Subgroups — D₃₁×#443 #16854597

Order 3968.

472 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#135×D₃₁	C₂	1	1	yes	maximal, characteristic
2	#131×D₃₁	C₂	1	2	yes	maximal, complemented
3	#14521167	C₂	1	1	yes	maximal, characteristic, complemented
4	#14521166	C₂	1	1	yes	maximal, characteristic, complemented
5	#14521165	C₂	1	1	yes	maximal, characteristic, complemented
6	#14521164	C₂	1	2	yes	maximal, complemented
7	#14521163	C₂	1	2	yes	maximal, complemented
8	#108×D₃₁	C₂	1	1	yes	maximal, characteristic, complemented
9	#104×D₃₁	C₂	1	2	yes	maximal, complemented
10	C₄×C₈×D₃₁	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃₁×#443	C₂	1	1	yes	maximal, characteristic, complemented
12	#41×D₃₁	C₂²	1	2	yes	—
13	C₃₁×#135	C₂²	1	1	yes	characteristic
14	C₃₁×#131	C₂²	1	2	yes	complemented
15	#11758527	C₂²	1	2	yes	—
16	#11758525	C₂²	1	2	yes	complemented
17	#11758496	C₄	1	1	yes	characteristic, complemented
18	Q₁₆×D₃₁	C₄	1	1	yes	characteristic, complemented
19	#11758493	C₄	1	2	yes	complemented
20	SD₁₆×D₃₁	C₄	1	2	yes	complemented
21	#11758489	C₄	1	1	yes	characteristic, complemented
22	D₈×D₃₁	C₄	1	1	yes	characteristic, complemented
23	#11758484	—	2	2	no	—
24	#11758484	C₂²	1	2	yes	complemented
25	M₁₆×D₃₁	—	2	2	no	—
26	M₁₆×D₃₁	C₂²	1	2	yes	—
27	#11758481	C₂²	1	1	yes	characteristic
28	#11758478	—	2	2	no	—
29	#11758478	C₂²	1	2	yes	complemented
30	C₂×C₈×D₃₁	—	2	2	no	—
31	C₂×C₈×D₃₁	C₂²	1	1	yes	characteristic
32	C₂×C₈×D₃₁	C₂²	1	1	yes	characteristic
33	C₄²×D₃₁	C₂²	1	1	yes	characteristic
34	C₃₁×#108	C₂²	1	1	yes	characteristic, complemented
35	C₃₁×#104	C₂²	1	2	yes	complemented
36	C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic, complemented
37	#11758425	C₂²	1	2	yes	complemented
38	#11758413	C₂²	1	2	yes	complemented
39	#11758410	C₂²	1	2	yes	complemented
40	#11758407	C₂²	1	1	yes	characteristic, complemented
41	C₈×Dic₃₁	C₂²	1	1	yes	characteristic, complemented
42	C₈×Dic₃₁	C₂²	1	1	yes	characteristic, complemented
43	#11758393	C₂²	1	2	yes	complemented
44	C₄×C₃₁⋊C₈	C₂²	1	1	yes	characteristic, complemented
45	#41×C₃₁	C₂³	1	2	yes	—
46	#92499	C₂×C₄	1	2	yes	—
47	Q₈×D₃₁	C₂×C₄	1	2	yes	—
48	#92497	—	2	2	no	—
49	#92497	C₂×C₄	1	2	yes	—
50	D₄×D₃₁	—	2	2	no	—
51	D₄×D₃₁	C₂×C₄	1	2	yes	—
52	#92495	—	2	2	no	—
53	#92495	C₂³	1	2	yes	—
54	C₂×C₄×D₃₁	—	2	1	no	—
55	C₂×C₄×D₃₁	—	2	2	no	—
56	C₂×C₄×D₃₁	C₂³	1	1	yes	characteristic
57	Q₁₆×C₃₁	C₂×C₄	1	1	yes	characteristic, complemented
58	SD₁₆×C₃₁	C₂×C₄	1	2	yes	complemented
59	D₈×C₃₁	C₂×C₄	1	1	yes	characteristic, complemented
60	M₁₆×C₃₁	—	2	2	no	—
61	M₁₆×C₃₁	C₂³	1	2	yes	—
62	C₂×C₈×C₃₁	—	2	2	no	—
63	C₂×C₈×C₃₁	C₂³	1	1	yes	characteristic
64	C₂×C₈×C₃₁	C₂³	1	1	yes	characteristic
65	C₄²×C₃₁	C₂³	1	1	yes	characteristic
66	#92482	C₂×C₄	1	2	yes	complemented
67	#92481	C₂×C₄	1	2	yes	complemented
68	#92480	C₂×C₄	1	2	yes	complemented
69	#92479	C₂×C₄	1	2	yes	complemented
70	C₄×Dic₃₁	—	2	1	no	—
71	C₄×Dic₃₁	C₂³	1	1	yes	characteristic
72	#92474	—	2	2	no	—
73	#92474	C₂³	1	2	yes	—
74	C₂×C₃₁⋊C₈	—	2	2	no	—
75	C₂×C₃₁⋊C₈	C₂³	1	1	yes	characteristic
76	C₂×C₃₁⋊C₈	C₂³	1	1	yes	characteristic
77	Dic₁₂₄	C₂×C₄	1	1	yes	characteristic, complemented
78	D₂₄₈	C₂×C₄	1	1	yes	characteristic, complemented
79	#92470	C₂×C₄	1	2	yes	complemented
80	#92469	—	2	2	no	—
81	#92469	—	2	2	no	—
82	#92469	—	2	2	no	—
83	C₈×D₃₁	—	2	2	no	—
84	C₈×D₃₁	—	2	2	no	—
85	C₈×D₃₁	—	2	2	no	—
86	C₈×D₃₁	C₂×C₄	1	1	yes	characteristic
87	C₈×D₃₁	C₂×C₄	1	1	yes	characteristic
88	C₈×D₃₁	C₂×C₄	1	1	yes	characteristic
89	C₈×D₃₁	C₂×C₄	1	1	yes	characteristic
90	C₈×D₃₁	D₄	1	1	yes	characteristic
91	C₈×D₃₁	D₄	1	1	yes	characteristic
92	C₈×D₃₁	D₄	1	1	yes	characteristic
93	C₈×D₃₁	D₄	1	1	yes	characteristic
94	C₂²×D₃₁	—	2	2	no	—
95	C₂²×D₃₁	#41	1	1	yes	characteristic
96	Q₈×C₃₁	C₂²×C₄	1	2	yes	—
97	D₄×C₃₁	—	2	2	no	—
98	D₄×C₃₁	C₂²×C₄	1	2	yes	—
99	C₂×C₄×C₃₁	—	2	1	no	—
100	C₂×C₄×C₃₁	—	2	2	no	—
101	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
102	C₂²⋊D₃₁	—	2	2	no	—
103	C₂²⋊D₃₁	—	2	2	no	—
104	C₂²⋊D₃₁	—	2	2	no	—
105	C₂×Dic₃₁	—	2	1	no	—
106	C₂×Dic₃₁	—	2	2	no	—
107	C₂×Dic₃₁	#41	1	1	yes	characteristic
108	D₁₂₄	—	2	2	no	—
109	D₁₂₄	C₂²×C₄	1	2	yes	—
110	C₄×D₃₁	—	2	4	no	—
111	C₄×D₃₁	—	2	2	no	—
112	C₄×D₃₁	—	2	2	no	—
113	C₄×D₃₁	—	2	2	no	—
114	C₄×D₃₁	C₂²×C₄	1	1	yes	characteristic
115	C₄×D₃₁	C₂²×C₄	1	1	yes	characteristic
116	C₄×D₃₁	C₂×D₄	1	1	yes	characteristic
117	C₄×D₃₁	C₂×D₄	1	1	yes	characteristic
118	Dic₆₂	—	2	2	no	—
119	Dic₆₂	C₂²×C₄	1	2	yes	—
120	C₈×C₃₁	—	2	2	no	—
121	C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
122	C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
123	C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
124	C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
125	C₃₁⋊C₈	—	2	2	no	—
126	C₃₁⋊C₈	C₂²×C₄	1	1	yes	characteristic
127	C₃₁⋊C₈	C₂²×C₄	1	1	yes	characteristic
128	C₃₁⋊C₈	C₂×D₄	1	1	yes	characteristic
129	C₃₁⋊C₈	C₂×D₄	1	1	yes	characteristic
130	C₂×#443	—	31	1	no	maximal, Sylow
131	C₂²×C₃₁	—	2	2	no	—
132	C₂²×C₃₁	C₂×#41	1	1	yes	characteristic
133	C₂×D₃₁	—	2	2	no	—
134	C₂×D₃₁	—	2	2	no	—
135	C₂×D₃₁	—	4	4	no	—
136	C₂×D₃₁	C₄×D₄	1	1	yes	characteristic
137	C₂×D₃₁	C₄×D₄	1	1	yes	characteristic
138	C₄×C₃₁	—	2	2	no	—
139	C₄×C₃₁	—	2	2	no	—
140	C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
141	C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
142	Dic₃₁	—	2	2	no	—
143	Dic₃₁	—	2	2	no	—
144	Dic₃₁	C₄×D₄	1	1	yes	characteristic
145	Dic₃₁	C₄×D₄	1	1	yes	characteristic
146	C₂×#135	—	31	1	no	—
147	C₂×#131	—	31	2	no	—
148	#443	—	31	1	no	—
149	#443	—	31	2	no	—
150	#443	—	31	1	no	—
151	#443	—	31	2	no	—
152	#443	—	31	1	no	—
153	#443	D₃₁	1	1	yes	characteristic, direct product factor
154	C₂×#108	—	31	1	no	—
155	C₂×#104	—	31	2	no	—
156	C₂×C₄×C₈	—	31	1	no	—
157	C₂×C₃₁	—	2	1	no	—
158	C₂×C₃₁	—	4	2	no	—
159	C₂×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
160	D₃₁	—	2	1	no	—
161	D₃₁	—	4	2	no	—
162	D₃₁	#443	1	2	yes	direct product factor
163	C₂×#41	—	31	2	no	—
164	#135	—	31	2	no	—
165	#135	—	31	1	no	—
166	#135	—	31	2	no	—
167	#135	—	31	1	no	—
168	#135	—	31	1	no	—
169	#135	C₂×D₃₁	1	1	yes	characteristic
170	C₂×Q₁₆	—	31	1	no	—
171	C₂×SD₁₆	—	31	2	no	—
172	C₂×D₈	—	31	1	no	—
173	#131	—	31	2	no	—
174	#131	—	31	2	no	—
175	#131	—	31	2	no	—
176	#131	—	62	2	no	—
177	#131	—	62	2	no	—
178	#131	C₂×D₃₁	1	2	yes	complemented
179	C₂×M₁₆	—	31	2	no	—
180	C₂×M₁₆	—	62	2	no	—
181	C₂²×C₈	—	31	1	no	—
182	C₂²×C₈	—	31	1	no	—
183	C₂²×C₈	—	62	2	no	—
184	C₂×C₄²	—	31	1	no	—
185	#108	—	31	2	no	—
186	#108	—	31	1	no	—
187	#108	C₂×D₃₁	1	1	yes	characteristic, complemented
188	#104	—	31	2	no	—
189	#104	—	31	2	no	—
190	#104	—	31	2	no	—
191	#104	C₂×D₃₁	1	2	yes	complemented
192	C₄×C₈	—	31	1	no	—
193	C₄×C₈	—	31	1	no	—
194	C₄×C₈	—	31	1	no	—
195	C₄×C₈	C₂×D₃₁	1	1	yes	characteristic, complemented
196	C₃₁	C₂×#443	1	1	yes	characteristic, complemented, Sylow
197	#41	—	31	2	no	—
198	#41	—	31	2	no	—
199	#41	—	31	2	no	—
200	#41	—	62	2	no	—
201	#41	—	62	2	no	—
202	#41	C₂²×D₃₁	1	2	yes	—
203	C₂×Q₈	—	31	2	no	—
204	C₂×D₄	—	31	2	no	—
205	C₂×D₄	—	62	2	no	—
206	C₂²×C₄	—	31	1	no	—
207	C₂²×C₄	—	62	2	no	—
208	C₂²×C₄	—	62	1	no	—
209	Q₁₆	—	31	2	no	—
210	Q₁₆	—	31	1	no	—
211	Q₁₆	C₄×D₃₁	1	1	yes	characteristic, complemented
212	SD₁₆	—	31	2	no	—
213	SD₁₆	—	31	2	no	—
214	SD₁₆	—	31	2	no	—
215	SD₁₆	C₄×D₃₁	1	2	yes	complemented
216	D₈	—	31	2	no	—
217	D₈	—	31	1	no	—
218	D₈	C₄×D₃₁	1	1	yes	characteristic, complemented
219	M₁₆	—	2	2	no	—
220	M₁₆	—	31	2	no	—
221	M₁₆	—	62	2	no	—
222	M₁₆	—	62	2	no	—
223	M₁₆	—	62	2	no	—
224	M₁₆	—	62	2	no	—
225	M₁₆	C₂²×D₃₁	1	2	yes	—
226	C₂×C₈	—	2	2	no	—
227	C₂×C₈	—	31	1	no	—
228	C₂×C₈	—	31	1	no	—
229	C₂×C₈	—	31	1	no	—
230	C₂×C₈	—	31	1	no	—
231	C₂×C₈	—	31	1	no	—
232	C₂×C₈	—	31	1	no	—
233	C₂×C₈	—	31	1	no	—
234	C₂×C₈	—	31	1	no	—
235	C₂×C₈	—	31	1	no	—
236	C₂×C₈	—	31	1	no	—
237	C₂×C₈	—	62	2	no	—
238	C₂×C₈	—	62	2	no	—
239	C₂×C₈	—	62	2	no	—
240	C₂×C₈	—	62	2	no	—
241	C₂×C₈	C₂²×D₃₁	1	1	yes	characteristic
242	C₂×C₈	C₂²×D₃₁	1	1	yes	characteristic
243	C₄²	—	31	1	no	—
244	C₄²	—	62	1	no	—
245	C₄²	C₂²×D₃₁	1	1	yes	characteristic
246	C₂³	—	31	1	no	—
247	C₂³	—	62	2	no	—
248	Q₈	—	31	2	no	—
249	Q₈	—	62	2	no	—
250	Q₈	C₂×C₄×D₃₁	1	2	yes	—
251	D₄	—	2	2	no	—
252	D₄	—	31	2	no	—
253	D₄	—	62	2	no	—
254	D₄	—	62	2	no	—
255	D₄	—	62	2	no	—
256	D₄	—	62	2	no	—
257	D₄	C₂×C₄×D₃₁	1	2	yes	—
258	C₂×C₄	—	2	1	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	31	1	no	—
261	C₂×C₄	—	31	1	no	—
262	C₂×C₄	—	31	1	no	—
263	C₂×C₄	—	31	1	no	—
264	C₂×C₄	—	31	1	no	—
265	C₂×C₄	—	62	2	no	—
266	C₂×C₄	—	62	2	no	—
267	C₂×C₄	—	62	4	no	—
268	C₂×C₄	—	62	2	no	—
269	C₂×C₄	—	62	1	no	—
270	C₂×C₄	—	62	2	no	—
271	C₂×C₄	C₂³×D₃₁	1	1	yes	characteristic
272	C₈	—	2	2	no	—
273	C₈	—	31	1	no	—
274	C₈	—	31	1	no	—
275	C₈	—	31	1	no	—
276	C₈	—	31	1	no	—
277	C₈	—	62	2	no	—
278	C₈	C₂×C₄×D₃₁	1	1	yes	characteristic
279	C₈	C₂×C₄×D₃₁	1	1	yes	characteristic
280	C₈	D₄×D₃₁	1	1	yes	characteristic
281	C₈	D₄×D₃₁	1	1	yes	characteristic
282	C₂²	—	2	2	no	—
283	C₂²	—	31	1	no	—
284	C₂²	—	31	1	no	—
285	C₂²	—	62	2	no	—
286	C₂²	—	62	2	no	—
287	C₂²	—	124	4	no	—
288	C₂²	#41×D₃₁	1	1	yes	characteristic
289	C₄	—	2	2	no	—
290	C₄	—	2	2	no	—
291	C₄	—	31	1	no	—
292	C₄	—	31	1	no	—
293	C₄	—	62	2	no	—
294	C₄	—	62	2	no	—
295	C₄	C₂²×C₄×D₃₁	1	1	yes	characteristic
296	C₄	C₂×D₄×D₃₁	1	1	yes	characteristic
297	C₂	—	2	1	no	—
298	C₂	—	4	2	no	—
299	C₂	—	31	2	no	—
300	C₂	—	62	1	no	—
301	C₂	—	124	2	no	—
302	C₂	C₄×D₄×D₃₁	1	1	yes	characteristic

	subgroup	G/H
center	C₈	D₄×D₃₁
commutator	C₄×C₃₁	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₃₁
Fitting	C₃₁×#443	C₂
socle	C₂×C₃₁	C₂×C₄×D₄
prime radical	#135×D₃₁	C₂
cosocle	C₂×C₄×C₃₁	C₂⁴
hypercenter	#443	D₃₁
hypocenter	C₃₁	C₂×#443
LCS	G ▷ C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₈ ◁ C₂×C₈ ◁ #443

Complements

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

N	H ≅ G/N	type
2. #131×D₃₁	C₂	semidirect
3. #14521167	C₂	semidirect
4. #14521166	C₂	semidirect
5. #14521165	C₂	semidirect
6. #14521164	C₂	semidirect
7. #14521163	C₂	semidirect
8. #108×D₃₁	C₂	semidirect
9. #104×D₃₁	C₂	semidirect
10. C₄×C₈×D₃₁	C₂	semidirect
11. C₃₁×#443	C₂	semidirect
14. C₃₁×#131	C₂²	semidirect
16. #11758525	C₂²	semidirect
17. #11758496	C₄	semidirect
18. Q₁₆×D₃₁	C₄	semidirect
19. #11758493	C₄	semidirect
20. SD₁₆×D₃₁	C₄	semidirect
21. #11758489	C₄	semidirect
22. D₈×D₃₁	C₄	semidirect
24. #11758484	C₂²	semidirect
29. #11758478	C₂²	semidirect
34. C₃₁×#108	C₂²	semidirect
35. C₃₁×#104	C₂²	semidirect
36. C₄×C₈×C₃₁	C₂²	semidirect
37. #11758425	C₂²	semidirect
38. #11758413	C₂²	semidirect
39. #11758410	C₂²	semidirect
40. #11758407	C₂²	semidirect
41. C₈×Dic₃₁	C₂²	semidirect
42. C₈×Dic₃₁	C₂²	semidirect
43. #11758393	C₂²	semidirect
44. C₄×C₃₁⋊C₈	C₂²	semidirect
57. Q₁₆×C₃₁	C₂×C₄	semidirect
58. SD₁₆×C₃₁	C₂×C₄	semidirect
59. D₈×C₃₁	C₂×C₄	semidirect
66. #92482	C₂×C₄	semidirect
67. #92481	C₂×C₄	semidirect
68. #92480	C₂×C₄	semidirect
69. #92479	C₂×C₄	semidirect
77. Dic₁₂₄	C₂×C₄	semidirect
78. D₂₄₈	C₂×C₄	semidirect
79. #92470	C₂×C₄	semidirect
153. #443	D₃₁	direct (N × H)
162. D₃₁	#443	direct (N × H)
178. #131	C₂×D₃₁	semidirect
187. #108	C₂×D₃₁	semidirect
191. #104	C₂×D₃₁	semidirect
195. C₄×C₈	C₂×D₃₁	semidirect
196. C₃₁	C₂×#443	semidirect
211. Q₁₆	C₄×D₃₁	semidirect
215. SD₁₆	C₄×D₃₁	semidirect
218. D₈	C₄×D₃₁	semidirect

Maximal subgroups

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

subgroup	index
#135×D₃₁	2
#131×D₃₁	2
#14521167	2
#14521166	2
#14521165	2
#14521164	2
#14521163	2
#108×D₃₁	2
#104×D₃₁	2
C₄×C₈×D₃₁	2
C₃₁×#443	2
C₂×#443	31

Subgroups — D31×#443 #16854597

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₃₁×#443 #16854597