Subgroups — C₃₁×#2568 #16842501

Order 3968.

468 conjugacy classes of subgroups.

The subgroup lattice has 162 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×Q₈×C₃₁	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#400	C₂	1	4	yes	maximal
4	C₃₁×#391	C₂	1	4	yes	maximal
5	C₃₁×#391	C₂	1	2	yes	maximal
6	C₃₁×#382	C₂	1	2	yes	maximal
7	C₄×C₄⋊C₄×C₃₁	C₂	1	1	yes	maximal, characteristic
8	C₂²×Q₈×C₃₁	C₂²	1	2	yes	—
9	C₂²×Q₈×C₃₁	C₂²	1	1	yes	characteristic
10	C₃₁×C₄⋊Q₈	—	2	4	no	—
11	C₄×Q₈×C₃₁	—	2	4	no	—
12	C₂×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
13	C₂×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
14	C₂×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
16	C₂×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
17	C₂×C₄²×C₃₁	C₂²	1	2	yes	—
18	C₂×C₄²×C₃₁	C₂²	1	2	yes	—
19	C₂×C₄²×C₃₁	C₂²	1	1	yes	characteristic
20	C₃₁×#95	C₂²	1	2	yes	—
21	C₃₁×#95	C₂²	1	8	yes	—
22	C₃₁×#95	C₂²	1	8	yes	—
23	C₂×Q₈×C₃₁	—	2	2	no	—
24	C₂×Q₈×C₃₁	—	2	4	no	—
25	C₂×Q₈×C₃₁	—	2	2	no	—
26	C₂×Q₈×C₃₁	—	2	8	no	—
27	C₂×Q₈×C₃₁	Q₈	1	4	yes	complemented
28	C₂²×C₄×C₃₁	C₂³	1	4	yes	—
29	C₂²×C₄×C₃₁	C₂³	1	2	yes	—
30	C₂²×C₄×C₃₁	C₂³	1	4	yes	—
31	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
32	C₂²×C₄×C₃₁	C₂³	1	2	yes	—
33	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
34	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
35	C₄⋊C₄×C₃₁	—	2	4	no	—
36	C₄⋊C₄×C₃₁	—	2	2	no	—
37	C₄⋊C₄×C₃₁	—	2	4	no	—
38	C₄⋊C₄×C₃₁	—	2	4	no	—
39	C₄⋊C₄×C₃₁	—	2	4	no	—
40	C₄²×C₃₁	—	2	4	no	—
41	C₄²×C₃₁	—	2	4	no	—
42	C₄²×C₃₁	D₄	1	4	yes	—
43	C₂³×C₃₁	C₂⁴	1	1	yes	characteristic
44	Q₈×C₃₁	—	2	8	no	—
45	Q₈×C₃₁	—	4	8	no	—
46	C₂×C₄×C₃₁	—	2	2	no	—
47	C₂×C₄×C₃₁	—	2	2	no	—
48	C₂×C₄×C₃₁	—	2	2	no	—
49	C₂×C₄×C₃₁	—	2	2	no	—
50	C₂×C₄×C₃₁	—	2	4	no	—
51	C₂×C₄×C₃₁	—	2	4	no	—
52	C₂×C₄×C₃₁	—	2	4	no	—
53	C₂×C₄×C₃₁	—	2	8	no	—
54	C₂×C₄×C₃₁	—	2	4	no	—
55	C₂×C₄×C₃₁	—	2	4	no	—
56	C₂×C₄×C₃₁	C₂×D₄	1	2	yes	—
57	C₂×C₄×C₃₁	C₂×D₄	1	2	yes	—
58	C₂×C₄×C₃₁	C₂×D₄	1	2	yes	—
59	C₂×C₄×C₃₁	C₂×Q₈	1	4	yes	—
60	C₂×C₄×C₃₁	C₂×Q₈	1	2	yes	—
61	C₂×C₄×C₃₁	#41	1	2	yes	—
62	C₂×C₄×C₃₁	#41	1	4	yes	—
63	#2568	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
64	C₂²×C₃₁	C₂²×D₄	1	1	yes	characteristic
65	C₂²×C₃₁	C₂²×Q₈	1	1	yes	characteristic
66	C₂²×C₃₁	C₂×#41	1	1	yes	characteristic
67	C₂²×C₃₁	C₂×#41	1	2	yes	—
68	C₂²×C₃₁	2₋¹⁺⁴	1	2	yes	—
69	C₄×C₃₁	—	2	4	no	—
70	C₄×C₃₁	—	2	2	no	—
71	C₄×C₃₁	—	4	4	no	—
72	C₄×C₃₁	—	4	2	no	—
73	C₄×C₃₁	—	4	4	no	—
74	C₄×C₃₁	C₂²⋊Q₈	1	4	yes	—
75	C₄×C₃₁	#124	1	4	yes	—
76	C₂×C₄⋊Q₈	C₂×C₃₁	1	1	yes	characteristic
77	C₂×C₄×Q₈	C₂×C₃₁	1	1	yes	characteristic
78	#400	C₂×C₃₁	1	4	yes	—
79	#391	C₂×C₃₁	1	4	yes	—
80	#391	C₂×C₃₁	1	2	yes	—
81	#382	C₂×C₃₁	1	2	yes	—
82	C₄×C₄⋊C₄	C₂×C₃₁	1	1	yes	characteristic
83	C₂×C₃₁	C₂×C₂²⋊Q₈	1	1	yes	characteristic
84	C₂×C₃₁	C₂×#124	1	1	yes	characteristic
85	C₂×C₃₁	#536	1	1	yes	characteristic
86	C₂×C₃₁	#556	1	2	yes	—
87	C₂×C₃₁	#557	1	2	yes	—
88	C₂²×Q₈	C₂²×C₃₁	1	2	yes	—
89	C₂²×Q₈	C₂²×C₃₁	1	1	yes	characteristic
90	C₄⋊Q₈	—	2	4	no	—
91	C₄×Q₈	—	2	4	no	—
92	C₂×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
93	C₂×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
94	C₂×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
95	C₂×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
96	C₂×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
97	C₂×C₄²	C₂²×C₃₁	1	2	yes	—
98	C₂×C₄²	C₂²×C₃₁	1	2	yes	—
99	C₂×C₄²	C₂²×C₃₁	1	1	yes	characteristic
100	#95	C₂²×C₃₁	1	2	yes	—
101	#95	C₂²×C₃₁	1	8	yes	—
102	#95	C₂²×C₃₁	1	8	yes	—
103	C₃₁	#2568	1	1	yes	characteristic, direct product factor, Sylow
104	C₂×Q₈	—	2	2	no	—
105	C₂×Q₈	—	2	4	no	—
106	C₂×Q₈	—	2	2	no	—
107	C₂×Q₈	—	2	8	no	—
108	C₂×Q₈	Q₈×C₃₁	1	4	yes	complemented
109	C₂²×C₄	C₂³×C₃₁	1	4	yes	—
110	C₂²×C₄	C₂³×C₃₁	1	2	yes	—
111	C₂²×C₄	C₂³×C₃₁	1	4	yes	—
112	C₂²×C₄	C₂³×C₃₁	1	1	yes	characteristic
113	C₂²×C₄	C₂³×C₃₁	1	2	yes	—
114	C₂²×C₄	C₂³×C₃₁	1	1	yes	characteristic
115	C₂²×C₄	C₂³×C₃₁	1	1	yes	characteristic
116	C₄⋊C₄	—	2	4	no	—
117	C₄⋊C₄	—	2	2	no	—
118	C₄⋊C₄	—	2	4	no	—
119	C₄⋊C₄	—	2	4	no	—
120	C₄⋊C₄	—	2	4	no	—
121	C₄²	—	2	4	no	—
122	C₄²	—	2	4	no	—
123	C₄²	D₄×C₃₁	1	4	yes	—
124	C₂³	C₂⁴×C₃₁	1	1	yes	characteristic
125	Q₈	—	2	8	no	—
126	Q₈	—	4	8	no	—
127	C₂×C₄	—	2	2	no	—
128	C₂×C₄	—	2	2	no	—
129	C₂×C₄	—	2	2	no	—
130	C₂×C₄	—	2	2	no	—
131	C₂×C₄	—	2	4	no	—
132	C₂×C₄	—	2	4	no	—
133	C₂×C₄	—	2	4	no	—
134	C₂×C₄	—	2	8	no	—
135	C₂×C₄	—	2	4	no	—
136	C₂×C₄	—	2	4	no	—
137	C₂×C₄	C₂×D₄×C₃₁	1	2	yes	—
138	C₂×C₄	C₂×D₄×C₃₁	1	2	yes	—
139	C₂×C₄	C₂×D₄×C₃₁	1	2	yes	—
140	C₂×C₄	C₂×Q₈×C₃₁	1	4	yes	—
141	C₂×C₄	C₂×Q₈×C₃₁	1	2	yes	—
142	C₂×C₄	#41×C₃₁	1	2	yes	—
143	C₂×C₄	#41×C₃₁	1	4	yes	—
144	C₂²	C₂²×D₄×C₃₁	1	1	yes	characteristic
145	C₂²	C₂²×Q₈×C₃₁	1	1	yes	characteristic
146	C₂²	C₂×#41×C₃₁	1	1	yes	characteristic
147	C₂²	C₂×#41×C₃₁	1	2	yes	—
148	C₂²	C₃₁×2₋¹⁺⁴	1	2	yes	—
149	C₄	—	2	4	no	—
150	C₄	—	2	2	no	—
151	C₄	—	4	4	no	—
152	C₄	—	4	2	no	—
153	C₄	—	4	4	no	—
154	C₄	C₃₁×C₂²⋊Q₈	1	4	yes	—
155	C₄	C₃₁×#124	1	4	yes	—
156	C₂	C₂×C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
157	C₂	C₂×C₃₁×#124	1	1	yes	characteristic
158	C₂	C₃₁×#536	1	1	yes	characteristic
159	C₂	C₃₁×#556	1	2	yes	—
160	C₂	C₃₁×#557	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
27. C₂×Q₈×C₃₁	Q₈	semidirect
63. #2568	C₃₁	direct (N × H)
103. C₃₁	#2568	direct (N × H)
108. C₂×Q₈	Q₈×C₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃₁×C₄⋊Q₈	2
C₂×C₄×Q₈×C₃₁	2
C₃₁×#400	2
C₃₁×#391	2
C₃₁×#382	2
C₄×C₄⋊C₄×C₃₁	2
#2568	31

Subgroups — C31×#2568 #16842501

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#2568 #16842501