Subgroups — C₃₁×#13549 #21470308

Order 7936.

238 conjugacy classes of subgroups.

The subgroup lattice has 130 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×M₆₄	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₃₁×C₃₂	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#2108	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#1397	C₂	1	2	yes	maximal, complemented
5	C₄×C₃₁×C₃₂	C₂	1	2	yes	maximal, complemented
6	C₂×C₃₁×M₃₂	C₄	1	1	yes	characteristic
7	C₂²×C₁₆×C₃₁	C₂²	1	1	yes	characteristic
8	C₃₁×#405	C₄	1	1	yes	characteristic
9	C₃₁×M₆₄	C₄	1	4	yes	complemented
10	C₂×C₃₁×C₃₂	C₄	1	4	yes	complemented
11	C₂×C₃₁×C₃₂	C₂²	1	2	yes	—
12	C₂×C₃₁×C₃₂	C₂²	1	1	yes	characteristic
13	C₂×C₃₁×C₃₂	C₂²	1	1	yes	characteristic
14	C₃₁×#346	C₄	1	2	yes	—
15	C₄×C₁₆×C₃₁	C₂²	1	2	yes	—
16	C₂×M₁₆×C₃₁	C₈	1	1	yes	characteristic
17	C₂²×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
18	C₃₁×#117	C₈	1	1	yes	characteristic
19	C₃₁×M₃₂	C₈	1	4	yes	—
20	C₂×C₁₆×C₃₁	C₂×C₄	1	1	yes	characteristic
21	C₂×C₁₆×C₃₁	C₂×C₄	1	2	yes	—
22	C₂×C₁₆×C₃₁	C₂×C₄	1	2	yes	—
23	C₂×C₁₆×C₃₁	C₂×C₄	1	2	yes	—
24	C₂×C₁₆×C₃₁	C₂³	1	1	yes	characteristic
25	C₃₁×#97	C₈	1	2	yes	—
26	C₄×C₈×C₃₁	C₂×C₄	1	2	yes	—
27	C₃₁×C₃₂	C₂×C₄	1	2	yes	—
28	C₃₁×C₃₂	C₂×C₄	1	2	yes	—
29	C₃₁×C₃₂	C₂×C₄	1	4	yes	—
30	C₂²×C₄×C₃₁	C₂×C₈	1	1	yes	characteristic
31	M₁₆×C₃₁	C₁₆	1	4	yes	—
32	C₂×C₈×C₃₁	C₄²	1	2	yes	—
33	C₂×C₈×C₃₁	C₂×C₈	1	1	yes	characteristic
34	C₂×C₈×C₃₁	C₂×C₈	1	2	yes	—
35	C₂×C₈×C₃₁	C₂×C₈	1	2	yes	—
36	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
37	C₄⋊C₄×C₃₁	C₁₆	1	2	yes	—
38	C₂²⋊C₄×C₃₁	C₁₆	1	2	yes	—
39	C₄²×C₃₁	C₂×C₈	1	2	yes	—
40	C₁₆×C₃₁	C₄²	1	2	yes	—
41	C₁₆×C₃₁	C₂×C₈	1	4	yes	—
42	C₁₆×C₃₁	C₂²×C₄	1	1	yes	characteristic
43	C₁₆×C₃₁	C₂²×C₄	1	1	yes	characteristic
44	#13549	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
45	C₂³×C₃₁	C₂×C₁₆	1	1	yes	characteristic
46	C₂×C₄×C₃₁	C₄×C₈	1	2	yes	—
47	C₂×C₄×C₃₁	C₂×C₁₆	1	4	yes	—
48	C₂×C₄×C₃₁	C₂×C₁₆	1	1	yes	characteristic
49	C₂×C₄×C₃₁	C₂×C₁₆	1	2	yes	—
50	C₂×C₄×C₃₁	C₂²×C₈	1	1	yes	characteristic
51	C₈×C₃₁	C₄×C₈	1	2	yes	—
52	C₈×C₃₁	C₂×C₁₆	1	4	yes	—
53	C₈×C₃₁	C₂×C₄²	1	1	yes	characteristic
54	C₈×C₃₁	C₂²×C₈	1	1	yes	characteristic
55	C₂×M₆₄	C₂×C₃₁	1	1	yes	characteristic
56	C₂²×C₃₂	C₂×C₃₁	1	1	yes	characteristic
57	#2108	C₂×C₃₁	1	1	yes	characteristic
58	#1397	C₂×C₃₁	1	2	yes	complemented
59	C₄×C₃₂	C₂×C₃₁	1	2	yes	complemented
60	C₂²×C₃₁	—	2	2	no	—
61	C₂²×C₃₁	C₄×C₁₆	1	2	yes	—
62	C₂²×C₃₁	C₂²×C₁₆	1	1	yes	characteristic
63	C₄×C₃₁	—	2	4	no	—
64	C₄×C₃₁	C₄×C₁₆	1	2	yes	—
65	C₄×C₃₁	C₂×C₄×C₈	1	1	yes	characteristic
66	C₄×C₃₁	C₂²×C₁₆	1	1	yes	characteristic
67	C₂×M₃₂	C₄×C₃₁	1	1	yes	characteristic
68	C₂²×C₁₆	C₂²×C₃₁	1	1	yes	characteristic
69	#405	C₄×C₃₁	1	1	yes	characteristic
70	M₆₄	C₄×C₃₁	1	4	yes	complemented
71	C₂×C₃₂	C₄×C₃₁	1	4	yes	complemented
72	C₂×C₃₂	C₂²×C₃₁	1	1	yes	characteristic
73	C₂×C₃₂	C₂²×C₃₁	1	2	yes	—
74	C₂×C₃₂	C₂²×C₃₁	1	1	yes	characteristic
75	#346	C₄×C₃₁	1	2	yes	—
76	C₄×C₁₆	C₂²×C₃₁	1	2	yes	—
77	C₂×C₃₁	—	2	2	no	—
78	C₂×C₃₁	C₂×C₄×C₁₆	1	1	yes	characteristic
79	C₂×C₃₁	#2258	1	2	yes	—
80	C₂×M₁₆	C₈×C₃₁	1	1	yes	characteristic
81	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
82	#117	C₈×C₃₁	1	1	yes	characteristic
83	M₃₂	C₈×C₃₁	1	4	yes	—
84	C₂×C₁₆	C₂×C₄×C₃₁	1	1	yes	characteristic
85	C₂×C₁₆	C₂×C₄×C₃₁	1	2	yes	—
86	C₂×C₁₆	C₂×C₄×C₃₁	1	2	yes	—
87	C₂×C₁₆	C₂×C₄×C₃₁	1	2	yes	—
88	C₂×C₁₆	C₂³×C₃₁	1	1	yes	characteristic
89	#97	C₈×C₃₁	1	2	yes	—
90	C₄×C₈	C₂×C₄×C₃₁	1	2	yes	—
91	C₃₂	C₂×C₄×C₃₁	1	2	yes	—
92	C₃₂	C₂×C₄×C₃₁	1	2	yes	—
93	C₃₂	C₂×C₄×C₃₁	1	4	yes	—
94	C₃₁	#13549	1	1	yes	characteristic, direct product factor, Sylow
95	C₂²×C₄	C₂×C₈×C₃₁	1	1	yes	characteristic
96	M₁₆	C₁₆×C₃₁	1	4	yes	—
97	C₂×C₈	C₄²×C₃₁	1	2	yes	—
98	C₂×C₈	C₂×C₈×C₃₁	1	1	yes	characteristic
99	C₂×C₈	C₂×C₈×C₃₁	1	2	yes	—
100	C₂×C₈	C₂×C₈×C₃₁	1	2	yes	—
101	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
102	C₄⋊C₄	C₁₆×C₃₁	1	2	yes	—
103	C₂²⋊C₄	C₁₆×C₃₁	1	2	yes	—
104	C₄²	C₂×C₈×C₃₁	1	2	yes	—
105	C₁₆	C₄²×C₃₁	1	2	yes	—
106	C₁₆	C₂×C₈×C₃₁	1	4	yes	—
107	C₁₆	C₂²×C₄×C₃₁	1	1	yes	characteristic
108	C₁₆	C₂²×C₄×C₃₁	1	1	yes	characteristic
109	C₂³	C₂×C₁₆×C₃₁	1	1	yes	characteristic
110	C₂×C₄	C₄×C₈×C₃₁	1	2	yes	—
111	C₂×C₄	C₂×C₁₆×C₃₁	1	4	yes	—
112	C₂×C₄	C₂×C₁₆×C₃₁	1	2	yes	—
113	C₂×C₄	C₂×C₁₆×C₃₁	1	1	yes	characteristic
114	C₂×C₄	C₂²×C₈×C₃₁	1	1	yes	characteristic
115	C₈	C₄×C₈×C₃₁	1	2	yes	—
116	C₈	C₂×C₁₆×C₃₁	1	4	yes	—
117	C₈	C₂×C₄²×C₃₁	1	1	yes	characteristic
118	C₈	C₂²×C₈×C₃₁	1	1	yes	characteristic
119	C₂²	—	2	2	no	—
120	C₂²	C₄×C₁₆×C₃₁	1	2	yes	—
121	C₂²	C₂²×C₁₆×C₃₁	1	1	yes	characteristic
122	C₄	—	2	4	no	—
123	C₄	C₄×C₁₆×C₃₁	1	2	yes	—
124	C₄	C₂×C₄×C₈×C₃₁	1	1	yes	characteristic
125	C₄	C₂²×C₁₆×C₃₁	1	1	yes	characteristic
126	C₂	—	2	2	no	—
127	C₂	C₂×C₄×C₁₆×C₃₁	1	1	yes	characteristic
128	C₂	C₃₁×#2258	1	2	yes	—

	subgroup	G/H
center	C₂×C₃₁×C₃₂	C₂²
commutator	C₂	C₂×C₄×C₁₆×C₃₁
Frattini, cosocle	C₂×C₁₆	C₂³×C₃₁
socle	C₂²×C₃₁	C₂²×C₁₆
prime radical	C₂³×C₃₁	C₂×C₁₆
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃₁×C₃₂ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₃₁×#1397	C₂	semidirect
5. C₄×C₃₁×C₃₂	C₂	semidirect
9. C₃₁×M₆₄	C₄	semidirect
10. C₂×C₃₁×C₃₂	C₄	semidirect
44. #13549	C₃₁	direct (N × H)
58. #1397	C₂×C₃₁	semidirect
59. C₄×C₃₂	C₂×C₃₁	semidirect
70. M₆₄	C₄×C₃₁	semidirect
71. C₂×C₃₂	C₄×C₃₁	semidirect
94. C₃₁	#13549	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃₁×M₆₄	2
C₂²×C₃₁×C₃₂	2
C₃₁×#2108	2
C₃₁×#1397	2
C₄×C₃₁×C₃₂	2
#13549	31

Subgroups — C31×#13549 #21470308

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#13549 #21470308