Subgroups — #16795014

Order 3328.

648 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₁₃⋊#432	C₂	1	1	yes	maximal, characteristic
2	D₁₃⋊#431	C₂	1	1	yes	maximal, characteristic
3	#13130410	C₂	1	4	yes	maximal, complemented
4	#13130407	C₂	1	4	yes	maximal, complemented
5	#13130388	C₂	1	4	yes	maximal, complemented
6	#13123466	C₂	1	1	yes	maximal, characteristic
7	C₂×C₄²×D₁₃	C₂²	1	1	yes	characteristic
8	#11719130	C₂²	1	2	yes	—
9	#11719128	—	2	1	no	—
10	#11719125	—	2	1	no	—
11	D₁₃⋊(C₂²⋊C₈)	—	2	1	no	—
12	D₁₃⋊(C₂²⋊C₈)	—	2	1	no	—
13	#11719119	C₂²	1	2	yes	—
14	#11719055	—	2	1	no	—
15	#11719054	C₂²	1	2	yes	—
16	#11719051	C₂²	1	2	yes	—
17	D₁₃⋊(C₄⋊C₈)	C₂²	1	2	yes	—
18	D₁₃⋊(C₄⋊C₈)	C₂²	1	2	yes	—
19	#11719049	—	2	1	no	—
20	#11719047	—	2	1	no	—
21	C₄×D₁₃⋊C₈	—	2	1	no	—
22	C₁₃×#432	C₄	1	1	yes	characteristic
23	#11718587	C₄	1	1	yes	characteristic
24	#11718498	C₄	1	2	yes	—
25	#11718494	C₂²	1	2	yes	—
26	#11718387	C₄	1	2	yes	—
27	#11718379	C₂²	1	2	yes	—
28	#11718368	C₂²	1	2	yes	—
29	#11718366	C₄	1	2	yes	—
30	#11718256	C₂²	1	4	yes	complemented
31	#11718255	C₂²	1	4	yes	complemented
32	#11718254	C₂²	1	4	yes	complemented
33	#11718253	C₂²	1	4	yes	complemented
34	C₂²×C₄×D₁₃	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₄×D₁₃	D₄	1	1	yes	characteristic
36	C₂²×C₄×D₁₃	Q₈	1	1	yes	characteristic
37	C₂×D₁₃⋊C₈	—	2	2	no	—
38	C₂×D₁₃⋊C₈	—	2	2	no	—
39	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄×Dic₁₃	D₄	1	1	yes	characteristic
41	C₂×C₄×Dic₁₃	Q₈	1	1	yes	characteristic
42	C₂×C₄×Dic₁₃	C₂³	1	1	yes	characteristic
43	C₂×#6094	C₂×C₄	1	4	yes	—
44	C₄²×D₁₃	C₂×C₄	1	1	yes	characteristic
45	C₄²×D₁₃	C₂×C₄	1	1	yes	characteristic
46	C₄²×D₁₃	D₄	1	1	yes	characteristic
47	C₄²×D₁₃	D₄	1	1	yes	characteristic
48	C₄²×D₁₃	Q₈	1	1	yes	characteristic
49	C₄²×D₁₃	Q₈	1	1	yes	characteristic
50	C₄²×D₁₃	C₂³	1	1	yes	characteristic
51	C₄²×D₁₃	C₂³	1	1	yes	characteristic
52	C₂²⋊#1057	—	2	2	no	—
53	C₂²⋊#1057	—	2	2	no	—
54	#88112	—	2	2	no	—
55	#88112	—	2	2	no	—
56	#88111	—	2	2	no	—
57	#88111	—	2	2	no	—
58	#88110	—	2	2	no	—
59	#88110	—	2	2	no	—
60	C₄⋊#1057	C₂³	1	4	yes	—
61	C₄⋊#1057	C₂³	1	4	yes	—
62	C₄×#1057	—	2	2	no	—
63	C₄×#1057	—	2	2	no	—
64	C₁₃×C₄⋊C₈	C₂×C₄	1	2	yes	—
65	C₁₃×C₂²⋊C₈	C₂×C₄	1	2	yes	—
66	C₁₃×#97	C₂×C₄	1	2	yes	—
67	C₂²⋊#1055	C₂×C₄	1	2	yes	—
68	#88060	C₂×C₄	1	4	yes	—
69	#88054	C₂×C₄	1	4	yes	—
70	C₈×Dic₁₃	C₂³	1	4	yes	—
71	C₄⋊#1055	C₂×C₄	1	2	yes	—
72	#88043	C₂×C₄	1	2	yes	—
73	#17810	—	13	1	no	maximal, Sylow
74	C₂³×D₁₃	C₄⋊C₄	1	1	yes	characteristic
75	C₂²×C₄×C₁₃	C₄⋊C₄	1	1	yes	characteristic
76	C₂²×C₄×C₁₃	C₄⋊C₄	1	1	yes	characteristic
77	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
78	C₂²×Dic₁₃	C₄⋊C₄	1	1	yes	characteristic
79	C₂²×Dic₁₃	C₂×D₄	1	1	yes	characteristic
80	C₂²×Dic₁₃	C₂×Q₈	1	1	yes	characteristic
81	C₂×C₄×D₁₃	—	2	4	no	—
82	C₂×C₄×D₁₃	—	4	2	no	—
83	C₂×C₄×D₁₃	—	4	2	no	—
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	C₄⋊C₄	1	1	yes	characteristic
90	C₂×C₄×D₁₃	C₂²×C₄	1	1	yes	characteristic
91	C₂×C₄×D₁₃	C₂²×C₄	1	1	yes	characteristic
92	C₂×C₄×D₁₃	C₂×D₄	1	1	yes	characteristic
93	C₂×C₄×D₁₃	C₂×D₄	1	1	yes	characteristic
94	C₂×C₄×D₁₃	C₂×Q₈	1	1	yes	characteristic
95	C₂×C₄×D₁₃	C₂×Q₈	1	1	yes	characteristic
96	C₂×#1057	—	2	4	no	—
97	C₂×#1057	—	2	4	no	—
98	D₁₃⋊C₈	—	4	4	no	—
99	D₁₃⋊C₈	—	4	4	no	—
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	C₂×#1055	C₂²×C₄	1	4	yes	—
118	#6094	—	2	8	no	—
119	#2920	—	13	1	no	—
120	C₂×#432	—	13	1	no	—
121	C₂×#431	—	13	1	no	—
122	#1573	—	13	4	no	—
123	#1572	—	13	4	no	—
124	#1565	—	13	4	no	—
125	C₂³×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
126	C₂²×D₁₃	—	2	2	no	—
127	C₂²×D₁₃	—	2	2	no	—
128	C₂²×D₁₃	—	4	1	no	—
129	C₂²×D₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
130	C₂²×D₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
131	C₂×C₄×C₁₃	—	2	2	no	—
132	C₂×C₄×C₁₃	—	4	1	no	—
133	C₂×C₄×C₁₃	—	4	1	no	—
134	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
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	C₂³×C₄	1	1	yes	characteristic
140	C₂×Dic₁₃	—	2	2	no	—
141	C₂×Dic₁₃	—	2	2	no	—
142	C₂×Dic₁₃	—	4	1	no	—
143	C₂×Dic₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
144	C₂×Dic₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
145	C₂×Dic₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
146	C₂×Dic₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
147	C₂×Dic₁₃	C₂²×D₄	1	1	yes	characteristic
148	C₂×Dic₁₃	C₂²×Q₈	1	1	yes	characteristic
149	C₄×D₁₃	—	2	2	no	—
150	C₄×D₁₃	—	2	2	no	—
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	—	4	1	no	—
157	C₄×D₁₃	—	4	1	no	—
158	C₄×D₁₃	—	4	1	no	—
159	C₄×D₁₃	—	4	1	no	—
160	C₄×D₁₃	#131	1	4	yes	—
161	#1057	—	4	4	no	—
162	#1057	—	4	4	no	—
163	C₈×C₁₃	—	2	4	no	—
164	#1055	—	2	4	no	—
165	C₂²×C₄²	—	13	1	no	—
166	#432	—	13	1	no	—
167	#432	—	13	2	no	—
168	#432	—	13	2	no	—
169	#432	—	13	2	no	—
170	#432	—	26	1	no	—
171	#432	—	26	1	no	—
172	#432	#261	1	1	yes	characteristic
173	#431	—	13	2	no	—
174	#431	—	13	2	no	—
175	#431	—	13	2	no	—
176	#431	—	13	2	no	—
177	#431	—	26	1	no	—
178	#431	—	26	1	no	—
179	#423	—	13	2	no	—
180	C₂×C₄⋊C₈	—	13	2	no	—
181	C₂×C₄⋊C₈	—	13	2	no	—
182	#407	—	13	2	no	—
183	C₂×C₂²⋊C₈	—	26	1	no	—
184	C₂×C₂²⋊C₈	—	26	1	no	—
185	C₄×M₁₆	—	13	2	no	—
186	C₂×#97	—	26	1	no	—
187	C₂×C₄×C₈	—	26	1	no	—
188	#335	—	13	4	no	—
189	#335	—	13	4	no	—
190	#334	—	13	4	no	—
191	#334	—	13	4	no	—
192	C₂²×C₁₃	—	2	1	no	—
193	C₂²×C₁₃	—	2	1	no	—
194	C₂²×C₁₃	—	2	1	no	—
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	—	4	4	no	—
201	C₂×D₁₃	#424	1	2	yes	—
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	—	2	1	no	—
208	C₄×C₁₃	C₂×#131	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	#424	1	2	yes	—
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	4	no	—
219	C₂²×C₈	—	26	2	no	—
220	C₂²×C₈	—	26	2	no	—
221	C₂×C₄²	—	13	1	no	—
222	C₂×C₄²	—	13	1	no	—
223	C₂×C₄²	—	13	1	no	—
224	C₂×C₄²	—	13	1	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	C₂×#261	1	1	yes	characteristic
233	C₄⋊C₈	—	13	4	no	—
234	C₄⋊C₈	—	13	4	no	—
235	C₄⋊C₈	—	13	4	no	—
236	C₄⋊C₈	—	13	2	no	—
237	C₄⋊C₈	—	26	2	no	—
238	C₄⋊C₈	—	26	2	no	—
239	C₄⋊C₈	C₂×#261	1	2	yes	—
240	C₂²⋊C₈	—	13	2	no	—
241	C₂²⋊C₈	—	13	4	no	—
242	C₂²⋊C₈	—	26	2	no	—
243	C₂²⋊C₈	—	26	2	no	—
244	C₂²⋊C₈	—	26	2	no	—
245	C₂²⋊C₈	—	26	2	no	—
246	C₂²⋊C₈	C₂×#261	1	2	yes	—
247	#97	—	13	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	—	26	2	no	—
253	C₄×C₈	—	26	2	no	—
254	C₂×C₁₃	—	4	1	no	—
255	C₂×C₁₃	C₂×#424	1	1	yes	characteristic
256	C₂×C₁₃	#2911	1	1	yes	characteristic
257	C₂×C₁₃	C₂×#430	1	1	yes	characteristic
258	D₁₃	—	2	2	no	—
259	D₁₃	—	2	2	no	—
260	C₂⁴	—	13	1	no	—
261	C₂²×C₄	—	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	—	26	4	no	—
277	C₂²×C₄	—	52	2	no	—
278	C₂²×C₄	—	52	2	no	—
279	C₂²×C₄	C₄⋊#261	1	1	yes	characteristic
280	C₂²×C₄	C₄⋊#261	1	1	yes	characteristic
281	C₂²×C₄	C₂²×#261	1	1	yes	characteristic
282	M₁₆	—	26	8	no	—
283	C₂×C₈	—	13	4	no	—
284	C₂×C₈	—	26	4	no	—
285	C₂×C₈	—	26	4	no	—
286	C₂×C₈	—	52	4	no	—
287	C₂×C₈	—	52	4	no	—
288	C₂×C₈	C₂²×#261	1	4	yes	—
289	C₄²	—	13	1	no	—
290	C₄²	—	13	1	no	—
291	C₄²	—	13	1	no	—
292	C₄²	—	13	1	no	—
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	C₄⋊#261	1	1	yes	characteristic
302	C₄²	C₄⋊#261	1	1	yes	characteristic
303	C₄²	C₂²×#261	1	1	yes	characteristic
304	C₄²	C₂²×#261	1	1	yes	characteristic
305	C₁₃	#17810	1	1	yes	characteristic, complemented, Sylow
306	C₂³	—	13	1	no	—
307	C₂³	—	13	1	no	—
308	C₂³	—	26	2	no	—
309	C₂³	—	26	2	no	—
310	C₂³	—	52	1	no	—
311	C₂³	C₂×C₄⋊#261	1	1	yes	characteristic
312	C₂×C₄	—	2	2	no	—
313	C₂×C₄	—	4	1	no	—
314	C₂×C₄	—	4	1	no	—
315	C₂×C₄	—	13	1	no	—
316	C₂×C₄	—	13	1	no	—
317	C₂×C₄	—	13	4	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	—	26	2	no	—
323	C₂×C₄	—	26	2	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	—	52	1	no	—
332	C₂×C₄	—	52	1	no	—
333	C₂×C₄	—	52	1	no	—
334	C₂×C₄	—	52	1	no	—
335	C₂×C₄	—	52	1	no	—
336	C₂×C₄	C₂×C₄⋊#261	1	1	yes	characteristic
337	C₂×C₄	C₂×C₄⋊#261	1	1	yes	characteristic
338	C₂×C₄	C₂×C₄⋊#261	1	1	yes	characteristic
339	C₂×C₄	C₂×C₄⋊#261	1	1	yes	characteristic
340	C₂×C₄	C₂×C₄⋊#261	1	1	yes	characteristic
341	C₂×C₄	C₂³×#261	1	1	yes	characteristic
342	C₈	—	2	4	no	—
343	C₈	—	26	4	no	—
344	C₈	—	52	4	no	—
345	C₈	—	52	4	no	—
346	C₂²	—	2	1	no	—
347	C₂²	—	2	1	no	—
348	C₂²	—	2	1	no	—
349	C₂²	—	13	2	no	—
350	C₂²	—	13	2	no	—
351	C₂²	—	26	1	no	—
352	C₂²	—	26	1	no	—
353	C₂²	—	26	1	no	—
354	C₂²	—	26	1	no	—
355	C₂²	—	52	4	no	—
356	C₂²	C₂²×C₄⋊#261	1	1	yes	characteristic
357	C₄	—	2	1	no	—
358	C₄	—	2	1	no	—
359	C₄	—	2	1	no	—
360	C₄	—	2	1	no	—
361	C₄	—	2	1	no	—
362	C₄	—	13	2	no	—
363	C₄	—	13	2	no	—
364	C₄	—	26	1	no	—
365	C₄	—	26	1	no	—
366	C₄	—	26	1	no	—
367	C₄	—	26	1	no	—
368	C₄	D₁₃⋊#131	1	2	yes	—
369	C₂	—	4	1	no	—
370	C₂	—	26	2	no	—
371	C₂	—	26	2	no	—
372	C₂	D₁₃⋊#424	1	1	yes	characteristic
373	C₂	#13131489	1	1	yes	characteristic
374	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	C₁₃×#432	C₄
socle	C₂²×C₁₃	C₂²×C₄⋊C₄
prime radical	C₂³×D₁₃	C₄⋊C₄
cosocle	C₄×Dic₁₃	C₂⁴
hypercenter	#432	#261
hypocenter	C₁₃	#17810
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #432

Complements

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

N	H ≅ G/N	type
3. #13130410	C₂	semidirect
4. #13130407	C₂	semidirect
5. #13130388	C₂	semidirect
30. #11718256	C₂²	semidirect
31. #11718255	C₂²	semidirect
32. #11718254	C₂²	semidirect
33. #11718253	C₂²	semidirect
305. C₁₃	#17810	semidirect

Maximal subgroups

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

subgroup	index
D₁₃⋊#432	2
D₁₃⋊#431	2
#13130410	2
#13130407	2
#13130388	2
#13123466	2
#17810	13