Subgroups — C₂³×C₃⁴×C₃₂ #72198321

Order 20736.

57452 conjugacy classes of subgroups.

The subgroup lattice has 165 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₃⁴×C₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₃⁴×C₃₂	C₂	1	14	yes	maximal, direct product factor
3	C₂³×C₃³×C₃₂	C₃	1	40	yes	maximal, direct product factor
4	C₂³×C₃⁴×C₈	C₄	1	1	yes	characteristic
5	C₂²×C₃⁴×C₁₆	C₄	1	7	yes	—
6	C₂²×C₃⁴×C₁₆	C₂²	1	7	yes	—
7	C₂×C₃⁴×C₃₂	C₂²	1	28	yes	direct product factor
8	C₂³×C₃³×C₁₆	C₂×C₃	1	40	yes	—
9	C₂²×C₃³×C₃₂	C₂×C₃	1	560	yes	direct product factor
10	C₂³×C₃⁴×C₄	C₈	1	1	yes	characteristic
11	C₂²×C₃⁴×C₈	C₈	1	7	yes	—
12	C₂²×C₃⁴×C₈	C₂×C₄	1	7	yes	—
13	C₂×C₃⁴×C₁₆	C₂×C₄	1	21	yes	—
14	C₂×C₃⁴×C₁₆	C₂³	1	7	yes	—
15	C₃⁴×C₃₂	C₂³	1	8	yes	direct product factor
16	C₂³×C₃²×C₃₂	C₃²	1	130	yes	direct product factor
17	C₂³×C₃³×C₈	C₃×C₄	1	40	yes	—
18	C₂²×C₃³×C₁₆	C₃×C₄	1	280	yes	—
19	C₂²×C₃³×C₁₆	C₂²×C₃	1	280	yes	—
20	C₂×C₃³×C₃₂	C₂²×C₃	1	1120	yes	direct product factor
21	C₂⁴×C₃⁴	C₁₆	1	1	yes	characteristic
22	C₂²×C₃⁴×C₄	C₁₆	1	7	yes	—
23	C₂²×C₃⁴×C₄	C₂×C₈	1	7	yes	—
24	C₂×C₃⁴×C₈	C₂×C₈	1	21	yes	—
25	C₂×C₃⁴×C₈	C₂²×C₄	1	7	yes	—
26	C₃⁴×C₁₆	C₂²×C₄	1	7	yes	—
27	C₃⁴×C₁₆	C₂⁴	1	1	yes	characteristic
28	C₂³×C₃²×C₁₆	C₂×C₃²	1	130	yes	—
29	C₂²×C₃²×C₃₂	C₂×C₃²	1	1820	yes	direct product factor
30	C₂³×C₃³×C₄	C₃×C₈	1	40	yes	—
31	C₂²×C₃³×C₈	C₃×C₈	1	280	yes	—
32	C₂²×C₃³×C₈	C₂×C₃×C₄	1	280	yes	—
33	C₂×C₃³×C₁₆	C₂×C₃×C₄	1	840	yes	—
34	C₂×C₃³×C₁₆	C₂³×C₃	1	280	yes	—
35	C₃³×C₃₂	C₂³×C₃	1	320	yes	direct product factor
36	C₂³×C₃×C₃₂	C₃³	1	40	yes	direct product factor
37	C₂³×C₃⁴	C₃₂	1	8	yes	direct product factor
38	C₂³×C₃⁴	C₂×C₁₆	1	7	yes	—
39	C₂×C₃⁴×C₄	C₂×C₁₆	1	21	yes	—
40	C₂×C₃⁴×C₄	C₂²×C₈	1	7	yes	—
41	C₃⁴×C₈	C₂²×C₈	1	7	yes	—
42	C₃⁴×C₈	C₂³×C₄	1	1	yes	characteristic
43	C₂³×C₃²×C₈	C₃²×C₄	1	130	yes	—
44	C₂²×C₃²×C₁₆	C₃²×C₄	1	910	yes	—
45	C₂²×C₃²×C₁₆	C₂²×C₃²	1	910	yes	—
46	C₂×C₃²×C₃₂	C₂²×C₃²	1	3640	yes	direct product factor
47	C₂⁴×C₃³	C₃×C₁₆	1	40	yes	—
48	C₂²×C₃³×C₄	C₃×C₁₆	1	280	yes	—
49	C₂²×C₃³×C₄	C₂×C₃×C₈	1	280	yes	—
50	C₂×C₃³×C₈	C₂×C₃×C₈	1	840	yes	—
51	C₂×C₃³×C₈	C₂²×C₃×C₄	1	280	yes	—
52	C₃³×C₁₆	C₂²×C₃×C₄	1	280	yes	—
53	C₃³×C₁₆	C₂⁴×C₃	1	40	yes	—
54	C₂³×C₃×C₁₆	C₂×C₃³	1	40	yes	—
55	C₂²×C₃×C₃₂	C₂×C₃³	1	560	yes	direct product factor
56	C₂²×C₃⁴	C₂×C₃₂	1	28	yes	direct product factor
57	C₂²×C₃⁴	C₂²×C₁₆	1	7	yes	—
58	C₃⁴×C₄	C₂²×C₁₆	1	7	yes	—
59	C₃⁴×C₄	C₂³×C₈	1	1	yes	characteristic
60	C₂³×C₃²×C₄	C₃²×C₈	1	130	yes	—
61	C₂²×C₃²×C₈	C₃²×C₈	1	910	yes	—
62	C₂²×C₃²×C₈	C₂×C₃²×C₄	1	910	yes	—
63	C₂×C₃²×C₁₆	C₂×C₃²×C₄	1	2730	yes	—
64	C₂×C₃²×C₁₆	C₂³×C₃²	1	910	yes	—
65	C₃²×C₃₂	C₂³×C₃²	1	1040	yes	direct product factor
66	C₂³×C₃₂	C₃⁴	1	1	yes	characteristic, direct product factor, Sylow
67	C₂³×C₃³	C₃×C₃₂	1	320	yes	direct product factor
68	C₂³×C₃³	C₂×C₃×C₁₆	1	280	yes	—
69	C₂×C₃³×C₄	C₂×C₃×C₁₆	1	840	yes	—
70	C₂×C₃³×C₄	C₂²×C₃×C₈	1	280	yes	—
71	C₃³×C₈	C₂²×C₃×C₈	1	280	yes	—
72	C₃³×C₈	C₂³×C₃×C₄	1	40	yes	—
73	C₂³×C₃×C₈	C₃³×C₄	1	40	yes	—
74	C₂²×C₃×C₁₆	C₃³×C₄	1	280	yes	—
75	C₂²×C₃×C₁₆	C₂²×C₃³	1	280	yes	—
76	C₂×C₃×C₃₂	C₂²×C₃³	1	1120	yes	direct product factor
77	C₂×C₃⁴	C₂²×C₃₂	1	14	yes	direct product factor
78	C₂×C₃⁴	C₂³×C₁₆	1	1	yes	characteristic
79	C₂⁴×C₃²	C₃²×C₁₆	1	130	yes	—
80	C₂²×C₃²×C₄	C₃²×C₁₆	1	910	yes	—
81	C₂²×C₃²×C₄	C₂×C₃²×C₈	1	910	yes	—
82	C₂×C₃²×C₈	C₂×C₃²×C₈	1	2730	yes	—
83	C₂×C₃²×C₈	C₂²×C₃²×C₄	1	910	yes	—
84	C₃²×C₁₆	C₂²×C₃²×C₄	1	910	yes	—
85	C₃²×C₁₆	C₂⁴×C₃²	1	130	yes	—
86	C₂³×C₁₆	C₂×C₃⁴	1	1	yes	characteristic
87	C₂²×C₃₂	C₂×C₃⁴	1	14	yes	direct product factor
88	C₂²×C₃³	C₂×C₃×C₃₂	1	1120	yes	direct product factor
89	C₂²×C₃³	C₂²×C₃×C₁₆	1	280	yes	—
90	C₃³×C₄	C₂²×C₃×C₁₆	1	280	yes	—
91	C₃³×C₄	C₂³×C₃×C₈	1	40	yes	—
92	C₂³×C₃×C₄	C₃³×C₈	1	40	yes	—
93	C₂²×C₃×C₈	C₃³×C₈	1	280	yes	—
94	C₂²×C₃×C₈	C₂×C₃³×C₄	1	280	yes	—
95	C₂×C₃×C₁₆	C₂×C₃³×C₄	1	840	yes	—
96	C₂×C₃×C₁₆	C₂³×C₃³	1	280	yes	—
97	C₃×C₃₂	C₂³×C₃³	1	320	yes	direct product factor
98	C₃⁴	C₂³×C₃₂	1	1	yes	characteristic, direct product factor, Sylow
99	C₂³×C₃²	C₃²×C₃₂	1	1040	yes	direct product factor
100	C₂³×C₃²	C₂×C₃²×C₁₆	1	910	yes	—
101	C₂×C₃²×C₄	C₂×C₃²×C₁₆	1	2730	yes	—
102	C₂×C₃²×C₄	C₂²×C₃²×C₈	1	910	yes	—
103	C₃²×C₈	C₂²×C₃²×C₈	1	910	yes	—
104	C₃²×C₈	C₂³×C₃²×C₄	1	130	yes	—
105	C₂³×C₈	C₃⁴×C₄	1	1	yes	characteristic
106	C₂²×C₁₆	C₃⁴×C₄	1	7	yes	—
107	C₂²×C₁₆	C₂²×C₃⁴	1	7	yes	—
108	C₂×C₃₂	C₂²×C₃⁴	1	28	yes	direct product factor
109	C₂×C₃³	C₂²×C₃×C₃₂	1	560	yes	direct product factor
110	C₂×C₃³	C₂³×C₃×C₁₆	1	40	yes	—
111	C₂⁴×C₃	C₃³×C₁₆	1	40	yes	—
112	C₂²×C₃×C₄	C₃³×C₁₆	1	280	yes	—
113	C₂²×C₃×C₄	C₂×C₃³×C₈	1	280	yes	—
114	C₂×C₃×C₈	C₂×C₃³×C₈	1	840	yes	—
115	C₂×C₃×C₈	C₂²×C₃³×C₄	1	280	yes	—
116	C₃×C₁₆	C₂²×C₃³×C₄	1	280	yes	—
117	C₃×C₁₆	C₂⁴×C₃³	1	40	yes	—
118	C₂²×C₃²	C₂×C₃²×C₃₂	1	3640	yes	direct product factor
119	C₂²×C₃²	C₂²×C₃²×C₁₆	1	910	yes	—
120	C₃²×C₄	C₂²×C₃²×C₁₆	1	910	yes	—
121	C₃²×C₄	C₂³×C₃²×C₈	1	130	yes	—
122	C₂³×C₄	C₃⁴×C₈	1	1	yes	characteristic
123	C₂²×C₈	C₃⁴×C₈	1	7	yes	—
124	C₂²×C₈	C₂×C₃⁴×C₄	1	7	yes	—
125	C₂×C₁₆	C₂×C₃⁴×C₄	1	21	yes	—
126	C₂×C₁₆	C₂³×C₃⁴	1	7	yes	—
127	C₃₂	C₂³×C₃⁴	1	8	yes	direct product factor
128	C₃³	C₂³×C₃×C₃₂	1	40	yes	direct product factor
129	C₂³×C₃	C₃³×C₃₂	1	320	yes	direct product factor
130	C₂³×C₃	C₂×C₃³×C₁₆	1	280	yes	—
131	C₂×C₃×C₄	C₂×C₃³×C₁₆	1	840	yes	—
132	C₂×C₃×C₄	C₂²×C₃³×C₈	1	280	yes	—
133	C₃×C₈	C₂²×C₃³×C₈	1	280	yes	—
134	C₃×C₈	C₂³×C₃³×C₄	1	40	yes	—
135	C₂×C₃²	C₂²×C₃²×C₃₂	1	1820	yes	direct product factor
136	C₂×C₃²	C₂³×C₃²×C₁₆	1	130	yes	—
137	C₂⁴	C₃⁴×C₁₆	1	1	yes	characteristic
138	C₂²×C₄	C₃⁴×C₁₆	1	7	yes	—
139	C₂²×C₄	C₂×C₃⁴×C₈	1	7	yes	—
140	C₂×C₈	C₂×C₃⁴×C₈	1	21	yes	—
141	C₂×C₈	C₂²×C₃⁴×C₄	1	7	yes	—
142	C₁₆	C₂²×C₃⁴×C₄	1	7	yes	—
143	C₁₆	C₂⁴×C₃⁴	1	1	yes	characteristic
144	C₂²×C₃	C₂×C₃³×C₃₂	1	1120	yes	direct product factor
145	C₂²×C₃	C₂²×C₃³×C₁₆	1	280	yes	—
146	C₃×C₄	C₂²×C₃³×C₁₆	1	280	yes	—
147	C₃×C₄	C₂³×C₃³×C₈	1	40	yes	—
148	C₃²	C₂³×C₃²×C₃₂	1	130	yes	direct product factor
149	C₂³	C₃⁴×C₃₂	1	8	yes	direct product factor
150	C₂³	C₂×C₃⁴×C₁₆	1	7	yes	—
151	C₂×C₄	C₂×C₃⁴×C₁₆	1	21	yes	—
152	C₂×C₄	C₂²×C₃⁴×C₈	1	7	yes	—
153	C₈	C₂²×C₃⁴×C₈	1	7	yes	—
154	C₈	C₂³×C₃⁴×C₄	1	1	yes	characteristic
155	C₂×C₃	C₂²×C₃³×C₃₂	1	560	yes	direct product factor
156	C₂×C₃	C₂³×C₃³×C₁₆	1	40	yes	—
157	C₂²	C₂×C₃⁴×C₃₂	1	28	yes	direct product factor
158	C₂²	C₂²×C₃⁴×C₁₆	1	7	yes	—
159	C₄	C₂²×C₃⁴×C₁₆	1	7	yes	—
160	C₄	C₂³×C₃⁴×C₈	1	1	yes	characteristic
161	C₃	C₂³×C₃³×C₃₂	1	40	yes	direct product factor
162	C₂	C₂²×C₃⁴×C₃₂	1	14	yes	direct product factor
163	C₂	C₂³×C₃⁴×C₁₆	1	1	yes	characteristic

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₁₆	C₂⁴×C₃⁴
socle, prime radical	C₂⁴×C₃⁴	C₁₆
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂²×C₃⁴×C₃₂	C₂	direct (N × H)
3. C₂³×C₃³×C₃₂	C₃	direct (N × H)
7. C₂×C₃⁴×C₃₂	C₂²	direct (N × H)
9. C₂²×C₃³×C₃₂	C₂×C₃	direct (N × H)
15. C₃⁴×C₃₂	C₂³	direct (N × H)
16. C₂³×C₃²×C₃₂	C₃²	direct (N × H)
20. C₂×C₃³×C₃₂	C₂²×C₃	direct (N × H)
29. C₂²×C₃²×C₃₂	C₂×C₃²	direct (N × H)
35. C₃³×C₃₂	C₂³×C₃	direct (N × H)
36. C₂³×C₃×C₃₂	C₃³	direct (N × H)
37. C₂³×C₃⁴	C₃₂	direct (N × H)
46. C₂×C₃²×C₃₂	C₂²×C₃²	direct (N × H)
55. C₂²×C₃×C₃₂	C₂×C₃³	direct (N × H)
56. C₂²×C₃⁴	C₂×C₃₂	direct (N × H)
65. C₃²×C₃₂	C₂³×C₃²	direct (N × H)
66. C₂³×C₃₂	C₃⁴	direct (N × H)
67. C₂³×C₃³	C₃×C₃₂	direct (N × H)
76. C₂×C₃×C₃₂	C₂²×C₃³	direct (N × H)
77. C₂×C₃⁴	C₂²×C₃₂	direct (N × H)
87. C₂²×C₃₂	C₂×C₃⁴	direct (N × H)
88. C₂²×C₃³	C₂×C₃×C₃₂	direct (N × H)
97. C₃×C₃₂	C₂³×C₃³	direct (N × H)
98. C₃⁴	C₂³×C₃₂	direct (N × H)
99. C₂³×C₃²	C₃²×C₃₂	direct (N × H)
108. C₂×C₃₂	C₂²×C₃⁴	direct (N × H)
109. C₂×C₃³	C₂²×C₃×C₃₂	direct (N × H)
118. C₂²×C₃²	C₂×C₃²×C₃₂	direct (N × H)
127. C₃₂	C₂³×C₃⁴	direct (N × H)
128. C₃³	C₂³×C₃×C₃₂	direct (N × H)
129. C₂³×C₃	C₃³×C₃₂	direct (N × H)
135. C₂×C₃²	C₂²×C₃²×C₃₂	direct (N × H)
144. C₂²×C₃	C₂×C₃³×C₃₂	direct (N × H)
148. C₃²	C₂³×C₃²×C₃₂	direct (N × H)
149. C₂³	C₃⁴×C₃₂	direct (N × H)
155. C₂×C₃	C₂²×C₃³×C₃₂	direct (N × H)
157. C₂²	C₂×C₃⁴×C₃₂	direct (N × H)
161. C₃	C₂³×C₃³×C₃₂	direct (N × H)
162. C₂	C₂²×C₃⁴×C₃₂	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₃⁴×C₁₆	2
C₂²×C₃⁴×C₃₂	2
C₂³×C₃³×C₃₂	3

Subgroups — C23×C34×C32 #72198321

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×C₃⁴×C₃₂ #72198321