Subgroups — D₃₁×#576 #16860459

Order 3968.

1512 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	H(5,𝔽₂)×D₃₁	C₂	1	2	yes	maximal, complemented
2	#14521854	C₂	1	1	yes	maximal, characteristic, complemented
3	#14521853	C₂	1	6	yes	maximal, complemented
4	#14521852	C₂	1	2	yes	maximal, complemented
5	#14521851	C₂	1	3	yes	maximal, complemented
6	#14521850	C₂	1	3	yes	maximal, complemented
7	AGL(1,ℤ/8)×D₃₁	C₂	1	6	yes	maximal, complemented
8	#135×D₃₁	C₂	1	3	yes	maximal, complemented
9	C₂×D₈×D₃₁	C₂	1	3	yes	maximal, complemented
10	#131×D₃₁	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃₁×#576	C₂	1	1	yes	maximal, characteristic, complemented
12	C₃₁×H(5,𝔽₂)	C₂²	1	2	yes	complemented
13	#11758566	—	2	2	no	—
14	#11758566	C₂²	1	6	yes	complemented
15	#11758566	C₂²	1	2	yes	complemented
16	#41×D₃₁	—	2	2	no	—
17	#41×D₃₁	C₂²	1	1	yes	characteristic
18	#41×D₃₁	C₂²	1	6	yes	complemented
19	#11758561	—	2	6	no	—
20	#11758561	C₂²	1	6	yes	complemented
21	C₂×D₄×D₃₁	—	2	6	no	—
22	C₂×D₄×D₃₁	C₂²	1	6	yes	complemented
23	C₃₁×AGL(1,ℤ/8)	C₂²	1	6	yes	complemented
24	C₃₁×#135	C₂²	1	3	yes	complemented
25	C₂×D₈×C₃₁	C₂²	1	3	yes	complemented
26	C₃₁×#131	C₂²	1	1	yes	characteristic, complemented
27	#11758527	C₂²	1	6	yes	complemented
28	#11758526	C₂²	1	6	yes	complemented
29	#11758525	C₂²	1	1	yes	characteristic, complemented
30	#11758509	C₂²	1	6	yes	complemented
31	C₂×#92479	C₂²	1	6	yes	complemented
32	#11758496	C₂²	1	3	yes	complemented
33	Q₁₆×D₃₁	C₂²	1	1	yes	characteristic, complemented
34	#11758493	C₂²	1	6	yes	complemented
35	#11758491	C₂²	1	12	yes	complemented
36	SD₁₆×D₃₁	C₂²	1	6	yes	complemented
37	#11758489	C₂²	1	3	yes	complemented
38	#11758488	C₂²	1	12	yes	complemented
39	D₈×D₃₁	C₂²	1	3	yes	complemented
40	D₈×D₃₁	C₂²	1	6	yes	complemented
41	D₈×D₃₁	C₂²	1	6	yes	complemented
42	D₈×D₃₁	C₂²	1	6	yes	complemented
43	#11758485	C₂²	1	6	yes	complemented
44	#11758484	C₂²	1	3	yes	complemented
45	M₁₆×D₃₁	C₂²	1	3	yes	complemented
46	#11758481	C₂²	1	3	yes	complemented
47	C₂×D₂₄₈	C₂²	1	3	yes	complemented
48	#11758478	C₂²	1	3	yes	complemented
49	C₂×C₈×D₃₁	C₂²	1	3	yes	complemented
50	C₂³×D₃₁	—	2	6	no	—
51	#41×C₃₁	—	2	2	no	—
52	#41×C₃₁	C₂³	1	1	yes	characteristic
53	#41×C₃₁	C₂³	1	6	yes	complemented
54	C₂×D₄×C₃₁	—	2	6	no	—
55	C₂×D₄×C₃₁	C₂³	1	6	yes	complemented
56	C₂×C₂²⋊D₃₁	—	2	12	no	—
57	C₂×C₂²⋊D₃₁	—	2	6	no	—
58	#92499	—	2	2	no	—
59	#92499	D₄	1	1	yes	characteristic
60	#92499	C₂³	1	6	yes	complemented
61	Q₈×D₃₁	D₄	1	1	yes	characteristic
62	Q₈×D₃₁	C₂³	1	2	yes	—
63	#92497	—	2	6	no	—
64	#92497	—	2	6	no	—
65	#92497	D₄	1	3	yes	—
66	#92497	C₂³	1	6	yes	complemented
67	D₄×D₃₁	—	2	6	no	—
68	D₄×D₃₁	—	2	6	no	—
69	D₄×D₃₁	—	2	12	no	—
70	D₄×D₃₁	—	2	6	no	—
71	D₄×D₃₁	—	2	6	no	—
72	D₄×D₃₁	—	2	6	no	—
73	D₄×D₃₁	D₄	1	3	yes	—
74	D₄×D₃₁	C₂³	1	6	yes	—
75	D₄×D₃₁	C₂³	1	12	yes	complemented
76	#92495	—	2	6	no	—
77	#92495	—	2	6	no	—
78	#92495	C₂³	1	3	yes	—
79	#92495	C₂³	1	6	yes	complemented
80	C₂×D₁₂₄	—	2	6	no	—
81	C₂×D₁₂₄	C₂³	1	6	yes	complemented
82	C₂×C₄×D₃₁	—	2	6	no	—
83	C₂×C₄×D₃₁	C₂³	1	3	yes	—
84	Q₁₆×C₃₁	C₂³	1	1	yes	characteristic, complemented
85	SD₁₆×C₃₁	C₂³	1	6	yes	complemented
86	D₈×C₃₁	C₂³	1	3	yes	complemented
87	D₈×C₃₁	C₂³	1	6	yes	complemented
88	M₁₆×C₃₁	C₂³	1	3	yes	complemented
89	C₂×C₈×C₃₁	C₂³	1	3	yes	complemented
90	#92482	C₂³	1	2	yes	complemented
91	#92481	C₂³	1	6	yes	complemented
92	#92480	C₂³	1	6	yes	complemented
93	#92479	C₂³	1	6	yes	complemented
94	#92479	C₂³	1	12	yes	complemented
95	#92474	C₂³	1	3	yes	complemented
96	C₂×C₃₁⋊C₈	C₂³	1	3	yes	complemented
97	Dic₁₂₄	C₂³	1	1	yes	characteristic, complemented
98	D₂₄₈	C₂³	1	3	yes	complemented
99	D₂₄₈	C₂³	1	6	yes	complemented
100	#92470	C₂³	1	6	yes	complemented
101	#92469	C₂³	1	6	yes	complemented
102	C₈×D₃₁	C₂³	1	1	yes	characteristic
103	C₈×D₃₁	C₂³	1	3	yes	—
104	C₈×D₃₁	C₂³	1	3	yes	complemented
105	C₈×D₃₁	C₂³	1	3	yes	complemented
106	C₂³×C₃₁	—	2	6	no	—
107	C₂²×D₃₁	—	2	6	no	—
108	C₂²×D₃₁	—	2	6	no	—
109	C₂²×D₃₁	—	2	12	no	—
110	C₂²×D₃₁	—	8	12	no	—
111	C₂²×D₃₁	C₂×D₄	1	3	yes	—
112	Q₈×C₃₁	C₂×D₄	1	1	yes	characteristic
113	Q₈×C₃₁	C₂⁴	1	2	yes	—
114	D₄×C₃₁	—	2	6	no	—
115	D₄×C₃₁	—	2	6	no	—
116	D₄×C₃₁	C₂×D₄	1	3	yes	—
117	D₄×C₃₁	C₂⁴	1	6	yes	—
118	C₂×C₄×C₃₁	—	2	6	no	—
119	C₂×C₄×C₃₁	C₂⁴	1	3	yes	—
120	C₂²⋊D₃₁	—	2	6	no	—
121	C₂²⋊D₃₁	—	2	6	no	—
122	C₂²⋊D₃₁	—	2	6	no	—
123	C₂²⋊D₃₁	—	2	12	no	—
124	C₂²⋊D₃₁	C₂×D₄	1	6	yes	—
125	C₂×Dic₃₁	—	2	6	no	—
126	C₂×Dic₃₁	C₂×D₄	1	3	yes	—
127	D₁₂₄	—	2	6	no	—
128	D₁₂₄	—	2	6	no	—
129	D₁₂₄	C₂×D₄	1	3	yes	—
130	D₁₂₄	C₂⁴	1	6	yes	—
131	C₄×D₃₁	—	2	2	no	—
132	C₄×D₃₁	—	2	6	no	—
133	C₄×D₃₁	—	2	6	no	—
134	C₄×D₃₁	C₂×D₄	1	3	yes	—
135	C₄×D₃₁	C₂×D₄	1	3	yes	—
136	C₄×D₃₁	C₂⁴	1	1	yes	characteristic
137	C₄×D₃₁	C₂⁴	1	3	yes	—
138	Dic₆₂	—	2	2	no	—
139	Dic₆₂	C₂×D₄	1	3	yes	—
140	Dic₆₂	C₂⁴	1	2	yes	—
141	C₈×C₃₁	C₂⁴	1	1	yes	characteristic
142	C₈×C₃₁	C₂⁴	1	3	yes	—
143	C₃₁⋊C₈	C₂⁴	1	1	yes	characteristic
144	C₃₁⋊C₈	C₂⁴	1	3	yes	—
145	C₂×#576	—	31	1	no	maximal, Sylow
146	C₂²×C₃₁	—	2	6	no	—
147	C₂²×C₃₁	—	8	6	no	—
148	C₂²×C₃₁	C₂²×D₄	1	3	yes	—
149	C₂×D₃₁	—	2	6	no	—
150	C₂×D₃₁	—	2	6	no	—
151	C₂×D₃₁	—	4	12	no	—
152	C₂×D₃₁	—	8	12	no	—
153	C₂×D₃₁	—	8	6	no	—
154	C₂×D₃₁	C₂²×D₄	1	3	yes	—
155	C₂×D₃₁	C₂²×D₄	1	1	yes	characteristic
156	C₄×C₃₁	—	2	2	no	—
157	C₄×C₃₁	C₂²×D₄	1	3	yes	—
158	C₄×C₃₁	C₂⁵	1	1	yes	characteristic
159	Dic₃₁	—	2	2	no	—
160	Dic₃₁	C₂²×D₄	1	1	yes	characteristic
161	Dic₃₁	C₂²×D₄	1	3	yes	—
162	C₂×H(5,𝔽₂)	—	31	2	no	—
163	#576	—	31	2	no	—
164	#576	—	31	6	no	—
165	#576	—	31	3	no	—
166	#576	—	31	3	no	—
167	#576	—	31	1	no	—
168	#576	D₃₁	1	1	yes	characteristic, direct product factor
169	C₂×AGL(1,ℤ/8)	—	31	6	no	—
170	C₂×#135	—	31	3	no	—
171	C₂²×D₈	—	31	3	no	—
172	C₂×#131	—	31	1	no	—
173	C₂×C₃₁	—	2	3	no	—
174	C₂×C₃₁	—	4	6	no	—
175	C₂×C₃₁	C₂³×D₄	1	1	yes	characteristic
176	D₃₁	—	2	3	no	—
177	D₃₁	—	4	6	no	—
178	D₃₁	#576	1	2	yes	direct product factor
179	H(5,𝔽₂)	—	31	2	no	—
180	H(5,𝔽₂)	—	31	6	no	—
181	H(5,𝔽₂)	—	31	6	no	—
182	H(5,𝔽₂)	—	62	6	no	—
183	H(5,𝔽₂)	—	62	2	no	—
184	H(5,𝔽₂)	C₂×D₃₁	1	2	yes	complemented
185	C₂×#41	—	31	1	no	—
186	C₂×#41	—	31	6	no	—
187	C₂×#41	—	62	2	no	—
188	C₂²×D₄	—	31	6	no	—
189	C₂²×D₄	—	62	6	no	—
190	AGL(1,ℤ/8)	—	31	6	no	—
191	AGL(1,ℤ/8)	—	31	6	no	—
192	AGL(1,ℤ/8)	—	31	12	no	—
193	AGL(1,ℤ/8)	—	31	12	no	—
194	AGL(1,ℤ/8)	—	31	6	no	—
195	AGL(1,ℤ/8)	C₂×D₃₁	1	6	yes	complemented
196	#135	—	31	6	no	—
197	#135	—	31	3	no	—
198	#135	—	31	6	no	—
199	#135	—	31	3	no	—
200	#135	—	31	3	no	—
201	#135	C₂×D₃₁	1	3	yes	complemented
202	C₂×Q₁₆	—	31	1	no	—
203	C₂×SD₁₆	—	31	6	no	—
204	C₂×D₈	—	31	6	no	—
205	C₂×D₈	—	31	3	no	—
206	C₂×D₈	—	31	6	no	—
207	C₂×D₈	—	31	6	no	—
208	C₂×D₈	—	31	6	no	—
209	C₂×D₈	—	31	3	no	—
210	C₂×D₈	C₂×D₃₁	1	3	yes	complemented
211	#131	—	31	1	no	—
212	#131	—	31	3	no	—
213	#131	—	31	3	no	—
214	#131	C₂×D₃₁	1	1	yes	characteristic, complemented
215	C₂×M₁₆	—	31	3	no	—
216	C₂²×C₈	—	31	3	no	—
217	C₃₁	C₂×#576	1	1	yes	characteristic, complemented, Sylow
218	C₂⁴	—	62	6	no	—
219	#41	—	2	2	no	—
220	#41	—	31	1	no	—
221	#41	—	31	3	no	—
222	#41	—	31	3	no	—
223	#41	—	31	6	no	—
224	#41	—	31	6	no	—
225	#41	—	31	6	no	—
226	#41	—	62	6	no	—
227	#41	—	62	6	no	—
228	#41	—	62	6	no	—
229	#41	—	62	6	no	—
230	#41	—	62	2	no	—
231	#41	C₂²×D₃₁	1	1	yes	characteristic
232	#41	C₂²×D₃₁	1	6	yes	complemented
233	C₂×Q₈	—	31	1	no	—
234	C₂×Q₈	—	31	2	no	—
235	C₂×D₄	—	2	6	no	—
236	C₂×D₄	—	31	3	no	—
237	C₂×D₄	—	31	6	no	—
238	C₂×D₄	—	31	12	no	—
239	C₂×D₄	—	31	6	no	—
240	C₂×D₄	—	62	6	no	—
241	C₂×D₄	—	62	6	no	—
242	C₂×D₄	—	62	6	no	—
243	C₂×D₄	—	62	6	no	—
244	C₂×D₄	—	62	12	no	—
245	C₂×D₄	—	62	6	no	—
246	C₂×D₄	—	62	6	no	—
247	C₂×D₄	—	62	12	no	—
248	C₂×D₄	—	62	6	no	—
249	C₂×D₄	C₂²×D₃₁	1	6	yes	complemented
250	C₂²×C₄	—	31	3	no	—
251	C₂²×C₄	—	62	6	no	—
252	Q₁₆	—	31	2	no	—
253	Q₁₆	—	31	1	no	—
254	Q₁₆	C₂²×D₃₁	1	1	yes	characteristic, complemented
255	SD₁₆	—	31	6	no	—
256	SD₁₆	—	31	6	no	—
257	SD₁₆	—	31	6	no	—
258	SD₁₆	C₂²×D₃₁	1	6	yes	complemented
259	D₈	—	31	6	no	—
260	D₈	—	31	12	no	—
261	D₈	—	31	3	no	—
262	D₈	—	31	6	no	—
263	D₈	C₂²×D₃₁	1	3	yes	complemented
264	D₈	C₂²×D₃₁	1	6	yes	complemented
265	M₁₆	—	31	3	no	—
266	M₁₆	—	31	6	no	—
267	M₁₆	C₂²×D₃₁	1	3	yes	complemented
268	C₂×C₈	—	31	3	no	—
269	C₂×C₈	—	31	1	no	—
270	C₂×C₈	—	31	3	no	—
271	C₂×C₈	—	31	3	no	—
272	C₂×C₈	—	31	3	no	—
273	C₂×C₈	C₂²×D₃₁	1	3	yes	complemented
274	C₂³	—	2	6	no	—
275	C₂³	—	31	3	no	—
276	C₂³	—	62	6	no	—
277	C₂³	—	62	12	no	—
278	C₂³	—	62	6	no	—
279	C₂³	—	248	12	no	—
280	Q₈	—	31	3	no	—
281	Q₈	—	31	2	no	—
282	Q₈	—	62	2	no	—
283	Q₈	D₄×D₃₁	1	1	yes	characteristic
284	Q₈	C₂³×D₃₁	1	2	yes	—
285	D₄	—	2	6	no	—
286	D₄	—	2	6	no	—
287	D₄	—	31	6	no	—
288	D₄	—	31	3	no	—
289	D₄	—	31	6	no	—
290	D₄	—	62	6	no	—
291	D₄	—	62	6	no	—
292	D₄	—	62	12	no	—
293	D₄	—	62	6	no	—
294	D₄	—	62	6	no	—
295	D₄	—	62	6	no	—
296	D₄	D₄×D₃₁	1	3	yes	—
297	D₄	C₂³×D₃₁	1	6	yes	—
298	C₂×C₄	—	2	6	no	—
299	C₂×C₄	—	31	3	no	—
300	C₂×C₄	—	31	3	no	—
301	C₂×C₄	—	31	3	no	—
302	C₂×C₄	—	31	1	no	—
303	C₂×C₄	—	31	3	no	—
304	C₂×C₄	—	62	6	no	—
305	C₂×C₄	—	62	2	no	—
306	C₂×C₄	—	62	6	no	—
307	C₂×C₄	—	62	6	no	—
308	C₂×C₄	C₂³×D₃₁	1	3	yes	—
309	C₈	—	31	1	no	—
310	C₈	—	31	3	no	—
311	C₈	C₂³×D₃₁	1	1	yes	characteristic
312	C₈	C₂³×D₃₁	1	3	yes	—
313	C₂²	—	2	6	no	—
314	C₂²	—	8	6	no	—
315	C₂²	—	31	1	no	—
316	C₂²	—	31	3	no	—
317	C₂²	—	62	6	no	—
318	C₂²	—	62	6	no	—
319	C₂²	—	124	12	no	—
320	C₂²	—	248	6	no	—
321	C₂²	—	248	12	no	—
322	C₂²	C₂×D₄×D₃₁	1	3	yes	—
323	C₄	—	2	2	no	—
324	C₄	—	31	1	no	—
325	C₄	—	31	3	no	—
326	C₄	—	62	2	no	—
327	C₄	C₂×D₄×D₃₁	1	3	yes	—
328	C₄	C₂⁴×D₃₁	1	1	yes	characteristic
329	C₂	—	2	3	no	—
330	C₂	—	4	6	no	—
331	C₂	—	31	2	no	—
332	C₂	—	62	3	no	—
333	C₂	—	124	6	no	—
334	C₂	C₂²×D₄×D₃₁	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂²×D₄×D₃₁
commutator, cosocle	C₄×C₃₁	C₂⁵
Frattini	C₄	C₂⁴×D₃₁
Fitting	C₃₁×#576	C₂
socle	C₂×C₃₁	C₂³×D₄
hypercenter	#576	D₃₁
hypocenter	C₃₁	C₂×#576
LCS	G ▷ C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂ ◁ #41 ◁ #576

Complements

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

N	H ≅ G/N	type
1. H(5,𝔽₂)×D₃₁	C₂	semidirect
2. #14521854	C₂	semidirect
3. #14521853	C₂	semidirect
4. #14521852	C₂	semidirect
5. #14521851	C₂	semidirect
6. #14521850	C₂	semidirect
7. AGL(1,ℤ/8)×D₃₁	C₂	semidirect
8. #135×D₃₁	C₂	semidirect
9. C₂×D₈×D₃₁	C₂	semidirect
10. #131×D₃₁	C₂	semidirect
11. C₃₁×#576	C₂	semidirect
12. C₃₁×H(5,𝔽₂)	C₂²	semidirect
14. #11758566	C₂²	semidirect
15. #11758566	C₂²	semidirect
18. #41×D₃₁	C₂²	semidirect
20. #11758561	C₂²	semidirect
22. C₂×D₄×D₃₁	C₂²	semidirect
23. C₃₁×AGL(1,ℤ/8)	C₂²	semidirect
24. C₃₁×#135	C₂²	semidirect
25. C₂×D₈×C₃₁	C₂²	semidirect
26. C₃₁×#131	C₂²	semidirect
27. #11758527	C₂²	semidirect
28. #11758526	C₂²	semidirect
29. #11758525	C₂²	semidirect
30. #11758509	C₂²	semidirect
31. C₂×#92479	C₂²	semidirect
32. #11758496	C₂²	semidirect
33. Q₁₆×D₃₁	C₂²	semidirect
34. #11758493	C₂²	semidirect
35. #11758491	C₂²	semidirect
36. SD₁₆×D₃₁	C₂²	semidirect
37. #11758489	C₂²	semidirect
38. #11758488	C₂²	semidirect
39. D₈×D₃₁	C₂²	semidirect
40. D₈×D₃₁	C₂²	semidirect
41. D₈×D₃₁	C₂²	semidirect
42. D₈×D₃₁	C₂²	semidirect
43. #11758485	C₂²	semidirect
44. #11758484	C₂²	semidirect
45. M₁₆×D₃₁	C₂²	semidirect
46. #11758481	C₂²	semidirect
47. C₂×D₂₄₈	C₂²	semidirect
48. #11758478	C₂²	semidirect
49. C₂×C₈×D₃₁	C₂²	semidirect
53. #41×C₃₁	C₂³	semidirect
55. C₂×D₄×C₃₁	C₂³	semidirect
60. #92499	C₂³	semidirect
66. #92497	C₂³	semidirect
75. D₄×D₃₁	C₂³	semidirect
79. #92495	C₂³	semidirect
81. C₂×D₁₂₄	C₂³	semidirect
84. Q₁₆×C₃₁	C₂³	semidirect
85. SD₁₆×C₃₁	C₂³	semidirect
86. D₈×C₃₁	C₂³	semidirect
87. D₈×C₃₁	C₂³	semidirect
88. M₁₆×C₃₁	C₂³	semidirect
89. C₂×C₈×C₃₁	C₂³	semidirect
90. #92482	C₂³	semidirect
91. #92481	C₂³	semidirect
92. #92480	C₂³	semidirect
93. #92479	C₂³	semidirect
94. #92479	C₂³	semidirect
95. #92474	C₂³	semidirect
96. C₂×C₃₁⋊C₈	C₂³	semidirect
97. Dic₁₂₄	C₂³	semidirect
98. D₂₄₈	C₂³	semidirect
99. D₂₄₈	C₂³	semidirect
100. #92470	C₂³	semidirect
101. #92469	C₂³	semidirect
104. C₈×D₃₁	C₂³	semidirect
105. C₈×D₃₁	C₂³	semidirect
168. #576	D₃₁	direct (N × H)
178. D₃₁	#576	direct (N × H)
184. H(5,𝔽₂)	C₂×D₃₁	semidirect
195. AGL(1,ℤ/8)	C₂×D₃₁	semidirect
201. #135	C₂×D₃₁	semidirect
210. C₂×D₈	C₂×D₃₁	semidirect
214. #131	C₂×D₃₁	semidirect
217. C₃₁	C₂×#576	semidirect
232. #41	C₂²×D₃₁	semidirect
249. C₂×D₄	C₂²×D₃₁	semidirect
254. Q₁₆	C₂²×D₃₁	semidirect
258. SD₁₆	C₂²×D₃₁	semidirect
263. D₈	C₂²×D₃₁	semidirect
264. D₈	C₂²×D₃₁	semidirect
267. M₁₆	C₂²×D₃₁	semidirect
273. C₂×C₈	C₂²×D₃₁	semidirect

Maximal subgroups

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

subgroup	index
H(5,𝔽₂)×D₃₁	2
#14521854	2
#14521853	2
#14521852	2
#14521851	2
#14521850	2
AGL(1,ℤ/8)×D₃₁	2
#135×D₃₁	2
C₂×D₈×D₃₁	2
#131×D₃₁	2
C₃₁×#576	2
C₂×#576	31

Subgroups — D31×#576 #16860459

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₃₁×#576 #16860459