Subgroups — C₃₁×#8659 #21465418

Order 7936.

394 conjugacy classes of subgroups.

The subgroup lattice has 266 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×Q₈×C₃₁	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#1848	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#1564	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#1487	C₂	1	2	yes	maximal
5	C₃₁×#1301	C₂	1	2	yes	maximal
6	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
7	C₈×Q₈×C₃₁	—	2	1	no	—
8	C₈×Q₈×C₃₁	—	2	1	no	—
9	C₈×Q₈×C₃₁	C₄	1	2	yes	—
10	C₈×Q₈×C₃₁	C₄	1	2	yes	—
11	C₃₁×#431	C₂²	1	2	yes	—
12	C₂×C₃₁×#108	—	2	1	no	—
13	C₂×C₃₁×#108	—	2	1	no	—
14	C₃₁×#423	C₂²	1	2	yes	—
15	C₂×C₃₁×C₄⋊C₈	—	2	1	no	—
16	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
17	C₂×C₄×C₈×C₃₁	—	2	1	no	—
18	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
19	C₃₁×#335	—	2	1	no	—
20	C₃₁×#334	—	2	1	no	—
21	C₃₁×Q₈⋊C₈	—	2	4	no	—
22	C₂²×Q₈×C₃₁	Q₈	1	1	yes	characteristic
23	C₂×M₁₆×C₃₁	—	2	2	no	—
24	C₂²×C₈×C₃₁	—	2	1	no	—
25	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
26	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
27	C₄×Q₈×C₃₁	—	2	1	no	—
28	C₄×Q₈×C₃₁	—	2	1	no	—
29	C₄×Q₈×C₃₁	C₂×C₄	1	2	yes	—
30	C₄×Q₈×C₃₁	D₄	1	2	yes	—
31	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
32	C₂×C₄⋊C₄×C₃₁	Q₈	1	1	yes	characteristic
33	C₂×C₄²×C₃₁	—	2	1	no	—
34	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
35	C₃₁×#108	—	4	2	no	—
36	C₃₁×#108	—	4	2	no	—
37	C₃₁×C₄⋊C₈	—	2	2	no	—
38	C₃₁×C₄⋊C₈	—	2	2	no	—
39	C₃₁×C₄⋊C₈	—	2	1	no	—
40	C₃₁×C₄⋊C₈	—	2	2	no	—
41	C₃₁×C₄⋊C₈	—	2	2	no	—
42	C₃₁×C₄⋊C₈	—	2	2	no	—
43	C₃₁×C₄⋊C₈	C₂×C₄	1	2	yes	—
44	C₃₁×C₂²⋊C₈	—	2	2	no	—
45	C₄×C₈×C₃₁	—	2	2	no	—
46	C₄×C₈×C₃₁	—	2	2	no	—
47	C₄×C₈×C₃₁	—	2	2	no	—
48	C₄×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
49	C₄×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
50	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
51	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
52	C₂×Q₈×C₃₁	—	2	4	no	—
53	C₂×Q₈×C₃₁	—	2	1	no	—
54	C₂×Q₈×C₃₁	C₄⋊C₄	1	2	yes	—
55	C₂²×C₄×C₃₁	—	2	1	no	—
56	C₂²×C₄×C₃₁	—	2	1	no	—
57	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄×C₃₁	C₂×Q₈	1	1	yes	characteristic
59	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
60	M₁₆×C₃₁	—	4	2	no	—
61	M₁₆×C₃₁	—	4	2	no	—
62	C₂×C₈×C₃₁	—	2	1	no	—
63	C₂×C₈×C₃₁	—	2	1	no	—
64	C₂×C₈×C₃₁	—	2	2	no	—
65	C₂×C₈×C₃₁	—	4	1	no	—
66	C₂×C₈×C₃₁	—	4	1	no	—
67	C₂×C₈×C₃₁	—	4	2	no	—
68	C₂×C₈×C₃₁	—	4	2	no	—
69	C₂×C₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
70	C₂×C₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
71	C₂×C₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
72	C₂×C₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
73	C₂×C₈×C₃₁	SD₁₆	1	2	yes	—
74	C₂×C₈×C₃₁	Q₁₆	1	2	yes	—
75	C₄⋊C₄×C₃₁	—	2	2	no	—
76	C₄⋊C₄×C₃₁	—	2	2	no	—
77	C₄⋊C₄×C₃₁	—	2	1	no	—
78	C₄⋊C₄×C₃₁	C₄⋊C₄	1	2	yes	—
79	C₄²×C₃₁	—	2	2	no	—
80	C₄²×C₃₁	—	2	2	no	—
81	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
82	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
83	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
84	C₄²×C₃₁	#41	1	1	yes	characteristic
85	#8659	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
86	C₂³×C₃₁	C₂²⋊Q₈	1	1	yes	characteristic
87	Q₈×C₃₁	—	2	4	no	—
88	Q₈×C₃₁	—	2	2	no	—
89	Q₈×C₃₁	#108	1	4	yes	complemented
90	C₂×C₄×C₃₁	—	2	4	no	—
91	C₂×C₄×C₃₁	—	2	2	no	—
92	C₂×C₄×C₃₁	—	2	2	no	—
93	C₂×C₄×C₃₁	—	2	2	no	—
94	C₂×C₄×C₃₁	—	4	1	no	—
95	C₂×C₄×C₃₁	—	4	2	no	—
96	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
97	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
98	C₂×C₄×C₃₁	#117	1	1	yes	characteristic
99	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
100	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
101	C₂×C₄×C₃₁	C₂²⋊Q₈	1	1	yes	characteristic
102	C₂×C₄×C₃₁	C₂×SD₁₆	1	1	yes	characteristic
103	C₂×C₄×C₃₁	C₂×Q₁₆	1	1	yes	characteristic
104	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	1	yes	characteristic
105	C₂×C₄×C₃₁	#137	1	1	yes	characteristic
106	C₈×C₃₁	—	2	1	no	—
107	C₈×C₃₁	—	2	1	no	—
108	C₈×C₃₁	—	4	2	no	—
109	C₈×C₃₁	—	8	2	no	—
110	C₈×C₃₁	—	8	2	no	—
111	C₈×C₃₁	#103	1	2	yes	—
112	C₈×C₃₁	#103	1	2	yes	—
113	C₂×C₈×Q₈	C₂×C₃₁	1	1	yes	characteristic
114	#1848	C₂×C₃₁	1	1	yes	characteristic
115	#1564	C₂×C₃₁	1	1	yes	characteristic
116	#1487	C₂×C₃₁	1	2	yes	—
117	#1301	C₂×C₃₁	1	2	yes	—
118	C₂²×C₃₁	—	2	1	no	—
119	C₂²×C₃₁	—	2	1	no	—
120	C₂²×C₃₁	#380	1	1	yes	characteristic
121	C₂²×C₃₁	#475	1	1	yes	characteristic
122	C₂²×C₃₁	#477	1	1	yes	characteristic
123	C₄×C₃₁	—	2	4	no	—
124	C₄×C₃₁	—	2	1	no	—
125	C₄×C₃₁	—	2	1	no	—
126	C₄×C₃₁	—	2	1	no	—
127	C₄×C₃₁	—	4	2	no	—
128	C₄×C₃₁	C₂×#103	1	1	yes	characteristic
129	C₄×C₃₁	#417	1	1	yes	characteristic
130	C₄×C₃₁	C₂×#108	1	2	yes	—
131	C₄×C₃₁	#460	1	1	yes	characteristic
132	C₄×C₃₁	#462	1	1	yes	characteristic
133	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
134	C₈×Q₈	—	2	1	no	—
135	C₈×Q₈	—	2	1	no	—
136	C₈×Q₈	C₄×C₃₁	1	2	yes	—
137	C₈×Q₈	C₄×C₃₁	1	2	yes	—
138	#431	C₂²×C₃₁	1	2	yes	—
139	C₂×#108	—	2	1	no	—
140	C₂×#108	—	2	1	no	—
141	#423	C₂²×C₃₁	1	2	yes	—
142	C₂×C₄⋊C₈	—	2	1	no	—
143	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
144	C₂×C₄×C₈	—	2	1	no	—
145	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
146	#335	—	2	1	no	—
147	#334	—	2	1	no	—
148	Q₈⋊C₈	—	2	4	no	—
149	C₂×C₃₁	—	2	1	no	—
150	C₂×C₃₁	—	2	1	no	—
151	C₂×C₃₁	#1803	1	1	yes	characteristic
152	C₂×C₃₁	#1810	1	1	yes	characteristic
153	C₂×C₃₁	#1814	1	1	yes	characteristic
154	C₂²×Q₈	Q₈×C₃₁	1	1	yes	characteristic
155	C₂×M₁₆	—	2	2	no	—
156	C₂²×C₈	—	2	1	no	—
157	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
158	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
159	C₄×Q₈	—	2	1	no	—
160	C₄×Q₈	—	2	1	no	—
161	C₄×Q₈	C₂×C₄×C₃₁	1	2	yes	—
162	C₄×Q₈	D₄×C₃₁	1	2	yes	—
163	C₂×C₄⋊C₄	—	2	1	no	—
164	C₂×C₄⋊C₄	Q₈×C₃₁	1	1	yes	characteristic
165	C₂×C₄²	—	2	1	no	—
166	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
167	#108	—	4	2	no	—
168	#108	—	4	2	no	—
169	C₄⋊C₈	—	2	2	no	—
170	C₄⋊C₈	—	2	2	no	—
171	C₄⋊C₈	—	2	1	no	—
172	C₄⋊C₈	—	2	2	no	—
173	C₄⋊C₈	—	2	2	no	—
174	C₄⋊C₈	—	2	2	no	—
175	C₄⋊C₈	C₂×C₄×C₃₁	1	2	yes	—
176	C₂²⋊C₈	—	2	2	no	—
177	C₄×C₈	—	2	2	no	—
178	C₄×C₈	—	2	2	no	—
179	C₄×C₈	—	2	2	no	—
180	C₄×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
181	C₄×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
182	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
183	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
184	C₃₁	#8659	1	1	yes	characteristic, direct product factor, Sylow
185	C₂×Q₈	—	2	4	no	—
186	C₂×Q₈	—	2	1	no	—
187	C₂×Q₈	C₄⋊C₄×C₃₁	1	2	yes	—
188	C₂²×C₄	—	2	1	no	—
189	C₂²×C₄	—	2	1	no	—
190	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
191	C₂²×C₄	C₂×Q₈×C₃₁	1	1	yes	characteristic
192	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
193	M₁₆	—	4	2	no	—
194	M₁₆	—	4	2	no	—
195	C₂×C₈	—	2	1	no	—
196	C₂×C₈	—	2	1	no	—
197	C₂×C₈	—	2	2	no	—
198	C₂×C₈	—	4	1	no	—
199	C₂×C₈	—	4	1	no	—
200	C₂×C₈	—	4	2	no	—
201	C₂×C₈	—	4	2	no	—
202	C₂×C₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
203	C₂×C₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
204	C₂×C₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
205	C₂×C₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
206	C₂×C₈	SD₁₆×C₃₁	1	2	yes	—
207	C₂×C₈	Q₁₆×C₃₁	1	2	yes	—
208	C₄⋊C₄	—	2	2	no	—
209	C₄⋊C₄	—	2	2	no	—
210	C₄⋊C₄	—	2	1	no	—
211	C₄⋊C₄	C₄⋊C₄×C₃₁	1	2	yes	—
212	C₄²	—	2	2	no	—
213	C₄²	—	2	2	no	—
214	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
215	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
216	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
217	C₄²	#41×C₃₁	1	1	yes	characteristic
218	C₂³	C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
219	Q₈	—	2	4	no	—
220	Q₈	—	2	2	no	—
221	Q₈	C₃₁×#108	1	4	yes	complemented
222	C₂×C₄	—	2	4	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	2	2	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	4	1	no	—
227	C₂×C₄	—	4	2	no	—
228	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
229	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
230	C₂×C₄	C₃₁×#117	1	1	yes	characteristic
231	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
232	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
233	C₂×C₄	C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
234	C₂×C₄	C₂×SD₁₆×C₃₁	1	1	yes	characteristic
235	C₂×C₄	C₂×Q₁₆×C₃₁	1	1	yes	characteristic
236	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
237	C₂×C₄	C₃₁×#137	1	1	yes	characteristic
238	C₈	—	2	1	no	—
239	C₈	—	2	1	no	—
240	C₈	—	4	2	no	—
241	C₈	—	8	2	no	—
242	C₈	—	8	2	no	—
243	C₈	C₃₁×#103	1	2	yes	—
244	C₈	C₃₁×#103	1	2	yes	—
245	C₂²	—	2	1	no	—
246	C₂²	—	2	1	no	—
247	C₂²	C₃₁×#380	1	1	yes	characteristic
248	C₂²	C₃₁×#475	1	1	yes	characteristic
249	C₂²	C₃₁×#477	1	1	yes	characteristic
250	C₄	—	2	4	no	—
251	C₄	—	2	1	no	—
252	C₄	—	2	1	no	—
253	C₄	—	2	1	no	—
254	C₄	—	4	2	no	—
255	C₄	C₂×C₃₁×#103	1	1	yes	characteristic
256	C₄	C₃₁×#417	1	1	yes	characteristic
257	C₄	C₂×C₃₁×#108	1	2	yes	—
258	C₄	C₃₁×#460	1	1	yes	characteristic
259	C₄	C₃₁×#462	1	1	yes	characteristic
260	C₂	—	2	1	no	—
261	C₂	—	2	1	no	—
262	C₂	C₃₁×#1803	1	1	yes	characteristic
263	C₂	C₃₁×#1810	1	1	yes	characteristic
264	C₂	C₃₁×#1814	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₃₁	#121
commutator	C₄²	C₂²×C₄×C₃₁
Frattini, cosocle	C₂×C₄²	C₂³×C₃₁
socle	C₂²×C₃₁	#380
prime radical	C₂³×C₃₁	C₂²⋊Q₈
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₃₁ ◁ C₂×C₄²×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
85. #8659	C₃₁	direct (N × H)
89. Q₈×C₃₁	#108	semidirect
184. C₃₁	#8659	direct (N × H)
221. Q₈	C₃₁×#108	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₈×Q₈×C₃₁	2
C₃₁×#1848	2
C₃₁×#1564	2
C₃₁×#1487	2
C₃₁×#1301	2
#8659	31

Subgroups — C31×#8659 #21465418

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#8659 #21465418