Subgroups — #16785713

Order 3328.

982 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#88101	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×#88101	—	2	4	no	—
11	C₂×#88101	—	2	4	no	—
12	C₂×#88101	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	#88101	—	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	#10861	—	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	SD₁₆	1	8	yes	—
65	C₂×C₄×D₁₃	C₂²×C₄	1	2	yes	—
66	C₂×C₄×D₁₃	C₂×D₄	1	1	yes	characteristic
67	C₂×C₄×D₁₃	C₂×D₄	1	2	yes	—
68	C₂×C₄×D₁₃	C₂×D₄	1	1	yes	characteristic
69	C₂×C₄×D₁₃	C₂×Q₈	1	1	yes	characteristic
70	C₂×C₄×D₁₃	C₂×Q₈	1	1	yes	characteristic
71	C₂×C₄×D₁₃	#41	1	2	yes	—
72	C₂×C₄×D₁₃	#41	1	2	yes	—
73	C₄⋊#261	—	2	8	no	—
74	C₄⋊#261	—	2	8	no	—
75	C₄⋊#261	—	4	4	no	—
76	C₄⋊#261	—	4	4	no	—
77	C₂×C₈×C₁₃	—	2	1	no	—
78	C₂×C₈×C₁₃	—	2	1	no	—
79	C₂×C₈×C₁₃	C₄²	1	2	yes	—
80	C₂×C₈×C₁₃	C₂²⋊C₄	1	2	yes	—
81	C₂×C₈×C₁₃	C₄⋊C₄	1	2	yes	—
82	C₂×C₈×C₁₃	C₄⋊C₄	1	2	yes	—
83	C₄²×C₁₃	—	2	1	no	—
84	C₄²×C₁₃	—	2	1	no	—
85	C₄×Dic₁₃	—	2	1	no	—
86	C₄×Dic₁₃	—	2	1	no	—
87	C₄×Dic₁₃	—	2	4	no	—
88	C₂×#1055	—	2	1	no	—
89	C₂×#1055	—	2	1	no	—
90	C₂×#1055	C₄²	1	2	yes	—
91	C₂×#1055	C₂²⋊C₄	1	2	yes	—
92	C₂×#1055	C₄⋊C₄	1	2	yes	—
93	C₂×#1055	C₄⋊C₄	1	2	yes	—
94	C₈×D₁₃	—	2	4	no	—
95	C₈×D₁₃	—	2	4	no	—
96	C₈×D₁₃	C₂²⋊C₄	1	8	yes	—
97	C₈×D₁₃	C₄⋊C₄	1	8	yes	—
98	C₂²×#106	—	13	2	no	—
99	C₂²×C₄×C₈	—	13	1	no	—
100	C₂×#340	—	13	2	no	—
101	C₄⋊#95	—	13	2	no	—
102	C₂³×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
103	C₂²×D₁₃	C₄×D₄	1	2	yes	—
104	C₂²×D₁₃	#121	1	2	yes	—
105	C₂²×D₁₃	#127	1	1	yes	characteristic
106	C₂²×D₁₃	C₄⋊Q₈	1	1	yes	characteristic
107	C₂²×D₁₃	C₂×SD₁₆	1	4	yes	—
108	C₂²×D₁₃	#135	1	4	yes	—
109	C₂×#261	—	4	8	no	—
110	C₂×#261	—	4	16	no	—
111	C₂×#261	—	8	8	no	—
112	C₂×#261	—	8	4	no	—
113	C₂×#261	—	8	4	no	—
114	C₂×C₄×C₁₃	—	2	2	no	—
115	C₂×C₄×C₁₃	—	2	2	no	—
116	C₂×C₄×C₁₃	#106	1	4	yes	complemented
117	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
118	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
119	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
120	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
121	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
122	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
123	C₂×Dic₁₃	—	2	2	no	—
124	C₂×Dic₁₃	—	2	2	no	—
125	C₂×Dic₁₃	#106	1	4	yes	complemented
126	C₂×Dic₁₃	C₄×Q₈	1	2	yes	—
127	C₂×Dic₁₃	C₂²⋊Q₈	1	2	yes	—
128	C₂×Dic₁₃	C₄⋊Q₈	1	1	yes	characteristic
129	C₂×Dic₁₃	C₄⋊Q₈	1	1	yes	characteristic
130	C₄×D₁₃	—	2	16	no	—
131	C₄×D₁₃	C₂×C₂²⋊C₄	1	2	yes	—
132	C₄×D₁₃	C₂×C₄⋊C₄	1	2	yes	—
133	C₄×D₁₃	C₄×D₄	1	2	yes	—
134	C₄×D₁₃	C₄×Q₈	1	2	yes	—
135	C₄×D₁₃	#121	1	2	yes	—
136	C₄×D₁₃	C₂²⋊Q₈	1	2	yes	—
137	C₄×D₁₃	#124	1	2	yes	—
138	C₄×D₁₃	#125	1	2	yes	—
139	C₈×C₁₃	—	2	2	no	—
140	C₈×C₁₃	#95	1	4	yes	complemented
141	#1055	—	2	2	no	—
142	#1055	#95	1	4	yes	complemented
143	C₂³×C₈	—	13	1	no	—
144	C₂³×C₈	—	13	1	no	—
145	C₂²×C₄⋊C₄	—	13	4	no	—
146	C₂²×C₄²	—	13	1	no	—
147	C₂×#106	—	13	8	no	—
148	C₂×#106	—	26	4	no	—
149	C₂×#106	—	26	4	no	—
150	C₂×C₄×C₈	—	13	1	no	—
151	C₂×C₄×C₈	—	13	1	no	—
152	C₂×C₄×C₈	—	13	1	no	—
153	C₂×C₄×C₈	—	26	4	no	—
154	C₂×C₄×C₈	#261	1	1	yes	characteristic, complemented
155	C₂×#95	—	26	4	no	—
156	#340	—	26	4	no	—
157	C₂²×C₁₃	C₄×C₄⋊C₄	1	1	yes	characteristic
158	C₂²×C₁₃	#380	1	1	yes	characteristic
159	C₂²×C₁₃	#384	1	1	yes	characteristic
160	C₂²×C₁₃	C₂×#106	1	2	yes	—
161	C₂²×C₁₃	#427	1	2	yes	—
162	C₂×D₁₃	#389	1	2	yes	—
163	C₂×D₁₃	#390	1	2	yes	—
164	C₂×D₁₃	C₄×SD₁₆	1	8	yes	—
165	C₂×D₁₃	#465	1	8	yes	—
166	C₂×D₁₃	#492	1	2	yes	—
167	C₂×D₁₃	#495	1	2	yes	—
168	C₂×D₁₃	#498	1	2	yes	—
169	C₂×D₁₃	#499	1	2	yes	—
170	#261	—	8	8	no	—
171	C₄×C₁₃	—	2	4	no	—
172	C₄×C₁₃	C₂×#95	1	1	yes	characteristic
173	C₄×C₁₃	C₄×C₄⋊C₄	1	1	yes	characteristic
174	C₄×C₁₃	#380	1	1	yes	characteristic
175	C₄×C₁₃	#382	1	1	yes	characteristic
176	Dic₁₃	—	2	4	no	—
177	Dic₁₃	#389	1	2	yes	—
178	Dic₁₃	#391	1	2	yes	—
179	C₂³×C₄	—	13	1	no	—
180	C₂³×C₄	—	13	2	no	—
181	C₂³×C₄	—	26	4	no	—
182	C₂²×C₈	—	13	1	no	—
183	C₂²×C₈	—	13	4	no	—
184	C₂²×C₈	—	13	2	no	—
185	C₂²×C₈	—	13	2	no	—
186	C₂²×C₈	—	13	4	no	—
187	C₂²×C₈	—	13	1	no	—
188	C₂²×C₈	—	13	2	no	—
189	C₂²×C₈	—	13	2	no	—
190	C₂²×C₈	—	26	2	no	—
191	C₂²×C₈	—	26	2	no	—
192	C₂²×C₈	C₂×#261	1	1	yes	characteristic
193	C₂²×C₈	C₂×#261	1	1	yes	characteristic
194	C₂×C₄⋊C₄	—	13	8	no	—
195	C₂×C₄⋊C₄	—	26	4	no	—
196	C₂×C₄⋊C₄	—	26	4	no	—
197	C₂×C₄⋊C₄	—	26	4	no	—
198	C₂×C₄⋊C₄	—	26	4	no	—
199	C₂×C₄⋊C₄	—	26	4	no	—
200	C₂×C₄²	—	13	2	no	—
201	C₂×C₄²	—	13	1	no	—
202	C₂×C₄²	—	26	2	no	—
203	C₂×C₄²	—	26	2	no	—
204	C₂×C₄²	C₂×#261	1	1	yes	characteristic
205	#106	—	26	16	no	—
206	C₄×C₈	—	2	2	no	—
207	C₄×C₈	—	26	2	no	—
208	C₄×C₈	—	26	2	no	—
209	C₄×C₈	—	26	2	no	—
210	#95	—	52	8	no	—
211	C₂×C₁₃	C₄⋊#95	1	1	yes	characteristic
212	C₂×C₁₃	C₄×#106	1	2	yes	—
213	C₂×C₁₃	#1807	1	2	yes	—
214	C₂×C₁₃	#1844	1	1	yes	characteristic
215	C₂×C₁₃	#1846	1	1	yes	characteristic
216	D₁₃	#1941	1	4	yes	—
217	D₁₃	#1968	1	4	yes	—
218	C₂⁴	—	13	1	no	—
219	C₂²×C₄	—	13	1	no	—
220	C₂²×C₄	—	13	2	no	—
221	C₂²×C₄	—	13	1	no	—
222	C₂²×C₄	—	13	1	no	—
223	C₂²×C₄	—	13	1	no	—
224	C₂²×C₄	—	13	1	no	—
225	C₂²×C₄	—	13	2	no	—
226	C₂²×C₄	—	13	2	no	—
227	C₂²×C₄	—	13	2	no	—
228	C₂²×C₄	—	13	2	no	—
229	C₂²×C₄	—	13	8	no	—
230	C₂²×C₄	—	26	4	no	—
231	C₂²×C₄	—	26	4	no	—
232	C₂²×C₄	—	26	8	no	—
233	C₂²×C₄	—	52	8	no	—
234	C₂²×C₄	—	52	4	no	—
235	C₂²×C₄	—	52	4	no	—
236	C₂²×C₄	—	52	8	no	—
237	C₂²×C₄	C₄⋊#261	1	2	yes	—
238	C₂²×C₄	C₂²×#261	1	1	yes	characteristic
239	C₂×C₈	—	2	1	no	—
240	C₂×C₈	—	2	1	no	—
241	C₂×C₈	—	13	2	no	—
242	C₂×C₈	—	13	2	no	—
243	C₂×C₈	—	13	2	no	—
244	C₂×C₈	—	13	8	no	—
245	C₂×C₈	—	13	8	no	—
246	C₂×C₈	—	13	2	no	—
247	C₂×C₈	—	26	4	no	—
248	C₂×C₈	—	26	4	no	—
249	C₂×C₈	—	26	1	no	—
250	C₂×C₈	—	26	1	no	—
251	C₂×C₈	C₄×#261	1	2	yes	—
252	C₂×C₈	C₄⋊#261	1	2	yes	—
253	C₂×C₈	C₄⋊#261	1	2	yes	—
254	C₂×C₈	C₂²⋊#261	1	2	yes	—
255	C₄⋊C₄	—	26	8	no	—
256	C₄⋊C₄	—	26	8	no	—
257	C₄⋊C₄	—	52	4	no	—
258	C₄⋊C₄	—	52	4	no	—
259	C₄²	—	2	1	no	—
260	C₄²	—	2	1	no	—
261	C₄²	—	26	4	no	—
262	C₄²	—	26	1	no	—
263	C₄²	—	26	1	no	—
264	C₁₃	#10861	1	1	yes	characteristic, complemented, Sylow
265	C₂³	—	13	2	no	—
266	C₂³	—	13	1	no	—
267	C₂³	—	13	1	no	—
268	C₂³	—	13	4	no	—
269	C₂³	—	13	4	no	—
270	C₂³	—	13	2	no	—
271	C₂³	C₂×C₄⋊#261	1	1	yes	characteristic
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	13	2	no	—
275	C₂×C₄	—	13	1	no	—
276	C₂×C₄	—	13	1	no	—
277	C₂×C₄	—	13	2	no	—
278	C₂×C₄	—	13	2	no	—
279	C₂×C₄	—	13	2	no	—
280	C₂×C₄	—	13	2	no	—
281	C₂×C₄	—	13	2	no	—
282	C₂×C₄	—	13	2	no	—
283	C₂×C₄	—	13	2	no	—
284	C₂×C₄	—	13	2	no	—
285	C₂×C₄	—	13	2	no	—
286	C₂×C₄	—	13	4	no	—
287	C₂×C₄	—	26	16	no	—
288	C₂×C₄	—	26	2	no	—
289	C₂×C₄	—	26	2	no	—
290	C₂×C₄	—	52	8	no	—
291	C₂×C₄	—	52	16	no	—
292	C₂×C₄	—	104	4	no	—
293	C₂×C₄	—	104	4	no	—
294	C₂×C₄	—	104	8	no	—
295	C₂×C₄	#88101	1	4	yes	complemented
296	C₂×C₄	C₂×C₄×#261	1	1	yes	characteristic
297	C₂×C₄	C₂×C₄⋊#261	1	1	yes	characteristic
298	C₂×C₄	C₂×C₄⋊#261	1	1	yes	characteristic
299	C₂×C₄	#88237	1	1	yes	characteristic
300	C₂×C₄	#88237	1	1	yes	characteristic
301	C₂×C₄	C₂×C₂²⋊#261	1	1	yes	characteristic
302	C₈	—	2	2	no	—
303	C₈	—	13	4	no	—
304	C₈	—	26	2	no	—
305	C₈	(C₂×#261)⋊C₄	1	4	yes	complemented
306	C₂²	—	13	2	no	—
307	C₂²	—	13	2	no	—
308	C₂²	—	13	2	no	—
309	C₂²	—	13	2	no	—
310	C₂²	—	13	2	no	—
311	C₂²	—	13	2	no	—
312	C₂²	—	13	8	no	—
313	C₂²	—	13	8	no	—
314	C₂²	C₄×C₄⋊#261	1	1	yes	characteristic
315	C₂²	#11719059	1	1	yes	characteristic
316	C₂²	C₂×#88101	1	2	yes	—
317	C₂²	#11719092	1	2	yes	—
318	C₂²	#11719136	1	1	yes	characteristic
319	C₄	—	2	4	no	—
320	C₄	—	13	2	no	—
321	C₄	—	13	2	no	—
322	C₄	—	26	4	no	—
323	C₄	—	104	8	no	—
324	C₄	C₄×C₄⋊#261	1	1	yes	characteristic
325	C₄	#11719060	1	1	yes	characteristic
326	C₄	C₂×(C₂×#261)⋊C₄	1	1	yes	characteristic
327	C₄	#11719136	1	1	yes	characteristic
328	C₂	—	13	4	no	—
329	C₂	—	13	4	no	—
330	C₂	#13130014	1	1	yes	characteristic
331	C₂	C₄×#88101	1	2	yes	—
332	C₂	#13130712	1	2	yes	—
333	C₂	#13130771	1	1	yes	characteristic
334	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	#10861
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×#88101	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
116. C₂×C₄×C₁₃	#106	semidirect
125. C₂×Dic₁₃	#106	semidirect
140. C₈×C₁₃	#95	semidirect
142. #1055	#95	semidirect
154. C₂×C₄×C₈	#261	semidirect
264. C₁₃	#10861	semidirect
295. C₂×C₄	#88101	semidirect
305. 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×#88101	2
C₂×#11718265	2
#13130014	2
C₂×C₄×C₈×D₁₃	2
#10861	13