Subgroups — C₃×C₄²×C₁₆×C₃₇ #72205373

Order 28416.

1116 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	C₃×C₄²×C₈×C₃₇	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₄×C₁₆×C₃₇	C₂	1	6	yes	maximal
3	C₃×C₄³×C₃₇	C₄	1	1	yes	characteristic
4	C₂×C₃×C₄×C₈×C₃₇	C₄	1	3	yes	—
5	C₂×C₃×C₄×C₈×C₃₇	C₂²	1	3	yes	—
6	C₂²×C₃×C₁₆×C₃₇	C₂²	1	4	yes	—
7	C₃×C₄×C₁₆×C₃₇	C₄	1	24	yes	direct product factor
8	C₂²×C₃×C₈×C₃₇	C₂×C₄	1	3	yes	—
9	C₂²×C₃×C₈×C₃₇	C₂³	1	1	yes	characteristic
10	C₂×C₃×C₄²×C₃₇	C₈	1	4	yes	—
11	C₂×C₃×C₄²×C₃₇	C₂×C₄	1	3	yes	—
12	C₃×C₄×C₈×C₃₇	C₈	1	12	yes	—
13	C₃×C₄×C₈×C₃₇	C₂×C₄	1	12	yes	—
14	C₂×C₃×C₁₆×C₃₇	C₂×C₄	1	24	yes	—
15	C₄²×C₁₆×C₃₇	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
16	C₂×C₄×C₁₆×C₃₇	C₂×C₃	1	6	yes	—
17	C₄²×C₈×C₃₇	C₂×C₃	1	1	yes	characteristic
18	C₂²×C₁₆×C₃₇	C₂²×C₃	1	4	yes	—
19	C₂×C₄×C₈×C₃₇	C₃×C₄	1	3	yes	—
20	C₂×C₄×C₈×C₃₇	C₂²×C₃	1	3	yes	—
21	C₄³×C₃₇	C₃×C₄	1	1	yes	characteristic
22	C₄×C₁₆×C₃₇	C₃×C₄	1	24	yes	direct product factor
23	C₂²×C₃×C₄×C₃₇	C₂×C₈	1	6	yes	—
24	C₂²×C₃×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
25	C₂×C₃×C₈×C₃₇	C₄²	1	6	yes	—
26	C₂×C₃×C₈×C₃₇	C₂×C₈	1	12	yes	—
27	C₂×C₃×C₈×C₃₇	C₂²×C₄	1	6	yes	—
28	C₃×C₄²×C₃₇	C₁₆	1	16	yes	direct product factor
29	C₃×C₄²×C₃₇	C₄²	1	6	yes	—
30	C₃×C₄²×C₃₇	C₂×C₈	1	6	yes	—
31	C₃×C₁₆×C₃₇	C₄²	1	16	yes	direct product factor
32	C₂²×C₈×C₃₇	C₂×C₃×C₄	1	3	yes	—
33	C₂²×C₈×C₃₇	C₂³×C₃	1	1	yes	characteristic
34	C₂×C₄²×C₃₇	C₃×C₈	1	4	yes	—
35	C₂×C₄²×C₃₇	C₂×C₃×C₄	1	3	yes	—
36	C₂×C₁₆×C₃₇	C₂×C₃×C₄	1	24	yes	—
37	C₄×C₈×C₃₇	C₃×C₈	1	12	yes	—
38	C₄×C₈×C₃₇	C₂×C₃×C₄	1	12	yes	—
39	C₂³×C₃×C₃₇	C₂²×C₈	1	1	yes	characteristic
40	C₂×C₃×C₄×C₃₇	C₄×C₈	1	12	yes	—
41	C₂×C₃×C₄×C₃₇	C₂×C₁₆	1	24	yes	—
42	C₂×C₃×C₄×C₃₇	C₂×C₄²	1	3	yes	—
43	C₂×C₃×C₄×C₃₇	C₂²×C₈	1	3	yes	—
44	C₃×C₈×C₃₇	C₄×C₈	1	12	yes	—
45	C₃×C₈×C₃₇	C₂×C₄²	1	4	yes	—
46	C₃×C₄²×C₁₆	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Hall
47	C₂²×C₄×C₃₇	C₂×C₃×C₈	1	6	yes	—
48	C₂²×C₄×C₃₇	C₂²×C₃×C₄	1	1	yes	characteristic
49	C₂×C₈×C₃₇	C₃×C₄²	1	6	yes	—
50	C₂×C₈×C₃₇	C₂×C₃×C₈	1	12	yes	—
51	C₂×C₈×C₃₇	C₂²×C₃×C₄	1	6	yes	—
52	C₄²×C₃₇	C₃×C₁₆	1	16	yes	direct product factor
53	C₄²×C₃₇	C₃×C₄²	1	6	yes	—
54	C₄²×C₃₇	C₂×C₃×C₈	1	6	yes	—
55	C₁₆×C₃₇	C₃×C₄²	1	16	yes	direct product factor
56	C₂²×C₃×C₃₇	C₂×C₄×C₈	1	3	yes	—
57	C₂²×C₃×C₃₇	C₂²×C₁₆	1	4	yes	—
58	C₃×C₄×C₃₇	C₄×C₁₆	1	24	yes	direct product factor
59	C₃×C₄×C₃₇	C₄³	1	1	yes	characteristic
60	C₃×C₄×C₃₇	C₂×C₄×C₈	1	3	yes	—
61	C₂×C₃×C₄×C₁₆	C₂×C₃₇	1	6	yes	—
62	C₃×C₄²×C₈	C₂×C₃₇	1	1	yes	characteristic
63	C₂³×C₃₇	C₂²×C₃×C₈	1	1	yes	characteristic
64	C₂×C₄×C₃₇	C₃×C₄×C₈	1	12	yes	—
65	C₂×C₄×C₃₇	C₂×C₃×C₁₆	1	24	yes	—
66	C₂×C₄×C₃₇	C₂×C₃×C₄²	1	3	yes	—
67	C₂×C₄×C₃₇	C₂²×C₃×C₈	1	3	yes	—
68	C₈×C₃₇	C₃×C₄×C₈	1	12	yes	—
69	C₈×C₃₇	C₂×C₃×C₄²	1	4	yes	—
70	C₄²×C₁₆	C₃×C₃₇	1	1	yes	characteristic, direct product factor, Sylow
71	C₂×C₃×C₃₇	C₄²×C₈	1	1	yes	characteristic
72	C₂×C₃×C₃₇	C₂×C₄×C₁₆	1	6	yes	—
73	C₂²×C₃×C₁₆	C₂²×C₃₇	1	4	yes	—
74	C₂×C₃×C₄×C₈	C₄×C₃₇	1	3	yes	—
75	C₂×C₃×C₄×C₈	C₂²×C₃₇	1	3	yes	—
76	C₃×C₄³	C₄×C₃₇	1	1	yes	characteristic
77	C₃×C₄×C₁₆	C₄×C₃₇	1	24	yes	direct product factor
78	C₂²×C₃₇	C₂×C₃×C₄×C₈	1	3	yes	—
79	C₂²×C₃₇	C₂²×C₃×C₁₆	1	4	yes	—
80	C₄×C₃₇	C₃×C₄×C₁₆	1	24	yes	direct product factor
81	C₄×C₃₇	C₃×C₄³	1	1	yes	characteristic
82	C₄×C₃₇	C₂×C₃×C₄×C₈	1	3	yes	—
83	C₂×C₄×C₁₆	C₂×C₃×C₃₇	1	6	yes	—
84	C₄²×C₈	C₂×C₃×C₃₇	1	1	yes	characteristic
85	C₃×C₃₇	C₄²×C₁₆	1	1	yes	characteristic, direct product factor, Hall
86	C₂²×C₃×C₈	C₂×C₄×C₃₇	1	3	yes	—
87	C₂²×C₃×C₈	C₂³×C₃₇	1	1	yes	characteristic
88	C₂×C₃×C₄²	C₈×C₃₇	1	4	yes	—
89	C₂×C₃×C₄²	C₂×C₄×C₃₇	1	3	yes	—
90	C₂×C₃×C₁₆	C₂×C₄×C₃₇	1	24	yes	—
91	C₃×C₄×C₈	C₈×C₃₇	1	12	yes	—
92	C₃×C₄×C₈	C₂×C₄×C₃₇	1	12	yes	—
93	C₂×C₃₇	C₃×C₄²×C₈	1	1	yes	characteristic
94	C₂×C₃₇	C₂×C₃×C₄×C₁₆	1	6	yes	—
95	C₂²×C₁₆	C₂²×C₃×C₃₇	1	4	yes	—
96	C₂×C₄×C₈	C₃×C₄×C₃₇	1	3	yes	—
97	C₂×C₄×C₈	C₂²×C₃×C₃₇	1	3	yes	—
98	C₄³	C₃×C₄×C₃₇	1	1	yes	characteristic
99	C₄×C₁₆	C₃×C₄×C₃₇	1	24	yes	direct product factor
100	C₂²×C₃×C₄	C₂×C₈×C₃₇	1	6	yes	—
101	C₂²×C₃×C₄	C₂²×C₄×C₃₇	1	1	yes	characteristic
102	C₂×C₃×C₈	C₄²×C₃₇	1	6	yes	—
103	C₂×C₃×C₈	C₂×C₈×C₃₇	1	12	yes	—
104	C₂×C₃×C₈	C₂²×C₄×C₃₇	1	6	yes	—
105	C₃×C₄²	C₁₆×C₃₇	1	16	yes	direct product factor
106	C₃×C₄²	C₄²×C₃₇	1	6	yes	—
107	C₃×C₄²	C₂×C₈×C₃₇	1	6	yes	—
108	C₃×C₁₆	C₄²×C₃₇	1	16	yes	direct product factor
109	C₃₇	C₃×C₄²×C₁₆	1	1	yes	characteristic, direct product factor, Sylow
110	C₂²×C₈	C₂×C₃×C₄×C₃₇	1	3	yes	—
111	C₂²×C₈	C₂³×C₃×C₃₇	1	1	yes	characteristic
112	C₂×C₄²	C₃×C₈×C₃₇	1	4	yes	—
113	C₂×C₄²	C₂×C₃×C₄×C₃₇	1	3	yes	—
114	C₂×C₁₆	C₂×C₃×C₄×C₃₇	1	24	yes	—
115	C₄×C₈	C₃×C₈×C₃₇	1	12	yes	—
116	C₄×C₈	C₂×C₃×C₄×C₃₇	1	12	yes	—
117	C₂³×C₃	C₂²×C₈×C₃₇	1	1	yes	characteristic
118	C₂×C₃×C₄	C₄×C₈×C₃₇	1	12	yes	—
119	C₂×C₃×C₄	C₂×C₁₆×C₃₇	1	24	yes	—
120	C₂×C₃×C₄	C₂×C₄²×C₃₇	1	3	yes	—
121	C₂×C₃×C₄	C₂²×C₈×C₃₇	1	3	yes	—
122	C₃×C₈	C₄×C₈×C₃₇	1	12	yes	—
123	C₃×C₈	C₂×C₄²×C₃₇	1	4	yes	—
124	C₂²×C₄	C₂×C₃×C₈×C₃₇	1	6	yes	—
125	C₂²×C₄	C₂²×C₃×C₄×C₃₇	1	1	yes	characteristic
126	C₂×C₈	C₃×C₄²×C₃₇	1	6	yes	—
127	C₂×C₈	C₂×C₃×C₈×C₃₇	1	12	yes	—
128	C₂×C₈	C₂²×C₃×C₄×C₃₇	1	6	yes	—
129	C₄²	C₃×C₁₆×C₃₇	1	16	yes	direct product factor
130	C₄²	C₃×C₄²×C₃₇	1	6	yes	—
131	C₄²	C₂×C₃×C₈×C₃₇	1	6	yes	—
132	C₁₆	C₃×C₄²×C₃₇	1	16	yes	direct product factor
133	C₂²×C₃	C₂×C₄×C₈×C₃₇	1	3	yes	—
134	C₂²×C₃	C₂²×C₁₆×C₃₇	1	4	yes	—
135	C₃×C₄	C₄×C₁₆×C₃₇	1	24	yes	direct product factor
136	C₃×C₄	C₄³×C₃₇	1	1	yes	characteristic
137	C₃×C₄	C₂×C₄×C₈×C₃₇	1	3	yes	—
138	C₂³	C₂²×C₃×C₈×C₃₇	1	1	yes	characteristic
139	C₂×C₄	C₂×C₃×C₁₆×C₃₇	1	24	yes	—
140	C₂×C₄	C₃×C₄×C₈×C₃₇	1	12	yes	—
141	C₂×C₄	C₂×C₃×C₄²×C₃₇	1	3	yes	—
142	C₂×C₄	C₂²×C₃×C₈×C₃₇	1	3	yes	—
143	C₈	C₃×C₄×C₈×C₃₇	1	12	yes	—
144	C₈	C₂×C₃×C₄²×C₃₇	1	4	yes	—
145	C₂×C₃	C₄²×C₈×C₃₇	1	1	yes	characteristic
146	C₂×C₃	C₂×C₄×C₁₆×C₃₇	1	6	yes	—
147	C₂²	C₂²×C₃×C₁₆×C₃₇	1	4	yes	—
148	C₂²	C₂×C₃×C₄×C₈×C₃₇	1	3	yes	—
149	C₄	C₃×C₄×C₁₆×C₃₇	1	24	yes	direct product factor
150	C₄	C₂×C₃×C₄×C₈×C₃₇	1	3	yes	—
151	C₄	C₃×C₄³×C₃₇	1	1	yes	characteristic
152	C₃	C₄²×C₁₆×C₃₇	1	1	yes	characteristic, direct product factor, Sylow
153	C₂	C₂×C₃×C₄×C₁₆×C₃₇	1	6	yes	—
154	C₂	C₃×C₄²×C₈×C₃₇	1	1	yes	characteristic

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₂²×C₈	C₂³×C₃×C₃₇
socle, prime radical	C₂³×C₃×C₃₇	C₂²×C₈
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
7. C₃×C₄×C₁₆×C₃₇	C₄	direct (N × H)
15. C₄²×C₁₆×C₃₇	C₃	direct (N × H)
22. C₄×C₁₆×C₃₇	C₃×C₄	direct (N × H)
28. C₃×C₄²×C₃₇	C₁₆	direct (N × H)
31. C₃×C₁₆×C₃₇	C₄²	direct (N × H)
46. C₃×C₄²×C₁₆	C₃₇	direct (N × H)
52. C₄²×C₃₇	C₃×C₁₆	direct (N × H)
55. C₁₆×C₃₇	C₃×C₄²	direct (N × H)
58. C₃×C₄×C₃₇	C₄×C₁₆	direct (N × H)
70. C₄²×C₁₆	C₃×C₃₇	direct (N × H)
77. C₃×C₄×C₁₆	C₄×C₃₇	direct (N × H)
80. C₄×C₃₇	C₃×C₄×C₁₆	direct (N × H)
85. C₃×C₃₇	C₄²×C₁₆	direct (N × H)
99. C₄×C₁₆	C₃×C₄×C₃₇	direct (N × H)
105. C₃×C₄²	C₁₆×C₃₇	direct (N × H)
108. C₃×C₁₆	C₄²×C₃₇	direct (N × H)
109. C₃₇	C₃×C₄²×C₁₆	direct (N × H)
129. C₄²	C₃×C₁₆×C₃₇	direct (N × H)
132. C₁₆	C₃×C₄²×C₃₇	direct (N × H)
135. C₃×C₄	C₄×C₁₆×C₃₇	direct (N × H)
149. C₄	C₃×C₄×C₁₆×C₃₇	direct (N × H)
152. C₃	C₄²×C₁₆×C₃₇	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃×C₄²×C₈×C₃₇	2
C₂×C₃×C₄×C₁₆×C₃₇	2
C₄²×C₁₆×C₃₇	3
C₃×C₄²×C₁₆	37

Subgroups — C3×C42×C16×C37 #72205373

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₄²×C₁₆×C₃₇ #72205373