Subgroups — C₂×M₁₆×Dic₃₁ #16854118

Order 3968.

656 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	C₂²×#92474	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄×Dic₃₁	C₂	1	1	yes	maximal, characteristic
3	M₁₆×Dic₃₁	C₂	1	8	yes	maximal, direct product factor
4	C₂×#11758403	C₂	1	2	yes	maximal, complemented
5	C₂×C₈×Dic₃₁	C₂	1	2	yes	maximal, complemented
6	C₂²×M₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic
7	C₂³×C₄×C₃₁	C₂²	1	1	yes	characteristic
8	C₂³×Dic₃₁	C₄	1	2	yes	—
9	C₂×M₁₆×C₃₁	C₄	1	8	yes	complemented
10	C₂×M₁₆×C₃₁	C₂²	1	4	yes	—
11	C₂²×C₈×C₃₁	C₂²	1	2	yes	—
12	C₂×C₄×Dic₃₁	C₄	1	4	yes	—
13	C₂×C₄×Dic₃₁	C₄	1	2	yes	—
14	C₂×C₄×Dic₃₁	C₂²	1	1	yes	characteristic
15	C₂×C₄×Dic₃₁	C₂²	1	1	yes	characteristic
16	C₂×C₄×Dic₃₁	C₂²	1	4	yes	—
17	C₂×#92474	C₄	1	8	yes	complemented
18	C₂×#92474	C₂²	1	4	yes	—
19	C₂²×C₃₁⋊C₈	C₂²	1	2	yes	—
20	#11758403	C₂²	1	8	yes	complemented
21	C₈×Dic₃₁	C₂²	1	8	yes	complemented
22	C₂⁴×C₃₁	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₄×C₃₁	C₂×C₄	1	2	yes	—
24	C₂²×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₄×C₃₁	C₂×C₄	1	4	yes	—
26	C₂²×C₄×C₃₁	C₂×C₄	1	4	yes	—
27	C₂²×C₄×C₃₁	C₂³	1	2	yes	—
28	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
29	C₂²×Dic₃₁	—	2	8	no	—
30	C₂²×Dic₃₁	C₂×C₄	1	2	yes	—
31	C₂²×Dic₃₁	C₂×C₄	1	8	yes	—
32	C₂²×Dic₃₁	C₂×C₄	1	2	yes	—
33	M₁₆×C₃₁	C₂×C₄	1	16	yes	complemented
34	C₂×C₈×C₃₁	C₂×C₄	1	8	yes	complemented
35	C₂×C₈×C₃₁	C₂³	1	4	yes	—
36	C₄×Dic₃₁	C₂×C₄	1	2	yes	—
37	C₄×Dic₃₁	C₂×C₄	1	8	yes	—
38	C₄×Dic₃₁	C₂×C₄	1	2	yes	—
39	C₄×Dic₃₁	C₂³	1	2	yes	—
40	C₄×Dic₃₁	C₂³	1	2	yes	—
41	#92474	C₂×C₄	1	16	yes	complemented
42	C₂×C₃₁⋊C₈	C₂×C₄	1	8	yes	complemented
43	C₂×C₃₁⋊C₈	C₂³	1	4	yes	—
44	C₂³×C₃₁	—	2	4	no	—
45	C₂³×C₃₁	C₄²	1	4	yes	—
46	C₂³×C₃₁	C₂²×C₄	1	2	yes	—
47	C₂³×C₃₁	C₂²×C₄	1	1	yes	characteristic
48	C₂×C₄×C₃₁	C₄²	1	8	yes	—
49	C₂×C₄×C₃₁	C₄²	1	4	yes	—
50	C₂×C₄×C₃₁	C₂²×C₄	1	2	yes	—
51	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₄×C₃₁	C₂²×C₄	1	2	yes	—
53	C₂×C₄×C₃₁	C₂²×C₄	1	4	yes	—
54	C₂×C₄×C₃₁	C₂²×C₄	1	2	yes	—
55	C₂×C₄×C₃₁	C₂²×C₄	1	4	yes	—
56	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
57	C₂×Dic₃₁	—	2	4	no	—
58	C₂×Dic₃₁	—	2	16	no	—
59	C₂×Dic₃₁	M₁₆	1	8	yes	direct product factor
60	C₂×Dic₃₁	C₂²×C₄	1	4	yes	—
61	C₂×Dic₃₁	C₂²×C₄	1	4	yes	—
62	C₈×C₃₁	C₂²×C₄	1	8	yes	complemented
63	C₃₁⋊C₈	C₂²×C₄	1	8	yes	complemented
64	C₂×C₄×M₁₆	—	31	1	no	maximal, Sylow
65	C₂²×C₃₁	—	2	4	no	—
66	C₂²×C₃₁	—	2	8	no	—
67	C₂²×C₃₁	C₂×C₄²	1	4	yes	—
68	C₂²×C₃₁	C₂×C₄²	1	2	yes	—
69	C₂²×C₃₁	C₂×M₁₆	1	4	yes	—
70	C₂²×C₃₁	C₂³×C₄	1	1	yes	characteristic
71	C₄×C₃₁	C₂×C₄²	1	2	yes	—
72	C₄×C₃₁	C₂×C₄²	1	4	yes	—
73	C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
74	C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
75	Dic₃₁	—	2	4	no	—
76	Dic₃₁	C₂×M₁₆	1	8	yes	direct product factor
77	C₂²×M₁₆	—	31	1	no	—
78	C₂²×M₁₆	D₃₁	1	1	yes	characteristic
79	C₂²×C₄²	—	31	1	no	—
80	C₄×M₁₆	—	31	8	no	—
81	C₂×#97	—	31	2	no	—
82	C₂×C₄×C₈	—	31	2	no	—
83	C₂×C₃₁	—	2	4	no	—
84	C₂×C₃₁	C₄×M₁₆	1	4	yes	complemented
85	C₂×C₃₁	C₂²×C₄²	1	1	yes	characteristic
86	C₂×C₃₁	C₂²×M₁₆	1	2	yes	—
87	C₂³×C₄	—	31	2	no	—
88	C₂³×C₄	C₂×D₃₁	1	1	yes	characteristic
89	C₂×M₁₆	—	31	8	no	—
90	C₂×M₁₆	—	31	4	no	—
91	C₂×M₁₆	Dic₃₁	1	8	yes	direct product factor
92	C₂×M₁₆	C₂×D₃₁	1	4	yes	—
93	C₂²×C₈	—	31	2	no	—
94	C₂²×C₈	C₂×D₃₁	1	2	yes	—
95	C₂×C₄²	—	31	2	no	—
96	C₂×C₄²	—	31	4	no	—
97	C₂×C₄²	—	31	4	no	—
98	C₂×C₄²	—	31	1	no	—
99	C₂×C₄²	—	31	1	no	—
100	#97	—	31	8	no	—
101	C₄×C₈	—	31	8	no	—
102	C₃₁	C₂×C₄×M₁₆	1	1	yes	characteristic, complemented, Sylow
103	C₂⁴	C₄×D₃₁	1	1	yes	characteristic
104	C₂²×C₄	—	31	2	no	—
105	C₂²×C₄	—	31	8	no	—
106	C₂²×C₄	—	31	2	no	—
107	C₂²×C₄	—	62	8	no	—
108	C₂²×C₄	C₄×D₃₁	1	1	yes	characteristic
109	C₂²×C₄	C₄×D₃₁	1	2	yes	—
110	C₂²×C₄	C₄×D₃₁	1	4	yes	—
111	C₂²×C₄	C₂×Dic₃₁	1	4	yes	—
112	C₂²×C₄	C₂²×D₃₁	1	2	yes	—
113	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
114	M₁₆	—	31	16	no	—
115	M₁₆	C₂×Dic₃₁	1	16	yes	direct product factor
116	C₂×C₈	—	31	8	no	—
117	C₂×C₈	—	31	4	no	—
118	C₂×C₈	C₂×Dic₃₁	1	8	yes	complemented
119	C₂×C₈	C₂²×D₃₁	1	4	yes	—
120	C₄²	—	31	2	no	—
121	C₄²	—	31	8	no	—
122	C₄²	—	31	2	no	—
123	C₄²	—	31	2	no	—
124	C₄²	—	31	2	no	—
125	C₂³	—	2	4	no	—
126	C₂³	C₄×Dic₃₁	1	4	yes	—
127	C₂³	C₂×C₄×D₃₁	1	2	yes	—
128	C₂³	C₂×C₄×D₃₁	1	1	yes	characteristic
129	C₂×C₄	—	31	8	no	—
130	C₂×C₄	—	31	4	no	—
131	C₂×C₄	—	31	4	no	—
132	C₂×C₄	—	62	16	no	—
133	C₂×C₄	—	62	4	no	—
134	C₂×C₄	C₄×Dic₃₁	1	8	yes	—
135	C₂×C₄	C₄×Dic₃₁	1	4	yes	—
136	C₂×C₄	C₂×C₄×D₃₁	1	2	yes	—
137	C₂×C₄	C₂×C₄×D₃₁	1	1	yes	characteristic
138	C₂×C₄	C₂×C₄×D₃₁	1	2	yes	—
139	C₂×C₄	C₂×C₄×D₃₁	1	4	yes	—
140	C₂×C₄	C₂²×Dic₃₁	1	2	yes	—
141	C₂×C₄	C₂²×Dic₃₁	1	4	yes	—
142	C₂×C₄	C₂³×D₃₁	1	1	yes	characteristic
143	C₈	—	31	8	no	—
144	C₈	C₂²×Dic₃₁	1	8	yes	complemented
145	C₂²	—	2	8	no	—
146	C₂²	—	2	4	no	—
147	C₂²	M₁₆×D₃₁	1	4	yes	—
148	C₂²	C₂×C₄×Dic₃₁	1	4	yes	—
149	C₂²	C₂×C₄×Dic₃₁	1	2	yes	—
150	C₂²	C₂²×C₄×D₃₁	1	1	yes	characteristic
151	C₄	—	31	8	no	—
152	C₄	—	62	4	no	—
153	C₄	C₂×C₄×Dic₃₁	1	4	yes	—
154	C₄	C₂×C₄×Dic₃₁	1	2	yes	—
155	C₄	C₂²×C₄×D₃₁	1	1	yes	characteristic
156	C₄	C₂³×Dic₃₁	1	1	yes	characteristic
157	C₂	—	2	4	no	—
158	C₂	M₁₆×Dic₃₁	1	4	yes	direct product factor
159	C₂	C₂²×C₄×Dic₃₁	1	1	yes	characteristic
160	C₂	C₂×M₁₆×D₃₁	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂²×D₃₁
commutator	C₂×C₃₁	C₂²×C₄²
Frattini	C₂×C₄	C₂³×D₃₁
Fitting	C₂²×M₁₆×C₃₁	C₂
socle	C₂³×C₃₁	C₂²×C₄
prime radical	C₂⁴×C₃₁	C₂×C₄
cosocle	C₂×C₄×C₃₁	C₂⁴
hypercenter	C₂²×M₁₆	D₃₁
hypocenter	C₃₁	C₂×C₄×M₁₆
LCS	G ▷ C₂×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×M₁₆

Complements

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

N	H ≅ G/N	type
3. M₁₆×Dic₃₁	C₂	direct (N × H)
4. C₂×#11758403	C₂	semidirect
5. C₂×C₈×Dic₃₁	C₂	semidirect
9. C₂×M₁₆×C₃₁	C₄	semidirect
17. C₂×#92474	C₄	semidirect
20. #11758403	C₂²	semidirect
21. C₈×Dic₃₁	C₂²	semidirect
33. M₁₆×C₃₁	C₂×C₄	semidirect
34. C₂×C₈×C₃₁	C₂×C₄	semidirect
41. #92474	C₂×C₄	semidirect
42. C₂×C₃₁⋊C₈	C₂×C₄	semidirect
59. C₂×Dic₃₁	M₁₆	direct (N × H)
62. C₈×C₃₁	C₂²×C₄	semidirect
63. C₃₁⋊C₈	C₂²×C₄	semidirect
76. Dic₃₁	C₂×M₁₆	direct (N × H)
84. C₂×C₃₁	C₄×M₁₆	semidirect
91. C₂×M₁₆	Dic₃₁	direct (N × H)
102. C₃₁	C₂×C₄×M₁₆	semidirect
115. M₁₆	C₂×Dic₃₁	direct (N × H)
118. C₂×C₈	C₂×Dic₃₁	semidirect
144. C₈	C₂²×Dic₃₁	semidirect
158. C₂	M₁₆×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
C₂²×#92474	2
C₂²×C₄×Dic₃₁	2
M₁₆×Dic₃₁	2
C₂×#11758403	2
C₂×C₈×Dic₃₁	2
C₂²×M₁₆×C₃₁	2
C₂×C₄×M₁₆	31

Subgroups — C2×M16×Dic31 #16854118

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×M₁₆×Dic₃₁ #16854118