Subgroups — #15832798

Order 3328.

686 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	#13124868	C₂	1	1	yes	maximal, characteristic
2	C₂×C₈×Dic₂₆	C₂	1	1	yes	maximal, characteristic, complemented
3	#13123549	C₂	1	1	yes	maximal, characteristic
4	#13113681	C₂	1	4	yes	maximal, complemented
5	C₈×#6095	C₂	1	4	yes	maximal, complemented
6	#13113607	C₂	1	1	yes	maximal, characteristic, complemented
7	#13113266	C₂	1	1	yes	maximal, characteristic, complemented
8	#13113236	C₂	1	1	yes	maximal, characteristic
9	C₁₃×#1450	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#88155	—	2	1	no	—
11	#11719182	C₂²	1	1	yes	characteristic
12	C₂×C₄×Dic₂₆	C₂²	1	1	yes	characteristic
13	C₁₃×#431	C₂²	1	1	yes	characteristic
14	C₁₃×#431	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
16	#11718771	—	2	1	no	—
17	C₂×#88054	—	2	1	no	—
18	C₂×C₈×Dic₁₃	—	2	1	no	—
19	#11718585	C₂²	1	1	yes	characteristic
20	C₂×C₄⋊#1055	C₂²	1	1	yes	characteristic
21	#11718512	C₄	1	2	yes	—
22	#11718509	C₄	1	2	yes	—
23	#11718399	C₄	1	2	yes	—
24	#11718396	C₄	1	2	yes	—
25	C₄×#6095	C₂²	1	4	yes	—
26	#11718349	—	2	1	no	—
27	#11718349	C₂²	1	2	yes	—
28	C₈×D₅₂	—	2	1	no	—
29	C₈×D₅₂	C₂²	1	2	yes	—
30	C₈×Dic₂₆	—	2	1	no	—
31	C₈×Dic₂₆	—	2	1	no	—
32	C₈×Dic₂₆	C₂²	1	2	yes	—
33	C₈×Dic₂₆	C₂²	1	2	yes	—
34	C₁₃×#322	C₂²	1	2	yes	—
35	C₈²×C₁₃	C₂²	1	2	yes	—
36	#11718051	C₂²	1	4	yes	—
37	#11718049	C₂²	1	4	yes	—
38	#11718048	C₂²	1	4	yes	—
39	C₂×#6127	—	2	1	no	—
40	C₂²×Dic₂₆	C₂×C₄	1	1	yes	characteristic
41	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
42	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
43	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
44	#88184	—	2	1	no	—
45	#88184	C₂×C₄	1	2	yes	—
46	C₂×C₄⋊Dic₁₃	C₂×C₄	1	1	yes	characteristic
47	C₂×#6101	—	2	1	no	—
48	C₂×C₄×Dic₁₃	—	2	1	no	—
49	C₂×#6099	—	2	1	no	—
50	C₂²×#1055	—	2	1	no	—
51	C₂×#6095	C₂×C₄	1	4	yes	—
52	#88155	—	4	4	no	—
53	C₂×#6094	—	2	2	no	—
54	C₂×C₈×D₁₃	—	2	2	no	—
55	#88129	—	2	1	no	—
56	#88128	C₂×C₄	1	1	yes	characteristic
57	C₄×D₅₂	—	2	1	no	—
58	C₄×D₅₂	C₂³	1	2	yes	—
59	#88123	C₂×C₄	1	1	yes	characteristic
60	C₄×Dic₂₆	—	2	1	no	—
61	C₄×Dic₂₆	—	2	1	no	—
62	C₄×Dic₂₆	C₂×C₄	1	2	yes	—
63	C₄×Dic₂₆	C₂³	1	2	yes	—
64	C₁₃×C₄⋊C₈	—	2	1	no	—
65	C₁₃×C₄⋊C₈	C₂×C₄	1	2	yes	—
66	C₁₃×C₂²⋊C₈	—	2	1	no	—
67	C₁₃×C₂²⋊C₈	C₂×C₄	1	2	yes	—
68	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
69	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
70	C₄×C₈×C₁₃	C₂³	1	2	yes	—
71	C₄×C₈×C₁₃	C₂³	1	2	yes	—
72	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
73	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
74	#88061	C₂×C₄	1	4	yes	—
75	#88060	—	2	2	no	—
76	#88060	—	2	2	no	—
77	#88057	C₂×C₄	1	4	yes	—
78	#88056	C₂×C₄	1	4	yes	—
79	#88054	—	2	2	no	—
80	#88054	—	2	2	no	—
81	C₈×Dic₁₃	—	2	2	no	—
82	C₈×Dic₁₃	—	2	2	no	—
83	C₄⋊#1055	—	2	1	no	—
84	C₄⋊#1055	—	2	1	no	—
85	C₄⋊#1055	C₂³	1	2	yes	—
86	C₄⋊#1055	C₂³	1	2	yes	—
87	#17901	—	13	1	no	maximal, Sylow
88	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
89	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
90	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
91	C₂×C₂²⋊D₁₃	—	2	2	no	—
92	C₂²×Dic₁₃	—	2	1	no	—
93	C₂²×Dic₁₃	—	2	1	no	—
94	#6127	—	4	2	no	—
95	#6127	—	4	2	no	—
96	C₂×D₅₂	—	2	1	no	—
97	C₂×D₅₂	—	2	1	no	—
98	C₂×D₅₂	C₂²×C₄	1	2	yes	—
99	C₂×C₄×D₁₃	—	2	2	no	—
100	C₂×Dic₂₆	—	2	1	no	—
101	C₂×Dic₂₆	—	2	4	no	—
102	C₂×Dic₂₆	—	2	1	no	—
103	C₂×Dic₂₆	C₂²×C₄	1	2	yes	—
104	C₂×C₈×C₁₃	—	2	2	no	—
105	C₂×C₈×C₁₃	—	2	2	no	—
106	C₂×C₈×C₁₃	SD₁₆	1	4	yes	complemented
107	C₂×C₈×C₁₃	C₂²×C₄	1	4	yes	—
108	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
109	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
110	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
111	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
112	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
113	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
114	C₄²×C₁₃	#41	1	1	yes	characteristic
115	C₄²×C₁₃	C₂⁴	1	1	yes	characteristic
116	#6103	—	2	2	no	—
117	#6103	—	2	2	no	—
118	C₄⋊Dic₁₃	—	2	1	no	—
119	C₄⋊Dic₁₃	—	2	1	no	—
120	C₄⋊Dic₁₃	C₂²×C₄	1	2	yes	—
121	C₄⋊Dic₁₃	C₂²×C₄	1	2	yes	—
122	#6101	—	2	2	no	—
123	#6101	—	2	2	no	—
124	C₄×Dic₁₃	—	2	2	no	—
125	C₄×Dic₁₃	—	2	2	no	—
126	#6099	—	4	2	no	—
127	C₂×#1055	—	2	2	no	—
128	C₂×#1055	—	2	2	no	—
129	C₂×#1055	—	4	2	no	—
130	#6095	—	2	8	no	—
131	#6094	—	4	4	no	—
132	C₈×D₁₃	—	4	4	no	—
133	#3103	—	13	1	no	—
134	C₂×C₈×Q₈	—	13	1	no	—
135	#2931	—	13	1	no	—
136	#1583	—	13	4	no	—
137	C₈×SD₁₆	—	13	4	no	—
138	#1570	—	13	1	no	—
139	#1487	—	13	1	no	—
140	#1480	—	13	1	no	—
141	#1450	D₁₃	1	1	yes	characteristic, complemented
142	C₂³×C₁₃	C₄×D₄	1	1	yes	characteristic
143	C₂²×D₁₃	—	2	2	no	—
144	C₂×C₄×C₁₃	—	2	2	no	—
145	C₂×C₄×C₁₃	—	4	1	no	—
146	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
147	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
148	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
149	C₂×C₄×C₁₃	C₂×SD₁₆	1	2	yes	—
150	C₂×C₄×C₁₃	#135	1	2	yes	—
151	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
152	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
153	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
154	C₂²⋊D₁₃	—	4	4	no	—
155	C₂×Dic₁₃	—	2	4	no	—
156	C₂×Dic₁₃	—	2	2	no	—
157	C₂×Dic₁₃	—	2	2	no	—
158	C₂×Dic₁₃	—	2	2	no	—
159	C₂×Dic₁₃	—	4	2	no	—
160	D₅₂	—	2	4	no	—
161	D₅₂	—	4	2	no	—
162	D₅₂	—	4	2	no	—
163	C₄×D₁₃	—	4	2	no	—
164	C₄×D₁₃	—	4	2	no	—
165	Dic₂₆	—	2	4	no	—
166	Dic₂₆	—	2	2	no	—
167	Dic₂₆	—	4	2	no	—
168	Dic₂₆	#131	1	4	yes	complemented
169	C₈×C₁₃	—	2	4	no	—
170	C₈×C₁₃	—	2	1	no	—
171	C₈×C₁₃	—	2	1	no	—
172	C₈×C₁₃	—	4	2	no	—
173	C₈×C₁₃	C₂×SD₁₆	1	2	yes	—
174	C₈×C₁₃	C₂×SD₁₆	1	2	yes	—
175	#1055	—	4	2	no	—
176	#1055	—	4	2	no	—
177	C₂×#131	—	26	1	no	—
178	#532	—	13	1	no	—
179	C₂×C₄×Q₈	—	13	1	no	—
180	#481	—	13	2	no	—
181	#475	—	13	2	no	—
182	#460	—	13	2	no	—
183	#448	—	13	2	no	—
184	C₈×Q₈	—	13	2	no	—
185	C₈×Q₈	—	13	2	no	—
186	C₈×Q₈	—	26	1	no	—
187	C₈×Q₈	—	26	1	no	—
188	C₄×SD₁₆	—	13	4	no	—
189	#436	—	13	2	no	—
190	#436	—	26	1	no	—
191	C₈×D₄	—	13	2	no	—
192	C₈×D₄	—	26	1	no	—
193	#431	C₂×D₁₃	1	1	yes	characteristic
194	#431	C₂×D₁₃	1	1	yes	characteristic
195	#423	—	13	1	no	—
196	C₂×C₄⋊C₈	—	13	1	no	—
197	C₂×C₄⋊C₈	—	26	1	no	—
198	#408	—	26	1	no	—
199	C₂×C₄×C₈	—	26	1	no	—
200	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
201	#334	—	13	4	no	—
202	Q₈⋊C₈	—	13	4	no	—
203	D₄⋊C₈	—	13	4	no	—
204	#322	C₂×D₁₃	1	2	yes	—
205	C₈²	C₂×D₁₃	1	2	yes	—
206	C₂²×C₁₃	—	2	1	no	—
207	C₂²×C₁₃	—	2	1	no	—
208	C₂²×C₁₃	C₄×SD₁₆	1	2	yes	—
209	C₂²×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
210	C₂×D₁₃	—	4	2	no	—
211	C₂×D₁₃	—	4	2	no	—
212	C₂×D₁₃	—	4	2	no	—
213	C₄×C₁₃	—	2	1	no	—
214	C₄×C₁₃	—	2	1	no	—
215	C₄×C₁₃	—	2	1	no	—
216	C₄×C₁₃	C₄×SD₁₆	1	2	yes	—
217	C₄×C₁₃	C₂×#131	1	2	yes	—
218	C₄×C₁₃	C₂²×SD₁₆	1	1	yes	characteristic
219	C₄×C₁₃	C₂×#135	1	1	yes	characteristic
220	Dic₁₃	—	2	4	no	—
221	Dic₁₃	—	4	2	no	—
222	Dic₁₃	—	4	2	no	—
223	C₂×#41	—	26	1	no	—
224	C₂²×Q₈	—	13	1	no	—
225	C₂×SD₁₆	—	13	4	no	—
226	#131	—	52	4	no	—
227	C₂×M₁₆	—	26	1	no	—
228	C₂×M₁₆	—	26	2	no	—
229	C₂²×C₈	—	26	1	no	—
230	C₂²×C₈	—	26	2	no	—
231	C₂²×C₈	D₅₂	1	1	yes	characteristic
232	C₂²×C₈	D₅₂	1	1	yes	characteristic
233	C₄⋊Q₈	—	13	1	no	—
234	#127	—	13	1	no	—
235	#124	—	26	1	no	—
236	#121	—	13	2	no	—
237	#121	—	26	1	no	—
238	C₄×Q₈	—	13	2	no	—
239	C₄×Q₈	—	13	2	no	—
240	C₄×Q₈	—	26	1	no	—
241	C₄×Q₈	—	26	1	no	—
242	C₄×D₄	—	13	2	no	—
243	C₄×D₄	—	26	1	no	—
244	C₂×C₄⋊C₄	—	13	1	no	—
245	C₂×C₄⋊C₄	—	26	1	no	—
246	C₂×C₄²	—	26	1	no	—
247	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
248	#106	—	13	4	no	—
249	C₄⋊C₈	—	2	1	no	—
250	C₄⋊C₈	—	13	2	no	—
251	C₄⋊C₈	—	13	2	no	—
252	C₄⋊C₈	—	26	2	no	—
253	C₄⋊C₈	—	26	2	no	—
254	C₄⋊C₈	—	26	1	no	—
255	C₄⋊C₈	—	26	1	no	—
256	C₄⋊C₈	C₄×D₁₃	1	2	yes	—
257	#103	—	13	4	no	—
258	#102	—	13	4	no	—
259	C₂²⋊C₈	—	2	1	no	—
260	C₂²⋊C₈	—	26	2	no	—
261	C₂²⋊C₈	—	26	2	no	—
262	C₂²⋊C₈	C₄×D₁₃	1	2	yes	—
263	C₄×C₈	—	26	2	no	—
264	C₄×C₈	—	26	2	no	—
265	C₄×C₈	D₅₂	1	1	yes	characteristic
266	C₄×C₈	D₅₂	1	1	yes	characteristic
267	C₄×C₈	C₂²×D₁₃	1	2	yes	—
268	C₄×C₈	C₂²×D₁₃	1	2	yes	—
269	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
270	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
271	C₂×C₁₃	—	2	2	no	—
272	C₂×C₁₃	#2929	1	1	yes	characteristic
273	C₂×C₁₃	C₂×C₄×SD₁₆	1	1	yes	characteristic
274	C₂×C₁₃	#2955	1	1	yes	characteristic
275	D₁₃	—	8	2	no	—
276	#41	—	52	2	no	—
277	#41	—	52	2	no	—
278	C₂×Q₈	—	13	2	no	—
279	C₂×Q₈	—	26	4	no	—
280	C₂×Q₈	—	26	1	no	—
281	C₂×Q₈	—	26	1	no	—
282	C₂×D₄	—	13	2	no	—
283	C₂×D₄	—	26	2	no	—
284	C₂×D₄	—	26	1	no	—
285	C₂×D₄	—	26	1	no	—
286	C₂²×C₄	—	26	1	no	—
287	C₂²×C₄	—	26	1	no	—
288	C₂²×C₄	—	26	2	no	—
289	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
290	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
291	C₂²×C₄	#6127	1	1	yes	characteristic
292	SD₁₆	—	26	8	no	—
293	M₁₆	—	52	2	no	—
294	M₁₆	—	52	4	no	—
295	C₂×C₈	—	2	2	no	—
296	C₂×C₈	—	2	2	no	—
297	C₂×C₈	—	26	2	no	—
298	C₂×C₈	—	26	2	no	—
299	C₂×C₈	—	52	2	no	—
300	C₂×C₈	—	52	4	no	—
301	C₂×C₈	#6095	1	4	yes	complemented
302	C₂×C₈	C₂×C₄×D₁₃	1	4	yes	—
303	C₂×C₈	C₂×D₅₂	1	1	yes	characteristic
304	C₂×C₈	C₂×D₅₂	1	1	yes	characteristic
305	C₂×C₈	C₂×D₅₂	1	1	yes	characteristic
306	C₂×C₈	C₂×D₅₂	1	1	yes	characteristic
307	C₄⋊C₄	—	13	2	no	—
308	C₄⋊C₄	—	13	2	no	—
309	C₄⋊C₄	—	26	1	no	—
310	C₄⋊C₄	—	26	2	no	—
311	C₄⋊C₄	—	26	1	no	—
312	C₄⋊C₄	—	26	2	no	—
313	C₂²⋊C₄	—	26	2	no	—
314	C₂²⋊C₄	—	26	2	no	—
315	C₄²	—	26	2	no	—
316	C₄²	—	26	2	no	—
317	C₄²	C₂×C₄×D₁₃	1	1	yes	characteristic
318	C₄²	C₂×D₅₂	1	1	yes	characteristic
319	C₄²	#6127	1	1	yes	characteristic
320	C₄²	C₂³×D₁₃	1	1	yes	characteristic
321	C₁₃	#17901	1	1	yes	characteristic, complemented, Sylow
322	C₂³	—	26	2	no	—
323	C₂³	C₄×D₅₂	1	1	yes	characteristic
324	Q₈	—	13	4	no	—
325	Q₈	—	26	4	no	—
326	Q₈	—	26	2	no	—
327	Q₈	—	52	2	no	—
328	D₄	—	26	4	no	—
329	D₄	—	52	2	no	—
330	D₄	—	52	4	no	—
331	D₄	—	52	2	no	—
332	C₂×C₄	—	2	2	no	—
333	C₂×C₄	—	4	1	no	—
334	C₂×C₄	—	26	4	no	—
335	C₂×C₄	—	26	2	no	—
336	C₂×C₄	—	26	2	no	—
337	C₂×C₄	—	26	2	no	—
338	C₂×C₄	—	52	2	no	—
339	C₂×C₄	—	52	2	no	—
340	C₂×C₄	—	52	2	no	—
341	C₂×C₄	C₄×D₅₂	1	1	yes	characteristic
342	C₂×C₄	C₄×D₅₂	1	1	yes	characteristic
343	C₂×C₄	C₄×D₅₂	1	1	yes	characteristic
344	C₂×C₄	C₂×#6095	1	2	yes	—
345	C₂×C₄	#88158	1	2	yes	—
346	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
347	C₂×C₄	C₂²×D₅₂	1	1	yes	characteristic
348	C₂×C₄	C₂×#6127	1	1	yes	characteristic
349	C₈	—	2	1	no	—
350	C₈	—	2	1	no	—
351	C₈	—	2	4	no	—
352	C₈	—	4	2	no	—
353	C₈	—	52	2	no	—
354	C₈	—	52	2	no	—
355	C₈	C₂×#6095	1	2	yes	—
356	C₈	C₂×#6095	1	2	yes	—
357	C₂²	—	2	1	no	—
358	C₂²	—	2	1	no	—
359	C₂²	—	52	2	no	—
360	C₂²	—	52	2	no	—
361	C₂²	—	52	2	no	—
362	C₂²	C₄×#6095	1	2	yes	—
363	C₂²	C₂×C₄×D₅₂	1	1	yes	characteristic
364	C₄	—	2	1	no	—
365	C₄	—	2	1	no	—
366	C₄	—	2	1	no	—
367	C₄	—	26	4	no	—
368	C₄	—	52	2	no	—
369	C₄	—	52	2	no	—
370	C₄	C₄×#6095	1	2	yes	—
371	C₄	C₂²×#6095	1	1	yes	characteristic
372	C₄	C₂×#88158	1	1	yes	characteristic
373	C₄	D₁₃×#131	1	2	yes	—
374	C₂	—	2	2	no	—
375	C₂	—	104	2	no	—
376	C₂	#13123532	1	1	yes	characteristic
377	C₂	C₂×C₄×#6095	1	1	yes	characteristic
378	C₂	#13123740	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂×D₅₂
commutator	C₂×C₄×C₁₃	C₂³×C₄
Frattini	C₄²	C₂³×D₁₃
Fitting	C₁₃×#1450	C₂
socle	C₂²×C₁₃	C₂×C₄×D₄
prime radical	#11719182	C₂²
cosocle	C₄²×C₁₃	C₂⁴
hypercenter	#1450	D₁₃
hypocenter	C₁₃	#17901
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ #1450

Complements

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

N	H ≅ G/N	type
2. C₂×C₈×Dic₂₆	C₂	semidirect
4. #13113681	C₂	semidirect
5. C₈×#6095	C₂	semidirect
6. #13113607	C₂	semidirect
7. #13113266	C₂	semidirect
9. C₁₃×#1450	C₂	semidirect
106. C₂×C₈×C₁₃	SD₁₆	semidirect
141. #1450	D₁₃	semidirect
168. Dic₂₆	#131	semidirect
301. C₂×C₈	#6095	semidirect
321. C₁₃	#17901	semidirect

Maximal subgroups

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

subgroup	index
#13124868	2
C₂×C₈×Dic₂₆	2
#13123549	2
#13113681	2
C₈×#6095	2
#13113607	2
#13113266	2
#13113236	2
C₁₃×#1450	2
#17901	13