Subgroups — #15833461

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	#13125404	C₂	1	1	yes	maximal, characteristic
2	#13123814	C₂	1	2	yes	maximal, complemented
3	#13113734	C₂	1	1	yes	maximal, characteristic, complemented
4	#13113724	C₂	1	2	yes	maximal, complemented
5	#13113642	C₂	1	1	yes	maximal, characteristic, complemented
6	#13113639	C₂	1	2	yes	maximal, complemented
7	D₈×#1055	C₂	1	1	yes	maximal, characteristic, complemented
8	#13113598	C₂	1	1	yes	maximal, characteristic, complemented
9	#13113293	C₂	1	2	yes	maximal, complemented
10	C₈²×D₁₃	C₂	1	1	yes	maximal, characteristic, complemented
11	C₁₃×#1589	C₂	1	1	yes	maximal, characteristic, complemented
12	#11719262	C₂²	1	2	yes	—
13	C₄×C₁₃×D₈	C₂²	1	1	yes	characteristic
14	C₁₃×#436	C₂²	1	2	yes	—
15	#11718818	C₄	1	1	yes	characteristic
16	#11718816	C₄	1	1	yes	characteristic
17	#11718788	—	2	2	no	—
18	#11718783	—	2	2	no	—
19	C₄×#6105	C₂²	1	2	yes	—
20	D₄×#1055	C₂²	1	2	yes	—
21	#11718548	C₄	1	1	yes	characteristic
22	#11718539	C₄	1	1	yes	characteristic
23	#11718496	C₂²	1	2	yes	—
24	#11718491	C₂²	1	2	yes	—
25	#11718439	C₄	1	2	yes	—
26	#11718429	C₄	1	2	yes	—
27	#11718386	—	2	2	no	—
28	#11718381	—	2	2	no	—
29	C₄×Dic₅₂	C₂²	1	1	yes	characteristic
30	C₄×C₈×D₁₃	C₂²	1	1	yes	characteristic
31	C₄×C₈×D₁₃	C₂²	1	1	yes	characteristic
32	C₄×C₈×D₁₃	C₂²	1	1	yes	characteristic
33	#11718340	C₂²	1	2	yes	—
34	C₁₃×#335	C₂²	1	1	yes	characteristic
35	C₁₃×D₄⋊C₈	C₂²	1	2	yes	—
36	C₈²×C₁₃	C₂²	1	1	yes	characteristic
37	D₄⋊#1055	C₂²	1	2	yes	—
38	#11718076	C₂²	1	2	yes	—
39	#11718072	C₂²	1	2	yes	—
40	#11718050	C₂²	1	1	yes	characteristic
41	#11718048	C₂²	1	2	yes	—
42	C₈×#1055	C₂²	1	1	yes	characteristic
43	C₈×#1055	C₂²	1	1	yes	characteristic
44	C₈×#1055	C₂²	1	1	yes	characteristic
45	C₂×C₁₃×D₈	C₂×C₄	1	1	yes	characteristic
46	C₂×C₁₃×M₁₆	—	2	2	no	—
47	C₄×D₄×C₁₃	C₂³	1	2	yes	—
48	#88192	C₂×C₄	1	2	yes	—
49	#88190	C₂×C₄	1	2	yes	—
50	C₂×#6105	C₂×C₄	1	2	yes	—
51	C₄×C₂²⋊D₁₃	—	2	2	no	—
52	#88180	—	2	2	no	—
53	C₂²×#1055	—	2	2	no	—
54	C₂×Dic₅₂	C₂×C₄	1	1	yes	characteristic
55	C₂×C₈×D₁₃	—	2	1	no	—
56	C₂×C₈×D₁₃	C₂×C₄	1	1	yes	characteristic
57	C₂×C₈×D₁₃	C₂×C₄	1	1	yes	characteristic
58	C₂×C₈×D₁₃	D₄	1	1	yes	characteristic
59	C₂×C₈×D₁₃	D₄	1	1	yes	characteristic
60	#88151	C₂×C₄	1	2	yes	—
61	#88144	C₂×C₄	1	2	yes	—
62	#88137	—	2	2	no	—
63	#88133	—	2	2	no	—
64	C₄²×D₁₃	C₂³	1	1	yes	characteristic
65	C₄×Dic₂₆	C₂³	1	2	yes	—
66	C₁₃×#107	C₂×C₄	1	1	yes	characteristic
67	C₁₃×C₄⋊C₈	C₂³	1	2	yes	—
68	C₁₃×#102	C₂×C₄	1	2	yes	—
69	C₁₃×C₂²⋊C₈	—	2	2	no	—
70	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
71	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
72	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
73	#88072	C₂×C₄	1	2	yes	—
74	C₂²⋊#1055	—	2	2	no	—
75	#88060	—	2	2	no	—
76	#88058	C₂×C₄	1	1	yes	characteristic
77	#88056	C₂×C₄	1	2	yes	—
78	#88055	—	2	2	no	—
79	#88055	—	2	2	no	—
80	#88054	—	2	2	no	—
81	#88054	—	2	2	no	—
82	C₈×Dic₁₃	—	2	1	no	—
83	C₈×Dic₁₃	—	2	2	no	—
84	C₈×Dic₁₃	C₂×C₄	1	1	yes	characteristic
85	C₈×Dic₁₃	C₂×C₄	1	1	yes	characteristic
86	C₈×Dic₁₃	D₄	1	1	yes	characteristic
87	C₈×Dic₁₃	D₄	1	1	yes	characteristic
88	#88050	C₂×C₄	1	2	yes	—
89	#88048	C₂×C₄	1	2	yes	—
90	C₄⋊#1055	C₂³	1	2	yes	—
91	C₄×#1055	C₂³	1	1	yes	characteristic
92	C₄×#1055	C₂³	1	1	yes	characteristic
93	C₄×#1055	C₂³	1	1	yes	characteristic
94	#17959	—	13	1	no	maximal, Sylow
95	C₂×D₄×C₁₃	C₂²×C₄	1	2	yes	—
96	C₂²×C₄×C₁₃	—	2	2	no	—
97	C₂×C₂²⋊D₁₃	—	2	2	no	—
98	C₂×C₄×D₁₃	C₂²×C₄	1	1	yes	characteristic
99	C₂×C₄×D₁₃	C₂×D₄	1	1	yes	characteristic
100	C₂×C₄×D₁₃	#41	1	1	yes	characteristic
101	C₂×Dic₂₆	C₂²×C₄	1	2	yes	—
102	C₁₃×D₈	—	2	2	no	—
103	C₁₃×M₁₆	—	4	4	no	—
104	C₂×C₈×C₁₃	—	2	1	no	—
105	C₂×C₈×C₁₃	—	2	2	no	—
106	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
107	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
108	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
109	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
110	C₁₃×C₄⋊C₄	C₂²×C₄	1	2	yes	—
111	C₁₃×C₂²⋊C₄	—	2	2	no	—
112	C₄²×C₁₃	C₂⁴	1	1	yes	characteristic
113	C₂²⋊Dic₁₃	—	2	2	no	—
114	C₂²⋊Dic₁₃	—	2	2	no	—
115	C₂²⋊Dic₁₃	—	2	2	no	—
116	#6105	—	2	4	no	—
117	#6103	—	2	2	no	—
118	C₄⋊Dic₁₃	—	2	2	no	—
119	C₄⋊Dic₁₃	—	2	2	no	—
120	C₄⋊Dic₁₃	C₂²×C₄	1	2	yes	—
121	#6101	—	2	2	no	—
122	#6101	—	2	2	no	—
123	C₄×Dic₁₃	—	2	2	no	—
124	C₄×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
125	C₄×Dic₁₃	C₂×D₄	1	1	yes	characteristic
126	C₄×Dic₁₃	#41	1	1	yes	characteristic
127	C₂×#1055	—	2	1	no	—
128	C₂×#1055	—	2	2	no	—
129	C₂×#1055	—	4	4	no	—
130	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
131	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
132	C₂×#1055	C₂×D₄	1	1	yes	characteristic
133	C₂×#1055	C₂×D₄	1	1	yes	characteristic
134	Dic₅₂	—	2	2	no	—
135	C₈×D₁₃	—	2	4	no	—
136	C₈×D₁₃	—	2	2	no	—
137	C₈×D₁₃	—	2	2	no	—
138	C₈×D₁₃	—	2	2	no	—
139	C₈×D₁₃	—	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	D₁₃	1	1	yes	characteristic, complemented
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	—	2	2	no	—
152	C₂²×D₁₃	C₄×D₄	1	1	yes	characteristic
153	D₄×C₁₃	—	2	2	no	—
154	D₄×C₁₃	—	2	2	no	—
155	C₂×C₄×C₁₃	—	2	2	no	—
156	C₂×C₄×C₁₃	—	4	2	no	—
157	C₂×C₄×C₁₃	—	4	2	no	—
158	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
159	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
160	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
161	C₂²⋊D₁₃	—	4	4	no	—
162	C₂×Dic₁₃	—	2	2	no	—
163	C₂×Dic₁₃	—	2	2	no	—
164	C₂×Dic₁₃	C₄×D₄	1	1	yes	characteristic
165	C₂×Dic₁₃	C₄×D₄	1	1	yes	characteristic
166	C₂×Dic₁₃	C₄×D₄	1	1	yes	characteristic
167	C₄×D₁₃	—	2	2	no	—
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	Dic₂₆	—	2	2	no	—
173	Dic₂₆	—	2	2	no	—
174	C₈×C₁₃	—	2	2	no	—
175	C₈×C₁₃	—	2	1	no	—
176	C₈×C₁₃	—	2	1	no	—
177	C₈×C₁₃	—	4	2	no	—
178	C₈×C₁₃	#131	1	2	yes	—
179	C₈×C₁₃	#135	1	2	yes	—
180	#1055	—	2	2	no	—
181	#1055	—	2	1	no	—
182	#1055	—	2	1	no	—
183	#1055	—	4	2	no	—
184	#1055	#131	1	2	yes	—
185	#1055	#135	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	—	13	1	no	—
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	C₂×D₁₃	1	1	yes	characteristic
198	#436	C₂×D₁₃	1	2	yes	—
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	C₂×D₁₃	1	1	yes	characteristic
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	C₂×D₁₃	1	2	yes	—
215	C₈²	—	13	1	no	—
216	C₈²	—	13	1	no	—
217	C₈²	—	13	1	no	—
218	C₈²	C₂×D₁₃	1	1	yes	characteristic
219	C₂²×C₁₃	—	4	2	no	—
220	C₂²×C₁₃	—	4	2	no	—
221	C₂²×C₁₃	—	4	2	no	—
222	C₂²×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
223	C₂×D₁₃	—	2	1	no	—
224	C₂×D₁₃	—	2	1	no	—
225	C₂×D₁₃	#443	1	2	yes	—
226	C₄×C₁₃	—	2	1	no	—
227	C₄×C₁₃	—	4	2	no	—
228	C₄×C₁₃	C₂×#131	1	1	yes	characteristic
229	C₄×C₁₃	C₂×#131	1	1	yes	characteristic
230	C₄×C₁₃	C₂×#135	1	1	yes	characteristic
231	C₄×C₁₃	C₂×#135	1	1	yes	characteristic
232	Dic₁₃	—	2	1	no	—
233	Dic₁₃	—	2	1	no	—
234	Dic₁₃	—	2	1	no	—
235	Dic₁₃	—	4	2	no	—
236	Dic₁₃	—	4	2	no	—
237	C₂×Q₁₆	—	13	1	no	—
238	C₂×SD₁₆	—	13	2	no	—
239	C₂×D₈	C₄×D₁₃	1	1	yes	characteristic
240	C₂×M₁₆	—	2	2	no	—
241	C₂²×C₈	—	13	1	no	—
242	C₂²×C₈	—	13	1	no	—
243	C₂²×C₈	—	13	1	no	—
244	C₂²×C₈	—	13	1	no	—
245	C₂²×C₈	—	26	1	no	—
246	C₂²×C₈	—	26	2	no	—
247	#126	—	26	2	no	—
248	#125	—	13	2	no	—
249	#124	—	13	2	no	—
250	#123	—	26	2	no	—
251	C₂²⋊Q₈	—	13	2	no	—
252	#121	—	13	2	no	—
253	C₄×Q₈	—	13	2	no	—
254	C₄×D₄	—	26	2	no	—
255	C₄×D₄	C₂²×D₁₃	1	2	yes	—
256	#117	—	26	2	no	—
257	C₂×C₄²	—	13	1	no	—
258	#107	—	13	1	no	—
259	#107	C₄×D₁₃	1	1	yes	characteristic
260	#106	—	13	2	no	—
261	C₄⋊C₈	—	13	2	no	—
262	C₄⋊C₈	—	26	2	no	—
263	C₄⋊C₈	—	26	2	no	—
264	C₄⋊C₈	C₂²×D₁₃	1	2	yes	—
265	#103	—	13	2	no	—
266	#103	—	13	2	no	—
267	#102	—	13	2	no	—
268	#102	C₄×D₁₃	1	2	yes	—
269	C₂²⋊C₈	—	2	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	—	26	1	no	—
282	C₄×C₈	—	26	2	no	—
283	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
284	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
285	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
286	C₂×C₁₃	—	8	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	—	2	2	no	—
291	C₂×Q₈	—	13	2	no	—
292	C₂×D₄	—	26	2	no	—
293	C₂×D₄	C₂×C₄×D₁₃	1	2	yes	—
294	C₂²×C₄	—	2	2	no	—
295	C₂²×C₄	—	13	1	no	—
296	C₂²×C₄	—	13	1	no	—
297	C₂²×C₄	—	13	1	no	—
298	Q₁₆	—	26	2	no	—
299	SD₁₆	—	26	4	no	—
300	D₈	—	2	2	no	—
301	M₁₆	—	4	4	no	—
302	C₂×C₈	—	2	1	no	—
303	C₂×C₈	—	2	2	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	—	26	2	no	—
309	C₂×C₈	—	26	2	no	—
310	C₂×C₈	—	26	4	no	—
311	C₂×C₈	—	26	1	no	—
312	C₂×C₈	—	26	2	no	—
313	C₂×C₈	—	26	2	no	—
314	C₂×C₈	—	26	2	no	—
315	C₂×C₈	—	52	4	no	—
316	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
317	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
318	C₂×C₈	D₄×D₁₃	1	1	yes	characteristic
319	C₂×C₈	D₄×D₁₃	1	1	yes	characteristic
320	C₄⋊C₄	—	13	2	no	—
321	C₄⋊C₄	—	26	2	no	—
322	C₄⋊C₄	—	26	2	no	—
323	C₄⋊C₄	—	26	2	no	—
324	C₄⋊C₄	—	26	2	no	—
325	C₄⋊C₄	C₂×C₄×D₁₃	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	—	26	2	no	—
335	C₄²	C₂³×D₁₃	1	1	yes	characteristic
336	C₁₃	#17959	1	1	yes	characteristic, complemented, Sylow
337	C₂³	—	2	2	no	—
338	C₂³	—	13	1	no	—
339	Q₈	—	26	2	no	—
340	Q₈	—	26	2	no	—
341	D₄	—	2	2	no	—
342	D₄	—	2	2	no	—
343	D₄	—	52	4	no	—
344	C₂×C₄	—	2	2	no	—
345	C₂×C₄	—	4	2	no	—
346	C₂×C₄	—	4	2	no	—
347	C₂×C₄	—	13	1	no	—
348	C₂×C₄	—	13	1	no	—
349	C₂×C₄	—	13	1	no	—
350	C₂×C₄	—	26	2	no	—
351	C₂×C₄	—	26	1	no	—
352	C₂×C₄	—	26	1	no	—
353	C₂×C₄	—	26	1	no	—
354	C₂×C₄	—	26	1	no	—
355	C₂×C₄	—	26	2	no	—
356	C₂×C₄	—	26	2	no	—
357	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
358	C₂×C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
359	C₂×C₄	#41×D₁₃	1	1	yes	characteristic
360	C₈	—	2	2	no	—
361	C₈	—	2	1	no	—
362	C₈	—	2	1	no	—
363	C₈	—	4	2	no	—
364	C₈	—	13	2	no	—
365	C₈	—	13	2	no	—
366	C₈	—	26	2	no	—
367	C₈	—	26	1	no	—
368	C₈	—	26	1	no	—
369	C₈	—	52	2	no	—
370	C₈	#88161	1	2	yes	—
371	C₈	#88166	1	2	yes	—
372	C₂²	—	4	2	no	—
373	C₂²	—	4	2	no	—
374	C₂²	—	4	2	no	—
375	C₂²	—	13	2	no	—
376	C₂²	—	26	1	no	—
377	C₂²	—	26	1	no	—
378	C₂²	C₄×D₄×D₁₃	1	1	yes	characteristic
379	C₄	—	2	1	no	—
380	C₄	—	4	2	no	—
381	C₄	—	26	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×#88161	1	1	yes	characteristic
387	C₄	D₁₃×#131	1	1	yes	characteristic
388	C₄	C₂×#88166	1	1	yes	characteristic
389	C₄	D₁₃×#135	1	1	yes	characteristic
390	C₂	—	8	2	no	—
391	C₂	—	26	2	no	—
392	C₂	#13123545	1	1	yes	characteristic
393	C₂	C₄×#88166	1	1	yes	characteristic
394	C₂	D₁₃×#443	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₁₃
commutator	C₂×C₄×C₁₃	C₂³×C₄
Frattini	C₄²	C₂³×D₁₃
Fitting	C₁₃×#1589	C₂
socle	C₂²×C₁₃	C₂×C₄×D₄
prime radical	#11718818	C₄
cosocle	C₄²×C₁₃	C₂⁴
hypercenter	#1589	D₁₃
hypocenter	C₁₃	#17959
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #1589

Complements

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

N	H ≅ G/N	type
2. #13123814	C₂	semidirect
3. #13113734	C₂	semidirect
4. #13113724	C₂	semidirect
5. #13113642	C₂	semidirect
6. #13113639	C₂	semidirect
7. D₈×#1055	C₂	semidirect
8. #13113598	C₂	semidirect
9. #13113293	C₂	semidirect
10. C₈²×D₁₃	C₂	semidirect
11. C₁₃×#1589	C₂	semidirect
144. #1589	D₁₃	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
#13125404	2
#13123814	2
#13113734	2
#13113724	2
#13113642	2
#13113639	2
D₈×#1055	2
#13113598	2
#13113293	2
C₈²×D₁₃	2
C₁₃×#1589	2
#17959	13