Subgroups — C₂×#41×Dic₃₁ #16858893

Order 3968.

1728 conjugacy classes of subgroups.

The subgroup lattice has 144 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#41×Dic₃₁	C₂	1	16	yes	maximal, direct product factor
2	C₂×Q₈×Dic₃₁	C₂	1	2	yes	maximal, complemented
3	C₂×D₄×Dic₃₁	C₂	1	6	yes	maximal, complemented
4	C₂×#11758503	C₂	1	3	yes	maximal, complemented
5	C₂²×C₄×Dic₃₁	C₂	1	3	yes	maximal, complemented
6	C₂²×#41×C₃₁	C₂	1	1	yes	maximal, characteristic
7	C₂×#41×C₃₁	C₄	1	16	yes	complemented
8	C₂×#41×C₃₁	C₂²	1	8	yes	—
9	C₂²×Q₈×C₃₁	C₂²	1	1	yes	characteristic
10	C₂²×D₄×C₃₁	C₂²	1	3	yes	—
11	C₂³×C₄×C₃₁	C₂²	1	3	yes	—
12	C₂³×Dic₃₁	C₂²	1	6	yes	—
13	C₂×C₂²⋊Dic₃₁	C₂²	1	6	yes	—
14	Q₈×Dic₃₁	C₂²	1	16	yes	complemented
15	D₄×Dic₃₁	C₂²	1	48	yes	complemented
16	#11758503	C₂²	1	24	yes	complemented
17	C₂×C₄⋊Dic₃₁	C₂²	1	6	yes	—
18	C₂×C₄×Dic₃₁	C₂²	1	1	yes	characteristic
19	C₂×C₄×Dic₃₁	C₂²	1	3	yes	—
20	C₂×C₄×Dic₃₁	C₂²	1	6	yes	—
21	C₂×C₄×Dic₃₁	C₂²	1	24	yes	complemented
22	C₂⁴×C₃₁	C₂³	1	3	yes	—
23	#41×C₃₁	C₂×C₄	1	64	yes	complemented
24	C₂×Q₈×C₃₁	C₂×C₄	1	8	yes	complemented
25	C₂×Q₈×C₃₁	C₂³	1	4	yes	—
26	C₂×D₄×C₃₁	C₂×C₄	1	24	yes	complemented
27	C₂×D₄×C₃₁	C₂³	1	12	yes	—
28	C₂²×C₄×C₃₁	C₂×C₄	1	24	yes	complemented
29	C₂²×C₄×C₃₁	C₂³	1	3	yes	—
30	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
31	C₂²×C₄×C₃₁	C₂³	1	12	yes	—
32	C₂²×Dic₃₁	—	2	24	no	—
33	C₂²×Dic₃₁	C₂³	1	6	yes	—
34	C₂²×Dic₃₁	C₂³	1	24	yes	—
35	C₂²×Dic₃₁	C₂³	1	2	yes	—
36	C₂²⋊Dic₃₁	C₂³	1	24	yes	—
37	C₄⋊Dic₃₁	C₂³	1	24	yes	—
38	C₄×Dic₃₁	C₂³	1	4	yes	—
39	C₄×Dic₃₁	C₂³	1	12	yes	—
40	C₄×Dic₃₁	C₂³	1	24	yes	—
41	C₂³×C₃₁	—	2	12	no	—
42	C₂³×C₃₁	C₂²×C₄	1	12	yes	—
43	C₂³×C₃₁	C₂⁴	1	6	yes	—
44	C₂³×C₃₁	C₂⁴	1	1	yes	characteristic
45	Q₈×C₃₁	C₂²×C₄	1	16	yes	complemented
46	D₄×C₃₁	C₂²×C₄	1	48	yes	complemented
47	C₂×C₄×C₃₁	C₂²×C₄	1	4	yes	—
48	C₂×C₄×C₃₁	C₂²×C₄	1	12	yes	—
49	C₂×C₄×C₃₁	C₂²×C₄	1	48	yes	complemented
50	C₂×C₄×C₃₁	C₂⁴	1	2	yes	—
51	C₂×C₄×C₃₁	C₂⁴	1	6	yes	—
52	C₂×Dic₃₁	—	2	12	no	—
53	C₂×Dic₃₁	—	2	48	no	—
54	C₂×Dic₃₁	#41	1	8	yes	direct product factor
55	C₂×Dic₃₁	C₂⁴	1	12	yes	—
56	C₂×Dic₃₁	C₂⁴	1	4	yes	—
57	C₂×C₄×#41	—	31	1	no	maximal, Sylow
58	C₂²×C₃₁	—	2	24	no	—
59	C₂²×C₃₁	—	2	12	no	—
60	C₂²×C₃₁	C₂³×C₄	1	12	yes	—
61	C₂²×C₃₁	C₂³×C₄	1	2	yes	—
62	C₂²×C₃₁	C₂×#41	1	4	yes	—
63	C₂²×C₃₁	C₂⁵	1	1	yes	characteristic
64	C₄×C₃₁	C₂³×C₄	1	4	yes	—
65	C₄×C₃₁	C₂³×C₄	1	12	yes	—
66	Dic₃₁	—	2	12	no	—
67	Dic₃₁	C₂×#41	1	8	yes	direct product factor
68	C₂²×#41	D₃₁	1	1	yes	characteristic
69	C₄×#41	—	31	16	no	—
70	C₂×C₄×Q₈	—	31	2	no	—
71	C₂×C₄×D₄	—	31	6	no	—
72	C₂×#117	—	31	3	no	—
73	C₂²×C₄²	—	31	3	no	—
74	C₂×C₃₁	—	2	12	no	—
75	C₂×C₃₁	C₄×#41	1	4	yes	complemented
76	C₂×C₃₁	C₂⁴×C₄	1	1	yes	characteristic
77	C₂×C₃₁	C₂²×#41	1	2	yes	—
78	C₂×#41	Dic₃₁	1	16	yes	direct product factor
79	C₂×#41	C₂×D₃₁	1	8	yes	—
80	C₂²×Q₈	C₂×D₃₁	1	1	yes	characteristic
81	C₂²×D₄	C₂×D₃₁	1	3	yes	—
82	C₂³×C₄	—	31	6	no	—
83	C₂³×C₄	C₂×D₃₁	1	3	yes	—
84	C₄×Q₈	—	31	16	no	—
85	C₄×D₄	—	31	48	no	—
86	#117	—	31	24	no	—
87	C₂×C₄⋊C₄	—	31	6	no	—
88	C₂×C₂²⋊C₄	—	31	6	no	—
89	C₂×C₄²	—	31	6	no	—
90	C₂×C₄²	—	31	24	no	—
91	C₂×C₄²	—	31	3	no	—
92	C₂×C₄²	—	31	1	no	—
93	C₃₁	C₂×C₄×#41	1	1	yes	characteristic, complemented, Sylow
94	C₂⁴	C₂²×D₃₁	1	3	yes	—
95	#41	C₂×Dic₃₁	1	64	yes	direct product factor
96	C₂×Q₈	C₂×Dic₃₁	1	8	yes	complemented
97	C₂×Q₈	C₂²×D₃₁	1	4	yes	—
98	C₂×D₄	C₂×Dic₃₁	1	24	yes	complemented
99	C₂×D₄	C₂²×D₃₁	1	12	yes	—
100	C₂²×C₄	—	31	2	no	—
101	C₂²×C₄	—	31	6	no	—
102	C₂²×C₄	—	31	24	no	—
103	C₂²×C₄	—	62	24	no	—
104	C₂²×C₄	C₂×Dic₃₁	1	24	yes	complemented
105	C₂²×C₄	C₂²×D₃₁	1	3	yes	—
106	C₂²×C₄	C₂²×D₃₁	1	12	yes	—
107	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
108	C₄⋊C₄	—	31	24	no	—
109	C₂²⋊C₄	—	31	24	no	—
110	C₄²	—	31	4	no	—
111	C₄²	—	31	12	no	—
112	C₄²	—	31	24	no	—
113	C₂³	—	2	12	no	—
114	C₂³	C₂²×Dic₃₁	1	12	yes	—
115	C₂³	C₂³×D₃₁	1	6	yes	—
116	C₂³	C₂³×D₃₁	1	1	yes	characteristic
117	Q₈	C₂²×Dic₃₁	1	16	yes	complemented
118	D₄	C₂²×Dic₃₁	1	48	yes	complemented
119	C₂×C₄	—	31	8	no	—
120	C₂×C₄	—	31	4	no	—
121	C₂×C₄	—	31	12	no	—
122	C₂×C₄	—	62	48	no	—
123	C₂×C₄	—	62	12	no	—
124	C₂×C₄	C₂²×Dic₃₁	1	4	yes	—
125	C₂×C₄	C₂²×Dic₃₁	1	12	yes	—
126	C₂×C₄	C₂²×Dic₃₁	1	48	yes	complemented
127	C₂×C₄	C₂³×D₃₁	1	2	yes	—
128	C₂×C₄	C₂³×D₃₁	1	6	yes	—
129	C₂²	—	2	12	no	—
130	C₂²	—	2	24	no	—
131	C₂²	#41×D₃₁	1	4	yes	—
132	C₂²	C₂³×Dic₃₁	1	12	yes	—
133	C₂²	C₂³×Dic₃₁	1	2	yes	—
134	C₂²	C₂⁴×D₃₁	1	1	yes	characteristic
135	C₄	—	31	8	no	—
136	C₄	—	62	12	no	—
137	C₄	C₂³×Dic₃₁	1	4	yes	—
138	C₄	C₂³×Dic₃₁	1	12	yes	—
139	C₂	—	2	12	no	—
140	C₂	#41×Dic₃₁	1	4	yes	direct product factor
141	C₂	C₂⁴×Dic₃₁	1	1	yes	characteristic
142	C₂	C₂×#41×D₃₁	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂²×D₃₁
commutator	C₂×C₃₁	C₂⁴×C₄
Frattini	C₂²	C₂⁴×D₃₁
Fitting, prime radical	C₂²×#41×C₃₁	C₂
socle	C₂³×C₃₁	C₂⁴
cosocle	C₂²×C₃₁	C₂⁵
hypercenter	C₂²×#41	D₃₁
hypocenter	C₃₁	C₂×C₄×#41
LCS	G ▷ C₂×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×#41

Complements

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

N	H ≅ G/N	type
1. #41×Dic₃₁	C₂	direct (N × H)
2. C₂×Q₈×Dic₃₁	C₂	semidirect
3. C₂×D₄×Dic₃₁	C₂	semidirect
4. C₂×#11758503	C₂	semidirect
5. C₂²×C₄×Dic₃₁	C₂	semidirect
7. C₂×#41×C₃₁	C₄	semidirect
14. Q₈×Dic₃₁	C₂²	semidirect
15. D₄×Dic₃₁	C₂²	semidirect
16. #11758503	C₂²	semidirect
21. C₂×C₄×Dic₃₁	C₂²	semidirect
23. #41×C₃₁	C₂×C₄	semidirect
24. C₂×Q₈×C₃₁	C₂×C₄	semidirect
26. C₂×D₄×C₃₁	C₂×C₄	semidirect
28. C₂²×C₄×C₃₁	C₂×C₄	semidirect
45. Q₈×C₃₁	C₂²×C₄	semidirect
46. D₄×C₃₁	C₂²×C₄	semidirect
49. C₂×C₄×C₃₁	C₂²×C₄	semidirect
54. C₂×Dic₃₁	#41	direct (N × H)
67. Dic₃₁	C₂×#41	direct (N × H)
75. C₂×C₃₁	C₄×#41	semidirect
78. C₂×#41	Dic₃₁	direct (N × H)
93. C₃₁	C₂×C₄×#41	semidirect
95. #41	C₂×Dic₃₁	direct (N × H)
96. C₂×Q₈	C₂×Dic₃₁	semidirect
98. C₂×D₄	C₂×Dic₃₁	semidirect
104. C₂²×C₄	C₂×Dic₃₁	semidirect
117. Q₈	C₂²×Dic₃₁	semidirect
118. D₄	C₂²×Dic₃₁	semidirect
126. C₂×C₄	C₂²×Dic₃₁	semidirect
140. C₂	#41×Dic₃₁	direct (N × H)

Maximal subgroups

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

subgroup	index
#41×Dic₃₁	2
C₂×Q₈×Dic₃₁	2
C₂×D₄×Dic₃₁	2
C₂×#11758503	2
C₂²×C₄×Dic₃₁	2
C₂²×#41×C₃₁	2
C₂×C₄×#41	31

Subgroups — C2×#41×Dic31 #16858893

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#41×Dic₃₁ #16858893