Subgroups — #261×#508 #16807190

Order 3328.

690 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#135×#261	C₂	1	2	yes	maximal, complemented
2	Q₃₂×#261	C₂	1	2	yes	maximal, complemented
3	SD₃₂×#261	C₂	1	4	yes	maximal, complemented
4	D₁₆×#261	C₂	1	2	yes	maximal, complemented
5	#13131187	C₂	1	1	yes	maximal, characteristic, complemented
6	#13131143	C₂	1	2	yes	maximal, complemented
7	C₂×C₁₆×#261	C₂	1	1	yes	maximal, characteristic, complemented
8	D₁₃×#508	C₂	1	1	yes	maximal, characteristic
9	#41×#261	—	2	2	no	—
10	D₁₃×#135	C₂²	1	2	yes	—
11	#11719165	—	2	2	no	—
12	Q₁₆×#261	C₂²	1	4	yes	—
13	SD₁₆×#261	—	2	4	no	—
14	D₈×#261	C₂²	1	4	yes	—
15	#11719092	C₂²	1	2	yes	—
16	C₂×C₈×#261	C₂²	1	1	yes	characteristic
17	C₂×D₁₃⋊C₁₆	C₂²	1	1	yes	characteristic
18	C₁₃×#508	C₄	1	1	yes	characteristic, complemented
19	#11718855	C₄	1	2	yes	complemented
20	#11718580	C₄	1	1	yes	characteristic, complemented
21	D₁₃×Q₃₂	C₂²	1	1	yes	characteristic
22	#11718577	C₄	1	2	yes	complemented
23	D₁₃×SD₃₂	C₂²	1	2	yes	—
24	#11718573	C₄	1	1	yes	characteristic, complemented
25	D₁₃×D₁₆	C₂²	1	1	yes	characteristic
26	#11718565	C₄	1	1	yes	characteristic, complemented
27	C₂×C₁₆×D₁₃	C₂²	1	1	yes	characteristic
28	#11718280	C₂²	1	4	yes	—
29	#11718276	C₂²	1	4	yes	—
30	#11718222	C₂²	1	2	yes	—
31	#11718221	C₂²	1	2	yes	—
32	C₁₆×#261	C₂²	1	2	yes	—
33	C₁₆×#261	C₂²	1	2	yes	—
34	#41×D₁₃	—	2	2	no	—
35	Q₈×#261	—	2	4	no	—
36	D₄×#261	—	2	4	no	—
37	D₄×#261	—	4	4	no	—
38	#88237	—	2	2	no	—
39	#88237	—	4	2	no	—
40	C₂×C₄×#261	—	4	2	no	—
41	C₂×C₄×#261	D₄	1	1	yes	characteristic
42	C₂×D₁₃⋊C₈	D₄	1	1	yes	characteristic
43	C₁₃×#135	C₂×C₄	1	2	yes	complemented
44	#88204	—	2	2	no	—
45	#88173	C₂×C₄	1	2	yes	complemented
46	Q₁₆×D₁₃	C₂³	1	2	yes	—
47	#88170	—	2	2	no	—
48	SD₁₆×D₁₃	—	2	2	no	—
49	#88166	C₂×C₄	1	2	yes	complemented
50	D₈×D₁₃	C₂³	1	2	yes	—
51	#88158	C₂×C₄	1	2	yes	complemented
52	C₂×C₈×D₁₃	C₂³	1	1	yes	characteristic
53	#88117	—	2	4	no	—
54	#88115	—	2	4	no	—
55	C₂×#6092	—	2	1	no	—
56	#88102	C₂³	1	4	yes	—
57	#88101	—	2	2	no	—
58	C₈×#261	D₄	1	2	yes	—
59	C₈×#261	C₂³	1	2	yes	—
60	D₁₃⋊C₁₆	—	2	1	no	—
61	D₁₃⋊C₁₆	C₂³	1	2	yes	—
62	C₁₃×Q₃₂	C₂×C₄	1	1	yes	characteristic, complemented
63	C₁₃×SD₃₂	C₂×C₄	1	2	yes	complemented
64	C₁₃×D₁₆	C₂×C₄	1	1	yes	characteristic, complemented
65	C₂×C₁₃×C₁₆	C₂×C₄	1	1	yes	characteristic, complemented
66	#88069	C₂×C₄	1	2	yes	complemented
67	#88068	C₂×C₄	1	2	yes	complemented
68	#88067	C₂×C₄	1	2	yes	complemented
69	#88066	C₂×C₄	1	2	yes	complemented
70	C₂×#6090	C₂×C₄	1	1	yes	characteristic, complemented
71	Dic₁₀₄	C₂×C₄	1	1	yes	characteristic, complemented
72	#88040	C₂×C₄	1	2	yes	complemented
73	D₂₀₈	C₂×C₄	1	1	yes	characteristic, complemented
74	C₁₆×D₁₃	C₂×C₄	1	1	yes	characteristic, complemented
75	C₁₆×D₁₃	C₂×C₄	1	1	yes	characteristic, complemented
76	C₁₆×D₁₃	C₂³	1	1	yes	characteristic
77	C₁₆×D₁₃	C₂³	1	1	yes	characteristic
78	C₄×#508	—	13	1	no	maximal, Sylow
79	C₂²×#261	—	4	4	no	—
80	C₂²×#261	D₈	1	2	yes	—
81	C₁₃×#41	—	2	2	no	—
82	#6131	—	2	2	no	—
83	Q₈×D₁₃	—	2	2	no	—
84	#6129	—	2	2	no	—
85	#6129	—	4	2	no	—
86	D₄×D₁₃	—	2	2	no	—
87	D₄×D₁₃	—	4	2	no	—
88	#6127	—	2	2	no	—
89	#6127	—	4	2	no	—
90	C₂×C₄×D₁₃	—	4	2	no	—
91	C₂×C₄×D₁₃	C₂×D₄	1	1	yes	characteristic
92	C₂²⋊#261	—	4	2	no	—
93	C₂²⋊#261	—	4	4	no	—
94	C₂×#1057	—	2	1	no	—
95	C₄⋊#261	—	2	4	no	—
96	C₄⋊#261	—	4	4	no	—
97	C₄×#261	—	4	4	no	—
98	C₄×#261	—	4	2	no	—
99	C₄×#261	D₈	1	1	yes	characteristic
100	C₄×#261	D₈	1	1	yes	characteristic
101	C₄×#261	C₂×D₄	1	2	yes	—
102	D₁₃⋊C₈	—	2	1	no	—
103	D₁₃⋊C₈	C₂×D₄	1	2	yes	—
104	C₁₃×Q₁₆	C₂²×C₄	1	2	yes	—
105	C₁₃×SD₁₆	—	2	2	no	—
106	C₁₃×D₈	C₂²×C₄	1	2	yes	—
107	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
108	#6107	—	2	2	no	—
109	#6106	—	2	2	no	—
110	#6105	—	2	2	no	—
111	#6104	—	2	2	no	—
112	C₂×#1055	#41	1	1	yes	characteristic
113	Dic₅₂	C₂²×C₄	1	2	yes	—
114	D₁₀₄	C₂²×C₄	1	2	yes	—
115	#6095	—	2	2	no	—
116	C₈×D₁₃	C₂²×C₄	1	1	yes	characteristic
117	C₈×D₁₃	C₂×D₄	1	1	yes	characteristic
118	C₈×D₁₃	#41	1	1	yes	characteristic
119	C₈×D₁₃	C₂⁴	1	1	yes	characteristic
120	#6092	—	2	2	no	—
121	C₁₃×C₁₆	C₂²×C₄	1	1	yes	characteristic
122	C₁₃×C₁₆	C₂²×C₄	1	1	yes	characteristic
123	#6090	C₂²×C₄	1	1	yes	characteristic
124	#6090	C₂²×C₄	1	1	yes	characteristic
125	C₂×#508	—	13	1	no	—
126	C₄×#135	—	13	2	no	—
127	C₄×Q₃₂	—	13	2	no	—
128	C₄×SD₃₂	—	13	4	no	—
129	C₄×D₁₆	—	13	2	no	—
130	#2158	—	13	1	no	—
131	#2138	—	13	2	no	—
132	C₂×C₄×C₁₆	—	13	1	no	—
133	C₂²×D₁₃	—	4	2	no	—
134	C₂²×D₁₃	C₂×D₈	1	1	yes	characteristic
135	C₂×#261	—	2	4	no	—
136	C₂×#261	—	4	4	no	—
137	C₂×#261	—	8	8	no	—
138	C₂×#261	C₂×D₈	1	2	yes	—
139	C₂×#261	C₂×D₈	1	2	yes	—
140	Q₈×C₁₃	—	2	2	no	—
141	D₄×C₁₃	—	2	2	no	—
142	D₄×C₁₃	—	4	2	no	—
143	C₂×C₄×C₁₃	—	4	2	no	—
144	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
145	C₂²⋊D₁₃	—	4	2	no	—
146	C₂²⋊D₁₃	—	4	2	no	—
147	C₂²⋊D₁₃	—	4	2	no	—
148	C₂×Dic₁₃	—	4	2	no	—
149	C₂×Dic₁₃	#135	1	1	yes	characteristic
150	D₅₂	—	2	2	no	—
151	D₅₂	—	4	2	no	—
152	C₄×D₁₃	—	4	2	no	—
153	C₄×D₁₃	—	4	2	no	—
154	C₄×D₁₃	—	4	2	no	—
155	C₄×D₁₃	C₄×D₄	1	1	yes	characteristic
156	C₄×D₁₃	C₂×D₈	1	1	yes	characteristic
157	C₄×D₁₃	#135	1	1	yes	characteristic
158	C₄×D₁₃	C₂²×D₄	1	1	yes	characteristic
159	Dic₂₆	—	2	2	no	—
160	Dic₂₆	—	4	2	no	—
161	#1057	—	2	2	no	—
162	C₈×C₁₃	C₄×D₄	1	1	yes	characteristic
163	C₈×C₁₃	C₂³×C₄	1	1	yes	characteristic
164	#1055	C₄×D₄	1	1	yes	characteristic
165	#1055	C₂×#41	1	1	yes	characteristic
166	C₂×#135	—	13	2	no	—
167	C₄×#41	—	26	2	no	—
168	#508	—	13	2	no	—
169	#508	—	13	1	no	—
170	#508	—	13	2	no	—
171	#508	—	13	1	no	—
172	#508	—	13	1	no	—
173	#508	#261	1	1	yes	characteristic, direct product factor
174	C₂×Q₃₂	—	13	1	no	—
175	C₂×SD₃₂	—	13	2	no	—
176	C₂×D₁₆	—	13	1	no	—
177	C₂²×C₁₆	—	13	1	no	—
178	C₂²×C₁₆	—	13	1	no	—
179	C₄×Q₁₆	—	13	4	no	—
180	C₄×SD₁₆	—	26	4	no	—
181	C₄×D₈	—	13	4	no	—
182	#427	—	13	2	no	—
183	#416	—	26	2	no	—
184	C₂×C₄×C₈	—	13	1	no	—
185	#367	—	13	2	no	—
186	#366	—	13	2	no	—
187	#358	—	13	4	no	—
188	#357	—	13	4	no	—
189	C₄×C₁₆	—	13	2	no	—
190	C₄×C₁₆	—	13	2	no	—
191	C₂²×C₁₃	—	4	2	no	—
192	C₂²×C₁₃	C₄×D₈	1	1	yes	characteristic
193	C₂×D₁₃	—	2	2	no	—
194	C₂×D₁₃	—	4	2	no	—
195	C₂×D₁₃	—	8	4	no	—
196	C₂×D₁₃	C₄×D₈	1	1	yes	characteristic
197	C₂×D₁₃	C₂²×D₈	1	1	yes	characteristic
198	#261	—	2	2	no	—
199	#261	—	8	4	no	—
200	#261	#508	1	4	yes	direct product factor
201	C₄×C₁₃	—	4	2	no	—
202	C₄×C₁₃	C₄×D₈	1	1	yes	characteristic
203	C₄×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
204	Dic₁₃	—	4	2	no	—
205	Dic₁₃	C₄×D₈	1	1	yes	characteristic
206	Dic₁₃	C₂×#135	1	1	yes	characteristic
207	C₂×#41	—	26	2	no	—
208	#135	—	13	2	no	—
209	#135	—	13	2	no	—
210	#135	—	13	2	no	—
211	#135	—	26	2	no	—
212	#135	—	26	2	no	—
213	#135	C₂×#261	1	2	yes	complemented
214	C₂×Q₁₆	—	13	2	no	—
215	C₂×SD₁₆	—	26	2	no	—
216	C₂×D₈	—	13	2	no	—
217	C₂²×C₈	—	13	1	no	—
218	C₂²×C₈	—	13	1	no	—
219	C₄×Q₈	—	26	4	no	—
220	C₄×D₄	—	26	4	no	—
221	C₄×D₄	—	52	4	no	—
222	#117	—	26	2	no	—
223	#117	—	52	2	no	—
224	C₂×C₄²	—	13	1	no	—
225	C₂×C₄²	—	52	2	no	—
226	Q₃₂	—	13	2	no	—
227	Q₃₂	—	13	1	no	—
228	Q₃₂	C₂×#261	1	1	yes	characteristic, complemented
229	SD₃₂	—	13	2	no	—
230	SD₃₂	—	13	2	no	—
231	SD₃₂	—	13	2	no	—
232	SD₃₂	C₂×#261	1	2	yes	complemented
233	D₁₆	—	13	2	no	—
234	D₁₆	—	13	1	no	—
235	D₁₆	C₂×#261	1	1	yes	characteristic, complemented
236	C₂×C₁₆	—	13	1	no	—
237	C₂×C₁₆	—	13	1	no	—
238	C₂×C₁₆	—	13	1	no	—
239	C₂×C₁₆	—	13	1	no	—
240	C₂×C₁₆	—	13	1	no	—
241	C₂×C₁₆	—	13	2	no	—
242	C₂×C₁₆	—	26	1	no	—
243	C₂×C₁₆	—	26	1	no	—
244	C₂×C₁₆	C₂×#261	1	1	yes	characteristic, complemented
245	#107	—	13	4	no	—
246	#106	—	26	2	no	—
247	#103	—	26	4	no	—
248	#102	—	26	4	no	—
249	C₄×C₈	—	13	2	no	—
250	C₄×C₈	—	13	2	no	—
251	C₂×C₁₃	—	2	1	no	—
252	C₂×C₁₃	—	8	2	no	—
253	C₂×C₁₃	C₂×C₄×D₈	1	1	yes	characteristic
254	D₁₃	—	2	1	no	—
255	D₁₃	—	8	2	no	—
256	D₁₃	C₂×#508	1	2	yes	—
257	#41	—	2	2	no	—
258	#41	—	26	2	no	—
259	#41	—	26	2	no	—
260	#41	—	26	2	no	—
261	#41	—	52	2	no	—
262	#41	—	52	2	no	—
263	C₂×Q₈	—	26	2	no	—
264	C₂×D₄	—	26	2	no	—
265	C₂×D₄	—	52	2	no	—
266	C₂²×C₄	—	13	1	no	—
267	C₂²×C₄	—	13	2	no	—
268	C₂²×C₄	—	52	2	no	—
269	C₂²×C₄	—	52	4	no	—
270	Q₁₆	—	13	2	no	—
271	Q₁₆	—	26	2	no	—
272	Q₁₆	C₂²×#261	1	2	yes	—
273	SD₁₆	—	2	2	no	—
274	SD₁₆	—	26	2	no	—
275	SD₁₆	—	26	2	no	—
276	SD₁₆	—	26	2	no	—
277	D₈	—	13	2	no	—
278	D₈	—	26	2	no	—
279	D₈	C₂²×#261	1	2	yes	—
280	C₂×C₈	—	13	1	no	—
281	C₂×C₈	—	13	1	no	—
282	C₂×C₈	—	13	1	no	—
283	C₂×C₈	—	13	1	no	—
284	C₂×C₈	—	13	1	no	—
285	C₂×C₈	—	13	2	no	—
286	C₂×C₈	—	26	1	no	—
287	C₂×C₈	—	26	1	no	—
288	C₂×C₈	C₂²×#261	1	1	yes	characteristic
289	C₄⋊C₄	—	26	4	no	—
290	C₄⋊C₄	—	52	4	no	—
291	C₂²⋊C₄	—	52	4	no	—
292	C₂²⋊C₄	—	52	2	no	—
293	C₄²	—	13	2	no	—
294	C₄²	—	13	1	no	—
295	C₄²	—	13	1	no	—
296	C₄²	—	52	2	no	—
297	C₄²	—	52	4	no	—
298	C₁₆	—	13	1	no	—
299	C₁₆	—	13	1	no	—
300	C₁₆	—	26	2	no	—
301	C₁₆	C₂²×#261	1	1	yes	characteristic
302	C₁₆	C₂²×#261	1	1	yes	characteristic
303	C₁₃	C₄×#508	1	1	yes	characteristic, complemented, Sylow
304	C₂³	—	13	1	no	—
305	C₂³	—	52	2	no	—
306	Q₈	—	2	2	no	—
307	Q₈	—	26	2	no	—
308	Q₈	—	52	2	no	—
309	D₄	—	2	2	no	—
310	D₄	—	4	2	no	—
311	D₄	—	26	2	no	—
312	D₄	—	52	2	no	—
313	D₄	—	52	2	no	—
314	D₄	—	52	2	no	—
315	D₄	—	52	2	no	—
316	C₂×C₄	—	4	2	no	—
317	C₂×C₄	—	13	2	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	—	13	1	no	—
323	C₂×C₄	—	13	2	no	—
324	C₂×C₄	—	26	4	no	—
325	C₂×C₄	—	52	2	no	—
326	C₂×C₄	—	52	2	no	—
327	C₂×C₄	—	52	2	no	—
328	C₂×C₄	—	52	2	no	—
329	C₂×C₄	—	52	4	no	—
330	C₂×C₄	—	104	8	no	—
331	C₂×C₄	D₄×#261	1	1	yes	characteristic
332	C₈	—	13	1	no	—
333	C₈	—	13	1	no	—
334	C₈	—	26	2	no	—
335	C₈	D₄×#261	1	1	yes	characteristic
336	C₈	C₂³×#261	1	1	yes	characteristic
337	C₂²	—	4	2	no	—
338	C₂²	—	13	1	no	—
339	C₂²	—	13	1	no	—
340	C₂²	—	26	2	no	—
341	C₂²	—	52	2	no	—
342	C₂²	—	104	4	no	—
343	C₂²	D₈×#261	1	1	yes	characteristic
344	C₄	—	4	2	no	—
345	C₄	—	13	4	no	—
346	C₄	—	13	1	no	—
347	C₄	—	13	1	no	—
348	C₄	—	26	2	no	—
349	C₄	—	52	2	no	—
350	C₄	—	104	4	no	—
351	C₄	D₈×#261	1	1	yes	characteristic
352	C₄	C₂×D₄×#261	1	1	yes	characteristic
353	C₂	—	2	1	no	—
354	C₂	—	8	2	no	—
355	C₂	—	13	2	no	—
356	C₂	—	26	1	no	—
357	C₂	—	104	2	no	—
358	C₂	C₂×D₈×#261	1	1	yes	characteristic

	subgroup	G/H
center	C₄	D₈×#261
commutator	C₈×C₁₃	C₂³×C₄
Frattini	C₈	C₂³×#261
Fitting	C₁₃×#508	C₄
socle	C₂×C₁₃	C₂×C₄×D₈
prime radical	D₁₃×#508	C₂
cosocle	C₈×D₁₃	C₂⁴
hypercenter	#508	#261
hypocenter	C₁₃	C₄×#508
LCS	G ▷ C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #508

Complements

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

N	H ≅ G/N	type
1. #135×#261	C₂	semidirect
2. Q₃₂×#261	C₂	semidirect
3. SD₃₂×#261	C₂	semidirect
4. D₁₆×#261	C₂	semidirect
5. #13131187	C₂	semidirect
6. #13131143	C₂	semidirect
7. C₂×C₁₆×#261	C₂	semidirect
18. C₁₃×#508	C₄	semidirect
19. #11718855	C₄	semidirect
20. #11718580	C₄	semidirect
22. #11718577	C₄	semidirect
24. #11718573	C₄	semidirect
26. #11718565	C₄	semidirect
43. C₁₃×#135	C₂×C₄	semidirect
45. #88173	C₂×C₄	semidirect
49. #88166	C₂×C₄	semidirect
51. #88158	C₂×C₄	semidirect
62. C₁₃×Q₃₂	C₂×C₄	semidirect
63. C₁₃×SD₃₂	C₂×C₄	semidirect
64. C₁₃×D₁₆	C₂×C₄	semidirect
65. C₂×C₁₃×C₁₆	C₂×C₄	semidirect
66. #88069	C₂×C₄	semidirect
67. #88068	C₂×C₄	semidirect
68. #88067	C₂×C₄	semidirect
69. #88066	C₂×C₄	semidirect
70. C₂×#6090	C₂×C₄	semidirect
71. Dic₁₀₄	C₂×C₄	semidirect
72. #88040	C₂×C₄	semidirect
73. D₂₀₈	C₂×C₄	semidirect
74. C₁₆×D₁₃	C₂×C₄	semidirect
75. C₁₆×D₁₃	C₂×C₄	semidirect
173. #508	#261	direct (N × H)
200. #261	#508	direct (N × H)
213. #135	C₂×#261	semidirect
228. Q₃₂	C₂×#261	semidirect
232. SD₃₂	C₂×#261	semidirect
235. D₁₆	C₂×#261	semidirect
244. C₂×C₁₆	C₂×#261	semidirect
303. C₁₃	C₄×#508	semidirect

Maximal subgroups

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

subgroup	index
#135×#261	2
Q₃₂×#261	2
SD₃₂×#261	2
D₁₆×#261	2
#13131187	2
#13131143	2
C₂×C₁₆×#261	2
D₁₃×#508	2
C₄×#508	13