Subgroups — #16795832

Order 3328.

630 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	#13131652	C₂	1	2	yes	maximal, complemented
2	#13130476	C₂	1	1	yes	maximal, characteristic, complemented
3	#13130470	C₂	1	2	yes	maximal, complemented
4	#13130468	C₂	1	1	yes	maximal, characteristic, complemented
5	Q₁₆×#1057	C₂	1	1	yes	maximal, characteristic, complemented
6	#13130420	C₂	1	2	yes	maximal, complemented
7	#13130417	C₂	1	1	yes	maximal, characteristic, complemented
8	#13130398	C₂	1	1	yes	maximal, characteristic, complemented
9	#13130189	C₂	1	2	yes	maximal, complemented
10	C₂×C₈×#1057	C₂	1	1	yes	maximal, characteristic, complemented
11	#13125407	C₂	1	1	yes	maximal, characteristic
12	#11719329	C₂²	1	2	yes	—
13	#11719158	C₂²	1	2	yes	—
14	Q₈×#1057	C₂²	1	2	yes	—
15	#11719130	C₂²	1	2	yes	—
16	C₂×C₄×#1057	C₂²	1	1	yes	characteristic
17	C₂×C₄×#1057	C₂²	1	1	yes	characteristic
18	#11719077	—	2	2	no	—
19	#11719076	C₂²	1	2	yes	—
20	#11719073	C₂²	1	2	yes	—
21	#11719072	—	2	2	no	—
22	#11719064	—	2	2	no	—
23	#11719063	—	2	2	no	—
24	C₁₃×#467	C₄	1	1	yes	characteristic
25	#11718852	C₄	1	1	yes	characteristic
26	Q₁₆×Dic₁₃	C₂²	1	1	yes	characteristic
27	#11718776	C₄	1	1	yes	characteristic
28	C₂×C₈×Dic₁₃	C₂²	1	1	yes	characteristic
29	#11718709	C₄	1	2	yes	—
30	#11718539	C₄	1	1	yes	characteristic
31	#11718533	C₂²	1	1	yes	characteristic
32	#11718461	C₄	1	2	yes	—
33	#11718449	C₂²	1	2	yes	—
34	Q₈⋊#1057	C₂²	1	2	yes	—
35	Q₈⋊#1057	C₂²	1	2	yes	—
36	#11718255	C₂²	1	2	yes	—
37	#11718254	C₂²	1	1	yes	characteristic
38	#11718254	C₂²	1	1	yes	characteristic
39	C₈×#1057	C₂²	1	1	yes	characteristic
40	C₈×#1057	C₂²	1	1	yes	characteristic
41	C₈×#1057	C₂²	1	1	yes	characteristic
42	C₈×#1057	C₂²	1	1	yes	characteristic
43	#11718243	C₂²	1	2	yes	—
44	#11718243	C₂²	1	2	yes	—
45	C₂²×#1057	—	2	1	no	—
46	C₂²×#1057	D₄	1	1	yes	characteristic
47	C₂²×#1057	D₄	1	1	yes	characteristic
48	C₂×#6118	—	2	2	no	—
49	C₂×D₁₃⋊C₈	—	2	2	no	—
50	C₂×C₁₃×Q₁₆	C₂×C₄	1	1	yes	characteristic
51	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
52	C₁₃×C₂²⋊Q₈	C₂×C₄	1	2	yes	—
53	#88201	C₂×C₄	1	2	yes	—
54	Q₈×Dic₁₃	C₂³	1	2	yes	—
55	C₂×#6106	C₂×C₄	1	2	yes	—
56	#88184	C₂×C₄	1	2	yes	—
57	C₂×C₄×Dic₁₃	C₂³	1	1	yes	characteristic
58	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
59	C₂×D₁₀₄	C₂×C₄	1	1	yes	characteristic
60	#88152	—	2	2	no	—
61	#88147	C₂³	1	2	yes	—
62	#88146	—	2	2	no	—
63	#88144	C₂×C₄	1	2	yes	—
64	#88140	—	2	2	no	—
65	#88135	—	2	2	no	—
66	C₂²⋊#1057	—	2	2	no	—
67	#88112	—	2	2	no	—
68	#88111	—	2	2	no	—
69	#88111	—	2	2	no	—
70	#88110	—	2	2	no	—
71	#88110	—	2	2	no	—
72	C₄⋊#1057	—	2	2	no	—
73	C₄⋊#1057	C₂³	1	2	yes	—
74	C₄⋊#1057	C₂³	1	2	yes	—
75	C₄×#1057	—	2	2	no	—
76	C₄×#1057	—	2	1	no	—
77	C₄×#1057	D₄	1	1	yes	characteristic
78	C₄×#1057	D₄	1	1	yes	characteristic
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	C₁₃×#107	C₂×C₄	1	1	yes	characteristic
84	C₁₃×#103	C₂×C₄	1	2	yes	—
85	#88075	C₂×C₄	1	2	yes	—
86	#88061	C₂×C₄	1	2	yes	—
87	#88058	C₂×C₄	1	1	yes	characteristic
88	C₈×Dic₁₃	C₂×C₄	1	1	yes	characteristic
89	C₈×Dic₁₃	C₂×C₄	1	1	yes	characteristic
90	C₈×Dic₁₃	C₂³	1	1	yes	characteristic
91	C₈×Dic₁₃	C₂³	1	1	yes	characteristic
92	#88049	C₂×C₄	1	2	yes	—
93	#88048	C₂×C₄	1	2	yes	—
94	#17959	—	13	1	no	maximal, Sylow
95	C₂×Q₈×C₁₃	C₂²×C₄	1	2	yes	—
96	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
97	C₂×C₂²⋊D₁₃	—	2	2	no	—
98	C₂²×Dic₁₃	C₂×D₄	1	1	yes	characteristic
99	C₂²×Dic₁₃	#41	1	1	yes	characteristic
100	C₂×D₅₂	C₂²×C₄	1	2	yes	—
101	C₂×C₄×D₁₃	—	2	2	no	—
102	C₂×#1057	—	2	4	no	—
103	C₂×#1057	—	2	1	no	—
104	C₂×#1057	—	2	1	no	—
105	C₂×#1057	—	2	2	no	—
106	C₂×#1057	—	2	2	no	—
107	C₂×#1057	—	2	2	no	—
108	C₂×#1057	—	2	2	no	—
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	C₂×#1057	C₂×D₄	1	1	yes	characteristic
113	#6118	—	4	4	no	—
114	D₁₃⋊C₈	—	4	4	no	—
115	C₁₃×Q₁₆	—	2	2	no	—
116	C₂×C₈×C₁₃	—	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×C₄⋊C₄	—	2	2	no	—
120	C₁₃×C₄⋊C₄	C₂²×C₄	1	2	yes	—
121	C₁₃×C₂²⋊C₄	—	2	2	no	—
122	#6106	—	2	4	no	—
123	#6103	—	2	2	no	—
124	#6103	—	2	2	no	—
125	#6103	—	2	2	no	—
126	#6103	—	2	2	no	—
127	C₄⋊Dic₁₃	—	2	2	no	—
128	C₄⋊Dic₁₃	—	2	2	no	—
129	C₄⋊Dic₁₃	C₂²×C₄	1	2	yes	—
130	#6101	—	2	2	no	—
131	C₄×Dic₁₃	—	2	2	no	—
132	C₄×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
133	C₄×Dic₁₃	C₂×D₄	1	1	yes	characteristic
134	C₄×Dic₁₃	#41	1	1	yes	characteristic
135	C₄×Dic₁₃	C₂⁴	1	1	yes	characteristic
136	C₂×#1055	—	2	2	no	—
137	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
138	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
139	D₁₀₄	—	2	2	no	—
140	#3168	—	13	1	no	—
141	#2966	—	13	2	no	—
142	#1591	—	13	1	no	—
143	#1590	—	13	2	no	—
144	#1589	—	13	1	no	—
145	C₈×Q₁₆	—	13	1	no	—
146	C₈×SD₁₆	—	13	2	no	—
147	C₈×D₈	—	13	1	no	—
148	#1568	—	13	1	no	—
149	#1494	—	13	2	no	—
150	C₂×C₈²	—	13	1	no	—
151	C₂³×C₁₃	C₄×D₄	1	1	yes	characteristic
152	C₂²×D₁₃	—	2	2	no	—
153	Q₈×C₁₃	—	2	2	no	—
154	Q₈×C₁₃	—	2	2	no	—
155	C₂×C₄×C₁₃	—	2	1	no	—
156	C₂×C₄×C₁₃	—	2	1	no	—
157	C₂×C₄×C₁₃	—	2	2	no	—
158	C₂×C₄×C₁₃	—	2	2	no	—
159	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
160	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
161	C₂²⋊D₁₃	—	4	4	no	—
162	C₂×Dic₁₃	—	2	2	no	—
163	C₂×Dic₁₃	—	2	1	no	—
164	C₂×Dic₁₃	—	2	1	no	—
165	C₂×Dic₁₃	—	2	2	no	—
166	C₂×Dic₁₃	C₄×D₄	1	1	yes	characteristic
167	C₂×Dic₁₃	C₄×D₄	1	1	yes	characteristic
168	C₂×Dic₁₃	C₂²×D₄	1	1	yes	characteristic
169	C₂×Dic₁₃	C₂×#41	1	1	yes	characteristic
170	D₅₂	—	2	2	no	—
171	D₅₂	—	2	2	no	—
172	C₄×D₁₃	—	4	2	no	—
173	C₄×D₁₃	—	4	2	no	—
174	#1057	—	2	2	no	—
175	#1057	—	2	2	no	—
176	#1057	—	2	1	no	—
177	#1057	—	2	1	no	—
178	#1057	—	4	2	no	—
179	#1057	—	4	2	no	—
180	#1057	#135	1	2	yes	—
181	#1057	#135	1	2	yes	—
182	C₈×C₁₃	—	2	1	no	—
183	C₈×C₁₃	#131	1	2	yes	—
184	#1055	—	2	1	no	—
185	#1055	#131	1	2	yes	—
186	#529	—	13	2	no	—
187	#499	—	13	1	no	—
188	#495	—	13	1	no	—
189	#488	—	13	2	no	—
190	#467	#261	1	1	yes	characteristic
191	#466	—	13	1	no	—
192	#465	—	13	2	no	—
193	#446	—	13	2	no	—
194	C₈×Q₈	—	13	2	no	—
195	C₄×Q₁₆	—	13	1	no	—
196	C₄×SD₁₆	—	13	2	no	—
197	C₄×D₈	—	13	1	no	—
198	#436	—	13	2	no	—
199	#435	—	26	2	no	—
200	C₈×D₄	—	13	2	no	—
201	C₈×D₄	—	26	2	no	—
202	#433	—	26	2	no	—
203	#432	—	13	2	no	—
204	#405	—	26	2	no	—
205	C₂×C₄×C₈	—	13	1	no	—
206	C₂×C₄×C₈	—	13	1	no	—
207	C₂×C₄×C₈	—	13	1	no	—
208	#335	—	13	1	no	—
209	#335	—	13	1	no	—
210	#334	—	13	2	no	—
211	Q₈⋊C₈	—	13	2	no	—
212	Q₈⋊C₈	—	13	2	no	—
213	D₄⋊C₈	—	13	2	no	—
214	D₄⋊C₈	—	13	2	no	—
215	C₈²	—	13	1	no	—
216	C₈²	—	13	1	no	—
217	C₈²	—	13	1	no	—
218	C₈²	—	13	1	no	—
219	C₂²×C₁₃	—	2	1	no	—
220	C₂²×C₁₃	—	2	1	no	—
221	C₂²×C₁₃	#443	1	2	yes	—
222	C₂²×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
223	C₂×D₁₃	—	4	2	no	—
224	C₂×D₁₃	—	4	2	no	—
225	C₂×D₁₃	—	4	2	no	—
226	C₄×C₁₃	—	2	1	no	—
227	C₄×C₁₃	—	4	2	no	—
228	C₄×C₁₃	—	4	2	no	—
229	C₄×C₁₃	C₂×#131	1	1	yes	characteristic
230	C₄×C₁₃	C₂×#131	1	1	yes	characteristic
231	Dic₁₃	—	2	1	no	—
232	Dic₁₃	—	2	1	no	—
233	Dic₁₃	—	2	1	no	—
234	Dic₁₃	—	4	2	no	—
235	Dic₁₃	C₂×#135	1	1	yes	characteristic
236	Dic₁₃	C₂×#135	1	1	yes	characteristic
237	C₂×Q₁₆	C₂×#261	1	1	yes	characteristic
238	C₂×SD₁₆	—	13	2	no	—
239	C₂×D₈	—	13	1	no	—
240	C₂×M₁₆	—	26	2	no	—
241	C₂²×C₈	—	13	1	no	—
242	C₂²×C₈	—	13	1	no	—
243	C₂²×C₈	—	13	1	no	—
244	C₂²×C₈	—	26	2	no	—
245	C₂²×C₈	—	26	1	no	—
246	C₂²×C₈	C₂×#261	1	1	yes	characteristic
247	#126	—	26	2	no	—
248	#125	—	13	2	no	—
249	#124	—	13	2	no	—
250	#123	—	26	2	no	—
251	C₂²⋊Q₈	C₂×#261	1	2	yes	—
252	#121	—	13	2	no	—
253	C₄×Q₈	—	13	2	no	—
254	C₄×D₄	—	13	2	no	—
255	C₄×D₄	—	26	2	no	—
256	#117	—	26	2	no	—
257	C₂×C₄²	—	13	1	no	—
258	#107	—	13	1	no	—
259	#107	C₂×#261	1	1	yes	characteristic
260	#106	—	13	2	no	—
261	C₄⋊C₈	—	13	2	no	—
262	C₄⋊C₈	—	13	2	no	—
263	C₄⋊C₈	—	26	2	no	—
264	C₄⋊C₈	—	26	2	no	—
265	#103	—	13	2	no	—
266	#103	C₂×#261	1	2	yes	—
267	#102	—	13	2	no	—
268	#102	—	13	2	no	—
269	C₂²⋊C₈	—	26	2	no	—
270	C₂²⋊C₈	—	26	2	no	—
271	C₂²⋊C₈	—	26	2	no	—
272	#97	—	26	2	no	—
273	#97	—	26	2	no	—
274	C₄×C₈	—	13	1	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	—	13	1	no	—
283	C₄×C₈	—	13	1	no	—
284	C₄×C₈	—	26	1	no	—
285	C₄×C₈	—	26	2	no	—
286	C₂×C₁₃	—	2	2	no	—
287	C₂×C₁₃	#2931	1	1	yes	characteristic
288	C₂×C₁₃	C₄×#135	1	1	yes	characteristic
289	C₂×C₁₃	C₂×#443	1	1	yes	characteristic
290	D₁₃	—	8	2	no	—
291	C₂×Q₈	C₂²×#261	1	2	yes	—
292	C₂×D₄	—	13	2	no	—
293	C₂×D₄	—	26	2	no	—
294	C₂²×C₄	—	13	1	no	—
295	C₂²×C₄	—	13	1	no	—
296	C₂²×C₄	—	26	2	no	—
297	C₂²×C₄	C₂²×#261	1	1	yes	characteristic
298	Q₁₆	—	2	2	no	—
299	SD₁₆	—	26	4	no	—
300	D₈	—	26	2	no	—
301	M₁₆	—	52	4	no	—
302	C₂×C₈	—	2	2	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	2	no	—
311	C₂×C₈	—	26	2	no	—
312	C₂×C₈	—	26	1	no	—
313	C₂×C₈	—	26	1	no	—
314	C₂×C₈	—	26	4	no	—
315	C₂×C₈	—	26	2	no	—
316	C₂×C₈	—	26	2	no	—
317	C₂×C₈	—	52	4	no	—
318	C₂×C₈	C₂²×#261	1	1	yes	characteristic
319	C₂×C₈	C₂²×#261	1	1	yes	characteristic
320	C₄⋊C₄	—	2	2	no	—
321	C₄⋊C₄	—	13	2	no	—
322	C₄⋊C₄	—	26	2	no	—
323	C₄⋊C₄	—	26	2	no	—
324	C₄⋊C₄	—	26	2	no	—
325	C₄⋊C₄	C₂²×#261	1	2	yes	—
326	C₂²⋊C₄	—	2	2	no	—
327	C₂²⋊C₄	—	26	2	no	—
328	C₂²⋊C₄	—	26	2	no	—
329	C₂²⋊C₄	—	26	2	no	—
330	C₂²⋊C₄	—	26	2	no	—
331	C₄²	—	13	1	no	—
332	C₄²	—	13	1	no	—
333	C₄²	—	13	1	no	—
334	C₄²	—	13	1	no	—
335	C₄²	—	26	2	no	—
336	C₁₃	#17959	1	1	yes	characteristic, complemented, Sylow
337	C₂³	—	26	2	no	—
338	C₂³	D₄×#261	1	1	yes	characteristic
339	Q₈	—	2	2	no	—
340	Q₈	—	2	2	no	—
341	D₄	—	26	2	no	—
342	D₄	—	26	2	no	—
343	D₄	—	52	4	no	—
344	C₂×C₄	—	2	1	no	—
345	C₂×C₄	—	2	1	no	—
346	C₂×C₄	—	2	2	no	—
347	C₂×C₄	—	2	2	no	—
348	C₂×C₄	—	13	1	no	—
349	C₂×C₄	—	13	1	no	—
350	C₂×C₄	—	13	1	no	—
351	C₂×C₄	—	13	1	no	—
352	C₂×C₄	—	26	1	no	—
353	C₂×C₄	—	26	1	no	—
354	C₂×C₄	—	26	2	no	—
355	C₂×C₄	—	26	2	no	—
356	C₂×C₄	—	52	2	no	—
357	C₂×C₄	—	52	2	no	—
358	C₂×C₄	D₄×#261	1	1	yes	characteristic
359	C₂×C₄	C₂³×#261	1	1	yes	characteristic
360	C₈	—	2	1	no	—
361	C₈	—	13	2	no	—
362	C₈	—	13	2	no	—
363	C₈	—	13	2	no	—
364	C₈	—	26	1	no	—
365	C₈	—	26	1	no	—
366	C₈	—	26	2	no	—
367	C₈	—	26	2	no	—
368	C₈	—	26	1	no	—
369	C₈	—	52	2	no	—
370	C₈	—	52	2	no	—
371	C₈	#88238	1	2	yes	—
372	C₂²	—	2	1	no	—
373	C₂²	—	2	1	no	—
374	C₂²	—	52	2	no	—
375	C₂²	—	52	2	no	—
376	C₂²	—	52	2	no	—
377	C₂²	#11719111	1	2	yes	—
378	C₂²	C₂×D₄×#261	1	1	yes	characteristic
379	C₄	—	2	1	no	—
380	C₄	—	4	2	no	—
381	C₄	—	4	2	no	—
382	C₄	—	13	1	no	—
383	C₄	—	13	1	no	—
384	C₄	—	26	1	no	—
385	C₄	—	26	1	no	—
386	C₄	—	26	1	no	—
387	C₄	—	52	2	no	—
388	C₄	C₂×#88238	1	1	yes	characteristic
389	C₄	D₁₃⋊#131	1	1	yes	characteristic
390	C₂	—	2	2	no	—
391	C₂	—	104	2	no	—
392	C₂	#13131559	1	1	yes	characteristic
393	C₂	#135×#261	1	1	yes	characteristic
394	C₂	C₂×#11719111	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×#467	C₄
socle	C₂²×C₁₃	C₂×C₄×D₄
prime radical	#11718776	C₄
cosocle	C₄×Dic₁₃	C₂⁴
hypercenter	#467	#261
hypocenter	C₁₃	#17959
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #467

Complements

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

N	H ≅ G/N	type
1. #13131652	C₂	semidirect
2. #13130476	C₂	semidirect
3. #13130470	C₂	semidirect
4. #13130468	C₂	semidirect
5. Q₁₆×#1057	C₂	semidirect
6. #13130420	C₂	semidirect
7. #13130417	C₂	semidirect
8. #13130398	C₂	semidirect
9. #13130189	C₂	semidirect
10. C₂×C₈×#1057	C₂	semidirect
336. C₁₃	#17959	semidirect

Maximal subgroups

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

subgroup	index
#13131652	2
#13130476	2
#13130470	2
#13130468	2
Q₁₆×#1057	2
#13130420	2
#13130417	2
#13130398	2
#13130189	2
C₂×C₈×#1057	2
#13125407	2
#17959	13