Subgroups — C₃₁×#12022 #21468781

Order 7936.

242 conjugacy classes of subgroups.

The subgroup lattice has 152 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#1567	C₂	1	2	yes	maximal
3	C₃₁×#1371	C₂	1	4	yes	maximal, complemented
4	C₂²×C₁₆×C₃₁	C₄	1	2	yes	—
5	C₃₁×#423	—	2	2	no	—
6	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
7	C₄×C₁₆×C₃₁	C₄	1	1	yes	characteristic
8	C₄×C₁₆×C₃₁	C₄	1	1	yes	characteristic
9	C₄×C₁₆×C₃₁	C₂²	1	2	yes	—
10	C₃₁×#335	—	2	2	no	—
11	C₃₁×#335	C₂²	1	4	yes	—
12	C₂×M₁₆×C₃₁	—	4	2	no	—
13	C₂²×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
14	C₂²×C₈×C₃₁	Q₈	1	1	yes	characteristic
15	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
16	C₂×C₁₆×C₃₁	—	2	4	no	—
17	C₂×C₁₆×C₃₁	C₂×C₄	1	2	yes	—
18	C₂×C₁₆×C₃₁	C₂×C₄	1	2	yes	—
19	C₃₁×C₄⋊C₈	—	2	4	no	—
20	C₃₁×C₄⋊C₈	—	4	2	no	—
21	C₄×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
22	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
23	C₄×C₈×C₃₁	Q₈	1	1	yes	characteristic
24	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
25	C₂²×C₄×C₃₁	C₄⋊C₄	1	1	yes	characteristic
26	C₂²×C₄×C₃₁	D₈	1	1	yes	characteristic
27	C₂²×C₄×C₃₁	Q₁₆	1	1	yes	characteristic
28	M₁₆×C₃₁	—	8	4	no	—
29	C₂×C₈×C₃₁	—	2	2	no	—
30	C₂×C₈×C₃₁	—	2	2	no	—
31	C₂×C₈×C₃₁	—	4	4	no	—
32	C₂×C₈×C₃₁	C₄⋊C₄	1	1	yes	characteristic
33	C₂×C₈×C₃₁	C₄⋊C₄	1	1	yes	characteristic
34	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
35	C₂×C₈×C₃₁	C₂×Q₈	1	1	yes	characteristic
36	C₄²×C₃₁	C₄⋊C₄	1	1	yes	characteristic
37	C₄²×C₃₁	D₈	1	1	yes	characteristic
38	C₄²×C₃₁	Q₁₆	1	1	yes	characteristic
39	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
40	C₁₆×C₃₁	—	2	2	no	—
41	C₁₆×C₃₁	M₁₆	1	4	yes	complemented
42	#12022	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
43	C₂³×C₃₁	#107	1	1	yes	characteristic
44	C₂×C₄×C₃₁	—	2	2	no	—
45	C₂×C₄×C₃₁	—	2	1	no	—
46	C₂×C₄×C₃₁	—	2	1	no	—
47	C₂×C₄×C₃₁	#107	1	1	yes	characteristic
48	C₂×C₄×C₃₁	#107	1	1	yes	characteristic
49	C₂×C₄×C₃₁	#107	1	1	yes	characteristic
50	C₂×C₄×C₃₁	D₁₆	1	2	yes	—
51	C₂×C₄×C₃₁	Q₃₂	1	2	yes	—
52	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
53	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
54	C₂×C₄×C₃₁	C₂×Q₁₆	1	1	yes	characteristic
55	C₈×C₃₁	—	2	1	no	—
56	C₈×C₃₁	—	2	1	no	—
57	C₈×C₃₁	—	8	4	no	—
58	C₈×C₃₁	#108	1	2	yes	—
59	C₈×C₃₁	C₂×M₁₆	1	2	yes	—
60	C₂×C₄×C₁₆	C₂×C₃₁	1	1	yes	characteristic
61	#1567	C₂×C₃₁	1	2	yes	—
62	#1371	C₂×C₃₁	1	4	yes	complemented
63	C₂²×C₃₁	—	2	1	no	—
64	C₂²×C₃₁	—	2	1	no	—
65	C₂²×C₃₁	#366	1	2	yes	—
66	C₂²×C₃₁	C₂×#107	1	1	yes	characteristic
67	C₄×C₃₁	—	2	1	no	—
68	C₄×C₃₁	—	2	1	no	—
69	C₄×C₃₁	#366	1	2	yes	—
70	C₄×C₃₁	#368	1	2	yes	—
71	C₄×C₃₁	#423	1	1	yes	characteristic
72	C₄×C₃₁	C₂×#108	1	1	yes	characteristic
73	C₄×C₃₁	C₂×D₁₆	1	1	yes	characteristic
74	C₄×C₃₁	C₂×Q₃₂	1	1	yes	characteristic
75	C₂²×C₁₆	C₄×C₃₁	1	2	yes	—
76	#423	—	2	2	no	—
77	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
78	C₄×C₁₆	C₄×C₃₁	1	1	yes	characteristic
79	C₄×C₁₆	C₄×C₃₁	1	1	yes	characteristic
80	C₄×C₁₆	C₂²×C₃₁	1	2	yes	—
81	#335	—	2	2	no	—
82	#335	C₂²×C₃₁	1	4	yes	—
83	C₂×C₃₁	—	2	2	no	—
84	C₂×C₃₁	#1567	1	1	yes	characteristic
85	C₂×C₃₁	C₂×#366	1	1	yes	characteristic
86	C₂×C₃₁	C₂×#368	1	1	yes	characteristic
87	C₂×M₁₆	—	4	2	no	—
88	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
89	C₂²×C₈	Q₈×C₃₁	1	1	yes	characteristic
90	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
91	C₂×C₁₆	—	2	4	no	—
92	C₂×C₁₆	C₂×C₄×C₃₁	1	2	yes	—
93	C₂×C₁₆	C₂×C₄×C₃₁	1	2	yes	—
94	C₄⋊C₈	—	2	4	no	—
95	C₄⋊C₈	—	4	2	no	—
96	C₄×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
97	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
98	C₄×C₈	Q₈×C₃₁	1	1	yes	characteristic
99	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
100	C₃₁	#12022	1	1	yes	characteristic, direct product factor, Sylow
101	C₂²×C₄	C₄⋊C₄×C₃₁	1	1	yes	characteristic
102	C₂²×C₄	D₈×C₃₁	1	1	yes	characteristic
103	C₂²×C₄	Q₁₆×C₃₁	1	1	yes	characteristic
104	M₁₆	—	8	4	no	—
105	C₂×C₈	—	2	2	no	—
106	C₂×C₈	—	2	2	no	—
107	C₂×C₈	—	4	4	no	—
108	C₂×C₈	C₄⋊C₄×C₃₁	1	1	yes	characteristic
109	C₂×C₈	C₄⋊C₄×C₃₁	1	1	yes	characteristic
110	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
111	C₂×C₈	C₂×Q₈×C₃₁	1	1	yes	characteristic
112	C₄²	C₄⋊C₄×C₃₁	1	1	yes	characteristic
113	C₄²	D₈×C₃₁	1	1	yes	characteristic
114	C₄²	Q₁₆×C₃₁	1	1	yes	characteristic
115	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
116	C₁₆	—	2	2	no	—
117	C₁₆	M₁₆×C₃₁	1	4	yes	complemented
118	C₂³	C₃₁×#107	1	1	yes	characteristic
119	C₂×C₄	—	2	1	no	—
120	C₂×C₄	—	2	1	no	—
121	C₂×C₄	—	2	2	no	—
122	C₂×C₄	C₃₁×#107	1	1	yes	characteristic
123	C₂×C₄	C₃₁×#107	1	1	yes	characteristic
124	C₂×C₄	C₃₁×#107	1	1	yes	characteristic
125	C₂×C₄	C₃₁×D₁₆	1	2	yes	—
126	C₂×C₄	C₃₁×Q₃₂	1	2	yes	—
127	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
128	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
129	C₂×C₄	C₂×Q₁₆×C₃₁	1	1	yes	characteristic
130	C₈	—	2	1	no	—
131	C₈	—	2	1	no	—
132	C₈	—	8	4	no	—
133	C₈	C₃₁×#108	1	2	yes	—
134	C₈	C₂×M₁₆×C₃₁	1	2	yes	—
135	C₂²	—	2	1	no	—
136	C₂²	—	2	1	no	—
137	C₂²	C₃₁×#366	1	2	yes	—
138	C₂²	C₂×C₃₁×#107	1	1	yes	characteristic
139	C₄	—	2	1	no	—
140	C₄	—	2	1	no	—
141	C₄	C₃₁×#366	1	2	yes	—
142	C₄	C₃₁×#368	1	2	yes	—
143	C₄	C₃₁×#423	1	1	yes	characteristic
144	C₄	C₂×C₃₁×#108	1	1	yes	characteristic
145	C₄	C₂×C₃₁×D₁₆	1	1	yes	characteristic
146	C₄	C₂×C₃₁×Q₃₂	1	1	yes	characteristic
147	C₂	—	2	2	no	—
148	C₂	C₃₁×#1567	1	1	yes	characteristic
149	C₂	C₂×C₃₁×#366	1	1	yes	characteristic
150	C₂	C₂×C₃₁×#368	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₃₁	C₂×D₈
commutator	C₂×C₈	C₂²×C₄×C₃₁
Frattini, cosocle	C₄×C₈	C₂³×C₃₁
socle	C₂²×C₃₁	C₂×#107
prime radical	C₂³×C₃₁	#107
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₃₁ ◁ C₂×C₄²×C₃₁ ◁ C₂×C₄×C₈×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₃₁×#1371	C₂	semidirect
41. C₁₆×C₃₁	M₁₆	semidirect
42. #12022	C₃₁	direct (N × H)
62. #1371	C₂×C₃₁	semidirect
100. C₃₁	#12022	direct (N × H)
117. C₁₆	M₁₆×C₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×C₁₆×C₃₁	2
C₃₁×#1567	2
C₃₁×#1371	2
#12022	31

Subgroups — C31×#12022 #21468781

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#12022 #21468781