Subgroups — C₃₁×#20572 #21477331

Order 7936.

914 conjugacy classes of subgroups.

The subgroup lattice has 180 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	2	yes	maximal
2	C₄²×Q₈×C₃₁	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#1770	C₂	1	6	yes	maximal
4	C₃₁×#1767	C₂	1	3	yes	maximal
5	C₃₁×#1744	C₂	1	3	yes	maximal
6	C₂×C₄×Q₈×C₃₁	C₄	1	2	yes	—
7	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
8	C₈×Q₈×C₃₁	C₄	1	16	yes	complemented
9	C₂×C₃₁×C₄⋊C₈	C₂²	1	6	yes	—
10	C₂×C₃₁×#97	C₂²	1	6	yes	—
11	C₂×C₄×C₈×C₃₁	C₂²	1	6	yes	—
12	C₄×C₄⋊C₄×C₃₁	C₄	1	6	yes	—
13	C₄×C₄⋊C₄×C₃₁	C₂²	1	3	yes	—
14	C₄³×C₃₁	C₂²	1	3	yes	—
15	C₃₁×#336	C₂²	1	1	yes	characteristic
16	C₃₁×#336	C₂²	1	3	yes	—
17	C₃₁×#336	C₂²	1	6	yes	—
18	C₂²×Q₈×C₃₁	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
20	C₂²×C₈×C₃₁	C₂³	1	6	yes	—
21	C₄×Q₈×C₃₁	—	2	8	no	—
22	C₄×Q₈×C₃₁	C₂×C₄	1	8	yes	—
23	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	6	yes	—
24	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	3	yes	—
25	C₂×C₄²×C₃₁	C₂×C₄	1	6	yes	—
26	C₂×C₄²×C₃₁	C₂×C₄	1	6	yes	—
27	C₂×C₄²×C₃₁	C₂³	1	3	yes	—
28	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
29	C₂×C₄²×C₃₁	C₂³	1	3	yes	—
30	C₃₁×C₄⋊C₈	C₂×C₄	1	24	yes	—
31	C₃₁×#97	—	4	6	no	—
32	C₄×C₈×C₃₁	C₂×C₄	1	24	yes	—
33	C₃₁×#95	C₂×C₄	1	6	yes	—
34	C₂×Q₈×C₃₁	C₄²	1	4	yes	—
35	C₂×Q₈×C₃₁	M₁₆	1	8	yes	—
36	C₂²×C₄×C₃₁	C₂²×C₄	1	6	yes	—
37	C₂²×C₄×C₃₁	C₂²×C₄	1	3	yes	—
38	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
39	C₂²×C₄×C₃₁	C₂²×C₄	1	3	yes	—
40	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
41	C₂×C₈×C₃₁	—	2	6	no	—
42	C₂×C₈×C₃₁	—	2	2	no	—
43	C₂×C₈×C₃₁	—	2	2	no	—
44	C₂×C₈×C₃₁	—	2	6	no	—
45	C₂×C₈×C₃₁	C₂²×C₄	1	4	yes	—
46	C₂×C₈×C₃₁	C₂²×C₄	1	12	yes	—
47	C₄⋊C₄×C₃₁	—	2	12	no	—
48	C₄⋊C₄×C₃₁	C₄²	1	12	yes	—
49	C₄²×C₃₁	—	2	12	no	—
50	C₄²×C₃₁	—	2	12	no	—
51	C₄²×C₃₁	—	2	3	no	—
52	C₄²×C₃₁	—	2	2	no	—
53	C₄²×C₃₁	—	2	3	no	—
54	C₄²×C₃₁	C₂²×C₄	1	12	yes	—
55	#20572	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
56	C₂³×C₃₁	C₂³×C₄	1	1	yes	characteristic
57	Q₈×C₃₁	#97	1	16	yes	complemented
58	C₂×C₄×C₃₁	—	2	2	no	—
59	C₂×C₄×C₃₁	—	2	2	no	—
60	C₂×C₄×C₃₁	—	2	6	no	—
61	C₂×C₄×C₃₁	—	2	6	no	—
62	C₂×C₄×C₃₁	—	2	6	no	—
63	C₂×C₄×C₃₁	—	2	6	no	—
64	C₂×C₄×C₃₁	C₂×C₄²	1	6	yes	—
65	C₂×C₄×C₃₁	C₂×C₄²	1	6	yes	—
66	C₂×C₄×C₃₁	C₂×M₁₆	1	12	yes	—
67	C₂×C₄×C₃₁	#131	1	4	yes	—
68	C₂×C₄×C₃₁	C₂³×C₄	1	2	yes	—
69	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
70	C₂×C₄×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
71	C₂×C₄×C₃₁	2₋¹⁺⁴	1	2	yes	—
72	C₈×C₃₁	—	2	4	no	—
73	C₈×C₃₁	—	2	12	no	—
74	C₂×C₈×Q₈	C₂×C₃₁	1	2	yes	—
75	C₄²×Q₈	C₂×C₃₁	1	1	yes	characteristic
76	#1770	C₂×C₃₁	1	6	yes	—
77	#1767	C₂×C₃₁	1	3	yes	—
78	#1744	C₂×C₃₁	1	3	yes	—
79	C₂²×C₃₁	C₂²×C₄²	1	1	yes	characteristic
80	C₂²×C₃₁	#519	1	1	yes	characteristic
81	C₂²×C₃₁	#520	1	1	yes	characteristic
82	C₂²×C₃₁	C₂²×M₁₆	1	2	yes	—
83	C₂²×C₃₁	C₂×#131	1	2	yes	—
84	C₄×C₃₁	—	2	6	no	—
85	C₄×C₃₁	—	2	4	no	—
86	C₄×C₃₁	—	4	6	no	—
87	C₄×C₃₁	C₂×#97	1	12	yes	—
88	C₄×C₃₁	#519	1	2	yes	—
89	C₄×C₃₁	#520	1	2	yes	—
90	C₂×C₄×Q₈	C₄×C₃₁	1	2	yes	—
91	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
92	C₈×Q₈	C₄×C₃₁	1	16	yes	complemented
93	C₂×C₄⋊C₈	C₂²×C₃₁	1	6	yes	—
94	C₂×#97	C₂²×C₃₁	1	6	yes	—
95	C₂×C₄×C₈	C₂²×C₃₁	1	6	yes	—
96	C₄×C₄⋊C₄	C₄×C₃₁	1	6	yes	—
97	C₄×C₄⋊C₄	C₂²×C₃₁	1	3	yes	—
98	C₄³	C₂²×C₃₁	1	3	yes	—
99	#336	C₂²×C₃₁	1	1	yes	characteristic
100	#336	C₂²×C₃₁	1	3	yes	—
101	#336	C₂²×C₃₁	1	6	yes	—
102	C₂×C₃₁	#2276	1	1	yes	characteristic
103	C₂×C₃₁	C₂²×#97	1	1	yes	characteristic
104	C₂×C₃₁	C₄×#131	1	1	yes	characteristic
105	C₂×C₃₁	#2984	1	2	yes	—
106	C₂×C₃₁	#2991	1	2	yes	—
107	C₂²×Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
108	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
109	C₂²×C₈	C₂³×C₃₁	1	6	yes	—
110	C₄×Q₈	—	2	8	no	—
111	C₄×Q₈	C₂×C₄×C₃₁	1	8	yes	—
112	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	6	yes	—
113	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	3	yes	—
114	C₂×C₄²	C₂×C₄×C₃₁	1	6	yes	—
115	C₂×C₄²	C₂×C₄×C₃₁	1	6	yes	—
116	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
117	C₂×C₄²	C₂³×C₃₁	1	3	yes	—
118	C₂×C₄²	C₂³×C₃₁	1	3	yes	—
119	C₄⋊C₈	C₂×C₄×C₃₁	1	24	yes	—
120	#97	—	4	6	no	—
121	C₄×C₈	C₂×C₄×C₃₁	1	24	yes	—
122	#95	C₂×C₄×C₃₁	1	6	yes	—
123	C₃₁	#20572	1	1	yes	characteristic, direct product factor, Sylow
124	C₂×Q₈	C₄²×C₃₁	1	4	yes	—
125	C₂×Q₈	M₁₆×C₃₁	1	8	yes	—
126	C₂²×C₄	C₂²×C₄×C₃₁	1	6	yes	—
127	C₂²×C₄	C₂²×C₄×C₃₁	1	3	yes	—
128	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
129	C₂²×C₄	C₂²×C₄×C₃₁	1	3	yes	—
130	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
131	C₂×C₈	—	2	6	no	—
132	C₂×C₈	—	2	2	no	—
133	C₂×C₈	—	2	2	no	—
134	C₂×C₈	—	2	6	no	—
135	C₂×C₈	C₂²×C₄×C₃₁	1	4	yes	—
136	C₂×C₈	C₂²×C₄×C₃₁	1	12	yes	—
137	C₄⋊C₄	—	2	12	no	—
138	C₄⋊C₄	C₄²×C₃₁	1	12	yes	—
139	C₄²	—	2	12	no	—
140	C₄²	—	2	12	no	—
141	C₄²	—	2	3	no	—
142	C₄²	—	2	2	no	—
143	C₄²	—	2	3	no	—
144	C₄²	C₂²×C₄×C₃₁	1	12	yes	—
145	C₂³	C₂³×C₄×C₃₁	1	1	yes	characteristic
146	Q₈	C₃₁×#97	1	16	yes	complemented
147	C₂×C₄	—	2	2	no	—
148	C₂×C₄	—	2	2	no	—
149	C₂×C₄	—	2	6	no	—
150	C₂×C₄	—	2	6	no	—
151	C₂×C₄	—	2	6	no	—
152	C₂×C₄	—	2	6	no	—
153	C₂×C₄	C₂×C₄²×C₃₁	1	6	yes	—
154	C₂×C₄	C₂×C₄²×C₃₁	1	6	yes	—
155	C₂×C₄	C₂×M₁₆×C₃₁	1	12	yes	—
156	C₂×C₄	C₃₁×#131	1	4	yes	—
157	C₂×C₄	C₂³×C₄×C₃₁	1	2	yes	—
158	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
159	C₂×C₄	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
160	C₂×C₄	C₃₁×2₋¹⁺⁴	1	2	yes	—
161	C₈	—	2	4	no	—
162	C₈	—	2	12	no	—
163	C₂²	C₂²×C₄²×C₃₁	1	1	yes	characteristic
164	C₂²	C₃₁×#519	1	1	yes	characteristic
165	C₂²	C₃₁×#520	1	1	yes	characteristic
166	C₂²	C₂²×M₁₆×C₃₁	1	2	yes	—
167	C₂²	C₂×C₃₁×#131	1	2	yes	—
168	C₄	—	2	6	no	—
169	C₄	—	2	4	no	—
170	C₄	—	4	6	no	—
171	C₄	C₂×C₃₁×#97	1	12	yes	—
172	C₄	C₃₁×#519	1	2	yes	—
173	C₄	C₃₁×#520	1	2	yes	—
174	C₂	C₃₁×#2276	1	1	yes	characteristic
175	C₂	C₂²×C₃₁×#97	1	1	yes	characteristic
176	C₂	C₄×C₃₁×#131	1	1	yes	characteristic
177	C₂	C₃₁×#2984	1	2	yes	—
178	C₂	C₃₁×#2991	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₃₁	C₂⁴
commutator	C₂²	C₂²×C₄²×C₃₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
socle, prime radical	C₂³×C₃₁	C₂³×C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
8. C₈×Q₈×C₃₁	C₄	semidirect
55. #20572	C₃₁	direct (N × H)
57. Q₈×C₃₁	#97	semidirect
92. C₈×Q₈	C₄×C₃₁	semidirect
123. C₃₁	#20572	direct (N × H)
146. Q₈	C₃₁×#97	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
C₄²×Q₈×C₃₁	2
C₃₁×#1770	2
C₃₁×#1767	2
C₃₁×#1744	2
#20572	31

Subgroups — C31×#20572 #21477331

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#20572 #21477331