Subgroups — #16794960

Order 3328.

712 conjugacy classes of subgroups.

The subgroup lattice has 380 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13131513	C₂	1	1	yes	maximal, characteristic
2	C₂×#11719054	C₂	1	1	yes	maximal, characteristic
3	#13130401	C₂	1	4	yes	maximal, complemented
4	#13130397	C₂	1	4	yes	maximal, complemented
5	#13130387	C₂	1	4	yes	maximal, complemented
6	C₄×M₁₆×D₁₃	C₂	1	1	yes	maximal, characteristic
7	C₂×M₁₆×D₁₃	C₄	1	2	yes	—
8	C₂×C₄²×D₁₃	C₂²	1	1	yes	characteristic
9	#11719130	—	2	1	no	—
10	#11719127	—	2	1	no	—
11	C₂×#88111	—	2	1	no	—
12	C₂×#88110	—	2	1	no	—
13	#11719119	C₂²	1	2	yes	—
14	C₂×C₄⋊#1057	C₂²	1	2	yes	—
15	#11719054	—	2	1	no	—
16	#11719054	—	2	1	no	—
17	#11719054	C₂²	1	2	yes	—
18	#11719054	C₂²	1	2	yes	—
19	#11719054	C₂²	1	2	yes	—
20	#11719052	C₂²	1	2	yes	—
21	#11719051	—	2	1	no	—
22	C₄×#6118	—	2	1	no	—
23	C₄×C₁₃×M₁₆	C₄	1	1	yes	characteristic
24	M₁₆×Dic₁₃	C₂²	1	2	yes	—
25	C₄×#6099	C₄	1	1	yes	characteristic
26	#11718368	C₂²	1	2	yes	—
27	D₁₃×#97	C₄	1	2	yes	—
28	C₄×#6094	C₄	1	2	yes	—
29	C₄×C₈×D₁₃	C₂²	1	2	yes	—
30	#11718254	C₂²	1	8	yes	complemented
31	#11718253	C₂²	1	8	yes	complemented
32	C₂²×C₄×D₁₃	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₄×D₁₃	D₄	1	1	yes	characteristic
34	C₂²×C₄×D₁₃	Q₈	1	1	yes	characteristic
35	C₂²×#1057	—	2	2	no	—
36	C₂×#6118	—	2	2	no	—
37	C₂×C₁₃×M₁₆	C₂×C₄	1	2	yes	—
38	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄×Dic₁₃	D₄	1	1	yes	characteristic
40	C₂×C₄×Dic₁₃	Q₈	1	1	yes	characteristic
41	C₂×C₄×Dic₁₃	C₂³	1	1	yes	characteristic
42	C₂×#6099	C₂×C₄	1	2	yes	—
43	M₁₆×D₁₃	—	2	8	no	—
44	C₂×#6094	C₂×C₄	1	4	yes	—
45	C₂×C₈×D₁₃	C₂×C₄	1	4	yes	—
46	C₄²×D₁₃	C₂×C₄	1	1	yes	characteristic
47	C₄²×D₁₃	C₂×C₄	1	1	yes	characteristic
48	C₄²×D₁₃	D₄	1	1	yes	characteristic
49	C₄²×D₁₃	D₄	1	1	yes	characteristic
50	C₄²×D₁₃	Q₈	1	1	yes	characteristic
51	C₄²×D₁₃	Q₈	1	1	yes	characteristic
52	C₄²×D₁₃	C₂³	1	1	yes	characteristic
53	C₄²×D₁₃	C₂³	1	1	yes	characteristic
54	C₂²⋊#1057	—	2	2	no	—
55	#88111	—	2	2	no	—
56	#88111	—	2	2	no	—
57	#88111	—	2	2	no	—
58	#88110	—	2	2	no	—
59	#88110	—	2	2	no	—
60	#88110	—	2	2	no	—
61	C₄⋊#1057	—	2	2	no	—
62	C₄⋊#1057	—	2	2	no	—
63	C₄⋊#1057	C₂³	1	4	yes	—
64	C₄⋊#1057	C₂³	1	4	yes	—
65	C₄×#1057	—	2	2	no	—
66	C₁₃×#97	C₂×C₄	1	2	yes	—
67	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
68	#88055	C₂×C₄	1	4	yes	—
69	C₈×Dic₁₃	C₂³	1	4	yes	—
70	#88043	C₂×C₄	1	2	yes	—
71	C₄×#1055	C₂×C₄	1	2	yes	—
72	#17797	—	13	1	no	maximal, Sylow
73	C₂³×D₁₃	C₄⋊C₄	1	1	yes	characteristic
74	C₂²×C₄×C₁₃	C₄⋊C₄	1	1	yes	characteristic
75	C₂²×C₄×C₁₃	C₄⋊C₄	1	1	yes	characteristic
76	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
77	C₂²×Dic₁₃	C₄⋊C₄	1	1	yes	characteristic
78	C₂²×Dic₁₃	C₂×D₄	1	1	yes	characteristic
79	C₂²×Dic₁₃	C₂×Q₈	1	1	yes	characteristic
80	C₂×C₄×D₁₃	—	2	4	no	—
81	C₂×C₄×D₁₃	—	4	2	no	—
82	C₂×C₄×D₁₃	—	4	2	no	—
83	C₂×C₄×D₁₃	C₄⋊C₄	1	1	yes	characteristic
84	C₂×C₄×D₁₃	C₄⋊C₄	1	1	yes	characteristic
85	C₂×C₄×D₁₃	C₄⋊C₄	1	1	yes	characteristic
86	C₂×C₄×D₁₃	C₄⋊C₄	1	1	yes	characteristic
87	C₂×C₄×D₁₃	C₄⋊C₄	1	1	yes	characteristic
88	C₂×C₄×D₁₃	C₄⋊C₄	1	1	yes	characteristic
89	C₂×C₄×D₁₃	C₂²×C₄	1	1	yes	characteristic
90	C₂×C₄×D₁₃	C₂²×C₄	1	1	yes	characteristic
91	C₂×C₄×D₁₃	C₂×D₄	1	1	yes	characteristic
92	C₂×C₄×D₁₃	C₂×D₄	1	1	yes	characteristic
93	C₂×C₄×D₁₃	C₂×Q₈	1	1	yes	characteristic
94	C₂×C₄×D₁₃	C₂×Q₈	1	1	yes	characteristic
95	C₂×#1057	—	2	4	no	—
96	C₂×#1057	—	2	4	no	—
97	C₂×#1057	—	4	4	no	—
98	#6118	—	4	4	no	—
99	C₁₃×M₁₆	M₁₆	1	8	yes	complemented
100	C₂×C₈×C₁₃	C₂²×C₄	1	4	yes	—
101	C₄²×C₁₃	C₄⋊C₄	1	1	yes	characteristic
102	C₄²×C₁₃	C₄⋊C₄	1	1	yes	characteristic
103	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
104	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
105	C₄×Dic₁₃	C₄⋊C₄	1	1	yes	characteristic
106	C₄×Dic₁₃	C₄⋊C₄	1	1	yes	characteristic
107	C₄×Dic₁₃	C₄⋊C₄	1	1	yes	characteristic
108	C₄×Dic₁₃	C₄⋊C₄	1	1	yes	characteristic
109	C₄×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
110	C₄×Dic₁₃	C₂×D₄	1	1	yes	characteristic
111	C₄×Dic₁₃	C₂×D₄	1	1	yes	characteristic
112	C₄×Dic₁₃	C₂×D₄	1	1	yes	characteristic
113	C₄×Dic₁₃	C₂×Q₈	1	1	yes	characteristic
114	C₄×Dic₁₃	C₂×Q₈	1	1	yes	characteristic
115	C₄×Dic₁₃	C₂×Q₈	1	1	yes	characteristic
116	C₄×Dic₁₃	C₂⁴	1	1	yes	characteristic
117	#6099	—	2	4	no	—
118	C₂×#1055	C₂²×C₄	1	4	yes	—
119	#6094	—	2	8	no	—
120	C₈×D₁₃	—	2	8	no	—
121	#2920	—	13	1	no	—
122	C₂×#432	—	13	1	no	—
123	C₂×C₄×M₁₆	—	13	1	no	—
124	#1569	—	13	4	no	—
125	#1568	—	13	4	no	—
126	#1565	—	13	4	no	—
127	C₂³×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
128	C₂²×D₁₃	—	2	2	no	—
129	C₂²×D₁₃	—	2	2	no	—
130	C₂²×D₁₃	—	4	1	no	—
131	C₂²×D₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
132	C₂²×D₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
133	C₂×C₄×C₁₃	—	4	1	no	—
134	C₂×C₄×C₁₃	—	4	1	no	—
135	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
136	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
137	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
138	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
139	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
140	C₂×C₄×C₁₃	C₂×M₁₆	1	4	yes	—
141	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
142	C₂×Dic₁₃	—	2	2	no	—
143	C₂×Dic₁₃	—	2	2	no	—
144	C₂×Dic₁₃	—	4	1	no	—
145	C₂×Dic₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
146	C₂×Dic₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
147	C₂×Dic₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
148	C₂×Dic₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
149	C₂×Dic₁₃	C₂²×D₄	1	1	yes	characteristic
150	C₂×Dic₁₃	C₂²×Q₈	1	1	yes	characteristic
151	C₄×D₁₃	—	2	2	no	—
152	C₄×D₁₃	—	2	2	no	—
153	C₄×D₁₃	—	2	2	no	—
154	C₄×D₁₃	—	2	2	no	—
155	C₄×D₁₃	—	2	2	no	—
156	C₄×D₁₃	—	2	2	no	—
157	C₄×D₁₃	—	2	2	no	—
158	C₄×D₁₃	—	2	2	no	—
159	C₄×D₁₃	—	4	1	no	—
160	C₄×D₁₃	—	4	1	no	—
161	C₄×D₁₃	—	4	1	no	—
162	C₄×D₁₃	—	4	1	no	—
163	#1057	—	4	4	no	—
164	#1057	—	4	4	no	—
165	C₈×C₁₃	C₂×M₁₆	1	8	yes	—
166	#1055	—	2	4	no	—
167	C₂²×M₁₆	—	13	2	no	—
168	C₂²×C₄²	—	13	1	no	—
169	#432	—	13	2	no	—
170	#432	—	13	2	no	—
171	#432	—	13	2	no	—
172	#432	—	26	1	no	—
173	#432	—	26	1	no	—
174	#432	—	26	1	no	—
175	#431	—	13	2	no	—
176	#431	—	26	1	no	—
177	#423	—	13	2	no	—
178	C₂×C₄⋊C₈	—	13	2	no	—
179	#407	—	26	1	no	—
180	C₂×C₂²⋊C₈	—	26	1	no	—
181	C₄×M₁₆	—	13	2	no	—
182	C₄×M₁₆	—	13	1	no	—
183	C₄×M₁₆	—	13	2	no	—
184	C₄×M₁₆	—	13	2	no	—
185	C₄×M₁₆	—	26	1	no	—
186	C₄×M₁₆	#261	1	1	yes	characteristic
187	C₂×#97	—	13	2	no	—
188	C₂×#97	—	26	1	no	—
189	C₂×C₄×C₈	—	13	2	no	—
190	#335	—	13	8	no	—
191	#334	—	13	8	no	—
192	C₂²×C₁₃	—	2	1	no	—
193	C₂²×C₁₃	—	2	1	no	—
194	C₂²×C₁₃	#423	1	2	yes	—
195	C₂²×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
196	C₂×D₁₃	—	2	1	no	—
197	C₂×D₁₃	—	2	1	no	—
198	C₂×D₁₃	—	2	1	no	—
199	C₂×D₁₃	—	2	1	no	—
200	C₂×D₁₃	—	2	1	no	—
201	C₂×D₁₃	—	4	4	no	—
202	C₂×D₁₃	#430	1	2	yes	—
203	C₄×C₁₃	—	2	1	no	—
204	C₄×C₁₃	—	2	1	no	—
205	C₄×C₁₃	—	2	1	no	—
206	C₄×C₁₃	—	2	1	no	—
207	C₄×C₁₃	#423	1	2	yes	—
208	C₄×C₁₃	C₂²×M₁₆	1	2	yes	—
209	Dic₁₃	—	2	1	no	—
210	Dic₁₃	—	2	1	no	—
211	Dic₁₃	—	2	1	no	—
212	Dic₁₃	—	2	1	no	—
213	Dic₁₃	—	2	1	no	—
214	Dic₁₃	#575	1	2	yes	—
215	C₂³×C₄	—	13	1	no	—
216	C₂³×C₄	—	13	1	no	—
217	C₂³×C₄	—	13	1	no	—
218	C₂×M₁₆	—	13	2	no	—
219	C₂×M₁₆	—	13	4	no	—
220	C₂×M₁₆	—	26	8	no	—
221	C₂×M₁₆	—	26	2	no	—
222	C₂×M₁₆	C₂×#261	1	2	yes	—
223	C₂²×C₈	—	13	4	no	—
224	C₂²×C₈	—	26	2	no	—
225	C₂×C₄²	—	13	1	no	—
226	C₂×C₄²	—	13	1	no	—
227	C₂×C₄²	—	13	1	no	—
228	C₂×C₄²	—	13	1	no	—
229	C₂×C₄²	—	13	1	no	—
230	C₂×C₄²	—	13	1	no	—
231	C₂×C₄²	—	13	1	no	—
232	C₂×C₄²	—	13	1	no	—
233	C₂×C₄²	—	13	1	no	—
234	C₂×C₄²	—	13	1	no	—
235	C₂×C₄²	—	13	1	no	—
236	C₂×C₄²	C₂×#261	1	1	yes	characteristic
237	C₄⋊C₈	—	13	4	no	—
238	C₄⋊C₈	—	13	4	no	—
239	C₄⋊C₈	—	26	2	no	—
240	C₄⋊C₈	—	26	2	no	—
241	C₂²⋊C₈	—	26	2	no	—
242	C₂²⋊C₈	—	26	2	no	—
243	C₂²⋊C₈	—	26	2	no	—
244	C₂²⋊C₈	—	26	2	no	—
245	#97	—	13	4	no	—
246	#97	—	13	2	no	—
247	#97	—	26	2	no	—
248	#97	—	26	2	no	—
249	#97	—	26	2	no	—
250	#97	C₂×#261	1	2	yes	—
251	C₄×C₈	—	13	4	no	—
252	C₄×C₈	—	13	2	no	—
253	C₄×C₈	—	26	2	no	—
254	C₄×C₈	C₂×#261	1	2	yes	—
255	C₂×C₁₃	—	2	2	no	—
256	C₂×C₁₃	C₂×#423	1	1	yes	characteristic
257	C₂×C₁₃	#2911	1	1	yes	characteristic
258	C₂×C₁₃	C₂×#430	1	1	yes	characteristic
259	D₁₃	—	2	2	no	—
260	D₁₃	—	4	1	no	—
261	C₂⁴	—	13	1	no	—
262	C₂²×C₄	—	13	1	no	—
263	C₂²×C₄	—	13	1	no	—
264	C₂²×C₄	—	13	1	no	—
265	C₂²×C₄	—	13	1	no	—
266	C₂²×C₄	—	13	1	no	—
267	C₂²×C₄	—	13	1	no	—
268	C₂²×C₄	—	13	1	no	—
269	C₂²×C₄	—	13	1	no	—
270	C₂²×C₄	—	13	1	no	—
271	C₂²×C₄	—	13	1	no	—
272	C₂²×C₄	—	13	1	no	—
273	C₂²×C₄	—	13	1	no	—
274	C₂²×C₄	—	13	1	no	—
275	C₂²×C₄	—	13	1	no	—
276	C₂²×C₄	—	13	1	no	—
277	C₂²×C₄	—	26	4	no	—
278	C₂²×C₄	—	52	2	no	—
279	C₂²×C₄	—	52	2	no	—
280	C₂²×C₄	C₄⋊#261	1	1	yes	characteristic
281	C₂²×C₄	C₄⋊#261	1	1	yes	characteristic
282	C₂²×C₄	C₂²×#261	1	1	yes	characteristic
283	M₁₆	—	26	4	no	—
284	M₁₆	—	26	8	no	—
285	M₁₆	—	52	4	no	—
286	M₁₆	#6118	1	8	yes	complemented
287	C₂×C₈	—	13	4	no	—
288	C₂×C₈	—	26	8	no	—
289	C₂×C₈	—	26	4	no	—
290	C₂×C₈	—	26	4	no	—
291	C₂×C₈	—	52	4	no	—
292	C₂×C₈	C₂²×#261	1	4	yes	—
293	C₄²	—	13	1	no	—
294	C₄²	—	13	1	no	—
295	C₄²	—	13	1	no	—
296	C₄²	—	13	1	no	—
297	C₄²	—	13	1	no	—
298	C₄²	—	13	1	no	—
299	C₄²	—	13	1	no	—
300	C₄²	—	13	1	no	—
301	C₄²	—	13	1	no	—
302	C₄²	—	13	1	no	—
303	C₄²	—	13	1	no	—
304	C₄²	—	13	1	no	—
305	C₄²	C₄⋊#261	1	1	yes	characteristic
306	C₄²	C₄⋊#261	1	1	yes	characteristic
307	C₄²	C₂²×#261	1	1	yes	characteristic
308	C₄²	C₂²×#261	1	1	yes	characteristic
309	C₁₃	#17797	1	1	yes	characteristic, complemented, Sylow
310	C₂³	—	13	1	no	—
311	C₂³	—	13	1	no	—
312	C₂³	—	26	2	no	—
313	C₂³	—	26	2	no	—
314	C₂³	—	52	1	no	—
315	C₂³	C₂×C₄⋊#261	1	1	yes	characteristic
316	C₂×C₄	—	4	1	no	—
317	C₂×C₄	—	4	1	no	—
318	C₂×C₄	—	13	1	no	—
319	C₂×C₄	—	13	1	no	—
320	C₂×C₄	—	13	1	no	—
321	C₂×C₄	—	13	1	no	—
322	C₂×C₄	—	13	1	no	—
323	C₂×C₄	—	13	1	no	—
324	C₂×C₄	—	26	2	no	—
325	C₂×C₄	—	26	2	no	—
326	C₂×C₄	—	26	2	no	—
327	C₂×C₄	—	26	2	no	—
328	C₂×C₄	—	26	2	no	—
329	C₂×C₄	—	26	2	no	—
330	C₂×C₄	—	26	2	no	—
331	C₂×C₄	—	26	2	no	—
332	C₂×C₄	—	26	2	no	—
333	C₂×C₄	—	26	2	no	—
334	C₂×C₄	—	52	1	no	—
335	C₂×C₄	—	52	1	no	—
336	C₂×C₄	—	52	1	no	—
337	C₂×C₄	—	52	1	no	—
338	C₂×C₄	—	52	1	no	—
339	C₂×C₄	C₂×#6118	1	4	yes	—
340	C₂×C₄	C₂×C₄⋊#261	1	1	yes	characteristic
341	C₂×C₄	C₂×C₄⋊#261	1	1	yes	characteristic
342	C₂×C₄	C₂×C₄⋊#261	1	1	yes	characteristic
343	C₂×C₄	C₂×C₄⋊#261	1	1	yes	characteristic
344	C₂×C₄	C₂×C₄⋊#261	1	1	yes	characteristic
345	C₂×C₄	C₂³×#261	1	1	yes	characteristic
346	C₈	—	26	4	no	—
347	C₈	—	52	4	no	—
348	C₈	—	52	4	no	—
349	C₈	C₂×#6118	1	8	yes	—
350	C₂²	—	2	1	no	—
351	C₂²	—	2	1	no	—
352	C₂²	—	13	2	no	—
353	C₂²	—	26	1	no	—
354	C₂²	—	26	1	no	—
355	C₂²	—	26	1	no	—
356	C₂²	—	26	1	no	—
357	C₂²	—	26	1	no	—
358	C₂²	—	52	4	no	—
359	C₂²	#11719052	1	2	yes	—
360	C₂²	C₂²×C₄⋊#261	1	1	yes	characteristic
361	C₄	—	2	1	no	—
362	C₄	—	2	1	no	—
363	C₄	—	2	1	no	—
364	C₄	—	2	1	no	—
365	C₄	—	13	2	no	—
366	C₄	—	26	1	no	—
367	C₄	—	26	1	no	—
368	C₄	—	26	1	no	—
369	C₄	—	26	1	no	—
370	C₄	—	26	1	no	—
371	C₄	#11719052	1	2	yes	—
372	C₄	C₂²×#6118	1	2	yes	—
373	C₂	—	2	2	no	—
374	C₂	—	26	2	no	—
375	C₂	—	52	1	no	—
376	C₂	C₂×#11719052	1	1	yes	characteristic
377	C₂	#13131489	1	1	yes	characteristic
378	C₂	D₁₃⋊#430	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×C₄⋊#261
commutator	C₂×C₄×C₁₃	C₂³×C₄
Frattini	C₂×C₄	C₂³×#261
Fitting	C₄×C₁₃×M₁₆	C₄
socle	C₂²×C₁₃	C₂²×C₄⋊C₄
prime radical	C₂³×D₁₃	C₄⋊C₄
cosocle	C₄×Dic₁₃	C₂⁴
hypercenter	C₄×M₁₆	#261
hypocenter	C₁₃	#17797
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄×M₁₆

Complements

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

N	H ≅ G/N	type
3. #13130401	C₂	semidirect
4. #13130397	C₂	semidirect
5. #13130387	C₂	semidirect
30. #11718254	C₂²	semidirect
31. #11718253	C₂²	semidirect
99. C₁₃×M₁₆	M₁₆	semidirect
286. M₁₆	#6118	semidirect
309. C₁₃	#17797	semidirect

Maximal subgroups

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

subgroup	index
#13131513	2
C₂×#11719054	2
#13130401	2
#13130397	2
#13130387	2
C₄×M₁₆×D₁₃	2
#17797	13