Subgroups — C₃₁×#8587 #21465346

Order 7936.

516 conjugacy classes of subgroups.

The subgroup lattice has 230 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₃₁×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#1842	C₂	1	2	yes	maximal
3	C₂×C₃₁×#333	C₂	1	2	yes	maximal
4	C₃₁×#1296	C₂	1	2	yes	maximal
5	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
6	C₂×C₄×Q₈×C₃₁	C₄	1	2	yes	—
7	C₂×C₃₁×C₄⋊C₈	—	2	2	no	—
8	C₂×C₃₁×C₄⋊C₈	—	2	2	no	—
9	C₂×C₃₁×C₄⋊C₈	C₂²	1	4	yes	—
10	C₃₁×#391	C₄	1	2	yes	—
11	C₃₁×#389	—	2	2	no	—
12	C₃₁×#384	C₂²	1	1	yes	characteristic
13	C₄×C₄⋊C₄×C₃₁	—	2	2	no	—
14	C₄³×C₃₁	C₂²	1	1	yes	characteristic
15	C₃₁×#333	—	2	4	no	—
16	C₂²×Q₈×C₃₁	C₂×C₄	1	2	yes	—
17	C₂²×C₈×C₃₁	—	2	4	no	—
18	C₃₁×C₄⋊Q₈	—	2	2	no	—
19	C₃₁×C₄⋊Q₈	D₄	1	2	yes	—
20	C₃₁×C₄⋊Q₈	Q₈	1	2	yes	—
21	C₄×Q₈×C₃₁	—	4	4	no	—
22	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
23	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
24	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
25	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
26	C₂×C₄²×C₃₁	—	2	2	no	—
27	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄²×C₃₁	D₄	1	2	yes	—
30	C₂×C₄²×C₃₁	D₄	1	2	yes	—
31	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
32	C₃₁×C₄⋊C₈	—	2	4	no	—
33	C₃₁×C₄⋊C₈	—	2	4	no	—
34	C₃₁×C₄⋊C₈	—	4	4	no	—
35	C₃₁×C₄⋊C₈	—	4	4	no	—
36	C₃₁×#95	—	2	4	no	—
37	C₂×Q₈×C₃₁	—	2	2	no	—
38	C₂×Q₈×C₃₁	—	4	2	no	—
39	C₂×Q₈×C₃₁	—	4	2	no	—
40	C₂×Q₈×C₃₁	C₄⋊C₄	1	4	yes	—
41	C₂²×C₄×C₃₁	—	2	2	no	—
42	C₂²×C₄×C₃₁	—	2	2	no	—
43	C₂²×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
44	C₂²×C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
45	C₂²×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
46	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
47	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
48	C₂×C₈×C₃₁	—	4	4	no	—
49	C₂×C₈×C₃₁	—	4	4	no	—
50	C₂×C₈×C₃₁	—	4	4	no	—
51	C₄⋊C₄×C₃₁	—	2	4	no	—
52	C₄⋊C₄×C₃₁	—	2	4	no	—
53	C₄⋊C₄×C₃₁	—	4	4	no	—
54	C₄⋊C₄×C₃₁	—	4	2	no	—
55	C₄⋊C₄×C₃₁	—	4	2	no	—
56	C₄⋊C₄×C₃₁	—	4	2	no	—
57	C₄²×C₃₁	—	2	2	no	—
58	C₄²×C₃₁	—	2	2	no	—
59	C₄²×C₃₁	—	2	2	no	—
60	C₄²×C₃₁	—	4	1	no	—
61	C₄²×C₃₁	—	4	4	no	—
62	C₄²×C₃₁	SD₁₆	1	4	yes	—
63	C₄²×C₃₁	Q₁₆	1	4	yes	—
64	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
65	C₄²×C₃₁	C₂×Q₈	1	1	yes	characteristic
66	C₄²×C₃₁	#41	1	2	yes	—
67	#8587	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
68	C₂³×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
69	Q₈×C₃₁	—	4	4	no	—
70	Q₈×C₃₁	—	4	4	no	—
71	C₂×C₄×C₃₁	—	2	1	no	—
72	C₂×C₄×C₃₁	—	2	2	no	—
73	C₂×C₄×C₃₁	—	2	1	no	—
74	C₂×C₄×C₃₁	—	2	2	no	—
75	C₂×C₄×C₃₁	—	2	4	no	—
76	C₂×C₄×C₃₁	—	2	1	no	—
77	C₂×C₄×C₃₁	—	2	2	no	—
78	C₂×C₄×C₃₁	—	4	2	no	—
79	C₂×C₄×C₃₁	—	4	2	no	—
80	C₂×C₄×C₃₁	—	4	2	no	—
81	C₂×C₄×C₃₁	—	4	4	no	—
82	C₂×C₄×C₃₁	#100	1	2	yes	—
83	C₂×C₄×C₃₁	#103	1	4	yes	—
84	C₂×C₄×C₃₁	#103	1	4	yes	—
85	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
86	C₂×C₄×C₃₁	#117	1	1	yes	characteristic
87	C₂×C₄×C₃₁	#121	1	2	yes	—
88	C₂×C₄×C₃₁	C₂²⋊Q₈	1	2	yes	—
89	C₂×C₄×C₃₁	C₂×SD₁₆	1	2	yes	—
90	C₂×C₄×C₃₁	C₂×Q₁₆	1	2	yes	—
91	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
92	C₂×C₄×C₃₁	#137	1	2	yes	—
93	C₈×C₃₁	—	8	4	no	—
94	C₄×C₄⋊Q₈	C₂×C₃₁	1	1	yes	characteristic
95	#1842	C₂×C₃₁	1	2	yes	—
96	C₂×#333	C₂×C₃₁	1	2	yes	—
97	#1296	C₂×C₃₁	1	2	yes	—
98	C₂²×C₃₁	#380	1	1	yes	characteristic
99	C₂²×C₃₁	C₂×#100	1	1	yes	characteristic
100	C₂²×C₃₁	#413	1	1	yes	characteristic
101	C₂²×C₃₁	C₂×#103	1	2	yes	—
102	C₂²×C₃₁	#417	1	2	yes	—
103	C₄×C₃₁	—	2	1	no	—
104	C₄×C₃₁	—	2	1	no	—
105	C₄×C₃₁	—	2	4	no	—
106	C₄×C₃₁	—	4	4	no	—
107	C₄×C₃₁	—	4	1	no	—
108	C₄×C₃₁	—	8	2	no	—
109	C₄×C₃₁	#333	1	4	yes	complemented
110	C₄×C₃₁	#460	1	2	yes	—
111	C₄×C₃₁	#462	1	2	yes	—
112	C₄×C₃₁	#475	1	2	yes	—
113	C₄×C₃₁	#477	1	2	yes	—
114	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
115	C₂×C₄×Q₈	C₄×C₃₁	1	2	yes	—
116	C₂×C₄⋊C₈	—	2	2	no	—
117	C₂×C₄⋊C₈	—	2	2	no	—
118	C₂×C₄⋊C₈	C₂²×C₃₁	1	4	yes	—
119	#391	C₄×C₃₁	1	2	yes	—
120	#389	—	2	2	no	—
121	#384	C₂²×C₃₁	1	1	yes	characteristic
122	C₄×C₄⋊C₄	—	2	2	no	—
123	C₄³	C₂²×C₃₁	1	1	yes	characteristic
124	#333	—	2	4	no	—
125	C₂×C₃₁	C₂×#333	1	1	yes	characteristic
126	C₂×C₃₁	#1550	1	2	yes	—
127	C₂×C₃₁	#1560	1	1	yes	characteristic
128	C₂×C₃₁	#1800	1	1	yes	characteristic
129	C₂×C₃₁	#1803	1	2	yes	—
130	C₂²×Q₈	C₂×C₄×C₃₁	1	2	yes	—
131	C₂²×C₈	—	2	4	no	—
132	C₄⋊Q₈	—	2	2	no	—
133	C₄⋊Q₈	D₄×C₃₁	1	2	yes	—
134	C₄⋊Q₈	Q₈×C₃₁	1	2	yes	—
135	C₄×Q₈	—	4	4	no	—
136	C₂×C₄⋊C₄	—	2	2	no	—
137	C₂×C₄⋊C₄	—	2	2	no	—
138	C₂×C₄⋊C₄	—	2	2	no	—
139	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
140	C₂×C₄²	—	2	2	no	—
141	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
142	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
143	C₂×C₄²	D₄×C₃₁	1	2	yes	—
144	C₂×C₄²	D₄×C₃₁	1	2	yes	—
145	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
146	C₄⋊C₈	—	2	4	no	—
147	C₄⋊C₈	—	2	4	no	—
148	C₄⋊C₈	—	4	4	no	—
149	C₄⋊C₈	—	4	4	no	—
150	#95	—	2	4	no	—
151	C₃₁	#8587	1	1	yes	characteristic, direct product factor, Sylow
152	C₂×Q₈	—	2	2	no	—
153	C₂×Q₈	—	4	2	no	—
154	C₂×Q₈	—	4	2	no	—
155	C₂×Q₈	C₄⋊C₄×C₃₁	1	4	yes	—
156	C₂²×C₄	—	2	2	no	—
157	C₂²×C₄	—	2	2	no	—
158	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
159	C₂²×C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
160	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
161	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
162	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
163	C₂×C₈	—	4	4	no	—
164	C₂×C₈	—	4	4	no	—
165	C₂×C₈	—	4	4	no	—
166	C₄⋊C₄	—	2	4	no	—
167	C₄⋊C₄	—	2	4	no	—
168	C₄⋊C₄	—	4	2	no	—
169	C₄⋊C₄	—	4	4	no	—
170	C₄⋊C₄	—	4	2	no	—
171	C₄⋊C₄	—	4	2	no	—
172	C₄²	—	2	2	no	—
173	C₄²	—	2	2	no	—
174	C₄²	—	2	2	no	—
175	C₄²	—	4	1	no	—
176	C₄²	—	4	4	no	—
177	C₄²	SD₁₆×C₃₁	1	4	yes	—
178	C₄²	Q₁₆×C₃₁	1	4	yes	—
179	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
180	C₄²	C₂×Q₈×C₃₁	1	1	yes	characteristic
181	C₄²	#41×C₃₁	1	2	yes	—
182	C₂³	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
183	Q₈	—	4	4	no	—
184	Q₈	—	4	4	no	—
185	C₂×C₄	—	2	1	no	—
186	C₂×C₄	—	2	2	no	—
187	C₂×C₄	—	2	1	no	—
188	C₂×C₄	—	2	2	no	—
189	C₂×C₄	—	2	1	no	—
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	2	4	no	—
192	C₂×C₄	—	4	2	no	—
193	C₂×C₄	—	4	2	no	—
194	C₂×C₄	—	4	2	no	—
195	C₂×C₄	—	4	4	no	—
196	C₂×C₄	C₃₁×#100	1	2	yes	—
197	C₂×C₄	C₃₁×#103	1	4	yes	—
198	C₂×C₄	C₃₁×#103	1	4	yes	—
199	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
200	C₂×C₄	C₃₁×#117	1	1	yes	characteristic
201	C₂×C₄	C₃₁×#121	1	2	yes	—
202	C₂×C₄	C₃₁×C₂²⋊Q₈	1	2	yes	—
203	C₂×C₄	C₂×SD₁₆×C₃₁	1	2	yes	—
204	C₂×C₄	C₂×Q₁₆×C₃₁	1	2	yes	—
205	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	2	yes	—
206	C₂×C₄	C₃₁×#137	1	2	yes	—
207	C₈	—	8	4	no	—
208	C₂²	C₃₁×#380	1	1	yes	characteristic
209	C₂²	C₂×C₃₁×#100	1	1	yes	characteristic
210	C₂²	C₃₁×#413	1	1	yes	characteristic
211	C₂²	C₂×C₃₁×#103	1	2	yes	—
212	C₂²	C₃₁×#417	1	2	yes	—
213	C₄	—	2	1	no	—
214	C₄	—	2	1	no	—
215	C₄	—	2	4	no	—
216	C₄	—	4	4	no	—
217	C₄	—	4	1	no	—
218	C₄	—	8	2	no	—
219	C₄	C₃₁×#333	1	4	yes	complemented
220	C₄	C₃₁×#460	1	2	yes	—
221	C₄	C₃₁×#462	1	2	yes	—
222	C₄	C₃₁×#475	1	2	yes	—
223	C₄	C₃₁×#477	1	2	yes	—
224	C₂	C₂×C₃₁×#333	1	1	yes	characteristic
225	C₂	C₃₁×#1550	1	2	yes	—
226	C₂	C₃₁×#1560	1	1	yes	characteristic
227	C₂	C₃₁×#1800	1	1	yes	characteristic
228	C₂	C₃₁×#1803	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₃₁	C₂×C₂²⋊C₄
commutator	C₂²×C₄	C₂²×C₄×C₃₁
Frattini, cosocle	C₂×C₄²	C₂³×C₃₁
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₄³×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
67. #8587	C₃₁	direct (N × H)
109. C₄×C₃₁	#333	semidirect
151. C₃₁	#8587	direct (N × H)
219. C₄	C₃₁×#333	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₃₁×C₄⋊Q₈	2
C₃₁×#1842	2
C₂×C₃₁×#333	2
C₃₁×#1296	2
#8587	31

Subgroups — C31×#8587 #21465346

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#8587 #21465346