	#88072	C₂×C₄	1	2	yes	—
88	#88058	C₂×C₄	1	1	yes	characteristic
89	#88058	C₂×C₄	1	1	yes	characteristic
90	#88057	C₂×C₄	1	2	yes	—
91	C₈×Dic₁₃	C₂³	1	1	yes	characteristic
92	C₈×Dic₁₃	C₂³	1	1	yes	characteristic
93	C₄×#1055	C₂×C₄	1	1	yes	characteristic
94	#18069	—	13	1	no	maximal, Sylow
95	C₂×Q₈×C₁₃	C₂²×C₄	1	2	yes	—
96	C₂×D₄×C₁₃	C₂²×C₄	1	2	yes	—
97	C₂×C₂²⋊D₁₃	—	2	2	no	—
98	C₂²×Dic₁₃	—	2	2	no	—
99	C₂×C₄×D₁₃	C₂²×C₄	1	1	yes	characteristic
100	C₂×C₄×D₁₃	C₂×D₄	1	1	yes	characteristic
101	C₂×C₄×D₁₃	#41	1	1	yes	characteristic
102	#6122	—	4	4	no	—
103	C₂×#1057	—	2	2	no	—
104	C₂×#1057	—	2	2	no	—
105	C₂×#1057	—	2	1	no	—
106	C₂×#1057	—	2	1	no	—
107	C₂×#1057	—	4	4	no	—
108	C₂×#1057	C₂×D₄	1	1	yes	characteristic
109	C₂×#1057	C₂×D₄	1	1	yes	characteristic
110	C₂×#1057	C₂×D₄	1	1	yes	characteristic
111	C₂×#1057	C₂×D₄	1	1	yes	characteristic
112	#6118	—	2	2	no	—
113	#6118	—	2	2	no	—
114	C₁₃×Q₁₆	—	2	2	no	—
115	C₁₃×SD₁₆	—	4	2	no	—
116	C₁₃×D₈	—	2	2	no	—
117	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
118	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
119	C₁₃×C₂²⋊C₄	—	2	2	no	—
120	C₁₃×C₂²⋊C₄	—	2	2	no	—
121	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
122	C₂²⋊Dic₁₃	—	2	2	no	—
123	C₂²⋊Dic₁₃	—	2	2	no	—
124	#6103	—	2	2	no	—
125	C₄⋊Dic₁₃	—	2	2	no	—
126	C₄⋊Dic₁₃	C₂²×C₄	1	2	yes	—
127	C₄⋊Dic₁₃	C₂²×C₄	1	2	yes	—
128	#6101	—	2	2	no	—
129	#6101	—	2	2	no	—
130	#6101	—	2	2	no	—
131	C₄×Dic₁₃	—	2	2	no	—
132	C₄×Dic₁₃	C₂×D₄	1	1	yes	characteristic
133	C₄×Dic₁₃	#41	1	1	yes	characteristic
134	C₄×Dic₁₃	C₂⁴	1	1	yes	characteristic
135	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
136	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
137	C₈×D₁₃	—	2	2	no	—
138	C₈×D₁₃	—	4	1	no	—
139	#3168	—	13	1	no	—
140	#2969	—	13	2	no	—
141	#1591	—	13	1	no	—
142	#1589	—	13	1	no	—
143	#1583	—	13	2	no	—
144	#1582	—	13	2	no	—
145	C₈×Q₁₆	—	13	1	no	—
146	C₈×D₈	—	13	1	no	—
147	#1564	—	13	1	no	—
148	#1482	—	13	2	no	—
149	C₈×M₁₆	—	13	1	no	—
150	C₂³×C₁₃	—	2	2	no	—
151	C₂²×D₁₃	C₄×D₄	1	1	yes	characteristic
152	Q₈×C₁₃	—	2	2	no	—
153	Q₈×C₁₃	—	2	2	no	—
154	D₄×C₁₃	—	2	2	no	—
155	D₄×C₁₃	—	2	2	no	—
156	C₂×C₄×C₁₃	—	2	2	no	—
157	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
158	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
159	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
160	C₂²⋊D₁₃	—	8	2	no	—
161	C₂×Dic₁₃	—	2	2	no	—
162	C₂×Dic₁₃	—	2	2	no	—
163	C₂×Dic₁₃	—	4	2	no	—
164	C₂×Dic₁₃	—	4	2	no	—
165	C₂×Dic₁₃	C₄×D₄	1	1	yes	characteristic
166	C₂×Dic₁₃	C₂²×D₄	1	1	yes	characteristic
167	C₂×Dic₁₃	C₂×#41	1	1	yes	characteristic
168	C₄×D₁₃	—	2	1	no	—
169	C₄×D₁₃	—	2	1	no	—
170	C₄×D₁₃	—	2	1	no	—
171	C₄×D₁₃	—	2	1	no	—
172	C₄×D₁₃	—	4	1	no	—
173	#1057	—	2	1	no	—
174	#1057	—	2	1	no	—
175	#1057	—	4	1	no	—
176	#1057	—	4	1	no	—
177	#1057	—	4	2	no	—
178	#1057	—	4	2	no	—
179	#1057	#135	1	2	yes	—
180	#1057	#135	1	2	yes	—
181	C₈×C₁₃	—	2	1	no	—
182	C₈×C₁₃	#131	1	2	yes	—
183	#1055	—	2	1	no	—
184	#1055	—	2	1	no	—
185	#529	—	13	2	no	—
186	#499	—	13	1	no	—
187	#495	#261	1	1	yes	characteristic
188	#488	—	13	2	no	—
189	#467	—	13	1	no	—
190	#466	—	13	1	no	—
191	#465	—	13	2	no	—
192	#446	—	13	2	no	—
193	C₈×Q₈	—	13	2	no	—
194	C₄×Q₁₆	—	13	1	no	—
195	C₄×SD₁₆	—	13	2	no	—
196	C₄×D₈	—	13	1	no	—
197	#436	—	13	2	no	—
198	#436	—	26	2	no	—
199	#435	—	26	2	no	—
200	C₈×D₄	—	13	2	no	—
201	#433	—	26	2	no	—
202	#431	—	13	1	no	—
203	#431	—	13	2	no	—
204	#405	—	26	2	no	—
205	C₄×M₁₆	—	13	1	no	—
206	C₂×C₄×C₈	—	13	1	no	—
207	#335	—	13	1	no	—
208	#335	—	13	1	no	—
209	#334	—	13	2	no	—
210	Q₈⋊C₈	—	13	2	no	—
211	Q₈⋊C₈	—	13	2	no	—
212	D₄⋊C₈	—	13	2	no	—
213	D₄⋊C₈	—	13	2	no	—
214	#322	—	13	1	no	—
215	#322	—	13	1	no	—
216	C₈²	—	13	1	no	—
217	C₈²	—	13	1	no	—
218	C₂²×C₁₃	—	4	2	no	—
219	C₂²×C₁₃	—	4	2	no	—
220	C₂²×C₁₃	—	4	2	no	—
221	C₂²×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
222	C₂×D₁₃	—	2	1	no	—
223	C₂×D₁₃	—	2	1	no	—
224	C₂×D₁₃	—	2	1	no	—
225	C₄×C₁₃	—	2	1	no	—
226	C₄×C₁₃	—	2	1	no	—
227	C₄×C₁₃	—	4	2	no	—
228	C₄×C₁₃	C₂×#131	1	1	yes	characteristic
229	C₄×C₁₃	#568	1	1	yes	characteristic
230	Dic₁₃	—	2	1	no	—
231	Dic₁₃	—	2	1	no	—
232	Dic₁₃	—	4	2	no	—
233	Dic₁₃	—	4	2	no	—
234	Dic₁₃	C₂×#135	1	1	yes	characteristic
235	Dic₁₃	C₂×#135	1	1	yes	characteristic
236	C₂×Q₁₆	C₂×#261	1	1	yes	characteristic
237	C₂×SD₁₆	C₂×#261	1	2	yes	—
238	C₂×D₈	C₂×#261	1	1	yes	characteristic
239	C₂×M₁₆	—	13	1	no	—
240	C₂×M₁₆	—	13	1	no	—
241	C₂×M₁₆	—	26	2	no	—
242	C₂²×C₈	—	13	1	no	—
243	C₂²×C₈	—	13	1	no	—
244	C₂²×C₈	—	26	2	no	—
245	#126	—	26	2	no	—
246	#125	—	13	2	no	—
247	#124	C₂×#261	1	2	yes	—
248	#123	—	26	2	no	—
249	C₂²⋊Q₈	—	13	2	no	—
250	#121	—	13	2	no	—
251	C₄×Q₈	—	13	2	no	—
252	C₄×D₄	—	13	2	no	—
253	C₄×D₄	—	26	2	no	—
254	#117	—	26	2	no	—
255	C₂×C₄²	—	13	1	no	—
256	#107	—	13	1	no	—
257	#107	—	13	1	no	—
258	#106	—	13	2	no	—
259	C₄⋊C₈	—	13	2	no	—
260	C₄⋊C₈	—	13	2	no	—
261	C₄⋊C₈	—	26	1	no	—
262	C₄⋊C₈	—	26	2	no	—
263	C₄⋊C₈	—	26	2	no	—
264	#103	—	13	2	no	—
265	#103	—	13	2	no	—
266	#102	—	13	2	no	—
267	#102	—	13	2	no	—
268	C₂²⋊C₈	—	26	2	no	—
269	C₂²⋊C₈	—	26	2	no	—
270	C₂²⋊C₈	—	26	1	no	—
271	C₂²⋊C₈	—	26	2	no	—
272	#97	—	13	1	no	—
273	#97	—	13	1	no	—
274	#97	—	26	2	no	—
275	C₄×C₈	—	13	1	no	—
276	C₄×C₈	—	13	1	no	—
277	C₄×C₈	—	13	1	no	—
278	C₄×C₈	—	13	1	no	—
279	C₄×C₈	—	13	1	no	—
280	C₄×C₈	—	13	1	no	—
281	C₄×C₈	—	13	1	no	—
282	C₄×C₈	—	26	2	no	—
283	C₄×C₈	—	26	2	no	—
284	C₄×C₈	C₂×#261	1	1	yes	characteristic
285	C₂×C₁₃	—	8	2	no	—
286	C₂×C₁₃	#2935	1	1	yes	characteristic
287	C₂×C₁₃	C₄×#135	1	1	yes	characteristic
288	C₂×C₁₃	#2957	1	1	yes	characteristic
289	D₁₃	—	4	1	no	—
290	C₂×Q₈	C₂²×#261	1	2	yes	—
291	C₂×D₄	—	26	2	no	—
292	C₂×D₄	C₂²×#261	1	2	yes	—
293	C₂²×C₄	—	13	1	no	—
294	C₂²×C₄	—	13	1	no	—
295	C₂²×C₄	—	13	1	no	—
296	C₂²×C₄	—	26	2	no	—
297	Q₁₆	—	2	2	no	—
298	SD₁₆	—	4	2	no	—
299	D₈	—	2	2	no	—
300	M₁₆	—	26	2	no	—
301	M₁₆	—	26	2	no	—
302	M₁₆	—	52	4	no	—
303	C₂×C₈	—	13	1	no	—
304	C₂×C₈	—	13	1	no	—
305	C₂×C₈	—	13	1	no	—
306	C₂×C₈	—	13	1	no	—
307	C₂×C₈	—	13	1	no	—
308	C₂×C₈	—	13	1	no	—
309	C₂×C₈	—	26	2	no	—
310	C₂×C₈	—	26	1	no	—
311	C₂×C₈	—	26	1	no	—
312	C₂×C₈	—	26	2	no	—
313	C₂×C₈	—	26	2	no	—
314	C₂×C₈	—	52	1	no	—
315	C₂×C₈	—	52	4	no	—
316	C₂×C₈	C₂²×#261	1	1	yes	characteristic
317	C₂×C₈	C₂²×#261	1	1	yes	characteristic
318	C₄⋊C₄	—	13	2	no	—
319	C₄⋊C₄	—	13	2	no	—
320	C₄⋊C₄	—	26	2	no	—
321	C₄⋊C₄	—	26	2	no	—
322	C₄⋊C₄	—	26	2	no	—
323	C₄⋊C₄	—	26	2	no	—
324	C₂²⋊C₄	—	2	2	no	—
325	C₂²⋊C₄	—	2	2	no	—
326	C₂²⋊C₄	—	26	2	no	—
327	C₂²⋊C₄	—	26	2	no	—
328	C₂²⋊C₄	—	26	2	no	—
329	C₄²	—	13	1	no	—
330	C₄²	—	13	1	no	—
331	C₄²	—	13	1	no	—
332	C₄²	—	26	2	no	—
333	C₄²	C₂²×#261	1	1	yes	characteristic
334	C₁₃	#18069	1	1	yes	characteristic, complemented, Sylow
335	C₂³	—	2	2	no	—
336	C₂³	—	13	1	no	—
337	Q₈	—	2	2	no	—
338	Q₈	—	2	2	no	—
339	D₄	—	2	2	no	—
340	D₄	—	2	2	no	—
341	D₄	—	104	2	no	—
342	C₂×C₄	—	2	2	no	—
343	C₂×C₄	—	13	1	no	—
344	C₂×C₄	—	13	1	no	—
345	C₂×C₄	—	13	1	no	—
346	C₂×C₄	—	26	1	no	—
347	C₂×C₄	—	26	1	no	—
348	C₂×C₄	—	26	1	no	—
349	C₂×C₄	—	26	1	no	—
350	C₂×C₄	—	26	2	no	—
351	C₂×C₄	—	26	2	no	—
352	C₂×C₄	—	52	1	no	—
353	C₂×C₄	—	52	2	no	—
354	C₂×C₄	—	52	2	no	—
355	C₂×C₄	D₄×#261	1	1	yes	characteristic
356	C₂×C₄	D₄×#261	1	1	yes	characteristic
357	C₂×C₄	C₂³×#261	1	1	yes	characteristic
358	C₈	—	2	1	no	—
359	C₈	—	13	2	no	—
360	C₈	—	13	2	no	—
361	C₈	—	26	1	no	—
362	C₈	—	26	1	no	—
363	C₈	—	26	1	no	—
364	C₈	—	26	1	no	—
365	C₈	—	52	2	no	—
366	C₈	—	52	2	no	—
367	C₈	—	52	1	no	—
368	C₈	—	52	1	no	—
369	C₈	#88238	1	2	yes	—
370	C₂²	—	4	2	no	—
371	C₂²	—	4	2	no	—
372	C₂²	—	4	2	no	—
373	C₂²	—	26	1	no	—
374	C₂²	—	26	1	no	—
375	C₂²	—	26	1	no	—
376	C₂²	C₂×D₄×#261	1	1	yes	characteristic
377	C₄	—	2	1	no	—
378	C₄	—	2	1	no	—
379	C₄	—	4	2	no	—
380	C₄	—	13	1	no	—
381	C₄	—	13	1	no	—
382	C₄	—	26	1	no	—
383	C₄	—	26	1	no	—
384	C₄	—	52	2	no	—
385	C₄	—	52	2	no	—
386	C₄	C₂×#88238	1	1	yes	characteristic
387	C₄	#11719621	1	1	yes	characteristic
388	C₂	—	8	2	no	—
389	C₂	—	52	1	no	—
390	C₂	#13131579	1	1	yes	characteristic
391	C₂	#135×#261	1	1	yes	characteristic
392	C₂	#13131613	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×#261
commutator	C₂×C₄×C₁₃	C₂³×C₄
Frattini	C₂×C₄	C₂³×#261
Fitting	C₁₃×#495	C₄
socle	C₂²×C₁₃	C₂×C₄×D₄
prime radical	#11718818	C₄
cosocle	C₄×Dic₁₃	C₂⁴
hypercenter	#495	#261
hypocenter	C₁₃	#18069
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #495

Complements

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

N	H ≅ G/N	type
1. #13131668	C₂	semidirect
2. #13130476	C₂	semidirect
3. #13130469	C₂	semidirect
4. #13130445	C₂	semidirect
5. #13130441	C₂	semidirect
6. Q₁₆×#1057	C₂	semidirect
7. D₈×#1057	C₂	semidirect
8. #13130386	C₂	semidirect
9. #13130148	C₂	semidirect
10. C₈×#6118	C₂	semidirect
334. C₁₃	#18069	semidirect

Maximal subgroups

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

subgroup	index
#13131668	2
#13130476	2
#13130469	2
#13130445	2
#13130441	2
Q₁₆×#1057	2
D₈×#1057	2
#13130386	2
#13130148	2
C₈×#6118	2
#13125403	2
#18069	13