Subgroups — C₃₁×#33950 #21490709

Order 7936.

448 conjugacy classes of subgroups.

The subgroup lattice has 210 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3383	C₂	1	2	yes	maximal
2	C₃₁×#2253	C₂	1	2	yes	maximal
3	C₃₁×#2252	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#2229	C₂	1	4	yes	maximal
5	C₃₁×#2225	C₂	1	2	yes	maximal
6	Q₈×C₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic
7	C₄×C₃₁×Q₃₂	C₂	1	1	yes	maximal, characteristic
8	C₄×C₃₁×Q₃₂	C₂	1	2	yes	maximal
9	C₃₁×#557	—	2	2	no	—
10	C₂×C₃₁×Q₃₂	C₂²	1	1	yes	characteristic
11	C₃₁×#500	C₂²	1	2	yes	—
12	C₃₁×#499	C₂²	1	4	yes	—
13	C₃₁×#479	—	2	4	no	—
14	C₃₁×#475	—	2	2	no	—
15	C₈×Q₈×C₃₁	C₂²	1	1	yes	characteristic
16	C₄×Q₁₆×C₃₁	C₂²	1	2	yes	—
17	C₄×Q₁₆×C₃₁	C₂²	1	4	yes	—
18	C₃₁×#367	C₂²	1	4	yes	—
19	C₃₁×#366	C₂²	1	2	yes	—
20	C₃₁×#366	C₂²	1	1	yes	characteristic
21	C₃₁×#366	C₂²	1	2	yes	—
22	C₃₁×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
23	C₃₁×C₄⋊C₁₆	C₂²	1	2	yes	—
24	C₃₁×#358	C₂²	1	4	yes	—
25	C₃₁×#358	C₂²	1	2	yes	—
26	C₄×C₁₆×C₃₁	C₂²	1	1	yes	characteristic
27	C₄×C₁₆×C₃₁	C₂²	1	2	yes	—
28	C₂×Q₁₆×C₃₁	C₂³	1	2	yes	—
29	C₃₁×C₄⋊Q₈	—	2	2	no	—
30	C₃₁×C₄⋊Q₈	—	4	2	no	—
31	C₃₁×#125	—	2	4	no	—
32	C₃₁×#125	—	4	4	no	—
33	C₄×Q₈×C₃₁	—	2	2	no	—
34	C₄×Q₈×C₃₁	—	2	4	no	—
35	C₄×Q₈×C₃₁	—	4	2	no	—
36	C₄×Q₈×C₃₁	D₄	1	1	yes	characteristic
37	C₃₁×Q₃₂	Q₈	1	4	yes	complemented
38	C₂×C₁₆×C₃₁	C₂³	1	1	yes	characteristic
39	C₂×C₁₆×C₃₁	C₂³	1	2	yes	—
40	C₂×C₁₆×C₃₁	C₂³	1	1	yes	characteristic
41	C₃₁×#107	—	2	2	no	—
42	C₃₁×#107	C₂³	1	2	yes	—
43	C₃₁×#107	C₂³	1	4	yes	—
44	C₃₁×#106	—	2	4	no	—
45	C₃₁×C₄⋊C₈	D₄	1	1	yes	characteristic
46	C₃₁×C₄⋊C₈	D₄	1	2	yes	—
47	C₃₁×#103	—	2	4	no	—
48	C₃₁×#103	—	2	2	no	—
49	C₄×C₈×C₃₁	C₂³	1	2	yes	—
50	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
51	C₂×Q₈×C₃₁	—	2	2	no	—
52	C₂×Q₈×C₃₁	—	4	2	no	—
53	C₂×Q₈×C₃₁	D₈	1	1	yes	characteristic
54	Q₁₆×C₃₁	—	2	2	no	—
55	Q₁₆×C₃₁	C₂×Q₈	1	4	yes	—
56	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
57	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
58	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
59	C₄⋊C₄×C₃₁	—	2	2	no	—
60	C₄⋊C₄×C₃₁	—	2	4	no	—
61	C₄⋊C₄×C₃₁	—	4	2	no	—
62	C₄⋊C₄×C₃₁	—	4	4	no	—
63	C₄⋊C₄×C₃₁	—	4	4	no	—
64	C₄⋊C₄×C₃₁	—	4	4	no	—
65	C₄⋊C₄×C₃₁	—	4	2	no	—
66	C₄⋊C₄×C₃₁	D₈	1	1	yes	characteristic
67	C₄⋊C₄×C₃₁	D₈	1	2	yes	—
68	C₄²×C₃₁	—	4	4	no	—
69	C₄²×C₃₁	—	4	2	no	—
70	C₄²×C₃₁	C₂×D₄	1	2	yes	—
71	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
72	C₁₆×C₃₁	—	2	1	no	—
73	C₁₆×C₃₁	—	2	2	no	—
74	C₁₆×C₃₁	C₂×Q₈	1	2	yes	—
75	#33950	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
76	Q₈×C₃₁	—	2	2	no	—
77	Q₈×C₃₁	—	2	4	no	—
78	Q₈×C₃₁	—	4	2	no	—
79	Q₈×C₃₁	—	8	4	no	—
80	C₂×C₄×C₃₁	—	4	2	no	—
81	C₂×C₄×C₃₁	—	4	2	no	—
82	C₂×C₄×C₃₁	—	4	4	no	—
83	C₂×C₄×C₃₁	C₂×D₈	1	2	yes	—
84	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
85	C₂×C₄×C₃₁	C₂×D₈	1	2	yes	—
86	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
87	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
88	C₈×C₃₁	—	2	2	no	—
89	C₈×C₃₁	—	2	1	no	—
90	C₈×C₃₁	C₂²×Q₈	1	1	yes	characteristic
91	C₈×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
92	#3383	C₂×C₃₁	1	2	yes	—
93	#2253	C₂×C₃₁	1	2	yes	—
94	#2252	C₂×C₃₁	1	1	yes	characteristic
95	#2229	C₂×C₃₁	1	4	yes	—
96	#2225	C₂×C₃₁	1	2	yes	—
97	Q₈×C₁₆	C₂×C₃₁	1	1	yes	characteristic
98	C₄×Q₃₂	C₂×C₃₁	1	1	yes	characteristic
99	C₄×Q₃₂	C₂×C₃₁	1	2	yes	—
100	C₂²×C₃₁	C₂²×D₈	1	1	yes	characteristic
101	C₄×C₃₁	—	2	2	no	—
102	C₄×C₃₁	—	2	2	no	—
103	C₄×C₃₁	—	2	1	no	—
104	C₄×C₃₁	—	4	4	no	—
105	C₄×C₃₁	—	8	4	no	—
106	C₄×C₃₁	—	8	2	no	—
107	C₄×C₃₁	#508	1	2	yes	—
108	C₄×C₃₁	D₄×Q₈	1	1	yes	characteristic
109	C₄×C₃₁	#578	1	1	yes	characteristic
110	#557	—	2	2	no	—
111	C₂×Q₃₂	C₂²×C₃₁	1	1	yes	characteristic
112	#500	C₂²×C₃₁	1	2	yes	—
113	#499	C₂²×C₃₁	1	4	yes	—
114	#479	—	2	4	no	—
115	#475	—	2	2	no	—
116	C₈×Q₈	C₂²×C₃₁	1	1	yes	characteristic
117	C₄×Q₁₆	C₂²×C₃₁	1	2	yes	—
118	C₄×Q₁₆	C₂²×C₃₁	1	4	yes	—
119	#367	C₂²×C₃₁	1	4	yes	—
120	#366	C₂²×C₃₁	1	2	yes	—
121	#366	C₂²×C₃₁	1	1	yes	characteristic
122	#366	C₂²×C₃₁	1	2	yes	—
123	C₄⋊C₁₆	C₂²×C₃₁	1	1	yes	characteristic
124	C₄⋊C₁₆	C₂²×C₃₁	1	2	yes	—
125	#358	C₂²×C₃₁	1	4	yes	—
126	#358	C₂²×C₃₁	1	2	yes	—
127	C₄×C₁₆	C₂²×C₃₁	1	1	yes	characteristic
128	C₄×C₁₆	C₂²×C₃₁	1	2	yes	—
129	C₂×C₃₁	Q₈×D₈	1	1	yes	characteristic
130	C₂×C₃₁	C₂×#508	1	1	yes	characteristic
131	C₂×C₃₁	#3417	1	1	yes	characteristic
132	C₂×Q₁₆	C₂³×C₃₁	1	2	yes	—
133	C₄⋊Q₈	—	2	2	no	—
134	C₄⋊Q₈	—	4	2	no	—
135	#125	—	2	4	no	—
136	#125	—	4	4	no	—
137	C₄×Q₈	—	2	2	no	—
138	C₄×Q₈	—	2	4	no	—
139	C₄×Q₈	—	4	2	no	—
140	C₄×Q₈	D₄×C₃₁	1	1	yes	characteristic
141	Q₃₂	Q₈×C₃₁	1	4	yes	complemented
142	C₂×C₁₆	C₂³×C₃₁	1	1	yes	characteristic
143	C₂×C₁₆	C₂³×C₃₁	1	2	yes	—
144	C₂×C₁₆	C₂³×C₃₁	1	1	yes	characteristic
145	#107	—	2	2	no	—
146	#107	C₂³×C₃₁	1	4	yes	—
147	#107	C₂³×C₃₁	1	2	yes	—
148	#106	—	2	4	no	—
149	C₄⋊C₈	D₄×C₃₁	1	1	yes	characteristic
150	C₄⋊C₈	D₄×C₃₁	1	2	yes	—
151	#103	—	2	4	no	—
152	#103	—	2	2	no	—
153	C₄×C₈	C₂³×C₃₁	1	2	yes	—
154	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
155	C₃₁	#33950	1	1	yes	characteristic, direct product factor, Sylow
156	C₂×Q₈	—	2	2	no	—
157	C₂×Q₈	—	4	2	no	—
158	C₂×Q₈	D₈×C₃₁	1	1	yes	characteristic
159	Q₁₆	—	2	2	no	—
160	Q₁₆	C₂×Q₈×C₃₁	1	4	yes	—
161	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
162	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
163	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
164	C₄⋊C₄	—	2	2	no	—
165	C₄⋊C₄	—	2	4	no	—
166	C₄⋊C₄	—	4	2	no	—
167	C₄⋊C₄	—	4	4	no	—
168	C₄⋊C₄	—	4	4	no	—
169	C₄⋊C₄	—	4	4	no	—
170	C₄⋊C₄	—	4	2	no	—
171	C₄⋊C₄	D₈×C₃₁	1	1	yes	characteristic
172	C₄⋊C₄	D₈×C₃₁	1	2	yes	—
173	C₄²	—	4	4	no	—
174	C₄²	—	4	2	no	—
175	C₄²	C₂×D₄×C₃₁	1	2	yes	—
176	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
177	C₁₆	—	2	1	no	—
178	C₁₆	—	2	2	no	—
179	C₁₆	C₂×Q₈×C₃₁	1	2	yes	—
180	Q₈	—	2	2	no	—
181	Q₈	—	2	4	no	—
182	Q₈	—	4	2	no	—
183	Q₈	—	8	4	no	—
184	C₂×C₄	—	4	2	no	—
185	C₂×C₄	—	4	2	no	—
186	C₂×C₄	—	4	4	no	—
187	C₂×C₄	C₂×D₈×C₃₁	1	2	yes	—
188	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
189	C₂×C₄	C₂×D₈×C₃₁	1	2	yes	—
190	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
191	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
192	C₈	—	2	2	no	—
193	C₈	—	2	1	no	—
194	C₈	C₂²×Q₈×C₃₁	1	1	yes	characteristic
195	C₈	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
196	C₂²	C₂²×D₈×C₃₁	1	1	yes	characteristic
197	C₄	—	2	2	no	—
198	C₄	—	2	2	no	—
199	C₄	—	2	1	no	—
200	C₄	—	4	4	no	—
201	C₄	—	8	4	no	—
202	C₄	—	8	2	no	—
203	C₄	C₃₁×#508	1	2	yes	—
204	C₄	D₄×Q₈×C₃₁	1	1	yes	characteristic
205	C₄	C₃₁×#578	1	1	yes	characteristic
206	C₂	Q₈×D₈×C₃₁	1	1	yes	characteristic
207	C₂	C₂×C₃₁×#508	1	1	yes	characteristic
208	C₂	C₃₁×#3417	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂²×C₃₁	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₃₁
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₄×Q₈×C₃₁ ◁ C₈×Q₈×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
37. C₃₁×Q₃₂	Q₈	semidirect
75. #33950	C₃₁	direct (N × H)
141. Q₃₂	Q₈×C₃₁	semidirect
155. C₃₁	#33950	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃₁×#3383	2
C₃₁×#2253	2
C₃₁×#2252	2
C₃₁×#2229	2
C₃₁×#2225	2
Q₈×C₁₆×C₃₁	2
C₄×C₃₁×Q₃₂	2
#33950	31

Subgroups — C31×#33950 #21490709

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#33950 #21490709