Subgroups — C₂³×C₄⋊C₈×C₇₃ #32969498

Order 18688.

3870 conjugacy classes of subgroups.

The subgroup lattice has 146 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×C₈×C₇₃	C₂	1	2	yes	maximal
2	C₂³×C₄²×C₇₃	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₄⋊C₈×C₇₃	C₂	1	28	yes	maximal, direct product factor
4	C₂⁴×C₄×C₇₃	C₄	1	1	yes	characteristic
5	C₂⁴×C₄×C₇₃	C₄	1	1	yes	characteristic
6	C₂⁴×C₄×C₇₃	C₂²	1	1	yes	characteristic
7	C₂³×C₈×C₇₃	—	2	16	no	—
8	C₂³×C₈×C₇₃	C₂²	1	28	yes	—
9	C₂²×C₄²×C₇₃	C₄	1	14	yes	—
10	C₂²×C₄²×C₇₃	C₂²	1	14	yes	—
11	C₂×C₄⋊C₈×C₇₃	C₂²	1	112	yes	direct product factor
12	C₂⁵×C₇₃	C₂×C₄	1	1	yes	characteristic
13	C₂³×C₄×C₇₃	—	2	8	no	—
14	C₂³×C₄×C₇₃	C₈	1	16	yes	complemented
15	C₂³×C₄×C₇₃	C₂×C₄	1	14	yes	—
16	C₂³×C₄×C₇₃	C₂×C₄	1	7	yes	—
17	C₂³×C₄×C₇₃	C₂×C₄	1	14	yes	—
18	C₂³×C₄×C₇₃	D₄	1	8	yes	—
19	C₂³×C₄×C₇₃	Q₈	1	8	yes	—
20	C₂³×C₄×C₇₃	C₂³	1	7	yes	—
21	C₂²×C₈×C₇₃	—	2	112	no	—
22	C₂²×C₈×C₇₃	C₂³	1	56	yes	—
23	C₂×C₄²×C₇₃	C₂×C₄	1	84	yes	—
24	C₂×C₄²×C₇₃	C₂³	1	28	yes	—
25	C₄⋊C₈×C₇₃	C₂³	1	64	yes	direct product factor
26	C₂⁴×C₇₃	C₄⋊C₄	1	8	yes	—
27	C₂⁴×C₇₃	C₂×C₈	1	8	yes	—
28	C₂⁴×C₇₃	M₁₆	1	8	yes	—
29	C₂⁴×C₇₃	C₂²×C₄	1	7	yes	—
30	C₂²×C₄×C₇₃	—	2	56	no	—
31	C₂²×C₄×C₇₃	C₄⋊C₄	1	56	yes	—
32	C₂²×C₄×C₇₃	C₂×C₈	1	112	yes	complemented
33	C₂²×C₄×C₇₃	C₂²×C₄	1	28	yes	—
34	C₂²×C₄×C₇₃	C₂²×C₄	1	21	yes	—
35	C₂²×C₄×C₇₃	C₂²×C₄	1	28	yes	—
36	C₂²×C₄×C₇₃	C₂×D₄	1	28	yes	—
37	C₂²×C₄×C₇₃	C₂×Q₈	1	28	yes	—
38	C₂²×C₄×C₇₃	C₂⁴	1	7	yes	—
39	C₂×C₈×C₇₃	—	2	112	no	—
40	C₂×C₈×C₇₃	C₂⁴	1	16	yes	—
41	C₄²×C₇₃	C₂²×C₄	1	56	yes	—
42	C₄²×C₇₃	C₂⁴	1	8	yes	—
43	C₂³×C₇₃	C₄⋊C₈	1	64	yes	direct product factor
44	C₂³×C₇₃	C₂×C₄⋊C₄	1	28	yes	—
45	C₂³×C₇₃	C₂²×C₈	1	28	yes	—
46	C₂³×C₇₃	C₂×M₁₆	1	28	yes	—
47	C₂³×C₇₃	C₂³×C₄	1	7	yes	—
48	C₂×C₄×C₇₃	—	2	56	no	—
49	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	84	yes	—
50	C₂×C₄×C₇₃	C₂²×C₈	1	112	yes	complemented
51	C₂×C₄×C₇₃	C₂³×C₄	1	8	yes	—
52	C₂×C₄×C₇₃	C₂³×C₄	1	7	yes	—
53	C₂×C₄×C₇₃	C₂³×C₄	1	8	yes	—
54	C₂×C₄×C₇₃	C₂²×D₄	1	14	yes	—
55	C₂×C₄×C₇₃	C₂²×Q₈	1	14	yes	—
56	C₂×C₄×C₇₃	C₂⁵	1	1	yes	characteristic
57	C₈×C₇₃	—	2	16	no	—
58	C₂²×C₇₃	C₂×C₄⋊C₈	1	112	yes	direct product factor
59	C₂²×C₇₃	C₂²×C₄⋊C₄	1	14	yes	—
60	C₂²×C₇₃	C₂³×C₈	1	14	yes	—
61	C₂²×C₇₃	C₂²×M₁₆	1	14	yes	—
62	C₂²×C₇₃	C₂⁴×C₄	1	1	yes	characteristic
63	C₄×C₇₃	—	2	8	no	—
64	C₄×C₇₃	C₂²×C₄⋊C₄	1	14	yes	—
65	C₄×C₇₃	C₂³×C₈	1	16	yes	complemented
66	C₄×C₇₃	C₂³×D₄	1	1	yes	characteristic
67	C₄×C₇₃	C₂³×Q₈	1	1	yes	characteristic
68	C₂³×C₄⋊C₈	C₇₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
69	C₂×C₇₃	C₂²×C₄⋊C₈	1	28	yes	direct product factor
70	C₂×C₇₃	C₂³×C₄⋊C₄	1	1	yes	characteristic
71	C₂×C₇₃	C₂⁴×C₈	1	1	yes	characteristic
72	C₂×C₇₃	C₂³×M₁₆	1	1	yes	characteristic
73	C₂⁴×C₈	C₂×C₇₃	1	2	yes	—
74	C₂³×C₄²	C₂×C₇₃	1	1	yes	characteristic
75	C₂²×C₄⋊C₈	C₂×C₇₃	1	28	yes	direct product factor
76	C₇₃	C₂³×C₄⋊C₈	1	1	yes	characteristic, direct product factor, Sylow
77	C₂⁴×C₄	C₄×C₇₃	1	1	yes	characteristic
78	C₂⁴×C₄	C₄×C₇₃	1	1	yes	characteristic
79	C₂⁴×C₄	C₂²×C₇₃	1	1	yes	characteristic
80	C₂³×C₈	—	2	16	no	—
81	C₂³×C₈	C₂²×C₇₃	1	28	yes	—
82	C₂²×C₄²	C₄×C₇₃	1	14	yes	—
83	C₂²×C₄²	C₂²×C₇₃	1	14	yes	—
84	C₂×C₄⋊C₈	C₂²×C₇₃	1	112	yes	direct product factor
85	C₂⁵	C₂×C₄×C₇₃	1	1	yes	characteristic
86	C₂³×C₄	—	2	8	no	—
87	C₂³×C₄	C₈×C₇₃	1	16	yes	complemented
88	C₂³×C₄	C₂×C₄×C₇₃	1	14	yes	—
89	C₂³×C₄	C₂×C₄×C₇₃	1	7	yes	—
90	C₂³×C₄	C₂×C₄×C₇₃	1	14	yes	—
91	C₂³×C₄	D₄×C₇₃	1	8	yes	—
92	C₂³×C₄	Q₈×C₇₃	1	8	yes	—
93	C₂³×C₄	C₂³×C₇₃	1	7	yes	—
94	C₂²×C₈	—	2	112	no	—
95	C₂²×C₈	C₂³×C₇₃	1	56	yes	—
96	C₂×C₄²	C₂×C₄×C₇₃	1	84	yes	—
97	C₂×C₄²	C₂³×C₇₃	1	28	yes	—
98	C₄⋊C₈	C₂³×C₇₃	1	64	yes	direct product factor
99	C₂⁴	C₄⋊C₄×C₇₃	1	8	yes	—
100	C₂⁴	C₂×C₈×C₇₃	1	8	yes	—
101	C₂⁴	M₁₆×C₇₃	1	8	yes	—
102	C₂⁴	C₂²×C₄×C₇₃	1	7	yes	—
103	C₂²×C₄	—	2	56	no	—
104	C₂²×C₄	C₄⋊C₄×C₇₃	1	56	yes	—
105	C₂²×C₄	C₂×C₈×C₇₃	1	112	yes	complemented
106	C₂²×C₄	C₂²×C₄×C₇₃	1	28	yes	—
107	C₂²×C₄	C₂²×C₄×C₇₃	1	21	yes	—
108	C₂²×C₄	C₂²×C₄×C₇₃	1	28	yes	—
109	C₂²×C₄	C₂×D₄×C₇₃	1	28	yes	—
110	C₂²×C₄	C₂×Q₈×C₇₃	1	28	yes	—
111	C₂²×C₄	C₂⁴×C₇₃	1	7	yes	—
112	C₂×C₈	—	2	112	no	—
113	C₂×C₈	C₂⁴×C₇₃	1	16	yes	—
114	C₄²	C₂²×C₄×C₇₃	1	56	yes	—
115	C₄²	C₂⁴×C₇₃	1	8	yes	—
116	C₂³	C₄⋊C₈×C₇₃	1	64	yes	direct product factor
117	C₂³	C₂×C₄⋊C₄×C₇₃	1	28	yes	—
118	C₂³	C₂²×C₈×C₇₃	1	28	yes	—
119	C₂³	C₂×M₁₆×C₇₃	1	28	yes	—
120	C₂³	C₂³×C₄×C₇₃	1	7	yes	—
121	C₂×C₄	—	2	56	no	—
122	C₂×C₄	C₂×C₄⋊C₄×C₇₃	1	84	yes	—
123	C₂×C₄	C₂²×C₈×C₇₃	1	112	yes	complemented
124	C₂×C₄	C₂³×C₄×C₇₃	1	8	yes	—
125	C₂×C₄	C₂³×C₄×C₇₃	1	7	yes	—
126	C₂×C₄	C₂³×C₄×C₇₃	1	8	yes	—
127	C₂×C₄	C₂²×D₄×C₇₃	1	14	yes	—
128	C₂×C₄	C₂²×Q₈×C₇₃	1	14	yes	—
129	C₂×C₄	C₂⁵×C₇₃	1	1	yes	characteristic
130	C₈	—	2	16	no	—
131	C₂²	C₂×C₄⋊C₈×C₇₃	1	112	yes	direct product factor
132	C₂²	C₂²×C₄⋊C₄×C₇₃	1	14	yes	—
133	C₂²	C₂³×C₈×C₇₃	1	14	yes	—
134	C₂²	C₂²×M₁₆×C₇₃	1	14	yes	—
135	C₂²	C₂⁴×C₄×C₇₃	1	1	yes	characteristic
136	C₄	—	2	8	no	—
137	C₄	C₂²×C₄⋊C₄×C₇₃	1	14	yes	—
138	C₄	C₂³×C₈×C₇₃	1	16	yes	complemented
139	C₄	C₂³×D₄×C₇₃	1	1	yes	characteristic
140	C₄	C₂³×Q₈×C₇₃	1	1	yes	characteristic
141	C₂	C₂²×C₄⋊C₈×C₇₃	1	28	yes	direct product factor
142	C₂	C₂³×C₄⋊C₄×C₇₃	1	1	yes	characteristic
143	C₂	C₂⁴×C₈×C₇₃	1	1	yes	characteristic
144	C₂	C₂³×M₁₆×C₇₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴×C₄×C₇₃	C₂²
commutator	C₂	C₂⁴×C₈×C₇₃
Frattini, cosocle	C₂×C₄	C₂⁵×C₇₃
socle, prime radical	C₂⁵×C₇₃	C₂×C₄
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₄×C₇₃ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂²×C₄⋊C₈×C₇₃	C₂	direct (N × H)
11. C₂×C₄⋊C₈×C₇₃	C₂²	direct (N × H)
14. C₂³×C₄×C₇₃	C₈	semidirect
25. C₄⋊C₈×C₇₃	C₂³	direct (N × H)
32. C₂²×C₄×C₇₃	C₂×C₈	semidirect
43. C₂³×C₇₃	C₄⋊C₈	direct (N × H)
50. C₂×C₄×C₇₃	C₂²×C₈	semidirect
58. C₂²×C₇₃	C₂×C₄⋊C₈	direct (N × H)
65. C₄×C₇₃	C₂³×C₈	semidirect
68. C₂³×C₄⋊C₈	C₇₃	direct (N × H)
69. C₂×C₇₃	C₂²×C₄⋊C₈	direct (N × H)
75. C₂²×C₄⋊C₈	C₂×C₇₃	direct (N × H)
76. C₇₃	C₂³×C₄⋊C₈	direct (N × H)
84. C₂×C₄⋊C₈	C₂²×C₇₃	direct (N × H)
87. C₂³×C₄	C₈×C₇₃	semidirect
98. C₄⋊C₈	C₂³×C₇₃	direct (N × H)
105. C₂²×C₄	C₂×C₈×C₇₃	semidirect
116. C₂³	C₄⋊C₈×C₇₃	direct (N × H)
123. C₂×C₄	C₂²×C₈×C₇₃	semidirect
131. C₂²	C₂×C₄⋊C₈×C₇₃	direct (N × H)
138. C₄	C₂³×C₈×C₇₃	semidirect
141. C₂	C₂²×C₄⋊C₈×C₇₃	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁴×C₈×C₇₃	2
C₂³×C₄²×C₇₃	2
C₂²×C₄⋊C₈×C₇₃	2
C₂³×C₄⋊C₈	73

Subgroups — C23×C4⋊C8×C73 #32969498

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×C₄⋊C₈×C₇₃ #32969498