Subgroups — C₂×C₃₁×#2254 #21490484

Order 7936.

558 conjugacy classes of subgroups.

The subgroup lattice has 140 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#500	C₂	1	2	yes	maximal
2	C₃₁×#2254	C₂	1	8	yes	maximal, direct product factor
3	C₂×C₃₁×#367	C₂	1	4	yes	maximal
4	C₂×C₄×C₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic
5	C₂×C₃₁×C₄⋊Q₈	—	2	2	no	—
6	C₂²×C₁₆×C₃₁	C₂²	1	2	yes	—
7	C₃₁×#500	—	2	4	no	—
8	C₃₁×#500	C₂²	1	8	yes	—
9	C₂×C₃₁×#107	—	2	2	no	—
10	C₂×C₃₁×#107	C₂²	1	4	yes	—
11	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
12	C₃₁×#367	C₂²	1	16	yes	—
13	C₄×C₁₆×C₃₁	C₂²	1	4	yes	—
14	C₂²×Q₈×C₃₁	—	4	2	no	—
15	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
16	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
17	C₃₁×C₄⋊Q₈	—	2	8	no	—
18	C₃₁×C₄⋊Q₈	—	4	4	no	—
19	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
20	C₂×C₄⋊C₄×C₃₁	—	4	2	no	—
21	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
22	C₂×C₁₆×C₃₁	Q₈	1	8	yes	complemented
23	C₂×C₁₆×C₃₁	C₂³	1	4	yes	—
24	C₃₁×#107	—	2	8	no	—
25	C₃₁×#107	—	2	4	no	—
26	C₃₁×#107	C₂³	1	8	yes	—
27	C₄×C₈×C₃₁	D₄	1	2	yes	—
28	C₄×C₈×C₃₁	C₂³	1	2	yes	—
29	C₂×Q₈×C₃₁	—	4	8	no	—
30	C₂×Q₈×C₃₁	—	8	8	no	—
31	C₂²×C₄×C₃₁	—	4	4	no	—
32	C₂²×C₄×C₃₁	D₈	1	2	yes	—
33	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
34	C₂×C₈×C₃₁	Q₁₆	1	4	yes	—
35	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
36	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
37	C₂×C₈×C₃₁	C₂×Q₈	1	4	yes	—
38	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
39	C₄⋊C₄×C₃₁	—	2	8	no	—
40	C₄⋊C₄×C₃₁	—	4	4	no	—
41	C₄⋊C₄×C₃₁	—	4	8	no	—
42	C₄²×C₃₁	D₈	1	2	yes	—
43	C₄²×C₃₁	C₂×D₄	1	2	yes	—
44	C₁₆×C₃₁	C₂×Q₈	1	8	yes	complemented
45	C₂×#2254	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
46	C₂³×C₃₁	C₂×D₈	1	1	yes	characteristic
47	Q₈×C₃₁	—	8	16	no	—
48	C₂×C₄×C₃₁	—	4	8	no	—
49	C₂×C₄×C₃₁	—	8	8	no	—
50	C₂×C₄×C₃₁	SD₃₂	1	8	yes	—
51	C₂×C₄×C₃₁	C₄⋊Q₈	1	2	yes	—
52	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
53	C₂×C₄×C₃₁	C₂×D₈	1	4	yes	—
54	C₂×C₄×C₃₁	C₂×Q₁₆	1	2	yes	—
55	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
56	C₈×C₃₁	C₄⋊Q₈	1	2	yes	—
57	C₈×C₃₁	C₂×Q₁₆	1	4	yes	—
58	C₈×C₃₁	C₂²×Q₈	1	2	yes	—
59	C₂×#500	C₂×C₃₁	1	2	yes	—
60	#2254	C₂×C₃₁	1	8	yes	direct product factor
61	C₂×#367	C₂×C₃₁	1	4	yes	—
62	C₂×C₄×C₁₆	C₂×C₃₁	1	1	yes	characteristic
63	C₂²×C₃₁	#500	1	2	yes	—
64	C₂²×C₃₁	C₂×SD₃₂	1	4	yes	—
65	C₂²×C₃₁	C₂²×D₈	1	1	yes	characteristic
66	C₄×C₃₁	—	8	8	no	—
67	C₄×C₃₁	#500	1	2	yes	—
68	C₄×C₃₁	C₂×SD₃₂	1	8	yes	—
69	C₄×C₃₁	C₂×C₄⋊Q₈	1	1	yes	characteristic
70	C₄×C₃₁	C₂²×Q₁₆	1	1	yes	characteristic
71	C₂×C₄⋊Q₈	—	2	2	no	—
72	C₂²×C₁₆	C₂²×C₃₁	1	2	yes	—
73	#500	—	2	4	no	—
74	#500	C₂²×C₃₁	1	8	yes	—
75	C₂×#107	—	2	2	no	—
76	C₂×#107	C₂²×C₃₁	1	4	yes	—
77	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
78	#367	C₂²×C₃₁	1	16	yes	—
79	C₄×C₁₆	C₂²×C₃₁	1	4	yes	—
80	C₂×C₃₁	#2254	1	4	yes	direct product factor
81	C₂×C₃₁	C₂×#500	1	1	yes	characteristic
82	C₂×C₃₁	C₂²×SD₃₂	1	2	yes	—
83	C₂²×Q₈	—	4	2	no	—
84	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
85	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
86	C₄⋊Q₈	—	2	8	no	—
87	C₄⋊Q₈	—	4	4	no	—
88	C₂×C₄⋊C₄	—	2	4	no	—
89	C₂×C₄⋊C₄	—	4	2	no	—
90	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
91	C₂×C₁₆	Q₈×C₃₁	1	8	yes	complemented
92	C₂×C₁₆	C₂³×C₃₁	1	4	yes	—
93	#107	—	2	8	no	—
94	#107	—	2	4	no	—
95	#107	C₂³×C₃₁	1	8	yes	—
96	C₄×C₈	D₄×C₃₁	1	2	yes	—
97	C₄×C₈	C₂³×C₃₁	1	2	yes	—
98	C₃₁	C₂×#2254	1	1	yes	characteristic, direct product factor, Sylow
99	C₂×Q₈	—	4	8	no	—
100	C₂×Q₈	—	8	8	no	—
101	C₂²×C₄	—	4	4	no	—
102	C₂²×C₄	D₈×C₃₁	1	2	yes	—
103	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
104	C₂×C₈	Q₁₆×C₃₁	1	4	yes	—
105	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
106	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
107	C₂×C₈	C₂×Q₈×C₃₁	1	4	yes	—
108	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
109	C₄⋊C₄	—	2	8	no	—
110	C₄⋊C₄	—	4	4	no	—
111	C₄⋊C₄	—	4	8	no	—
112	C₄²	D₈×C₃₁	1	2	yes	—
113	C₄²	C₂×D₄×C₃₁	1	2	yes	—
114	C₁₆	C₂×Q₈×C₃₁	1	8	yes	complemented
115	C₂³	C₂×D₈×C₃₁	1	1	yes	characteristic
116	Q₈	—	8	16	no	—
117	C₂×C₄	—	4	8	no	—
118	C₂×C₄	—	8	8	no	—
119	C₂×C₄	C₃₁×SD₃₂	1	8	yes	—
120	C₂×C₄	C₃₁×C₄⋊Q₈	1	2	yes	—
121	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
122	C₂×C₄	C₂×D₈×C₃₁	1	4	yes	—
123	C₂×C₄	C₂×Q₁₆×C₃₁	1	2	yes	—
124	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
125	C₈	C₃₁×C₄⋊Q₈	1	2	yes	—
126	C₈	C₂×Q₁₆×C₃₁	1	4	yes	—
127	C₈	C₂²×Q₈×C₃₁	1	2	yes	—
128	C₂²	C₃₁×#500	1	2	yes	—
129	C₂²	C₂×C₃₁×SD₃₂	1	4	yes	—
130	C₂²	C₂²×D₈×C₃₁	1	1	yes	characteristic
131	C₄	—	8	8	no	—
132	C₄	C₃₁×#500	1	2	yes	—
133	C₄	C₂×C₃₁×SD₃₂	1	8	yes	—
134	C₄	C₂×C₃₁×C₄⋊Q₈	1	1	yes	characteristic
135	C₄	C₂²×Q₁₆×C₃₁	1	1	yes	characteristic
136	C₂	C₃₁×#2254	1	4	yes	direct product factor
137	C₂	C₂×C₃₁×#500	1	1	yes	characteristic
138	C₂	C₂²×C₃₁×SD₃₂	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₃₁	C₂×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₃₁
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×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
2. C₃₁×#2254	C₂	direct (N × H)
22. C₂×C₁₆×C₃₁	Q₈	semidirect
44. C₁₆×C₃₁	C₂×Q₈	semidirect
45. C₂×#2254	C₃₁	direct (N × H)
60. #2254	C₂×C₃₁	direct (N × H)
80. C₂×C₃₁	#2254	direct (N × H)
91. C₂×C₁₆	Q₈×C₃₁	semidirect
98. C₃₁	C₂×#2254	direct (N × H)
114. C₁₆	C₂×Q₈×C₃₁	semidirect
136. C₂	C₃₁×#2254	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×#500	2
C₃₁×#2254	2
C₂×C₃₁×#367	2
C₂×C₄×C₁₆×C₃₁	2
C₂×#2254	31

Subgroups — C2×C31×#2254 #21490484

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃₁×#2254 #21490484