Subgroups — D₃₁×#475 #16855472

Order 3968.

416 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊Q₈×D₃₁	C₂	1	1	yes	maximal, characteristic
2	C₄×Q₈×D₃₁	C₂	1	1	yes	maximal, characteristic
3	#14521335	C₂	1	1	yes	maximal, characteristic, complemented
4	#14521334	C₂	1	1	yes	maximal, characteristic, complemented
5	#14521333	C₂	1	2	yes	maximal, complemented
6	#14521332	C₂	1	2	yes	maximal, complemented
7	#14521331	C₂	1	1	yes	maximal, characteristic, complemented
8	#106×D₃₁	C₂	1	2	yes	maximal
9	C₄⋊C₈×D₃₁	C₂	1	1	yes	maximal, characteristic
10	#103×D₃₁	C₂	1	2	yes	maximal
11	C₃₁×#475	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×Q₈×D₃₁	—	2	1	no	—
13	C₂×Q₈×D₃₁	C₂²	1	1	yes	characteristic
14	C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
15	C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
16	Q₈×Dic₃₁	C₂²	1	1	yes	characteristic
17	#11758521	—	2	1	no	—
18	C₂×C₈×D₃₁	C₂²	1	2	yes	—
19	C₄⋊C₄×D₃₁	—	2	1	no	—
20	C₄⋊C₄×D₃₁	—	2	1	no	—
21	C₄⋊C₄×D₃₁	C₂²	1	1	yes	characteristic
22	C₄⋊C₄×D₃₁	C₂²	1	2	yes	—
23	#11758465	—	2	1	no	—
24	#11758465	C₂²	1	2	yes	—
25	#11758464	—	2	1	no	—
26	#11758464	C₂²	1	1	yes	characteristic
27	C₄²×D₃₁	—	2	1	no	—
28	C₄²×D₃₁	C₂²	1	1	yes	characteristic
29	#11758446	C₂²	1	1	yes	characteristic
30	C₄×Dic₆₂	—	2	1	no	—
31	C₄×Dic₆₂	C₂²	1	1	yes	characteristic
32	C₃₁×#106	C₂²	1	2	yes	—
33	C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
34	C₃₁×#103	C₂²	1	2	yes	—
35	#11758423	C₂²	1	2	yes	—
36	#11758405	C₂²	1	2	yes	—
37	#11758404	C₂²	1	2	yes	—
38	Dic₃₁⋊C₈	C₂²	1	2	yes	—
39	#11758398	C₂²	1	2	yes	—
40	#11758396	C₂²	1	2	yes	—
41	#11758396	C₂²	1	2	yes	—
42	C₄⋊(C₃₁⋊C₈)	C₂²	1	1	yes	characteristic
43	C₂×Q₈×C₃₁	—	2	1	no	—
44	C₂×Q₈×C₃₁	C₂³	1	1	yes	characteristic
45	Q₈×D₃₁	—	2	2	no	—
46	Q₈×D₃₁	—	4	4	no	—
47	Q₈×D₃₁	Q₈	1	4	yes	complemented
48	C₂×C₄×D₃₁	—	2	1	no	—
49	C₂×C₄×D₃₁	—	2	1	no	—
50	C₂×C₄×D₃₁	—	2	2	no	—
51	C₂×C₄×D₃₁	D₄	1	1	yes	characteristic
52	C₂×C₄×D₃₁	D₄	1	1	yes	characteristic
53	C₂×C₄×D₃₁	C₂³	1	1	yes	characteristic
54	C₂×Dic₆₂	—	2	1	no	—
55	C₂×Dic₆₂	—	2	1	no	—
56	C₂×Dic₆₂	—	2	2	no	—
57	C₂×Dic₆₂	C₂³	1	1	yes	characteristic
58	C₂×C₈×C₃₁	C₂³	1	2	yes	—
59	C₄⋊C₄×C₃₁	—	2	1	no	—
60	C₄⋊C₄×C₃₁	—	2	1	no	—
61	C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
62	C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
63	C₄²×C₃₁	—	2	1	no	—
64	C₄²×C₃₁	C₂³	1	1	yes	characteristic
65	C₄⋊Dic₃₁	—	2	1	no	—
66	C₄⋊Dic₃₁	—	2	1	no	—
67	C₄⋊Dic₃₁	C₂³	1	1	yes	characteristic
68	C₄⋊Dic₃₁	C₂³	1	2	yes	—
69	Dic₃₁⋊C₄	—	2	1	no	—
70	Dic₃₁⋊C₄	—	2	1	no	—
71	Dic₃₁⋊C₄	—	2	1	no	—
72	Dic₃₁⋊C₄	—	2	2	no	—
73	Dic₃₁⋊C₄	—	2	2	no	—
74	C₄×Dic₃₁	—	2	1	no	—
75	C₄×Dic₃₁	—	2	1	no	—
76	C₄×Dic₃₁	—	2	1	no	—
77	C₄×Dic₃₁	D₄	1	1	yes	characteristic
78	C₄×Dic₃₁	D₄	1	1	yes	characteristic
79	C₄×Dic₃₁	C₂³	1	1	yes	characteristic
80	C₂×C₃₁⋊C₈	C₂³	1	2	yes	—
81	C₈×D₃₁	—	2	4	no	—
82	C₂²×D₃₁	C₂×D₄	1	1	yes	characteristic
83	Q₈×C₃₁	—	2	1	no	—
84	Q₈×C₃₁	—	4	2	no	—
85	Q₈×C₃₁	C₂×Q₈	1	2	yes	complemented
86	C₂×C₄×C₃₁	—	2	1	no	—
87	C₂×C₄×C₃₁	—	2	1	no	—
88	C₂×C₄×C₃₁	—	2	2	no	—
89	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
90	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
91	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
92	C₂×Dic₃₁	—	2	1	no	—
93	C₂×Dic₃₁	—	2	1	no	—
94	C₂×Dic₃₁	—	2	2	no	—
95	C₂×Dic₃₁	C₂×D₄	1	1	yes	characteristic
96	C₂×Dic₃₁	C₂×D₄	1	1	yes	characteristic
97	C₂×Dic₃₁	C₂×D₄	1	1	yes	characteristic
98	C₄×D₃₁	—	2	4	no	—
99	C₄×D₃₁	—	2	2	no	—
100	C₄×D₃₁	—	4	4	no	—
101	C₄×D₃₁	—	4	2	no	—
102	C₄×D₃₁	SD₁₆	1	4	yes	—
103	C₄×D₃₁	C₂×Q₈	1	2	yes	—
104	C₄×D₃₁	#41	1	2	yes	—
105	Dic₆₂	—	2	1	no	—
106	Dic₆₂	—	2	4	no	—
107	Dic₆₂	—	4	2	no	—
108	Dic₆₂	—	4	4	no	—
109	Dic₆₂	C₂×Q₈	1	2	yes	complemented
110	C₈×C₃₁	—	2	2	no	—
111	C₃₁⋊C₈	—	2	2	no	—
112	C₂×#475	—	31	1	no	maximal, Sylow
113	C₂²×C₃₁	C₂²×D₄	1	1	yes	characteristic
114	C₂×D₃₁	C₂²⋊Q₈	1	2	yes	—
115	C₂×D₃₁	C₂×SD₁₆	1	2	yes	—
116	C₂×D₃₁	#137	1	2	yes	—
117	C₄×C₃₁	—	2	2	no	—
118	C₄×C₃₁	—	2	1	no	—
119	C₄×C₃₁	—	4	2	no	—
120	C₄×C₃₁	—	4	1	no	—
121	C₄×C₃₁	C₂×SD₁₆	1	2	yes	—
122	C₄×C₃₁	C₂²×Q₈	1	1	yes	characteristic
123	C₄×C₃₁	C₂×#41	1	1	yes	characteristic
124	Dic₃₁	—	2	2	no	—
125	Dic₃₁	—	2	1	no	—
126	Dic₃₁	—	4	2	no	—
127	Dic₃₁	—	4	1	no	—
128	Dic₃₁	C₂²⋊Q₈	1	2	yes	—
129	Dic₃₁	C₂×SD₁₆	1	2	yes	—
130	C₂×C₄⋊Q₈	—	31	1	no	—
131	C₂×C₄×Q₈	—	31	1	no	—
132	#475	—	31	1	no	—
133	#475	—	31	1	no	—
134	#475	—	31	2	no	—
135	#475	—	31	1	no	—
136	#475	—	31	2	no	—
137	#475	D₃₁	1	1	yes	characteristic, direct product factor
138	C₂×#106	—	31	2	no	—
139	C₂×C₄⋊C₈	—	31	1	no	—
140	C₂×#103	—	31	2	no	—
141	C₂×C₃₁	C₂×C₂²⋊Q₈	1	1	yes	characteristic
142	C₂×C₃₁	C₂²×SD₁₆	1	1	yes	characteristic
143	C₂×C₃₁	C₂×#137	1	1	yes	characteristic
144	D₃₁	#475	1	4	yes	direct product factor
145	C₂²×Q₈	—	31	1	no	—
146	C₂²×Q₈	—	62	1	no	—
147	C₂²×C₈	—	31	2	no	—
148	C₄⋊Q₈	—	31	2	no	—
149	C₄⋊Q₈	—	31	1	no	—
150	C₄⋊Q₈	—	62	1	no	—
151	C₄⋊Q₈	—	62	1	no	—
152	C₄⋊Q₈	C₂×D₃₁	1	1	yes	characteristic
153	C₄×Q₈	—	31	1	no	—
154	C₄×Q₈	—	31	1	no	—
155	C₄×Q₈	—	31	1	no	—
156	C₄×Q₈	—	62	1	no	—
157	C₄×Q₈	—	62	1	no	—
158	C₄×Q₈	C₂×D₃₁	1	1	yes	characteristic
159	C₂×C₄⋊C₄	—	31	1	no	—
160	C₂×C₄⋊C₄	—	31	2	no	—
161	C₂×C₄⋊C₄	—	62	1	no	—
162	C₂×C₄⋊C₄	—	62	1	no	—
163	C₂×C₄²	—	31	1	no	—
164	C₂×C₄²	—	62	1	no	—
165	#106	—	31	2	no	—
166	#106	—	31	2	no	—
167	#106	—	31	2	no	—
168	#106	C₂×D₃₁	1	2	yes	—
169	C₄⋊C₈	—	31	2	no	—
170	C₄⋊C₈	—	31	1	no	—
171	C₄⋊C₈	C₂×D₃₁	1	1	yes	characteristic
172	#103	—	31	2	no	—
173	#103	—	31	2	no	—
174	#103	—	31	2	no	—
175	#103	C₂×D₃₁	1	2	yes	—
176	C₃₁	C₂×#475	1	1	yes	characteristic, complemented, Sylow
177	C₂×Q₈	—	2	1	no	—
178	C₂×Q₈	—	31	4	no	—
179	C₂×Q₈	—	31	1	no	—
180	C₂×Q₈	—	62	1	no	—
181	C₂×Q₈	—	62	2	no	—
182	C₂×Q₈	—	62	2	no	—
183	C₂×Q₈	—	62	1	no	—
184	C₂×Q₈	—	124	4	no	—
185	C₂×Q₈	C₂²×D₃₁	1	1	yes	characteristic
186	C₂²×C₄	—	31	1	no	—
187	C₂²×C₄	—	31	1	no	—
188	C₂²×C₄	—	31	1	no	—
189	C₂²×C₄	—	62	1	no	—
190	C₂²×C₄	—	62	1	no	—
191	C₂²×C₄	—	62	2	no	—
192	C₂×C₈	—	31	2	no	—
193	C₂×C₈	—	62	4	no	—
194	C₂×C₈	C₂²×D₃₁	1	2	yes	—
195	C₄⋊C₄	—	2	1	no	—
196	C₄⋊C₄	—	2	1	no	—
197	C₄⋊C₄	—	31	2	no	—
198	C₄⋊C₄	—	31	1	no	—
199	C₄⋊C₄	—	62	1	no	—
200	C₄⋊C₄	—	62	2	no	—
201	C₄⋊C₄	—	62	1	no	—
202	C₄⋊C₄	—	62	2	no	—
203	C₄⋊C₄	—	62	1	no	—
204	C₄⋊C₄	—	62	1	no	—
205	C₄⋊C₄	—	62	1	no	—
206	C₄⋊C₄	C₂²×D₃₁	1	2	yes	—
207	C₄⋊C₄	C₂²×D₃₁	1	1	yes	characteristic
208	C₄²	—	2	1	no	—
209	C₄²	—	31	1	no	—
210	C₄²	—	31	1	no	—
211	C₄²	—	31	1	no	—
212	C₄²	—	62	1	no	—
213	C₄²	—	62	1	no	—
214	C₄²	—	62	1	no	—
215	C₄²	C₂²×D₃₁	1	1	yes	characteristic
216	C₂³	—	31	1	no	—
217	Q₈	—	2	1	no	—
218	Q₈	—	4	2	no	—
219	Q₈	—	31	2	no	—
220	Q₈	—	62	4	no	—
221	Q₈	—	62	1	no	—
222	Q₈	—	124	4	no	—
223	Q₈	—	124	2	no	—
224	Q₈	Q₈×D₃₁	1	2	yes	complemented
225	C₂×C₄	—	2	1	no	—
226	C₂×C₄	—	2	1	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	31	1	no	—
229	C₂×C₄	—	31	1	no	—
230	C₂×C₄	—	31	4	no	—
231	C₂×C₄	—	31	2	no	—
232	C₂×C₄	—	31	2	no	—
233	C₂×C₄	—	31	1	no	—
234	C₂×C₄	—	62	2	no	—
235	C₂×C₄	—	62	4	no	—
236	C₂×C₄	—	62	1	no	—
237	C₂×C₄	—	62	1	no	—
238	C₂×C₄	—	62	2	no	—
239	C₂×C₄	—	124	2	no	—
240	C₂×C₄	—	124	4	no	—
241	C₂×C₄	D₄×D₃₁	1	1	yes	characteristic
242	C₂×C₄	D₄×D₃₁	1	1	yes	characteristic
243	C₂×C₄	C₂³×D₃₁	1	1	yes	characteristic
244	C₈	—	2	2	no	—
245	C₈	—	62	2	no	—
246	C₂²	—	31	2	no	—
247	C₂²	—	31	2	no	—
248	C₂²	—	31	2	no	—
249	C₂²	C₂×D₄×D₃₁	1	1	yes	characteristic
250	C₄	—	2	2	no	—
251	C₄	—	2	1	no	—
252	C₄	—	4	2	no	—
253	C₄	—	4	1	no	—
254	C₄	—	31	2	no	—
255	C₄	—	31	2	no	—
256	C₄	—	62	2	no	—
257	C₄	—	62	1	no	—
258	C₄	—	124	2	no	—
259	C₄	—	124	1	no	—
260	C₄	SD₁₆×D₃₁	1	2	yes	—
261	C₄	C₂×Q₈×D₃₁	1	1	yes	characteristic
262	C₄	#41×D₃₁	1	1	yes	characteristic
263	C₂	—	31	4	no	—
264	C₂	C₂²⋊Q₈×D₃₁	1	1	yes	characteristic
265	C₂	C₂×SD₁₆×D₃₁	1	1	yes	characteristic
266	C₂	#137×D₃₁	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₃₁
commutator, cosocle	C₂×C₄×C₃₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₃₁
Fitting	C₃₁×#475	C₂
socle	C₂²×C₃₁	C₂²×D₄
prime radical	C₂²×D₃₁	C₂×D₄
hypercenter	#475	D₃₁
hypocenter	C₃₁	C₂×#475
LCS	G ▷ C₂×C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #475

Complements

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

N	H ≅ G/N	type
3. #14521335	C₂	semidirect
4. #14521334	C₂	semidirect
5. #14521333	C₂	semidirect
6. #14521332	C₂	semidirect
7. #14521331	C₂	semidirect
11. C₃₁×#475	C₂	semidirect
47. Q₈×D₃₁	Q₈	semidirect
85. Q₈×C₃₁	C₂×Q₈	semidirect
109. Dic₆₂	C₂×Q₈	semidirect
137. #475	D₃₁	direct (N × H)
144. D₃₁	#475	direct (N × H)
176. C₃₁	C₂×#475	semidirect
224. Q₈	Q₈×D₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₄⋊Q₈×D₃₁	2
C₄×Q₈×D₃₁	2
#14521335	2
#14521334	2
#14521333	2
#14521332	2
#14521331	2
#106×D₃₁	2
C₄⋊C₈×D₃₁	2
#103×D₃₁	2
C₃₁×#475	2
C₂×#475	31

Subgroups — D31×#475 #16855472

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₃₁×#475 #16855472