Subgroups — #16785715

Order 3328.

982 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#88102	C₂	1	2	yes	maximal
2	C₂×#11718265	C₂	1	2	yes	maximal
3	#13130014	C₂	1	2	yes	maximal
4	C₂×C₄×C₈×D₁₃	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₄⋊#261	C₂²	1	4	yes	—
6	C₂²×C₈×D₁₃	C₂²	1	1	yes	characteristic
7	C₂²×C₈×D₁₃	C₂²	1	1	yes	characteristic
8	C₂×C₄²×D₁₃	C₂²	1	1	yes	characteristic
9	C₂×(C₂×#261)⋊C₄	—	2	4	no	—
10	C₂×#88102	—	2	4	no	—
11	C₂×#88102	—	2	4	no	—
12	C₂×#88102	C₄	1	8	yes	complemented
13	C₂×C₄×C₈×C₁₃	C₄	1	1	yes	characteristic, complemented
14	C₂×C₈×Dic₁₃	C₄	1	1	yes	characteristic, complemented
15	C₂×C₈×Dic₁₃	C₄	1	1	yes	characteristic, complemented
16	C₂×C₄×#1055	C₄	1	1	yes	characteristic, complemented
17	C₄×C₈×D₁₃	—	2	4	no	—
18	#11718265	—	2	4	no	—
19	C₂³×#261	—	2	4	no	—
20	C₂²×C₄×D₁₃	D₄	1	2	yes	—
21	C₂²×C₄×D₁₃	C₂³	1	1	yes	characteristic
22	C₂×C₄⋊#261	—	2	4	no	—
23	C₂×C₄⋊#261	—	2	4	no	—
24	C₂×C₄⋊#261	—	2	4	no	—
25	C₂×C₄⋊#261	—	2	4	no	—
26	C₂×C₄⋊#261	—	2	4	no	—
27	C₂×C₄⋊#261	C₂×C₄	1	8	yes	—
28	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄×Dic₁₃	Q₈	1	2	yes	—
33	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
34	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
35	C₂×C₈×D₁₃	—	2	2	no	—
36	C₂×C₈×D₁₃	—	2	2	no	—
37	C₂×C₈×D₁₃	C₂×C₄	1	4	yes	—
38	C₂×C₈×D₁₃	D₄	1	2	yes	—
39	C₂×C₈×D₁₃	D₄	1	2	yes	—
40	C₂×C₈×D₁₃	D₄	1	4	yes	—
41	C₂×C₈×D₁₃	Q₈	1	2	yes	—
42	C₂×C₈×D₁₃	Q₈	1	2	yes	—
43	C₄²×D₁₃	—	2	2	no	—
44	C₄²×D₁₃	—	2	2	no	—
45	(C₂×#261)⋊C₄	—	4	8	no	—
46	#88102	—	2	16	no	—
47	C₄×C₈×C₁₃	—	2	2	no	—
48	C₈×Dic₁₃	—	2	2	no	—
49	C₈×Dic₁₃	—	2	2	no	—
50	C₄×#1055	—	2	2	no	—
51	#10862	—	13	1	no	maximal, Sylow
52	C₂³×D₁₃	C₂×D₄	1	1	yes	characteristic
53	C₂²×#261	—	2	8	no	—
54	C₂²×#261	—	4	4	no	—
55	C₂²×#261	—	4	4	no	—
56	C₂²×#261	—	4	8	no	—
57	C₂²×#261	—	4	8	no	—
58	C₂²×C₄×C₁₃	C₄⋊C₄	1	2	yes	—
59	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
60	C₂²×Dic₁₃	C₄⋊C₄	1	2	yes	—
61	C₂²×Dic₁₃	C₂×Q₈	1	1	yes	characteristic
62	C₂×C₄×D₁₃	—	2	4	no	—
63	C₂×C₄×D₁₃	—	2	4	no	—
64	C₂×C₄×D₁₃	D₈	1	4	yes	—
65	C₂×C₄×D₁₃	Q₁₆	1	4	yes	—
66	C₂×C₄×D₁₃	C₂²×C₄	1	2	yes	—
67	C₂×C₄×D₁₃	C₂×D₄	1	1	yes	characteristic
68	C₂×C₄×D₁₃	C₂×D₄	1	2	yes	—
69	C₂×C₄×D₁₃	C₂×D₄	1	1	yes	characteristic
70	C₂×C₄×D₁₃	C₂×Q₈	1	1	yes	characteristic
71	C₂×C₄×D₁₃	C₂×Q₈	1	1	yes	characteristic
72	C₂×C₄×D₁₃	#41	1	2	yes	—
73	C₂×C₄×D₁₃	#41	1	2	yes	—
74	C₄⋊#261	—	2	8	no	—
75	C₄⋊#261	—	2	8	no	—
76	C₄⋊#261	—	4	4	no	—
77	C₄⋊#261	—	4	4	no	—
78	C₂×C₈×C₁₃	—	2	1	no	—
79	C₂×C₈×C₁₃	—	2	1	no	—
80	C₂×C₈×C₁₃	C₄²	1	2	yes	—
81	C₂×C₈×C₁₃	C₂²⋊C₄	1	2	yes	—
82	C₂×C₈×C₁₃	C₄⋊C₄	1	2	yes	—
83	C₂×C₈×C₁₃	C₄⋊C₄	1	2	yes	—
84	C₄²×C₁₃	—	2	1	no	—
85	C₄²×C₁₃	—	2	1	no	—
86	C₄×Dic₁₃	—	2	1	no	—
87	C₄×Dic₁₃	—	2	1	no	—
88	C₄×Dic₁₃	—	2	4	no	—
89	C₂×#1055	—	2	1	no	—
90	C₂×#1055	—	2	1	no	—
91	C₂×#1055	C₄²	1	2	yes	—
92	C₂×#1055	C₂²⋊C₄	1	2	yes	—
93	C₂×#1055	C₄⋊C₄	1	2	yes	—
94	C₂×#1055	C₄⋊C₄	1	2	yes	—
95	C₈×D₁₃	—	2	4	no	—
96	C₈×D₁₃	—	2	4	no	—
97	C₈×D₁₃	C₂²⋊C₄	1	8	yes	—
98	C₈×D₁₃	C₄⋊C₄	1	8	yes	—
99	C₂²×#107	—	13	2	no	—
100	C₂²×C₄×C₈	—	13	1	no	—
101	C₂×#340	—	13	2	no	—
102	C₄⋊#95	—	13	2	no	—
103	C₂³×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
104	C₂²×D₁₃	C₄×D₄	1	2	yes	—
105	C₂²×D₁₃	#121	1	2	yes	—
106	C₂²×D₁₃	#127	1	1	yes	characteristic
107	C₂²×D₁₃	C₄⋊Q₈	1	1	yes	characteristic
108	C₂²×D₁₃	C₂×D₈	1	2	yes	—
109	C₂²×D₁₃	C₂×Q₁₆	1	2	yes	—
110	C₂²×D₁₃	#135	1	4	yes	—
111	C₂×#261	—	4	8	no	—
112	C₂×#261	—	4	16	no	—
113	C₂×#261	—	8	8	no	—
114	C₂×#261	—	8	4	no	—
115	C₂×#261	—	8	4	no	—
116	C₂×C₄×C₁₃	—	2	2	no	—
117	C₂×C₄×C₁₃	—	2	2	no	—
118	C₂×C₄×C₁₃	#107	1	4	yes	complemented
119	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
120	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
121	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
122	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
123	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
124	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
125	C₂×Dic₁₃	—	2	2	no	—
126	C₂×Dic₁₃	—	2	2	no	—
127	C₂×Dic₁₃	#107	1	4	yes	complemented
128	C₂×Dic₁₃	C₄×Q₈	1	2	yes	—
129	C₂×Dic₁₃	C₂²⋊Q₈	1	2	yes	—
130	C₂×Dic₁₃	C₄⋊Q₈	1	1	yes	characteristic
131	C₂×Dic₁₃	C₄⋊Q₈	1	1	yes	characteristic
132	C₄×D₁₃	—	2	8	no	—
133	C₄×D₁₃	—	2	8	no	—
134	C₄×D₁₃	C₂×C₂²⋊C₄	1	2	yes	—
135	C₄×D₁₃	C₂×C₄⋊C₄	1	2	yes	—
136	C₄×D₁₃	C₄×D₄	1	2	yes	—
137	C₄×D₁₃	C₄×Q₈	1	2	yes	—
138	C₄×D₁₃	#121	1	2	yes	—
139	C₄×D₁₃	C₂²⋊Q₈	1	2	yes	—
140	C₄×D₁₃	#124	1	2	yes	—
141	C₄×D₁₃	#125	1	2	yes	—
142	C₈×C₁₃	—	2	2	no	—
143	C₈×C₁₃	#95	1	4	yes	complemented
144	#1055	—	2	2	no	—
145	#1055	#95	1	4	yes	complemented
146	C₂³×C₈	—	13	1	no	—
147	C₂³×C₈	—	13	1	no	—
148	C₂²×C₄⋊C₄	—	13	4	no	—
149	C₂²×C₄²	—	13	1	no	—
150	C₂×#107	—	13	8	no	—
151	C₂×#107	—	26	4	no	—
152	C₂×#107	—	26	4	no	—
153	C₂×C₄×C₈	—	13	1	no	—
154	C₂×C₄×C₈	—	13	1	no	—
155	C₂×C₄×C₈	—	13	1	no	—
156	C₂×C₄×C₈	—	26	4	no	—
157	C₂×C₄×C₈	#261	1	1	yes	characteristic, complemented
158	C₂×#95	—	26	4	no	—
159	#340	—	26	4	no	—
160	C₂²×C₁₃	C₄×C₄⋊C₄	1	1	yes	characteristic
161	C₂²×C₁₃	#380	1	1	yes	characteristic
162	C₂²×C₁₃	#384	1	1	yes	characteristic
163	C₂²×C₁₃	C₂×#107	1	2	yes	—
164	C₂²×C₁₃	#427	1	2	yes	—
165	C₂×D₁₃	#389	1	2	yes	—
166	C₂×D₁₃	#390	1	2	yes	—
167	C₂×D₁₃	C₄×D₈	1	4	yes	—
168	C₂×D₁₃	C₄×Q₁₆	1	4	yes	—
169	C₂×D₁₃	#466	1	4	yes	—
170	C₂×D₁₃	#467	1	4	yes	—
171	C₂×D₁₃	#493	1	1	yes	characteristic
172	C₂×D₁₃	#494	1	1	yes	characteristic
173	C₂×D₁₃	#495	1	2	yes	—
174	C₂×D₁₃	#499	1	2	yes	—
175	C₂×D₁₃	#500	1	2	yes	—
176	#261	—	8	8	no	—
177	C₄×C₁₃	—	2	4	no	—
178	C₄×C₁₃	C₂×#95	1	1	yes	characteristic
179	C₄×C₁₃	C₄×C₄⋊C₄	1	1	yes	characteristic
180	C₄×C₁₃	#380	1	1	yes	characteristic
181	C₄×C₁₃	#382	1	1	yes	characteristic
182	Dic₁₃	—	2	4	no	—
183	Dic₁₃	#389	1	2	yes	—
184	Dic₁₃	#391	1	2	yes	—
185	C₂³×C₄	—	13	1	no	—
186	C₂³×C₄	—	13	2	no	—
187	C₂³×C₄	—	26	4	no	—
188	C₂²×C₈	—	13	1	no	—
189	C₂²×C₈	—	13	4	no	—
190	C₂²×C₈	—	13	2	no	—
191	C₂²×C₈	—	13	2	no	—
192	C₂²×C₈	—	13	4	no	—
193	C₂²×C₈	—	13	1	no	—
194	C₂²×C₈	—	13	2	no	—
195	C₂²×C₈	—	13	2	no	—
196	C₂²×C₈	—	26	2	no	—
197	C₂²×C₈	—	26	2	no	—
198	C₂²×C₈	C₂×#261	1	1	yes	characteristic
199	C₂²×C₈	C₂×#261	1	1	yes	characteristic
200	C₂×C₄⋊C₄	—	13	8	no	—
201	C₂×C₄⋊C₄	—	26	4	no	—
202	C₂×C₄⋊C₄	—	26	4	no	—
203	C₂×C₄⋊C₄	—	26	4	no	—
204	C₂×C₄⋊C₄	—	26	4	no	—
205	C₂×C₄⋊C₄	—	26	4	no	—
206	C₂×C₄²	—	13	2	no	—
207	C₂×C₄²	—	13	1	no	—
208	C₂×C₄²	—	26	2	no	—
209	C₂×C₄²	—	26	2	no	—
210	C₂×C₄²	C₂×#261	1	1	yes	characteristic
211	#107	—	26	16	no	—
212	C₄×C₈	—	2	2	no	—
213	C₄×C₈	—	26	2	no	—
214	C₄×C₈	—	26	2	no	—
215	C₄×C₈	—	26	2	no	—
216	#95	—	52	8	no	—
217	C₂×C₁₃	C₄⋊#95	1	1	yes	characteristic
218	C₂×C₁₃	C₄×#107	1	2	yes	—
219	C₂×C₁₃	#1808	1	2	yes	—
220	C₂×C₁₃	#1845	1	1	yes	characteristic
221	C₂×C₁₃	#1846	1	1	yes	characteristic
222	D₁₃	#1942	1	4	yes	—
223	D₁₃	#1969	1	2	yes	—
224	D₁₃	#1970	1	2	yes	—
225	C₂⁴	—	13	1	no	—
226	C₂²×C₄	—	13	1	no	—
227	C₂²×C₄	—	13	2	no	—
228	C₂²×C₄	—	13	1	no	—
229	C₂²×C₄	—	13	1	no	—
230	C₂²×C₄	—	13	1	no	—
231	C₂²×C₄	—	13	1	no	—
232	C₂²×C₄	—	13	2	no	—
233	C₂²×C₄	—	13	2	no	—
234	C₂²×C₄	—	13	2	no	—
235	C₂²×C₄	—	13	2	no	—
236	C₂²×C₄	—	13	4	no	—
237	C₂²×C₄	—	13	4	no	—
238	C₂²×C₄	—	26	4	no	—
239	C₂²×C₄	—	26	4	no	—
240	C₂²×C₄	—	26	8	no	—
241	C₂²×C₄	—	52	8	no	—
242	C₂²×C₄	—	52	4	no	—
243	C₂²×C₄	—	52	4	no	—
244	C₂²×C₄	—	52	8	no	—
245	C₂²×C₄	C₄⋊#261	1	2	yes	—
246	C₂²×C₄	C₂²×#261	1	1	yes	characteristic
247	C₂×C₈	—	2	1	no	—
248	C₂×C₈	—	2	1	no	—
249	C₂×C₈	—	13	2	no	—
250	C₂×C₈	—	13	2	no	—
251	C₂×C₈	—	13	2	no	—
252	C₂×C₈	—	13	8	no	—
253	C₂×C₈	—	13	8	no	—
254	C₂×C₈	—	13	2	no	—
255	C₂×C₈	—	26	4	no	—
256	C₂×C₈	—	26	4	no	—
257	C₂×C₈	—	26	1	no	—
258	C₂×C₈	—	26	1	no	—
259	C₂×C₈	C₄×#261	1	2	yes	—
260	C₂×C₈	C₄⋊#261	1	2	yes	—
261	C₂×C₈	C₄⋊#261	1	2	yes	—
262	C₂×C₈	C₂²⋊#261	1	2	yes	—
263	C₄⋊C₄	—	26	8	no	—
264	C₄⋊C₄	—	26	8	no	—
265	C₄⋊C₄	—	52	4	no	—
266	C₄⋊C₄	—	52	4	no	—
267	C₄²	—	2	1	no	—
268	C₄²	—	2	1	no	—
269	C₄²	—	26	4	no	—
270	C₄²	—	26	1	no	—
271	C₄²	—	26	1	no	—
272	C₁₃	#10862	1	1	yes	characteristic, complemented, Sylow
273	C₂³	—	13	2	no	—
274	C₂³	—	13	1	no	—
275	C₂³	—	13	1	no	—
276	C₂³	—	13	2	no	—
277	C₂³	—	13	2	no	—
278	C₂³	—	13	4	no	—
279	C₂³	—	13	2	no	—
280	C₂³	C₂×C₄⋊#261	1	1	yes	characteristic
281	C₂×C₄	—	2	2	no	—
282	C₂×C₄	—	2	2	no	—
283	C₂×C₄	—	13	2	no	—
284	C₂×C₄	—	13	1	no	—
285	C₂×C₄	—	13	1	no	—
286	C₂×C₄	—	13	2	no	—
287	C₂×C₄	—	13	2	no	—
288	C₂×C₄	—	13	2	no	—
289	C₂×C₄	—	13	2	no	—
290	C₂×C₄	—	13	2	no	—
291	C₂×C₄	—	13	2	no	—
292	C₂×C₄	—	13	2	no	—
293	C₂×C₄	—	13	2	no	—
294	C₂×C₄	—	13	2	no	—
295	C₂×C₄	—	13	4	no	—
296	C₂×C₄	—	26	8	no	—
297	C₂×C₄	—	26	8	no	—
298	C₂×C₄	—	26	2	no	—
299	C₂×C₄	—	26	2	no	—
300	C₂×C₄	—	52	8	no	—
301	C₂×C₄	—	52	16	no	—
302	C₂×C₄	—	104	4	no	—
303	C₂×C₄	—	104	4	no	—
304	C₂×C₄	—	104	8	no	—
305	C₂×C₄	#88102	1	4	yes	complemented
306	C₂×C₄	C₂×C₄×#261	1	1	yes	characteristic
307	C₂×C₄	C₂×C₄⋊#261	1	1	yes	characteristic
308	C₂×C₄	C₂×C₄⋊#261	1	1	yes	characteristic
309	C₂×C₄	#88237	1	1	yes	characteristic
310	C₂×C₄	#88237	1	1	yes	characteristic
311	C₂×C₄	C₂×C₂²⋊#261	1	1	yes	characteristic
312	C₈	—	2	2	no	—
313	C₈	—	13	4	no	—
314	C₈	—	26	2	no	—
315	C₈	(C₂×#261)⋊C₄	1	4	yes	complemented
316	C₂²	—	13	1	no	—
317	C₂²	—	13	1	no	—
318	C₂²	—	13	2	no	—
319	C₂²	—	13	2	no	—
320	C₂²	—	13	2	no	—
321	C₂²	—	13	2	no	—
322	C₂²	—	13	2	no	—
323	C₂²	—	13	4	no	—
324	C₂²	—	13	4	no	—
325	C₂²	—	13	4	no	—
326	C₂²	—	13	4	no	—
327	C₂²	C₄×C₄⋊#261	1	1	yes	characteristic
328	C₂²	#11719059	1	1	yes	characteristic
329	C₂²	C₂×#88102	1	2	yes	—
330	C₂²	#11719092	1	2	yes	—
331	C₂²	#11719136	1	1	yes	characteristic
332	C₄	—	2	4	no	—
333	C₄	—	13	2	no	—
334	C₄	—	13	2	no	—
335	C₄	—	26	4	no	—
336	C₄	—	104	8	no	—
337	C₄	C₄×C₄⋊#261	1	1	yes	characteristic
338	C₄	#11719060	1	1	yes	characteristic
339	C₄	C₂×(C₂×#261)⋊C₄	1	1	yes	characteristic
340	C₄	#11719136	1	1	yes	characteristic
341	C₂	—	13	2	no	—
342	C₂	—	13	4	no	—
343	C₂	—	13	2	no	—
344	C₂	#13130014	1	1	yes	characteristic
345	C₂	C₄×#88102	1	2	yes	—
346	C₂	#13130713	1	2	yes	—
347	C₂	#13130772	1	1	yes	characteristic
348	C₂	#13130773	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×C₄⋊#261
commutator	C₂×C₄×C₁₃	C₂×C₄²
Frattini	C₂²×C₄	C₂²×#261
Fitting	C₂×C₄×C₈×C₁₃	C₄
socle	C₂³×C₁₃	C₂×C₄⋊C₄
prime radical	C₂³×D₁₃	C₂×D₄
cosocle	C₂²×C₄×D₁₃	C₂³
hypercenter	C₂×C₄×C₈	#261
hypocenter	C₁₃	#10862
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
12. C₂×#88102	C₄	semidirect
13. C₂×C₄×C₈×C₁₃	C₄	semidirect
14. C₂×C₈×Dic₁₃	C₄	semidirect
15. C₂×C₈×Dic₁₃	C₄	semidirect
16. C₂×C₄×#1055	C₄	semidirect
118. C₂×C₄×C₁₃	#107	semidirect
127. C₂×Dic₁₃	#107	semidirect
143. C₈×C₁₃	#95	semidirect
145. #1055	#95	semidirect
157. C₂×C₄×C₈	#261	semidirect
272. C₁₃	#10862	semidirect
305. C₂×C₄	#88102	semidirect
315. C₈	(C₂×#261)⋊C₄	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#88102	2
C₂×#11718265	2
#13130014	2
C₂×C₄×C₈×D₁₃	2
#10862	13