Subgroups — D₃₁×#533 #16858996

Order 3968.

1168 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14521606	C₂	1	1	yes	maximal, characteristic, complemented
2	#14521605	C₂	1	2	yes	maximal, complemented
3	#14521604	C₂	1	2	yes	maximal, complemented
4	#14521603	C₂	1	4	yes	maximal, complemented
5	#14521602	C₂	1	2	yes	maximal, complemented
6	#14521601	C₂	1	4	yes	maximal, complemented
7	C₄⋊Q₈×D₃₁	C₂	1	2	yes	maximal, complemented
8	#125×D₃₁	C₂	1	2	yes	maximal, complemented
9	C₂²⋊Q₈×D₃₁	C₂	1	4	yes	maximal
10	C₄×Q₈×D₃₁	C₂	1	4	yes	maximal, complemented
11	#117×D₃₁	C₂	1	2	yes	maximal
12	C₂×C₄²×D₃₁	C₂	1	1	yes	maximal, characteristic
13	C₃₁×#533	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂×Q₈×D₃₁	C₂²	1	4	yes	—
15	C₂²×C₄×D₃₁	C₂²	1	2	yes	—
16	C₂²×C₄×D₃₁	C₂²	1	1	yes	characteristic
17	C₃₁×C₄⋊Q₈	C₂²	1	2	yes	complemented
18	C₃₁×#125	C₂²	1	2	yes	complemented
19	C₃₁×C₂²⋊Q₈	C₂²	1	4	yes	—
20	C₄×Q₈×C₃₁	C₂²	1	4	yes	complemented
21	C₃₁×#117	C₂²	1	2	yes	—
22	C₂×C₄²×C₃₁	C₂²	1	1	yes	characteristic
23	#11758523	C₂²	1	4	yes	—
24	Q₈×Dic₃₁	C₂²	1	4	yes	complemented
25	#11758521	C₂²	1	4	yes	complemented
26	#11758503	C₂²	1	2	yes	—
27	#11758501	C₂²	1	4	yes	—
28	C₂×C₄×Dic₃₁	C₂²	1	2	yes	—
29	C₂×C₄×Dic₃₁	C₂²	1	1	yes	characteristic
30	#11758474	C₂²	1	4	yes	—
31	#11758473	C₂²	1	8	yes	—
32	#11758469	C₂²	1	4	yes	—
33	C₄⋊C₄×D₃₁	C₂²	1	4	yes	—
34	C₄⋊C₄×D₃₁	C₂²	1	8	yes	—
35	C₄⋊C₄×D₃₁	C₂²	1	4	yes	—
36	#11758467	C₂²	1	4	yes	complemented
37	#11758466	C₂²	1	8	yes	complemented
38	#11758465	C₂²	1	4	yes	complemented
39	#11758465	C₂²	1	4	yes	complemented
40	#11758464	C₂²	1	4	yes	complemented
41	#11758464	C₂²	1	8	yes	complemented
42	C₂²⋊C₄×D₃₁	C₂²	1	4	yes	—
43	#11758455	C₂²	1	8	yes	—
44	#11758454	C₂²	1	4	yes	—
45	#11758449	C₂²	1	4	yes	—
46	C₄²×D₃₁	C₂²	1	2	yes	—
47	C₄²×D₃₁	C₂²	1	1	yes	characteristic
48	C₄²×D₃₁	C₂²	1	1	yes	characteristic
49	C₄²×D₃₁	C₂²	1	2	yes	—
50	C₄²×D₃₁	C₂²	1	1	yes	characteristic
51	C₄²×D₃₁	C₂²	1	1	yes	characteristic
52	C₄²×D₃₁	C₂²	1	4	yes	—
53	#11758447	C₂²	1	2	yes	complemented
54	#11758446	C₂²	1	2	yes	complemented
55	C₄×Dic₆₂	C₂²	1	8	yes	complemented
56	C₄×Dic₆₂	C₂²	1	4	yes	complemented
57	C₂³×D₃₁	C₂³	1	1	yes	characteristic
58	C₂×Q₈×C₃₁	C₂³	1	4	yes	—
59	C₂²×C₄×C₃₁	C₂³	1	2	yes	—
60	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
61	C₂²×Dic₃₁	C₂³	1	2	yes	—
62	C₂²×Dic₃₁	C₂³	1	1	yes	characteristic
63	Q₈×D₃₁	—	2	16	no	—
64	C₂×C₄×D₃₁	—	2	8	no	—
65	C₂×C₄×D₃₁	Q₈	1	8	yes	complemented
66	C₂×C₄×D₃₁	C₂³	1	2	yes	—
67	C₂×C₄×D₃₁	C₂³	1	2	yes	—
68	C₂×C₄×D₃₁	C₂³	1	2	yes	—
69	C₂×C₄×D₃₁	C₂³	1	8	yes	—
70	C₂×C₄×D₃₁	C₂³	1	1	yes	characteristic
71	C₂×C₄×D₃₁	C₂³	1	1	yes	characteristic
72	C₂×C₄×D₃₁	C₂³	1	1	yes	characteristic
73	C₂×C₄×D₃₁	C₂³	1	1	yes	characteristic
74	C₂×C₄×D₃₁	C₂³	1	2	yes	—
75	C₂×Dic₆₂	C₂³	1	4	yes	—
76	C₂×Dic₆₂	C₂³	1	8	yes	—
77	C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
78	C₄⋊C₄×C₃₁	C₂³	1	8	yes	—
79	C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
80	C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
81	C₄²×C₃₁	C₂³	1	2	yes	—
82	C₄²×C₃₁	C₂³	1	4	yes	—
83	C₄²×C₃₁	C₂³	1	1	yes	characteristic
84	C₄²×C₃₁	C₂³	1	1	yes	characteristic
85	C₂²⋊Dic₃₁	C₂³	1	4	yes	—
86	(C₂×D₃₁)⋊C₄	C₂³	1	8	yes	—
87	C₄⋊Dic₃₁	C₂³	1	4	yes	—
88	C₄⋊Dic₃₁	C₂³	1	8	yes	—
89	C₄⋊Dic₃₁	C₂³	1	4	yes	—
90	Dic₃₁⋊C₄	C₂³	1	8	yes	—
91	Dic₃₁⋊C₄	C₂³	1	8	yes	—
92	Dic₃₁⋊C₄	C₂³	1	8	yes	—
93	Dic₃₁⋊C₄	C₂³	1	8	yes	—
94	C₄×Dic₃₁	C₂³	1	2	yes	—
95	C₄×Dic₃₁	C₂³	1	4	yes	—
96	C₄×Dic₃₁	C₂³	1	2	yes	—
97	C₄×Dic₃₁	C₂³	1	2	yes	—
98	C₄×Dic₃₁	C₂³	1	1	yes	characteristic
99	C₄×Dic₃₁	C₂³	1	2	yes	—
100	C₄×Dic₃₁	C₂³	1	2	yes	—
101	C₄×Dic₃₁	C₂³	1	1	yes	characteristic
102	C₄×Dic₃₁	C₂³	1	8	yes	—
103	C₂³×C₃₁	C₂⁴	1	1	yes	characteristic
104	C₂²×D₃₁	—	2	4	no	—
105	C₂²×D₃₁	C₂×Q₈	1	4	yes	—
106	C₂²×D₃₁	C₂⁴	1	1	yes	characteristic
107	C₂²×D₃₁	C₂⁴	1	1	yes	characteristic
108	Q₈×C₃₁	—	2	8	no	—
109	C₂×C₄×C₃₁	—	2	4	no	—
110	C₂×C₄×C₃₁	C₂×Q₈	1	4	yes	complemented
111	C₂×C₄×C₃₁	C₂⁴	1	2	yes	—
112	C₂×C₄×C₃₁	C₂⁴	1	8	yes	—
113	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
114	C₂×C₄×C₃₁	C₂⁴	1	2	yes	—
115	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
116	C₂×Dic₃₁	—	2	4	no	—
117	C₂×Dic₃₁	C₂×Q₈	1	4	yes	complemented
118	C₂×Dic₃₁	C₂⁴	1	2	yes	—
119	C₂×Dic₃₁	C₂⁴	1	8	yes	—
120	C₂×Dic₃₁	C₂⁴	1	1	yes	characteristic
121	C₂×Dic₃₁	C₂⁴	1	1	yes	characteristic
122	C₂×Dic₃₁	C₂⁴	1	2	yes	—
123	C₄×D₃₁	—	2	4	no	—
124	C₄×D₃₁	—	2	16	no	—
125	C₄×D₃₁	—	2	4	no	—
126	C₄×D₃₁	—	2	4	no	—
127	C₄×D₃₁	C₂×Q₈	1	4	yes	—
128	C₄×D₃₁	C₂×Q₈	1	4	yes	—
129	C₄×D₃₁	C₂×Q₈	1	4	yes	complemented
130	C₄×D₃₁	C₂×Q₈	1	4	yes	complemented
131	C₄×D₃₁	#41	1	8	yes	—
132	Dic₆₂	—	2	8	no	—
133	Dic₆₂	—	2	16	no	—
134	C₂×#533	—	31	1	no	maximal, Sylow
135	C₂²×C₃₁	—	2	2	no	—
136	C₂²×C₃₁	C₂²×Q₈	1	2	yes	—
137	C₂²×C₃₁	C₂⁵	1	1	yes	characteristic
138	C₂×D₃₁	—	2	2	no	—
139	C₂×D₃₁	—	2	8	no	—
140	C₂×D₃₁	C₂²×Q₈	1	2	yes	—
141	C₂×D₃₁	C₂²×Q₈	1	2	yes	—
142	C₂×D₃₁	C₂×#41	1	4	yes	—
143	C₄×C₃₁	—	2	8	no	—
144	C₄×C₃₁	—	2	2	no	—
145	C₄×C₃₁	C₂²×Q₈	1	2	yes	—
146	C₄×C₃₁	C₂²×Q₈	1	2	yes	—
147	C₄×C₃₁	C₂×#41	1	4	yes	—
148	Dic₃₁	—	2	8	no	—
149	Dic₃₁	—	2	2	no	—
150	Dic₃₁	C₂²×Q₈	1	2	yes	—
151	Dic₃₁	C₂²×Q₈	1	2	yes	—
152	Dic₃₁	C₂×#41	1	4	yes	—
153	#533	—	31	2	no	—
154	#533	—	31	2	no	—
155	#533	—	31	4	no	—
156	#533	—	31	4	no	—
157	#533	—	31	2	no	—
158	#533	—	31	1	no	—
159	#533	D₃₁	1	1	yes	characteristic, direct product factor
160	C₂×C₄⋊Q₈	—	31	2	no	—
161	C₂×#125	—	31	2	no	—
162	C₂×C₂²⋊Q₈	—	31	4	no	—
163	C₂×C₄×Q₈	—	31	4	no	—
164	C₂×#117	—	31	2	no	—
165	C₂²×C₄²	—	31	1	no	—
166	C₂×C₃₁	—	2	2	no	—
167	C₂×C₃₁	C₂³×Q₈	1	1	yes	characteristic
168	C₂×C₃₁	C₂²×#41	1	2	yes	—
169	D₃₁	—	2	2	no	—
170	D₃₁	#533	1	4	yes	direct product factor
171	C₂²×Q₈	—	31	4	no	—
172	C₂³×C₄	—	31	1	no	—
173	C₂³×C₄	—	31	2	no	—
174	C₄⋊Q₈	—	31	4	no	—
175	C₄⋊Q₈	—	31	4	no	—
176	C₄⋊Q₈	—	31	4	no	—
177	C₄⋊Q₈	—	31	2	no	—
178	C₄⋊Q₈	C₂×D₃₁	1	2	yes	complemented
179	#125	—	31	8	no	—
180	#125	—	31	4	no	—
181	#125	—	31	2	no	—
182	#125	C₂×D₃₁	1	2	yes	complemented
183	C₂²⋊Q₈	—	31	4	no	—
184	C₂²⋊Q₈	—	31	4	no	—
185	C₂²⋊Q₈	—	31	8	no	—
186	C₂²⋊Q₈	—	31	4	no	—
187	C₂²⋊Q₈	—	31	8	no	—
188	C₂²⋊Q₈	C₂×D₃₁	1	4	yes	—
189	C₄×Q₈	—	31	4	no	—
190	C₄×Q₈	—	31	4	no	—
191	C₄×Q₈	—	31	8	no	—
192	C₄×Q₈	—	31	8	no	—
193	C₄×Q₈	—	31	4	no	—
194	C₄×Q₈	C₂×D₃₁	1	4	yes	complemented
195	#117	—	31	2	no	—
196	#117	—	31	4	no	—
197	#117	—	31	4	no	—
198	#117	—	31	4	no	—
199	#117	C₂×D₃₁	1	2	yes	—
200	C₂×C₄⋊C₄	—	31	4	no	—
201	C₂×C₄⋊C₄	—	31	8	no	—
202	C₂×C₄⋊C₄	—	31	4	no	—
203	C₂×C₂²⋊C₄	—	31	4	no	—
204	C₂×C₄²	—	31	2	no	—
205	C₂×C₄²	—	31	1	no	—
206	C₂×C₄²	—	31	4	no	—
207	C₂×C₄²	—	31	2	no	—
208	C₂×C₄²	—	31	1	no	—
209	C₂×C₄²	—	31	1	no	—
210	C₂×C₄²	—	31	1	no	—
211	C₂×C₄²	—	31	1	no	—
212	C₂×C₄²	—	31	2	no	—
213	C₂×C₄²	C₂×D₃₁	1	1	yes	characteristic
214	C₃₁	C₂×#533	1	1	yes	characteristic, complemented, Sylow
215	C₂⁴	—	31	1	no	—
216	C₂×Q₈	—	31	8	no	—
217	C₂×Q₈	—	31	4	no	—
218	C₂×Q₈	—	62	16	no	—
219	C₂×Q₈	C₂²×D₃₁	1	4	yes	—
220	C₂²×C₄	—	31	1	no	—
221	C₂²×C₄	—	31	8	no	—
222	C₂²×C₄	—	31	1	no	—
223	C₂²×C₄	—	31	2	no	—
224	C₂²×C₄	—	31	1	no	—
225	C₂²×C₄	—	31	1	no	—
226	C₂²×C₄	—	31	1	no	—
227	C₂²×C₄	—	31	2	no	—
228	C₂²×C₄	—	31	2	no	—
229	C₂²×C₄	—	31	2	no	—
230	C₂²×C₄	—	31	2	no	—
231	C₂²×C₄	—	31	8	no	—
232	C₂²×C₄	—	62	8	no	—
233	C₂²×C₄	C₂²×D₃₁	1	2	yes	—
234	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
235	C₄⋊C₄	—	31	8	no	—
236	C₄⋊C₄	—	31	8	no	—
237	C₄⋊C₄	—	31	8	no	—
238	C₄⋊C₄	—	31	8	no	—
239	C₄⋊C₄	—	31	4	no	—
240	C₄⋊C₄	—	31	8	no	—
241	C₄⋊C₄	—	31	4	no	—
242	C₄⋊C₄	C₂²×D₃₁	1	4	yes	—
243	C₄⋊C₄	C₂²×D₃₁	1	8	yes	—
244	C₄⋊C₄	C₂²×D₃₁	1	4	yes	—
245	C₂²⋊C₄	—	31	4	no	—
246	C₂²⋊C₄	—	31	8	no	—
247	C₂²⋊C₄	C₂²×D₃₁	1	4	yes	—
248	C₄²	—	31	2	no	—
249	C₄²	—	31	2	no	—
250	C₄²	—	31	2	no	—
251	C₄²	—	31	1	no	—
252	C₄²	—	31	2	no	—
253	C₄²	—	31	1	no	—
254	C₄²	—	31	8	no	—
255	C₄²	—	31	2	no	—
256	C₄²	—	31	4	no	—
257	C₄²	C₂²×D₃₁	1	1	yes	characteristic
258	C₄²	C₂²×D₃₁	1	1	yes	characteristic
259	C₄²	C₂²×D₃₁	1	2	yes	—
260	C₄²	C₂²×D₃₁	1	4	yes	—
261	C₂³	—	31	1	no	—
262	C₂³	—	31	4	no	—
263	C₂³	—	31	1	no	—
264	C₂³	—	62	4	no	—
265	C₂³	C₂³×D₃₁	1	1	yes	characteristic
266	Q₈	—	2	8	no	—
267	Q₈	—	62	16	no	—
268	Q₈	—	62	8	no	—
269	C₂×C₄	—	2	4	no	—
270	C₂×C₄	—	31	4	no	—
271	C₂×C₄	—	31	1	no	—
272	C₂×C₄	—	31	1	no	—
273	C₂×C₄	—	31	2	no	—
274	C₂×C₄	—	31	4	no	—
275	C₂×C₄	—	31	4	no	—
276	C₂×C₄	—	31	4	no	—
277	C₂×C₄	—	31	4	no	—
278	C₂×C₄	—	31	8	no	—
279	C₂×C₄	—	31	2	no	—
280	C₂×C₄	—	31	8	no	—
281	C₂×C₄	—	62	4	no	—
282	C₂×C₄	—	62	4	no	—
283	C₂×C₄	—	62	16	no	—
284	C₂×C₄	—	62	4	no	—
285	C₂×C₄	—	62	4	no	—
286	C₂×C₄	Q₈×D₃₁	1	4	yes	complemented
287	C₂×C₄	C₂³×D₃₁	1	2	yes	—
288	C₂×C₄	C₂³×D₃₁	1	8	yes	—
289	C₂×C₄	C₂³×D₃₁	1	1	yes	characteristic
290	C₂×C₄	C₂³×D₃₁	1	2	yes	—
291	C₂×C₄	C₂³×D₃₁	1	1	yes	characteristic
292	C₂²	—	2	2	no	—
293	C₂²	—	31	2	no	—
294	C₂²	—	31	4	no	—
295	C₂²	—	31	2	no	—
296	C₂²	—	62	8	no	—
297	C₂²	—	62	2	no	—
298	C₂²	C₂×Q₈×D₃₁	1	2	yes	—
299	C₂²	C₂⁴×D₃₁	1	1	yes	characteristic
300	C₄	—	2	2	no	—
301	C₄	—	2	8	no	—
302	C₄	—	31	2	no	—
303	C₄	—	31	2	no	—
304	C₄	—	31	4	no	—
305	C₄	—	62	2	no	—
306	C₄	—	62	8	no	—
307	C₄	C₂×Q₈×D₃₁	1	2	yes	—
308	C₄	C₂×Q₈×D₃₁	1	2	yes	—
309	C₄	#41×D₃₁	1	4	yes	—
310	C₂	—	2	2	no	—
311	C₂	—	31	4	no	—
312	C₂	—	62	2	no	—
313	C₂	C₂²×Q₈×D₃₁	1	1	yes	characteristic
314	C₂	C₂×#41×D₃₁	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂³×D₃₁
commutator, socle, cosocle	C₂²×C₃₁	C₂⁵
Frattini	C₂²	C₂⁴×D₃₁
Fitting	C₃₁×#533	C₂
prime radical	C₂³×D₃₁	C₂³
hypercenter	#533	D₃₁
hypocenter	C₃₁	C₂×#533
LCS	G ▷ C₂²×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂×C₄ ◁ #533

Complements

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

N	H ≅ G/N	type
1. #14521606	C₂	semidirect
2. #14521605	C₂	semidirect
3. #14521604	C₂	semidirect
4. #14521603	C₂	semidirect
5. #14521602	C₂	semidirect
6. #14521601	C₂	semidirect
7. C₄⋊Q₈×D₃₁	C₂	semidirect
8. #125×D₃₁	C₂	semidirect
10. C₄×Q₈×D₃₁	C₂	semidirect
13. C₃₁×#533	C₂	semidirect
17. C₃₁×C₄⋊Q₈	C₂²	semidirect
18. C₃₁×#125	C₂²	semidirect
20. C₄×Q₈×C₃₁	C₂²	semidirect
24. Q₈×Dic₃₁	C₂²	semidirect
25. #11758521	C₂²	semidirect
36. #11758467	C₂²	semidirect
37. #11758466	C₂²	semidirect
38. #11758465	C₂²	semidirect
39. #11758465	C₂²	semidirect
40. #11758464	C₂²	semidirect
41. #11758464	C₂²	semidirect
53. #11758447	C₂²	semidirect
54. #11758446	C₂²	semidirect
55. C₄×Dic₆₂	C₂²	semidirect
56. C₄×Dic₆₂	C₂²	semidirect
65. C₂×C₄×D₃₁	Q₈	semidirect
110. C₂×C₄×C₃₁	C₂×Q₈	semidirect
117. C₂×Dic₃₁	C₂×Q₈	semidirect
129. C₄×D₃₁	C₂×Q₈	semidirect
130. C₄×D₃₁	C₂×Q₈	semidirect
159. #533	D₃₁	direct (N × H)
170. D₃₁	#533	direct (N × H)
178. C₄⋊Q₈	C₂×D₃₁	semidirect
182. #125	C₂×D₃₁	semidirect
194. C₄×Q₈	C₂×D₃₁	semidirect
214. C₃₁	C₂×#533	semidirect
286. C₂×C₄	Q₈×D₃₁	semidirect

Maximal subgroups

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

subgroup	index
#14521606	2
#14521605	2
#14521604	2
#14521603	2
#14521602	2
#14521601	2
C₄⋊Q₈×D₃₁	2
#125×D₃₁	2
C₂²⋊Q₈×D₃₁	2
C₄×Q₈×D₃₁	2
#117×D₃₁	2
C₂×C₄²×D₃₁	2
C₃₁×#533	2
C₂×#533	31

Subgroups — D31×#533 #16858996

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₃₁×#533 #16858996