Subgroups — C₂²×C₃×C₈²×C₃₁ #72201110

Order 23808.

2684 conjugacy classes of subgroups.

The subgroup lattice has 156 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×C₄×C₈×C₃₁	C₂	1	3	yes	maximal
2	C₂×C₃×C₈²×C₃₁	C₂	1	12	yes	maximal, direct product factor
3	C₂²×C₃×C₄²×C₃₁	C₂²	1	1	yes	characteristic
4	C₂³×C₃×C₈×C₃₁	C₄	1	6	yes	—
5	C₂×C₃×C₄×C₈×C₃₁	C₄	1	18	yes	—
6	C₂×C₃×C₄×C₈×C₃₁	C₂²	1	18	yes	—
7	C₃×C₈²×C₃₁	C₂²	1	16	yes	direct product factor
8	C₂³×C₃×C₄×C₃₁	C₂×C₄	1	3	yes	—
9	C₂²×C₃×C₈×C₃₁	C₈	1	48	yes	direct product factor
10	C₂²×C₃×C₈×C₃₁	C₂×C₄	1	36	yes	—
11	C₂×C₃×C₄²×C₃₁	C₂×C₄	1	9	yes	—
12	C₂×C₃×C₄²×C₃₁	C₂³	1	3	yes	—
13	C₃×C₄×C₈×C₃₁	C₂×C₄	1	36	yes	—
14	C₃×C₄×C₈×C₃₁	C₂³	1	12	yes	—
15	C₂²×C₈²×C₃₁	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
16	C₂²×C₄×C₈×C₃₁	C₂×C₃	1	3	yes	—
17	C₂×C₈²×C₃₁	C₂×C₃	1	12	yes	direct product factor
18	C₂³×C₈×C₃₁	C₃×C₄	1	6	yes	—
19	C₂²×C₄²×C₃₁	C₂²×C₃	1	1	yes	characteristic
20	C₂×C₄×C₈×C₃₁	C₃×C₄	1	18	yes	—
21	C₂×C₄×C₈×C₃₁	C₂²×C₃	1	18	yes	—
22	C₈²×C₃₁	C₂²×C₃	1	16	yes	direct product factor
23	C₂⁴×C₃×C₃₁	C₄²	1	1	yes	characteristic
24	C₂²×C₃×C₄×C₃₁	C₄²	1	9	yes	—
25	C₂²×C₃×C₄×C₃₁	C₂×C₈	1	24	yes	—
26	C₂²×C₃×C₄×C₃₁	C₂²×C₄	1	9	yes	—
27	C₂×C₃×C₈×C₃₁	C₂×C₈	1	144	yes	direct product factor
28	C₂×C₃×C₈×C₃₁	C₂²×C₄	1	24	yes	—
29	C₃×C₄²×C₃₁	C₄²	1	6	yes	—
30	C₃×C₄²×C₃₁	C₂²×C₄	1	9	yes	—
31	C₃×C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
32	C₂³×C₄×C₃₁	C₂×C₃×C₄	1	3	yes	—
33	C₂²×C₈×C₃₁	C₃×C₈	1	48	yes	direct product factor
34	C₂²×C₈×C₃₁	C₂×C₃×C₄	1	36	yes	—
35	C₂×C₄²×C₃₁	C₂×C₃×C₄	1	9	yes	—
36	C₂×C₄²×C₃₁	C₂³×C₃	1	3	yes	—
37	C₄×C₈×C₃₁	C₂×C₃×C₄	1	36	yes	—
38	C₄×C₈×C₃₁	C₂³×C₃	1	12	yes	—
39	C₂²×C₃×C₈²	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Hall
40	C₂³×C₃×C₃₁	C₄×C₈	1	12	yes	—
41	C₂³×C₃×C₃₁	C₂×C₄²	1	3	yes	—
42	C₂×C₃×C₄×C₃₁	C₄×C₈	1	36	yes	—
43	C₂×C₃×C₄×C₃₁	C₂×C₄²	1	9	yes	—
44	C₂×C₃×C₄×C₃₁	C₂²×C₈	1	36	yes	—
45	C₂×C₃×C₄×C₃₁	C₂³×C₄	1	3	yes	—
46	C₃×C₈×C₃₁	C₂²×C₈	1	48	yes	direct product factor
47	C₂⁴×C₃₁	C₃×C₄²	1	1	yes	characteristic
48	C₂²×C₄×C₃₁	C₃×C₄²	1	9	yes	—
49	C₂²×C₄×C₃₁	C₂×C₃×C₈	1	24	yes	—
50	C₂²×C₄×C₃₁	C₂²×C₃×C₄	1	9	yes	—
51	C₂×C₈×C₃₁	C₂×C₃×C₈	1	144	yes	direct product factor
52	C₂×C₈×C₃₁	C₂²×C₃×C₄	1	24	yes	—
53	C₄²×C₃₁	C₃×C₄²	1	6	yes	—
54	C₄²×C₃₁	C₂²×C₃×C₄	1	9	yes	—
55	C₄²×C₃₁	C₂⁴×C₃	1	1	yes	characteristic
56	C₂²×C₃×C₄×C₈	C₂×C₃₁	1	3	yes	—
57	C₂×C₃×C₈²	C₂×C₃₁	1	12	yes	direct product factor
58	C₂²×C₃×C₃₁	C₈²	1	16	yes	direct product factor
59	C₂²×C₃×C₃₁	C₂×C₄×C₈	1	18	yes	—
60	C₂²×C₃×C₃₁	C₂²×C₄²	1	1	yes	characteristic
61	C₃×C₄×C₃₁	C₂×C₄×C₈	1	18	yes	—
62	C₃×C₄×C₃₁	C₂³×C₈	1	6	yes	—
63	C₂²×C₈²	C₃×C₃₁	1	1	yes	characteristic, direct product factor, Sylow
64	C₂³×C₃₁	C₃×C₄×C₈	1	12	yes	—
65	C₂³×C₃₁	C₂×C₃×C₄²	1	3	yes	—
66	C₂×C₄×C₃₁	C₃×C₄×C₈	1	36	yes	—
67	C₂×C₄×C₃₁	C₂×C₃×C₄²	1	9	yes	—
68	C₂×C₄×C₃₁	C₂²×C₃×C₈	1	36	yes	—
69	C₂×C₄×C₃₁	C₂³×C₃×C₄	1	3	yes	—
70	C₈×C₃₁	C₂²×C₃×C₈	1	48	yes	direct product factor
71	C₂³×C₃×C₈	C₄×C₃₁	1	6	yes	—
72	C₂²×C₃×C₄²	C₂²×C₃₁	1	1	yes	characteristic
73	C₂×C₃×C₄×C₈	C₄×C₃₁	1	18	yes	—
74	C₂×C₃×C₄×C₈	C₂²×C₃₁	1	18	yes	—
75	C₃×C₈²	C₂²×C₃₁	1	16	yes	direct product factor
76	C₂×C₃×C₃₁	C₂×C₈²	1	12	yes	direct product factor
77	C₂×C₃×C₃₁	C₂²×C₄×C₈	1	3	yes	—
78	C₂²×C₄×C₈	C₂×C₃×C₃₁	1	3	yes	—
79	C₂×C₈²	C₂×C₃×C₃₁	1	12	yes	direct product factor
80	C₂²×C₃₁	C₃×C₈²	1	16	yes	direct product factor
81	C₂²×C₃₁	C₂×C₃×C₄×C₈	1	18	yes	—
82	C₂²×C₃₁	C₂²×C₃×C₄²	1	1	yes	characteristic
83	C₄×C₃₁	C₂×C₃×C₄×C₈	1	18	yes	—
84	C₄×C₃₁	C₂³×C₃×C₈	1	6	yes	—
85	C₂³×C₃×C₄	C₂×C₄×C₃₁	1	3	yes	—
86	C₂²×C₃×C₈	C₈×C₃₁	1	48	yes	direct product factor
87	C₂²×C₃×C₈	C₂×C₄×C₃₁	1	36	yes	—
88	C₂×C₃×C₄²	C₂×C₄×C₃₁	1	9	yes	—
89	C₂×C₃×C₄²	C₂³×C₃₁	1	3	yes	—
90	C₃×C₄×C₈	C₂×C₄×C₃₁	1	36	yes	—
91	C₃×C₄×C₈	C₂³×C₃₁	1	12	yes	—
92	C₃×C₃₁	C₂²×C₈²	1	1	yes	characteristic, direct product factor, Hall
93	C₂³×C₈	C₃×C₄×C₃₁	1	6	yes	—
94	C₂²×C₄²	C₂²×C₃×C₃₁	1	1	yes	characteristic
95	C₂×C₄×C₈	C₃×C₄×C₃₁	1	18	yes	—
96	C₂×C₄×C₈	C₂²×C₃×C₃₁	1	18	yes	—
97	C₈²	C₂²×C₃×C₃₁	1	16	yes	direct product factor
98	C₂×C₃₁	C₂×C₃×C₈²	1	12	yes	direct product factor
99	C₂×C₃₁	C₂²×C₃×C₄×C₈	1	3	yes	—
100	C₂⁴×C₃	C₄²×C₃₁	1	1	yes	characteristic
101	C₂²×C₃×C₄	C₄²×C₃₁	1	9	yes	—
102	C₂²×C₃×C₄	C₂×C₈×C₃₁	1	24	yes	—
103	C₂²×C₃×C₄	C₂²×C₄×C₃₁	1	9	yes	—
104	C₂×C₃×C₈	C₂×C₈×C₃₁	1	144	yes	direct product factor
105	C₂×C₃×C₈	C₂²×C₄×C₃₁	1	24	yes	—
106	C₃×C₄²	C₄²×C₃₁	1	6	yes	—
107	C₃×C₄²	C₂²×C₄×C₃₁	1	9	yes	—
108	C₃×C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
109	C₂³×C₄	C₂×C₃×C₄×C₃₁	1	3	yes	—
110	C₂²×C₈	C₃×C₈×C₃₁	1	48	yes	direct product factor
111	C₂²×C₈	C₂×C₃×C₄×C₃₁	1	36	yes	—
112	C₂×C₄²	C₂×C₃×C₄×C₃₁	1	9	yes	—
113	C₂×C₄²	C₂³×C₃×C₃₁	1	3	yes	—
114	C₄×C₈	C₂×C₃×C₄×C₃₁	1	36	yes	—
115	C₄×C₈	C₂³×C₃×C₃₁	1	12	yes	—
116	C₃₁	C₂²×C₃×C₈²	1	1	yes	characteristic, direct product factor, Sylow
117	C₂³×C₃	C₄×C₈×C₃₁	1	12	yes	—
118	C₂³×C₃	C₂×C₄²×C₃₁	1	3	yes	—
119	C₂×C₃×C₄	C₄×C₈×C₃₁	1	36	yes	—
120	C₂×C₃×C₄	C₂×C₄²×C₃₁	1	9	yes	—
121	C₂×C₃×C₄	C₂²×C₈×C₃₁	1	36	yes	—
122	C₂×C₃×C₄	C₂³×C₄×C₃₁	1	3	yes	—
123	C₃×C₈	C₂²×C₈×C₃₁	1	48	yes	direct product factor
124	C₂⁴	C₃×C₄²×C₃₁	1	1	yes	characteristic
125	C₂²×C₄	C₃×C₄²×C₃₁	1	9	yes	—
126	C₂²×C₄	C₂×C₃×C₈×C₃₁	1	24	yes	—
127	C₂²×C₄	C₂²×C₃×C₄×C₃₁	1	9	yes	—
128	C₂×C₈	C₂×C₃×C₈×C₃₁	1	144	yes	direct product factor
129	C₂×C₈	C₂²×C₃×C₄×C₃₁	1	24	yes	—
130	C₄²	C₃×C₄²×C₃₁	1	6	yes	—
131	C₄²	C₂²×C₃×C₄×C₃₁	1	9	yes	—
132	C₄²	C₂⁴×C₃×C₃₁	1	1	yes	characteristic
133	C₂²×C₃	C₈²×C₃₁	1	16	yes	direct product factor
134	C₂²×C₃	C₂×C₄×C₈×C₃₁	1	18	yes	—
135	C₂²×C₃	C₂²×C₄²×C₃₁	1	1	yes	characteristic
136	C₃×C₄	C₂×C₄×C₈×C₃₁	1	18	yes	—
137	C₃×C₄	C₂³×C₈×C₃₁	1	6	yes	—
138	C₂³	C₃×C₄×C₈×C₃₁	1	12	yes	—
139	C₂³	C₂×C₃×C₄²×C₃₁	1	3	yes	—
140	C₂×C₄	C₃×C₄×C₈×C₃₁	1	36	yes	—
141	C₂×C₄	C₂×C₃×C₄²×C₃₁	1	9	yes	—
142	C₂×C₄	C₂²×C₃×C₈×C₃₁	1	36	yes	—
143	C₂×C₄	C₂³×C₃×C₄×C₃₁	1	3	yes	—
144	C₈	C₂²×C₃×C₈×C₃₁	1	48	yes	direct product factor
145	C₂×C₃	C₂×C₈²×C₃₁	1	12	yes	direct product factor
146	C₂×C₃	C₂²×C₄×C₈×C₃₁	1	3	yes	—
147	C₂²	C₃×C₈²×C₃₁	1	16	yes	direct product factor
148	C₂²	C₂×C₃×C₄×C₈×C₃₁	1	18	yes	—
149	C₂²	C₂²×C₃×C₄²×C₃₁	1	1	yes	characteristic
150	C₄	C₂×C₃×C₄×C₈×C₃₁	1	18	yes	—
151	C₄	C₂³×C₃×C₈×C₃₁	1	6	yes	—
152	C₃	C₂²×C₈²×C₃₁	1	1	yes	characteristic, direct product factor, Sylow
153	C₂	C₂×C₃×C₈²×C₃₁	1	12	yes	direct product factor
154	C₂	C₂²×C₃×C₄×C₈×C₃₁	1	3	yes	—

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₄²	C₂⁴×C₃×C₃₁
socle, prime radical	C₂⁴×C₃×C₃₁	C₄²
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₃×C₈²×C₃₁	C₂	direct (N × H)
7. C₃×C₈²×C₃₁	C₂²	direct (N × H)
9. C₂²×C₃×C₈×C₃₁	C₈	direct (N × H)
15. C₂²×C₈²×C₃₁	C₃	direct (N × H)
17. C₂×C₈²×C₃₁	C₂×C₃	direct (N × H)
22. C₈²×C₃₁	C₂²×C₃	direct (N × H)
27. C₂×C₃×C₈×C₃₁	C₂×C₈	direct (N × H)
33. C₂²×C₈×C₃₁	C₃×C₈	direct (N × H)
39. C₂²×C₃×C₈²	C₃₁	direct (N × H)
46. C₃×C₈×C₃₁	C₂²×C₈	direct (N × H)
51. C₂×C₈×C₃₁	C₂×C₃×C₈	direct (N × H)
57. C₂×C₃×C₈²	C₂×C₃₁	direct (N × H)
58. C₂²×C₃×C₃₁	C₈²	direct (N × H)
63. C₂²×C₈²	C₃×C₃₁	direct (N × H)
70. C₈×C₃₁	C₂²×C₃×C₈	direct (N × H)
75. C₃×C₈²	C₂²×C₃₁	direct (N × H)
76. C₂×C₃×C₃₁	C₂×C₈²	direct (N × H)
79. C₂×C₈²	C₂×C₃×C₃₁	direct (N × H)
80. C₂²×C₃₁	C₃×C₈²	direct (N × H)
86. C₂²×C₃×C₈	C₈×C₃₁	direct (N × H)
92. C₃×C₃₁	C₂²×C₈²	direct (N × H)
97. C₈²	C₂²×C₃×C₃₁	direct (N × H)
98. C₂×C₃₁	C₂×C₃×C₈²	direct (N × H)
104. C₂×C₃×C₈	C₂×C₈×C₃₁	direct (N × H)
110. C₂²×C₈	C₃×C₈×C₃₁	direct (N × H)
116. C₃₁	C₂²×C₃×C₈²	direct (N × H)
123. C₃×C₈	C₂²×C₈×C₃₁	direct (N × H)
128. C₂×C₈	C₂×C₃×C₈×C₃₁	direct (N × H)
133. C₂²×C₃	C₈²×C₃₁	direct (N × H)
144. C₈	C₂²×C₃×C₈×C₃₁	direct (N × H)
145. C₂×C₃	C₂×C₈²×C₃₁	direct (N × H)
147. C₂²	C₃×C₈²×C₃₁	direct (N × H)
152. C₃	C₂²×C₈²×C₃₁	direct (N × H)
153. C₂	C₂×C₃×C₈²×C₃₁	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×C₃×C₄×C₈×C₃₁	2
C₂×C₃×C₈²×C₃₁	2
C₂²×C₈²×C₃₁	3
C₂²×C₃×C₈²	31

Subgroups — C22×C3×C82×C31 #72201110

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₃×C₈²×C₃₁ #72201110